Mercurial > hg > mpdl-group
changeset 18:dc5e9fcb3fdc
Erstellung
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.classpath Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,9 @@ +<?xml version="1.0" encoding="UTF-8"?> +<classpath> + <classpathentry kind="src" path="src"/> + <classpathentry kind="con" path="org.eclipse.jdt.launching.JRE_CONTAINER"/> + <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>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.externalToolBuilders/New_Builder (1).launch Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,12 @@ +<?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="${workspace_loc:/mpiwg-mpdl-xml/build/build.xml}"/> +<stringAttribute key="org.eclipse.ui.externaltools.ATTR_RUN_BUILD_KINDS" value="full,incremental,"/> +<booleanAttribute key="org.eclipse.ui.externaltools.ATTR_TRIGGERS_CONFIGURED" value="true"/> +</launchConfiguration>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.project Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,27 @@ +<?xml version="1.0" encoding="UTF-8"?> +<projectDescription> + <name>mpiwg-mpdl-xml</name> + <comment></comment> + <projects> + </projects> + <buildSpec> + <buildCommand> + <name>org.eclipse.jdt.core.javabuilder</name> + <arguments> + </arguments> + </buildCommand> + <buildCommand> + <name>org.eclipse.ui.externaltools.ExternalToolBuilder</name> + <triggers>full,incremental,</triggers> + <arguments> + <dictionary> + <key>LaunchConfigHandle</key> + <value><project>/.externalToolBuilders/New_Builder (1).launch</value> + </dictionary> + </arguments> + </buildCommand> + </buildSpec> + <natures> + <nature>org.eclipse.jdt.core.javanature</nature> + </natures> +</projectDescription>
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/.settings/org.eclipse.core.resources.prefs Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,3 @@ +#Mon Nov 07 15:08:16 CET 2011 +eclipse.preferences.version=1 +encoding//src/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.java=UTF-8
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/build/build.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,41 @@ +<!DOCTYPE project> +<project name="mpiwg-mpdl-xml" default="dist" basedir="../"> + <description>mpiwg-mpdl-xml</description> + <!-- global properties --> + <property name="src" location="src"/> + <property name="lib" location="lib"/> + <property name="build" location="build/classes"/> + <property name="dist" location="dist"/> + + <path id="classpath"> + <fileset dir="${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"> + <javac srcdir="${src}" destdir="${build}" classpathref="classpath" includeantruntime="false"/> + <copy todir="${build}"> + <fileset dir="${src}"> + <include name="**/*.xsl"/> + <include name="**/*.properties"/> + </fileset> + </copy> + </target> + + <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-lt/lib"/> + </target> + + <target name="clean" description="clean" > + <delete dir="${build}"/> + <delete file="${dist}/mpiwg-mpdl-xml.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-xml/build/classes/constants.properties Wed Nov 09 15:27:46 2011 +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/build/classes/de/mpg/mpiwg/berlin/mpdl/exception/ApplicationException.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/util/FileUtil.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/util/Util.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/exception/ApplicationException.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/general/Constants.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/tmp/GetFragmentContentHandler.class has changed
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/tmp/XmlExamples.class has changed
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--- /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/getFragment.xsl Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,58 @@ +<?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> + +<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::*"/> + +<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:when> + <xsl:otherwise><xsl:apply-templates/></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: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/getFragmentSaxonFast.xsl Wed Nov 09 15:27:46 2011 +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>
Binary file software/mpdl-services/mpiwg-mpdl-xml/build/classes/de/mpg/mpiwg/berlin/mpdl/xml/xquery/XQueryEvaluator.class has changed
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/Aristoteles_1585_XSY685ZD-Ids.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,1299 @@ +<?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 ſit artis Mechanicæ facultas.</head> + <p> + <s xml:id="s1" xml:space="preserve">MIraculo ſune ea quidem, quæ natura <lb/>contingunt, quorum ignorantur cau <lb/>ſæ: </s> + <s xml:id="s2" xml:space="preserve">illa uerò, quæ pręter 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">qđ <lb/>nobis uſui eſſe poteſt, contrarium facit: </s> + <s xml:id="s7" xml:space="preserve">natura <lb/>etenim eundem ſemper habet modum, & </s> + <s xml:id="s8" xml:space="preserve">ſimpli <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 ſua hæſitationem praſtat, ar-<lb/>teq́ indiget: </s> + <s xml:id="s11" xml:space="preserve">quam ob rem eam artis partẽ, quæ <lb/>huiuſmodi ſuccurrit difficulcatibus, Mechanicã <lb/>appellamus. </s> + <s xml:id="s12" xml:space="preserve">Quemad modum enim Antipho ſcri <lb/>bit poeta, ſic ſe res habet: </s> + <s xml:id="s13" xml:space="preserve">arte enim ſuperamus <lb/>da, à quibus natura uincimur. </s> + <s xml:id="s14" xml:space="preserve">Huiuſmodi autem <lb/>ſunt, in quibus & </s> + <s xml:id="s15" xml:space="preserve">minora ſuperant 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 ſeiugata valde: </s> + <s xml:id="s23" xml:space="preserve">verùm mathema-<lb/>ticarũ contemplationũ naturaliumq́; </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, admirandis̈ 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">Abſurdum enim eſſe videtur, ma-<lb/>gnum moueri pondus ab exigua virtute, cũ 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 quiſpiã <lb/>mouere nõ poteſt, idipſum ponderis citius mo-<lb/>uet, uectis ad illud pondus adiungens. </s> + <s xml:id="s31" xml:space="preserve">Oĩum ãt <lb/>huiuſmodi cauſæ principiũ habet circulus, Iſtud <lb/>verò ratione contingit; </s> + <s xml:id="s32" xml:space="preserve">ex ad mirabili etem̄ mi-<lb/>randum accidere quippiam, non eſt ab ſurdum.</s> + <s xml:id="s33" xml:space="preserve"/> + </p> + <p> + <s xml:id="s34" xml:space="preserve">Maximè autem eſt admirãdum ſimul contra-<lb/>ria fieri. </s> + <s xml:id="s35" xml:space="preserve">Circulus vero ex huiuſmodi eſt conſtitu <lb/>tus: </s> + <s xml:id="s36" xml:space="preserve">ſtatim enim ex commoto effectus eſt, & </s> + <s xml:id="s37" xml:space="preserve">mæ <lb/>nente, quorũ natura ad ſe inuicem eſt cõtraria. <lb/></s> + <s xml:id="s38" xml:space="preserve">quamobrẽ iſthæc 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 ſcilicet, <lb/>& </s> + <s xml:id="s40" xml:space="preserve">curuum. </s> + <s xml:id="s41" xml:space="preserve">hæc aũt eo à ſeinuicem diſtant mõ, <lb/>quo magnum & </s> + <s xml:id="s42" xml:space="preserve">paruum, illorũ etenim medium <lb/>eſt æquale: </s> + <s xml:id="s43" xml:space="preserve">horum verò rectũ quapropter cùm, <lb/>ad ſeinuicem cõmutantur, illa quidem prius æ-<lb/>qualia fieri neceſſe eſt, quàm extremorum vtrũ-<lb/>libet: </s> + <s xml:id="s44" xml:space="preserve">lineam verò rectam, qñ eſt curua concaua, <lb/>ut ex huiu ſmoà<unsure/>i rurſum curua fit, & </s> + <s xml:id="s45" xml:space="preserve">circularis +<pb o="506" file="0506" n="3" rhead="Ariſt. Quæſtiones"/> +Vnum quidem igitur iſtuc abſurdum ineſt circti<unsure/> <lb/>lo. </s> + <s xml:id="s46" xml:space="preserve">Alterum autem, ꝙ ſimul contrarijs mouetur <lb/>motionibus: </s> + <s xml:id="s47" xml:space="preserve">ſimul enim ad anteriorem mouetur <lb/>locum, & </s> + <s xml:id="s48" xml:space="preserve">ad poſter<unsure/>iorem. </s> + <s xml:id="s49" xml:space="preserve">Et ea, quæ circulũ de-<lb/>ſcribit, linea eodem ſe 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 rurſus <lb/>efficitur primum. </s> + <s xml:id="s56" xml:space="preserve">quamobrem manifeſtum, ꝙ in <lb/>de mutatnm eſt. </s> + <s xml:id="s57" xml:space="preserve">quapropter (vt dictum eſt prius) <lb/>non eſt incõueniens ipſ<unsure/>u<unsure/>m miracolorum oĩum <lb/>eſſe principlum. </s> + <s xml:id="s58" xml:space="preserve">Ea igi ur, quæ circa libram fiũt, <lb/>ad circulum referuntur: </s> + <s xml:id="s59" xml:space="preserve">quæ vero circa vectem, <lb/>ad ipſam librã: </s> + <s xml:id="s60" xml:space="preserve">alia autem ferè omn<unsure/>ia, quæ circa <lb/>męchanicas ſunt motiones, ad vectern<unsure/>. </s> + <s xml:id="s61" xml:space="preserve">Prætereà <lb/>etiã, quoniam vnica exiſtente, quæ ex cehtro eſt, <lb/>linea, nullum aliud alij, quæ in illa ſunt puncto-<lb/>rum æqua velocitate feratur; </s> + <s xml:id="s62" xml:space="preserve">ſed cirius ſemper <lb/>quod à manente termino eſt remo ius, pleraque <lb/>miraculorum accidũt in circuli motionibus: </s> + <s xml:id="s63" xml:space="preserve">de <lb/>quibus in ijs, quæ poſthac addacentur, quęſtioni <lb/>bus erit manifeſtum. </s> + <s xml:id="s64" xml:space="preserve">Quoniam autem ſecundũ <lb/>contrarias fimul motiones mouetur circulus: </s> + <s xml:id="s65" xml:space="preserve">& </s> + <s xml:id="s66" xml:space="preserve"><lb/>alterũ 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 ſimul moueantur circuli: </s> + <s xml:id="s69" xml:space="preserve">quemadmodum, <lb/>ſunt illi, quos in locis proponũt ſacris, 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 eſt 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">rurſus ipſe conti-<lb/>guum ubi EF, in contrarium ſibijpſi mouebitur <lb/>propter eandem cauſam. </s> + <s xml:id="s76" xml:space="preserve">Eodem ẽt modo ſi plu- +<pb o="507" file="0507" n="4" rhead="Mechanicæ."/> +res fuerint, idẽ facient, uno ſolo commoto. </s> + <s xml:id="s77" xml:space="preserve">Hane <lb/>igitur in circulo exiſtentem animaduertens na-<lb/>turam architecti, in ſtrumentũ fabricant, celãtes <lb/>principium, vt machinæ ſolum manifeftum ſit il <lb/>lud, quod admirationẽ pręſtat, cauſa 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, quamnàm ob cauſam exactiores <lb/>minoribus maiores ſunt libræ. </s> + <s xml:id="s80" xml:space="preserve">huius autem rei <lb/>principium eſt, quamobrem in ipſo circulo, quæ <lb/>plus à cent<unsure/>ro diſter linea, eadẽ ui cõmota cit<unsure/>ius <lb/>fertur, quàm illa, quæ minus diſtar. </s> + <s xml:id="s81" xml:space="preserve">Citius enim <lb/>bifariam dicitur: </s> + <s xml:id="s82" xml:space="preserve">ſiue enim in minori tempore <lb/>zqualem pertranſi@ locũ, citius feciſſe dicimus: <lb/></s> + <s xml:id="s83" xml:space="preserve">ſeu in æquali maiorem. </s> + <s xml:id="s84" xml:space="preserve">Maior autem in æquali <lb/>tempore maiorem deſcribit circulum: </s> + <s xml:id="s85" xml:space="preserve">qui enim <lb/>extra eſt, maior eo qui intus eft. </s> + <s xml:id="s86" xml:space="preserve">Horum autẽ cau <lb/>fa, qm̄ duas fertur lationesea, quæ circulum de-<lb/>ſcribit, linea. </s> + <s xml:id="s87" xml:space="preserve">Quandoquidem igitur in propor-<lb/>tione fertur aliqua id, quod fertur, ſuper 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 huiuſmodi proportio <lb/>ne coaptant̃, lineę. </s> + <s xml:id="s90" xml:space="preserve">Sit enim proportio ſecũdum <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 verſus B, A B uerò ſubterferatur <lb/>verſus M C: </s> + <s xml:id="s93" xml:space="preserve">latum aut<unsure/>em ſit A quidem ad D. </s> + <s xml:id="s94" xml:space="preserve">ubi <lb/>autem eſt A B, uerſus 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/>eſt & </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 eſt 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 eſt 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 oſtendet̃ mo-<lb/>do, vbicu<unsure/>nq; </s> + <s xml:id="s103" xml:space="preserve">latio deprehendatur; </s> + <s xml:id="s104" xml:space="preserve">ſemper enim <lb/>ſupra diametrum erit. </s> + <s xml:id="s105" xml:space="preserve">Manifeſtum igitur, quòd <lb/>id, quod ſecundum diametrum duabus fertur la <lb/>tionibus, neceſſariò ſecundum laterum propor- +<pb o="508" file="0508" n="5" rhead="Ariſt. Quæſtiones."/> +tionem fertur. </s> + <s xml:id="s106" xml:space="preserve">Sienim ſecundum 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 ſecundum duas lationes <lb/>nullo in tempore, rectã eſſe lationem eſt impoſ-<lb/>ſibile. </s> + <s xml:id="s108" xml:space="preserve">Sit enim recta. </s> + <s xml:id="s109" xml:space="preserve">Poſita igitur hac pro diame <lb/>tro, & </s> + <s xml:id="s110" xml:space="preserve">circum reple is lateribus, illud quod fer-<lb/>tur, ſecundum laterum pro portionem ferri n<unsure/>e-<lb/>ceſſe eſt: </s> + <s xml:id="s111" xml:space="preserve">hoc enim demonſtratum eſt prius. </s> + <s xml:id="s112" xml:space="preserve">non <lb/>igitur rectam efficiet id, quod ſecundùm 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 eſt tem pus rectam eſ <lb/>ſe lationem, per ea, quæ retro ſunt dicta. </s> + <s xml:id="s115" xml:space="preserve">Quam-<lb/>obrem circulare eſtid, quod ſecundũ nullam, <lb/>proportionem nullo in tempore duas fertur la-<lb/>tiones Quòd quidem igitur ea, quæ circulum do<unsure/> <lb/>ſcribir, duas ſimul feratur lationes, manifeſtum <lb/>eſt cùm ex iſtis, tum quia ẜ m rectum lata, aſt ꝑ. <lb/></s> + <s xml:id="s116" xml:space="preserve">pendiculum peruenit, vt ſit rurſus ipſa 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 eſt B, feratur ad ipſum D: </s> + <s xml:id="s119" xml:space="preserve">peruenit ſa <lb/>hè aliquando ad ipſum 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 ſecundùm 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 conſentaneum eſt <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, q́uæ ex centro circulos deſcri <lb/>bunt. </s> + <s xml:id="s126" xml:space="preserve">quoniam enim proprius eſt manenti, eius <lb/>quæ minor eſt, extrem um, quàm id quod eſt ma <lb/>oris, veluti retractum in contrariũ ad medium, <lb/>ardius fertur minoris extremũ. </s> + <s xml:id="s127" xml:space="preserve">Omni quidem +<pb o="509" file="0509" n="6" rhead="Mechanicæ."/> +igitur circulum deſcribenti iſtuc accidit: </s> + <s xml:id="s128" xml:space="preserve">fertur-<lb/>q́ue eam quæ ẜm naturam eſt lationem, ſecun-<lb/>dum circunferentiam: </s> + <s xml:id="s129" xml:space="preserve">illam verò quę pręter na <lb/>turam, in tranſuerſum, & </s> + <s xml:id="s130" xml:space="preserve">ſecundùm centrum. <lb/></s> + <s xml:id="s131" xml:space="preserve">maiorem aut<unsure/>ẽ ſemper eam, quæ præter naturã <lb/>eſt, ipſa minor fertur: </s> + <s xml:id="s132" xml:space="preserve">quia enim centro eſt 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 eſt mouetur ipſa mi-<lb/>nor, quàm maior illarũ, quæ ex centro circulos <lb/>deſcribũt ex ijs eſt manifeſtum. </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 ſuppleatur DKRC: </s> + <s xml:id="s139" xml:space="preserve">ſi quidem AB circu-<lb/>lum deſcribẽs ad id perueniet, vnde eſt egreſſa; </s> + <s xml:id="s140" xml:space="preserve"><lb/>manifeſtum eſt quod ad ipſam fertur A B. </s> + <s xml:id="s141" xml:space="preserve">Simi <lb/>literetiam A M ad ipſam A M perueniet. </s> + <s xml:id="s142" xml:space="preserve">Tar-<lb/>dius autem fertur A M, quàru A B quemadmo-<lb/>dum dictum eſt: </s> + <s xml:id="s143" xml:space="preserve">quia maior fit repulſio & </s> + <s xml:id="s144" xml:space="preserve">magis <lb/>rerrahitur A M. </s> + <s xml:id="s145" xml:space="preserve">Ducatur igitur ipſa A L F, & </s> + <s xml:id="s146" xml:space="preserve">ab <lb/>ipſo L perpendiculum ad ip ſam A B, ipſa L Q in <lb/>minore circulo: </s> + <s xml:id="s147" xml:space="preserve">& </s> + <s xml:id="s148" xml:space="preserve">rurſum ab L ducatur iuxta A <lb/>B L S, & </s> + <s xml:id="s149" xml:space="preserve">S T ad ipſam 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 ſunt S T, & </s> + <s xml:id="s153" xml:space="preserve">L Q, æquales: </s> + <s xml:id="s154" xml:space="preserve"><lb/>ipſa ergo B T minor eſt, quàm 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 ſectionẽ in maioribus circulis, eſt autem <lb/>ipſa S T æqualis ipſi L Q. </s> + <s xml:id="s156" xml:space="preserve">In quanto autem tema <lb/>pore ipſa A L ipſam M L lata eſt, in tanto tem@ <lb/>poris ſpatio in maiori circulo maiorem, quàm <lb/>fit B S, latum erit extremum ipſius B A. </s> + <s xml:id="s157" xml:space="preserve">Latio <lb/>quidem igitur ſecundùm naturam æqualis: </s> + <s xml:id="s158" xml:space="preserve">ea <lb/>a<unsure/>utem quæ præter naturam eſt minor, videlices +<pb o="510" file="0510" n="7" rhead="Ariſt. Quæſtiones."/> +B T, quàm M Q. </s> + <s xml:id="s159" xml:space="preserve">Oportet autem proportionabi-<lb/>liter eſſe, ſicut quod eſt ſecundũ naturã, ita quod <lb/>eſt pręter naturam, ad id quod eſt pręter naturã; <lb/></s> + <s xml:id="s160" xml:space="preserve">maiorem igituĩ circunferentiam pertranſiuit, ꝗ̃ <lb/>ſit ipſa S B. </s> + <s xml:id="s161" xml:space="preserve">Neceſſe autem eſt ipſam F B in hoc <lb/>rẽpore pertran ſiſſe: </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 eſt præter naturã, <lb/>ad id quod eſt ſecundùm naturam. </s> + <s xml:id="s166" xml:space="preserve">Si igitur ma-<lb/>ius eſt, quod ſecundum naturam in maiori, & </s> + <s xml:id="s167" xml:space="preserve"><lb/>quod eſt 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 ſit latum per ipſam <lb/>B F in tanto tempore, in quo M punctum per <lb/>ipſam M L. </s> + <s xml:id="s170" xml:space="preserve">hic enim ſecundùm naturam quidẽ <lb/>figno B fit X F: </s> + <s xml:id="s171" xml:space="preserve">eſt enim ab ipſo F perpendiculũ: </s> + <s xml:id="s172" xml:space="preserve"><lb/>præter naturam verò ad ipſam X B. </s> + <s xml:id="s173" xml:space="preserve">Eſt autem <lb/>quemadmodum F X ad X B, ſic L Q. </s> + <s xml:id="s174" xml:space="preserve">ad M Q. </s> + <s xml:id="s175" xml:space="preserve">ma <lb/>nifeſtum autem, ſi coniunguntur abipſa B M, <lb/>ad F L. </s> + <s xml:id="s176" xml:space="preserve">Si autem minor, aut maior quàm ſit F B, <lb/>erit illa quam latum eſt B, non ſimiliter erit, ne-<lb/>que proportionale in vtriſque quod eſt ſecundũ <lb/>naturam, ad id quod eſt præter naturam. </s> + <s xml:id="s177" xml:space="preserve">Quam <lb/>igicur ob cauſam ab eadem potentia celerius fe@ <lb/>tur id quod plus à centro diſtat punctum, ex ijs, <lb/>quæ dicta ſunt, eſt manifeſtum.</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 aũt maiores libræ exactiores <lb/> +<anchor type="note" xlink:label="note-0510-01a" xlink:href="note-0510-01"/> +ſunt minotib<emph style="sub">9</emph>. </s> + <s xml:id="s180" xml:space="preserve">palã eſt ex ijs. </s> + <s xml:id="s181" xml:space="preserve">Spar <lb/>tum. </s> + <s xml:id="s182" xml:space="preserve">n fit centrũ: </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 aũt libræ vtrinq; </s> + <s xml:id="s186" xml:space="preserve">eſt, exeun <lb/>tes à centro. </s> + <s xml:id="s187" xml:space="preserve">Ab eodẽ igit̃ põdere <lb/>citius moueri neceſſe eſt extre-<lb/>mũ libræ, quo plus à ſparto diſceſſerit. </s> + <s xml:id="s188" xml:space="preserve">Et nõnul <lb/>la quidẽ in paruis libris impoſita nõ manifeſta +<pb o="511" file="0511" n="8" rhead="Mechanicæ."/> +fenſui ſunt pondera: </s> + <s xml:id="s189" xml:space="preserve">in magis aũt maniſeſta: </s> + <s xml:id="s190" xml:space="preserve">ni <lb/>hil enim prohibet minorem moueri magnitudi <lb/>nem, ꝗ̃ vt viſioni ſit maniſeſta. </s> + <s xml:id="s191" xml:space="preserve">In magna aũt li-<lb/>bra idem pondus viſibile efficit magnitudo. </s> + <s xml:id="s192" xml:space="preserve">Quę <lb/>dam verò manifeſta quidem ſunt in vtriſq;</s> + <s xml:id="s193" xml:space="preserve">, ſed <lb/>multo magis in maioribus, qm̄ multo maior in <lb/>clinationis fit magnitudo ab eodem ꝑondere <lb/>in maioribus. </s> + <s xml:id="s194" xml:space="preserve">Quamobrem machinanturij, qui <lb/>purpuram vendunt, vt pendendo defraudent, tũ <lb/>ad medium ſpartum 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 gđ ad radicẽ vergebat, in eam quam de <lb/>ferri vola<unsure/>nt, partem conſtituentes; </s> + <s xml:id="s197" xml:space="preserve">aut ſi nodum <lb/>habuerit: </s> + <s xml:id="s198" xml:space="preserve">ligni enim grauior illa eſt pars, in qua <lb/>eſt radix, nodus vero radix quædam eſt.</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 ſiquidẽ ſurſum ſuerit ſpartũ, qñ deorsũ lato <lb/> +<anchor type="note" xlink:label="note-0511-01a" xlink:href="note-0511-01"/> +põdere, quiſpiã id admouet, rurſum aſcendit li-<lb/>bra: </s> + <s xml:id="s201" xml:space="preserve">ſi autem deorſum conſtitutum ſuerit, non <lb/>aſcendit, ſed manet? </s> + <s xml:id="s202" xml:space="preserve">An quia ſurſum quidẽ ſpar <lb/>to exiſtente, plus libræ extra perpendiculum fit: <lb/></s> + <s xml:id="s203" xml:space="preserve">ſpartum enim eſt perpendiculum: </s> + <s xml:id="s204" xml:space="preserve">quare neceſſe <lb/>eſt deorſum ferri id quod plus eſt, donec aſcen-<lb/>dat quæ bifariam libram diuidit, ad ipſum per-<lb/>pendiculũ, cùm onus incumbat ad libræ partem <lb/>furſum raptam. </s> + <s xml:id="s205" xml:space="preserve">Sit libra recta vbi B C, ſpartum <lb/>autem A D. </s> + <s xml:id="s206" xml:space="preserve">hoc igitur deorſum proiecto perpen <lb/>diculum erit vbi A D M. </s> + <s xml:id="s207" xml:space="preserve">Si igitur in ipſo 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 ſecat, pri-<lb/>mo quidem erit D M ipſius 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 ipſius vbi <lb/>E H, quod extra perpendiculum eſt A M, vbi eſt <lb/>P Q, maius eſt dimidio. </s> + <s xml:id="s211" xml:space="preserve">Si igitur amoueatur <lb/>onus abipſo E, neceſſe eſt deorſum ferri H, mi-<lb/>@@s enim eſt ipſum E. </s> + <s xml:id="s212" xml:space="preserve">Si quidem igitur ſurſum +<pb o="512" file="0512" n="9" rhead="Ariſt. Quæſtiones"/> +habuerit ſpartum, rurſum propter hoc aſcendit<unsure/> <lb/>libta. </s> + <s xml:id="s213" xml:space="preserve">Si autẽ deorſum fuerit, id, quod ſubſtat, cõ <lb/>trarium facit: </s> + <s xml:id="s214" xml:space="preserve">plus enim dimidio fit libræ, quæ <lb/>deorſum eſt pars, quàm quod perpendiculum ſe <lb/>cet, quapropter nõ aſcẽdit: </s> + <s xml:id="s215" xml:space="preserve">eleuata enim pars le <lb/>uior eſt. </s> + <s xml:id="s216" xml:space="preserve">Sit libra recta vbi N O, perpendiculum <lb/>autem K L M bifariã igitur ſecatu@ K G. </s> + <s xml:id="s217" xml:space="preserve">Impo-<lb/>ſito autem onere in ipſo N, erit quidẽ N vbi O, <lb/>ipſum autem G vbi R, K L autem vbi L P. </s> + <s xml:id="s218" xml:space="preserve">quare <lb/>maius eſt K O, quàm L R, ipſo P K L. </s> + <s xml:id="s219" xml:space="preserve">Et ablato <lb/>igitur onere, neceſſe eſt manere: </s> + <s xml:id="s220" xml:space="preserve">in cumbit enim <lb/>ceu onus exceſſus medietatis eius vbi eſt 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 exiguę vires (quem admodum à principio <lb/> +<anchor type="note" xlink:label="note-0512-01a" xlink:href="note-0512-01"/> +dictum eſt) vecte magna mouent pondera, vecis <lb/>inſuper onus accipien tis? </s> + <s xml:id="s223" xml:space="preserve">cùm ſacilius ſit mino-<lb/>grem mouere grauitatem: </s> + <s xml:id="s224" xml:space="preserve">minor autẽ eſt ſine ve-<lb/>cte. </s> + <s xml:id="s225" xml:space="preserve">An quoniã ipſe vectis eſt in cauſa hbra exi-<lb/>ſtens, ſpartũ infernè habens, in inæqualia diui-<lb/>ſa. </s> + <s xml:id="s226" xml:space="preserve">hypomochlion enim eſt ſpartum. </s> + <s xml:id="s227" xml:space="preserve">a mbo nan <lb/>que ſtamt, vt centrum. </s> + <s xml:id="s228" xml:space="preserve">quoniam autem ab ęqua-<lb/>ſi pondere celerius mouetur maior earum, q̃ à <lb/>centro ſunt: </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">ſem <lb/>per autem quanto ab hy pomochlio diſtabit ma <lb/>gis, tanto facilius mo<unsure/>uebit. </s> + <s xml:id="s233" xml:space="preserve">Cauſa autem eſt, quę <lb/>retro cõmemorata eſt: </s> + <s xml:id="s234" xml:space="preserve">quoniam quę plus à cen-<lb/>@ro diſtat, maiorem deſcribit circuium: </s> + <s xml:id="s235" xml:space="preserve">quare ab <lb/>eadem potentia plus ſeparabitur mouens illud, <lb/>quod plus ab hypomochlio diſtabit. </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 eſt 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 ſunt 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 eſt: </s> + <s xml:id="s243" xml:space="preserve">hy +<pb o="513" file="0513" n="10" rhead="Mechanicæ."/> +pomochlion auteni fit ſcalmus, ſtat enim ille ẜ <lb/>pondus verò mare eſt, quod propellit remus: </s> + <s xml:id="s244" xml:space="preserve">ve-<lb/>ctem autem mouens ipſe eſt remex. </s> + <s xml:id="s245" xml:space="preserve">Semper aũt <lb/>plus mouet ponderis, quanto magis ab hypomo <lb/>chlio diſtabit, 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">ſcalmus autem hypomo-<lb/>chlion exiſtens, centrum eſt. </s> + <s xml:id="s248" xml:space="preserve">In medio autem na <lb/>uis plurimum remi intus eſt: </s> + <s xml:id="s249" xml:space="preserve">ilia enim parte la-<lb/>tiſſima eſt 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 ſecus eſt. </s> + <s xml:id="s253" xml:space="preserve">Moue-<lb/>tur autem nauis, quoniam appellente ad mare <lb/>remo, extremum illius quod intus eſt, in ante <lb/>prom ouctur. </s> + <s xml:id="s254" xml:space="preserve">nauem verò ſcalmo alligatam ſi-<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 eſt. </s> + <s xml:id="s256" xml:space="preserve">plurimum autem di@i <lb/>dit, vbi pars plurima remia ſcalmo eſt. </s> + <s xml:id="s257" xml:space="preserve">Et eam <lb/>cauſam remiges, qui in media ſunt naui, mouent <lb/>illam maximè. </s> + <s xml:id="s258" xml:space="preserve">maxima enim remi pars à ſcalmo <lb/>in nauis medio intus eſt.</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 exiſtens 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, magnę nauigiorum moueantur <lb/>moles? </s> + <s xml:id="s263" xml:space="preserve">An quoniam gubernaculum vectis eſt: </s> + <s xml:id="s264" xml:space="preserve">o-<lb/>@us autem mare: </s> + <s xml:id="s265" xml:space="preserve">gubernator verò mouens eſt. <lb/></s> + <s xml:id="s266" xml:space="preserve">Non autem ſecundum latitudinẽ, veluti remus, <lb/>mare accipit gubernaculum: </s> + <s xml:id="s267" xml:space="preserve">non enim in antè <lb/>nauigium mouet, ſed ipſum commotum mare <lb/>accipiens inclinat obliquè: </s> + <s xml:id="s268" xml:space="preserve">quoniam enim pon-<lb/>dus eſt mare, contrario innixum modo nauem <lb/>inclinat. </s> + <s xml:id="s269" xml:space="preserve">hypomochlion enim in contrarium <lb/>verſatur: </s> + <s xml:id="s270" xml:space="preserve">mare vero interius, ſi illud exterius. </s> + <s xml:id="s271" xml:space="preserve"><lb/>illud autem ſequitur nauis, quoniam illi eſt ai<unsure/>li-<lb/>@ata. </s> + <s xml:id="s272" xml:space="preserve">Et remus quidem ſecundùm latitudinem +<pb o="514" file="0514" n="11" rhead="Ariſt. Quæſtiones."/> +onus propellens, & </s> + <s xml:id="s273" xml:space="preserve">ab eodem repulſus, in rectũ <lb/>propellit: </s> + <s xml:id="s274" xml:space="preserve">gubernaculum autem, ut obliquum ia <lb/>cet, hinc inde in obliquum motionē facit. </s> + <s xml:id="s275" xml:space="preserve">In ex <lb/>tremo autē, & </s> + <s xml:id="s276" xml:space="preserve">non in medio iacet, quoniam mo <lb/>uenti facilimū eſt 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, ſic ipſius cōtinui in fine imbeciliſſima eſt la-<lb/>tio. </s> + <s xml:id="s279" xml:space="preserve">imbeciliſſima autem ad expellendum eſt 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, quoniā parua ibi motione <lb/>ſacta, multò maius interuallum fit in ultimo: </s> + <s xml:id="s282" xml:space="preserve">ꝗa <lb/>æqualis angulus ſemper maiorem ſpectat: </s> + <s xml:id="s283" xml:space="preserve">tan-<lb/>toq́ue 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 manifeſtum eſt, quam ob <lb/>cauſſam magis in contrarium procedit nauigiũ, <lb/>quàm remi ipſius palmula. </s> + <s xml:id="s286" xml:space="preserve">eadem enim magni <lb/>tudoijſdem mota uiribus, in aete plus, quàm in <lb/>aqua progreditur. </s> + <s xml:id="s287" xml:space="preserve">Si enim A B remus, Guerò <lb/>ſcalmus: </s> + <s xml:id="s288" xml:space="preserve">A autem in nauigio ſit remi principiũ, <lb/>B uerò in mari palmula. </s> + <s xml:id="s289" xml:space="preserve">Si igitur A ubi D tranſ. </s> + <s xml:id="s290" xml:space="preserve"><lb/>latum eſt, non erit B ubi E: </s> + <s xml:id="s291" xml:space="preserve">æqualis enim B E ipſi <lb/>A C. </s> + <s xml:id="s292" xml:space="preserve">æquale igitur tranſlatū erit, ſed minus. </s> + <s xml:id="s293" xml:space="preserve">erit <lb/>igitur ubi eſt F. </s> + <s xml:id="s294" xml:space="preserve">minor enim C F, ipſa A D. </s> + <s xml:id="s295" xml:space="preserve">quare <lb/>ipſa G F, ipſa D F. </s> + <s xml:id="s296" xml:space="preserve">ſimiles enim ſunt trianguli. </s> + <s xml:id="s297" xml:space="preserve"><lb/>Stans autem erit medium, ubi eſt C. </s> + <s xml:id="s298" xml:space="preserve">in contrariũ <lb/>enim ipſi, quod in mari eſt, extremo uidelicet B <lb/>procedit, ubi extremum in nauigio eſt A. </s> + <s xml:id="s299" xml:space="preserve">Non <lb/>procedere autēubi eſt D. </s> + <s xml:id="s300" xml:space="preserve">niſi commouere@ur na <lb/>uigium, & </s> + <s xml:id="s301" xml:space="preserve">eo transferretur, ubi remi eſt princi-<lb/>plum Idipſnm etiam facit gubernaculam, niſi ꝙ <lb/>(ut dictum eſt terro) nihil nauigio ad id, quod-<lb/>antè eſt, confert, ſed ſolùm puppim in obliquũ <lb/>pell@t, ubicunq; </s> + <s xml:id="s302" xml:space="preserve">fuerit in con@rariū enim eo mo <lb/>do uergit prora. </s> + <s xml:id="s303" xml:space="preserve">Vbi igitur applicatum eſt 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">quemadmodũ ſcalmus <lb/>remo. </s> + <s xml:id="s306" xml:space="preserve">Medium autem procedit ſecundùm quod <lb/>gubernaculum trãsfertur. </s> + <s xml:id="s307" xml:space="preserve">ſi quidē introrſus 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 exiſtente 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 ſublimior fuerit, ijſdem <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 ferũtur 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 ſedes, in qua collocatur: </s> + <s xml:id="s316" xml:space="preserve">pondus autem, <lb/>quod moueri debet, ipſum nauigium: </s> + <s xml:id="s317" xml:space="preserve">mou@s ve <lb/>ro is, qui vela tendit, ſpiri@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 ſublata antenna, velum à mali ſede, quæ hypo <lb/>mochlion eſt, 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, conſtringunt: </s> + <s xml:id="s322" xml:space="preserve">illam ve-<lb/>rò quæ proram verſus eſt, pedem facientes rela <lb/>xant? </s> + <s xml:id="s323" xml:space="preserve">An quia retrahere quidem multo exiſten <lb/>tevento gubernaculum non poteſt: </s> + <s xml:id="s324" xml:space="preserve">pauco autẽ <lb/>poteſt, quem conſtringunt. </s> + <s xml:id="s325" xml:space="preserve">Propellit quidem igi <lb/>tur ipſe ventus: </s> + <s xml:id="s326" xml:space="preserve">in puppim vero illum cōſtituit, <lb/>gubernaculum retrahens, & </s> + <s xml:id="s327" xml:space="preserve">mare compellẽs: </s> + <s xml:id="s328" xml:space="preserve">ſi-<lb/>mul & </s> + <s xml:id="s329" xml:space="preserve">nautæ ipſi cum vento contendunt: </s> + <s xml:id="s330" xml:space="preserve">in cõ-<lb/>trariam enim ſe 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">rotũdę ſunt <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 abſidem <lb/>centro ſimul moto, quemadmodum paluſtriver <lb/>titur rota: </s> + <s xml:id="s337" xml:space="preserve">aut circa manens centrū, veluti tro-<lb/>chleæ ſtante centro: </s> + <s xml:id="s338" xml:space="preserve">autin pauimento manente <lb/>@ẽtro, ſicut figuli rota conuertitur. </s> + <s xml:id="s339" xml:space="preserve">An celer@ima +<pb o="516" file="0516" n="13" rhead="Ariſt. Quæſtiones"/> +quidem huiuſmodi ſunt, quoniam parua ſuipa@ <lb/>te planum contingunt, veluti circulas ſecūdum <lb/>punctum: </s> + <s xml:id="s340" xml:space="preserve">& </s> + <s xml:id="s341" xml:space="preserve">quoniam non offenſant: </s> + <s xml:id="s342" xml:space="preserve">à terra em̄ <lb/>ſemotus eſt angulus. </s> + <s xml:id="s343" xml:space="preserve">Pręterea etiam cui obuiam <lb/>fiunt corpori, id rurſum ſecundùm puſillam tan <lb/>gunt. </s> + <s xml:id="s344" xml:space="preserve">ſi autem rectilineum eſſet, rectitudine ſua <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/>ſuper plano circuli fuerit diameter, planū ſecun <lb/>dùm punctum, contingēte circulo æquale vtriu-<lb/>que pondus difterminat diameter. </s> + <s xml:id="s347" xml:space="preserve">cùm autē 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">ſi quidem difficulter ad cōtrarium nutus ſui mo <lb/>uetur modum. </s> + <s xml:id="s351" xml:space="preserve">Præterea nonnulli autumn@nt ꝙ <lb/>circuli linea in perpeti verſatur motu, quemad-<lb/>modum manentia propter contrarium nixũ ma <lb/>nent: </s> + <s xml:id="s352" xml:space="preserve">ſicur maioribus contingit circulis ad mino <lb/>res. </s> + <s xml:id="s353" xml:space="preserve">Celerius enim ab æquali mouentur potentia <lb/>maiores circuli, mouentq́ue 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">ſicut diameter ad <lb/>diametrum, ita maior circulus ad minorem. </s> + <s xml:id="s356" xml:space="preserve">it fi <lb/>niti autem ſunt minores. </s> + <s xml:id="s357" xml:space="preserve">Si autem ad alte@um <lb/>nutum habet circulus, ſimiliter eſt bene mobi-<lb/>lis. </s> + <s xml:id="s358" xml:space="preserve">Et aliam ſanè habet inclinationem circulus, <lb/>& </s> + <s xml:id="s359" xml:space="preserve">ea quæ à circulo mouentur, licèt planitiẽ abſi-<lb/>de non contingat, ſed aut iuxta planitiẽ, aut velu <lb/>tì trocleæ. </s> + <s xml:id="s360" xml:space="preserve">etenim hoc ſe 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 ſui portione cùm tangit, tum offenſat cir-<lb/>culus, ſed aliam ob cauſam. </s> + <s xml:id="s363" xml:space="preserve">ea autem eſt, quæ di <lb/>cta eſt prius, quod circulus ſcilicet ex duabus ef-<lb/>fectus eſt 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 ſemper 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 ſecundùm circun ferentiam illum mouerint: <lb/></s> + <s xml:id="s367" xml:space="preserve">latam enim ipſam mouent. </s> + <s xml:id="s368" xml:space="preserve">Eam quidem igitur, <lb/>quæ in obliquum eſt, motionem ipſum impellit <lb/>mouens: </s> + <s xml:id="s369" xml:space="preserve">ſecundum vero illam, quæ ſuper diame <lb/>trum eſt, ſeipſum 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 quàm minoribus, & </s> + <s xml:id="s374" xml:space="preserve"><lb/>fcytalis ſimiliter? </s> + <s xml:id="s375" xml:space="preserve">An quoniã, qnanto maior fue-<lb/>ri@@illa quæ à centro eſt, in ęquali tempore maius <lb/>mouetur ſpatium. </s> + <s xml:id="s376" xml:space="preserve">Quam ob rem æquali inexiſtẽ <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">ſpar <lb/>tum enim in illis centrum eſt: </s> + <s xml:id="s380" xml:space="preserve">libræ autem vtrin <lb/>que partes, quæ ex centro ſunt, exiſtunt.</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 ſine pondere eſt, moue-<lb/> +<anchor type="note" xlink:label="note-0517-02a" xlink:href="note-0517-02"/> +tur libra, quàm cum pondus habet? </s> + <s xml:id="s383" xml:space="preserve">Similiq́ mo <lb/>do rota, & </s> + <s xml:id="s384" xml:space="preserve">huiuſmodi quippiam, quod grauius <lb/>quidem eſt, maius autem minore & </s> + <s xml:id="s385" xml:space="preserve">leuiore? </s> + <s xml:id="s386" xml:space="preserve">An <lb/>quia non ſolùm in contrarium quod graue eſt, <lb/>ſed 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 eſt: </s> + <s xml:id="s388" xml:space="preserve">quò autem vergit, eſt 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 ſuper ſcytalas facilius portantur onera, <lb/> +<anchor type="note" xlink:label="note-0517-03a" xlink:href="note-0517-03"/> +quã ſ@per currus, cùm tamẽ ij magnas habeant <lb/>rotas, illæ verò puſillas? </s> + <s xml:id="s392" xml:space="preserve">An qui in ſcytalis nulla <lb/>eſt offen ſatio, in curribus autem axis eſt, ad quẽ <lb/>offenſant. </s> + <s xml:id="s393" xml:space="preserve">deſuper enim illum premunt, & </s> + <s xml:id="s394" xml:space="preserve">late-<lb/>ribus. </s> + <s xml:id="s395" xml:space="preserve">quod autem eſt in ſcycalis, ad iſthæc duo <lb/>mouetur, & </s> + <s xml:id="s396" xml:space="preserve">inſernè ſubſtrato ſpatio, & </s> + <s xml:id="s397" xml:space="preserve">onere ſu <lb/>per impoſito. </s> + <s xml:id="s398" xml:space="preserve">in vtriſque 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="Ariſt. Quæſtiones"/> + <p> + <s xml:id="s401" xml:space="preserve">Curlongius feruntur miſſilia funda, quàm ma <lb/> +<anchor type="note" xlink:label="note-0518-01a" xlink:href="note-0518-01"/> +numiſſa, cùm alioquin proiector mann magis <lb/>pondus comprehendat, quàm cùmillud ſuſpen-<lb/>dit? </s> + <s xml:id="s402" xml:space="preserve">Præterea ſic 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 aũt modo ſolum <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/>ſubinde rotans, id iaculatur: </s> + <s xml:id="s407" xml:space="preserve">ex manu autem à <lb/>quiete eſt initium: </s> + <s xml:id="s408" xml:space="preserve">omnia autem cùm in motu <lb/>ſunt, quàm cùm quieſcunt, facilius mouẽtur. </s> + <s xml:id="s409" xml:space="preserve">An <lb/>& </s> + <s xml:id="s410" xml:space="preserve">eam ob cauſam e@t, ſed nec minus etiam, quia <lb/>in fundæ vſu manus quidem fit centrum: </s> + <s xml:id="s411" xml:space="preserve">ſunda <lb/>vero, quod à centro exit. </s> + <s xml:id="s412" xml:space="preserve">quanto aũt productius <lb/>ſueritid, quod à cētro eſt, tanto citius mouetur. <lb/></s> + <s xml:id="s413" xml:space="preserve">iactus aūt qui māu fit, fundę reſpectu breuis eſt.</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, quàm minores mouentur? </s> + <s xml:id="s416" xml:space="preserve">& </s> + <s xml:id="s417" xml:space="preserve">item ſuculæ, <lb/>quæ graciliores ſunt, ab eadẽ viꝗ̃ craſſiores? </s> + <s xml:id="s418" xml:space="preserve">An <lb/>quía ſecula quidem, & </s> + <s xml:id="s419" xml:space="preserve">iugum cētrum eſt: </s> + <s xml:id="s420" xml:space="preserve">pro-<lb/>minentes autem longitudines eæ, quæ ſunt à cē-<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 ſunt circulorum ab eadem vi, quàm quę mi <lb/>norum. </s> + <s xml:id="s423" xml:space="preserve">ab eadem enim vi plus transfertur id <lb/>extremum, quod longius à centro diſtat. </s> + <s xml:id="s424" xml:space="preserve">Quam <lb/>ob rem ad iugum quidem inſtrumenta faciunt <lb/>collopas, quibus facilius verſant: </s> + <s xml:id="s425" xml:space="preserve">in gracilibus <lb/>autem ſuculis 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 eiuſdem magnitudinis lignum facilius <lb/> +<anchor type="note" xlink:label="note-0518-03a" xlink:href="note-0518-03"/> +genus frangitur, ſi quiſpiam æquì diductis mani <lb/>bus extrema comprehendens fregerit, quàm ſi <lb/>iuxta genu: </s> + <s xml:id="s429" xml:space="preserve">& </s> + <s xml:id="s430" xml:space="preserve">ſi terræ illud applicans pede ſu-<lb/>perimpoſito manu longè diducta confregerit, ꝗ̃ <lb/>prope? </s> + <s xml:id="s431" xml:space="preserve">An quiaibi quidem genu centrũ eſt, hic <lb/>vero ipſe pes. </s> + <s xml:id="s432" xml:space="preserve">quanto autem remotius à centro +<pb o="519" file="0519" n="16" rhead="Mechanicæ."/> +ſuerit, faciliùs mouetur quodcunq;</s> + <s xml:id="s433" xml:space="preserve">. moueriau-<lb/>tem quod frangitur, neceſſe eſt.</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 ſunt figura, cùm alioquin à principio ex <lb/>magnis ſint lapidibus, oſtreiſve? </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">intcruallũ <lb/>verò ea, quæ à centro. </s> + <s xml:id="s439" xml:space="preserve">Sem per autem maior ab <lb/>æquali motione maiorem deſcribit circulum. <lb/></s> + <s xml:id="s440" xml:space="preserve">quod autem maius in æquali pertranſit tẽpore, <lb/>celerius fertur: </s> + <s xml:id="s441" xml:space="preserve">quæ autem celeriùs ex ę quali fe <lb/>runtur ſpatio, uehementiùs im petunt: </s> + <s xml:id="s442" xml:space="preserve">quę 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 diſtant, cōfringi neceſ <lb/>ſe eſt, id autem cùm patiantur, rotunda fieri eſt <lb/>neceſſarium. </s> + <s xml:id="s445" xml:space="preserve">Crocis autem propter maris mo-<lb/>tum, quoniam ſimul cum illo agitantur, in perpe <lb/>tua eſſe accidit motione, eoq́; </s> + <s xml:id="s446" xml:space="preserve">verſatas mō ſem <lb/>per offenſare. </s> + <s xml:id="s447" xml:space="preserve">id autem ipſis maximè extremis <lb/>contingere partibus eſt 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 ſuntligna, 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">ſi tollantur, in flectuntur magis: <lb/></s> + <s xml:id="s452" xml:space="preserve">tametſi quod breue quidem eſt, ceu bicubitum, <lb/>fuerit tenue: </s> + <s xml:id="s453" xml:space="preserve">quod verò cubitorum centum, craſ <lb/>ſum? </s> + <s xml:id="s454" xml:space="preserve">An ꝗa & </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 ipſa 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 eſt, pōdus. </s> + <s xml:id="s460" xml:space="preserve">quam ob rem quanto exten. </s> + <s xml:id="s461" xml:space="preserve"><lb/>ſius fuerit id, quod ab hypocomochlio eſt, tanto <lb/>inflecti neceſſe ẽ magis. </s> + <s xml:id="s462" xml:space="preserve">quo enim plus ab hypo-<lb/>mochlio diſtat, eo magis incuruari neceſſe eſt. </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, ipſum inflecti magis <lb/>cùm extollitur neceſſe eſt: </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 vltimũ eſt, quie- +<pb o="520" file="0520" n="17" rhead="Ariſt. Quæſtiones"/> +ſcenti hy pomochlio deprope ſit.</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 ſcindun-<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 ſunt vectes ſibi <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 ipſius 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">quoniā mo <lb/>tum mouet, pſa celeritate valentius fit. </s> + <s xml:id="s479" xml:space="preserve">paruo au <lb/>tem exiſtente vecte, magnæ illum conſequuntur <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 ſcinditur, D E F G. </s> + <s xml:id="s481" xml:space="preserve">vectis igitur ſit ipſa <lb/>A B, pondus verò ipſius 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">ſcindit enim ipſum B.</s> + <s xml:id="s484" xml:space="preserve"/> + </p> + <p> + <s xml:id="s485" xml:space="preserve">Cur ſi quiſpiam trochleas componens duas in <lb/> +<anchor type="note" xlink:label="note-0520-01a" xlink:href="note-0520-01"/> +tignis duobus ad ſe inuicem inuctis cõtrario ad <lb/>trochleas modo circulo funem circunduxerit, <lb/>cuius alterum quidem caput tigonrum appenda <lb/>tur alteri, alterum varò troclheis ſit 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 ſi mouetur, vecte <lb/>medio trãsfertur magis, quàm 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 ſi vna faci-<lb/>lius traher, & </s> + <s xml:id="s490" xml:space="preserve">ab vnico tractu multò grauius tra-<lb/>het, quàm facere poſſit manus, idip ſum duæ tro-<lb/>chleæ plus quàm in dupla velocitate leuabunt. <lb/></s> + <s xml:id="s491" xml:space="preserve">Minus enim altera trahit, quàm ſi ip ſa per ſeip-<lb/>ſam traheret, quando circa alteram iniectus ſue <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">Pariq́; </s> + <s xml:id="s494" xml:space="preserve">modo ſi ad plures iniectus fuerit funis <lb/>in paucis trochleis, multa ſit 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, traducũt <lb/>enim ab una trochlea ad aliam, & </s> + <s xml:id="s497" xml:space="preserve">rurſus ab illa <lb/>ad ſuculas, & </s> + <s xml:id="s498" xml:space="preserve">vectes. </s> + <s xml:id="s499" xml:space="preserve">hoc autem idem eſt, ac ſi <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 ſi quis ſuper lignum magnã imponat ſe-<lb/> +<anchor type="note" xlink:label="note-0521-01a" xlink:href="note-0521-01"/> +carim, deſuperq; </s> + <s xml:id="s502" xml:space="preserve">illi magnum adijciat pondus, <lb/>ligni quippiam quod curandum ſit, non diuidit: <lb/></s> + <s xml:id="s503" xml:space="preserve">ſiverò ſecurim extoll@s percutiat, illud ſcindit, <lb/>cùm alioquin multò minus habeat ponderis id <lb/>quod percutit, quàm id quod ſuperiacet, & </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/>ipſum grauitatis magis aſſumit motum dũ mo-<lb/>uetur, ꝗ̃ dum quieſcit Incumbensigitur conua-<lb/>tum graui motionem non mouetur, motum ve-<lb/>rò & </s> + <s xml:id="s508" xml:space="preserve">ſecundum hanc mouetur, & </s> + <s xml:id="s509" xml:space="preserve">ſecundùm eã <lb/>quæ eſt percutientis. </s> + <s xml:id="s510" xml:space="preserve">Pręterea ſecuris ipſa effici-<lb/>tur cuneus: </s> + <s xml:id="s511" xml:space="preserve">paruus autẽ exiſtens cuneus magna <lb/>diuidit cum ex duobus ſit vectibus contrario ad <lb/>ſeſe inuicem modo conſtitutis</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 ſtatera, 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, ſolùm ſuſpenditur lanx: </s> + <s xml:id="s514" xml:space="preserve">in altera verò par <lb/>te, ſola eſt ſtatera. </s> + <s xml:id="s515" xml:space="preserve">An quia ſimul libram & </s> + <s xml:id="s516" xml:space="preserve">vectẽ <lb/>ipſam contingit eſſe ſtateram. </s> + <s xml:id="s517" xml:space="preserve">libram quidem, <lb/>vbi ſpartorum quodcunque ſtateræ 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 ſi 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">manifeſtũ enim, <lb/>quòd tantundem tradit ponderis ei quod in alte <lb/>raiacet lance. </s> + <s xml:id="s522" xml:space="preserve">Quemadmodum autem ſi vna li-<lb/>bra multæ ſint libræ, ſic talia inſunt ſparta mul.</s> + <s xml:id="s523" xml:space="preserve"> +<pb o="522" file="0522" n="19" rhead="Ariſt. Quæſtiones"/> +tain eiuſmodi libra: </s> + <s xml:id="s524" xml:space="preserve">quorum vniuſcuiuſq; </s> + <s xml:id="s525" xml:space="preserve">quod <lb/>intrinſecus eſt ad appẽdiculum, ſtateræ eſt dimi-<lb/>dium. </s> + <s xml:id="s526" xml:space="preserve">& </s> + <s xml:id="s527" xml:space="preserve">omnino iſtud libra eſt, vnam quidẽ ha-<lb/>benslancem, in qua pondus appẽditur: </s> + <s xml:id="s528" xml:space="preserve">alteram <lb/>verò, vbi in ſtatera æquipondium. </s> + <s xml:id="s529" xml:space="preserve">Quam ob rem <lb/>appendiculum ad alteram ſui partem eſt ſtatera. <lb/></s> + <s xml:id="s530" xml:space="preserve">huiuſmodi autem exiſtēs multæ ſunt libræ, totq́; </s> + <s xml:id="s531" xml:space="preserve"><lb/>quot fuerint ſparta. </s> + <s xml:id="s532" xml:space="preserve">Semper autem quod lanci <lb/>propinquius eſt ſpartum, appenſoq́ue oneri, ma-<lb/>ius trahit pōdus: </s> + <s xml:id="s533" xml:space="preserve">quoniam fit quidem omnis ſta-<lb/>tera inuerſus vectis: </s> + <s xml:id="s534" xml:space="preserve">hypomochlion nãq; </s> + <s xml:id="s535" xml:space="preserve">vnum-<lb/>quodque ſpartum ſuperne exiſtens: </s> + <s xml:id="s536" xml:space="preserve">pondus verũ <lb/>id quod lanci ineſt. </s> + <s xml:id="s537" xml:space="preserve">Quanto autem productior ve <lb/>ctis ſuerit longitudo ab ipfo hypomochlio, tãto <lb/>ibi quidem faciliùs mouet: </s> + <s xml:id="s538" xml:space="preserve">hic autem æquilibri-<lb/>brium facit, pondusq́ue ſtateræ 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 ſacilius dentes extrahūt dentifor-<lb/>cipis onere adiecto, quàm ſi ſola vtantur manu? <lb/></s> + <s xml:id="s541" xml:space="preserve">An quia ex manu magis quàm ex dentiforcipe <lb/>lubricus elabitur dẽs? </s> + <s xml:id="s542" xml:space="preserve">An ferro id potius accidit <lb/>quàm digitis, quoniam vndique dentem non cō-<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/>@iſorceps duo contrarij vectes, vnicum habentes <lb/>hypom ochlion, eius ſcilicet inſtrumenti 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 eſt 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 ſimul comprehendentes mouent. </s> + <s xml:id="s558" xml:space="preserve"><lb/>quando autem commotus fuerit, faciliùs manu <lb/>trahitur, quàm inſtrumento.</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 abſque ictu facilè cõfrigun tinſtru-<lb/>mẽtis, quæ ad eum fiunt vſum? </s> + <s xml:id="s561" xml:space="preserve">multumeinim au <lb/>fertur virium, motionis ſcilicet, & </s> + <s xml:id="s562" xml:space="preserve">violẽ@iæ. </s> + <s xml:id="s563" xml:space="preserve">Prę-<lb/>terea duro & </s> + <s xml:id="s564" xml:space="preserve">graui comprimens inſtrumento ci <lb/>tiùs conſringet, quàm ligneo & </s> + <s xml:id="s565" xml:space="preserve">leui. </s> + <s xml:id="s566" xml:space="preserve">An quia ſic <lb/>vtrinqne à duobus comprimitur vectibus ipſa <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 inſtrumẽtum ex duobus componitur vecti <lb/>bus idem habentibus hypomochlion, connexio-<lb/>nem videlicet @pſam vbi eſt A. </s> + <s xml:id="s569" xml:space="preserve">Quemadmodum <lb/>igitur fuere diductę ſecundũ extrema motis C D <lb/>ipſæ F E, ſic à parua faciliter potentia conducun <lb/>tur. </s> + <s xml:id="s570" xml:space="preserve">Quod igitur cum percuſſione ſeciſſet 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 eſt K. </s> + <s xml:id="s574" xml:space="preserve">Hanc etiam ob cauſam <lb/>quanto vicinius fuerit K ipſum A, confringitur <lb/>celerius. </s> + <s xml:id="s575" xml:space="preserve">quanto enim ab hypomochlio plus di-<lb/>ſtat 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">Eſt igitur A quidem hypomochlion: </s> + <s xml:id="s578" xml:space="preserve">ipſa 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 ipſum K vicinius ſuerit angulo ipſius A, tãto <lb/>vicinius fit connexioni vbi eſt A. </s> + <s xml:id="s582" xml:space="preserve">hoc autem eſt <lb/>hypomochlion. </s> + <s xml:id="s583" xml:space="preserve">Ab eadem igitur potentia appli-<lb/>cante F E plus extolli neceſſe eſt. </s> + <s xml:id="s584" xml:space="preserve">Quam ob rem <lb/>quoniã ex contrario eſt eleuatio, neceſſe eſt 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 ſi duo extrema in rhombo puncta dua-<lb/>bus ferantur lationibus, haudquaquam æquale <lb/>vtrunque eorũ pertianſit rectam, ſed multò plus <lb/>alterum ?</s> + <s xml:id="s588" xml:space="preserve">idem autem eſt ſermo, cur quod ſuper <lb/>latus fertur, minus pertranſit, quàm ipſum 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 ſer +<pb o="524" file="0524" n="21" rhead="Ariſt Quæſtiones."/> +t<unsure/>ur lationibus. </s> + <s xml:id="s592" xml:space="preserve">Feratur enim ex ipſa A B, A qui-<lb/>dem ad ipſum B, B vero ad ipſum D eadem cele <lb/>ritate. </s> + <s xml:id="s593" xml:space="preserve">feratur aurẽ & </s> + <s xml:id="s594" xml:space="preserve">ipſa A B in ipſa A C iuxta <lb/>C D eadem celeritate cum illis. </s> + <s xml:id="s595" xml:space="preserve">neceſſe igitur eſt <lb/>A quidem in ipſa A D diametro ferri, B verò in <lb/>ipſa 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">ſimul pertranſiſſe, & </s> + <s xml:id="s599" xml:space="preserve">ipſa@@ <lb/>A B ipſum la<unsure/>tus A C. </s> + <s xml:id="s600" xml:space="preserve">latum enim ſit ipſum A ip <lb/>ſam A E, A B autem ipſam A E, & </s> + <s xml:id="s601" xml:space="preserve">proiecta ſit <lb/>F G iuxta ipſum A B, & </s> + <s xml:id="s602" xml:space="preserve">ab ipſo E ſimiliter replea <lb/>tur. </s> + <s xml:id="s603" xml:space="preserve">Simile igitur fit, quod repletum eſt ipſi toti. <lb/></s> + <s xml:id="s604" xml:space="preserve">æqualis igirur A F ipſi A E ipſa autem A B ipſam <lb/>A F lata erit. </s> + <s xml:id="s605" xml:space="preserve">in diametro igitur erit ſecūdū K. </s> + <s xml:id="s606" xml:space="preserve">& </s> + <s xml:id="s607" xml:space="preserve"><lb/>fem per neceſſe eſt ipſum ferri ſecundùm diame <lb/>trum. </s> + <s xml:id="s608" xml:space="preserve">& </s> + <s xml:id="s609" xml:space="preserve">ſimul A B latus pertranfit latus A C, & </s> + <s xml:id="s610" xml:space="preserve"><lb/>ipſum A diametrum pertranſit A D. </s> + <s xml:id="s611" xml:space="preserve">Similiter ẽt <lb/>demon ſtrabitur & </s> + <s xml:id="s612" xml:space="preserve">ipſum B in ipſa B C diametro <lb/>latum. </s> + <s xml:id="s613" xml:space="preserve">æqualis enim eſt ipſa B E ipſi B G Reple. </s> + <s xml:id="s614" xml:space="preserve"><lb/>to igitur ab ipſo G. </s> + <s xml:id="s615" xml:space="preserve">quod intus eſt, toti eſt ſimi-<lb/>le: </s> + <s xml:id="s616" xml:space="preserve">& </s> + <s xml:id="s617" xml:space="preserve">ipſum B in ipſa diame@ro erit ſecundùm la <lb/>terum connexionem. </s> + <s xml:id="s618" xml:space="preserve">Et ſimul latus pertranſit <lb/>latus, & </s> + <s xml:id="s619" xml:space="preserve">B ipſum B C diametrum. </s> + <s xml:id="s620" xml:space="preserve">Simuligitur A <lb/>multo plus ipſa A B pertrāſit: </s> + <s xml:id="s621" xml:space="preserve">& </s> + <s xml:id="s622" xml:space="preserve">ipſum latus, mi-<lb/>nus latus eadem lata celeritate: </s> + <s xml:id="s623" xml:space="preserve">& </s> + <s xml:id="s624" xml:space="preserve">ipſum latus <lb/>maiorem, quàm B, pertranſiuit vna latum latio-<lb/>ne. </s> + <s xml:id="s625" xml:space="preserve">Quanto enim acutior fuerit rhombus, diame <lb/>ter quidem minor ſit. </s> + <s xml:id="s626" xml:space="preserve">A C autẽ maior: </s> + <s xml:id="s627" xml:space="preserve">latus ve-<lb/>ro ipſius B C minus. </s> + <s xml:id="s628" xml:space="preserve">Abſurdū eſt enim (vt dictũ) <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">vtriſq; </s> + <s xml:id="s632" xml:space="preserve">poſitis <lb/>æquali velocitate punctis, alterū pertranſire ma <lb/>iorem. </s> + <s xml:id="s633" xml:space="preserve">Cauſa autem eſt, quoniã ei quod ab obt@ <lb/>ſo ſertur angulo ambæ ferè cōtrariæ fiunt latio-<lb/>n<unsure/>es, & </s> + <s xml:id="s634" xml:space="preserve">illa ſecūdū quā ipſum fertur, & </s> + <s xml:id="s635" xml:space="preserve">illa ſecun <lb/>dũ quã ipſum à latere defertur. </s> + <s xml:id="s636" xml:space="preserve">Ei autẽ quod ab <lb/>@cuto fertur, accidit vt ad idē feratur, coadiuuat +<pb o="525" file="0525" n="22" rhead="Mechanicæ."/> +@nim quæ ipſius eſt lateris, illã quæ eſt ſuper 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 obtuſum magis: </s> + <s xml:id="s639" xml:space="preserve">hæc quidē tar-<lb/>dior erit, illa verò celerior. </s> + <s xml:id="s640" xml:space="preserve">Hæ quidẽ igitur ma-<lb/>gis contrariæ fiunt, qm̄ obtuſior fit angulus: </s> + <s xml:id="s641" xml:space="preserve">illæ <lb/>verò ad idẽ magis, qm̄ lineæ coarctantur: </s> + <s xml:id="s642" xml:space="preserve">ipſum <lb/>enim A ferè ad idem fertur ſecundum 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 ſanè <lb/>acutior fuerit angulus, tanto magis ipſum A ad <lb/>contrarium: </s> + <s xml:id="s646" xml:space="preserve">ipſum. </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/>fertipſum ad D. </s> + <s xml:id="s650" xml:space="preserve">Et quanto ſanè obtuſior 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">ſi autẽ omnino recta fieret, <lb/>penitus vtique eſſent contrariæ latus verò ſecun <lb/>dùm vnicam latuum lationem à nullo præpedi-<lb/>tur: </s> + <s xml:id="s653" xml:space="preserve">rationabiliter igitur maiorem pertranſit.</s> + <s xml:id="s654" xml:space="preserve"/> + </p> + <p> + <s xml:id="s655" xml:space="preserve">Dubitatur quam ob cauſam maior circulus ę-<lb/>qualem minori circulo conuoluit<unsure/>ur lineam, qū<unsure/> <lb/>circa idem centrum fuerint poſiti: </s> + <s xml:id="s656" xml:space="preserve">ſeorſum autẽ <lb/>reuoluti, quemadmodum alterius magnitudo ad <lb/>magnitudiuem ſe habet alterius, fic & </s> + <s xml:id="s657" xml:space="preserve">illorũ ad <lb/>ſeinuicem fiunt lineæ. </s> + <s xml:id="s658" xml:space="preserve">Præterea vno et@ã & </s> + <s xml:id="s659" xml:space="preserve">eodē <lb/>v<unsure/>triſque exiſtente centro, aliquando quidem tan <lb/>ta ſit linea, quam conuoluuntur, quantam minor <lb/>per ſe 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, manifeſtum eſt. </s> + <s xml:id="s662" xml:space="preserve">angulus enim <lb/>ſen ſu videtur eſſe cuiuſque 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 ſecundùm ſenſum ad ſe lineæ. </s> + <s xml:id="s664" xml:space="preserve">ſecun-<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 poſiti centrum, manifeſtũ eſt: </s> + <s xml:id="s666" xml:space="preserve">& </s> + <s xml:id="s667" xml:space="preserve"><lb/>ſic fiunt aliquando quidē æquales lineæ, ſecun- +<pb o="526" file="0526" n="23" rhead="Ariſt Quæſtiones"/> +dum quam maior conuoluitur circulus, aliquan <lb/>do verò ſecundùm 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">vtriſq́; </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 ſe magnus conuoluitur, ſit vbi F I: </s> + <s xml:id="s672" xml:space="preserve">quã <lb/>vero per ſe minor vbi GK, æqualis A. </s> + <s xml:id="s673" xml:space="preserve">Si igitur mi <lb/>norem mouero, idem mouens centū vbi A, ma-<lb/>ior autem ſit annexus. </s> + <s xml:id="s674" xml:space="preserve">qñ igitur A B fuerit recta <lb/>ad ipſam G K, ſimul & </s> + <s xml:id="s675" xml:space="preserve">A C fit recta ad ipſam F I. <lb/></s> + <s xml:id="s676" xml:space="preserve">quam ob rem ęqualem ſemper tranſlata erit ip-<lb/>ſam quidem G K, vbi eſt G B circunferentia: </s> + <s xml:id="s677" xml:space="preserve">ip-<lb/>ſam verò FL, quæ eſt vbi F C. </s> + <s xml:id="s678" xml:space="preserve">Si aũt quarta pars <lb/>æqualem conuoluitur, manifeſtum eſt, ꝙ totus <lb/>circulus toti circulo æqualem conuoluetur. </s> + <s xml:id="s679" xml:space="preserve">qua <lb/>re qñ B C, linea ad ipſum peruenerit K, & </s> + <s xml:id="s680" xml:space="preserve">ipſa <lb/>F C circunferentia erit in ipſa C L: </s> + <s xml:id="s681" xml:space="preserve">& </s> + <s xml:id="s682" xml:space="preserve">vniuerſus <lb/>crit conuolutus circulus. </s> + <s xml:id="s683" xml:space="preserve">Similiq́; </s> + <s xml:id="s684" xml:space="preserve">modo ſi ma-<lb/>gnum mouero, illi paruuū annectens, eodem exi <lb/>ftente cẽtro, ſimul cum A C ipſi 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 ipſam F I, illa <lb/>verò ad G M. </s> + <s xml:id="s687" xml:space="preserve">Quam ob rem quando hæc quidẽ <lb/>æqualem ipſi G M pert<unsure/>rãſiuerit, illa verò ipſi F I, <lb/>& </s> + <s xml:id="s688" xml:space="preserve">rurſum facta fuerit recta ipſa F A ad ipſam <lb/>F L. </s> + <s xml:id="s689" xml:space="preserve">& </s> + <s xml:id="s690" xml:space="preserve">ipſa A C rurſum recta, velut à principio <lb/>erant in ipſis 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">ſ. </s> + <s xml:id="s693" xml:space="preserve">per <lb/>aliquod temporis ſpatium ſtaret in eodem pun-<lb/>cto: </s> + <s xml:id="s694" xml:space="preserve">neque tranſiliente minore aliquod punctū, <lb/>maiorem quidem æqualem minori pertranſire, <lb/>hunc a@tem maiori, abſurdum eſt. </s> + <s xml:id="s695" xml:space="preserve">Præterea vni-<lb/>ea etiam ſemper exiſtente motione, centrū mo-<lb/>rum interdum quidem magnam, nonnunquàm <lb/>verò minorem conuerti, admirandum eſt. </s> + <s xml:id="s696" xml:space="preserve">Idem <lb/>enim celeritate eadem latus æqualẽ natum hoc <lb/>eft pertranſire: </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/>ſumendum eſt circa iſtorum cauſam, ꝙ 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 ſeipſo moueri natũ non ſit, <lb/>ſi ſimul & </s> + <s xml:id="s701" xml:space="preserve">illud mouerit, quod natũ eſt moueri. <lb/></s> + <s xml:id="s702" xml:space="preserve">tardius mouebitur, quàm ſi ipſum per ſe moue-<lb/>retur. </s> + <s xml:id="s703" xml:space="preserve">Et ſi quidem natum fuerit moueri, non ſi-<lb/>mul aũt moueatur, ſimiliter ſe habebit. </s> + <s xml:id="s704" xml:space="preserve">Et im<unsure/>poſ <lb/>ſibile certè eſt plus moueri, quàm mouens: </s> + <s xml:id="s705" xml:space="preserve">non <lb/>enim ſuam ipſius mouetur motionem. </s> + <s xml:id="s706" xml:space="preserve">Si igitur<unsure/> <lb/>circulus maior vbi A, minor autem vbi B, ſi mi-<lb/>nor maiorem impellet non reuolutum ex ſe, ma<unsure/> <lb/>nifeſtum eſt, ꝙ tantum ipſius rectæ maior pertrã <lb/>ſit, quantum eſt impulſus: </s> + <s xml:id="s707" xml:space="preserve">tantum aũt eſt impul-<lb/>ſus, quantum paruus eſt motus: </s> + <s xml:id="s708" xml:space="preserve">æqualem igitur <lb/>ipſius rectæ pertranſiuerunt. </s> + <s xml:id="s709" xml:space="preserve">Neceſſe igitur eſt <lb/>ſi reuolutus minor maiorem impellet, reuol@i ſi<unsure/> <lb/>mul cum impulſione: </s> + <s xml:id="s710" xml:space="preserve">tantum aūt quantum mi-<lb/>nor reuolutus eſt, ſi nihil ipſe ſub ipſius motio-<lb/>ne mouetur. </s> + <s xml:id="s711" xml:space="preserve">Qūo enim, & </s> + <s xml:id="s712" xml:space="preserve">quantum mouit, tan-<lb/>tum motũ eſſe neceſſe eſt, quod mouetur ab ill@ <lb/>ſed profectò paruus circulus tãtum ſeipſum cir-<lb/>culariter mouit, quantum eſt pedalis quantitas: </s> + <s xml:id="s713" xml:space="preserve"><lb/>(tantum enim ſit id quod motus eſt) & </s> + <s xml:id="s714" xml:space="preserve">magnus <lb/>igitur tantum motus erit. </s> + <s xml:id="s715" xml:space="preserve">Similiq́; </s> + <s xml:id="s716" xml:space="preserve">modo ſi ma-<lb/>gnus paruū mouebit, motus erit paruus, quēad-<lb/>modum maior. </s> + <s xml:id="s717" xml:space="preserve">Per ſe aūt motus illorum vtrum-<lb/>libet ſiue celeriter, ſeu tardè eadem velocitate, <lb/>ſtatim quantum maior natus eſt circunſerri li-<lb/>neam: </s> + <s xml:id="s718" xml:space="preserve">quod difficultatē facit, quod non ſimiliter <lb/>ſaciunt, quando fuerint connexi. </s> + <s xml:id="s719" xml:space="preserve">hoc autem eſt, <lb/>ſi alteri ab altero moueatur, non quã natus eſt, <lb/>@eque peculiarem motionem. </s> + <s xml:id="s720" xml:space="preserve">Nihil enim referr <lb/>citcũponere, & </s> + <s xml:id="s721" xml:space="preserve">annectere, aut cōiungere vtrum- +<pb o="528" file="0528" n="25" rhead="Ariſt Queſtiones"/> +liber alteri. </s> + <s xml:id="s722" xml:space="preserve">ſimiliter enim quando hic quidem <lb/>mouer, ille vero mouetur ab iſto; </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 propenſus, <lb/>non ſemper conueluitur: </s> + <s xml:id="s726" xml:space="preserve">quādo verò circa idem <lb/>poſi@i fuerint cẽtrum, alterum ab altero ſemper <lb/>conuolui neceſſe eſt. </s> + <s xml:id="s727" xml:space="preserve">Sed nihilominus non ſuam <lb/>ipſius motionem mouetur alter, ſed veluti nullã <lb/>haberet motionem: </s> + <s xml:id="s728" xml:space="preserve">& </s> + <s xml:id="s729" xml:space="preserve">ſi habuerit, illa autem nõ <lb/>vtatur, tan@undem accidit. </s> + <s xml:id="s730" xml:space="preserve">Quandoquidem igi-<lb/>tur màgnus moueri ſibi alligatum paruum, par-<lb/>uus mouetur quãtum ille: </s> + <s xml:id="s731" xml:space="preserve">quādo autem paruus, <lb/>rurſus magnus quantum iſte: </s> + <s xml:id="s732" xml:space="preserve">ſeparatus aũt vter-<lb/>que ſeipſum mouet. </s> + <s xml:id="s733" xml:space="preserve">Quod autem eodem exiſten <lb/>te centro, & </s> + <s xml:id="s734" xml:space="preserve">mouente eadem velocitate, accidit <lb/>inæqualem illos pertrāſire lineam, paralogiſmo <lb/>fophiſticę vtitur is, qui dubitat. </s> + <s xml:id="s735" xml:space="preserve">idem enim am-<lb/>bobus eſt centrũ, verùm per accidens, velúti mu <lb/>ſicum & </s> + <s xml:id="s736" xml:space="preserve">album. </s> + <s xml:id="s737" xml:space="preserve">eſſe enim vtriuſque circuli cen-<lb/>tro nō 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 ſim pliciter mouet, ſed eſt quomodo.</s> + <s xml:id="s742" xml:space="preserve"/> + </p> + <p> + <s xml:id="s743" xml:space="preserve">Cur lect@lorum ſpondas ſecūdum duplam fa-<lb/> +<anchor type="note" xlink:label="note-0528-01a" xlink:href="note-0528-01"/> +ciunt proportionem, hanc quidem ſex pedū vel <lb/>paulo am pliorem, illam vero trium: </s> + <s xml:id="s744" xml:space="preserve">curve non <lb/>ſecundùm diametrum illos reſtibus extendunt? <lb/></s> + <s xml:id="s745" xml:space="preserve">An tantos quidem magnitudine faciunt, vt cor-<lb/>poribus ſint proportionem habẽtes: </s> + <s xml:id="s746" xml:space="preserve">fiunt enim <lb/>ſic ſecundum ſpondas dupli, longitudine quidẽ <lb/>cubitorum quatuor, latitudine vero duorũ. </s> + <s xml:id="s747" xml:space="preserve">Ex-<lb/>tendunt autem illos nō ſecundùm diametrum, <lb/>ſed ex oppoſito, vt & </s> + <s xml:id="s748" xml:space="preserve">ligna minus diſtra hantur. </s> + <s xml:id="s749" xml:space="preserve"><lb/>celertrimè enim ſcindũtur ſecundum naturam <lb/>diuiſa, & </s> + <s xml:id="s750" xml:space="preserve">eodem modo diſtenta laborant ma- +<pb o="529" file="0529" n="26" rhead="Mechanicæ."/> +@imè. </s> + <s xml:id="s751" xml:space="preserve">Amplius, quoniam opus eſt, vt reſtes pon-<lb/>dus ferre poſſint, ſic certè pondere impoſito mi-<lb/>nus laborabunt, ſi tranſuerſim, quàm ſi obliquè <lb/>extendantur. </s> + <s xml:id="s752" xml:space="preserve">Præterea hoc etiam modo minus <lb/>abſumitur reſtium. </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 ipſa F G ſecundùm B, æqua-<lb/>lia certè foramina ſunt in ipſa F B, & </s> + <s xml:id="s755" xml:space="preserve">in ipſa F A. <lb/></s> + <s xml:id="s756" xml:space="preserve">latera enim ſunt æqualia. </s> + <s xml:id="s757" xml:space="preserve">nam totum F G duplũ <lb/>eſt. </s> + <s xml:id="s758" xml:space="preserve">Extendunt autem, vt deſcriptum eſt, ab ipſo <lb/>A ad ipſum B ita vbi eſt C, ita vbi D: </s> + <s xml:id="s759" xml:space="preserve">ita vbi H, <lb/>poſtea vbi E, & </s> + <s xml:id="s760" xml:space="preserve">eodem ſemper modo, donec ad <lb/>angulum peruenerint alium. </s> + <s xml:id="s761" xml:space="preserve">Duo enim anguli <lb/>reſtis habent capi a. </s> + <s xml:id="s762" xml:space="preserve">æquales autem ſunt reſtes <lb/>ſecundum curuaturas, videlicet A B; </s> + <s xml:id="s763" xml:space="preserve">& </s> + <s xml:id="s764" xml:space="preserve">B C ipſis <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 ſimili ſe habet modo, quo-<lb/>niam eadem demonſtratio. </s> + <s xml:id="s768" xml:space="preserve">ipſa enim AB æqua-<lb/>lis eſt ipſi H E: </s> + <s xml:id="s769" xml:space="preserve">æqualia enim ſunt latera ſpatij <lb/>B G, M A, & </s> + <s xml:id="s770" xml:space="preserve">foramina æquè diſtant, ipſa autem <lb/>B G æquelis eſt ipſi M A: </s> + <s xml:id="s771" xml:space="preserve">angulus enim B æqua-<lb/>lis eſt 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 eſt ſemirectus: </s> + <s xml:id="s775" xml:space="preserve"><lb/>eſt enim F B æqualis ipſi F A: </s> + <s xml:id="s776" xml:space="preserve">& </s> + <s xml:id="s777" xml:space="preserve">angulus vbi F, re <lb/>ctus eſt. </s> + <s xml:id="s778" xml:space="preserve">B autẽ angulus æqualis ei vbi eſt G, quo-<lb/>niam quadratum altera parte longius duplũ eſt: </s> + <s xml:id="s779" xml:space="preserve"><lb/>& </s> + <s xml:id="s780" xml:space="preserve">ad medium eſt curuatura. </s> + <s xml:id="s781" xml:space="preserve">quamobrẽ A D ip-<lb/>ſi E G eſt æqualis: </s> + <s xml:id="s782" xml:space="preserve">huic verò ipſa H M. </s> + <s xml:id="s783" xml:space="preserve">Similiq́ue <lb/>modo demonſtrantur aliæ, quoniã æquales ſunt <lb/>duæ, quæ ſecũdùm curuataras ſunt, duabus. </s> + <s xml:id="s784" xml:space="preserve">Qua <lb/>re manifeſtum eſt, ꝙ tot ſunt reſtes in lectulo, <lb/>quot ſunt quatuor, ſicut A B. </s> + <s xml:id="s785" xml:space="preserve">Quanta autem fora <lb/>minũ eſt multirudo in ipſo F G latere, & </s> + <s xml:id="s786" xml:space="preserve">in eius <lb/>dimidio F B eſt medietas. </s> + <s xml:id="s787" xml:space="preserve">Quamobrem in dimi-<lb/>diato lectulo tantæ reſtium magnitudines erũt, <lb/>quantum eſt A B: </s> + <s xml:id="s788" xml:space="preserve">multitudine verò tot, quot in <lb/>B G ſunt foramina. </s> + <s xml:id="s789" xml:space="preserve">Hoc autem nihil refert dice +<pb o="530" file="0530" n="27" rhead="Ariſt. Quæſtiones"/> +re, quàm quot ſunt in ipſis A F & </s> + <s xml:id="s790" xml:space="preserve">B F ſimul <lb/>ſumpris. </s> + <s xml:id="s791" xml:space="preserve">Si autem ſecundum diametrum exten-<lb/>dantur reſtes, quemad modum ſe habet in lectu-<lb/>lo A B C D, dimidia non tot ſunt, 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, ſunt foramina: </s> + <s xml:id="s793" xml:space="preserve">maiores autem ſunt</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 exiſtentes, quàm A B, <lb/>quare reſtis in tantum maior, quantum ambola <lb/>@era diame@ro ſunt maiora.</s> + <s xml:id="s795" xml:space="preserve"/> + </p> + <p> + <s xml:id="s796" xml:space="preserve">Cur difficilius eſt longa ligna ab extremo ſu-<lb/> +<anchor type="note" xlink:label="note-0530-01a" xlink:href="note-0530-01"/> +per humeros ferre, quàm ſecundum medium, æ-<lb/>quali exiſtente pondere? </s> + <s xml:id="s797" xml:space="preserve">An quia vibrato ligno <lb/>ipſum 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 eſt ab extremo, <lb/>quoniam faciliùs ex medio eleuatur, quàm ab <lb/>extremo, & </s> + <s xml:id="s799" xml:space="preserve">ideo ſic ferre eſt facilius: </s> + <s xml:id="s800" xml:space="preserve">Cauſa aũt, <lb/>qm̄ ſecundùm medium quidem eleuato ligno, <lb/>ſemper ſeſe inuicem ſuſpendunt extrema, & </s> + <s xml:id="s801" xml:space="preserve">al-<lb/>tera pars alteram bene ſubleuat medium enim <lb/>veluti centrum fit, vbi habet is, qui eleuat, aut <lb/>ſert. </s> + <s xml:id="s802" xml:space="preserve">Extremorũ igitur v@@rũq; </s> + <s xml:id="s803" xml:space="preserve">deorſum vergens, <lb/>ſurſum ſuſpenditur. </s> + <s xml:id="s804" xml:space="preserve">ꝙ ſi ab extremo eleuetur, <lb/>aut feratur, non ſanè facit: </s> + <s xml:id="s805" xml:space="preserve">ſed vniuerſum 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 ſecundùm A, ipſum q́uidem B <lb/>deorſum @utans, ſurſum eleuat C: </s> + <s xml:id="s808" xml:space="preserve">ipſum autem <lb/>C deorſum nutans, B ſurſum eleuat: </s> + <s xml:id="s809" xml:space="preserve">ambo au-<lb/>tem ſurſum 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 ſi valde procerum fuerit idẽ pondus diffi-<lb/> +<anchor type="note" xlink:label="note-0530-02a" xlink:href="note-0530-02"/> +cilius ſuper humeros geſtatur, etiam ſi medium <lb/>quiſpiam illud ferat, quàm ſi breuius ſit? </s> + <s xml:id="s812" xml:space="preserve">Quod <lb/>@nim dudum dictum eſt, cauſa nõ eſt: </s> + <s xml:id="s813" xml:space="preserve">ſed vibra- +<pb o="531" file="0531" n="28" rhead="Mechanicę."/> +tio nunc eſt cauſa. </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 geſtare. </s> + <s xml:id="s818" xml:space="preserve">Vibrationis aũt cã eſt, quoniã <lb/>ab eadẽ motione magis transferuntur extrema, <lb/>quanto procerius fuerit lignũ. </s> + <s xml:id="s819" xml:space="preserve">Humerus quidẽ <lb/>fit centrũ 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 aũt AB, AC, quę <lb/>ſunt ex centro. </s> + <s xml:id="s824" xml:space="preserve">quanto aũt maius fuerit id quod <lb/>ex centro eſt, ſiue A B, ſeu A C, plus transfertur <lb/>ſpatij. </s> + <s xml:id="s825" xml:space="preserve">Demonſtratum autem eſt 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 viſun <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 adiũgunt pondus: </s> + <s xml:id="s831" xml:space="preserve">cũ <lb/>alioquin vas ipſum & </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 qm̄ duobus tem poribus hau@ien <lb/>di, diuiſo 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 ſur-<lb/>ſum trahere) continget demittere quidẽ vacuum <lb/>faciliter, trahere vero plenum difficulter. </s> + <s xml:id="s838" xml:space="preserve">Cōmo-<lb/>dum igitur eſt paulò tardius illud demittere, cū <lb/>multò leuius effectum ſuſtolatur 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 ſit pondus, ꝗ <lb/>fi ſolummodo vacuum oportet demittere: </s> + <s xml:id="s841" xml:space="preserve">cum <lb/>verò plenum fuerit, ſurſum 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 ſunt, quàm 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 ſuper ligno, aut hmõi quopiam duo <lb/> +<anchor type="note" xlink:label="note-0531-02a" xlink:href="note-0531-02"/> +portauerint homines æquale pondus, non ſimili <lb/>ter premuntur, ſi ad vnum non declinet pondus, <lb/>ſeu magis, quanto vicinius fuerit geſtãtibus? </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 eſt pon-<lb/>deri ex ijs, qui illud geſtant, id quod mouetur: </s> + <s xml:id="s847" xml:space="preserve">al <lb/>ter verò portantiũ, quod mouet. </s> + <s xml:id="s848" xml:space="preserve">Quanto igitur <lb/>plus diſtat à pondere, tanto facilius mouet, & </s> + <s xml:id="s849" xml:space="preserve"><lb/>alerum premit magis inferius, veluti contrani-<lb/>@te pondere impoſito, quod hypomochlion +<pb o="532" file="0532" n="29" rhead="Ariſt. Quæſtiones."/> +factum Eſt. </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">ſed <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 ſurgentes omnes, femori crus ad acutum <lb/> +<anchor type="note" xlink:label="note-0532-01a" xlink:href="note-0532-01"/> +cõſtituentes angulũ, & </s> + <s xml:id="s854" xml:space="preserve">thoraci ſimiliter femur, <lb/>ſurgunt? </s> + <s xml:id="s855" xml:space="preserve">quod ſi non, haudquaquam ſurgere po-<lb/>terunt. </s> + <s xml:id="s856" xml:space="preserve">An quia id quod æquale eſt, quietis vbiq; <lb/></s> + <s xml:id="s857" xml:space="preserve">eſt cauſa: </s> + <s xml:id="s858" xml:space="preserve">rectus autẽ angulus æqualitatis eſt, ſta-<lb/>tionemq́; </s> + <s xml:id="s859" xml:space="preserve">facit: </s> + <s xml:id="s860" xml:space="preserve">quamobrẽ ad ſimiles fertur an-<lb/>gulos ipſi terræ circumferentiæ: </s> + <s xml:id="s861" xml:space="preserve">non enim quod <lb/>ad rectum eſt ipſi pauimento. </s> + <s xml:id="s862" xml:space="preserve">An q̇m̄ ſurgens fit <lb/>rectus, ſtantẽ vero neceſſe eſt perpẽdiculum eſſe <lb/>ad terram. </s> + <s xml:id="s863" xml:space="preserve">ſiquidẽ igitur ad rectũ debet eſſe, hoc <lb/>aũt eſt caput ſecundũ pedes habere, & </s> + <s xml:id="s864" xml:space="preserve">fieri opor <lb/>tet, cùm ſurgit. </s> + <s xml:id="s865" xml:space="preserve">Quandoquidem igitur fuerit @e <lb/>dens, ſecundum 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 ſit A, thorax A B, fe-<lb/>mur B C, crura C D. </s> + <s xml:id="s869" xml:space="preserve">ad rectum autem ſit & </s> + <s xml:id="s870" xml:space="preserve">tho-<lb/>rãx vbi A B, ipſi femori, & </s> + <s xml:id="s871" xml:space="preserve">cruri ſemur ſic feden-<lb/>te. </s> + <s xml:id="s872" xml:space="preserve">quamobrem eo ſe habẽtem modo ſurgere eſt <lb/>impoſſibile. </s> + <s xml:id="s873" xml:space="preserve">neceſſe autem eſt crus reclinare, pe-<lb/>desque conſtituere ſub capite. </s> + <s xml:id="s874" xml:space="preserve">hoc autem erit, ſi <lb/>C D fiet, vbi C F, & </s> + <s xml:id="s875" xml:space="preserve">fimul ſurgere continget, & </s> + <s xml:id="s876" xml:space="preserve">in <lb/>eadem ęquali habere caput & </s> + <s xml:id="s877" xml:space="preserve">pedes. </s> + <s xml:id="s878" xml:space="preserve">ipſa autem <lb/>C F a cutum facit angulum ad ipſam 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, quàm 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, quàm mo-<lb/>ueri in cipientes. </s> + <s xml:id="s882" xml:space="preserve">An quia difficillimũ eſt pondus <lb/>mouere, quod in cõtrariũ 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 eſt tardiorẽ eſſe impulſio <lb/>nem illius quod repellitur. </s> + <s xml:id="s885" xml:space="preserve">Secundo autẽloco ſi <lb/>quieuerit: </s> + <s xml:id="s886" xml:space="preserve">reſiſtit enim ipſum quieſcẽs. </s> + <s xml:id="s887" xml:space="preserve">Quod aũt <lb/>mouetur ad id ipſum ad quod impellitur, impel-<lb/>lenti ſimile facit, ceu ſi quiſpiam 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 ipſum tacit ex ſe 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 deſinit potentia, v@l propter retra <lb/>ctionem, vel propter rei proiectę inclinationem, <lb/>quando ea valentior fuerit, quam proijcientis vi <lb/>res. </s> + <s xml:id="s893" xml:space="preserve">Aut iſthæc ambigere, principium relinquen-<lb/>tes, abſurdum eſt.</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 ſibi fertur latio <lb/> +<anchor type="note" xlink:label="note-0533-02a" xlink:href="note-0533-02"/> +nem, impulfore alioquin non conſequente? </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 rurſum, v@ alterum? </s> + <s xml:id="s898" xml:space="preserve">Ceſſat au-<lb/>tem quando non poteſt amplius facere primum <lb/>impellens id quod fertur, vt impellat, & </s> + <s xml:id="s899" xml:space="preserve">qñ ip-<lb/>ſius lati grauitas nut@ ſuo declinat magis, quàm <lb/>impellentis in antè ſit 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, ſed commẽſurationem quandã <lb/>illa habere oportet ad id quod proijcit? </s> + <s xml:id="s901" xml:space="preserve">An quia <lb/>neceſſe eſt 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 ſua nihil cedit, aut imbellicitate nihil contra-<lb/>nititur non efficit proiectionem, neque impulſio <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 eſt 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, quantũ 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 iſthæc 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 exiſtunt: </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 vorticoſis 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="Ariſt. Quæſtiones Mech."/> +ob rem illius extrema in duobus ſunt circulis, <lb/>hoc quidem minori illo verò maiori: </s> + <s xml:id="s917" xml:space="preserve">quare ma-<lb/>ior diſtrahit: </s> + <s xml:id="s918" xml:space="preserve">quoniã celerius fertur, & </s> + <s xml:id="s919" xml:space="preserve">tranſuer-<lb/>ſum impellit illud ad minorem: </s> + <s xml:id="s920" xml:space="preserve">quoniam autem <lb/>id quod ſertur, latitudinem habet, & </s> + <s xml:id="s921" xml:space="preserve">iſte rurſus <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, ſimili <lb/>ſe habet modo ad omnes circulos propter me-<lb/>dium. </s> + <s xml:id="s924" xml:space="preserve">medium enim in vnoquoque circulo ęqua <lb/>liter diſtat. </s> + <s xml:id="s925" xml:space="preserve">An quia quorum quidem circũauctæ <lb/>quæ latio non ſuperior propter magnitudinem, <lb/>ſed grauitate ſua circuli celeritatẽ excellunt, ea <lb/>neceſſe eſt relinqui, & </s> + <s xml:id="s926" xml:space="preserve">tardiùs ferri, tardius aũt <lb/>minor circulus fertur. </s> + <s xml:id="s927" xml:space="preserve">non idem enim in tempo-<lb/>re æquali magnus cũ paruo reuoluitur circulus, <lb/>quando circa idem ſuerint medium. </s> + <s xml:id="s928" xml:space="preserve">quamobrẽ <lb/>in minori circulo relinqui eſt neceſſe, donec ad <lb/>medium perueniant. </s> + <s xml:id="s929" xml:space="preserve">Quorum cunque autem ſu-<lb/>perior à principio fuerit latio, ea finiensidem eſ <lb/>ficiet. </s> + <s xml:id="s930" xml:space="preserve">oportet enim @ãc quidem ſtatim, alterum <lb/>verò celeritate ſuperare grauitatem, quam obrẽ <lb/>ad interiorẽ ſemper circulum relinquetur quod-<lb/>cunque. </s> + <s xml:id="s931" xml:space="preserve">Neceſſe enim eſt, quod non ſuperatur, <lb/>aut in exteriori, aut in interiori moueri: </s> + <s xml:id="s932" xml:space="preserve">in illo <lb/>autem in quo eſt, impoſſibile eſt ferri, quod non <lb/>ſuperatur adhuc vero minus in exteriori: </s> + <s xml:id="s933" xml:space="preserve">cele-<lb/>rior enim exterioris circuli eſt latio: </s> + <s xml:id="s934" xml:space="preserve">reſtat igitur, <lb/>vtid, quod non ſuperatur, ad interiorem tranſ-<lb/>feratur. </s> + <s xml:id="s935" xml:space="preserve">ſemper autem vnumquodque proficit, vt <lb/>non ſuperetur. </s> + <s xml:id="s936" xml:space="preserve">Quoniam verò peruenire ad me-<lb/>dium, finem quidem efficit, vt quippiam nõ mo-<lb/>ueatur, ſtat autem ſolummodo ipſum centrum: <lb/></s> + <s xml:id="s937" xml:space="preserve">ad hoc ſanè omnia congregari neceſſe eſt.</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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/Aristoteles_1585_XSY685ZD.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,1258 @@ +<?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 ſit artis Mechanicæ facultas.</head> +<p> + <s xml:space="preserve">MIraculo ſune ea quidem, quæ natura <lb/>contingunt, quorum ignorantur cau <lb/>ſæ: </s> + <s xml:space="preserve">illa uerò, quæ pręter 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">qđ <lb/>nobis uſui eſſe poteſt, contrarium facit: </s> + <s xml:space="preserve">natura <lb/>etenim eundem ſemper habet modum, & </s> + <s xml:space="preserve">ſimpli <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 ſua hæſitationem praſtat, ar-<lb/>teq́ indiget: </s> + <s xml:space="preserve">quam ob rem eam artis partẽ, quæ <lb/>huiuſmodi ſuccurrit difficulcatibus, Mechanicã <lb/>appellamus. </s> + <s xml:space="preserve">Quemad modum enim Antipho ſcri <lb/>bit poeta, ſic ſe res habet: </s> + <s xml:space="preserve">arte enim ſuperamus <lb/>da, à quibus natura uincimur. </s> + <s xml:space="preserve">Huiuſmodi autem <lb/>ſunt, in quibus & </s> + <s xml:space="preserve">minora ſuperant 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 ſeiugata valde: </s> + <s xml:space="preserve">verùm mathema-<lb/>ticarũ contemplationũ naturaliumq́; </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, admirandis̈ 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">Abſurdum enim eſſe videtur, ma-<lb/>gnum moueri pondus ab exigua virtute, cũ plu-<lb/>ri præſertim pondere. </s> + <s xml:space="preserve">quod. </s> + <s xml:space="preserve">n. </s> + <s xml:space="preserve">una vecte quiſpiã <lb/>mouere nõ poteſt, idipſum ponderis citius mo-<lb/>uet, uectis ad illud pondus adiungens. </s> + <s xml:space="preserve">Oĩum ãt <lb/>huiuſmodi cauſæ principiũ habet circulus, Iſtud <lb/>verò ratione contingit; </s> + <s xml:space="preserve">ex ad mirabili etem̄ mi-<lb/>randum accidere quippiam, non eſt ab ſurdum.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">Maximè autem eſt admirãdum ſimul contra-<lb/>ria fieri. </s> + <s xml:space="preserve">Circulus vero ex huiuſmodi eſt conſtitu <lb/>tus: </s> + <s xml:space="preserve">ſtatim enim ex commoto effectus eſt, & </s> + <s xml:space="preserve">mæ <lb/>nente, quorũ natura ad ſe inuicem eſt cõtraria. <lb/></s> + <s xml:space="preserve">quamobrẽ iſthæc 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 ſcilicet, <lb/>& </s> + <s xml:space="preserve">curuum. </s> + <s xml:space="preserve">hæc aũt eo à ſeinuicem diſtant mõ, <lb/>quo magnum & </s> + <s xml:space="preserve">paruum, illorũ etenim medium <lb/>eſt æquale: </s> + <s xml:space="preserve">horum verò rectũ quapropter cùm, <lb/>ad ſeinuicem cõmutantur, illa quidem prius æ-<lb/>qualia fieri neceſſe eſt, quàm extremorum vtrũ-<lb/>libet: </s> + <s xml:space="preserve">lineam verò rectam, qñ eſt curua concaua, <lb/>ut ex huiu ſmoà<unsure/>i rurſum curua fit, & </s> + <s xml:space="preserve">circularis +<pb o="506" file="0506" n="3" rhead="Ariſt. Quæſtiones"/> +Vnum quidem igitur iſtuc abſurdum ineſt circti<unsure/> <lb/>lo. </s> + <s xml:space="preserve">Alterum autem, ꝙ ſimul contrarijs mouetur <lb/>motionibus: </s> + <s xml:space="preserve">ſimul enim ad anteriorem mouetur <lb/>locum, & </s> + <s xml:space="preserve">ad poſter<unsure/>iorem. </s> + <s xml:space="preserve">Et ea, quæ circulũ de-<lb/>ſcribit, linea eodem ſe 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 rurſus <lb/>efficitur primum. </s> + <s xml:space="preserve">quamobrem manifeſtum, ꝙ in <lb/>de mutatnm eſt. </s> + <s xml:space="preserve">quapropter (vt dictum eſt prius) <lb/>non eſt incõueniens ipſ<unsure/>u<unsure/>m miracolorum oĩum <lb/>eſſe principlum. </s> + <s xml:space="preserve">Ea igi ur, quæ circa libram fiũt, <lb/>ad circulum referuntur: </s> + <s xml:space="preserve">quæ vero circa vectem, <lb/>ad ipſam librã: </s> + <s xml:space="preserve">alia autem ferè omn<unsure/>ia, quæ circa <lb/>męchanicas ſunt motiones, ad vectern<unsure/>. </s> + <s xml:space="preserve">Prætereà <lb/>etiã, quoniam vnica exiſtente, quæ ex cehtro eſt, <lb/>linea, nullum aliud alij, quæ in illa ſunt puncto-<lb/>rum æqua velocitate feratur; </s> + <s xml:space="preserve">ſed cirius ſemper <lb/>quod à manente termino eſt remo ius, pleraque <lb/>miraculorum accidũt in circuli motionibus: </s> + <s xml:space="preserve">de <lb/>quibus in ijs, quæ poſthac addacentur, quęſtioni <lb/>bus erit manifeſtum. </s> + <s xml:space="preserve">Quoniam autem ſecundũ <lb/>contrarias fimul motiones mouetur circulus: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve"><lb/>alterũ 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 ſimul moueantur circuli: </s> + <s xml:space="preserve">quemadmodum, <lb/>ſunt illi, quos in locis proponũt ſacris, 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 eſt 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">rurſus ipſe conti-<lb/>guum ubi EF, in contrarium ſibijpſi mouebitur <lb/>propter eandem cauſam. </s> + <s xml:space="preserve">Eodem ẽt modo ſi plu- +<pb o="507" file="0507" n="4" rhead="Mechanicæ."/> +res fuerint, idẽ facient, uno ſolo commoto. </s> + <s xml:space="preserve">Hane <lb/>igitur in circulo exiſtentem animaduertens na-<lb/>turam architecti, in ſtrumentũ fabricant, celãtes <lb/>principium, vt machinæ ſolum manifeftum ſit il <lb/>lud, quod admirationẽ pręſtat, cauſa verò lateat.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">In primis igitur, quæ accidunt circa librã, du-<lb/>bitare faciunt, quamnàm ob cauſam exactiores <lb/>minoribus maiores ſunt libræ. </s> + <s xml:space="preserve">huius autem rei <lb/>principium eſt, quamobrem in ipſo circulo, quæ <lb/>plus à cent<unsure/>ro diſter linea, eadẽ ui cõmota cit<unsure/>ius <lb/>fertur, quàm illa, quæ minus diſtar. </s> + <s xml:space="preserve">Citius enim <lb/>bifariam dicitur: </s> + <s xml:space="preserve">ſiue enim in minori tempore <lb/>zqualem pertranſi@ locũ, citius feciſſe dicimus: <lb/></s> + <s xml:space="preserve">ſeu in æquali maiorem. </s> + <s xml:space="preserve">Maior autem in æquali <lb/>tempore maiorem deſcribit circulum: </s> + <s xml:space="preserve">qui enim <lb/>extra eſt, maior eo qui intus eft. </s> + <s xml:space="preserve">Horum autẽ cau <lb/>fa, qm̄ duas fertur lationesea, quæ circulum de-<lb/>ſcribit, linea. </s> + <s xml:space="preserve">Quandoquidem igitur in propor-<lb/>tione fertur aliqua id, quod fertur, ſuper rectam <lb/>ferri neceſſe: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">hæc diameter efficitur figuræ, <lb/>quam faciunt illæ, quæ in huiuſmodi proportio <lb/>ne coaptant̃, lineę. </s> + <s xml:space="preserve">Sit enim proportio ſecũdum <lb/>quam latum fertur, quam habet A B ad AC: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">A <lb/>quidem fertur verſus B, A B uerò ſubterferatur <lb/>verſus M C: </s> + <s xml:space="preserve">latum aut<unsure/>em ſit A quidem ad D. </s> + <s xml:space="preserve">ubi <lb/>autem eſt A B, uerſus E: </s> + <s xml:space="preserve">quoniã igitur la@ionis <lb/>erat proportio, quam A B habet ad A C, neceſſe <lb/>eſt & </s> + <s xml:space="preserve">A D ad A E hanc habere proportionem. </s> + <s xml:space="preserve">Si-<lb/>mile igitur eſt proportione paruum quadrilate-<lb/>rum maiori; </s> + <s xml:space="preserve">quamobrẽ & </s> + <s xml:space="preserve">eadẽ illorum eſt dia-<lb/>meter: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">A erit ad F. </s> + <s xml:space="preserve">Eodem etiam oſtendet̃ mo-<lb/>do, vbicu<unsure/>nq; </s> + <s xml:space="preserve">latio deprehendatur; </s> + <s xml:space="preserve">ſemper enim <lb/>ſupra diametrum erit. </s> + <s xml:space="preserve">Manifeſtum igitur, quòd <lb/>id, quod ſecundum diametrum duabus fertur la <lb/>tionibus, neceſſariò ſecundum laterum propor- +<pb o="508" file="0508" n="5" rhead="Ariſt. Quæſtiones."/> +tionem fertur. </s> + <s xml:space="preserve">Sienim ſecundum aliam quam-<lb/>piam, nõ fertur ẜ m dia metrum. </s> + <s xml:space="preserve">Si autem in nul <lb/>la fertur proportione ſecundum duas lationes <lb/>nullo in tempore, rectã eſſe lationem eſt impoſ-<lb/>ſibile. </s> + <s xml:space="preserve">Sit enim recta. </s> + <s xml:space="preserve">Poſita igitur hac pro diame <lb/>tro, & </s> + <s xml:space="preserve">circum reple is lateribus, illud quod fer-<lb/>tur, ſecundum laterum pro portionem ferri n<unsure/>e-<lb/>ceſſe eſt: </s> + <s xml:space="preserve">hoc enim demonſtratum eſt prius. </s> + <s xml:space="preserve">non <lb/>igitur rectam efficiet id, quod ſecundùm 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 eſt tem pus rectam eſ <lb/>ſe lationem, per ea, quæ retro ſunt dicta. </s> + <s xml:space="preserve">Quam-<lb/>obrem circulare eſtid, quod ſecundũ nullam, <lb/>proportionem nullo in tempore duas fertur la-<lb/>tiones Quòd quidem igitur ea, quæ circulum do<unsure/> <lb/>ſcribir, duas ſimul feratur lationes, manifeſtum <lb/>eſt cùm ex iſtis, tum quia ẜ m rectum lata, aſt ꝑ. <lb/></s> + <s xml:space="preserve">pendiculum peruenit, vt ſit rurſus ipſa à centro <lb/>perpendiculum. </s> + <s xml:space="preserve">Sit circulas A B C D: </s> + <s xml:space="preserve">extremum <lb/>autem vbi eſt B, feratur ad ipſum D: </s> + <s xml:space="preserve">peruenit ſa <lb/>hè aliquando ad ipſum C. </s> + <s xml:space="preserve">Si quidẽ igit<unsure/>ur in pro <lb/>portione feratur, quam haber BE, EC fertur vti <lb/>que ſecundùm 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 conſentaneum eſt <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, q́uæ ex centro circulos deſcri <lb/>bunt. </s> + <s xml:space="preserve">quoniam enim proprius eſt manenti, eius <lb/>quæ minor eſt, extrem um, quàm id quod eſt ma <lb/>oris, veluti retractum in contrariũ ad medium, <lb/>ardius fertur minoris extremũ. </s> + <s xml:space="preserve">Omni quidem +<pb o="509" file="0509" n="6" rhead="Mechanicæ."/> +igitur circulum deſcribenti iſtuc accidit: </s> + <s xml:space="preserve">fertur-<lb/>q́ue eam quæ ẜm naturam eſt lationem, ſecun-<lb/>dum circunferentiam: </s> + <s xml:space="preserve">illam verò quę pręter na <lb/>turam, in tranſuerſum, & </s> + <s xml:space="preserve">ſecundùm centrum. <lb/></s> + <s xml:space="preserve">maiorem aut<unsure/>ẽ ſemper eam, quæ præter naturã <lb/>eſt, ipſa minor fertur: </s> + <s xml:space="preserve">quia enim centro eſt vici-<lb/>nior, quod trahit, vincitur magis<unsure/>: </s> + <s xml:space="preserve">Quòd autem-<unsure/> <lb/>magis quod præter natui<unsure/>ã eſt mouetur ipſa mi-<lb/>nor, quàm maior illarũ, quæ ex centro circulos <lb/>deſcribũt ex ijs eſt manifeſtum. </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 ſuppleatur DKRC: </s> + <s xml:space="preserve">ſi quidem AB circu-<lb/>lum deſcribẽs ad id perueniet, vnde eſt egreſſa; </s> + <s xml:space="preserve"><lb/>manifeſtum eſt quod ad ipſam fertur A B. </s> + <s xml:space="preserve">Simi <lb/>literetiam A M ad ipſam A M perueniet. </s> + <s xml:space="preserve">Tar-<lb/>dius autem fertur A M, quàru A B quemadmo-<lb/>dum dictum eſt: </s> + <s xml:space="preserve">quia maior fit repulſio & </s> + <s xml:space="preserve">magis <lb/>rerrahitur A M. </s> + <s xml:space="preserve">Ducatur igitur ipſa A L F, & </s> + <s xml:space="preserve">ab <lb/>ipſo L perpendiculum ad ip ſam A B, ipſa L Q in <lb/>minore circulo: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">rurſum ab L ducatur iuxta A <lb/>B L S, & </s> + <s xml:space="preserve">S T ad ipſam 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 ſunt S T, & </s> + <s xml:space="preserve">L Q, æquales: </s> + <s xml:space="preserve"><lb/>ipſa ergo B T minor eſt, quàm 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 ſectionẽ in maioribus circulis, eſt autem <lb/>ipſa S T æqualis ipſi L Q. </s> + <s xml:space="preserve">In quanto autem tema <lb/>pore ipſa A L ipſam M L lata eſt, in tanto tem@ <lb/>poris ſpatio in maiori circulo maiorem, quàm <lb/>fit B S, latum erit extremum ipſius B A. </s> + <s xml:space="preserve">Latio <lb/>quidem igitur ſecundùm naturam æqualis: </s> + <s xml:space="preserve">ea <lb/>a<unsure/>utem quæ præter naturam eſt minor, videlices +<pb o="510" file="0510" n="7" rhead="Ariſt. Quæſtiones."/> +B T, quàm M Q. </s> + <s xml:space="preserve">Oportet autem proportionabi-<lb/>liter eſſe, ſicut quod eſt ſecundũ naturã, ita quod <lb/>eſt pręter naturam, ad id quod eſt pręter naturã; <lb/></s> + <s xml:space="preserve">maiorem igituĩ circunferentiam pertranſiuit, ꝗ̃ <lb/>ſit ipſa S B. </s> + <s xml:space="preserve">Neceſſe autem eſt ipſam F B in hoc <lb/>rẽpore pertran ſiſſe: </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 eſt præter naturã, <lb/>ad id quod eſt ſecundùm naturam. </s> + <s xml:space="preserve">Si igitur ma-<lb/>ius eſt, quod ſecundum naturam in maiori, & </s> + <s xml:space="preserve"><lb/>quod eſt præter naturam. </s> + <s xml:space="preserve">magis vtique hic coin <lb/>cidit vno modo: </s> + <s xml:space="preserve">ita quod B ſit latum per ipſam <lb/>B F in tanto tempore, in quo M punctum per <lb/>ipſam M L. </s> + <s xml:space="preserve">hic enim ſecundùm naturam quidẽ <lb/>figno B fit X F: </s> + <s xml:space="preserve">eſt enim ab ipſo F perpendiculũ: </s> + <s xml:space="preserve"><lb/>præter naturam verò ad ipſam X B. </s> + <s xml:space="preserve">Eſt autem <lb/>quemadmodum F X ad X B, ſic L Q. </s> + <s xml:space="preserve">ad M Q. </s> + <s xml:space="preserve">ma <lb/>nifeſtum autem, ſi coniunguntur abipſa B M, <lb/>ad F L. </s> + <s xml:space="preserve">Si autem minor, aut maior quàm ſit F B, <lb/>erit illa quam latum eſt B, non ſimiliter erit, ne-<lb/>que proportionale in vtriſque quod eſt ſecundũ <lb/>naturam, ad id quod eſt præter naturam. </s> + <s xml:space="preserve">Quam <lb/>igicur ob cauſam ab eadem potentia celerius fe@ <lb/>tur id quod plus à centro diſtat punctum, ex ijs, <lb/>quæ dicta ſunt, eſt manifeſtum.</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 aũt maiores libræ exactiores <lb/> +<anchor type="note" xlink:label="note-0510-01a" xlink:href="note-0510-01"/> +ſunt minotib<emph style="sub">9</emph>. </s> + <s xml:space="preserve">palã eſt ex ijs. </s> + <s xml:space="preserve">Spar <lb/>tum. </s> + <s xml:space="preserve">n fit centrũ: </s> + <s xml:space="preserve">id pãq; </s> + <s xml:space="preserve">manet. <lb/></s> + <s xml:space="preserve">quod aũt libræ vtrinq; </s> + <s xml:space="preserve">eſt, exeun <lb/>tes à centro. </s> + <s xml:space="preserve">Ab eodẽ igit̃ põdere <lb/>citius moueri neceſſe eſt extre-<lb/>mũ libræ, quo plus à ſparto diſceſſerit. </s> + <s xml:space="preserve">Et nõnul <lb/>la quidẽ in paruis libris impoſita nõ manifeſta +<pb o="511" file="0511" n="8" rhead="Mechanicæ."/> +fenſui ſunt pondera: </s> + <s xml:space="preserve">in magis aũt maniſeſta: </s> + <s xml:space="preserve">ni <lb/>hil enim prohibet minorem moueri magnitudi <lb/>nem, ꝗ̃ vt viſioni ſit maniſeſta. </s> + <s xml:space="preserve">In magna aũt li-<lb/>bra idem pondus viſibile efficit magnitudo. </s> + <s xml:space="preserve">Quę <lb/>dam verò manifeſta quidem ſunt in vtriſq;</s> + <s xml:space="preserve">, ſed <lb/>multo magis in maioribus, qm̄ multo maior in <lb/>clinationis fit magnitudo ab eodem ꝑondere <lb/>in maioribus. </s> + <s xml:space="preserve">Quamobrem machinanturij, qui <lb/>purpuram vendunt, vt pendendo defraudent, tũ <lb/>ad medium ſpartum non ponentes: </s> + <s xml:space="preserve">tum plum-<lb/>bum in alteraruram libræ partem infundentes: <lb/></s> + <s xml:space="preserve">aut ligni gđ ad radicẽ vergebat, in eam quam de <lb/>ferri vola<unsure/>nt, partem conſtituentes; </s> + <s xml:space="preserve">aut ſi nodum <lb/>habuerit: </s> + <s xml:space="preserve">ligni enim grauior illa eſt pars, in qua <lb/>eſt radix, nodus vero radix quædam eſt.</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 ſiquidẽ ſurſum ſuerit ſpartũ, qñ deorsũ lato <lb/> +<anchor type="note" xlink:label="note-0511-01a" xlink:href="note-0511-01"/> +põdere, quiſpiã id admouet, rurſum aſcendit li-<lb/>bra: </s> + <s xml:space="preserve">ſi autem deorſum conſtitutum ſuerit, non <lb/>aſcendit, ſed manet? </s> + <s xml:space="preserve">An quia ſurſum quidẽ ſpar <lb/>to exiſtente, plus libræ extra perpendiculum fit: <lb/></s> + <s xml:space="preserve">ſpartum enim eſt perpendiculum: </s> + <s xml:space="preserve">quare neceſſe <lb/>eſt deorſum ferri id quod plus eſt, donec aſcen-<lb/>dat quæ bifariam libram diuidit, ad ipſum per-<lb/>pendiculũ, cùm onus incumbat ad libræ partem <lb/>furſum raptam. </s> + <s xml:space="preserve">Sit libra recta vbi B C, ſpartum <lb/>autem A D. </s> + <s xml:space="preserve">hoc igitur deorſum proiecto perpen <lb/>diculum erit vbi A D M. </s> + <s xml:space="preserve">Si igitur in ipſo 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 ſecat, pri-<lb/>mo quidem erit D M ipſius perpendiculi: </s> + <s xml:space="preserve">incum <lb/>bente autẽ onere erit D G. </s> + <s xml:space="preserve">quare libra ipſius vbi <lb/>E H, quod extra perpendiculum eſt A M, vbi eſt <lb/>P Q, maius eſt dimidio. </s> + <s xml:space="preserve">Si igitur amoueatur <lb/>onus abipſo E, neceſſe eſt deorſum ferri H, mi-<lb/>@@s enim eſt ipſum E. </s> + <s xml:space="preserve">Si quidem igitur ſurſum +<pb o="512" file="0512" n="9" rhead="Ariſt. Quæſtiones"/> +habuerit ſpartum, rurſum propter hoc aſcendit<unsure/> <lb/>libta. </s> + <s xml:space="preserve">Si autẽ deorſum fuerit, id, quod ſubſtat, cõ <lb/>trarium facit: </s> + <s xml:space="preserve">plus enim dimidio fit libræ, quæ <lb/>deorſum eſt pars, quàm quod perpendiculum ſe <lb/>cet, quapropter nõ aſcẽdit: </s> + <s xml:space="preserve">eleuata enim pars le <lb/>uior eſt. </s> + <s xml:space="preserve">Sit libra recta vbi N O, perpendiculum <lb/>autem K L M bifariã igitur ſecatu@ K G. </s> + <s xml:space="preserve">Impo-<lb/>ſito autem onere in ipſo N, erit quidẽ N vbi O, <lb/>ipſum autem G vbi R, K L autem vbi L P. </s> + <s xml:space="preserve">quare <lb/>maius eſt K O, quàm L R, ipſo P K L. </s> + <s xml:space="preserve">Et ablato <lb/>igitur onere, neceſſe eſt manere: </s> + <s xml:space="preserve">in cumbit enim <lb/>ceu onus exceſſus medietatis eius vbi eſt 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 exiguę vires (quem admodum à principio <lb/> +<anchor type="note" xlink:label="note-0512-01a" xlink:href="note-0512-01"/> +dictum eſt) vecte magna mouent pondera, vecis <lb/>inſuper onus accipien tis? </s> + <s xml:space="preserve">cùm ſacilius ſit mino-<lb/>grem mouere grauitatem: </s> + <s xml:space="preserve">minor autẽ eſt ſine ve-<lb/>cte. </s> + <s xml:space="preserve">An quoniã ipſe vectis eſt in cauſa hbra exi-<lb/>ſtens, ſpartũ infernè habens, in inæqualia diui-<lb/>ſa. </s> + <s xml:space="preserve">hypomochlion enim eſt ſpartum. </s> + <s xml:space="preserve">a mbo nan <lb/>que ſtamt, vt centrum. </s> + <s xml:space="preserve">quoniam autem ab ęqua-<lb/>ſi pondere celerius mouetur maior earum, q̃ à <lb/>centro ſunt: </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">ſem <lb/>per autem quanto ab hy pomochlio diſtabit ma <lb/>gis, tanto facilius mo<unsure/>uebit. </s> + <s xml:space="preserve">Cauſa autem eſt, quę <lb/>retro cõmemorata eſt: </s> + <s xml:space="preserve">quoniam quę plus à cen-<lb/>@ro diſtat, maiorem deſcribit circuium: </s> + <s xml:space="preserve">quare ab <lb/>eadem potentia plus ſeparabitur mouens illud, <lb/>quod plus ab hypomochlio diſtabit. </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 eſt 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 ſunt 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 eſt: </s> + <s xml:space="preserve">hy +<pb o="513" file="0513" n="10" rhead="Mechanicæ."/> +pomochlion auteni fit ſcalmus, ſtat enim ille ẜ <lb/>pondus verò mare eſt, quod propellit remus: </s> + <s xml:space="preserve">ve-<lb/>ctem autem mouens ipſe eſt remex. </s> + <s xml:space="preserve">Semper aũt <lb/>plus mouet ponderis, quanto magis ab hypomo <lb/>chlio diſtabit, quicunque id mouet. </s> + <s xml:space="preserve">maior enim <lb/>ita fit, quæ ex centro. </s> + <s xml:space="preserve">ſcalmus autem hypomo-<lb/>chlion exiſtens, centrum eſt. </s> + <s xml:space="preserve">In medio autem na <lb/>uis plurimum remi intus eſt: </s> + <s xml:space="preserve">ilia enim parte la-<lb/>tiſſima eſt 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 ſecus eſt. </s> + <s xml:space="preserve">Moue-<lb/>tur autem nauis, quoniam appellente ad mare <lb/>remo, extremum illius quod intus eſt, in ante <lb/>prom ouctur. </s> + <s xml:space="preserve">nauem verò ſcalmo alligatam ſi-<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 eſt. </s> + <s xml:space="preserve">plurimum autem di@i <lb/>dit, vbi pars plurima remia ſcalmo eſt. </s> + <s xml:space="preserve">Et eam <lb/>cauſam remiges, qui in media ſunt naui, mouent <lb/>illam maximè. </s> + <s xml:space="preserve">maxima enim remi pars à ſcalmo <lb/>in nauis medio intus eſt.</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 exiſtens 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, magnę nauigiorum moueantur <lb/>moles? </s> + <s xml:space="preserve">An quoniam gubernaculum vectis eſt: </s> + <s xml:space="preserve">o-<lb/>@us autem mare: </s> + <s xml:space="preserve">gubernator verò mouens eſt. <lb/></s> + <s xml:space="preserve">Non autem ſecundum latitudinẽ, veluti remus, <lb/>mare accipit gubernaculum: </s> + <s xml:space="preserve">non enim in antè <lb/>nauigium mouet, ſed ipſum commotum mare <lb/>accipiens inclinat obliquè: </s> + <s xml:space="preserve">quoniam enim pon-<lb/>dus eſt mare, contrario innixum modo nauem <lb/>inclinat. </s> + <s xml:space="preserve">hypomochlion enim in contrarium <lb/>verſatur: </s> + <s xml:space="preserve">mare vero interius, ſi illud exterius. </s> + <s xml:space="preserve"><lb/>illud autem ſequitur nauis, quoniam illi eſt ai<unsure/>li-<lb/>@ata. </s> + <s xml:space="preserve">Et remus quidem ſecundùm latitudinem +<pb o="514" file="0514" n="11" rhead="Ariſt. Quæſtiones."/> +onus propellens, & </s> + <s xml:space="preserve">ab eodem repulſus, in rectũ <lb/>propellit: </s> + <s xml:space="preserve">gubernaculum autem, ut obliquum ia <lb/>cet, hinc inde in obliquum motionē facit. </s> + <s xml:space="preserve">In ex <lb/>tremo autē, & </s> + <s xml:space="preserve">non in medio iacet, quoniam mo <lb/>uenti facilimū eſt 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, ſic ipſius cōtinui in fine imbeciliſſima eſt la-<lb/>tio. </s> + <s xml:space="preserve">imbeciliſſima autem ad expellendum eſt fa <lb/>cilis. </s> + <s xml:space="preserve">Propter hæcigitur in puppi gubernaculum <lb/>ponitur. </s> + <s xml:space="preserve">Nec minus, quoniā parua ibi motione <lb/>ſacta, multò maius interuallum fit in ultimo: </s> + <s xml:space="preserve">ꝗa <lb/>æqualis angulus ſemper maiorem ſpectat: </s> + <s xml:space="preserve">tan-<lb/>toq́ue magis. </s> + <s xml:space="preserve">quanto maiores fuerint illæ, quæ <lb/>con@inẽt. </s> + <s xml:space="preserve">Ex ijs etiam manifeſtum eſt, quam ob <lb/>cauſſam magis in contrarium procedit nauigiũ, <lb/>quàm remi ipſius palmula. </s> + <s xml:space="preserve">eadem enim magni <lb/>tudoijſdem mota uiribus, in aete plus, quàm in <lb/>aqua progreditur. </s> + <s xml:space="preserve">Si enim A B remus, Guerò <lb/>ſcalmus: </s> + <s xml:space="preserve">A autem in nauigio ſit remi principiũ, <lb/>B uerò in mari palmula. </s> + <s xml:space="preserve">Si igitur A ubi D tranſ. </s> + <s xml:space="preserve"><lb/>latum eſt, non erit B ubi E: </s> + <s xml:space="preserve">æqualis enim B E ipſi <lb/>A C. </s> + <s xml:space="preserve">æquale igitur tranſlatū erit, ſed minus. </s> + <s xml:space="preserve">erit <lb/>igitur ubi eſt F. </s> + <s xml:space="preserve">minor enim C F, ipſa A D. </s> + <s xml:space="preserve">quare <lb/>ipſa G F, ipſa D F. </s> + <s xml:space="preserve">ſimiles enim ſunt trianguli. </s> + <s xml:space="preserve"><lb/>Stans autem erit medium, ubi eſt C. </s> + <s xml:space="preserve">in contrariũ <lb/>enim ipſi, quod in mari eſt, extremo uidelicet B <lb/>procedit, ubi extremum in nauigio eſt A. </s> + <s xml:space="preserve">Non <lb/>procedere autēubi eſt D. </s> + <s xml:space="preserve">niſi commouere@ur na <lb/>uigium, & </s> + <s xml:space="preserve">eo transferretur, ubi remi eſt princi-<lb/>plum Idipſnm etiam facit gubernaculam, niſi ꝙ <lb/>(ut dictum eſt terro) nihil nauigio ad id, quod-<lb/>antè eſt, confert, ſed ſolùm puppim in obliquũ <lb/>pell@t, ubicunq; </s> + <s xml:space="preserve">fuerit in con@rariū enim eo mo <lb/>do uergit prora. </s> + <s xml:space="preserve">Vbi igitur applicatum eſt 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">quemadmodũ ſcalmus <lb/>remo. </s> + <s xml:space="preserve">Medium autem procedit ſecundùm quod <lb/>gubernaculum trãsfertur. </s> + <s xml:space="preserve">ſi quidē introrſus 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 exiſtente 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 ſublimior fuerit, ijſdem <lb/> +<anchor type="note" xlink:label="note-0515-01a" xlink:href="note-0515-01"/> +uelis, & </s> + <s xml:space="preserve">uento eodẽ celerius ferũtur nauigia? </s> + <s xml:space="preserve">An <lb/>quia malus quidem fit uectis: </s> + <s xml:space="preserve">hypomochlion ue <lb/>rò mali ſedes, in qua collocatur: </s> + <s xml:space="preserve">pondus autem, <lb/>quod moueri debet, ipſum nauigium: </s> + <s xml:space="preserve">mou@s ve <lb/>ro is, qui vela tendit, ſpiri@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 ſublata antenna, velum à mali ſede, quæ hypo <lb/>mochlion eſt, 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, conſtringunt: </s> + <s xml:space="preserve">illam ve-<lb/>rò quæ proram verſus eſt, pedem facientes rela <lb/>xant? </s> + <s xml:space="preserve">An quia retrahere quidem multo exiſten <lb/>tevento gubernaculum non poteſt: </s> + <s xml:space="preserve">pauco autẽ <lb/>poteſt, quem conſtringunt. </s> + <s xml:space="preserve">Propellit quidem igi <lb/>tur ipſe ventus: </s> + <s xml:space="preserve">in puppim vero illum cōſtituit, <lb/>gubernaculum retrahens, & </s> + <s xml:space="preserve">mare compellẽs: </s> + <s xml:space="preserve">ſi-<lb/>mul & </s> + <s xml:space="preserve">nautæ ipſi cum vento contendunt: </s> + <s xml:space="preserve">in cõ-<lb/>trariam enim ſe 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">rotũdę ſunt <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 abſidem <lb/>centro ſimul moto, quemadmodum paluſtriver <lb/>titur rota: </s> + <s xml:space="preserve">aut circa manens centrū, veluti tro-<lb/>chleæ ſtante centro: </s> + <s xml:space="preserve">autin pauimento manente <lb/>@ẽtro, ſicut figuli rota conuertitur. </s> + <s xml:space="preserve">An celer@ima +<pb o="516" file="0516" n="13" rhead="Ariſt. Quæſtiones"/> +quidem huiuſmodi ſunt, quoniam parua ſuipa@ <lb/>te planum contingunt, veluti circulas ſecūdum <lb/>punctum: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">quoniam non offenſant: </s> + <s xml:space="preserve">à terra em̄ <lb/>ſemotus eſt angulus. </s> + <s xml:space="preserve">Pręterea etiam cui obuiam <lb/>fiunt corpori, id rurſum ſecundùm puſillam tan <lb/>gunt. </s> + <s xml:space="preserve">ſi autem rectilineum eſſet, rectitudine ſua <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/>ſuper plano circuli fuerit diameter, planū ſecun <lb/>dùm punctum, contingēte circulo æquale vtriu-<lb/>que pondus difterminat diameter. </s> + <s xml:space="preserve">cùm autē 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">ſi quidem difficulter ad cōtrarium nutus ſui mo <lb/>uetur modum. </s> + <s xml:space="preserve">Præterea nonnulli autumn@nt ꝙ <lb/>circuli linea in perpeti verſatur motu, quemad-<lb/>modum manentia propter contrarium nixũ ma <lb/>nent: </s> + <s xml:space="preserve">ſicur maioribus contingit circulis ad mino <lb/>res. </s> + <s xml:space="preserve">Celerius enim ab æquali mouentur potentia <lb/>maiores circuli, mouentq́ue 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">ſicut diameter ad <lb/>diametrum, ita maior circulus ad minorem. </s> + <s xml:space="preserve">it fi <lb/>niti autem ſunt minores. </s> + <s xml:space="preserve">Si autem ad alte@um <lb/>nutum habet circulus, ſimiliter eſt bene mobi-<lb/>lis. </s> + <s xml:space="preserve">Et aliam ſanè habet inclinationem circulus, <lb/>& </s> + <s xml:space="preserve">ea quæ à circulo mouentur, licèt planitiẽ abſi-<lb/>de non contingat, ſed aut iuxta planitiẽ, aut velu <lb/>tì trocleæ. </s> + <s xml:space="preserve">etenim hoc ſe habentes modo facilli-<lb/>me mouentur, & </s> + <s xml:space="preserve">onera commouent. </s> + <s xml:space="preserve">An nõ quia <lb/>parua ſui portione cùm tangit, tum offenſat cir-<lb/>culus, ſed aliam ob cauſam. </s> + <s xml:space="preserve">ea autem eſt, quæ di <lb/>cta eſt prius, quod circulus ſcilicet ex duabus ef-<lb/>fectus eſt lationibus: </s> + <s xml:space="preserve">quam ob rem illarum alte- +<pb o="517" file="0517" n="14" rhead="Mechanicæ."/> +ram pro nutu ſemper habet, & </s> + <s xml:space="preserve">veluti continuo <lb/>motum illum mòuent, quicunq; </s> + <s xml:space="preserve">mouent, quan-<lb/>do ſecundùm circun ferentiam illum mouerint: <lb/></s> + <s xml:space="preserve">latam enim ipſam mouent. </s> + <s xml:space="preserve">Eam quidem igitur, <lb/>quæ in obliquum eſt, motionem ipſum impellit <lb/>mouens: </s> + <s xml:space="preserve">ſecundum vero illam, quæ ſuper diame <lb/>trum eſt, ſeipſum 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 quàm minoribus, & </s> + <s xml:space="preserve"><lb/>fcytalis ſimiliter? </s> + <s xml:space="preserve">An quoniã, qnanto maior fue-<lb/>ri@@illa quæ à centro eſt, in ęquali tempore maius <lb/>mouetur ſpatium. </s> + <s xml:space="preserve">Quam ob rem æquali inexiſtẽ <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">ſpar <lb/>tum enim in illis centrum eſt: </s> + <s xml:space="preserve">libræ autem vtrin <lb/>que partes, quæ ex centro ſunt, exiſtunt.</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 ſine pondere eſt, moue-<lb/> +<anchor type="note" xlink:label="note-0517-02a" xlink:href="note-0517-02"/> +tur libra, quàm cum pondus habet? </s> + <s xml:space="preserve">Similiq́ mo <lb/>do rota, & </s> + <s xml:space="preserve">huiuſmodi quippiam, quod grauius <lb/>quidem eſt, maius autem minore & </s> + <s xml:space="preserve">leuiore? </s> + <s xml:space="preserve">An <lb/>quia non ſolùm in contrarium quod graue eſt, <lb/>ſed in obliquum etiam difficulter mouetur. </s> + <s xml:space="preserve">In <lb/>contrarium enim ei, ad quod vergit onus, moue-<lb/>re difficile eſt: </s> + <s xml:space="preserve">quò autem vergit, eſt 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 ſuper ſcytalas facilius portantur onera, <lb/> +<anchor type="note" xlink:label="note-0517-03a" xlink:href="note-0517-03"/> +quã ſ@per currus, cùm tamẽ ij magnas habeant <lb/>rotas, illæ verò puſillas? </s> + <s xml:space="preserve">An qui in ſcytalis nulla <lb/>eſt offen ſatio, in curribus autem axis eſt, ad quẽ <lb/>offenſant. </s> + <s xml:space="preserve">deſuper enim illum premunt, & </s> + <s xml:space="preserve">late-<lb/>ribus. </s> + <s xml:space="preserve">quod autem eſt in ſcycalis, ad iſthæc duo <lb/>mouetur, & </s> + <s xml:space="preserve">inſernè ſubſtrato ſpatio, & </s> + <s xml:space="preserve">onere ſu <lb/>per impoſito. </s> + <s xml:space="preserve">in vtriſque 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="Ariſt. Quæſtiones"/> +<p> + <s xml:space="preserve">Curlongius feruntur miſſilia funda, quàm ma <lb/> +<anchor type="note" xlink:label="note-0518-01a" xlink:href="note-0518-01"/> +numiſſa, cùm alioquin proiector mann magis <lb/>pondus comprehendat, quàm cùmillud ſuſpen-<lb/>dit? </s> + <s xml:space="preserve">Præterea ſic quidem duo mouet pondera, <lb/>fundæ videlicet, & </s> + <s xml:space="preserve">miſſilis: </s> + <s xml:space="preserve">illo aũt modo ſolum <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/>ſubinde rotans, id iaculatur: </s> + <s xml:space="preserve">ex manu autem à <lb/>quiete eſt initium: </s> + <s xml:space="preserve">omnia autem cùm in motu <lb/>ſunt, quàm cùm quieſcunt, facilius mouẽtur. </s> + <s xml:space="preserve">An <lb/>& </s> + <s xml:space="preserve">eam ob cauſam e@t, ſed nec minus etiam, quia <lb/>in fundæ vſu manus quidem fit centrum: </s> + <s xml:space="preserve">ſunda <lb/>vero, quod à centro exit. </s> + <s xml:space="preserve">quanto aũt productius <lb/>ſueritid, quod à cētro eſt, tanto citius mouetur. <lb/></s> + <s xml:space="preserve">iactus aūt qui māu fit, fundę reſpectu breuis eſt.</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, quàm minores mouentur? </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">item ſuculæ, <lb/>quæ graciliores ſunt, ab eadẽ viꝗ̃ craſſiores? </s> + <s xml:space="preserve">An <lb/>quía ſecula quidem, & </s> + <s xml:space="preserve">iugum cētrum eſt: </s> + <s xml:space="preserve">pro-<lb/>minentes autem longitudines eæ, quæ ſunt à cē-<lb/>tro. </s> + <s xml:space="preserve">Celerius autem & </s> + <s xml:space="preserve">plus mouentur quæ maio <lb/>rum ſunt circulorum ab eadem vi, quàm quę mi <lb/>norum. </s> + <s xml:space="preserve">ab eadem enim vi plus transfertur id <lb/>extremum, quod longius à centro diſtat. </s> + <s xml:space="preserve">Quam <lb/>ob rem ad iugum quidem inſtrumenta faciunt <lb/>collopas, quibus facilius verſant: </s> + <s xml:space="preserve">in gracilibus <lb/>autem ſuculis 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 eiuſdem magnitudinis lignum facilius <lb/> +<anchor type="note" xlink:label="note-0518-03a" xlink:href="note-0518-03"/> +genus frangitur, ſi quiſpiam æquì diductis mani <lb/>bus extrema comprehendens fregerit, quàm ſi <lb/>iuxta genu: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ſi terræ illud applicans pede ſu-<lb/>perimpoſito manu longè diducta confregerit, ꝗ̃ <lb/>prope? </s> + <s xml:space="preserve">An quiaibi quidem genu centrũ eſt, hic <lb/>vero ipſe pes. </s> + <s xml:space="preserve">quanto autem remotius à centro +<pb o="519" file="0519" n="16" rhead="Mechanicæ."/> +ſuerit, faciliùs mouetur quodcunq;</s> + <s xml:space="preserve">. moueriau-<lb/>tem quod frangitur, neceſſe eſt.</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 ſunt figura, cùm alioquin à principio ex <lb/>magnis ſint lapidibus, oſtreiſve? </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">intcruallũ <lb/>verò ea, quæ à centro. </s> + <s xml:space="preserve">Sem per autem maior ab <lb/>æquali motione maiorem deſcribit circulum. <lb/></s> + <s xml:space="preserve">quod autem maius in æquali pertranſit tẽpore, <lb/>celerius fertur: </s> + <s xml:space="preserve">quæ autem celeriùs ex ę quali fe <lb/>runtur ſpatio, uehementiùs im petunt: </s> + <s xml:space="preserve">quę 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 diſtant, cōfringi neceſ <lb/>ſe eſt, id autem cùm patiantur, rotunda fieri eſt <lb/>neceſſarium. </s> + <s xml:space="preserve">Crocis autem propter maris mo-<lb/>tum, quoniam ſimul cum illo agitantur, in perpe <lb/>tua eſſe accidit motione, eoq́; </s> + <s xml:space="preserve">verſatas mō ſem <lb/>per offenſare. </s> + <s xml:space="preserve">id autem ipſis maximè extremis <lb/>contingere partibus eſt 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 ſuntligna, 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">ſi tollantur, in flectuntur magis: <lb/></s> + <s xml:space="preserve">tametſi quod breue quidem eſt, ceu bicubitum, <lb/>fuerit tenue: </s> + <s xml:space="preserve">quod verò cubitorum centum, craſ <lb/>ſum? </s> + <s xml:space="preserve">An ꝗa & </s> + <s xml:space="preserve">vectis, & </s> + <s xml:space="preserve">onus, & </s> + <s xml:space="preserve">hypomochlion <lb/>in leuando ipſa 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 eſt, pōdus. </s> + <s xml:space="preserve">quam ob rem quanto exten. </s> + <s xml:space="preserve"><lb/>ſius fuerit id, quod ab hypocomochlio eſt, tanto <lb/>inflecti neceſſe ẽ magis. </s> + <s xml:space="preserve">quo enim plus ab hypo-<lb/>mochlio diſtat, eo magis incuruari neceſſe eſt. </s> + <s xml:space="preserve"><lb/>neceſſario igitur extrema vectis eleuantur. </s> + <s xml:space="preserve">Siigi <lb/>tur flexilis fuerit vectis, ipſum inflecti magis <lb/>cùm extollitur neceſſe eſt: </s> + <s xml:space="preserve">quod lõgis accidit li <lb/>gnis. </s> + <s xml:space="preserve">in breuibus autem quod vltimũ eſt, quie- +<pb o="520" file="0520" n="17" rhead="Ariſt. Quæſtiones"/> +ſcenti hy pomochlio deprope ſit.</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 ſcindun-<lb/>tur pondera, & </s> + <s xml:space="preserve">corporum moles, validaque fit <lb/>impreſſio? </s> + <s xml:space="preserve">An quia cuneus duo ſunt vectes ſibi <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 ipſius latio pondus, quod <lb/>percutit & </s> + <s xml:space="preserve">mouet, magnum facit: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">quoniā mo <lb/>tum mouet, pſa celeritate valentius fit. </s> + <s xml:space="preserve">paruo au <lb/>tem exiſtente vecte, magnæ illum conſequuntur <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 ſcinditur, D E F G. </s> + <s xml:space="preserve">vectis igitur ſit ipſa <lb/>A B, pondus verò ipſius 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">ſcindit enim ipſum B.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">Cur ſi quiſpiam trochleas componens duas in <lb/> +<anchor type="note" xlink:label="note-0520-01a" xlink:href="note-0520-01"/> +tignis duobus ad ſe inuicem inuctis cõtrario ad <lb/>trochleas modo circulo funem circunduxerit, <lb/>cuius alterum quidem caput tigonrum appenda <lb/>tur alteri, alterum varò troclheis ſit 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 ſi mouetur, vecte <lb/>medio trãsfertur magis, quàm à manu: </s> + <s xml:space="preserve">trochlea <lb/>autem idem vecti facit. </s> + <s xml:space="preserve">quam ob rem ſi vna faci-<lb/>lius traher, & </s> + <s xml:space="preserve">ab vnico tractu multò grauius tra-<lb/>het, quàm facere poſſit manus, idip ſum duæ tro-<lb/>chleæ plus quàm in dupla velocitate leuabunt. <lb/></s> + <s xml:space="preserve">Minus enim altera trahit, quàm ſi ip ſa per ſeip-<lb/>ſam traheret, quando circa alteram iniectus ſue <lb/>rit funis: </s> + <s xml:space="preserve">illa nanque minus etiam pondus effe-<lb/>cit. </s> + <s xml:space="preserve">Pariq́; </s> + <s xml:space="preserve">modo ſi ad plures iniectus fuerit funis <lb/>in paucis trochleis, multa ſit 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, traducũt <lb/>enim ab una trochlea ad aliam, & </s> + <s xml:space="preserve">rurſus ab illa <lb/>ad ſuculas, & </s> + <s xml:space="preserve">vectes. </s> + <s xml:space="preserve">hoc autem idem eſt, ac ſi <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 ſi quis ſuper lignum magnã imponat ſe-<lb/> +<anchor type="note" xlink:label="note-0521-01a" xlink:href="note-0521-01"/> +carim, deſuperq; </s> + <s xml:space="preserve">illi magnum adijciat pondus, <lb/>ligni quippiam quod curandum ſit, non diuidit: <lb/></s> + <s xml:space="preserve">ſiverò ſecurim extoll@s percutiat, illud ſcindit, <lb/>cùm alioquin multò minus habeat ponderis id <lb/>quod percutit, quàm id quod ſuperiacet, & </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/>ipſum grauitatis magis aſſumit motum dũ mo-<lb/>uetur, ꝗ̃ dum quieſcit Incumbensigitur conua-<lb/>tum graui motionem non mouetur, motum ve-<lb/>rò & </s> + <s xml:space="preserve">ſecundum hanc mouetur, & </s> + <s xml:space="preserve">ſecundùm eã <lb/>quæ eſt percutientis. </s> + <s xml:space="preserve">Pręterea ſecuris ipſa effici-<lb/>tur cuneus: </s> + <s xml:space="preserve">paruus autẽ exiſtens cuneus magna <lb/>diuidit cum ex duobus ſit vectibus contrario ad <lb/>ſeſe inuicem modo conſtitutis</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 ſtatera, 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, ſolùm ſuſpenditur lanx: </s> + <s xml:space="preserve">in altera verò par <lb/>te, ſola eſt ſtatera. </s> + <s xml:space="preserve">An quia ſimul libram & </s> + <s xml:space="preserve">vectẽ <lb/>ipſam contingit eſſe ſtateram. </s> + <s xml:space="preserve">libram quidem, <lb/>vbi ſpartorum quodcunque ſtateræ 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 ſi quis alterain apponeret lan-<lb/>cem, & </s> + <s xml:space="preserve">ilii pondus imponeret. </s> + <s xml:space="preserve">manifeſtũ enim, <lb/>quòd tantundem tradit ponderis ei quod in alte <lb/>raiacet lance. </s> + <s xml:space="preserve">Quemadmodum autem ſi vna li-<lb/>bra multæ ſint libræ, ſic talia inſunt ſparta mul.</s> + <s xml:space="preserve"> +<pb o="522" file="0522" n="19" rhead="Ariſt. Quæſtiones"/> +tain eiuſmodi libra: </s> + <s xml:space="preserve">quorum vniuſcuiuſq; </s> + <s xml:space="preserve">quod <lb/>intrinſecus eſt ad appẽdiculum, ſtateræ eſt dimi-<lb/>dium. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">omnino iſtud libra eſt, vnam quidẽ ha-<lb/>benslancem, in qua pondus appẽditur: </s> + <s xml:space="preserve">alteram <lb/>verò, vbi in ſtatera æquipondium. </s> + <s xml:space="preserve">Quam ob rem <lb/>appendiculum ad alteram ſui partem eſt ſtatera. <lb/></s> + <s xml:space="preserve">huiuſmodi autem exiſtēs multæ ſunt libræ, totq́; </s> + <s xml:space="preserve"><lb/>quot fuerint ſparta. </s> + <s xml:space="preserve">Semper autem quod lanci <lb/>propinquius eſt ſpartum, appenſoq́ue oneri, ma-<lb/>ius trahit pōdus: </s> + <s xml:space="preserve">quoniam fit quidem omnis ſta-<lb/>tera inuerſus vectis: </s> + <s xml:space="preserve">hypomochlion nãq; </s> + <s xml:space="preserve">vnum-<lb/>quodque ſpartum ſuperne exiſtens: </s> + <s xml:space="preserve">pondus verũ <lb/>id quod lanci ineſt. </s> + <s xml:space="preserve">Quanto autem productior ve <lb/>ctis ſuerit longitudo ab ipfo hypomochlio, tãto <lb/>ibi quidem faciliùs mouet: </s> + <s xml:space="preserve">hic autem æquilibri-<lb/>brium facit, pondusq́ue ſtateræ 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 ſacilius dentes extrahūt dentifor-<lb/>cipis onere adiecto, quàm ſi ſola vtantur manu? <lb/></s> + <s xml:space="preserve">An quia ex manu magis quàm ex dentiforcipe <lb/>lubricus elabitur dẽs? </s> + <s xml:space="preserve">An ferro id potius accidit <lb/>quàm digitis, quoniam vndique dentem non cō-<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/>@iſorceps duo contrarij vectes, vnicum habentes <lb/>hypom ochlion, eius ſcilicet inſtrumenti 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 eſt 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 ſimul comprehendentes mouent. </s> + <s xml:space="preserve"><lb/>quando autem commotus fuerit, faciliùs manu <lb/>trahitur, quàm inſtrumento.</s> + <s xml:space="preserve"/> +</p> +<pb o="523" file="0523" n="20" rhead="Mechanicæ"/> +<p> + <s xml:space="preserve">Cur nuces abſque ictu facilè cõfrigun tinſtru-<lb/>mẽtis, quæ ad eum fiunt vſum? </s> + <s xml:space="preserve">multumeinim au <lb/>fertur virium, motionis ſcilicet, & </s> + <s xml:space="preserve">violẽ@iæ. </s> + <s xml:space="preserve">Prę-<lb/>terea duro & </s> + <s xml:space="preserve">graui comprimens inſtrumento ci <lb/>tiùs conſringet, quàm ligneo & </s> + <s xml:space="preserve">leui. </s> + <s xml:space="preserve">An quia ſic <lb/>vtrinqne à duobus comprimitur vectibus ipſa <lb/>nux: </s> + <s xml:space="preserve">a vecte autem facile diuelluntur onera. </s> + <s xml:space="preserve">id <lb/>enim inſtrumẽtum ex duobus componitur vecti <lb/>bus idem habentibus hypomochlion, connexio-<lb/>nem videlicet @pſam vbi eſt A. </s> + <s xml:space="preserve">Quemadmodum <lb/>igitur fuere diductę ſecundũ extrema motis C D <lb/>ipſæ F E, ſic à parua faciliter potentia conducun <lb/>tur. </s> + <s xml:space="preserve">Quod igitur cum percuſſione ſeciſſet 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 eſt K. </s> + <s xml:space="preserve">Hanc etiam ob cauſam <lb/>quanto vicinius fuerit K ipſum A, confringitur <lb/>celerius. </s> + <s xml:space="preserve">quanto enim ab hypomochlio plus di-<lb/>ſtat vectis, facilius & </s> + <s xml:space="preserve">plus mouet ab eadẽ poten-<lb/>tia. </s> + <s xml:space="preserve">Eſt igitur A quidem hypomochlion: </s> + <s xml:space="preserve">ipſa 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 ipſum K vicinius ſuerit angulo ipſius A, tãto <lb/>vicinius fit connexioni vbi eſt A. </s> + <s xml:space="preserve">hoc autem eſt <lb/>hypomochlion. </s> + <s xml:space="preserve">Ab eadem igitur potentia appli-<lb/>cante F E plus extolli neceſſe eſt. </s> + <s xml:space="preserve">Quam ob rem <lb/>quoniã ex contrario eſt eleuatio, neceſſe eſt 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 ſi duo extrema in rhombo puncta dua-<lb/>bus ferantur lationibus, haudquaquam æquale <lb/>vtrunque eorũ pertianſit rectam, ſed multò plus <lb/>alterum ?</s> + <s xml:space="preserve">idem autem eſt ſermo, cur quod ſuper <lb/>latus fertur, minus pertranſit, quàm ipſum 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 ſer +<pb o="524" file="0524" n="21" rhead="Ariſt Quæſtiones."/> +t<unsure/>ur lationibus. </s> + <s xml:space="preserve">Feratur enim ex ipſa A B, A qui-<lb/>dem ad ipſum B, B vero ad ipſum D eadem cele <lb/>ritate. </s> + <s xml:space="preserve">feratur aurẽ & </s> + <s xml:space="preserve">ipſa A B in ipſa A C iuxta <lb/>C D eadem celeritate cum illis. </s> + <s xml:space="preserve">neceſſe igitur eſt <lb/>A quidem in ipſa A D diametro ferri, B verò in <lb/>ipſa B C. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">vtranq; </s> + <s xml:space="preserve">ſimul pertranſiſſe, & </s> + <s xml:space="preserve">ipſa@@ <lb/>A B ipſum la<unsure/>tus A C. </s> + <s xml:space="preserve">latum enim ſit ipſum A ip <lb/>ſam A E, A B autem ipſam A E, & </s> + <s xml:space="preserve">proiecta ſit <lb/>F G iuxta ipſum A B, & </s> + <s xml:space="preserve">ab ipſo E ſimiliter replea <lb/>tur. </s> + <s xml:space="preserve">Simile igitur fit, quod repletum eſt ipſi toti. <lb/></s> + <s xml:space="preserve">æqualis igirur A F ipſi A E ipſa autem A B ipſam <lb/>A F lata erit. </s> + <s xml:space="preserve">in diametro igitur erit ſecūdū K. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve"><lb/>fem per neceſſe eſt ipſum ferri ſecundùm diame <lb/>trum. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ſimul A B latus pertranfit latus A C, & </s> + <s xml:space="preserve"><lb/>ipſum A diametrum pertranſit A D. </s> + <s xml:space="preserve">Similiter ẽt <lb/>demon ſtrabitur & </s> + <s xml:space="preserve">ipſum B in ipſa B C diametro <lb/>latum. </s> + <s xml:space="preserve">æqualis enim eſt ipſa B E ipſi B G Reple. </s> + <s xml:space="preserve"><lb/>to igitur ab ipſo G. </s> + <s xml:space="preserve">quod intus eſt, toti eſt ſimi-<lb/>le: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ipſum B in ipſa diame@ro erit ſecundùm la <lb/>terum connexionem. </s> + <s xml:space="preserve">Et ſimul latus pertranſit <lb/>latus, & </s> + <s xml:space="preserve">B ipſum B C diametrum. </s> + <s xml:space="preserve">Simuligitur A <lb/>multo plus ipſa A B pertrāſit: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ipſum latus, mi-<lb/>nus latus eadem lata celeritate: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ipſum latus <lb/>maiorem, quàm B, pertranſiuit vna latum latio-<lb/>ne. </s> + <s xml:space="preserve">Quanto enim acutior fuerit rhombus, diame <lb/>ter quidem minor ſit. </s> + <s xml:space="preserve">A C autẽ maior: </s> + <s xml:space="preserve">latus ve-<lb/>ro ipſius B C minus. </s> + <s xml:space="preserve">Abſurdū eſt enim (vt dictũ) <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">vtriſq; </s> + <s xml:space="preserve">poſitis <lb/>æquali velocitate punctis, alterū pertranſire ma <lb/>iorem. </s> + <s xml:space="preserve">Cauſa autem eſt, quoniã ei quod ab obt@ <lb/>ſo ſertur angulo ambæ ferè cōtrariæ fiunt latio-<lb/>n<unsure/>es, & </s> + <s xml:space="preserve">illa ſecūdū quā ipſum fertur, & </s> + <s xml:space="preserve">illa ſecun <lb/>dũ quã ipſum à latere defertur. </s> + <s xml:space="preserve">Ei autẽ quod ab <lb/>@cuto fertur, accidit vt ad idē feratur, coadiuuat +<pb o="525" file="0525" n="22" rhead="Mechanicæ."/> +@nim quæ ipſius eſt lateris, illã quæ eſt ſuper dia-<lb/>metrum. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">quanto hunc quidem acutiorem fe-<lb/>ceris, illum verò obtuſum magis: </s> + <s xml:space="preserve">hæc quidē tar-<lb/>dior erit, illa verò celerior. </s> + <s xml:space="preserve">Hæ quidẽ igitur ma-<lb/>gis contrariæ fiunt, qm̄ obtuſior fit angulus: </s> + <s xml:space="preserve">illæ <lb/>verò ad idẽ magis, qm̄ lineæ coarctantur: </s> + <s xml:space="preserve">ipſum <lb/>enim A ferè ad idem fertur ſecundum ambas la <lb/>tiones. </s> + <s xml:space="preserve">Coadiuuatur igitur altera: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">quanto ſanè <lb/>acutior fuerit angulus, tanto magis ipſum A ad <lb/>contrarium: </s> + <s xml:space="preserve">ipſum. </s> + <s xml:space="preserve">n. </s> + <s xml:space="preserve">ad B fertur: </s> + <s xml:space="preserve">latus autẽ de-<lb/>fertipſum ad D. </s> + <s xml:space="preserve">Et quanto ſanè obtuſior fuerit <lb/>angulus, magis contrariæ fiunt lationes: </s> + <s xml:space="preserve">rectior <lb/>enim efficitur linea. </s> + <s xml:space="preserve">ſi autẽ omnino recta fieret, <lb/>penitus vtique eſſent contrariæ latus verò ſecun <lb/>dùm vnicam latuum lationem à nullo præpedi-<lb/>tur: </s> + <s xml:space="preserve">rationabiliter igitur maiorem pertranſit.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">Dubitatur quam ob cauſam maior circulus ę-<lb/>qualem minori circulo conuoluit<unsure/>ur lineam, qū<unsure/> <lb/>circa idem centrum fuerint poſiti: </s> + <s xml:space="preserve">ſeorſum autẽ <lb/>reuoluti, quemadmodum alterius magnitudo ad <lb/>magnitudiuem ſe habet alterius, fic & </s> + <s xml:space="preserve">illorũ ad <lb/>ſeinuicem fiunt lineæ. </s> + <s xml:space="preserve">Præterea vno et@ã & </s> + <s xml:space="preserve">eodē <lb/>v<unsure/>triſque exiſtente centro, aliquando quidem tan <lb/>ta ſit linea, quam conuoluuntur, quantam minor <lb/>per ſe 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, manifeſtum eſt. </s> + <s xml:space="preserve">angulus enim <lb/>ſen ſu videtur eſſe cuiuſque 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 ſecundùm ſenſum ad ſe lineæ. </s> + <s xml:space="preserve">ſecun-<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 poſiti centrum, manifeſtũ eſt: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve"><lb/>ſic fiunt aliquando quidē æquales lineæ, ſecun- +<pb o="526" file="0526" n="23" rhead="Ariſt Quæſtiones"/> +dum quam maior conuoluitur circulus, aliquan <lb/>do verò ſecundùm 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">vtriſq́; </s> + <s xml:space="preserve">autem centrum A. </s> + <s xml:space="preserve">Et quam qui-<lb/>dem per ſe magnus conuoluitur, ſit vbi F I: </s> + <s xml:space="preserve">quã <lb/>vero per ſe minor vbi GK, æqualis A. </s> + <s xml:space="preserve">Si igitur mi <lb/>norem mouero, idem mouens centū vbi A, ma-<lb/>ior autem ſit annexus. </s> + <s xml:space="preserve">qñ igitur A B fuerit recta <lb/>ad ipſam G K, ſimul & </s> + <s xml:space="preserve">A C fit recta ad ipſam F I. <lb/></s> + <s xml:space="preserve">quam ob rem ęqualem ſemper tranſlata erit ip-<lb/>ſam quidem G K, vbi eſt G B circunferentia: </s> + <s xml:space="preserve">ip-<lb/>ſam verò FL, quæ eſt vbi F C. </s> + <s xml:space="preserve">Si aũt quarta pars <lb/>æqualem conuoluitur, manifeſtum eſt, ꝙ totus <lb/>circulus toti circulo æqualem conuoluetur. </s> + <s xml:space="preserve">qua <lb/>re qñ B C, linea ad ipſum peruenerit K, & </s> + <s xml:space="preserve">ipſa <lb/>F C circunferentia erit in ipſa C L: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">vniuerſus <lb/>crit conuolutus circulus. </s> + <s xml:space="preserve">Similiq́; </s> + <s xml:space="preserve">modo ſi ma-<lb/>gnum mouero, illi paruuū annectens, eodem exi <lb/>ftente cẽtro, ſimul cum A C ipſi A B perpendicu <lb/>lum & </s> + <s xml:space="preserve">recta eri@: </s> + <s xml:space="preserve">hæc quidem ad ipſam F I, illa <lb/>verò ad G M. </s> + <s xml:space="preserve">Quam ob rem quando hæc quidẽ <lb/>æqualem ipſi G M pert<unsure/>rãſiuerit, illa verò ipſi F I, <lb/>& </s> + <s xml:space="preserve">rurſum facta fuerit recta ipſa F A ad ipſam <lb/>F L. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ipſa A C rurſum recta, velut à principio <lb/>erant in ipſis M I. </s> + <s xml:space="preserve">Hoc autem neque aliqua in-<lb/>tercedente mora maioris ad minorem, vt. </s> + <s xml:space="preserve">ſ. </s> + <s xml:space="preserve">per <lb/>aliquod temporis ſpatium ſtaret in eodem pun-<lb/>cto: </s> + <s xml:space="preserve">neque tranſiliente minore aliquod punctū, <lb/>maiorem quidem æqualem minori pertranſire, <lb/>hunc a@tem maiori, abſurdum eſt. </s> + <s xml:space="preserve">Præterea vni-<lb/>ea etiam ſemper exiſtente motione, centrū mo-<lb/>rum interdum quidem magnam, nonnunquàm <lb/>verò minorem conuerti, admirandum eſt. </s> + <s xml:space="preserve">Idem <lb/>enim celeritate eadem latus æqualẽ natum hoc <lb/>eft pertranſire: </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/>ſumendum eſt circa iſtorum cauſam, ꝙ 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 ſeipſo moueri natũ non ſit, <lb/>ſi ſimul & </s> + <s xml:space="preserve">illud mouerit, quod natũ eſt moueri. <lb/></s> + <s xml:space="preserve">tardius mouebitur, quàm ſi ipſum per ſe moue-<lb/>retur. </s> + <s xml:space="preserve">Et ſi quidem natum fuerit moueri, non ſi-<lb/>mul aũt moueatur, ſimiliter ſe habebit. </s> + <s xml:space="preserve">Et im<unsure/>poſ <lb/>ſibile certè eſt plus moueri, quàm mouens: </s> + <s xml:space="preserve">non <lb/>enim ſuam ipſius mouetur motionem. </s> + <s xml:space="preserve">Si igitur<unsure/> <lb/>circulus maior vbi A, minor autem vbi B, ſi mi-<lb/>nor maiorem impellet non reuolutum ex ſe, ma<unsure/> <lb/>nifeſtum eſt, ꝙ tantum ipſius rectæ maior pertrã <lb/>ſit, quantum eſt impulſus: </s> + <s xml:space="preserve">tantum aũt eſt impul-<lb/>ſus, quantum paruus eſt motus: </s> + <s xml:space="preserve">æqualem igitur <lb/>ipſius rectæ pertranſiuerunt. </s> + <s xml:space="preserve">Neceſſe igitur eſt <lb/>ſi reuolutus minor maiorem impellet, reuol@i ſi<unsure/> <lb/>mul cum impulſione: </s> + <s xml:space="preserve">tantum aūt quantum mi-<lb/>nor reuolutus eſt, ſi nihil ipſe ſub ipſius motio-<lb/>ne mouetur. </s> + <s xml:space="preserve">Qūo enim, & </s> + <s xml:space="preserve">quantum mouit, tan-<lb/>tum motũ eſſe neceſſe eſt, quod mouetur ab ill@ <lb/>ſed profectò paruus circulus tãtum ſeipſum cir-<lb/>culariter mouit, quantum eſt pedalis quantitas: </s> + <s xml:space="preserve"><lb/>(tantum enim ſit id quod motus eſt) & </s> + <s xml:space="preserve">magnus <lb/>igitur tantum motus erit. </s> + <s xml:space="preserve">Similiq́; </s> + <s xml:space="preserve">modo ſi ma-<lb/>gnus paruū mouebit, motus erit paruus, quēad-<lb/>modum maior. </s> + <s xml:space="preserve">Per ſe aūt motus illorum vtrum-<lb/>libet ſiue celeriter, ſeu tardè eadem velocitate, <lb/>ſtatim quantum maior natus eſt circunſerri li-<lb/>neam: </s> + <s xml:space="preserve">quod difficultatē facit, quod non ſimiliter <lb/>ſaciunt, quando fuerint connexi. </s> + <s xml:space="preserve">hoc autem eſt, <lb/>ſi alteri ab altero moueatur, non quã natus eſt, <lb/>@eque peculiarem motionem. </s> + <s xml:space="preserve">Nihil enim referr <lb/>citcũponere, & </s> + <s xml:space="preserve">annectere, aut cōiungere vtrum- +<pb o="528" file="0528" n="25" rhead="Ariſt Queſtiones"/> +liber alteri. </s> + <s xml:space="preserve">ſimiliter enim quando hic quidem <lb/>mouer, ille vero mouetur ab iſto; </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 propenſus, <lb/>non ſemper conueluitur: </s> + <s xml:space="preserve">quādo verò circa idem <lb/>poſi@i fuerint cẽtrum, alterum ab altero ſemper <lb/>conuolui neceſſe eſt. </s> + <s xml:space="preserve">Sed nihilominus non ſuam <lb/>ipſius motionem mouetur alter, ſed veluti nullã <lb/>haberet motionem: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">ſi habuerit, illa autem nõ <lb/>vtatur, tan@undem accidit. </s> + <s xml:space="preserve">Quandoquidem igi-<lb/>tur màgnus moueri ſibi alligatum paruum, par-<lb/>uus mouetur quãtum ille: </s> + <s xml:space="preserve">quādo autem paruus, <lb/>rurſus magnus quantum iſte: </s> + <s xml:space="preserve">ſeparatus aũt vter-<lb/>que ſeipſum mouet. </s> + <s xml:space="preserve">Quod autem eodem exiſten <lb/>te centro, & </s> + <s xml:space="preserve">mouente eadem velocitate, accidit <lb/>inæqualem illos pertrāſire lineam, paralogiſmo <lb/>fophiſticę vtitur is, qui dubitat. </s> + <s xml:space="preserve">idem enim am-<lb/>bobus eſt centrũ, verùm per accidens, velúti mu <lb/>ſicum & </s> + <s xml:space="preserve">album. </s> + <s xml:space="preserve">eſſe enim vtriuſque circuli cen-<lb/>tro nō 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 ſim pliciter mouet, ſed eſt quomodo.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">Cur lect@lorum ſpondas ſecūdum duplam fa-<lb/> +<anchor type="note" xlink:label="note-0528-01a" xlink:href="note-0528-01"/> +ciunt proportionem, hanc quidem ſex pedū vel <lb/>paulo am pliorem, illam vero trium: </s> + <s xml:space="preserve">curve non <lb/>ſecundùm diametrum illos reſtibus extendunt? <lb/></s> + <s xml:space="preserve">An tantos quidem magnitudine faciunt, vt cor-<lb/>poribus ſint proportionem habẽtes: </s> + <s xml:space="preserve">fiunt enim <lb/>ſic ſecundum ſpondas dupli, longitudine quidẽ <lb/>cubitorum quatuor, latitudine vero duorũ. </s> + <s xml:space="preserve">Ex-<lb/>tendunt autem illos nō ſecundùm diametrum, <lb/>ſed ex oppoſito, vt & </s> + <s xml:space="preserve">ligna minus diſtra hantur. </s> + <s xml:space="preserve"><lb/>celertrimè enim ſcindũtur ſecundum naturam <lb/>diuiſa, & </s> + <s xml:space="preserve">eodem modo diſtenta laborant ma- +<pb o="529" file="0529" n="26" rhead="Mechanicæ."/> +@imè. </s> + <s xml:space="preserve">Amplius, quoniam opus eſt, vt reſtes pon-<lb/>dus ferre poſſint, ſic certè pondere impoſito mi-<lb/>nus laborabunt, ſi tranſuerſim, quàm ſi obliquè <lb/>extendantur. </s> + <s xml:space="preserve">Præterea hoc etiam modo minus <lb/>abſumitur reſtium. </s> + <s xml:space="preserve">Sit enim lectulus A F G K, & </s> + <s xml:space="preserve"><lb/>bifariam diuidatur ipſa F G ſecundùm B, æqua-<lb/>lia certè foramina ſunt in ipſa F B, & </s> + <s xml:space="preserve">in ipſa F A. <lb/></s> + <s xml:space="preserve">latera enim ſunt æqualia. </s> + <s xml:space="preserve">nam totum F G duplũ <lb/>eſt. </s> + <s xml:space="preserve">Extendunt autem, vt deſcriptum eſt, ab ipſo <lb/>A ad ipſum B ita vbi eſt C, ita vbi D: </s> + <s xml:space="preserve">ita vbi H, <lb/>poſtea vbi E, & </s> + <s xml:space="preserve">eodem ſemper modo, donec ad <lb/>angulum peruenerint alium. </s> + <s xml:space="preserve">Duo enim anguli <lb/>reſtis habent capi a. </s> + <s xml:space="preserve">æquales autem ſunt reſtes <lb/>ſecundum curuaturas, videlicet A B; </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">B C ipſis <lb/>C D, & </s> + <s xml:space="preserve">D H. </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">aliæ ſimili ſe habet modo, quo-<lb/>niam eadem demonſtratio. </s> + <s xml:space="preserve">ipſa enim AB æqua-<lb/>lis eſt ipſi H E: </s> + <s xml:space="preserve">æqualia enim ſunt latera ſpatij <lb/>B G, M A, & </s> + <s xml:space="preserve">foramina æquè diſtant, ipſa autem <lb/>B G æquelis eſt ipſi M A: </s> + <s xml:space="preserve">angulus enim B æqua-<lb/>lis eſt 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 eſt ſemirectus: </s> + <s xml:space="preserve"><lb/>eſt enim F B æqualis ipſi F A: </s> + <s xml:space="preserve">& </s> + <s xml:space="preserve">angulus vbi F, re <lb/>ctus eſt. </s> + <s xml:space="preserve">B autẽ angulus æqualis ei vbi eſt G, quo-<lb/>niam quadratum altera parte longius duplũ eſt: </s> + <s xml:space="preserve"><lb/>& </s> + <s xml:space="preserve">ad medium eſt curuatura. </s> + <s xml:space="preserve">quamobrẽ A D ip-<lb/>ſi E G eſt æqualis: </s> + <s xml:space="preserve">huic verò ipſa H M. </s> + <s xml:space="preserve">Similiq́ue <lb/>modo demonſtrantur aliæ, quoniã æquales ſunt <lb/>duæ, quæ ſecũdùm curuataras ſunt, duabus. </s> + <s xml:space="preserve">Qua <lb/>re manifeſtum eſt, ꝙ tot ſunt reſtes in lectulo, <lb/>quot ſunt quatuor, ſicut A B. </s> + <s xml:space="preserve">Quanta autem fora <lb/>minũ eſt multirudo in ipſo F G latere, & </s> + <s xml:space="preserve">in eius <lb/>dimidio F B eſt medietas. </s> + <s xml:space="preserve">Quamobrem in dimi-<lb/>diato lectulo tantæ reſtium magnitudines erũt, <lb/>quantum eſt A B: </s> + <s xml:space="preserve">multitudine verò tot, quot in <lb/>B G ſunt foramina. </s> + <s xml:space="preserve">Hoc autem nihil refert dice +<pb o="530" file="0530" n="27" rhead="Ariſt. Quæſtiones"/> +re, quàm quot ſunt in ipſis A F & </s> + <s xml:space="preserve">B F ſimul <lb/>ſumpris. </s> + <s xml:space="preserve">Si autem ſecundum diametrum exten-<lb/>dantur reſtes, quemad modum ſe habet in lectu-<lb/>lo A B C D, dimidia non tot ſunt, 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, ſunt foramina: </s> + <s xml:space="preserve">maiores autem ſunt</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 exiſtentes, quàm A B, <lb/>quare reſtis in tantum maior, quantum ambola <lb/>@era diame@ro ſunt maiora.</s> + <s xml:space="preserve"/> +</p> +<p> + <s xml:space="preserve">Cur difficilius eſt longa ligna ab extremo ſu-<lb/> +<anchor type="note" xlink:label="note-0530-01a" xlink:href="note-0530-01"/> +per humeros ferre, quàm ſecundum medium, æ-<lb/>quali exiſtente pondere? </s> + <s xml:space="preserve">An quia vibrato ligno <lb/>ipſum 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 eſt ab extremo, <lb/>quoniam faciliùs ex medio eleuatur, quàm ab <lb/>extremo, & </s> + <s xml:space="preserve">ideo ſic ferre eſt facilius: </s> + <s xml:space="preserve">Cauſa aũt, <lb/>qm̄ ſecundùm medium quidem eleuato ligno, <lb/>ſemper ſeſe inuicem ſuſpendunt extrema, & </s> + <s xml:space="preserve">al-<lb/>tera pars alteram bene ſubleuat medium enim <lb/>veluti centrum fit, vbi habet is, qui eleuat, aut <lb/>ſert. </s> + <s xml:space="preserve">Extremorũ igitur v@@rũq; </s> + <s xml:space="preserve">deorſum vergens, <lb/>ſurſum ſuſpenditur. </s> + <s xml:space="preserve">ꝙ ſi ab extremo eleuetur, <lb/>aut feratur, non ſanè facit: </s> + <s xml:space="preserve">ſed vniuerſum 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 ſecundùm A, ipſum q́uidem B <lb/>deorſum @utans, ſurſum eleuat C: </s> + <s xml:space="preserve">ipſum autem <lb/>C deorſum nutans, B ſurſum eleuat: </s> + <s xml:space="preserve">ambo au-<lb/>tem ſurſum 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 ſi valde procerum fuerit idẽ pondus diffi-<lb/> +<anchor type="note" xlink:label="note-0530-02a" xlink:href="note-0530-02"/> +cilius ſuper humeros geſtatur, etiam ſi medium <lb/>quiſpiam illud ferat, quàm ſi breuius ſit? </s> + <s xml:space="preserve">Quod <lb/>@nim dudum dictum eſt, cauſa nõ eſt: </s> + <s xml:space="preserve">ſed vibra- +<pb o="531" file="0531" n="28" rhead="Mechanicę."/> +tio nunc eſt cauſa. </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 geſtare. </s> + <s xml:space="preserve">Vibrationis aũt cã eſt, quoniã <lb/>ab eadẽ motione magis transferuntur extrema, <lb/>quanto procerius fuerit lignũ. </s> + <s xml:space="preserve">Humerus quidẽ <lb/>fit centrũ 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 aũt AB, AC, quę <lb/>ſunt ex centro. </s> + <s xml:space="preserve">quanto aũt maius fuerit id quod <lb/>ex centro eſt, ſiue A B, ſeu A C, plus transfertur <lb/>ſpatij. </s> + <s xml:space="preserve">Demonſtratum autem eſt 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 viſun <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 adiũgunt pondus: </s> + <s xml:space="preserve">cũ <lb/>alioquin vas ipſum & </s> + <s xml:space="preserve">plenum, & </s> + <s xml:space="preserve">vacuum pon-<lb/>dus habeat? </s> + <s xml:space="preserve">An qm̄ duobus tem poribus hau@ien <lb/>di, diuiſo opere, (intingere. </s> + <s xml:space="preserve">n. </s> + <s xml:space="preserve">oportet, & </s> + <s xml:space="preserve">id ſur-<lb/>ſum trahere) continget demittere quidẽ vacuum <lb/>faciliter, trahere vero plenum difficulter. </s> + <s xml:space="preserve">Cōmo-<lb/>dum igitur eſt paulò tardius illud demittere, cū <lb/>multò leuius effectum ſuſtolatur 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 ſit pondus, ꝗ <lb/>fi ſolummodo vacuum oportet demittere: </s> + <s xml:space="preserve">cum <lb/>verò plenum fuerit, ſurſum id rapit plumbum, <lb/>aut quicquid ponderis inerit. </s> + <s xml:space="preserve">Quam ob rem faci <lb/>liora hoc modo ambo ſunt, quàm 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 ſuper ligno, aut hmõi quopiam duo <lb/> +<anchor type="note" xlink:label="note-0531-02a" xlink:href="note-0531-02"/> +portauerint homines æquale pondus, non ſimili <lb/>ter premuntur, ſi ad vnum non declinet pondus, <lb/>ſeu magis, quanto vicinius fuerit geſtãtibus? </s> + <s xml:space="preserve">An <lb/>quoniam vectis quidem lignum efficitur, põdus <lb/>verò hypomochlion: </s> + <s xml:space="preserve">qui autem propior eſt pon-<lb/>deri ex ijs, qui illud geſtant, id quod mouetur: </s> + <s xml:space="preserve">al <lb/>ter verò portantiũ, quod mouet. </s> + <s xml:space="preserve">Quanto igitur <lb/>plus diſtat à pondere, tanto facilius mouet, & </s> + <s xml:space="preserve"><lb/>alerum premit magis inferius, veluti contrani-<lb/>@te pondere impoſito, quod hypomochlion +<pb o="532" file="0532" n="29" rhead="Ariſt. Quæſtiones."/> +factum Eſt. </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">ſed <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 ſurgentes omnes, femori crus ad acutum <lb/> +<anchor type="note" xlink:label="note-0532-01a" xlink:href="note-0532-01"/> +cõſtituentes angulũ, & </s> + <s xml:space="preserve">thoraci ſimiliter femur, <lb/>ſurgunt? </s> + <s xml:space="preserve">quod ſi non, haudquaquam ſurgere po-<lb/>terunt. </s> + <s xml:space="preserve">An quia id quod æquale eſt, quietis vbiq; <lb/></s> + <s xml:space="preserve">eſt cauſa: </s> + <s xml:space="preserve">rectus autẽ angulus æqualitatis eſt, ſta-<lb/>tionemq́; </s> + <s xml:space="preserve">facit: </s> + <s xml:space="preserve">quamobrẽ ad ſimiles fertur an-<lb/>gulos ipſi terræ circumferentiæ: </s> + <s xml:space="preserve">non enim quod <lb/>ad rectum eſt ipſi pauimento. </s> + <s xml:space="preserve">An q̇m̄ ſurgens fit <lb/>rectus, ſtantẽ vero neceſſe eſt perpẽdiculum eſſe <lb/>ad terram. </s> + <s xml:space="preserve">ſiquidẽ igitur ad rectũ debet eſſe, hoc <lb/>aũt eſt caput ſecundũ pedes habere, & </s> + <s xml:space="preserve">fieri opor <lb/>tet, cùm ſurgit. </s> + <s xml:space="preserve">Quandoquidem igitur fuerit @e <lb/>dens, ſecundum 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 ſit A, thorax A B, fe-<lb/>mur B C, crura C D. </s> + <s xml:space="preserve">ad rectum autem ſit & </s> + <s xml:space="preserve">tho-<lb/>rãx vbi A B, ipſi femori, & </s> + <s xml:space="preserve">cruri ſemur ſic feden-<lb/>te. </s> + <s xml:space="preserve">quamobrem eo ſe habẽtem modo ſurgere eſt <lb/>impoſſibile. </s> + <s xml:space="preserve">neceſſe autem eſt crus reclinare, pe-<lb/>desque conſtituere ſub capite. </s> + <s xml:space="preserve">hoc autem erit, ſi <lb/>C D fiet, vbi C F, & </s> + <s xml:space="preserve">fimul ſurgere continget, & </s> + <s xml:space="preserve">in <lb/>eadem ęquali habere caput & </s> + <s xml:space="preserve">pedes. </s> + <s xml:space="preserve">ipſa autem <lb/>C F a cutum facit angulum ad ipſam 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, quàm 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, quàm mo-<lb/>ueri in cipientes. </s> + <s xml:space="preserve">An quia difficillimũ eſt pondus <lb/>mouere, quod in cõtrariũ 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 eſt tardiorẽ eſſe impulſio <lb/>nem illius quod repellitur. </s> + <s xml:space="preserve">Secundo autẽloco ſi <lb/>quieuerit: </s> + <s xml:space="preserve">reſiſtit enim ipſum quieſcẽs. </s> + <s xml:space="preserve">Quod aũt <lb/>mouetur ad id ipſum ad quod impellitur, impel-<lb/>lenti ſimile facit, ceu ſi quiſpiam 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 ipſum tacit ex ſe 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 deſinit potentia, v@l propter retra <lb/>ctionem, vel propter rei proiectę inclinationem, <lb/>quando ea valentior fuerit, quam proijcientis vi <lb/>res. </s> + <s xml:space="preserve">Aut iſthæc ambigere, principium relinquen-<lb/>tes, abſurdum eſt.</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 ſibi fertur latio <lb/> +<anchor type="note" xlink:label="note-0533-02a" xlink:href="note-0533-02"/> +nem, impulfore alioquin non conſequente? </s> + <s xml:space="preserve">An <lb/>videlicet quoniam primum id efficit, @@al@erum <lb/>impellat: </s> + <s xml:space="preserve">illudque rurſum, v@ alterum? </s> + <s xml:space="preserve">Ceſſat au-<lb/>tem quando non poteſt amplius facere primum <lb/>impellens id quod fertur, vt impellat, & </s> + <s xml:space="preserve">qñ ip-<lb/>ſius lati grauitas nut@ ſuo declinat magis, quàm <lb/>impellentis in antè ſit 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, ſed commẽſurationem quandã <lb/>illa habere oportet ad id quod proijcit? </s> + <s xml:space="preserve">An quia <lb/>neceſſe eſt 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 ſua nihil cedit, aut imbellicitate nihil contra-<lb/>nititur non efficit proiectionem, neque impulſio <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 eſt im-<lb/>berillius. </s> + <s xml:space="preserve">nihil contranititur. </s> + <s xml:space="preserve">An quia tãtum fer-<lb/>tur id quod fertur, quantũ 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 iſthæc ha <lb/>bere. </s> + <s xml:space="preserve">Valde enim magnum, & </s> + <s xml:space="preserve">valde paruum, ceù <lb/>non mota exiſtunt: </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 vorticoſis 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="Ariſt. Quæſtiones Mech."/> +ob rem illius extrema in duobus ſunt circulis, <lb/>hoc quidem minori illo verò maiori: </s> + <s xml:space="preserve">quare ma-<lb/>ior diſtrahit: </s> + <s xml:space="preserve">quoniã celerius fertur, & </s> + <s xml:space="preserve">tranſuer-<lb/>ſum impellit illud ad minorem: </s> + <s xml:space="preserve">quoniam autem <lb/>id quod ſertur, latitudinem habet, & </s> + <s xml:space="preserve">iſte rurſus <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, ſimili <lb/>ſe habet modo ad omnes circulos propter me-<lb/>dium. </s> + <s xml:space="preserve">medium enim in vnoquoque circulo ęqua <lb/>liter diſtat. </s> + <s xml:space="preserve">An quia quorum quidem circũauctæ <lb/>quæ latio non ſuperior propter magnitudinem, <lb/>ſed grauitate ſua circuli celeritatẽ excellunt, ea <lb/>neceſſe eſt relinqui, & </s> + <s xml:space="preserve">tardiùs ferri, tardius aũt <lb/>minor circulus fertur. </s> + <s xml:space="preserve">non idem enim in tempo-<lb/>re æquali magnus cũ paruo reuoluitur circulus, <lb/>quando circa idem ſuerint medium. </s> + <s xml:space="preserve">quamobrẽ <lb/>in minori circulo relinqui eſt neceſſe, donec ad <lb/>medium perueniant. </s> + <s xml:space="preserve">Quorum cunque autem ſu-<lb/>perior à principio fuerit latio, ea finiensidem eſ <lb/>ficiet. </s> + <s xml:space="preserve">oportet enim @ãc quidem ſtatim, alterum <lb/>verò celeritate ſuperare grauitatem, quam obrẽ <lb/>ad interiorẽ ſemper circulum relinquetur quod-<lb/>cunque. </s> + <s xml:space="preserve">Neceſſe enim eſt, quod non ſuperatur, <lb/>aut in exteriori, aut in interiori moueri: </s> + <s xml:space="preserve">in illo <lb/>autem in quo eſt, impoſſibile eſt ferri, quod non <lb/>ſuperatur adhuc vero minus in exteriori: </s> + <s xml:space="preserve">cele-<lb/>rior enim exterioris circuli eſt latio: </s> + <s xml:space="preserve">reſtat igitur, <lb/>vtid, quod non ſuperatur, ad interiorem tranſ-<lb/>feratur. </s> + <s xml:space="preserve">ſemper autem vnumquodque proficit, vt <lb/>non ſuperetur. </s> + <s xml:space="preserve">Quoniam verò peruenire ad me-<lb/>dium, finem quidem efficit, vt quippiam nõ mo-<lb/>ueatur, ſtat autem ſolummodo ipſum centrum: <lb/></s> + <s xml:space="preserve">ad hoc ſanè omnia congregari neceſſe eſt.</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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/Benedetti_1585-Ids.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,15419 @@ +<?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 Philoſophi.</head> + <head xml:id="head2" xml:space="preserve"><emph style="it">DIVERSARVM SPECVLATIONVM <lb/>Mathematicarum, & Phyſicarum</emph> <lb/>Liber.</head> + <head xml:id="head3" xml:space="preserve">Quarum ſeriem ſequens 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 perſpectiuæ.</head> + <head xml:id="head9" xml:space="preserve">De Mechanicis.</head> + <head xml:id="head10" xml:space="preserve">Diſputationes de quibuſdam placitis Ariſt.</head> + <head xml:id="head11" xml:space="preserve">In quintum Euclidis librum.</head> + <head xml:id="head12" xml:space="preserve">Phyſica, & Mathematica reſponſa per Epiſtolas.</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 Celſitudinis, ac-<lb/>cerſitus ex vrbe Parmenſi 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 deſeruiendi, tam vebe-<lb/>menti cupiditate fui accenſus, vt ſub eius ditione quodſuper-<lb/>eßet vitæ agere conſtituerem. </s> + <s xml:id="s3" xml:space="preserve">Cuius in me benignitas, mea <lb/>in illum obſeruantia mirum in modum mutuo vſu, & conſue-<lb/>tudine eſt adaucta, vt idem Dux me ſecum dum ruſticaretur <lb/>eße vellet, ſæpè etiam ſecum pernoctare; </s> + <s xml:id="s4" xml:space="preserve">quo quidem tempo-<lb/>re de Matbematicis ſcientijs mecum agebat, in quibus perdi-<lb/>ſcendis mea opera vtebatur, quæſtiones, Arithmeticam, Geo<lb/>metriam, Opticen, Muſicam, aut Astrologiam ſpectantes <lb/>proponens. </s> + <s xml:id="s5" xml:space="preserve">Cui vt quod in me eßet ſatisfacerem, acrius <lb/>quàm anteainea studia (adquætamen ſemper fui propenſißi-<lb/>mus) incubui. </s> + <s xml:id="s6" xml:space="preserve"><reg norm="Illiusque" type="simple">Illiusq́ꝫ</reg> imitatione (vt ferècæteri Principum <lb/>studiaimitantur) non pauci aut præſentes, aut per litter as me <lb/>de his, atque illis Mathematicis quæstionibus conſuluerunt. <lb/></s> + <s xml:id="s7" xml:space="preserve">Cùmque ego nunquam laborem amicorum cauſa defugerim, <lb/>euenit vt post tot annorum curricula, mea ſcrinia ſcrutatus, <lb/>inuenerim tot abſolutas quæſtiones, vt ex eis corpus mediocre <lb/>effici poſſe videretur. </s> + <s xml:id="s8" xml:space="preserve">Quas, cùm rationibus in epiſtola ſub-<lb/>ſequenti allatis edere constituiſſem, non ſub cuiuſque alte-<lb/>rius nomine, & auſpicijs quam tuæ Celſitudinis 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 Celſitudine paternam in me fouendo, & augendo<unsure/> <lb/>benignit atem ineße ſemper ſum expertus, tum quòd tuæ Celſi-<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 ſemper in <lb/>his dedic ationibus ſpectandum put aui, tuam Celſitudinem 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/>ſarum Regum gloriam æquauit, & nos veluti in ſpem certam <lb/>fælicitatis buius ſæculi induxit, ſi verum eſt Platonis va-<lb/>ticinium, beat am eam futuram Rempublic am in qua <lb/>Principes Philoſophentur. </s> + <s xml:id="s12" xml:space="preserve">Tua igitur celſi-<lb/>tudo libellum tot ei nominibus debitum, <lb/>ea qua ſolet bumanitate accipe-<lb/>re nè grauetur. </s> + <s xml:id="s13" xml:space="preserve">Deus tuas <lb/>omnes cogitationes, <lb/>& conatus ad <lb/>fœlicißi-<lb/>mos <lb/>ſemper exitus perducat, <lb/><reg norm="teque" type="simple">teq́ꝫ</reg> diutißimè ſer-<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 diuerſis <lb/>diſciplinis contemplatus ſim, partim à præ-<lb/>ſtantibus viris patronis ac amicis meis exci-<lb/>tatus, quiſuper eis ſententiam meam exquire-<lb/>bant, partim, abingenito mihi deſiderio, ra-<lb/>tionem, & cauſam eorum percipiendi, com-<lb/>mittendum non putaui, quin qualiacunque <lb/>meaſcripta in illis ſcientijs, ſtudioſis impartirer, <lb/>non dubitans quin illis aliquid commodi atque vtilitatis allatura ſint, prę <lb/>ſertim cum in eiuſmodi quæſtionibus inueſtigandis atque perpendendis, <lb/>nemo ( quod ſciam ) hactenus elaborauerit. </s> + <s xml:id="s15" xml:space="preserve">Nihil enim his libris à me <lb/>traditum eſt, quod aut legiſſe, aut ab alijs audiuiſſe meminerim, nam ſi <lb/>aliena attigi, ea, aut cum aliqua differentia demonſtrationis, aut diluci-<lb/>dius ſcripſi, quod ſi 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 Ariſtoteles ipſe ſenſit facilè fieri poteſt, vt pluribus, eædem opinio-<lb/>nes in mentem veniant. </s> + <s xml:id="s17" xml:space="preserve">Immo multa ſcribenti euenire poteſt, vt cum <lb/>iamdiu aliquid ſcripſerit, iam oblitus, idem repetat, quod mihi etiam <lb/>nonnunquam accidit. </s> + <s xml:id="s18" xml:space="preserve">In his autemlibris non ſuſcepi munus integræ ali <lb/>cuius ſcientiæ tradendæ, ne, quæ abalijs iam tradita ſunt, ipſe inutiliter re <lb/>peterem, mihiq́ue viderer exalienis laboribus laudem voluiſſe comparare. <lb/></s> + <s xml:id="s19" xml:space="preserve">Singularum enim ſcientiarum volumina, iam ab alijs collecta, at-<lb/>que in ordinemſunt digeſta, & ſi pauciſſimi ſint libri quorum omnes <lb/>ſententiæ, omniaq́ue inuenta vnius ſint authoris, excipio Archime-<lb/>dis volumina. </s> + <s xml:id="s20" xml:space="preserve">Cumque multi ſint, qui vel vnam rem à ſe inuentam <lb/>in publicum proferre non dubitent, multo magis mihi qui multa ex-<lb/>cogitaui, & ſi inter ſe hætereogenea, atque vtcunque expreſſa, idem <lb/>licere ſum arbitratus. </s> + <s xml:id="s21" xml:space="preserve">In his autem meditandis, ex Arithmeticis autho-<lb/>ribus quos inſpexi, præcipuus fuit Nicolaus Tartalea, quippe quem fe-<lb/>rè omnia ab alijs ſcripta collegiſſe conſtat, nec alios ex præcipuis, quos le-<lb/>gere potui omittendos duxi, inter quos ſunt Hieronymus Cardanus, Mi-<lb/>chael Stifelius, Gemma Friſius, Ioannes Nouiomagus, Cuthebertus <lb/>Tonſtallus, <reg norm="cæterique" type="simple">cæteriq́;</reg> huiuſinodi. </s> + <s xml:id="s22" xml:space="preserve">Quorundam tamen volumina illorum <lb/>qui à Tartalea citantur, vt Leonardi Piſani, Proſdocimi, Ioannis Infor-<lb/>tunati, Fratris Lucæ, Petri Borgi, aliorumq́ue aliquot inſpiciendorum, + <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 nonnullę inueniantur <lb/>propoſitiones, quæ diſiunctam ab alijs habeant rationem, eæ non ſper-<lb/>nendæ tamen ſunt, viam fortaſſe alicui aperient vlterius progrediendi. <lb/></s> + <s xml:id="s24" xml:space="preserve">Quemadmodum enim, exempli gratia, ex ſub contraria coni ſectione, <lb/>ſumpta poſtea fuit diuina illa Planisferijdelineation, quæ ſub Ptolomæi no-<lb/>mine legitur, & ſicuti ex <ref id="ref-0001">penultima primi Euclidis</ref>, quam Pythagoras <lb/>excogitauit propè innumeræ pulchræ conſequentiæ in Aſtronomia, in <lb/>Architectura, in <reg norm="multisque" type="simple">multisq́;</reg> alijs ſcientijs deſumptæ ſunt, immo quemad-<lb/>modum ex ſingulis propoſitionibus à noſtris maioribus excogitatis mul-<lb/>ta egregia ſunt deducta, ita fortaſſe continget, vt ex mearum muentio-<lb/>num aliqua, <reg norm="nonnihil" type="context">nõnihil</reg> in poſterum vtilitatis deſumatur. </s> + <s xml:id="s25" xml:space="preserve">Si quid verò, hic in-<lb/>ueneris, quod tuo genio non arrideat, illa prudentiſſimi hominis ſen-<lb/>tentia in mentem veniat. </s> + <s xml:id="s26" xml:space="preserve"><emph style="it">Quot capita, tot ſententiæ</emph>, ac per raro con-<lb/>tingere, vt idem omnibus probari, atque placere queat, & perdifficulter <lb/>inueniri hominem cui placeant omnia quæ alteri ſatisfaciunt. </s> + <s xml:id="s27" xml:space="preserve">Nec te mo <lb/>ueat, quodhęc Theoremata ſiue excogitationes non videas ordine illo di-<lb/>ſpoſitas, quo collocari debere exiſtimaueris, tum in Arithmeticis, tum in <lb/>cæteris. </s> + <s xml:id="s28" xml:space="preserve">Cum enim in huiuſmodi rebus ordo non ſit neceſſarr<unsure/>us, vi-<lb/>ſum eſt mihi poſſe me, ſine repræhenſione, illum negligere, cum ſpe-<lb/>culationi, ſiue inuentioni preęcipuè adeo mihi incumbendum decreuerim <lb/>vtin collocatione operam ponere, & tempus abſumere operæpretium <lb/>non duxerim, quod idem in epiſtolarum collocatione feci, in quibus per-<lb/>ſonarum ad quas ſcribo nullus ferè graduum ordo ſeruatus eſt, nec tem-<lb/>poris, quo ſunt ſcriptæ, quæſitorum tantummodo ratione habita. </s> + <s xml:id="s29" xml:space="preserve">Nec <lb/>admirari quenquam velim, quod in ſpeculandis 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 eſt abſtractus, <emph style="it">quoniam oportet in-<lb/>telligentem phantaſmata ſpeculari</emph>, cum pręterea perſpicuum ſit, diſcretum <lb/>omne, ex continui diuiſione aliquo modo oriri, ſiue actu, ſiue potentia. <lb/></s> + <s xml:id="s30" xml:space="preserve">Deinde ſi forte meis in deinonſtrationibus tibi videbor aliquando bre-<lb/>uior, illud in cauſa fuiſſe ſcias, quod ibi ad viros ſcribebam in his diſcipli-<lb/>nis exercitatos, quibus ſatis fuit rem ſignificare. </s> + <s xml:id="s31" xml:space="preserve">Libuit autem mihi om-<lb/>nes voluminis Arithmetici propoſitiones potius vocabulo theorema-<lb/>tum appellare, quam problematum, quia pars earum ſpeculatiua tan-<lb/>tum mea eſt, & ſi ex varijs eiuſmodi propoſitionibus etiam operatiuam <lb/>adinuenerim. </s> + <s xml:id="s32" xml:space="preserve">Quoniam verò multis in locis accidit, vt veritatis iudi-<lb/>candæ cauſa neceſſe mihi fuerit quorundam ſententijs aduerſari 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">meq́;</reg> hoc nomine carptorem <reg norm="maledicumque" type="simple">maledicumq́;</reg> ha-<lb/>bere quod alienos errores aperiam, cum potius habenda ſit mihi gratia, <lb/>quod in ijs interdum laborans (quę Antiſthenes in diſciplinis magis ne-<lb/>ceſſaria eſſe dixit, <emph style="it">vt mala ſcilicet prius dediſcantur<sic comment="closing parenthesis in italics">)</sic></emph> falſas opiniones euel-<lb/>lere ſtudeam, <reg norm="veritatemque" type="simple">veritatemq́;</reg> oſtendere, quam omnis philoſophus, Ariſto-<lb/>telis exemplo, pluris quam cuiuſuis hominis authoritatem, aut gratiam <lb/>facere debet. </s> + <s xml:id="s33" xml:space="preserve">Cumq́ue in hoc volumine aliquid eiuſmodi legeris <lb/>te oratum volo, vt in iudicando, affectum omnem exuas, <lb/>Salluſtianum 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> conſultant, ab odio amicitia, ira, atque <lb/>miſericordia vacuos eſſe decet. </s> + <s xml:id="s35" xml:space="preserve">Hinc fiet, vt <lb/>non perſonæ (vt multiſolent) ſed <lb/>veritati, quę ſummo ſtudio di-<lb/>gniſſima eſt, ſemper po <lb/>tius faueas. </s> + <s xml:id="s36" xml:space="preserve">Vale <lb/>noſtrisq́ue <lb/>labo-<lb/>ribus vtere, ſi quem inde fructum, <lb/>ſicuti ſpero tuleris, illi præ-<lb/>cipuè habeas gratiam à <lb/>quo omnes fluunt <lb/>ſcientiæ.</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 philoſophi de nu<lb/>meris eorumq́ue effectibus excogitata poſteris tradide-<lb/>runt, quorum cum vix vllam rationem reddiderint, aut <lb/>certè per exiguam, occaſione diuerſorum problematum <lb/>mihi à Sereniſſimo Sabaudiæ Duce propoſitorum præbi-<lb/>ta, de ijs quæ ab antiquis propoſita fuerunt contemplanda <lb/>nonnulla occurrerunt, quæ poſteritati comendare non <lb/>inutile arbitratus fum, ne hæ meæ cogitationes intercide-<lb/>rent, & occaſionem præberem quamplurimis abſtruſa hęc <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 queſita, 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 cognoſci poſ-<lb/>ſet ſcientificè & ſpeculatiue (vt dicitur) productum ex duobus fractis numeris, <lb/>quolibet producentium minus eſſe. </s> + <s xml:id="s40" xml:space="preserve">Cui reſpondi, mente & cogitatione conci-<lb/>piendum eſſe fractos producentes cum fractis productis, non vnius eiuſdemq́ue 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 ſint <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 ſuperficiali <var>.d.b.</var> generato ex <lb/>vno in aliud totorum linearium, nam ſi conſtitueretur <var>.a.i.</var> octauum ipſius <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 ipſius <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">relatiuũ</reg> ipſius <var>.c.i.</var> non aliquod totum producentium. <lb/></s> + <s xml:id="s43" xml:space="preserve">Mirum itaque non eſt ſi productum <var>.c.i.</var> minus videatur fuis producentibus, cum <lb/>toto, diuerſæ naturæ à primis conferatur, fractum fiquidem ab integro eiuſdem <lb/>naturæ, linearis, ſuperficialis, aut corporeæ denominatur.</s> + </p> + <p> + <s xml:id="s44" xml:space="preserve">Quòd ſi amplioris cognitionis gratia ex ſcientiæ præceptis ſpeculari 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 ſit in ſuo integro <var>.d.b.</var> fracto <var>.a.i.</var> in <lb/>ſuo integro <var>.a.b.</var> aut fracto <var>.a.c.</var> in ſuo integro <var>.a.d.</var> conſideret is quo pacto pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor ſit 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/>nifeſtum eſt ex <ref id="ref-0003">prima ſexti de quantitate <lb/>continua</ref>, aut <ref id="ref-0004">.18. ſeptimi Euclidis</ref> de diſcre <lb/> + <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> + ta, proportionem ipſius <var>.d.i.</var> ad <var>.d.b.</var> eſſe ſi-<lb/>cut <var>.a.i.</var> ad <var>.a.b.</var> & cum <var>.c.i.</var> minor ſit <var>.d.i.</var> <lb/>velut pars ſuo 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="eiuſdem" type="context">eiuſdẽ</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 eiſdem cauſis, medio <var>.c.b</var>. </s> + <s xml:id="s47" xml:space="preserve">Ex quibus pa-<lb/>tet ratio, cur fracti diuerſarum denomina-<lb/>tionum ad vnicam reducantur. </s> + <s xml:id="s48" xml:space="preserve">Cur etiam <lb/>numeros integros in partes fractis ſimiles <lb/>frangere liceat, quæ omnia ex ſubſequenti <lb/>figura facilè cognoſci 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> ſit ratio, cur hi, qui numeros, fractos diuerſarum denominationum col-<lb/>ligere volunt, & in ſummam redigere, multiplicent vnum ex numerantibus <lb/>per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum <lb/>vltimum productum, commune eſt denominans duorum priorum productorum, <lb/>quæ collecta in ſummam efficiunt quod quærebatur.</s> + </p> + <p> + <s xml:id="s50" xml:space="preserve">Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius <reg norm="eiuſdem- q́ue" type="context">eiuſdẽ-<lb/>q́ue</reg> magnitudinis quantitatis continuæ, linearum (verbigratia) <var>a.b.</var> et <var>.a.d.</var> <reg norm="æqualium" type="context">æqualiũ</reg> <lb/>in longitudine, <reg norm="quarum" type="context">quarũ</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 ſi colli-<lb/>gere voluerimus duo tertia cum tribus quartis, multiplicabimus <var>.a.c.</var> duo tertia, <lb/>cum <var>.a.b.</var> diuiſa in 4. partes, produceturq́ue <var>.c.b.</var> octo partium ſuperficialium, de-<lb/>hinc multiplicando <var>.a.i.</var> tres quartas cum <var>.a.d.</var> diuiſa in .3. partes producetur <var>.i.d.</var> pri <lb/>mis ſingulis æqualis, nouem partium ſuper <lb/>ficialium, multiplicata deinde <var>a.b.</var> diui-<lb/> + <anchor type="figure" xlink:label="fig-0014-02a" xlink:href="fig-0014-02"/> + ſa in .4. partes per <var>.a.d.</var> in .3. diuiſa, produ-<lb/>cetur quadratum <var>.d.b.</var> in continuo, in 12. <lb/>partes diuiſum, quod erit totum commune <lb/>ſingulis productis, quorum primum erat <var>.c.<lb/>b</var>. </s> + <s xml:id="s52" xml:space="preserve">Quare <var>.c.b.</var> ita ſe habet ad totum <var>.d.b.</var> ſi-<lb/>cut <var>.a.c.</var> ad <var>.a.d.</var> ex prima ſexti in continuis, <lb/>aut .18. ſeptimi in diſcretis quantitatibus, <lb/>et <var>.d.i.</var> ad <var>.d.b.</var> ſicut <var>.a.i.</var> ad <var>.a.b.</var> ex eiſdem <lb/>propoſitionibus. </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> manifeſtè depræhendetur eiuſmodi <lb/>ſummam componi ex partibus vnius totius <lb/>communis ſingulis 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 ſit fractus aliquis numerus reſpectu 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 ſi cauſam noſce vis, ſume <var>.o.i.</var> & <var>.o.u.</var> pro totis denominatoribus, tum <var>.o.e.</var> <lb/>& <var>.o.a.</var> pro numeratoribus (exempli cauſa) ſit <var>.o.i.</var> ſenarius <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 noſce vis quæ ſint tres quartę partes quinque ſextarum, <lb/>patet ex regulis practicis oriri quindecim vigeſimaſquartas. </s> + <s xml:id="s58" xml:space="preserve">Id quomodo fiat, ex <lb/>ſubſcripta ſigura depræhendetur, memores tamen eſſe oportet, quodlibet <reg norm="productum" type="context">productũ</reg> <lb/>conſiderari <reg norm="tanquam" type="context">tanquã</reg> ſuperficiem, producentia <reg norm="autem" type="context">autẽ</reg> tan-<lb/>quam lineas. </s> + <s xml:id="s59" xml:space="preserve">In hac igitur ſigura productum ex totis <lb/> + <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> + linearibus eſt <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 ſe habebit <lb/>ad productum totale <var>.u.i.</var> ſicut <var>.o.e.</var> ad <var>o.i.</var> ex prima <lb/>ſexti aut .18. ſeptimi, ita <var>.u.e.</var> erunt quinque ſextæ par <lb/>tes <var>.u.i.</var> quarum in propoſito exemplo, tres quartæ <lb/><reg norm="quæruntur" type="context">quærũtur</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">cũ</reg> <var>.o.a.</var> orietur <lb/>productum <var>.a.e.</var> ita <reg norm="proportionatum" type="context">proportionatũ</reg> ad <var>.u.e.</var> ſicut <var>.o.a.</var> ad <lb/><var>o.u.</var> reperitur, ex prædictis rationibus. </s> + <s xml:id="s62" xml:space="preserve">Quòd ſi <reg norm="ſtatutum" type="context">ſtatutũ</reg> <lb/>eſt <var>.o.a.</var> tres quartas partes eſſe ipſius <var>.u.o.</var> <reg norm="etiam" type="context">etiã</reg> <var>.a.e.</var> tres <lb/>quartæ partes <reg norm="erunt" type="context">erũt</reg> <var>.u.e.</var> ſed <var>.u.e.</var> quinque ſextæ ſunt ip-<lb/>ſius <var>.u.i.</var> ex quo ſequitur bonum eſſe huiuſmodi 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, partianturq́ue 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 ſubſequenti figura, numerum integrorum <lb/>tanquam lineam <var>.a.e.</var> qui, verbigratia, ſit denarius, quorum vnuſquiſque ſit æqualis <lb/><var>a.i.</var> cogiteturq́ue productum ipſius <var>.a.e.</var> in <var>.a.i.</var> ſitq́ue <var>.u.e.</var> quod quidem erit dena-<lb/>rius ſuperficialis, conſtituta prius <var>.a.u.</var> æqualis <var>.a.i.</var> & <var>.a.o.</var> ſint duæ tertiæ <var>.a.u.</var> <reg norm="quarum" type="context">quarũ</reg> <lb/>duarum tertiarum productum in numerum <var>.a.e.</var> ſit <var>.o.e.</var> pariter <var>.u.i.</var> vnitas ſit ſuper-<lb/>ficialis prout <var>.a.i.</var> vnitas eſt linearis, quam <var>.u.i.</var> reſpicere debet productum <var>.o.e.</var> ex <lb/>quo integer ſuperficialis <var>.u.i.</var> erit tanquam ternarius, & productum <var>.o.i.</var> tanquam bi <lb/>narius, & quia quælibet pars è viginti ipſius <var>.o.e.</var> æqualis eſt tertiæ parti <var>.u.i.</var> vnita-<lb/>tis ſuperficialis; </s> + <s xml:id="s65" xml:space="preserve">ſi cupiamus ſcire quot integræ vnitates ſint in partibus <var>.o.e.</var> conſul-<lb/>tum eſt eaſdem diuidere per denominantem <var>.u.i.</var> compoſitum ex tribus partibus ſu <lb/>perficialibus, & cum tam linea <var>u.a.</var> quam ſuperficies <var>.u.i.</var> diuidatur in 3. partes <reg norm="aequa les" type="simple">ęqua<lb/>les</reg> noſce peroportunum eſt eiuſmodi partitionem numeri <var>.o.e.</var> fieri per numerum <lb/>ipſius <var>.u.i.</var> non <var>.u.a.</var> ex prædictis cauſis.</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 cauſa, ſpeculando inno-<lb/>teſcere poteſt, cuius rei gratia for-<lb/>metur ſequens figura <var>.e.o.a.u.n.</var> <lb/>eiuſmodi, vt <var>a.e.</var> ſit numerus li-<lb/>linearis integrorum, & <var>o.e.</var> produ-<lb/>ctum numerantis ipſorum <reg norm="fractorum" type="context">fractorũ</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">linearũ</reg> ſingulę ſtatuuntur æqua <lb/>les vnitati lineari, ſuperficies autem <lb/>parallelogramma <var>.u.n.</var> conſtituatur <lb/>æqualis magnitudinis ſuperficiei <var>.o.<lb/>e.</var> ex quo <var>.u.n.</var> erit nobis cognita ſu-<lb/>perficies. </s> + <s xml:id="s67" xml:space="preserve">Cognoſcetur pariter quan <lb/>titas partium <var>.a.u.</var> quam in propoſi-<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 ſi <var>.u.a.</var> dat <var>.a.e.</var> ſine dubio <var>.o.<lb/>a.</var> dabit <var>.a.n.</var> numerum linearem. <lb/>quæ regula ex 15. ſexti in continuis, <lb/>& ex 20. ſeptimi in diſcretis, 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 ſpeculatione cognoſci poteſt hoc rectè fieri, mul-<lb/>tiplicantes enim has duas tertias per decem, debemus conſide-<lb/> + <anchor type="figure" xlink:label="fig-0016-02a" xlink:href="fig-0016-02"/> + rare quantitatem duarum <reg norm="tertiarum" type="context">tertiarũ</reg> decies produci, ex quo oriuntur <lb/>20. tertia, quandoquidem ſingulæ vnitates, </s> + <s xml:id="s71" xml:space="preserve">tunc pro duobus ter-<lb/>tijs ſumuntur, ſed c<unsure/>um quilibet integer tria fragmenta contineat, <lb/>ideo ex ratione partiendi quoties ternarius ingrediatur viginti, <lb/>ſtatim cognoſcemus 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 ipſum accideret ſi integri in eiuſmodi ſpecie fractorum diui-<lb/>derentur. quo facto hi multiplicandi eſſent cum numerante propo <lb/>ſito, & <reg norm="partiendum" type="context">partiendũ</reg> productum per quadratum denominantis.</s> + </p> + <p> + <s xml:id="s73" xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s> + <s xml:id="s74" xml:space="preserve">Sit linea <var>.a.e.</var> conſtans ex <reg norm="quinque" type="simple">quinq;</reg> <lb/>integris numeris, quorum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> æqualis ſit <var>.a.u.</var> vel <var>.a.i.</var> & <var>.a.o.</var> <lb/>ſint 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 ſua <reg norm="fragmenta" type="context">fragmẽta</reg> linearia, quę in propoſito exemplo <lb/>erunt 15. multiplicatis iam 15. cum propoſitis, videlicet <var>a.o.</var> orie-<lb/>tur productum <var>.o.e.</var> triginta fragmentorum ſuperficialium, <reg norm="quorum" type="context">quorũ</reg> <lb/>in ſingulos integros ſuperficiales <reg norm="cadunt" type="context">cadũt</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, propoſitum con-<lb/>ſequemur.</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 ſpecies 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 cauſa) ſi 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 diuiſum per <lb/>productum ternarijin quaternarium, videlicet per duode <lb/>cim, quatuor integri proferentur cum ſeptem 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 ſubſequens figura in qua linea <var>a.i.</var> æqualis ſit li-<lb/>neæ <var>.u.a.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <reg norm="conſideretur" type="context">cõſideretur</reg> pro integro nume <lb/>ro: </s> + <s xml:id="s80" xml:space="preserve"><reg norm="cogiteturque" type="simple">cogiteturq́;</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">cũ</reg> <lb/>duobus tertijs, & <var>a.e.</var> æquipollens duobus integris & tri-<lb/>bus quartis. </s> + <s xml:id="s82" xml:space="preserve">Iam ſi hæ duæ lineæ in ſuos fractos redu-<lb/>cantur, multiplicata (vt in ſequenti figura apparet.) <var>a.o.</var> <reg norm="cum" type="context">cũ</reg> <lb/><var>a.e.</var> orietur productum <var>o.e.</var> fractorum ſuperficialium <lb/><reg norm="quinquagintaquinque" type="simple">quinquagintaquinq;</reg>, quorum integer ſuperficialis va-<lb/>let duodecim, ſcilicet <var>.u.i.</var> vt cuique manifeſtum eſt, 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 ſi fractiad vnam eandemq́ue denominationem reduceren-<lb/>tur, qui poſtmodum ſimul multiplicarentur, productumq́ue partiremur per qua-<lb/>dratum denominantis communis.</s> + </p> + <p> + <s xml:id="s84" xml:space="preserve">Exempli cauſa, ſint eadem quinque tertia, & vndecim quarta adinuicem multi-<lb/>plicanda, quæ ſi reducantur ad vnam & eandem denominationem quinarius <lb/>numerans vnius, multiplicabitur cum quaternario deno-<lb/>minante alterius, & vndenarius ſecundi 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 ſi multiplicentur vi <lb/>ginti cum trigintatribus, dabuntur 660. fracti, quorum in-<lb/>teger erit quadratum duodenarij, nempe 144. quibus qui-<lb/>dem 660. diuiſis per 144. proferentur quatuor integri & <lb/>ſeptem 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 ſit in ſubſcripta 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> ſit vndecim <reg norm="partium" type="context">partiũ</reg> <lb/>talium qualium <var>.a.i.</var> eſt quatuor, & <var>.a.o.</var> ſit quinque pro-<lb/>ut <var>.a.u.</var> eſt 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 ſuperficialium. </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">trigintatriũ</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> conſtabit ex duode-<lb/>cim partibus eiuſdem rationis cum reliquis duobus <lb/>productis, quod quadratum <var>.u.i.</var> vnitas eſt ſuperficia-<lb/>lis, & communis denominans duorum productorum. <lb/></s> + <s xml:id="s90" xml:space="preserve">quod ſi in præſentiarum cogitabimus lineam <var>.c.d.</var> tri-<lb/>gintatrium partium æqualium, et <var>.c.t.</var> duodecim ſimi-<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 ſuperficies <var>.f.d.</var> 660. fractorum <lb/>ſuperficialium, quorum vnitas integra ſuperficialis <lb/>erit quadratum <var>.n.t.</var> 144. partium cuiuſmodi <var>.f.d.</var> <lb/>partes habet .660. diuiſo itaque <var>.f.d.</var> per <var>.n.t.</var> pro-<lb/>poſitum conſequetur. </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 eſt 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 ſexti vel 18. ſeptimi, ſed vt <var>.f.d.</var> ad id <reg norm="quod" type="simple">ꝙ</reg> <lb/>fit ex <var>.f.c.</var> in <var>.c.t.</var> eſt vt <var>.c.d.</var> ad <var>.c.t.</var> & vt eius <reg norm="quod" type="simple">ꝙ</reg> fit ex <lb/><var>f.c.</var> in <var>.c.t.</var> ad <var>.n.t.</var> eſt vt <var>.f.c.</var> ad <var>.c.n.</var> ex dictis pro-<lb/>poſitionibus </s> + <s xml:id="s92" xml:space="preserve">quare ex æqua proportionalitate, eodem <lb/>modo diſcurrendo in figura <var>.o.a.e.</var> ita ſe 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 conſiderata fuerunt, apertè <lb/>ratio deprehenditur, cur productum, ſingulis producen <lb/>tibus ſemper minus ſit, cum producta ſint ſuperficialia <lb/>producentia verò ſemper linearia, omiſſis productis <lb/>corporeis, quæ omnia ad ſuperficialia 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 diuiſione, animaduertendum eſt, denominantes numeros <lb/>ſemper æquales inuicem eſſe debere, vnius ſcilicet ſpeciei, quòd ſi æquales non <lb/>fuerint, neceſſe eſt via multiplicationis ipſorum denominantium adinuicem effice-<lb/>re æquales vt ſint, ex quo productum oritur eiuſmodi, vt aptum ſit habere partes <lb/>fractorum, quæ deſiderabantur.</s> + </p> + <p> + <s xml:id="s95" xml:space="preserve">Exempli gratia, ſi proponerentur diuidenda ſeptem 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 propoſita ori-<lb/>tur triginta duarum partium commune denominans, cuius duo numerantes ſunt 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 ſeptem octaua particularum vniformium, prout ope <lb/>primæ ſexti aut decimæoctauæ ſeptimi in ſubſcripta figura cognoſci poteſt.</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"/> + ſit <var>.u.i.</var> trigintaduarum particularum <lb/>fuperficialium fimilium & <reg norm="æqualium" type="context">æqualiũ</reg> ad-<lb/>inuicem. </s> + <s xml:id="s99" xml:space="preserve">fit deinde <var>.a.e.</var> ſeptem <reg norm="partium" type="context">partiũ</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 ſuperficialium vigintiocto <lb/>& productum <var>.a.o.</var> in <var>.a.i.</var> erit <var>.o.i.</var> par <lb/>ticularum <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg> vigintiquatuor <lb/>eiuſdem 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> ſeu diuidere vno numero alium numerum, eſt etiam quodammodo <lb/>eiuſmodi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis, <lb/>cuiuſmodi eſt vnitas in diuidente refpectu totius diuidentis, partem inquam numeri <lb/>diuiſibilis ſic ſe habentem ad totum numerum diuiſibilem ſicut vnitas ad totum di-<lb/>uidentem, quod ſimiliter ex regula de tribus præſtamus dicentes, ſi tantus numerus <lb/>diuidens dat <reg norm="vnitatem" type="context">vnitatẽ</reg>, quid dabit numerus diuifibilis, quemadmodum ex <ref id="ref-0006">.15. ſexti</ref> <lb/>ſeu <ref id="ref-0007">.20. ſeptimi</ref> licet ſpeculari, Idcircò quotieſcunque minorem numerum per <lb/>maiorem diuidimus, ſemper qui prouenit fractus eſt.</s> + </p> + <p> + <s xml:id="s103" xml:space="preserve">Exempli gratia, ſi cogitaremus lineam <var>.a.e.</var> diuiſam in octo partes æquales, qua <lb/>rum vna ſcilicet vnitas effet <var>.a.i.</var> & cupere-<lb/>mus eam diuidere in nouem partes, ac ſcire <lb/> + <anchor type="figure" xlink:label="fig-0019-02a" xlink:href="fig-0019-02"/> + quan a ſit nona illius pars; </s> + <s xml:id="s104" xml:space="preserve">manifeſtum eſſet, <lb/>nonam partem ipſius <var>.a.e.</var> minorem futuram <lb/>ipſa <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 innoteſcat, hoc <lb/>etiam modo licebit videre ſitlinea <var>.n.c.</var> no-<lb/>nupla ad <var>.a.i.</var> & parallela ad <var>.a.e.</var> dubium non <lb/>eſt quin <var>.n.c.</var> maior futura ſit ipſa <var>.a.e.</var> iam ſi <lb/>earum extrema congiungantur medijs duabus <lb/>lineis <var>.n.a.</var> et <var>.c.e.</var> quæ ſimul concurrant in <lb/>puncto <var>.o.</var> (quod eſt 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/>ipſius <var>.n.c.</var> quæ <var>.n.t.</var> æqualis erit <var>.a.i.</var> ex præſuppoſito. </s> + <s xml:id="s107" xml:space="preserve">ducatur deinde <var>.o.t.</var> quę <lb/>interſecet <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 eſt <var>.n.c.</var> neque enim dubium eſſe poteſt 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> ſint æquales inuicem quandoqui-<lb/> + <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> + dem vnaquæque earum ex triangulorum ſimi <lb/>litudine æqualis eſt 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 eſt <var>.a.i.</var> tanto maior erit <var>.a.x.</var> <lb/>quanto <var>.n.c.</var> maior eſt <var>.a.e.</var> vnde ficut <var>.a.e.</var> con-<lb/>ſtat octo nonis ipſius <var>.n.c.</var> ita pars <var>.a.x.</var> ipſius <var>.<lb/>a.e.</var> octo nonis conſtabit ipſius <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 eſt præterea diuidere numerum <lb/>per numerum: </s> + <s xml:id="s111" xml:space="preserve">eſſe inuenire <reg norm="alterum" type="context">alterũ</reg> latus à quo <lb/>producitur, ſuppoſito ſemper quòd numerus <lb/>diuifibilis ſuperſicialis ſit, & rectangulus.</s> + </p> + <p> + <s xml:id="s112" xml:space="preserve">Exempli gratia, ſi proponantur triginta diuidenda per quinarium, nihil aliud erit <lb/>hæc diuiſio, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-<lb/>cat triginta ſuperficies rectangulas, huiuſmodi verò eſt ſenarius, cuius ſingulæ vnita-<lb/>tes ſuperficiales erunt.</s> + </p> + <p> + <s xml:id="s113" xml:space="preserve">Cuius rei gratia ſit ſubſcriptum rectangulum <var>.a.e.</var> triginta vnitatum <reg norm="ſuperſicialium" type="context">ſuperſicialiũ</reg>, <lb/>cuius latus <var>.e.n.</var> ſit quinque vnitatum. </s> + <s xml:id="s114" xml:space="preserve">hinc latus <var>.a.n.</var> erit ſex 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 ſex vnitatum. <lb/></s> + <s xml:id="s116" xml:space="preserve">Sin verò diuiſerimus 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 ſicut diuidentis ad vnita-<lb/>tem, ex prima ſexti, aut .18. vel .19. ſeptimi, & permutatim <lb/>ita ſe habebit diuiſibile ad diuidentem, ſicut 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 eſt, quam inuenire latus rectanguli, quod productum in <lb/>diuidente, numerum diuiſibilem compl at, ex quo numerus diuiſibilis ſuperficialis <lb/>eſt, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuſcemodi <lb/>numerum diuiſibilem. </s> + <s xml:id="s119" xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-<lb/>tem ex multiplicatione laterum ſiue linearum generatur ſuperficies, ex diuiſione po-<lb/>ſtea ipſius ſuperficiei inuenitur alterum latus. </s> + <s xml:id="s120" xml:space="preserve">quare mirum non eſt, ſi proueniens ex <lb/>vna diuiſione (via fractorum) ſit ſemper maius numero diuiſibili.</s> + </p> + <p> + <s xml:id="s121" xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reſultat vnus integer nu <lb/>merus cum dimidio pro numero qui oritur. </s> + <s xml:id="s122" xml:space="preserve">Sit itaque dimidium ſuperſiciale diuiſi-<lb/>bile <var>.b.c.</var> cuius totum ſit <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 ſit <var>.b.d.</var> quærendum nobis eſt latus <var>.b.s.</var> quod cum latere <var>.b.n.</var> producat re <lb/>ctangulum <var>.n.s.</var> æquale dimidio ſuperſiciali propoſito <var>.b.c.</var> quod ſi ſiat, ex .15. ſexti, <lb/>aut .20. ſeptimi. erit eadem proportio <var>.b.n.</var> ad <var>.b.q.</var> quæ eſt <var>.q.c.</var> ad <var>.b.s.</var> dicemus itaque <lb/>ſi <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> ſed <var>.n.b.</var> eſt tertium lineare et <var>.b.q.</var> lineare <reg norm="in- tegrum" type="context">in-<lb/>tegrũ</reg>, & <var>b.s.</var> proueniens lineare. </s> + <s xml:id="s125" xml:space="preserve">& quia <var>.b.c.</var> dimidium ſuperficiale, 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> ſit ęqualis <var>.b.c.</var> & productum ex <var>.<lb/>b.n.</var> minori <var>.q.c.</var> neceſſe eſt, vt producatur in <var>.b.s.</var> maiore <var>.q.b.</var> quod <var>.q.b.</var> maius eſt <var>.q.c.</var> <lb/>quod quidem <var>.q.c.</var> ita appellatur ſicut <var>.b.c</var>. </s> + <s xml:id="s127" xml:space="preserve">quare mirum non eſt ſi proueniens per fra-<lb/>ctos numeros ex diuiſione, maior ſit numero diuiſibili.</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 manifeſte patet quamlibet <reg norm="diuiſionem" type="context">diuiſionẽ</reg> aut partitionem oriri ex regula de tri-<lb/>bus, quandoquidem ſinguli diuidentes æquipollent vni integro, & loco illius ſu-<lb/>muntur. </s> + <s xml:id="s129" xml:space="preserve">Perinde enim eſt diuidere centum per viginti, ac <reg norm="regulam" type="context">regulã</reg> obſeruare de tri-<lb/>bus <reg norm="dicentes" type="context">dicẽtes</reg>, ſi viginti æquipollent vni, quibus <reg norm="ęquiualebunt" type="context">ęquiualebũt</reg> <reg norm="centum" type="context">cẽtum</reg>? </s> + <s xml:id="s130" xml:space="preserve">Hoc autem ex ſub <lb/>ſequenti figura facile deprehendetur, in qua linea <var>.a.b.</var> ſignificat 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">centũ</reg>: </s> + <s xml:id="s131" xml:space="preserve">o.c. verò centum vnitates ſuperficiales, <lb/>et <var>.a.d.</var> <reg norm="quinque" type="simple">quinq;</reg> vnitates lineares, et <var>.d.b.</var> centum vnitates ſuperficiales, ex quo manife-<lb/>ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eſt, quam inueni <lb/>re <reg norm="productum" type="context">productũ</reg> ex duobus lateribus propoſitis, it a partiri nihil aliud eſt, quam da-<lb/>to vno latere inuenire aliud latus producti propoſiti.</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="quotieſcunque" type="simple">quotieſcunq;</reg> <reg norm="ratiocinantes" type="context">ratiocinãtes</reg> dicimus tantundem numeri, immediate produci <lb/>mus ſuperficiem, <reg norm="mediante" type="context">mediãte</reg> vnitate in huiuſmodi numero, qui numerus <reg norm="antequam" type="context">antequã</reg> pro-<lb/>ducatur in vnitatem, mente concipiendus eſt tanqua m linearis, tanquam linea in-<lb/>quam diuiſa in totidem particulas lineares, ſingulas continuas & æquales vnitati <lb/>propoſitæ. </s> + <s xml:id="s133" xml:space="preserve"><reg norm="Cum" type="context">Cũ</reg> verò productus fuerit numerus in vnitate ſuperficialis, erit ac ſi tot eſ-<lb/>ſent vnitates quadratæ, quod ſi ita non eſſet, nulla mentio facienda eſſet quo-<lb/>rumuis <reg norm="fractorum" type="context">fractorũ</reg>. </s> + <s xml:id="s134" xml:space="preserve">Ex <reg norm="eadem" type="context">eadẽ</reg> regula de tribus reduci poteſtad praxim <reg norm="tertium" type="context">tertiũ</reg> theorema.</s> + </p> + <p> + <s xml:id="s135" xml:space="preserve">Quare cupientes ſcire quæ ſint illæ partes, quæ ſunt tres quartę, ipſarum quin-<lb/>que ſextarum, dicemus ſi quatuor dant tria, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> ſextæ? </s> + <s xml:id="s136" xml:space="preserve">dabunt .15. <lb/>vigeſimas quartas, quæ quindecim ſunt tres quartæ ipſius .20. viginti <reg norm="autem" type="context">autẽ</reg> <reg norm="quinque" type="simple">quinq;</reg> ſex <lb/>tæ vigintiquatuor, quandoquidem nos numerum quęrimus, cui ita proportionentur <lb/><reg norm="quinque" type="simple">quinq;</reg> ſextæ alterius numeri, ſicut quatuor ad tria, vnde ſic ſe habent .20. ad .15. ſi-<lb/>cut .4. ad .3. ipſe autem .20. <reg norm="quinque" type="simple">quinq;</reg> ſextę partes ſunt vigintiquatuor, vt per ſe <reg norm="notum" type="context">notũ</reg> eſt.</s> + </p> + <p> + <s xml:id="s137" xml:space="preserve">Ex eadem regula de tribus, huiuſmodi quęſito reſponderi poteſt, ſi conſtituamus <lb/>prædictas <reg norm="quinque" type="simple">quinq;</reg> ſextas eſſe numerum, cuius tres quartæ quęrantur, dicentes, ſi vnus <lb/>integer dat tres quartas, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> ſextæ? </s> + <s xml:id="s138" xml:space="preserve">quare ſequentes regulam de <lb/>tribus, dabuntur quindecim vigeſimæ 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/>ſcire poſſit, quæ pars aut partes numeri propoſiti ſit aliquis numerus.</s> + </p> + <p> + <s xml:id="s141" xml:space="preserve">Exempli gratia, ſcire cupienti, quæ pars aut partes ipſius vigintiquatuor ſint ſex-<lb/>decim, conſtituentur .24. tanquam vnum totum, cuius pars aut partes ſint ſexdecim, <lb/>dicemus igitur ſi .24. dant ſexdecim, quid dabit vnum? </s> + <s xml:id="s142" xml:space="preserve">ſexdecim videlicet vigeſi-<lb/>masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ. <lb/></s> + <s xml:id="s143" xml:space="preserve">Eadem ratione qui ſcire uellet, quæ partes aut pars eſſent tres quartæ, octo no-<lb/>narum, diceret, ſi octo nonæ danttres quartas, quid dabit vnum? </s> + <s xml:id="s144" xml:space="preserve">prouenient .27. <lb/>trigeſimęſecundæ.</s> + </p> + <p> + <s xml:id="s145" xml:space="preserve">Subſeruit pariter ad <reg norm="ſciendum" type="context">ſciendũ</reg> <reg norm="naturam" type="context">naturã</reg> <reg norm="partium" type="context">partiũ</reg> numeri propoſiti. </s> + <s xml:id="s146" xml:space="preserve">Exempli cauſa, ſi quis <lb/>quærat, cuius numeri, duodecim ſint duæ tertiæ partes. </s> + <s xml:id="s147" xml:space="preserve">Dicet ſi 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 ſcilicet, <lb/></s> + <s xml:id="s149" xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita ſe <lb/>habeant duodecim, ſicut duo ad tria. </s> + <s xml:id="s150" xml:space="preserve">Ita etiam ſi quis quærat, cuius numeri duo <lb/>tertia ſint tres quintę, dicet, ſi 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> quęrere numerum ad quem ſic ſe <lb/>habeant duo tertia ſicut tria ad <reg norm="quinque" type="simple">quinq;</reg>, quod manifeſtum eſt per ſe.</s> + </p> + <p> + <s xml:id="s153" xml:space="preserve">Eadem ratione qui ſcire vellet, cuius numeri duæ ſeptimæ, eſſent octo integra-<lb/>rum cum duabus quintis, diceret, ſi duo dant ſeptem 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 ſi proponerentur vnde cim tertiædecimæ vnius totius, toto diui-<lb/>ſo in .13. partes, <reg norm="deſideraremusque" type="simple">deſideraremusq́;</reg> ſcire, quot partes totius <reg norm="eſsent" type="context">eſsẽt</reg> vndecim <reg norm="tertiaedeci- mæ" type="simple">tertiędeci-<lb/>mæ</reg>, toto in .4. partes diuiſo, diceremus ſi .13. dant .11. quid dabunt quatuor? </s> + <s xml:id="s157" xml:space="preserve">nem <lb/>pe <reg norm="dabunt" type="context">dabũt</reg> tres quartas <reg norm="cum" type="context">cũ</reg> <reg norm="quinque" type="simple">quinq;</reg> tertijsdecimis unius quartæ, hoc verò nihil aliud eſt <lb/>quam querere numerum, ad quem ſic ſe habeat totum in 4. partes diuiſum, ſicut <lb/>idem totum diuiſum in tredecim ſe habet ad undecim tertiasdecimas, Porrò ad <lb/>alia etiam multa hæc regula accommodata eſt.</s> + </p> + <p> + <s xml:id="s158" xml:space="preserve">Hæc enim <reg norm="non" type="context">nõ</reg> ſine propoſito dicta ſunt, ſed ut <reg norm="quiſque" type="simple">quiſq;</reg> videat cauſam ſimilium ope-<lb/>rationum, quæ à practicis circa fractos numeros ſcriptæ ſunt, omnem à diuina illa <lb/>regula de tribus originem trahere ut etiam in ſequentibus 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, ſemper æquale eſt numero <lb/>diuiſibili ſi queras ita accipe.</s> + </p> + <p> + <s xml:id="s160" xml:space="preserve">Sit numerus diuiſibilis <var>.b.</var> quod oritur ſit <var>.c.</var> diuidens <var>.d.</var> & vnitas diuidentis <var>.t.</var> cum <lb/>igitur, vt in præcedenti theoremate dictum <lb/>fuit, eadem ſit proportio <var>.b.</var> ad <var>.c.</var> quæ eſt <var>.d.</var> <lb/> + <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> + ad <var>.t.</var> manifeſte deprehenditur ex .20. ſepti <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 ipſum alia ratione contemplari licet.</s> + </p> + <p> + <s xml:id="s162" xml:space="preserve">Numerus diuiſibilis ſignificetur 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> ſit ſuperficies <var>.u.a</var>. </s> + <s xml:id="s164" xml:space="preserve">Dico ſuperficiem <var>.u.a.</var> componi <lb/>ex tot vnitatibus ſuperficialibus quot linearibus conſtat linea <var>.n.e.</var> nam ex ijs quæ <lb/>diuidendi ratione notauimus, <reg norm="conſtituitur" 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"/> + quę eſt <var>.a.e.</var> ad <var>.o.e</var>. </s> + <s xml:id="s165" xml:space="preserve">At ex prima ſexti aut <lb/>18. ſeptimi ſic ſe habet totale <reg norm="productum" type="context">productũ</reg> <var>.<lb/>u.a.</var> ad partiale <var>.u.o.</var> ſicut <var>.a.e.</var> ad <var>.o.e</var>. <lb/></s> + <s xml:id="s166" xml:space="preserve">quare ſic ſe habebit <var>.u.a.</var> ad <var>.u.o.</var> ſicut <var>.n.<lb/>e.</var> ad <var>.u.e.</var> ſed <var>.u.e.</var> et <var>.u.o.</var> numero non differunt, cum ſint vnius & eiuſdem ſpeciei, (ta-<lb/>met ſi numerus <var>.u.o.</var> ſit ſuperficialis 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 diuiſibilem per proueniens, oritur numerus diui-<lb/>dens?</s> + </p> + <p> + <s xml:id="s169" xml:space="preserve">Sit ſubſcriptus rectangulus <var>.o.e.</var> numerus diuiſi <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 ſi <var>.a.o.</var> diuidens fuerit <var>.a.e.</var> proue-<lb/>niens erit, ſi 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">HOcipſum, alia <reg norm="quoque" type="simple">quoq;</reg> uia licebit ſpeculari.</s> + </p> + <p> + <s xml:id="s172" xml:space="preserve">Sit linea <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> numerum diuiſibilem, et <var>.o.</var> primi prouenientis linea <var>.e.</var> pri <lb/>mi diuidentis <var>.u.</var> ſecundi prouenientis ideſt cum <var>.o.</var> pro diuidente ſumetur. </s> + <s xml:id="s173" xml:space="preserve">Iam ex <lb/>indicata definitione diuiſionis nono theoremate huius libri, dabitur proportio <var>.a.</var> <lb/>ad <var>.o.</var> prout datur <var>.e.</var> ad vnitatem ſignificatam li-<lb/>nea <var>.i.</var> & permutatim <var>.a.</var> ad <var>.e.</var> ſicut <var>.o.</var> ad <var>.i.</var> ſed <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0023-02a" xlink:href="fig-0023-02"/> + ad <var>.u.</var> ſic ſe habet prout <var>.o.</var> ad <var>.i.</var> ex eadem definitio-<lb/>ne diuiſionis, <reg norm="itaque" type="simple">itaq;</reg> ſic ſe habebit <var>.a.</var> ad <var>.u.</var> ſicut <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 cognoſcere cuius numeri quatuor quintæ par-<lb/>tes, ſint duæ tertię, aut quid ſimile, <reg norm="conſultiſſime" type="context">cõſultiſſime</reg> faciat, ſi ad unam <reg norm="eandemque" type="simple">eandemq;</reg> <lb/>denominationem reduxerit.</s> + </p> + <p> + <s xml:id="s175" xml:space="preserve">Prout in propoſito exemplo, <reg norm="cum" type="context">cũ</reg> <reg norm="denominans" type="context">denominãs</reg> <reg norm="communis" type="context">cõmunis</reg> ſit quindecim, cuius duæ ter <lb/>tiæ ſunt <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 ſit per quatuor quintas, ſcilicet duodecim, & productum diuidendum per <lb/>duas tertias, hoc eſt 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">numeratorũ</reg> ad vnam & eandem denominationem attinet, <lb/>ea de cauſa fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-<lb/>minis indiget, quo quartus à prędictis dependens, inueniri poſſit, quandoquidem <lb/>bini illi reſpectus, tribus terminis comprehendi <reg norm="poſsunt" type="context">poſsũt</reg>. </s> + <s xml:id="s177" xml:space="preserve">At quod ad multiplicatio-<lb/>nem ſpectat denominantis <reg norm="communis" type="context">cõmunis</reg> <reg norm="cum" type="context">cũ</reg> numerante denominantis in cogniti & diui-<lb/>ſionem producti per numerantem <reg norm="cognitum" type="context">cognitũ</reg> illę nihil aliud ſunt, quam <reg norm="quartum" type="context">quartũ</reg> <reg norm="terminum" type="context">terminũ</reg> <lb/>inuenire, ita proportionatum tertio, vt ſecundus primo.</s> + </p> + <p> + <s xml:id="s178" xml:space="preserve">Excmpli gratia, ſit <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> nume-<lb/>rantem denominantis cogniti, qui ſigni <lb/> + <anchor type="figure" xlink:label="fig-0023-03a" xlink:href="fig-0023-03"/> + ficetur linea <var>.o.</var> et <var>.e.</var> ſit denominantis in-<lb/>cogniti numerans, denotati linea <var>.u.</var> imò <lb/>verò & cogniti <var>.o.</var> nempe quatuor <lb/>quintæ, Iam ſi <var>.o.</var> cum <var>.e.</var> multiplicemus, & productum per <var>.a.</var> diuidemus dabitur <var>.u.</var> <lb/>ſic ſe habens ad <var>.e.</var> ſicut <var>.o.</var> ad <var>.a.</var> ex .20. ſeptimi.</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, ſint quatuor quintę partes, mul<lb/>tiplicandę eſſent duæ tertiæ per denominantem communem, & productum diui-<lb/>dendum per quatuor quintas ipſius de-<lb/>nominantis. </s> + <s xml:id="s180" xml:space="preserve">Ac ſi quis diceret ſi <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/>propoſito exemplo, terminus <var>.a.</var> loco <var>.e.</var> <lb/>duos ſortietur denominantes, cognitum <lb/>videlicet <var>.o.</var> et <var>.u.</var> incognitum quod po-<lb/>ſtea cognitum oritur ex regula de tribus, vt dictum eſt.</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 cognoſci poterit proportionem quantitatis cenſicæ cenſicæ ad <lb/>ſimilem quantitatem quadruplam eſſe ad eam, quæ eſt ſuarum 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="ſciendus" type="context">ſciẽdus</reg> eſt modus <reg norm="productionis" type="simple">ꝓductionis</reg> <reg norm="harum" type="context">harũ</reg> <reg norm="dignitatum" type="context">dignitatũ</reg> qui oritur ex produ-<lb/>ctione primæ radicis in ſeipſam, prout qui <reg norm="cubum" type="context">cubũ</reg> requirit, ducat radicé in ſuo quadra-<lb/>to, & orietur cubus, hæc poſtea ducta in cubum, <reg norm="quantitatem" type="context">quantitatẽ</reg> cenſicam <reg norm="cenſicam" type="context">cenſicã</reg>, et in <lb/>hanc, prædictam radicem, dabit quantitatem primam relatam. </s> + <s xml:id="s185" xml:space="preserve">Quod vbi ſciueri-<lb/>mus, meminiſſe oportet Euclidem decimaoctaua ſexti aut .11. octaui docere, pro-<lb/>portionem quadrati ad <reg norm="quadratum" type="context">quadratũ</reg>, duplam eſſe proportioni ſuarum radicum, & .36. <lb/>vndecimi aut .11. octaui, cubi ad <reg norm="cubum" type="context">cubũ</reg> triplam eſſe, ego verò nunc aſſero, cenſici cen <lb/>ſici 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 ſpeculationis gratia, detur linea <var>.d.</var> quæ cubum maiorem ſignificet. et <var>.b.</var> <lb/>minorem <var>.c.</var> verò ſit radixipſius <var>.d.</var> et <var>.e.</var> ipſius <var>.b.</var> ita ordinate adinuicem, vt in ſub-<lb/>ſcripta figura cernitur. </s> + <s xml:id="s187" xml:space="preserve">Iam <var>.c.</var> cum <var>.d.</var> producatur <reg norm="proueniatque" type="simple">proueniatq́;</reg> <var>.q.</var> cenſicum cenſi-<lb/>cum, tum producatur <var>.e.</var> cum <var>.b.</var> et dabitur <var>.p.</var> alterum cenſicum cenſicum. </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/>cauſa 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. ſexti, aut quinta octaui <lb/>depręhenditur. </s> + <s xml:id="s189" xml:space="preserve">Quare <reg norm="cum" type="context">cũ</reg> proportio <var>.d.</var> ad <var>.<lb/>b.</var> proportioni <var>.c.</var> ad <var>.e.</var> tripla ſit, 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, ſumptis ſemper <var>.d</var> et <var>.b.</var> pro duo-<lb/>bus cenſibus cenſuum, 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 ſit diuiſionis vnius radicis per alteram?</s> + </p> + <p> + <s xml:id="s193" xml:space="preserve">Sint exempli gratia duę lineæ <var>.b.q.</var> et <var>.f.g.</var> quæ ſignificent duas radices cuiuſuis <lb/>dignitatis; </s> + <s xml:id="s194" xml:space="preserve"><reg norm="demusque" type="simple">demusq́;</reg> eſſe radices duorum quadratorum, <reg norm="quadratumque" type="simple">quadratumq́;</reg> ipſius <var>b.q.</var> <lb/>per quadratum ipſius <var>.f.g.</var> diuidatur; </s> + <s xml:id="s195" xml:space="preserve">quadrataq́ue radix prouenientis ſit <var>.d.q.</var> <lb/>vnitas verò linearis ſit <var>.i.g</var>. </s> + <s xml:id="s196" xml:space="preserve">Dico ipſam <var>.d.q.</var> eſſe proueniens ex diuiſione <var>.b.q.</var> <lb/>per <var>.f.g</var>. </s> + <s xml:id="s197" xml:space="preserve">Patet enim ex definitione diuiſionis 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 ipſius <var>.d.q.</var> talem eſſe partem quadrati ipſius <var>.b.q.</var> qualis quadratum ipſius <var>.g.i.</var> <lb/>eſt quadrati ipſius <var>.f.g</var>. </s> + <s xml:id="s198" xml:space="preserve">Scimus pręterea ex .19. ſexti, aut vndecima octaui, propor-<lb/>tioné quadrati ipſius <var>.b.q.</var> ad <reg norm="quadratum" type="context">quadratũ</reg> ipſius <var>.d.q.</var> duplam eſſe proportioni <var>.b.q.</var> ad <var>.<lb/>d.q.</var> ſuarum radicum (cuborum enim tripla eſſet & cenſuum cenſuum, quadrupla, <lb/><reg norm="atque" type="simple">atq;</reg> ita deinceps ex præcedenti theoremate) Id ipſum dico de dignitatibus ipſius <var>.<lb/>f.g.</var> et <var>.i.g.</var> reſpectu radicum <var>.f.g.</var> et <var>.i.g</var>. </s> + <s xml:id="s199" xml:space="preserve">Vnde <lb/>cum proportio dignitatis ipſius <var>.b.q.</var> ad il-<lb/>lam <var>.d.q.</var> ęqualis ſit proportioni dignitatis <lb/> + <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/> + ipſius <var>.f.g.</var> ad illam <var>.g.i.</var> ex communi ſcien-<lb/>tia apertè cognoſcemus ſimplices propor-<lb/>tiones eſſe interſe æquales, nempe eam quę <lb/>eſt <var>.b.q.</var> ad <var>.d.q.</var> æqualem eſſe ei, quæ eſt <var>.f.<lb/>g.</var> ad <var>.i.g.</var> <reg norm="itaque" type="simple">itaq;</reg> ſequitur ex definitione diuiſionis <var>.d.q.</var> eſſe proueniens ex diuiſione <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, eſt quadrata radix, producti <lb/>ſuorum quadratorum ſimul?</s> + </p> + <p> + <s xml:id="s201" xml:space="preserve">In cuius rei gratiam, ſint duo quadrata <var>.d.a.</var> et <var>n.o.</var> coniuncta ſimul, prout in ſub-<lb/>ſcripta figura apparet, ita tamen vtangulus <var>.a.n.u.</var> ſitre <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">coniũgentur</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 ipſius <var>.a.n.</var> in <var>.n.u.</var> duarum videlicet radicum <lb/>quadratarum ſimul, dabitur ex prima ſexti, aut de-<lb/>cimaottaua ſeptimi, productum <var>.a.u.</var> medium propor <lb/>tionale inter quadratum <var>.a.d.</var> et <var>.u.c.</var> quod ſi cogi-<lb/>temus has tres ſuperficies, tres numeros eſſe, pate-<lb/>bit ex vigeſimaprima ſeptimi productum <var>.a.u.</var> in ſe-<lb/>ipſum, quadratum ſcilicet <var>.a.u.</var> æquale eſſe producto <var>.<lb/>a.d.</var> in <var>.u.c.</var> ex quo propoſiti euidentia conſequetur.</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 ipſum in cubis cognoſci poterit. <lb/></s> + <s xml:id="s205" xml:space="preserve">Sit cubus <var>.l.b.</var> & cubus <var>.o.p.</var> quorum productum ſit <var>.u.g.</var> quod aſſero eſle <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 demonſtra-<lb/>bo eſſe numeri æqualis numero <var>.m.q.</var> <lb/>qui <var>.m.q.</var> productum eſt ipſius <var>.m.e.</var> in <var>.e.<lb/>q.</var> radicum propoſitorum 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">productũ</reg> ſit 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, tantę 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">vocenturq́;</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 ſeptimi, eadem erit proportio <var>.a.g.</var> ad <var>.c.g.</var> <lb/>quæ eſt <var>.l.b.</var> ad <var>.l.x.</var> corporeum, ſed ex .25. vndecimi & prima ſexti, ita ſe habet <var>.a.K.</var> <lb/>ad <var>.K.c.</var> vnitatem linearé ſicut <var>.a.g.</var> ad <var>.c.g.</var> & ex <reg norm="eiſdem" type="context">eiſdẽ</reg> ita ſe habebit <var>.b.e.</var> ad <var>.e.x.</var> vnita-<lb/>tem linearem, ſicut <var>.l.b.</var> ad quadratum <var>.l.x.</var> corporeum. </s> + <s xml:id="s209" xml:space="preserve">Itaque ſic ſe habebit <var>.b.e.</var> ad <lb/>vnitatem linearem <var>.e.x.</var> videlicet <var>.K.c.</var> ſicut <var>.a.K.</var> ad ipſam <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> & conſequenter æqualis <var>.m.e</var>. </s> + <s xml:id="s211" xml:space="preserve">Iam verò ſit <var>.u.g.</var> productum <var>.l.b.</var> <lb/>cubi, in cubum <var>.o.p.</var> vt ſupra dictum eſt, Hinc patebit ex quauis duarum propoſitio-<lb/>num, decimaoctaua, aut decimanona ſeptimi, eandem futuram proportionem <var>.u.g.</var> <lb/>ad <var>.a.g.</var> quæ eſt <var>.o.p.</var> ad <var>.x.p.</var> quadratum corporeum. </s> + <s xml:id="s212" xml:space="preserve">Quare ex poſtremis, dictis ratio-<lb/>nibus, eadem erit proportio <var>.u.K.</var> ad <var>.a.K.</var> quæ eſt <var>.o.e.</var> ad vnitatem linearem <var>.e.x.</var> at <lb/>ex dictis decimaoctaua & decimanona ſeptimi, ita ſe habet <reg norm="numerus" type="simple">numerꝰ</reg> <var>.m.q.</var> ad <reg norm="numerum" type="context">numerũ</reg> <lb/><reg norm="ſuperficialem" type="context">ſuperficialẽ</reg> <var>.m.e.</var> qui <reg norm="producitur" type="simple">ꝓducitur</reg> à lineari <var>.m.e.</var> in vnitaté <reg norm="linearem" type="context">linearẽ</reg> ipſius <var>.e.q.</var> ſicut nume <lb/>rus <var>.q.e.</var> ad ſuam vnitaté, ſed <reg norm="cum" type="context">cũ</reg> numerus <var>.a.K.</var> æqualis ſit numero <var>.m.e.</var> vt <reg norm="probatum" type="context">probatũ</reg> eſt <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 eſt 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 ſequitur <var>.u.g.</var> cubum eſſe, & <var>f.g.</var> radicem ipſius, 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 uniuerſum ſciri poſſit totum <reg norm="infinitum" type="context">infinitũ</reg> dignitatum, hoc eſt radicem <lb/>producti duarum dignitatum ſimilium, 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 ſit quadratum <var>.m.</var> radix cuius ſit <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">ꝓducto</reg> <var>.q.p.</var> in <var>.g.k.</var> <reg norm="quod" type="wordlist">qđ</reg> <reg norm="autem" type="context">autẽ</reg> ſit <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 ſuam vnitatem linearem, ex decimaoctaua, aut decima-<lb/>nona ſeptimi, hæc vero vnitas linearis ſit <var>.t.</var> cuius ſuperficialis ſit <var>.u.</var> vnitas ſcilicet to-<lb/>ties in ſeipſam multiplicata quoties propoſita dignitas patitur, tametſi in præſen <lb/>ti exemplo quadrata dignitas ſumatur. </s> + <s xml:id="s218" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex eiſdem propoſitionibus decimaocta <lb/>ua aut decimanona, ſic ſe habet <var>.m.</var> ad <var>.n.</var> ſicut <var>.i.</var> ad <var>.u</var>. </s> + <s xml:id="s219" xml:space="preserve">Scimus pręterea <reg norm="proportionem" type="context">proportionẽ</reg> <var>.<lb/>m.</var> ad <var>.n.</var> (eo quod in propoſito exemplo ſint quadrata) duplam eſſe proportioni <var>.b.<lb/>d.</var> ad <var>.q.p.</var> et ipſius <var>.i.</var> ad <var>.u.</var> pariter duplam proportioni <var>.g.k.</var> ad <var>.t.</var> iam autem dictum <lb/>fuit ſic ſe habere <var>.m.</var> ad <var>.n.</var> ſicut <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> ſic ſe habebit ad <var>.q.p.</var> ſicut <var>.g.k.</var> ad <var>.t.</var> <lb/> + <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> + quandoquidem ſic ſe habeattotum ad <reg norm="to- tum" type="context">to-<lb/>tũ</reg>, ſicut pars ad <reg norm="partem" type="context">partẽ</reg>, <reg norm="dum" type="context">dũ</reg> ſimiles ſint, proba <lb/><reg norm="tum" type="context">tũ</reg> <reg norm="autem" type="context">autẽ</reg> eſt ſuperius ita ſe habere <var>.o.</var> ad <var>.q.p.</var> <lb/>ſicut <var>.g.k.</var> ad <var>.t.</var> <reg norm="itaque" type="simple">itaq;</reg> <var>.o.</var> ſic ſe habebit ad <var>.q.p.</var> <lb/>ſicut <var>.b.d.</var> ad <var>.q.p.</var> vnde <var>.o.</var> æqualis erit <var>.b.d.</var> <lb/>Hocipſum cęteris dignitatibus conueniet, <lb/>mutatis tantummodo proportionibus <var>.m.<lb/>n.</var> ad proportionem <var>.b.d</var>: <var>q.p.</var> ſic propor-<lb/>tionibus duarum dignitatum <var>.i.u.</var> ad pro-<lb/>portionem ſuarum 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 ſi quilibet numerus in duas partes inæquales diuiſus <lb/>fuerit, <reg norm="totumque" type="simple">totumq́</reg> diuiſum per <reg norm="vnam" type="context">vnã</reg> partium, & per eandem pars altera diuiſa fue-<lb/>rit: </s> + <s xml:id="s222" xml:space="preserve">differentia prouenientium ſemper vnitas erit. </s> + <s xml:id="s223" xml:space="preserve">quodquidem veriſſimum eſt.</s> + </p> + <p> + <s xml:id="s224" xml:space="preserve">Detur enim <var>.b.d.</var> propoſitus numerus in duas partes inæquales diuiſus <var>.b.c.</var> et <var>.c.d.</var> <lb/>& in primis <reg norm="totum" type="context">totũ</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">.ꝑ</reg> <var>.i.o.</var> ſigni-<lb/>ficetur, tum pars ipſa <var>.b.c.</var> <reg norm="per" type="simple punctuation">ꝑ.</reg> <reg norm="eandem" type="context">eãdem</reg> <var>.c.d.</var> diuidatur, <reg norm="ſitque" type="simple">ſitq́;</reg> <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a</var>. </s> + <s xml:id="s225" xml:space="preserve">Sanè ex defini-<lb/>tione diuiſionis, eadem erit proportio <var>.b.d.</var> ad <var>.e.o.</var> quæ eſt <var>.c.d.</var> ad <var>.i.o.</var> et ita <var>.b.c.</var> ad <var>.a.</var> <lb/>ſicut <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 ſe habet <var>.b.c.</var> ad <var>.e.i.</var> ſicut <var>.b.d.</var> ad <var>.e.o.</var> at <var>.b.d.</var> <lb/>ad <var>.e.o.</var> ſic ſe habet ſicut <var>.c.d.</var> ad <var>.i.o.</var> hoc eſt ſicut <var>.b.c.</var> ad <var>.a</var>. </s> + <s xml:id="s227" xml:space="preserve">Quare ex .11. quinti ſic ſe <lb/>habebit <var>.b.c.</var> ad <var>.e.i.</var> ſicut .ad <var>.a.</var> ex quo ex .9. <reg norm="praedi cti" type="simple">prędi<lb/>cti</reg> <var>.a.</var> æqualis erit <var>.e.i.</var> ſed <var>.e.i.</var> minor eſt <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 ſequitur propoſitum verum eſ<lb/>ſe. </s> + <s xml:id="s229" xml:space="preserve">Quod ipſum pauciſſimis verbis ſic definiri <lb/>poteſt, ſi dixerimus, eiuſmodi diuidens .in par-<lb/>te diuiſibili, <reg norm="quam" type="context">quã</reg> in toto, ſemel minus ingredi, <lb/>quandoquidem altera pars eſt, 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">HOcipſum alia ratione contemplari po<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 diuiſus <var>.a.u.</var> et <var>.u.e.</var> totius autem diuidens ſit <var>.u.e.</var> & partis alterius <var>.a.u.</var> <lb/>totius verò <reg norm="proueniens" type="context">proueniẽs</reg> ſit <var>.a.c.</var> partis <reg norm="autem" type="context">autẽ</reg>, ſit <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a.n.</var> tum differentia ſit <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, ſit <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 eſt <var>.u.e.</var> ad <var>.a.i.</var> ex definitione diuiſionis, et eandem <lb/>eſſe <var>.a.u.</var> ad <var>.a.n.</var> quæ eſt <var>.u.e.</var> ad <var>.a.i.</var> vnde ex .<lb/>11. quinti ſic ſe habebit <var>.a.e.</var> ad <var>.a.c.</var> ſicut <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. eiuſdem ſic ſe habe-<lb/>bit <var>.u.e.</var> ad <var>.n.c.</var> ſicut <var>.a.e.</var> ad <var>.a.c.</var> ſed. ſic ſe <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, ſic ſe habebit <var>.u.e.</var> ad <var>.n.c.</var> ſicut ad <var>.a.<lb/>i</var>. </s> + <s xml:id="s234" xml:space="preserve">Quare ex .9. eiuſdem <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 diuiſis adinuicem, <reg norm="multiplicatisque" type="simple">multiplicatisq́</reg> prouenien <lb/>tibus ſimul, productum, ſemper eſt vnitas ſuperficialis? </s> + <s xml:id="s236" xml:space="preserve">Nempe ex .20. ſeptimi, <lb/>quoniam vnitas linearis ſemper media proportionalis eſt inter bina prouenientia. <lb/></s> + <s xml:id="s237" xml:space="preserve">Quodita ſpecularilicet.</s> + </p> + <p> + <s xml:id="s238" xml:space="preserve"><reg norm="Significentur" type="context">Significẽtur</reg> duo propoſiti numeri per <var>.b.p.</var> et <var>.b.d.</var> mutuo diuiſi, proueniens au-<lb/>tem <var>.b.p.</var> per <var>.b.d.</var> diuiſum ſit <var>.b.n.</var> tum proueniens <var>.b.d.</var> diuiſum per <var>.b.p.</var> ſit <var>.b.a.</var> <lb/>et <var>.b.t.</var> ſit 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 diuiſionis, dabitur eadem proportio <var>.b.p.</var> ad <var>.b.n.</var> quæ eſt <var>.b.d.</var> <lb/>ad <var>.b.e.</var> et proportio <var>.b.d.</var> ad <var>.b.a.</var> quæ eſt <var>.b.p.</var> ad <var>.b.t</var>. </s> + <s xml:id="s240" xml:space="preserve">Sed cum ſic ſe habeat <var>.b.<lb/>p.</var> ad <var>.b.n.</var> ſicut <var>.b.d.</var> ad <var>.b.e.</var> permutando ſic ſe habebit <var>.b.p.</var> ad <var>.b.d.</var> ſicut <var>.b.n.</var> ad <var>.b.<lb/>e.</var> hoc eſt ad <var>.b.t.</var> et cum ſic ſe habeat <var>.b.d.</var> ad <var>.b.a.</var> ſicut <var>.b.p.</var> ad <var>.b.t</var>: permutando ſic ſe <lb/>habebit <var>.b.d.</var> ad <var>.b.p.</var> ſicut <var>.b.a.</var> ad <var>.b.t</var>. <lb/></s> + <s xml:id="s241" xml:space="preserve">Quare euerſim ſic ſe 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> ſicut <var>.b.t.</var> ad <var>.b.a.</var> ſed <var>.b.n.</var> ad <var>.b.t.</var> ſic <lb/>ſe 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/>quintiſic ſe habebit <var>.b.n.</var> ad <var>.b.t.</var> ſicut <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 eſt <var>.b.e.</var> et <var>.b.t.</var> idem omnino eſſe. </s> + <s xml:id="s244" xml:space="preserve">Quare ex .20. ſeptimi pro-<lb/>poſiti veritas innoteſcet.</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">IDipſum & 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> ſignificentur mutuo diuiſi, proueniens <reg norm="autem" type="context">autẽ</reg> <var>.o.</var> per <var>.<lb/>u.</var> ſit <var>.e.</var> et proueniens <var>.u.</var> per <var>.o.</var> ſit <var>.x.</var> vnitas uerò per <var>.i.</var> ſignificetur, quas tamen quanti-<lb/>tates ſubſcripto modo ad inuicem diſponi-<lb/>to. </s> + <s xml:id="s247" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex definitione diuiſionis, eadem erit <lb/> + <anchor type="figure" xlink:label="fig-0028-04a" xlink:href="fig-0028-04"/> + proportio <var>.o.</var> ad <var>.e.</var> quę eſt <var>.u.</var> ad <var>.i.</var> et <var>.o.</var> ad <var>.i.</var> quę <lb/>eſt <var>.u.</var> ad <var>.x</var>. </s> + <s xml:id="s248" xml:space="preserve">Quare ex æqualitate <reg norm="proportionum" type="context">proportionũ</reg> <var>.<lb/>c.</var> ad <var>.i.</var> ſic ſe habebit ſicut <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/>ſeptimi propoſitum concludetur. </s> + <s xml:id="s249" xml:space="preserve">Huiuſmodi rei cauſa etiam eſt, quod proueniens <lb/>diuiſionis vnius eſt numerator æqualis denominatori diuiſionis 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 diuiſis, <reg norm="sumptis" type="context">sũptis</reg> deinde prouenientibus ſimul et adinui <lb/>cem, & per hanc ſummam, diuiſa ſumma quadratorum dictorum <reg norm="propoſitorum" type="context">propoſitorũ</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 ſimul.</s> + </p> + <p> + <s xml:id="s251" xml:space="preserve">Sint exempli gratia propoſiti numeri .2. et .8. qui mutuo diuiſi in primis dent pro <lb/>uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-<lb/>cta dabunt ſummam quatuor integrorum et quartæ partis vnius, ſumma autem qua <lb/>dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar <lb/>tam partem vnius diuiſus 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 ſpeculatio, ſint duæ lineæ <var>.o.e.</var> et <var>.o.n.</var> quæ duos numeros pro-<lb/>poſitos ſignificent, inuicem ad angulum rectum <var>.o.</var> coniunctæ, quarum quadrata <lb/>ſint <var>.o.a.</var> et <var>.o.p.</var> ipſorum productum ſit <var>.n.e.</var> tum <var>.o.t.</var> ſit proueniens ex diuiſione <var>.o.e.</var> <lb/>per <var>.o.n</var>. </s> + <s xml:id="s253" xml:space="preserve">Hęc ſingulatim conſideremus (<reg norm="nam" type="context">nã</reg> ſi in partibus ſimplicibus quod dicimus ac <lb/>ciderit, id ipſum in compoſitis conſequenter eueniet) quamobrem ex definitione di <lb/>uiſionis dabitur eadem proportio <var>.o.e.</var> ad <var>.o.t.</var> quæ eft <var>.o.n.</var> ad vnitatem, quæ ſit <var>.o.<lb/>x</var>. </s> + <s xml:id="s254" xml:space="preserve">Nunc cogitemus <reg norm="ſuperficiem" type="context">ſuperficiẽ</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 diuiſione numeri quadrati <var>.o.a.</var> per <reg norm="numerum" type="context">numerũ</reg> <var>.o.t.</var> <reg norm="eritque" type="simple">eritq́</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio <var>.c.t.</var> ad <var>.o.e.</var> quæ eſt <var>.o.e.</var> ad <var>.o.t.</var> ex ſecunda parte quintæ decimæ ſexti, <lb/>aut .20. ſeptimi. </s> + <s xml:id="s256" xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> <reg norm="autem" type="context">autẽ</reg> dictum eſt <var>.o.e.</var> ad <var>.o.t.</var> ſic ſe habere ſicut <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 ſic ſe habebit <var>.c.t.</var> ad <var>.o.e.</var> ſicut <var>.o.n.</var> ad <var>.o.x</var>. </s> + <s xml:id="s258" xml:space="preserve">Sed ex prima ſexti, aut .18. vel .<lb/>19. ſeptimi, ſic ſe habet <reg norm="productum" type="simple">ꝓductum</reg> <var>.n.e.</var> ad <var>.e.x.</var> ſicut <var>.o.n.</var> ad <var>.o.x</var>. </s> + <s xml:id="s259" xml:space="preserve">quare denuo ſic ſe ha-<lb/>bebit numerus <var>.c.t.</var> ad numerum <var>.o.e.</var> ſicut 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> ſpecie idem eſt, 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 ipſum de quadrato ipſius <var>.o.n.</var> videlicet <var>.p.o.</var> <lb/>dico. </s> + <s xml:id="s262" xml:space="preserve">Nam ſi proueniens <var>.o.n.</var> diuiſo per <var>.o.e.</var> ideſt <var>.<lb/>o.i.</var> proportionale reſpondens ad <var>.o.t.</var> cum <var>.o.t.</var> <lb/><reg norm="coniunctum" type="context">coniunctũ</reg> fuerit, et per <reg norm="hanc" type="context">hãc</reg> ſummam diuiſa ſumma <lb/>quadratorum <var>.o.a.</var> et <var>.o.p.</var> patet per ſe proueniens <lb/>futurum eiuſdem numeri <var>.c.t.</var> <reg norm="ipſumque" type="simple">ipſumq́</reg> <var>.c.t.</var> proue-<lb/>niens ſemper ſuturum.</s> + </p> + <p> + <s xml:id="s263" xml:space="preserve">Quo autem lucidius res hæc innoteſcat. </s> + <s xml:id="s264" xml:space="preserve">Cogi <lb/>temus proueniens quadrati <var>.o.p.</var> diuiſi ab <var>.o.i.</var> re-<lb/><reg norm="ſpondentisque" type="simple">ſpondentisq;</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 conſe-<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 compoſito 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/>poſito <var>.t.u.</var> <reg norm="quam" type="context">quã</reg> ex ſimplici <var>.o.c.</var> ex quo propoſiti ſe <lb/>ſe 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, ſed quod tamen citra al-<lb/>gebraticam effectionem, aut neſcierunt, aut noluerunt diſſoluere, quod nihi-<lb/>lominus facillimum eſt.</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 eiuſmodi partes diuidendum, vt ſumma qua-<lb/>dratorum dictarum partium, alteri numero poſsibili propoſito æqualis ſit, poſſi-<lb/>bili inquam, etenim ſi eiuſmodi numerus propoſitus, minor eſſet producto totius <lb/>primi in ſuum dimidium, eſſet huiuſmodi factum impoſſibile. </s> + <s xml:id="s267" xml:space="preserve">Quod nos exequi <lb/>cupientes, ſumamus primum <reg norm="numerum" type="context">numerũ</reg> propoſitum, quem in ſe ipſum multiplice-<lb/>mus. </s> + <s xml:id="s268" xml:space="preserve">ab hoc quadrato deducamus ſecundum numerum propoſitum, tum quod re-<lb/>manſerit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-<lb/>hi, accepta poſtea radice quadrata reſidui & dempta ex priori numero propoſito, <lb/></s> + <s xml:id="s269" xml:space="preserve">tunc dimidium reſidui 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 eiuſmodi partes, vt ſumma <lb/>quadratorum ipſarum partium æqualis ſit .272. qui numerus maior eſt .200. maior <lb/>inquam dimidio quadrati .400. ipſorum .20. hic autem numerus .272. è quadra-<lb/>to .400. deducatur, <reg norm="remanebunt" type="context">remanebũt</reg> enim .128. quod duplicari iubeo, <reg norm="producentur" type="context">producẽtur</reg> <reg norm="ſiquidem" type="context">ſiquidẽ</reg> <num value="256">.<lb/>256.</num> quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem ſumi <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 propoſito .20. detra-<lb/>hetur, <reg norm="remanebitque" type="simple">remanebitq́</reg> .16. pro altera parte.</s> + </p> + <p> + <s xml:id="s271" xml:space="preserve">Cuius demonſtrationis cauſa, in primis cogitemus quadratum <var>.a.c.</var> cognitum nu-<lb/>meri <var>.a.b.</var> primò propoſiti, qui cogitetur diuiſus in duo quadrata <var>.d.e.</var> et <var>.e.b.</var> <reg norm="duo- que" type="simple">duo-<lb/>q́</reg> ſupplementa <var>.a.e.</var> et <var>.e.c.</var> numerus autem ſummæ duorum quadratorum <var>.d.e.<lb/>b.</var> pro ſecundo propoſito datur; </s> + <s xml:id="s272" xml:space="preserve">ex quo, ſumma duorum ſupplementorum <var>.a.e.c.</var> <lb/>conſequenter erit cognita, quę cum duplicata fuerit, & quatuor hæc ſupplementa<unsure/> <lb/>cogitatione accommodata, prout in <lb/>quadrato <var>.f.g.</var> apparet (<reg norm="quanuis" type="context">quãuis</reg> idipſum <lb/> + <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> + proueniret ſi modo Eucl. octaua <reg norm="ſecundi" type="context">ſecũdi</reg> <lb/>aptaretur) æquali quadrato <var>.a.c.</var> ita vt <lb/>cogitatis quatuor ſupplementis numeri <lb/>cogniti in quadrato <var>.f.g.</var> ex conſequen-<lb/>ti cognoſcetur 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 eſt) primo propoſiti, 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 ſupplementi, cum in <lb/>præcedenti theoremate dictum fuerit, <lb/>qua ratione manifeſtetur duplum ſupple-<lb/>menti ipſius.</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 ſubſcripta figura lineam <var>.<lb/>a.b.</var> tanquam primum numerum propoſi-<lb/>tum, & productum <var>.a.e.</var> ſupplemento <var>.a.e.</var> primæ præcedentis figuræ æquale ſit, <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 ſecundi Eucli. dabit quod propoſitum 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 eſt, cur ſubtracto duplo producti duorum numerorum ad inui-<lb/>cem <reg norm="multiplicatorum" type="context">multiplicatorũ</reg> ex ſumma ſuorum quadratorum, ſemper quod ſuper <lb/>eſt duorum numerorum quadratum differentiæ ſit?</s> + </p> + <p> + <s xml:id="s277" xml:space="preserve">Exempli gratia ſi proponerentur duo numeri .16. et .4. duplum producti eorum <lb/>eſſet .128. quò detracto ex ſumma ſuorum 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 vtſciamus, duo numeri propoſiti, duabus lineis ſignificentur, maiore <var>.q.g.</var> <lb/>et minore <var>.g.p.</var> directè coniunctis, ſuper 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">dabũtur</reg> <var>.q.n.</var> et <var>.n.u.</var> ſingula 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 propoſi-<lb/>torum, quod ſatis <reg norm="ſuperque" type="simple">ſuperq́</reg> , probatur quarta ſecundi 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 ſecundi, 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 ipſo 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> ſed hæ duæ quantitates, ſunt par-<lb/>tes duorum quadratorum dictorum, & quæ ſuper <lb/>eſt <var>.m.e.</var> quadratum differentiæ vnius numeri pro-<lb/>poſiti ab altero, prout in ſubſcripta figura licebit cui <lb/>libet conſiderare. </s> + <s xml:id="s281" xml:space="preserve">Itaque veritas hæc manifeſta <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 propoſitis maiorem per minorem diuidunt, ſi <lb/>proueniens per maiorem numerum multiplicauerint, productum æquale <lb/>erit prouenienti ex diuiſione quadrati maioris numeri per minorem?</s> + </p> + <p> + <s xml:id="s283" xml:space="preserve">Exempli gratia ſi proponantur duo numeri .20. et .4. <reg norm="ipſeque" type="simple">ipſeq́</reg> .20. per .4. diui-<lb/>datur, dabit quinque, tum .400. quadrato .20. diuiſo 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 ſpeculationis cauſa, ſint duo numeri, qui lineis <var>.x.u.</var> et <var>.x.s.</var> maiore <reg norm="atque" type="simple">atq;</reg> mi-<lb/>nore ſignificétur, tum <var>.u.x.</var> numerus per <var>.s.x.</var> di-<lb/>uidatur, ſitq́ue proueniens <var>.x.n.</var> poſtmodum qua-<lb/> + <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/> + dratum <var>.u.x.</var> ſit <var>.x.o.</var> et productum ex <var>.n.x.</var> in <var>.u.<lb/>x.</var> ſit <var>.x.e.</var> quod æquale eſſe dico prouenienti ex <lb/>diuiſione quadrati <var>.o.x.</var> per <var>.s.x.</var> quod ſit <var>.m</var>. </s> + <s xml:id="s285" xml:space="preserve">Patet <lb/>enim ex definitione diuiſionis, talem futuram pro-<lb/>portionem <var>.u.x.</var> ad <var>.n.x.</var> qualis eſt <var>.s.x.</var> ad vnitatem, <lb/>& quadratum <var>.o.x.</var> ad rectangulum <var>.e.x.</var> ita ſe 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, ſicut <var>.u.x.</var> ad <var>.n.x.</var> ex prima ſexti aut .18. vel .19. ſeptimi, </s> + <s xml:id="s286" xml:space="preserve">quare ex 11. <lb/>quinti ita ſe habebit <var>.o.x.</var> ad <var>.e.x.</var> ſicut <var>.s.x.</var> ad vnitatem; </s> + <s xml:id="s287" xml:space="preserve">ſed ſicut ſe habet <var>.s.x.</var> ad. <lb/>vnitatem, ita ſe habet pariter <var>.o.x.</var> ad <var>.m</var>. </s> + <s xml:id="s288" xml:space="preserve">vnde ex .11. prædicta ita ſe habebit <var>.o.<lb/>x.</var> ad <var>.m.</var> ſicut idipſum <var>.o.x.</var> ad <var>.e.x.</var> itaq́ue 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 propoſito aliquo numero in duas partes inæquales diuiſo, ſi rurſus per <lb/>quamlibet ipſarum diuidatur, prouenientia tantumdem coniuncta quantum <lb/>multiplicata efficiant.</s> + </p> + <p> + <s xml:id="s290" xml:space="preserve">Exempli gratia, ſit denarius prop oſitus numerus, per binarium & octonarium <lb/>diuiſus, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta <lb/>crunt .6. cum quarta parte lineari, quæ ſi mul multiplicata, pariter erunt .6. cum <lb/>quarta parte ſuperficiali.</s> + </p> + <p> + <s xml:id="s291" xml:space="preserve">Cuius ſpeculationis cauſa, totalis numerns, linea <var>.q.p.</var> ſignificetur, eius duæ <lb/>partes, per <var>.k.</var> maiorem et <var>.u.</var> minorem, ipſa vnitas per .t: proueniens ex diuiſio-<lb/>ne <var>.q.p.</var> per <var>.k.</var> ſit <var>.q.i.</var> proueniens autem ipſius <var>.q.p.</var> per <var>.u.</var> ſit <var>.q.f.</var> </s> + <s xml:id="s292" xml:space="preserve">quare ex defini-<lb/>tione diuiſionis ita ſe habebit <var>.q.p.</var> ad <var>.q.i.</var> ſicut <var>.k.</var> ad <var>.t.</var> et <var>.q.p.</var> ad <var>.q.f.</var> ſicut <var>.u.</var> ad <var>.t.</var> <lb/>hoc eſt <var>.q.f.</var> ad <var>.q.p.</var> ſicut <var>.t.</var> ad <var>.u.</var> vnde ex æqualitate <reg norm="proportionum" type="context">proportionũ</reg> ſic ſe habebit <var>.q.f.</var> <lb/>ad <var>.q.i.</var> ſicut <var>.k.</var> ad <var>.u.</var> et conuerſim. </s> + <s xml:id="s293" xml:space="preserve">Ad hæc in linea <var>.q.p.</var> vnitas, per lineam <var>.q.o.</var> ſigni-<lb/>ficetur, quo facto, dicamus, ſi <var>.q.p.</var> ad <var>.q.i.</var> ſic ſe habet vt <var>.k.</var> ad <var>.q.o.</var> itaque permu-<lb/>tando, ſic ſe habebit <var>.q.p.</var> ad <var>.k.</var> ſicut <var>.q.i.</var> ad <var>.q.o.</var> hoc eſt <var>.k.u.</var> ad <var>.k.</var> ſicut <var>.i.q.f.</var> ad <var>.<lb/>q.f.</var> (nam <var>.k.u.</var> partes ſunt integrales totius <var>.q.p.</var> et <var>.k.u.</var> ad <var>.k.</var> eſt ſicut <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> ſicut <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, ſit <var>.x.f.</var> quod probabo <lb/>æquale eſſe ſummæ <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 ſit <var>.q.o.</var> ſigneturq́ue 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 eſt <var>.q.x.</var> <lb/>ad <var>.q.s.</var> ex prima ſexti, aut .18. vel 19. <lb/>ſeptimi, hoc eſt, ſicut <var>.q.i.</var> ad <var>.q.o.</var> et <lb/>ex .11. quinti (vt dictum eſt) ſicut <var>.i.q.<lb/>f.</var> ad <var>.q.f.</var> ſed numerus <var>.s.f.</var> fuperficia-<lb/>lis tantus eſt, quantus linearis <var>.q.f</var>. <lb/></s> + <s xml:id="s297" xml:space="preserve">quare ex .9. quinti tantus erit (ſu-<lb/>perficialiter) numerus <var>.x.f.</var> quantus <lb/>(lineariter). <var>f.q.i.</var> quod erat pro-<lb/>poſitum.</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 diuiſo, ſi rurſus diuidatur per <lb/>ſingulas partes, ſumma duorum prouenientium per binarium, ſemper ma-<lb/>ior ſit ſumma prouenientium ex diuiſione vnius partis per alteram.</s> + </p> + <p> + <s xml:id="s299" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, ſi 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. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-<lb/>niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-<lb/>tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-<lb/>niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma 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 conſiderationis cauſa, propoſitus numerus linea <var>.q.p.</var> ſignificetur, eius duę <lb/>partes lineis <var>.q.x.</var> et <var>.x.p.</var> <reg norm="tum" type="context">tũ</reg> <var>.q.f.</var> ſit proueniens ex diuiſione totius <var>.q.p.</var> per <var>.x.p.</var> et <var>.<lb/>q.i.</var> ſit proueniens ex diuiſione eiuſdem <var>.q.p.</var> per <var>.q.x.</var> adhæc <var>.h.m.</var> ſit proueniens, <lb/>ex diuiſione <var>.q.x.</var> per <var>x.p.</var> et <var>.h.k.</var> proue-<lb/>niensex diuiſione <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 propoſitum.</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 eſt <lb/>proportionalis inter numerum <lb/> + <anchor type="figure" xlink:label="fig-0033-02a" xlink:href="fig-0033-02"/> + ſui 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 propoſitus, <lb/>qui linea <var>.a.u.</var> ſignificetur, cuiusqua-<lb/>dratum ſit <var>.u.n.</var> vnitas linearis ſit <var>.i.a.</var> <lb/>et ſuperficialis <var>.o.</var> patebit ex .18. ſexti <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> ſed <var>.i.a.</var> e<unsure/>t.o. eadem (ſpecie) <lb/>res <reg norm="sunt" type="context">sũt</reg>, tanta ſcilicet <var>.a.i.</var> quanta <var>.o.</var> vni <lb/> + <anchor type="figure" xlink:label="fig-0033-03a" xlink:href="fig-0033-03"/> + tas eſt, 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> ipſum quod diximus & alia ratione ſpeculari licebit.</s> + </p> + <p> + <s xml:id="s306" xml:space="preserve">Propoſitus numerus, nunc etiam per <var>.a.u.</var> ſignificetur, eius quadratum per <var>.<lb/>u.n.</var> vnitas linearis per <var>.a.i.</var> <reg norm="productumque" type="simple">productumq́;</reg> <var>.a.u.</var> in <var>.a.i.</var> terminetur, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.n.i</var>. </s> + <s xml:id="s307" xml:space="preserve">quare <lb/><var>n.i.</var> conſtabit numero íuperficiali æquali numero lineari <var>.a.u.</var> & ex prima fexti aut .<lb/>18. vel .19. ſeptimi, eadem erit proportio <var>.u.n.</var> ad <var>.i.n.</var> quæ eſt <var>.a.u.</var> ad <var>.a.i.</var> ſed nu-<lb/>merus <var>.a.u.</var> cum numero <var>.n.i.</var> idem ſpecie eſt. </s> + <s xml:id="s308" xml:space="preserve">Itaque medius eſt 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, & diuiſus, medius eſt propor-<lb/>tionalis inter productum multiplicationis, & proueniens diaiſionis.</s> + </p> + <p> + <s xml:id="s310" xml:space="preserve">Exempli gratia, ſi .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 eſt propor-<lb/>tionalis.</s> + </p> + <p> + <s xml:id="s311" xml:space="preserve">Hoc vt ſpeculemur, proponatur numerus multiplicandus & diuidendus, qui ſi-<lb/>gnificetur linea <var>.u.e.</var> multiplicans autem & diuidens linea <var>.a.u.</var> multiplicationis <lb/>productum ſit <var>.e.a.</var> proueniens ex diuiſione ſit <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/>merũ</reg> <var>.a.u.</var> diuidentem multiplicetur, cuius multiplicationis productum ſit <var>.e.i.</var> <lb/>quare, eadem erit proportio numeri <var>.a.e.</var> <lb/>ad numerum <var>.e.i.</var> quæ eſt 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 ſextiaut .18. vel <lb/>19. ſeptimi. </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 ſit. </s> + <s xml:id="s314" xml:space="preserve">verum eſſe, quod propoſi-<lb/>tum fuit conſequetur.</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 propoſitum numerum ita multiplicare & diuidere cupiunt, vt pro <lb/>ductum multiplicationis, tam ſit multiplex prouenienti ex diuiſione, quam <lb/>quæritur, rectè ſumant aliquem numerum pro multiplicante & diuidente, qui ſit ra <lb/>dix quadrata denominantis quęſitę 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 ſit prouenienti ex diuiſione, nempè, vt pro-<lb/>ueniens, nona pars ſit eiuſmodi producti, </s> + <s xml:id="s317" xml:space="preserve">quare quadratam radicem ipſorum no-<lb/>uem, ideſt denominantis ſumunt, tria ſcilicet, 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/>propoſitum ſequitur.</s> + </p> + <p> + <s xml:id="s319" xml:space="preserve">Cuius ſpeculationis cauſa, ſignificetur numerus propoſitus linea <var>.u.e.</var> multipli-<lb/>cans autem & diuidens linea <var>.u.a.</var> productum ſit <var>.e.a.</var> proueniens <var>.e.o.</var> quadratum <lb/>verò <var>.a.u.</var> ſit <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">terminenturq́;</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 eſt <var>.a.e.</var> ad <var>.u.e.</var> numerus enim <var>.e.i.</var> (quamuis ſuperficialis) idem eſt cum nume-<lb/>ro lineari <var>.u.e.</var> ſed <var>.a.e.</var> ad <var>.e.i.</var> ſic ſe habet ſicut <var>.a.u.</var> ad <var>.u.i.</var> ex prima ſexti aut .18. <lb/>vel .19. ſeptimi, (quod ipſum 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/>eſt <var>.x.u.</var> ęqualis erit <reg norm="proportioni" type="simple">ꝓportioni</reg> <var>.a.e.</var> ad <var>.u.e.</var> at trigeſimotertio & trigeſimoquarto theo <lb/>remate probatum eſt proportionem numeri <var>.a.x.</var> ad vnitatem, duplam eſſe propor-<lb/>tioni eiuſdem numeri <var>.a.x.</var> ad <var>.u.x.</var> ſequitur <lb/>igitur cum dimidia ſint æqualia, tota etiam <lb/>æqualia eſſe: </s> + <s xml:id="s323" xml:space="preserve">hoc eſt 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/>ſumitur numerus <var>.a.u.</var> eiuſmodi vt <reg norm="quadratum" type="context">quadratũ</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"/> + ipſius <var>.a.x.</var> tam ſit 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 ſummam colle-<lb/>cta, æqualia ſint numero propoſito, & ijſdem numeris multiplicatis ad-<lb/>inuicem, productum alteri numero propoſito ſit æquale, rectè ſumant dimidium <lb/>primi numeri propoſiti, cui ſumma quadratorum æquari debet, <reg norm="hocque" type="simple">hocq́;</reg> dimidium <lb/>in ſeipſum multiplicent, vnà etiam alterum numerum propoſitum in ſeipſum <lb/>multiplicent, quod quadratum detrahunt de primo, & reſidui quadratam radicem, <lb/>dimidio primi numeri propoſiti coniungunt, ex qua ſumma, quadratam radicem <lb/><reg norm="eruunt" type="context">eruũt</reg>, quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero <lb/>detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum <reg norm="quae- ſitorum" type="simple">quę-<lb/>ſitorum</reg>.</s> + </p> + <p> + <s xml:id="s326" xml:space="preserve">Exempli gratia, ſi proponerentur .34. pro primo numero cui æquari de-<lb/>beret ſumma duorum quadratorum, quorum radicum productum æquale eſſe de-<lb/>beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi <lb/>numeri in ſeipſum multiplicari, cuius dimidij quadratum erit .289. è quo ſi detra-<lb/>has quadratum ſecundi numeri, nempe .225. remanebit .64. <reg norm="atque" type="simple">atq;</reg> huius ſi quadra-<lb/>tam radicem ſumas 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. ſcilicet, quorum <lb/>radices .5. et .3. eſſent ij numeri, qui quæruntur.</s> + </p> + <p> + <s xml:id="s327" xml:space="preserve">Cuius ſpeculationis gratia, cogitemus primum numerum, cui quadratorum fum <lb/>ma æquari debet, ſignificari linea <var>.a.n.</var> tum concipiamus quæſita quadrata ſignifi-<lb/>cari, <reg norm="coniungique" type="simple">coniungiq́</reg> modo ſubſcripto <var>.t.b.k.</var> ſecundum porrò numerum propoſitum, <lb/>ſignificari producto <var>.d.b</var>. </s> + <s xml:id="s328" xml:space="preserve">Iam nil ſupereſt 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> ſummæ quadratorum numerus detur, quadratum di-<lb/>midij <var>.o.a.</var> ſit <var>.s.a.</var> quod nobis erit cognitum; </s> + <s xml:id="s330" xml:space="preserve">ſit 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> ſit <reg norm="ſe- cundus" type="context">ſe-<lb/>cũdus</reg> numerus <lb/>propoſitus, quæ <lb/>minor erit <var>.a.s.</var> ex quinta ſecundi, aut ſeptima conſequentia poſt .16. noni Eucli-<lb/>dis. </s> + <s xml:id="s333" xml:space="preserve">Iam ſubtracta quantitate <var>.z.a.</var> è quadrato <var>.a.s.</var> cognoſcetur quadratum <var>.t.x.</var> <lb/>cuius radix æqualis erit <var>.o.u.</var> ex poſtremo adductis, Itaque cognoſcemus <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 cognoſcetur, & eorum radices conſequenter.</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 ipſum & alia ratione perfici poteſt, nempe, iuncta ſumma <var>.k.b</var>: <var>b.d</var>: ec<unsure/> <var>.<lb/>b.t.</var> alteri rectangulo æquali <var>.b.d.</var> quod ſit <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 conſequenter eius radicem <var>.d.k.</var> cognoſcemus, cuius <lb/>ope ac producti <var>.d.b.</var> cognoſcemus <var>.d.p.</var> et <var>.p.k.</var> prout ex theoremate quadrageſi-<lb/>moquinto huius libri patebit.</s> + </p> + <p> + <s xml:id="s335" xml:space="preserve">Michael Stifelius, vndecimo cap. tertij libri, problema eiuſmodi proponit, <lb/>quod tamen ipſe via algebræ diſsoluit.</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 propoſito æquetur, & quadratorum eorundem differentia alteri nu-<lb/>mero propoſito æqualis ſir. </s> + <s xml:id="s337" xml:space="preserve">Rectè dimidium ſecundi numeri propoſiti in ſeipſum <lb/>multiplicent, cui quidem numero differentia quadratorum æquari debet; </s> + <s xml:id="s338" xml:space="preserve">porrò <lb/>huic quadrato primi propoſiti numeri, cui æquandum eſt productum numerorum <lb/>quæſitorum, quadratum adiungant; </s> + <s xml:id="s339" xml:space="preserve">tum radicem quadratam huius ſummæ co-<lb/>pulet dimidio ſecundi numeri propoſiti, ei inquam, cui differentia quadratorum <lb/>æqualis eſſe debet, ex quo quadratum maius conſurgit, à quo, detracto ſecundo <lb/>numero, ſupereſt quadratum minus.</s> + </p> + <p> + <s xml:id="s340" xml:space="preserve">Exempli gratia, ſi proponeretur primo loco numerus .8. cui æquandum eſt <lb/>productum numerorum quæſitorum, tum proponeretur numerus .12. cui, detra-<lb/>cto minore à maiore, differentia quadratorum vtriuſque quæſiti numeri æqualis <lb/>eſſe debet, oportet huius vltimi numeri .12. dimidium in ſeipſum multiplicare, <reg norm="fient- q́ue" type="context">fiẽt-<lb/>q́ue</reg> .36. quadratum dimidij, vnde in ſummam colligeremus quadratum primi <lb/>numeri .8. quod eſſet .64. quæ cum .36. efficerent .100. cuius centenarij radice, nem <lb/>pe .10. collecta in ſummam cum dimidio ſecundi numeri, nempe .6. daretur qua-<lb/>dratum maius, nempe .16. ex quo, detracto ſecundo numero, nempe .12. rema-<lb/>neret quadratum minus .4.</s> + </p> + <p> + <s xml:id="s341" xml:space="preserve">Cuius ſpeculationis cauſa, maius quadratum <lb/> + <anchor type="figure" xlink:label="fig-0036-02a" xlink:href="fig-0036-02"/> + incognitum ſignificetur 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> ſua dimidia; </s> + <s xml:id="s343" xml:space="preserve">tunc cogite-<lb/>tur quadratum <var>.y.g.</var> ſuper <var>.b.g.</var> et <reg norm="parallelogram- mum" type="context">parallelogrã-<lb/>mum</reg> rectangu<unsure/>lum <var>.g.r.</var> deſignatum, & ita etiam <lb/>gnomon <var>.u.g.t.</var> prout ſexta ſecundi Euclidis pro <lb/>ponitur, ex quo quadratum <var>.b.i.</var> nempe <var>.u.t.</var> co-<lb/>gnitum erit, ſed gnomon æqualis eſt rectangulo <var>.g.r.</var> ex prædicta, aut ex .8. poſt .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">hocq́;</reg> rectangulum <var>.g.r.</var> quadratum eſt primi numeri propoſiti 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> cognoſcetur, <lb/>quare totum quadratum <var>.g.y.</var> <reg norm="eiusque" type="simple">eiusq́;</reg> radix <var>.b.g.</var> manifęſta erit, cui coniuncta <var>.q.b.</var> <lb/>data, maius quadratum <var>.q.g.</var> cognoſcetur, ex qua <var>.b.g.</var> detracta <var>.b.i.</var> data, cogno-<lb/>ſcetur <var>.i.g.</var> quadratum minus conſequenter, 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 idipſum definiri poteſt, prætermiſſa antiquorum via, <lb/>nempe multiplicatis in ſemetipſis primo & ſecundo, numeris propoſitis, qua-<lb/><reg norm="druplicatoque" type="simple">druplicatoq́;</reg> quadrato primi, qua ſumma coniuncta cum quadrato ſecundi nume-<lb/>ri, & ex hac altera ſumma eruta radice quadrata, ex qua detracto ſecundo nume-<lb/>ro, & è reliquo ſumpto dimidio, quod erit <reg norm="quadratum" type="context">quadratũ</reg> minus, quo detracto ex radi-<lb/>ce poſtremo iuncta, ſupererit quadrarum maius.</s> + </p> + <p> + <s xml:id="s346" xml:space="preserve">Exempli gratia, ſi proponeretur numerus .8. cui productum duorum numerorum <lb/>quæſitorum æquandum eſt, 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 ſe <lb/>ipſum multiplicari, ex quo exurget .64. pro numero ſui quadrati, quod quadru-<lb/>plicari volo, <reg norm="eritque" type="simple">eritq́;</reg> productum .256. quod cenſeo <reg norm="coniungendum" type="context">coniũgendum</reg> cum quadrato ſe-<lb/>cundi numeri propoſiti, nempe .144. <reg norm="eritque" type="simple">eritq́;</reg> ſumma .400. ex quaſumetur radix, ſci <lb/>licet .20. & ex hac detrahetur ſecundus numerus .12. <reg norm="reſiduique" type="simple">reſiduiq́;</reg> dimidium, nempe <num value="4">.<lb/>4.</num> pro quadrato minore, quo in ſummam collecto cum, 12. dabit quadratum <lb/>maius .16.</s> + </p> + <p> + <s xml:id="s348" xml:space="preserve">Cuius ſpeculationis cauſa, quadratum maius per lineam <var>.q.g.</var> minus per <var>.g.p.</var> ſi-<lb/>gnificetur: </s> + <s xml:id="s349" xml:space="preserve">ſuper integram autem <var>.q.p.</var> erigatur quadratum integrum <var>.d.p.</var> diuiſum, <lb/>vt quadratum <var>.f.g.</var> vigeſimiſeptimi theorematis huius libri, (idipſum accideret di-<lb/>uiſo quadrato modo octauæ ſecundi Euclidis) quæ quidem diuiſio, eſt via quatuor <lb/>productorum <var>.q.g.</var> in <var>.g.p.</var> è quibus vnum ſit <var>.g.r.</var> quod erit cognitum ex .19. theore <lb/>mate cum ſit <reg norm="quadratum" type="context">quadratũ</reg> primi numeri ppoſiti, ex quo illa quatuor cognita <reg norm="erunt" type="context">erũt</reg>. </s> + <s xml:id="s350" xml:space="preserve">Iam <lb/>verò ſi cogitemus <var>.q.p.</var> ſectam in puncto <var>.t.</var> ita vt <var>.q.t.</var> æqualis ſit <var>.p.g.</var> dabitur differen <lb/>tia <var>.t.g.</var> cognita, vt radix quadrati <var>.e.o.</var> cum ex præſup-<lb/>poſito <var>.r.n.</var> æqualis ſit <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> ipſius <var>.t.g.</var> cum quadruplo <var>.g.r</var>: cognitum erit <lb/>quadratum <var>.d.p.</var> ipſius <var>.q.p.</var> </s> + <s xml:id="s352" xml:space="preserve">quare cognoſcetur <var>.q.p.</var> de <lb/>quo numero detracta differétia quadratorum cognita <var>.<lb/>t.g.</var> ſupererit aggregatum <var>.p.g.</var> et <var>.q.t.</var> cognitum. </s> + <s xml:id="s353" xml:space="preserve">Qua-<lb/>re ex conſequenti, dimidium aggregati, nempe <var>.g.p.</var> <lb/>cognoſcetur, 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 eiuſmodi numeros inuenire, vt eorum maior mi-<lb/>norem, numero propoſito ſuperet, & productum vnius in alterum, alteri nu-<lb/>mero propoſito adęquetur, conſultiſsimum ſit dimidium primi numeri propoſiti, + <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 eſt, in ſeipſum <lb/>multiplicare, atque huic quadrato, ſecundum numerum propoſitum iungere, cui, <lb/>productum numerorum quæſitorum æquale eſſe debet, & ex hac ſumma eruere qua <lb/>dratam radicem, quæ coniuncta dimidio primi numeri propoſiti, 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, ſi proponeretur .12. cui differentia vnius numeri ab altero æqua-<lb/>ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-<lb/>torum ſimul <reg norm="æquandum" type="context">æquãdum</reg> eſſet. </s> + <s xml:id="s357" xml:space="preserve">Dimidium primi numeri in ſeipſum multiplicaremus, <lb/><reg norm="proueniretque" type="simple">proueniretq́;</reg> <reg norm="quadratum" type="context">quadratũ</reg> .36. cui coniuncto ſecundo, nempe .64. totum eſſet .100. <lb/>ex quo detracta quadrata radice .10. etipſi coniuncto ſenario, 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 ſpeculatio hæc eſt. </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 ſiue cognitum ſit <var>.a.s</var>: conſide-<lb/>remus nunc <var>.e.i.</var> dimidium <var>.e.o.</var> datæ differentiæ, & ex compoſito <var>.a.i.</var> imaginetur <lb/>quadratum <var>.a.x.</var> in quo protracta ſit <var>.t.u.</var> æquidiſtans lateri <var>.a.i.</var> & tam ab ipſa <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 ſcilicet 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 ſe conſiderare, vnde ſequitur 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, ſi coniunctus fuerit quadrato <var>.e.t.</var> cognito <lb/>ex radice <var>.e.i.</var> cognita (vt dimidia toralis differentię <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 conſequenter quæ quidem ſpeculatio eadem eſt <lb/>quæ .6. ſecundi ſeu .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, hocipſum 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 propoſito numero, inuenturi ſunt duos numeros inter ſe <lb/>differentes, quorum quadratorum ſumma altero numero propoſito æqualis <lb/>ſit, rectè primum numerum propoſitum in ſeipſum multiplicant, quod quadratum <lb/>exſecundo numero <reg norm="detrahunt" type="context">detrahũt</reg>, & dimidium reſidui ſumunt, quod productum erit <lb/>multiplicationis duorum numerorum interſe, in reliquis præcedentis theorematis <lb/>ordinem ſequuntur.</s> + </p> + <p> + <s xml:id="s362" xml:space="preserve">Exempli gratia, ſi proponeretur .12. tanquam numerus, cui differentia duorum <lb/>numerorum quæſitorum æquanda eſt, proponerentur præterea .272. quibus ſum-<lb/>ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret ſanè <lb/>primum numerum, nempe .12. in ſeipſum multiplicare, cuius <reg norm="quadratum" type="context">quadratũ</reg> hoc loco <lb/>eſſet .144. atque hoc detrahere ex ſecundo numero, ſupereſſet .128. ſumpto <lb/>deinde dimidio huiuſce numeri, népe .64. producto in quam duorum numerorum <lb/><reg norm="quæſitorum" type="context">quæſitorũ</reg>. </s> + <s xml:id="s363" xml:space="preserve">Cum hoc .64. proſtea et duodenario primo propoſito numero, præceden <lb/>tis theorematis ordinem ſequeremur.</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 ſpeculemus, conſideremus ſubſcriptam figuram, vigefiminoni theore-<lb/>matis figuræ ſimilem, 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">rũ</reg> quadrata <reg norm="erunt" type="context">erũt</reg> <var>.r.c.</var> et <var>.g.s.</var> <reg norm="quorum" type="context">quorũ</reg> <reg norm="summa" type="context">sũma</reg> <reg norm="iterum" type="context">iterũ</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">duorũ</reg> <lb/>numerorum primo propofita fit <var>.q.i.</var> eius verò qua-<lb/>dratum <var>.m.e.</var> quod cognitum eſt ex ſua radice <var>.q.i</var>. <lb/></s> + <s xml:id="s366" xml:space="preserve">quare gnomon <var>.e.n.m.</var> ſimul cum quadrato minori <var>.<lb/>g.s.</var> cognitus erit, quæ ſumma æqualis eſt duplo <var>.g.r.</var> <lb/>producto datorum numerorum. </s> + <s xml:id="s367" xml:space="preserve">Itaque & ipſa <var>.g.<lb/>r.</var> cognoſcetur, nunc ſi præcedentis theorematis ſpe-<lb/>culationem in reliquis conſuluerimus propoſitum <lb/>conſequemur.</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 idipſum conſequi poſſemus, non conſulto qua-<lb/>drageſimo theoremate. </s> + <s xml:id="s369" xml:space="preserve">Nam ſubtracto quadrato differentiæ, numeri primi <lb/>(<reg norm="inquam" type="context">inquã</reg>) propoſiti, ex <reg norm="summa" type="context">sũma</reg> duorum quadratorum, nempe ex ſecundo numero pro-<lb/>poſito colligendum eſſet reſiduum in ſummam cum prædicto ſecundo numero, & <lb/>ex ſumma hac deſumenda quadrata radix, quæ duorum numerorum ſumma erit, <lb/>de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius <lb/>dimidio addito primo numero propoſito, aut detracto minore inuento ex radice <lb/>poſtremo inuenta, dabitur numerus maior, qui quæritur.</s> + </p> + <p> + <s xml:id="s370" xml:space="preserve">Exempli gratia, cum ſuperfuerint .128. hæc ſi cum ſecundo numero <reg norm="nempe" type="context">nẽpe</reg> .272. <lb/>iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo <lb/>propoſito, nempe .12. ſupererunt .8. quorum <reg norm="dimidium" type="context">dimidiũ</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 reſiduum detractionis <lb/>quadrati <var>.m.e.</var> ex ſumma <reg norm="duorum" type="context">duorũ</reg> quadratorum <var>.r.c.</var> et <var>.g.s.</var> numerum præbet æqua-<lb/>lem duobus ſupplementis <var>.q.n.</var> et <var>.n.u.</var> ex .8. ſecundi Euclidis. qui coniunctus duo-<lb/>bus quadratis (quorum ſumma ſecundo propoſita fuit) cognitionem profert qua-<lb/>drati <var>.q.u.</var> & eius radicis <var>.q.p.</var> de qua, detracto primo dato numero, ſcilicet <var>.q.i.</var> ſu-<lb/>pereſt <var>.i.p.</var> cuius dimidium nempe <var>.g.p.</var> minor eſt numerus qui quęritur; </s> + <s xml:id="s373" xml:space="preserve">reſiduum <lb/>verò totius <var>.g.q.</var> maior ſcilicet.</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 ſumma æqualis propo-<lb/>fito alicui numero futura ſit, & ſumma quadratorum maior eorum produ-<lb/>cto per quantitatem alterius propoſiti numeri, rectè dimidium primi dati numeri in <lb/>ſeipſum multiplicant, quod quadratum ex <reg norm="ſecundo" type="context">ſecũdo</reg> dato numero detrahunt, ſumunt<lb/>q́ue tertię partis refidui quadratam radicem, quam dimidio primi numeri coniun-<lb/>gunt, ex quo maior numerus <reg norm="duorum" type="context">duorũ</reg> <reg norm="quæſitorum" type="context">quæſitorũ</reg> datur, quo ex toto primo detracto, ſu-<lb/>pererit minor.</s> + </p> + <p> + <s xml:id="s375" xml:space="preserve">Exempli gratia, propoſito numero .20. cui æquanda eſt ſumma duorum nume-<lb/>rorum quæſitorum, <reg norm="datoque" type="simple">datoq́;</reg> ſecundo numero .208. qui ſemper 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 ſpeculatione huiuſmodi operis cognoſcetur, <reg norm="cuiæquanda" type="context">cuiæquãda</reg> <lb/>eſt <reg norm="differentia" type="context">differẽtia</reg> inter <reg norm="ſummam" type="context">ſummã</reg> <reg norm="quadratorum" type="context">quadratorũ</reg> <reg norm="duorum" type="context">duorũ</reg> qui <reg norm="quæruntur" type="context">quærũtur</reg> <reg norm="numerorum" type="context">numerorũ</reg>, ſimul <reg norm="cum" type="context">cũ</reg> pro <lb/>ducto <reg norm="eorum" type="context">eorũ</reg> radicum. </s> + <s xml:id="s376" xml:space="preserve">Dimidium numeri .20. in ſeipſum multiplicandum eſſet, qua-<lb/><reg norm="dratumque" type="simple">dratumq́;</reg> detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua <lb/>drata radix eſſet .6. quæ ſi 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 ſpeculationis cauſa, datus primus numerus ſignificetur linea <var>.g.h.</var> in qua <lb/>maior numerus incognitus ſit <var>.g.h.</var> minor verò <var>.b.h.</var> quorum quadrata ſint <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> ſit <var>.g.c.</var> <reg norm="cogitenturque" type="simple">cogitenturq́;</reg> duo <lb/>diametri <var>.q.h.</var> et <var>.g.p.</var> diuiſi per medium in puncto <var>.o.</var> per quod duę 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">vnumquodq́;</reg>, æquale erit quadrato <var>.<lb/>g.f.</var> dimidij ipſius <var>.g.h.</var> datę, </s> + <s xml:id="s379" xml:space="preserve">quare eorum <reg norm="vnumquodque" type="simple">vnumquodq́;</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 diſtan-<lb/>tem à <var>.g.k.</var> quantum <var>.y.l.</var> ab <var>.g.h.</var> diſtare 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 ſubtractis è <lb/>duobus quadratis ſuperius dictis <var>.a.t.y.x.</var> et <var>.b.l.</var> <lb/>producto <var>.y.b.</var> æqualibus, ſupererunt <var>.k.d.</var> et <var>.a.c.<lb/>x.</var> cognita, tanquam æqualia dato ſecundo nu-<lb/>mero, ſed <var>.k.d.</var> quadratum eſt medietatis <var>.g.f.</var> <lb/>cognitæ, cognoſcetur igitur reſiduum <var>.a.c.x.</var> vnà <lb/>etiam ſingulæ 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> ſingularum, <lb/>qua coniuncta dimidio <var>.g.f.</var> <reg norm="rurfusque" type="simple">rurfusq́;</reg> ab <reg norm="eodem" type="context">eodẽ</reg> de-<lb/>tracta, propoſitum conſequemur.</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 ſi quis cupiat numerum propoſitum in duas eiuſmodi partes diuidere, vt <lb/>quadratum maioris, quadratum minoris ſuperet quantitate alterius numeri <lb/>propoſiti, rectè primum numerum in ſeipſum multiplicabit, & ab eodem ſecun-<lb/>dum numerum detrahet, reſiduum 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, ſi proponantur .20. diuiſa in duas eiuſmodi partes, vt <reg norm="quadratum" type="context">quadratũ</reg> <lb/>maioris ſuperet quadratum minoris numero æquali ipſi .240. oportebit primum <lb/>numerum, qui quadratus cum fuerit, erit .400. in ſeipſum multiplicare, & ex hoc <lb/>quadrato ſecundum numerum nempe .240. detrahere, </s> + <s xml:id="s386" xml:space="preserve">tunc remanebunt .160. quę <lb/>diuiſa per .40. <reg norm="numerum" type="context">numerũ</reg> <reg norm="duplum" type="context">duplũ</reg> primo, dabuntur quatuor pro minori numero, à reſi-<lb/>duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s> + </p> + <p> + <s xml:id="s387" xml:space="preserve">Quod vt exactè conſideremus, primus numerus propoſitus ſignificetur 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">Poſtmodum eriga <lb/><gap extent="2"/>r quadratum <var>.q.e.</var> diuiſum diametro <var>.f.h.</var> <reg norm="ductisque" type="simple">ductisq́;</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 reſiduum 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 ſit ſecundo loco data, cogitemus <lb/>detrahi è toto quadrato cognito <var>.q.e.</var> ex quo <lb/>ſumma duorum ſupplementorum <var>.q.o.</var> et <var>.o.e.</var> <lb/>cognoſcetur, vnà cum quadratis <var>.u.n.</var> et <var>.p.a.</var> du <lb/>plo ſcilicet <var>.q.a.</var> quo diuiſo per duplum <var>.q.h.</var> aut <lb/>ſimplex <var>.q.a.</var> per <var>.q.h.</var> ſimplicem, 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 propoſitum in duas eiuſmodi partes, vt pro <lb/>ductum vnius in alteram, alteri numero propoſito æquetur, rectè dimidium <lb/>primi dati numeri in ſeipſum multiplicant, ex quo quadrato ſecundum datum nu-<lb/>merum detrahunt, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem ſumunt, qua coniuncta vni dimidio primi nu-<lb/>meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeſtabit.</s> + </p> + <p> + <s xml:id="s391" xml:space="preserve">Exempli gratia, ſi 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/>ſeipſum multiplicaremus, cuius quadratum eſſet .289. de quo detracto ſecundo 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">detractoq́;</reg> ex .17. ſupereſſet .2. pars <lb/>inquam minor.</s> + </p> + <p> + <s xml:id="s393" xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus propoſitus ſignificetur linea <var>.a.d.</var> cu-<lb/>ius dimidium <var>.c.d.</var> cognitum erit, vnà etiam eius quadratum <var>.c.f.</var> quo diuiſo per dia <lb/>metrum <var>.e.d.</var> ſupponantur partes ignotæ <lb/> + <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/> + ipſius <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 ſimi <lb/>li figuræ quintæ ſecundi 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 propoſitis tribus numeris, quorum prior in duas eiuſmodi partes diui-<lb/>dendus ſit, ut mutuò diuiſæ, & per ſummam prouenientium diuiſo ſecundo <lb/>numero, proueniens vltimum ſit æquale tertio numerorum propoſitorum. </s> + <s xml:id="s396" xml:space="preserve">Conſul <lb/>tiſsimum ſit ſecundum numerum per tertium diuidere, ex quo proueniens ſit ſum-<lb/>ma prouenientium è duabus partibus mutuò diuiſis, quam ſummam ſi quis velit di-<lb/>ſtinguere, rectè poſſit medio operationis <reg norm="pręcedentis" type="context">pręcedẽtis</reg> theorematis <reg norm="sumpta" type="context">sũpta</reg> vnitate ſuper <lb/>ficiali pro ſecundo numero diſtinctis poſtmodum 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, ſi ſumma vnius dictorum prouenientium cum vnitate dat primum numerum, <lb/>quid ipſa eadem vnitas dabit? </s> + <s xml:id="s397" xml:space="preserve">ex quo propoſitum oriatur.</s> + </p> + <p> + <s xml:id="s398" xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. ſecundus .34. tertius .8. <lb/>Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prębeant duo <lb/>prouenientia, quorum ſumma tanta ſit vt per eam diuiſo .34. proueniat numerus <lb/>æqualis tertio numero .8. </s> + <s xml:id="s399" xml:space="preserve">Quod vt præſtemus iubet regula ſecundum .34. per <reg norm="tertium" type="context">tertiũ</reg> <num value="8">.<lb/>8.</num> diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit ſumma pro <lb/>uenientium ex diuiſione duarum partium quæſitarum, quæ ſi diſtinguere volueri-<lb/>mus, præcedentis theorematis methodum ſequemur, vnitate ſuperficiali pro ſecun <lb/>do numero propoſito ſumpta, ac ſi diceremus, diuidatur .4. cum vna quarta parte <lb/>in duas eiuſmodi partes, vt productum vnius in alteram ſit vnitas ſuperficialis, cer-<lb/>tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemſeptem <lb/>quartæ lineares, verum cum neceſſe ſit, ex præcedenti theoremate, dimidium in <lb/>ſeipſum multiplicare, <reg norm="eſſetque" type="simple">eſſetq́;</reg> dimidium .8. quartarum partium cum octaua, com-<lb/>modius totum conſtituetur .34. octauarum, quarum dimidium, nempe decemſep-<lb/>tem octauæ, in ſeipſum multiplicatum erunt .289. ſexageſimæ quartæ vnius integri <lb/>ſuperficialis, quandoquidem <reg norm="integrum" type="context">integrũ</reg> ſuperficiale, cuius vnitas linearis in .8. partes <lb/>diuiditur eſt .64. vt ex primo theoremate huius libri depræhendi poteſt. </s> + <s xml:id="s400" xml:space="preserve">Nunc vni-<lb/>tate hac ſuperficiali, nempe .64. ex .289. detracta, ſupererit .225. cuius radix qua-<lb/>drata, ſcilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit <lb/>maius proueniens .32. <reg norm="detractaque" type="simple">detractaq́;</reg> ex altero dimidio, dabit proueniens minus .2. hoc <lb/>eſt pro maiore proueniente .32. octauas, & pro minore duas, quatuor ſcilicet inte-<lb/>gros pro maiore, & quartam partem vnius integri pro minore. </s> + <s xml:id="s401" xml:space="preserve">Nunc ſi ex regula <lb/>de tribus dixerimus, ſi .4. iuncta vni, nempe .5. dant .20. primum numerum, quid <lb/>dabunt .4. integra (proueniens inquam maius) <reg norm="dabunt" type="context">dabũt</reg> certè .16. partem maiorem. <lb/></s> + <s xml:id="s402" xml:space="preserve">Tum ſi dixerimus, ſi quarta pars coniuncta vnitati dat .20: </s> + <s xml:id="s403" xml:space="preserve">quid dabit quarta illa <lb/>pars (hoc eſt proueniens minus) dabit <reg norm="profectò" type="simple">ꝓfectò</reg> quatuor ſcilicet <reg norm="minorem" type="context">minorẽ</reg> partem, quod <lb/>ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-<lb/>ſcientes ijs vti ad inueniendas duas primi numeri partes.</s> + </p> + <p> + <s xml:id="s404" xml:space="preserve">Cuius ſpeculationis gratia, demus primum numerum ſignificari linea <var>.e.u.</var> cuius <lb/>partes <var>.e.a.</var> & <var>a.u.</var> ſint quæ quæruntur, alter verò numerus ſignificetur linea <var>.b.<lb/>d.</var> tertius linea <var>.g.f.</var> proueniens <reg norm="autem" type="wordlist">aũt</reg> diuiſionis <var>.e.a.</var> per <var>.a.u.</var> ſit <var>.n.t.</var> diuiſionis <reg norm="autem" type="wordlist">aũt</reg> <var>.a.u.</var> <lb/>per <var>.a.e.</var> ſit <var>.t.o.</var> ſumma 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 ſi numerus <var>.f.g.</var> tertiò <lb/>propoſitus ex diuiſione ſecundi per <var>.o.t.n.</var> proferri debet. </s> + <s xml:id="s406" xml:space="preserve">Ex .13. theoremate patet, <lb/>quòd ſi <var>.b.d.</var> per <var>.g.f.</var> diuiſerimus, proferetur <var>.o.t.n.</var> qui cum fuerit inuentus, <reg norm="ſummam" type="context">ſummã</reg> <lb/>eſſe oportet <reg norm="duorum" type="context">duorũ</reg> <reg norm="prouenientium" type="context">prouenientiũ</reg>, ex diuiſione mutua <reg norm="duorum" type="context">duorũ</reg> numerorum, nempe <var>.<lb/>a.e.</var> per <var>.a.u.</var> et <var>.a.u.</var> per <var>.a.e.</var> deinde manifeſtum eſt ex .24. aut .25. theoremate <reg norm="eorum" type="context">eorũ</reg> <lb/>productum (multiplicatis prouenientibus adinuicem) vnitatem ſuperficialem 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/>ſolam vnitatem ſuperficialem <reg norm="contineat" type="context">cõtineat</reg>, quo <lb/> + <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> + facto, ſi, vt antedictum eſt, cogitauerimus <var>.n.<lb/>t.</var> <reg norm="proueniens" type="context">proueniẽs</reg> eſſe ex diuiſione <var>.e.a.</var> per <var>.a.u.</var> et <var>.<lb/>t.o.</var> proueniens ex diuiſione <var>.a.u.</var> per <var>.a.e.</var> pa-<lb/>tebit ex definitione diuiſionis, quod eadem <lb/>erit proportio <var>.a.e.</var> ad <var>.n.t.</var> quæ eſt <var>.a.u.</var> ad vni-<lb/>tatem <var>.n.i.</var> et <var>.a.u.</var> ad <var>.o.t.</var> eadem quæ eſt <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">permutandoq́;</reg> <var>.e.a.</var> ad <var>.a.u.</var> ſicut <var>.t.n.</var> ad <var>.n.i.</var> & componendo <var>.e.a.u.</var> <lb/>ad <var>a.u.</var> ſicut <var>.t.n.i.</var> ad <var>.n.i</var>: & euerſim <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. ſepti <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 ſit. </s> + <s xml:id="s410" xml:space="preserve">Non mirum tamen ſi huiuſmodi 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 diuiſis, ſi per ſummam prouenientium, produ-<lb/>ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-<lb/>tn<gap extent="2"/>m duorum numerorum æquale futurum ſit.</s> + </p> + <p> + <s xml:id="s412" xml:space="preserve">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit <num value="4">.<lb/>4.</num> integrorum cum quarta parte, qua ſumma multiplicata cum producto <reg norm="primorum" type="context">primorũ</reg> <lb/>numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-<lb/>torum duorum numerorum æquantur.</s> + </p> + <p> + <s xml:id="s413" xml:space="preserve">Hoc vt conſideremus, duo numeri partibus <var>.a.e.</var> et <var>.e.i.</var> in linea <var>.a.i.</var> ſignificentur, <lb/>quorum productum ſit <var>.e.d.</var> & <reg norm="quadratum" type="context">quadratũ</reg> ipſius <var>.a.e.</var> ſit <var>.e.p</var>: ipſius verò <var>.e.i.</var> ſit <var>.e.q.</var> pro-<lb/>ueniens <reg norm="autem" type="wordlist">aũt</reg> ex diuiſione <var>.e.i.</var> per <var>.a.e.</var> ſit <var>.o.u.</var> proueniens <reg norm="autem" type="wordlist">aũt</reg> <var>.a.e.</var> per <var>.e.i.</var> ſit <var>.o.t.</var> quo-<lb/>rum ſumma ſit <var>.o.u.t.</var> tum productum <var>.e.d</var>: linea <var>.u.n.</var> ſignificetur ad angulum <reg norm="rectum" type="context">rectũ</reg> <lb/>coniuncta in puncto <var>.u.</var> extremo ipſius <var>.o.u.t.</var> productum <reg norm="autem" type="wordlist">aũt</reg> <var>.u.o.t.</var> in <var>.u.n.</var> ſit <var>.n.t</var>. </s> + <s xml:id="s414" xml:space="preserve">Iam <lb/>probandum nobis eſt <var>.n.t.</var> æqualem eſſe ſummæ duorum quadratorum <var>.q.e.p</var>. </s> + <s xml:id="s415" xml:space="preserve">Quod <lb/>ſingillatim probo, & aſſero productum <var>.o.n.</var> æquale eſſe quadrato <var>.q.e.</var> & <reg norm="productum" type="context">productũ</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 præſuppoſito ab <var>.e.a.</var> multiplicetur <lb/>& diuidatur, cuius multiplicationis produ-<lb/>ctum eſt <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"/> + diuiſione eſt <var>.o.u</var>: </s> + <s xml:id="s417" xml:space="preserve">quare ex dicto theorema-<lb/>te <var>.e.i.</var> media proportionalis eſt 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 eſt qua-<lb/>drato <var>.e.q.</var> ex .16. ſexti vel .20. ſeptimi. </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 eſt <var>.d.e.</var> nempe <var>o.s.</var> & <lb/>proueniens ex diuiſione <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/>eſt. </s> + <s xml:id="s421" xml:space="preserve">Quare ex allatis propoſitionibus <reg norm="productum" type="context">productũ</reg> <var>.s.t.</var> æquale eſt quadrato <var>.e.p.</var> ſed <reg norm="totum" type="context">totũ</reg> <lb/>productum <var>.n.t.</var> ſumma eſt duorum productorum <var>.o.n.</var> et <var>.s.t.</var> ex prima ſecundi Eucli. <lb/></s> + <s xml:id="s422" xml:space="preserve">Itaque verum eſſe quod dictum eſt, conſequitur.</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 ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium, <lb/>per minorem diuidat, <reg norm="maioremque" type="simple">maioremq́;</reg> per proueniens multiplicet, productum, <lb/><reg norm="summæ" type="context">sũmæ</reg> ipſius maioris cum eodem proueniente æquale erit.</s> + </p> + <p> + <s xml:id="s424" xml:space="preserve">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-<lb/>to per proueniens, ipſo nempe .10: </s> + <s xml:id="s425" xml:space="preserve">datur productum .11. cum nona parte, tantum ſci + <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 ſumma eſt maioris cum proueniente.</s> + </p> + <p> + <s xml:id="s426" xml:space="preserve">Cuius ſpeculationis cauſa, maior numerus ſignificetur <var>.a.i.</var> et minor linea <var>.a.o.</var> ex <lb/>quo ex præſupoſito <var>.o.i.</var> vnitas erit. </s> + <s xml:id="s427" xml:space="preserve">Sit autem proueniens ex diuiſione <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 ipſi <var>.a.i.</var> et productum <var>.a.i.</var> in <var>.a.e.</var> ſit <var>.u.i</var>. </s> + <s xml:id="s429" xml:space="preserve">Probabo <lb/>numerum ſuperficialem <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 ſi numerus diuiſibilis per pro-<lb/>ueniens diuidatur, proueniens futurus ſit numerus diuidens, </s> + <s xml:id="s431" xml:space="preserve">quare <var>.a.o.</var> erit pro-<lb/>ueniens ex diuiſione <var>.a.i.</var> per <var>.a.e.</var> & ex deſinitione diuiſionis ita ſe habebit <var>.e.a.</var> ad <var>.<lb/>a.i.</var> ſicut <var>.o.i.</var> ad <var>.o.a.</var> & componondo ita <var>.e.i.</var> ad <var>.a.i.</var> ſicut <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> ſed <var>.a.i.</var> non modò diuiſa <reg norm="nunc" type="context">nũc</reg> cogitatur ab <var>.e.a.</var> ex <lb/>quo ſit proueniens <var>.a.o.</var> ſed 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 eſt 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> ſicut <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 propoſitum.</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">IDipſtim etiam alia ratione conſiderari poteſt.</s> + </p> + <p> + <s xml:id="s437" xml:space="preserve">Linea <var>.u.a.</var> ſecetur in puncto <var>.t.</var> ita vt <var>.a.t.</var> æqualis ſit vnitati <var>.o.i.</var> & media paral <lb/>lela <var>.t.n.</var> terminetur productum <var>.t.i.</var> quod conſtabit æquali numero, quamuis ſuperfi-<lb/>ciali, numero <var>.a.i.</var> tametſi lineari. </s> + <s xml:id="s438" xml:space="preserve">Tumparallela ducatur à puncto <var>.o.</var> ipſi <var>.a.u.</var> termi <lb/><reg norm="neturque" type="simple">neturq́;</reg> productum <var>.o.u.</var> ex quo bina producta dabuntur <var>.u.o.</var> et <var>.t.i.</var> inter ſe æqualia <lb/>ex .15. ſexti aut .20. ſeptimi cum ita ſe habeat <var>.a.i.</var> ad <var>.a.u.</var> ſicut <var>.a.o.</var> ad <var>.a.t.</var> ſed <var>.a.i.</var> ad <var>.<lb/>a.o.</var> permutando ſic ſe habet ſicut <var>.a.u.</var> ad <var>.a.t.</var> & ex prima ſexti aut .18. vel .19. ſepti-<lb/>mi ſic ſe habet <var>.u.i.</var> ad <var>.u.o.</var> ſicut <var>.a.i.</var> ad <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0044-02a" xlink:href="fig-0044-02"/> + o. hoc eſt <var>.u.i.</var> ad <var>.t.i.</var> ope .11. quinti. </s> + <s xml:id="s439" xml:space="preserve">Iam <lb/>ex definitione diuiſionis ita ſe habet <var>.a.e.</var> <lb/>ad <var>.a.i.</var> ſicut <var>.o.i.</var> ad <var>.o.a.</var> & componendo <var>.<lb/>e.i.</var> ad <var>.a.i.</var> ſicut <var>.i.a.</var> ad <var>.o.a</var>. </s> + <s xml:id="s440" xml:space="preserve">Itaque ex præ-<lb/>dicta .11. ſic ſe habebit <var>.e.i.</var> ad <var>.i.a.</var> ſicut <var>.u.<lb/>i.</var> ad <var>.t.i.</var> ſed <var>.t.i.</var> numero conſtat 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 propoſitum in duas eiuſmodi partes, vt <reg norm="productum" type="context">productũ</reg> <lb/>vnius in alteram cum i pſarum differentia in ſummam collectum, æquale ſit <lb/>alicui alteri numero maiori primo. </s> + <s xml:id="s443" xml:space="preserve">Rectè primum ex ſecundo detrahunt, reſiduum <lb/>verò conſeruant, tum ex primo ſemper binarium deſumunt, <reg norm="dimidiumque" type="simple">dimidiumq́;</reg> conſer-<lb/>uant, alterum verò dimidium in ſeipſo multiplicant, & ex quadrato numerum con <lb/>ſeruatum eruunt, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem ex dimidio conſeruato, quod vltimum reſi-<lb/>duum propoſiti numeri quæſita pars minor eſt.</s> + </p> + <p> + <s xml:id="s444" xml:space="preserve">Exempli gratia, ſi proponatur numerus .20. ita <reg norm="diuidendus" type="context">diuidẽdus</reg>, vt <reg norm="productum" type="context">productũ</reg> vnius partis <lb/>in alteram, cum partium differentia collectum in ſummam, æquale ſit propoſito + <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 ſecundo, <lb/>nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi <lb/>narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi <lb/>dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81. <lb/>primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9. <lb/>tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua <lb/>dratum, ſic ſupererit .6. hoc eſt .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 eſt ſpeculatio. </s> + <s xml:id="s446" xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis <lb/>ſignificetur linea <var>.q.g.</var> maior vero linea <var>.x.</var> tum cogitemus <var>.q.g.</var> diuiſam, cuius maior <lb/>pars ſit <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> ſit autem produ-<lb/>ctum <var>.b.o</var>. </s> + <s xml:id="s447" xml:space="preserve">Oportet igitur, ut <var>.b.o.</var> ſimul cum differentia <var>.q.p.</var> æquale ſit numero <var>.x.</var> ſe-<lb/>cundò propoſito, qui notus eſt, </s> + <s xml:id="s448" xml:space="preserve">quare etiam ſumma producti <var>.b.o.</var> cum differentia <lb/><var>q.p.</var> cognita erit, ex qua detracto primo numero <var>.q.g.</var> reſiduum cognitum erit, nunc <lb/>igitur quodnam erit hoc reſiduum? </s> + <s xml:id="s449" xml:space="preserve">attendamus qua ratione ex ſumma <var>.b.o.</var> et <var>.q.p.</var> <lb/>detrahenda ſit <var>.q.g</var>. </s> + <s xml:id="s450" xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma <var>.q.p.</var> quę pars eſt <var>.q.g.</var> <lb/>ſupererit detrahenda <var>.p.g.</var> ex <var>.b.o.</var> pars inquam ipſius <var>.q.g.</var> quod fiet quotieſcunque <lb/>cogitauerimus <var>.q.o.</var> duabus vnitatibus diminutam, et per <var>.o.g.</var> multiplicatam, ſit au-<lb/>tem productum <var>.b.e.</var> nam cum <var>.o.g.</var> toties <var>.b.o.</var> ingrediatur, quot ſunt in <var>.q.o.</var> vnitates <lb/>ex prima ſexti aut .18. vel .19. ſeptimi, <reg norm="detrahendaque" type="simple">detrahendaq́;</reg> ſit <var>.p.g.</var> ex <var>.b.o.</var> quæ <var>.p.g.</var> dupla <lb/>eſt <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> ijſdem <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> cognoſcamus <lb/>ſimul cum <var>.q.i</var>: Sivoluerimus partes <var>.q.e.</var> <lb/>et <var>.e.i.</var> cognoſcere, vtemur .45. theorema-<lb/>te huius libri, & propoſitum obtinebimus, nam cognoſcemus <var>.e.i.</var> & ex conſequen-<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 eiuſmodi partes, quæ pro medio proportionali <lb/>alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni <lb/>hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra <lb/>to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod <lb/>tamen .45. theoremate fuit à nobis ſpeculatum.</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 poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-<lb/>dum per tertium multiplicetur, atque ſecundum hoc productum <reg norm="corporeum" type="context">corporeũ</reg>, <lb/>per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-<lb/>cundi in tertium.</s> + </p> + <p> + <s xml:id="s454" xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12. <reg norm="multiplicenturque" type="simple">multiplicenturq́;</reg> .10. <reg norm="cum" type="context">cũ</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. diuiſum dabit .132. numerum æqualem producto <lb/>ſecundi in tertium numerorum propoſitorum, ſcilicet .132.</s> + </p> + <p> + <s xml:id="s456" xml:space="preserve">Hoc vt ſpeculemur, primus numerus ſignificetur line <var>a.o.u.</var> ſecundus <var>.e.o.</var> tertius <var>.<lb/>e.a.</var> productum verò <var>.o.u.</var> in <var>.o.e.</var> ſit <var>.o.i.</var> ipſius ve <lb/>rò <var>.o.i.</var> per <var>.e.a.</var> <reg norm="productum" type="context">productũ</reg> <reg norm="corporeum" type="context">corporeũ</reg> ſit <var>.i.c.</var> tum <lb/> + <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> + <reg norm="productum" type="context">productũ</reg> <var>.e.o.</var> in <var>.e.a.</var> ſit <var>.e.c</var>. </s> + <s xml:id="s457" xml:space="preserve">Dico <reg norm="nunc" type="context">nũc</reg> quod di-<lb/>uiſo numero corporeo <var>.i.c.</var> per <reg norm="primum" type="context">primũ</reg> <var>.o.u.</var> <reg norm="proue" type="simple">ꝓue</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 eſt, 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/>ſequitur quòd diuiſo <var>.i.c.</var> per <var>o.u.</var> proueniens ſit <lb/><var>e.c.</var> corporeum, æquale nihilominus producto <var>.e.c.</var> ſuperficiali.</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 propoſitum numerum in tres partes ſic ſe habentes vt produ-<lb/>ctum primi in ſecundam, in tertia <reg norm="multiplicatum" type="context">multiplicatũ</reg>, præbeat numerum alteri nu-<lb/>mero propoſito æqualem. </s> + <s xml:id="s461" xml:space="preserve">Rectè ſecundum 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 ſumma cogni <lb/>ta erit, detracto numero diuidente ex primo dato, quam quidem ſi diſtinguere <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 ſic ſe <lb/>habeant, ut productum primæ in ſecundam in tertia multiplicatum det .90. itaque <lb/>ſumenda erit pro prima vna pars ipſius .20. quæcunque illa ſit, verbi gratia .2. qua <lb/>ſecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum <lb/>cæterarum partium inter ſe, quarum ſumma eſt .18. quam ſummam ſi diſtinguere <lb/>volueris in cęteris duabus partibus ſeparatis, vteris .45. theoremate, vt quàm citiſ-<lb/>ſimè quod cupis exequaris, erunt autem partes .3. et .15.</s> + </p> + <p> + <s xml:id="s463" xml:space="preserve">In cuius ſpeculationis gratiam nihil aliud occurrit, quàm quod præcedenti theo-<lb/>remate, & ſuperiore .45. allatum eſt.</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. eiuſmodi partes, vt quadratum vnius ſit æquale <lb/>producto reliquarum duarum inter ſe, idem omnino eſt cum 51. theoremate. <lb/></s> + <s xml:id="s465" xml:space="preserve">Nam qui ſumet quamlibet partem propoſiti numeri, quæ tertia parte maior tamen <lb/>non ſit, <reg norm="reſiduumque" type="simple">reſiduumq́</reg> in duas tales partes diuiſerit, vt prima ſumpta, media proportio <lb/>nalis ſit ex probatione .51. theoremate allata, propoſitum conſequetur.</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 ipſum alia ratione ab ea diuerſa <reg norm="quam" type="context">quã</reg> .51. theoremate adduximus, <reg norm="profici" type="simple">ꝓfici</reg> poteſt.</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, cuiuſcunque denique pro <lb/>portionalitatis, qui in ſummam colligantur, ac poſtmodum, regula de trib. dica-<lb/>mus. </s> + <s xml:id="s468" xml:space="preserve">Si ſumma hæc primo numero propoſito in tres partes diuidendo reſpondet, <lb/>cuireſpondebit vna ex tribus partibus huiuſcę <reg norm="summæ" type="context">sũmæ</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, ſi proponatur numerus .57. diuidendus in tres continuas partes <lb/>proportionales proportione ſeſquialtera, tres numeros in eiuſmodi proportio-<lb/>nalitate diſtinctos ſumemus, vt potè .4. 6. 9. qui in ſummam collecti dabunt <reg norm="ſum- mam" type="context">ſum-<lb/>mã</reg> .19. <reg norm="dicemusque" type="simple">dicemusq́;</reg> ſi .19. dant .4. quid <reg norm="dabunt" type="context">dabũt</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 ſi dicamus, ſi .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">Poſtremò, ſi <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 ſe.</s> + </p> + <p> + <s xml:id="s476" xml:space="preserve">Quod vt ſciamus, numerus propoſitus in tres quaſlibet partes diuidendus ſi-<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 ſe coniunctis denotentur. </s> + <s xml:id="s477" xml:space="preserve">Cogitemus pariter lineam <var>.d.a.</var> in <lb/>tres partes diuiſam <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 ſuarum partium, quæ eſt <var>.e.h.</var> ad reſponden <lb/>tem ipſius in <var>.a.d</var>: Verbi gratia reſpondentem <var>.a.b.</var> ipſi <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 ſe habebit <var>.a.d.</var> ad <var>.c.d.</var> ſicut <var>.e.h.</var> ad <var>.g.h</var>. </s> + <s xml:id="s480" xml:space="preserve">Nam cum ſic ſe habeat <var>.a.<lb/>b.</var> ad <var>.b.c.</var> ſicut <var>.e.f.</var> ad <var>.f.g.</var> ex præſuppoſito, permutando ſic ſe habebit <var>.a.b.</var> ad <var>.e.f.</var> ſi-<lb/>cut <var>.b.c.</var> ad <var>.f.g.</var> & eadem ratione ſic ſe habe-<lb/>bit <var>.c.d.</var> ad <var>.g.h.</var> ſicut <var>.b.c.</var> ad <var>.f.g.</var> & <reg norm="conſequen- ter" type="context">cõſequen-<lb/> + <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> + ter</reg> ſicut <var>.a.b.</var> ad <var>.e.f.</var> ex quo ex .13. quinti ſic <lb/>ſe habebit tota <var>.a.d.</var> ad totam <var>.e.h.</var> ſicut <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 propoſitum manifeſtum erit, ipſum autem productum <var>.a.b.</var> in <var>.c.b.</var> <lb/>æquale erit quadrato <var>.b.c.</var> ex .15. fexti aut .20. ſeptimi.</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 propoſuerunt, quod ex <lb/>more ratione à me definietur, eſt autem eiuſmodi.</s> + </p> + <p> + <s xml:id="s482" xml:space="preserve">Quomodo propoſitus numerus in tres eiuſmodi partes diuidatur, vt <reg norm="quadratum" type="context">quadratũ</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 ſeparata ſumamus, <reg norm="quorum" type="context">quorũ</reg> <reg norm="vnum" type="context">vnũ</reg> æquale ſit reliquis <lb/>duobus; </s> + <s xml:id="s484" xml:space="preserve"><reg norm="eorum" type="context">eorũ</reg> <reg norm="autem" type="context">autẽ</reg> radices in ſummam ſimul colligantur, tum regulam de tribus ſe <lb/>quemur, ratione præcedenti theoremate demonſtrata, & rectè vt infra docebimus, <lb/>quod autem dico de quadratis, etiam de cubis, & quibuſuis dignitatibus aſſero.</s> + </p> + <p> + <s xml:id="s485" xml:space="preserve">Exempli gratia, ſi numerus diuiſibilis proponatur .30. in tres eiuſmodi partes di <lb/>uidendus, vt quadratum vnius æquale ſit ſummæ quadratorum reliquarum duarum <lb/>partium, in primis radices trium quadratorum ſumemus, ſic quomodocunque ſe <lb/>habentes, vt maius ipſorum æquale ſit ſummæ reliquorum duorum, verbi gratia .25. <lb/>16. et .9. nempe .5. 4. et .3. quæ ſi colligantur in ſummam efficiunt .12. </s> + <s xml:id="s486" xml:space="preserve">Tum ex regu-<lb/>la de tribus dicemus, ſi .12. reſpondet .30: </s> + <s xml:id="s487" xml:space="preserve">cui, 5. radix maior reſpondebit? </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 ſi dixerimus ſi .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 ſumma 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 ſummæ quadratorum duarum par <lb/>tium, nempe .100. cum .55.</s> + </p> + <p> + <s xml:id="s492" xml:space="preserve">Hoc vt <reg norm="demonſtremus" type="context">demõſtremus</reg>, numerus diuiſibilis propoſitus ſignificetur linea <var>.a.d.</var> & ſum <lb/>ma radicum, noſtro modo ſumptarum, linea <var>.e.h.</var> quarum prima & maior ſit <var>.e.f.</var> ſe-<lb/>cunda <var>.f.g.</var> tertia <var>.g.h.</var> cogitemus etiam lineam <var>.a.d.</var> ea ratione diuiſam eſſe qua <var>.e.h.</var> <lb/>patebit cnim ex modo præcedentis theorematis vnamquanque partium <var>.a.d.</var> ita ſe <lb/>habituram ad ſuum totum ſicut ſe habent ſingulæ <var>.e.h.</var> ad ſuum. </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 ſic ſe habebit <var>.a.b.</var> ad <var>.b.c.</var> ſicut <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. ſexti aut .11. octaui, eadem erit proportio quadrati <var>.a.b.</var> ad <reg norm="quadratum" type="context">quadratũ</reg> <var>.<lb/>b.c.</var> quæ quadrati <var>.e.f.</var> ad quadratum <var>.f.g.</var> tota enim ſunt æqualia, cum eorum partes <lb/>ſimiles inter ſe ſunt æquales. </s> + <s xml:id="s497" xml:space="preserve">Idem dico de proportione qu@drati <var>.a.b.</var> nempe ita <lb/>ſe habere ad <var>.c.d.</var> ſicut 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 ſummam quadratorum duarum partium <var>.b.c.</var> et <var>.c.d.</var> ſic ſe ha <lb/>bebit ut quadrati <var>.e.f.</var> ad ſummam 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"/> + eſt ſummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> igitur <lb/>ſic etiam ſe 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">Idipſum de cæ <lb/>teris dignitatibus dices, <reg norm="vterisque" type="simple">vterisq́;</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 eiuſ-<lb/>modi eſt.</s> + </p> + <p> + <s xml:id="s501" xml:space="preserve">An numerus aliquis in tres eiuſmodi partes di@idi poſſit, vt quadratum vnius æ-<lb/>quale ſit ſummæ quadratorum cæterarum duarum partium ſimul cum producto <lb/>vnius in alteram.</s> + </p> + <p> + <s xml:id="s502" xml:space="preserve">Exempli gratia, ſi proponatur numerus .50. vt iam dictum eſt diuidendus, repe <lb/>riendus erit alius quilibet numerus, qui tamen ſumma ſit trium radicum ſic ſe ha-<lb/>bentium, vt quadratum vnius æquale ſit ſummæ quadratorum duarum partium ſi-<lb/>mul cum producto vnius in alteram, eum autem qui primò occurrit ſumamus, utpo <lb/>tè .30. qui ſumma eſt numerorum .6. 10. 14. partium ſic ſe habentium, vt quadratum <lb/>ipſius .14. æquale ſit ſummæ quadratorum cæterarum partium ſimul cum produ-<lb/>cto vnius in alteram, agamusq́ue regula de tribus, ac dicamus, ſi .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. abſque @ractis, <lb/>ex quo quadratum primæ erit .544. cum .4. nonis, ſecundæ .277. cum ſeptem nonis, <lb/>tertiæ .100. & productum ſecundæ in tertiam .166. cum .6. nonis, quod productum, <lb/>cum quadratis ſecundæ & tertiæ collectum erit .544. cum .4. nonis.</s> + </p> + <p> + <s xml:id="s504" xml:space="preserve">Huius rei ſpeculatio eadem eſt, quę fuit præcedentis theorematis vſquequo no-<lb/>ueris eandem proportionem eſſe quadrati <var>.a.b.</var> ad ſummam quadratorum <var>.b.c.</var> et <var>.c.<lb/>d.</var> quæ quadrati <var>.e.f.</var> ad ſummam 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 ſummæ 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 eſt, è contrario, ſubſequentes figuræ cogitandæ erunt, qua-<lb/>rum <var>.i.</var> ſit quadratum <var>.a.b</var>: l. ſit 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. ſit productum <var>.b.c.</var> in <var>.c.d</var>: m. ſit 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 ſpeculatione præcedentis theorematis, eadem erit proportio <var>.n.<lb/>t.</var> ad <var>.o.u.</var> quæ eſt <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 ſimile erit <lb/> + <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> + producto <var>.m.</var> cum vtraque ſint 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. ſexti 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 ſic ſe ha-<lb/>bebit <var>.k.</var> ad <var>.i.</var> ſicut <var>.m.</var> ad <var>.l.</var> ſed <var>.x.p.</var> ad <var>.i.</var> <lb/>ſicſe habere probatum eſt vt <var>.y.q.</var> ad <var>.l</var>. <lb/></s> + <s xml:id="s509" xml:space="preserve">Quare ex eadem .24. quinti ſic ſe habe <lb/>bit <var>.x.p.k.</var> ad <var>.i.</var> ſicut <var>.y.q.m.</var> ad <var>.l.</var> ſed <var>.y.q.<lb/>m.</var> æqualis eſt <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 propoſuerunt, <lb/>nempe an aliquis numerus in .4. eiuſmodi partes diuidi poſſit, vt ſumma qua-<lb/>dratorum duarum partium dupla ſit ſummæ quadratorum reliquarum duarum.</s> + </p> + <p> + <s xml:id="s512" xml:space="preserve">Verum huius effectio & ſpeculatio non erit difficilis, <reg norm="cum" type="context">cũ</reg> ſit eadem quæ præmiſsis <lb/>proximè duobus theorematibus allata fuit, ſumpta nempe ſumma radicum quarun <lb/>cunque ſic ſe 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">tũc</reg> progrediemur regula de tribus dicentes. </s> + <s xml:id="s514" xml:space="preserve">Si .44 numerum propoſi-<lb/>tum valet, quid .16. pars maior? </s> + <s xml:id="s515" xml:space="preserve">nempe valebit partem maiorem numeri propoſi-<lb/>ti reſpondentem .16. idem de cæteris dico.</s> + </p> + <p> + <s xml:id="s516" xml:space="preserve">Porrò ſpeculatio eadem eſt cum ſuperioribus.</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 propoſitum numerum in duas eiuſmodi partes, vt productum <lb/>radicum quadratarum ipſarum partium æquale ſit alteri numero propoſito, <lb/>cuius <reg norm="tamen" type="context">tamẽ</reg> quadratum maius <reg norm="non" type="context">nõ</reg> ſit quadrato dimidij primi numeri propoſiti. </s> + <s xml:id="s518" xml:space="preserve">Rectè <lb/>ſecundum numerum propoſitum in ſeipſum multiplicat, & <reg norm="eundem" type="context">eundẽ</reg> ex quadrato di-<lb/>midij primi detrahit, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> quadratam radicem ſubtrahit ex dimidio ipſius pri-<lb/>mi, ex quo datur minor pars quæſita, quaipſi dimidio coniuncta, maior pars ha-<lb/>betur.</s> + </p> + <p> + <s xml:id="s519" xml:space="preserve">Exempli gratia, ſi proponatur numerus, 20. propoſito modo, in duas partes <lb/>eiuſmodi diuidendus, vt productum radicum æquale ſit (verbigratia) 8. </s> + <s xml:id="s520" xml:space="preserve">Dimi-<lb/>dium priminumeri in ſeipſum multiplicabimus, cuius quadratum erit .100. ex <lb/>quo quadratum ſecundi numeri, nempe .64. detrahemus, <reg norm="remanebitque" type="simple">remanebitq́;</reg> .36. cuius radi <lb/>ce quadrata coniuncta .10. dimidio inquam primi numeri propoſiti, dabitur nume <lb/>rus .16. pars maior, & ſubtracta 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 demonſtremus, primus nu-<lb/> + <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> + merus linea <var>.a.b.</var> ſignificetur, quam di-<lb/>uiſam 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 ſignificetur per <var>.d.</var> et <var>.e.</var> <lb/>productum autem radicum cognitum <var>.<lb/>f.</var> quandoquidem datum eſt, 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 ſit <var>.x.</var> <reg norm="itaque" type="simple">itaq;</reg> cognitum, quo facto, doctrinam .45. theorematis libri huius ſecuti, <lb/>propoſitum conſequemur.</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 ſummam ipſarum, ſemper <lb/>differentia ſit quadratorum ipſarum radicum.</s> + </p> + <p> + <s xml:id="s524" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, quoslibet duos numeros pro radicibus ſumpſerimus, vt potè .3. et <num value="5">.<lb/>5.</num> quorum differentia eſt .2. certè ſi differentiam hanc per ſummam radicum ſcili-<lb/>cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eſt <lb/>ſuorum quadratorum, nempeinter .9. et .25.</s> + </p> + <p> + <s xml:id="s525" xml:space="preserve">Hoc vt ſpeculemur, duæ radices in linea <var>.n.i.</var> ſignificentur, quarum vna ſit <var>.n.c.</var> & <lb/>altera <var>.c.i.</var> ipſarum 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">ductaq́</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/>ipſi <var>.n.i.</var> & à puncto <var>.a.</var> quarta <var>.x.a.e.</var> parallela ipſi <var>.<lb/>o.</var> inueniemus <var>.b.n.</var> productum eſſe differentiæ <var>.n.<lb/>t.</var> in ſumma radicum <var>.n.i.</var> & cum <var>.d.o.</var> et <var>.a.o.</var> ſint <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 ſit <var>.<lb/>x.u.</var> ex quo gnomon <var>.e.d.u.</var> æqualis erit producto <var>.<lb/>b.n.</var> quod ſcire 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 propoſitum aliquem numerum diuiſuri in duas eiuſmodi partes, vt diffe-<lb/>rentia radicum quadratarum æqualis ſit alteri numero propoſito, cuius ta-<lb/>men quadratum dimidij primi quadratum non excedat. </s> + <s xml:id="s529" xml:space="preserve">Rectè ſecundum numerum <lb/>in ſeipſum multiplicant, productum verò ex primo numero detrahunt, <reg norm="rurſusque" type="simple">rurſusq́;</reg> di <lb/>midium reſidui quadrant, & quadratum hoc ex quadrato dimidij primi ſubtrahunt, <lb/>atque ita radice quadrata reſidui, dimidio primi coniuncta, pars maior datur, qua <lb/>ex ipſo dimidio detracta, pars minor relinquitur.</s> + </p> + <p> + <s xml:id="s530" xml:space="preserve">Exempli gratia, propoſito numero .20. ita ut propoſitum eſt, diuidendo, nem-<lb/>pe vt differentia radicum quadratarum dictarum partium æqualis ſit binario, bina-<lb/>rium hocin ſeipſum 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="ſupereritque" type="simple">ſupereritq́;</reg> numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-<lb/>cabimus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> numerus .64. qui cum ex quadrato dimidij primi detractus fue-<lb/>rit, nempe ex .100. & reſiduo .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 ſpeculationis cauſa, primus numerus <lb/>propoſitus ſigniſicetur linea <var>.x.y.</var> pro voto diui-<lb/> + <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> + ſa in puncto <var>.c.</var> et <var>.x.t.</var> productum ſit ipſius <var>.x.<lb/>c.</var> in <var>.c.y.</var> pariter etiam <var>.q.p.</var> ſit ſumma radicum <lb/>quadratarum, nempe <var>.q.g.</var> ipſius <var>.t.c.</var> et <var>.g.p.</var> ip-<lb/>ſius <var>.c.y</var>. </s> + <s xml:id="s532" xml:space="preserve">Tum ſuper <var>.q.p.</var> extruatur & diuidatur <lb/>quadratum <var>.q.u.</var> ea ratione qua .41. theoremate <lb/>aut .29. diuiſimus, in quo ſanè quadrato, quadra <lb/>tum ipſius <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 cognoſcimus productum <var>.g.r.</var> (vſi <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">cognitũ</reg>, ac proinde cum no <lb/>uerimus <var>.x.y.</var> ſi rationem ſequemur .45. theore <lb/>mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed <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 propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia <lb/><reg norm="ſuarum" type="context">ſuarũ</reg> <reg norm="radicum" type="context">radicũ</reg> <reg norm="quadratarum" type="context">quadratarũ</reg> æqualis ſit alteri numero propoſito. </s> + <s xml:id="s534" xml:space="preserve">Cuius <reg norm="tamen" type="wordlist">tamẽ</reg> <reg norm="qua- dratum" type="context">qua-<lb/>dratũ</reg> maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </s> + <s xml:id="s535" xml:space="preserve">Rectè <reg norm="etiam" type="context">etiã</reg> <lb/><reg norm="quadratum" type="context">quadratũ</reg> dimidij ſecundi numeri ex dimidio primi <reg norm="detrahunt" type="context">detrahũt</reg>, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem per <lb/>ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum <lb/>pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</s> + </p> + <p> + <s xml:id="s536" xml:space="preserve">Exempli gratia, ſi 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 ſecundi numeri eſſet .9. eo detra <lb/>cto ex dimidio primi, remaneret .16. cu <lb/>ius radix .4. ſcilicet per totum ſecundum <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="reſidum" type="context">reſidũ</reg> .50. hoc eſt .49. <lb/>radices autem erunt .1. et .7. differentes <lb/>inter ſe, numero ſenario.</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 ſciamus, duo numeri lineis <reg norm="ſi- gnificentur" type="context">ſi-<lb/>gnificẽtur</reg>, primus linea .b: <reg norm="ſecundus" type="context">ſecũdus</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/>ſi <var>.c.</var> æqualis & co gnita ſit <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> ſit <var>.a.i.</var> et <var>.t.i.</var> æqualis <var>.a.i.</var> et <var>.l.i.</var> pariter <lb/>ſecetur 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="ſupereritque" type="simple">ſupereritq́;</reg> quadratum <var>.q.u.</var> differentiæ <var>.a.h.</var> cognitum, hoc verò cogi-<lb/>temus diuiſum 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 cognoſcetur, & <reg norm="quadratum" type="context">quadratũ</reg> erit ipſius <var>.a.K.</var> aut ipſius <var>.K.h.</var> dimidij <var>.a.h</var>. </s> + <s xml:id="s540" xml:space="preserve">Quòd <lb/>ſi aliquod iſtorum quadratorum detrahere voluerimus, nempe <var>.n.r.</var> ex dimidio ſum <lb/>mæ <var>.b.</var> duorum quadratorum <var>.q.i.</var> et <var>.i.d.</var> cognitæ, hac via procedemus, primum con <lb/>ſiderabimus <var>.t.r.</var> coniunctam <var>.t.i.</var> quæ quantitates erunt ſumma dimidij <reg norm="duorum" type="context">duorũ</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">dimidiũ</reg> eſt quadrati <var>.t.l.</var> et <var>.t.i.</var> <reg norm="dimidium" type="context">dimidiũ</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 ipſius <var>.K.h.</var> nem <lb/>pe <var>.n.r</var>: </s> + <s xml:id="s541" xml:space="preserve">quare quod ſupereſt cognitum <lb/>erit nempe <var>.y.s.</var> cum <var>.n.i.</var> ſed <var>.y.m.</var> æqualis <lb/>eſt <var>.n.i.</var> et <var>.y.m.</var> cum <var>.y.s.</var> conſtituunt 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/>conſequenter <var>.p.s.</var> eius radix cognoſce-<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="eſtque" type="simple">eſtq́;</reg> produ-<lb/>ctum huiuſmodi ſemper 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> cognoſcetur, conſequen-<lb/>ter <var>.i.</var> @q. tanquam reſiduum ex <var>.b.</var> & eo-<lb/>rum radices quadratæ cognoſcentur <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 poteſt. </s> + <s xml:id="s545" xml:space="preserve">Secundus numerus in <reg norm="ſuum" type="context">ſuũ</reg> dimi <lb/><reg norm="dium" type="context">diũ</reg> multiplicetur, <reg norm="productum" type="context">productũ</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="ſicque" type="simple">ſicq́;</reg> huius ſummæ quadrata radix erit ſumma radicum quadratarum dictarum <lb/>partium, cui iuncto producto ex quadrageſimoquinto theoremate ſingulæ radices <lb/>proferentur.</s> + </p> + <p> + <s xml:id="s546" xml:space="preserve">Exempli gratia, primus numerus diuiſibilis erat .50. alter verò .6. </s> + <s xml:id="s547" xml:space="preserve">Iam ſi multi-<lb/>plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-<lb/>mi, nempe .25. detracto, ſupererit .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 ſcilicet .8. erit ſumma radicum duarum partium quæſitarum, <lb/>qua & producto .7. ex quadrag eſimoquinto theoremate dictæ radices diſtinguen, <lb/>tur, quarum vna erit .7. & altera <var>.I</var>.</s> + </p> + <p> + <s xml:id="s548" xml:space="preserve">Vtautem hocſpeculemur, præcedenti figura vti poterimus, in qua patet <var>.t.r.</var> pro <lb/>ductum eſſe ſecundi numeri <var>.c.</var> nempe <var>.a.h.</var> hoc eſt <var>.t.u.</var> in dimidio <var>.a.e.</var> ſcilicet <var>.p.t.</var> re-<lb/>ſiduum autem dimidij primi <var>.b.</var> eſſe <var>.t.i.</var> nempe <var>.a.i.</var> productum radicum, quod ſupple + <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 eſt quadrati <var>.q.d.</var> totalis. </s> + <s xml:id="s549" xml:space="preserve">Quare duplicato <var>.a.i.</var> & coniuncto <var>.b.</var> cognoſci-<lb/>mustotum <var>.q.d.</var> & conſequenter <var>.a.d.</var> ſuam radicem, hoc eſt ſummam duarum radi <lb/>cum <var>.a.g.</var> et <var>.g.d.</var> quæ medio <var>.a.i.</var> cognito, & quadrageſimoquinto theoremate ſingu-<lb/>læ cognoſcuntur.</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 propoſitum aliquem num erum in duas eiuſmodi partes diuiſuri, vt ſum-<lb/>ma radicum dictarum partium æqualis ſit alteri numero propoſito. </s> + <s xml:id="s551" xml:space="preserve">Rectè ſe-<lb/>cundum numerum in ſeipſum multiplicant, ex quo quadrato, primum datum nu-<lb/>merum detrahunt, <reg norm="rurſusque" type="simple">rurſusq́;</reg> reſiduum in ſeipſum multiplicant, & ex eo quadrato <lb/>quartam partem deſumunt, <reg norm="quam" type="context">quã</reg> ex quadrato dimidij primi numeri detrahunt, radi-<lb/>cemq́ue qua dratam reſidui cum iunxerint, & ex dimidio primi numeri detraxerint, <lb/>partes quæſitæ proferuntur.</s> + </p> + <p> + <s xml:id="s552" xml:space="preserve">Exempli gratia, ſi proponeretur primus numerus .20. diuidendus et .6. ſecundus <lb/>pro ſumma radicum, hunc ſecundum .6. in ſeipſum multiplicabimus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> nu-<lb/>merus .36. ex quo quadrato primus numerus detrahetur, <reg norm="ſupereritque" type="simple">ſupereritq́;</reg> numerus .16. <lb/>qui quadratus dabit .256. cuius numeri quarta pars ſumetur, nempe .64. quæ ex qua <lb/>drato dimidij primi numeri detrahetur, nempe .100. <reg norm="ſupereritque" type="simple">ſupereritq́;</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 ſpeculatio, primus numerus diuiſibilis ſignificetur linea <var>.a.b.</var> diui-<lb/>ſa in puncto <var>.e.</var> in partes adhuc incognitas, et <var>.a.c.</var> ſit productum <var>.a.e.</var> in <var>.e.b.</var> item <var>.q.<lb/>p.</var> ſecundum numerum ſignificet, æqualem ſummæ radicum, quæ puncto <var>.n.</var> diſtin-<lb/>guantur. </s> + <s xml:id="s554" xml:space="preserve">Poſtmodum totum quadratum <var>.p.d.</var> erigatur (quod nobis eſt cognitum), <lb/>in duo quadrata diuiſum <var>.o.p.</var> et <var>.o.d.</var> quorum ſumma <var>.a.b.</var> cum detur, cognita rema-<lb/>net ſumma <reg norm="duorum" type="context">duorũ</reg> <reg norm="ſupplementorum" type="context">ſupplementorũ</reg> <var>.o.u.</var> et <var>.o.q.</var> qua quadrata <reg norm="cum" type="context">cũ</reg> fuerit dabit quadru <lb/><reg norm="plum" type="context">plũ</reg> quadrati <reg norm="ſupplementi" type="context">ſupplemẽti</reg> <var>.o.q.</var> <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quadruplum" type="context">quadruplũ</reg> producti <var>.a.c.</var> etenim <var>.a.c.</var> ex .19. theo <lb/>remate huius libri quadratum eft ipſius <var>.q.o.</var> <reg norm="ſicque" type="simple">ſicq́;</reg> poterant etiam veteres quadrare <lb/>dimidium differentiæ <var>.a.b.</var> ab <var>.p.d.</var> nempe quadrato tantummodo ſupplemento <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 ſingu <lb/>læ partes cognoſcentur.</s> + </p> + <p> + <s xml:id="s556" xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito ſupplemento <var>.<lb/>q.o.</var> diſtinguendæ radices <var>q.n.</var> et <var>.n.p.</var> ex .45. theoremate, quibus cognitis, eorum <lb/>etiam quadrata cognoſcuntur.</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 propoſito numero in tres qualeſcunque partes diuiſo, ſi prima in <lb/>tertiam multiplicetur, & huic producto, ſecundæ in primam productum <lb/>coniungatur, <reg norm="itemque" type="simple">itemq́;</reg> ſecundæ in tertiam, hæc ſumma duplicata æqualis ſit ſummæ <lb/>productorum ſingularum in cæteras duas.</s> + </p> + <p> + <s xml:id="s558" xml:space="preserve">Exempli gratia, ſi proponatur .20. diuiſus in tres partes nempe .12. 5. 3. multipli-<lb/>cato primo .12. per .3. tertiam partem dabitur .36. ſecunda verò multiplicata per re <lb/>liquas duas, hoc eſt .5. per .12. et .3. in primis dabitur .60. poſtea .15. <reg norm="quorum" type="context">quorũ</reg> <reg norm="trium" type="context">triũ</reg> pro <lb/>ductorum ſumma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-<lb/>citur ſummæ productorum ſingularum partium in reliquas duas, nempe ſummæ .60. <lb/>36. 60. 15. 36. 15. hoc eſt ipſis .222.</s> + </p> + <p> + <s xml:id="s559" xml:space="preserve">Cuius rei per ſe patet ſpeculatio, cum in his ſex vltimis productis, ſingula 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 propoſito numero in .3. qualeſcunque partes diuiſo, ſi in reliquas duas ſin-<lb/>gulæ multiplicentur, & hæc producta cum ſumma ſuorum quadratorum con-<lb/>iungantur, tota ſumma hæc vltima æqualis erit quadrato totali propoſiti numeri.</s> + </p> + <p> + <s xml:id="s561" xml:space="preserve">Exempli gratia, ſi fuerit idem numerus .20. in .3. partes diuiſus .12. 5. 3. </s> + <s xml:id="s562" xml:space="preserve">Si .12. in <lb/>5. et .3. producatur, ſumma productorum erit .96. at .5. in .12. et .3. erit .75. poſtmo-<lb/>dum .3. in .12. et .5. erit .51. nempe in vniuerſum .222. quadratorum porrò ſumma <lb/>erit .178 quæ coniuncta .222. dabit .400. quadratum ipſius .20.</s> + </p> + <p> + <s xml:id="s563" xml:space="preserve">Erit autem huiuſce rei facillima ſpeculatio, ſi ſequentem figuram mente conce-<lb/>perimus, in qua <var>.a.b.</var> propoſitum numerum ſignificet, cuius partes diſtinctæ ſint me-<lb/>dio <var>.e.</var> et <var>.c</var>. </s> + <s xml:id="s564" xml:space="preserve">Ip ſum autem <var>.q.b.</var> ſit quadratum <lb/>totale parallelis <var>.e.s.</var> et <var>.c.x.</var> diuiſum, 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> compoſitum ex producto <var>.a.<lb/>e.</var> in ſemetipſam, nempe quadratum <var>.o.e.</var> & <lb/>ex producto eiuſdem <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 ſingularum 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 ſex 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 propoſuerunt, quod tamen à <lb/>nobis determinabitur.</s> + </p> + <p> + <s xml:id="s567" xml:space="preserve">Cur diuiſuri propoſitum numerum in duas eiuſmodi partes, vt mutuò diuiſis, & <lb/>per ſummam prouenientium diuiſa ſumma qua dratorum partium, oriatur proue-<lb/>niens alter numerus propoſitus.</s> + </p> + <p> + <s xml:id="s568" xml:space="preserve">Propoſito deinde tertio quolibet numero diuidendo per ſingulas 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 ſimul prouenientibus in ſummam collectis huius fummæ ad primum nume-<lb/>rum propoſitum proportio futura ſit ea quæ eſt tertij ad ſecundum. </s> + <s xml:id="s569" xml:space="preserve">Rectè dimidium <lb/>primi numeri in ſeipſum multiplicant, ex quo quadrato ſecundum numerum detra <lb/>hunt, tum reſidui radicem ſumunt, quam iungentes, & detrahentes ex dimidio <lb/>primi, partes quæſitas habent, cætera ex neceſsitate ſubſequuntur, 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/>ſtea mutuò diuiſis, & per ſummam prouenientium diuiſa ſumma quadratorum, <lb/>dent <reg norm="ſecundum" type="context">ſecundũ</reg> numerum propoſitum .36. nam reliqua conſequuntur. </s> + <s xml:id="s571" xml:space="preserve">Itaque .10. <lb/>dimidium primi in ſeipſum multiplicatur, & ex quadrato .100. eruitur numerus .36. <lb/>nempe ſecundus propoſitus reſidui porrò .64. quadrata radix .8. fumitur, quam con <lb/>iungimus & detrahimus ex dimidio primi ſcilicet .10. ex quo partes quæſitæ dabun <lb/>tur .18. et .2. quæ mutuo diuiſæ dabunt ſuorum prouenientium ſummam .9. cum no-<lb/>na parte, per quam diuidentes .328. ſummam quadratorum ipſarum partium, <lb/>exactè dabitur numerus .36. qui fuit ſecundò propoſitus. </s> + <s xml:id="s572" xml:space="preserve">Tum ſi per ſingu-<lb/>las iam inuentas partes quilibet numerus diuiſus fuerit, verbi gratia .72. ſumma pro <lb/>uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. ſer <lb/>uabit, quam tertius propoſitus .72. cum ſecundo .36.</s> + </p> + <p> + <s xml:id="s573" xml:space="preserve">Quod vt ſpeculemur, primus numerus ſignificetur linea <var>.n.e.</var> ita diuidendus à <lb/>puncto <var>.o.</var> vt diuiſa parte <var>.n.o.</var> per <var>.o.e.</var> et <var>.o.e.</var> per <var>.n.o.</var> & per ſummam prouenien-<lb/>tium diuiſa ſumma quadratorum <var>.n.o.</var> et <var>.o.e.</var> detur ſecundus 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 ſe futurum, nempe producto <var>.n.o.</var> in <var>.o.<lb/>e.</var> quod ſignificetur rectangulo <var>.n.e</var>. </s> + <s xml:id="s575" xml:space="preserve">Itaque datis <var>.n.e.</var> et <var>.q.K.</var> ſi .45. theorema conſu-<lb/>luerimus, partes <var>.n.o.</var> et <var>.o.e.</var> cognoſcemus.</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 ſi 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="ſummam" type="context">ſummã</reg> duorum <lb/>horum prouenientium, ſic primo nume-<lb/>ro <var>.n.e.</var> dato proportionatam eſſe, ſicut <lb/>tertius <var>.x.</var> <reg norm="ſecundo" type="context">ſecũdo</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 ſit <var>.o.b.</var> ex <lb/>quo <var>.q.d.</var> erit ſumma 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 ſit <var>.o.e.</var> <reg norm="termineturque" type="simple">termineturq́</reg> rectangulum <var>.<lb/>q.u.</var> quod tertio numero propoſito <var>.x.</var> vt <lb/>patet, æquale erit, </s> + <s xml:id="s580" xml:space="preserve">quare ex .15. ſexti aut .<lb/>20. ſeptimi 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> ſicut <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 eſt <var>.<lb/>b.o.</var> ad <var>.o.n.</var> nempe ſicut <var>.b.e.</var> ad <var>.e.n.</var> ſuperficialem, ex prima ſexti aut .18. vel .19. <lb/>ſeptimi, ſed rectangulum <var>.e.n.</var> conſtitutum fuit æquale numero <var>.q.K</var>. </s> + <s xml:id="s581" xml:space="preserve">itaque verum <lb/>eſt propoſitum.</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 diuiſo, <reg norm="productoque" type="simple">productoq́;</reg> duorum numerorum per pro-<lb/>ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua <lb/>dratum exiſtat.</s> + </p> + <p> + <s xml:id="s583" xml:space="preserve">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo <lb/>bus numeris multiplicato, nempe .20. habe <lb/>bimus .100. quadratum numeri diuiſi.</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 ſint <var>.a.</var> et <var>.e.</var> por <lb/>rò <var>.a.</var> per <var>.e.</var> diuiſo detur <var>.u.</var> tum <var>.o.</var> produ-<lb/>ctum <var>.a.</var> in <var>.e.</var> eſſe conſtituatur, quo per <var>.u.</var> <lb/>multiplicato dabitur <var>.x.</var> quadratum <var>.a.</var> pro-<lb/>ptereà quòd <var>.a.</var> medium eſt 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. ſexti aut .20. ſeptimi, propoſiti veri-<lb/>tas eluceſcet.</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 & diuiſo, ſi per horum duo-<lb/>rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-<lb/>timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</s> + </p> + <p> + <s xml:id="s587" xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden <lb/>dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336. <lb/>productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-<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 ſignificetur linea <var>.q.b.</var> multiplicandus & diuiden-<lb/>dus numeris deſignatis per <var>.k.m.</var> et <var>.y.m.</var> productorum ſumma ſit <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> ſit <var>.f.<lb/>m</var>. </s> + <s xml:id="s589" xml:space="preserve">Dico quòd ſi <var>.k.z.</var> per <var>.f.m.</var> diuiſerimus proueni et <var>.a.e</var>. </s> + <s xml:id="s590" xml:space="preserve">Quod cum ſic fuerit, erit <lb/>quoque verum quòd diuiſa <var>.k.z.</var> per <var>.a.e.</var> proueniet <var>.f.m.</var> numerus ſcilicet æqualis <lb/>numero <var>.f.m.</var> ex .13. theoremate huius. </s> + <s xml:id="s591" xml:space="preserve">Itaque quotieſcunque probauero quòd di-<lb/>uiſa <var>.k.z.</var> per <var>.a.e.</var> proueniat numerus æqualis ipſi <var>.f.m.</var> propoſitum verum eſſe con <lb/>ſequetur. ex .13. theoremate. </s> + <s xml:id="s592" xml:space="preserve">Quòd ſi proueniens ex diuiſione <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 ipſum <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> ſuper 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. ſexti, aut, 20. ſeptimi, numerus autem <var>.k.p.</var> erit <lb/>proueniens, quod probandum eſt æquale eſſe <var>.f.m</var>.</s> + </p> + <p> + <s xml:id="s594" xml:space="preserve">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato 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 eſt ſic ſe habere <var>.k.g.</var> ad <var>.k.u.</var> ſicut <var>.k.p.</var> ad <var>.k.y.</var> ſufficiet igitur pro-<lb/>bare ſic ſe habere <var>.k.g.</var> ad <var>.k.u.</var> ſicut <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. ſatis erit igitur probare ita ſe habere <var>.q.b.</var> ad <var>.a.e.</var> ſicut <var>.f.m.</var> ad <var>.k.y</var>. </s> + <s xml:id="s598" xml:space="preserve">Scimus au-<lb/>tem quòd eadem eſt proportio <var>.q.b.</var> ad <var>.a.o.</var> quæ <var>.m.k.</var> ad vnitatem, quæ ſit <var>.x.</var> & quod <lb/>proportio <var>.o.e.</var> ad <var>.q.b.</var> eadem eſt, quæ <var>.x.</var> ad <var>.m.y.</var> ex definitione diuiſionis. </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 ſic ſe habebit <var>.k.y.</var> ad <var>.m.y.</var> ſicut <var>.e.a.</var> ad <var>.o.a.</var> & permutando <var>.k.y.</var> ad <var>.e.<lb/>a.</var> ſicut <var>.m.y.</var> ad <var>.o.a.</var> & ex .19. quinti ita <var>.k.m.</var> ad <var>.e.o.</var> ſicut <var>.k.y.</var> ad <var>.e.a.</var> & permutando <var>.<lb/>k.m.</var> ad <var>.k.y.</var> ſicut <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 ſit <var>.k.y.</var> <reg norm="produ- ctaque" type="simple">produ-<lb/>ctaq́;</reg> <var>.m.t.</var> done <var>c.t.s.</var> æqualis ſit vnitati <var>.x.</var> <reg norm="termineturque" type="simple">termineturq́;</reg> rectangulum <var>.s.i.</var> ex quo da-<lb/>bitur proportio numeri <var>.f.m.</var> ad numerum <var>.s.i.</var> compoſita ex <var>.m.t.</var> ad <var>.t.s.</var> et <var>.f.t.</var> ad <var>.t.i.</var> <lb/>ex .24. ſexti, aut quinta octaui, ſed ita etiam proportio <var>.q.b.</var> ad <var>.a.e.</var> componitur ex <lb/>eiſdem 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 eſt ad <reg norm="numerum" type="context">numerũ</reg> <lb/>ipſius <var>.k.y.</var> ęqualis eſt 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 eſt <var>.k.p.</var> ad <lb/><var>x.y.</var> ex quo ſequitur <var>.k.p.</var> conſtare numero ęquali <var>.f.m.</var> proueniens igitur ex diuiſione <lb/>numeri <var>.k.z.</var> per <var>.f.m.</var> æquale eſt numero ipſius <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 concluſio alia etiam via demonſtrari poteſt.</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> ſint <var>.k.m.</var> et <var>.m.y.</var> prouenientia ex diuiſione ſint <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 ſumma ſit <var>.a.e</var>: productum <lb/>autem <var>.b.a.</var> in <var>.k.m.</var> ſit <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> ſit <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">ueniensq́;</reg> ſit <var>.a.c.</var> cui, <var>a.c</var>: <reg norm="productum" type="context">productũ</reg> <var>.<lb/>a.s.</var> <reg norm="eandem" type="context context">eãdẽ</reg> <reg norm="proportionem" type="context">proportionẽ</reg> ſeruabit, <reg norm="quam" type="context">quã</reg> <lb/><var>k.y.</var> rectangulum ad vnitatem ex <lb/>definitione diuiſionis, hoc autem <lb/>proueniens <var>.a.c.</var> <reg norm="conſtare" type="context">cõſtare</reg> numero æ-<lb/>quali aſſero ſummæ <var>.a.e</var>. </s> + <s xml:id="s604" xml:space="preserve">Primum <lb/>enim ex dicta definitione diuiſio-<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 ſic ſe habet <var>.b.a.</var> <lb/>ad <var>.o.e.</var> ſicut <var>.k.m.</var> ad eandem vnita <lb/>tem. </s> + <s xml:id="s605" xml:space="preserve">Itaque vnitas hæc linearis ſi-<lb/>gnificetur per <var>.m.x.</var> in ſingulis late-<lb/>ribus <var>.k.m.</var> et <var>.m.y.</var> producentibus rectangulum <var>.k.y</var>: ſuperficialis autem vnitas ſit. + <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">cogiteturq́;</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 eſt <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> ſed ex prima ſexti aut .18. vel .19. ſeptimi, ſic ſe habet rectangu-<lb/>lum <var>.k.y.</var> ad <var>.x.y.</var> ſicut <var>.k.m.</var> ad <var>.m.x.</var> </s> + <s xml:id="s607" xml:space="preserve">quare ſicut <var>.b.a.</var> ad <var>.o.e.</var> ex .11. quinti, & eiuſdem <lb/>rectanguli <var>.k.y.</var> ad rectangulum <var>.k.x.</var> ſicut <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 ſcientia, ſic ſe habebit duplum rectanguli <var>.k.y.</var> ad ſummam <var>.y.x.</var> cum <var>.<lb/>k.x.</var> rectangulorum, ſicut duplum <var>.b.a.</var> ad ſummam <var>.a.o.e.</var> et proportio ſummæ re-<lb/>ctangulorum <var>.y.x.</var> et <var>.k.x.</var> duplo <var>.g.m.</var> ſicut duplum <var>.b.a.</var> ad <var>.a.o.e</var>. </s> + <s xml:id="s609" xml:space="preserve">Igitur ſumma 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 ſuperſicialis <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> ſicut dupli <var>.b.a.</var> ad <var>.a.e.</var> ex propoſitioni-<lb/>bus notatis, ſexti aut ſeptimi. </s> + <s xml:id="s611" xml:space="preserve">Quare etiam ſicut dupli rectanguli <var>.k.y.</var> ad <reg norm="ſummam" type="context">ſummã</reg> <lb/>rectangulorum <var>.y.x.</var> et <var>.k.x</var>. </s> + <s xml:id="s612" xml:space="preserve">Iam verò ſi conſtituatur <var>.e.c.</var> pro vnitate lineari ipſius <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/>eſt <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 hypotheſi, nempe quòd <var>.<lb/>e.r.</var> æqualis ſit numero <var>.k.m.y.</var> <lb/> + <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> + hoc eſt 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/>ſcientia <reg norm="medium" type="context">mediũ</reg> proportionale erit <lb/>inter duplum <var>.a.s.</var> & duplum <var>.a.c.</var> <reg norm="ea demque" type="context simple">ea<lb/>dẽq́;</reg> <reg norm="proportio" type="simple">ꝓportio</reg> dupli prędicti <var>.a.s.</var> ad <lb/>duplum <var>.a.c.</var> ex æqualitate propor-<lb/>tionum ſimul collectarum, eadem <lb/>erit qùæ proportio dupli rectangu-<lb/>li <var>.k.y.</var> ad duplum <var>.m.g.</var> hoc eſt <var>.a.s.</var> <lb/>ſimplicis ad ſimplicem <var>.a.c.</var> quæ ſim <lb/>plicis rectanguli <var>.k.y.</var> ad ſimplicem <lb/>vnitatem <var>.g.m.</var> ſic enim ſe habet ſim <lb/>plex ad ſimplex, ſicut duplum ad <lb/>duplum. </s> + <s xml:id="s614" xml:space="preserve">Sed pariter ita ſe habet <var>.a.s.</var> ad <var>.a.</var> c<unsure/>. cogitato <var>.a.c.</var> tamquam proueniente <lb/>ex diuiſione <var>.a.s.</var> per rectangulum <var>.k.y.</var> vt conſtitutum eſt, ſicut <var>.k.y.</var> ad <var>.m.g.</var> ex defi-<lb/>nitione diuiſionis vt iam dictum eſt, </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 propoſitis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-<lb/>dis, ſi <reg norm="adinuicem" type="context">adinuicẽ</reg> diuiſi fuerint, <reg norm="duoque" type="simple">duoq́;</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 ſeruetur, ſi deinde ijdem numeri verſa vice mutuo diuiſi fue <lb/>rint, & inter ſe multiplicata prouenientia, <reg norm="productum" type="context">productũ</reg> hoc, primo ſeruato numero <lb/>æquale erit.</s> + </p> + <p> + <s xml:id="s617" xml:space="preserve">Exempli gratia propoſitis his .4. numeris .20. 30. 5. 10. duo autem .20. ſcilicet <lb/>et .30. ſint 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 ſimul multiplicati <reg norm="proferent" type="context">proferẽt</reg> <num value="12">.<lb/>12.</num> tum .20. per .10. d iuiſo et .30. per .5. prouenientia erunt .2. 6. quæ inter ſe 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 ſi quæris, ſignificentur .4. numeri lineis, <var>a.e.o.u.</var> <reg norm="diuidaturque" type="simple">diuidaturq́;</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> diuiſo per <var>.o.</var> oriatur <var>.z.</var> & per <var>.u.</var> <lb/>proueniat <var>.f.</var> tum <var>.n.</var> ſit 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/>ſionis 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 ſe ha-<lb/>bet <var>.a.</var> ad <var>.y.</var> et <var>.e.</var> ad <var>.f.</var> ſicut <var>.u.</var> ad <var>.x.</var> ex <lb/>quo ſic ſe habebit <var>.s.</var> ad <var>.a.</var> ſicut <var>.z.</var> ad <lb/>e. et <var>.a.</var> ad. y, ſicut <var>.e.</var> ad <var>.f</var>. </s> + <s xml:id="s622" xml:space="preserve">Itaque ex <lb/>æqualitate proportionum ſic ſe ha-<lb/>bebit s. ad <var>.y.</var> ſicut <var>.z.</var> ad <var>.f</var>. </s> + <s xml:id="s623" xml:space="preserve">Igitur ex <lb/>15. ſexti aut .20. ſeptimi 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 eſt, nempe vt proponantur .4. <lb/>numeri qualeſcunque tandem, quorum duo diuiſibiles ſint, tertius diuiſor <lb/>vnius è duobus pro libito, <reg norm="quæramusque" type="simple">quæramusq́;</reg> alterius diuidentem, qui ſic ſe habeat vt pro <lb/>ductum duorum prouenientium quarto numero propoſito ſit 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/>ſint diuiſibiles 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. eiuſmodi vt productum prouenientium æquale ſit .12. </s> + <s xml:id="s627" xml:space="preserve">Diuidam itaque <num value="20">.<lb/>20.</num> per .5. <reg norm="prouenietque" type="simple">prouenietq́;</reg> 4. quem per .48. multiplicabo, nempe per alterum diuiſibi-<lb/>lem, <reg norm="ſicque" type="simple">ſicq́;</reg> proueniet .192. quod productum per quartum numerum nempe .12. diui-<lb/>fum dabit .16. qui erit diuidens quæſitus, quo diuiſo .48. proueniet .3. ſecundum ſci <lb/>licet proueniens, quo per alterum hoc eſt .4. multiplicato producetur quartus nu-<lb/>merus .12.</s> + </p> + <p> + <s xml:id="s628" xml:space="preserve">Quod vt ſciamus, primus nume-<lb/>rus diuiſibilis ſignificetur <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> ſecundus 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> ſecundus diuidens la <lb/>tere <var>.e.u.</var> (hic autem eſt quem quæri-<lb/>mus) tum alterum proueniens ſigni <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 ſint 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 cognoſcetur, <lb/>cætera in ſubſcripta 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 eſt, nempe ſi duæ radices quadratæ in <lb/>ſummam collectæ fuerint, & ex dimidio eiuſmodi ſummæ detracta fuerit mi <lb/>nor radix, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> quadratum duplicatum <reg norm="eique" type="simple">eiq́;</reg> ſummæ coniungatur du-<lb/>plum producti ipſius reſidui in dimidium ſummæ radicum, atque huic ſummæ du-<lb/>plum producti eiuſdem reſidui in radicem minorem coniunctum fuerit; </s> + <s xml:id="s631" xml:space="preserve">vltima hæc <lb/>ſumma differentia erit duorum quadratorum propoſitorum.</s> + </p> + <p> + <s xml:id="s632" xml:space="preserve">Exempli gratia duæ radices quadraræ ſint .5. et .11. harum ſumma erit .16. & dimi <lb/>dium .8. differentia minoris ab ipſo dimidio erit .3: duplum quadrati huius differen <lb/>tiæ erit .18: </s> + <s xml:id="s633" xml:space="preserve">duplum producti huius differentię in dimidium ſummę radicum erit .48. <lb/>item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium <lb/>ſumma erit .96. tantaq́ue erit differentia ſuorum quadratorum, quorum vnum <lb/>erit .25. alterum verò .121.</s> + </p> + <p> + <s xml:id="s634" xml:space="preserve">Pro cuius rei ſcientia, duæ quadratæ radices ſint <var>.h.o.</var> et <var>.o.d.</var> directæ inter ſe con-<lb/>iunctæ, quæ ſumma 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">ducanturq́;</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> ſimul cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex .12. ſecundi Eucli. </s> + <s xml:id="s636" xml:space="preserve">Sed ex .13. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> quadratum <var>.b.d.</var> <lb/>minus eſt 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/>eiuſmodi producti æquale eſt 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 eiuſdem, itaque duo quadrata ſcili-<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">cũ</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> ſummæ 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 ſi ex ſingulis duabus ſummis quadratorum demptum fuerit quadratum <var>.o.b.</var> <lb/>eadem producta & quadrata ipſius <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 ſumma duorum <reg norm="extremorum" type="context">extremorũ</reg> quatuor terminorum <reg norm="proportionalium" type="context">proportionaliũ</reg> arith-<lb/>meticè, æqualis eſt ſummæ duorum mediorum, vbi nota hac in re neceſſa-<lb/>rium non eſſe proportionalitatem continuam exiſtere.</s> + </p> + <p> + <s xml:id="s639" xml:space="preserve">Exempli gratia, ſi darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea <lb/>dem eſſet primi ad ſecundum quæ tertij ad quartum, ſumma primi cum quarto eſſet <lb/>26. <reg norm="tantaque" type="simple">tantaq́;</reg> ſecundi cum tertio.</s> + </p> + <p> + <s xml:id="s640" xml:space="preserve">Cuius ſpeculationis cauſa, primus <reg norm="maiorque" type="simple">maiorq́;</reg> numerus ſignificetur linea <var>.e.o.</var> ſecun-<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> ſit <var>.i.o.</var> quæ æqualis <lb/>erit differentiæ <var>.r.c.</var> qua quartus à tertio ſuperatur ex hypotheſi. </s> + <s xml:id="s641" xml:space="preserve">Itaque aſſero ſum <lb/>mam <var>.e.o.</var> cum <var>.g.t.</var> nempe <var>.a.o.</var> æqualem eſſe ſummę <var>.q.s.</var> et <var>.u.c.</var> <reg norm="ſitque" type="simple">ſitq́;</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 tertiusq́ue terminus reperiuntur, eſt <lb/> + <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> + enim ſecundus <var>.e.i.</var> tertius <var>.i.o.</var> et <var>.e.a.</var> quando-<lb/>quidem ex præſuppoſito <var>.e.i.</var> æqualis eſt <var>.s.q.</var> et <lb/><var>i.o.</var> æqualis <var>.r.c.</var> et <var>.a.e.</var> cum ſit æqualis <var>.g.t.</var> cui <lb/>pariter æqualis eſt <var>.r.u.</var> ex quo <var>.a.e.</var> æqualis <lb/>eſt <var>.u.r</var>. </s> + <s xml:id="s643" xml:space="preserve">Itaque illud ſequitur <var>.a.o.</var> ipſi <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 ſumma duorum terminorum extremorum imparium arithmeticæ pro-<lb/>portionalitatis ſemper duplo medij termini æqualis eſt.</s> + </p> + <p> + <s xml:id="s645" xml:space="preserve">Exempli gratia, ſunt hitres termini proportionalitatis arithmeticæ .20. 15. 10 <lb/>ſumma duorum extremorum erit .30. quæ duplum eſt medij termini .15.</s> + </p> + <p> + <s xml:id="s646" xml:space="preserve">Quod vt ſpeculemur, tres termini, tribus lineis <var>.b.d</var>: <var>n.u.</var> et <var>.q.p.</var> <reg norm="ſignificentur" type="context">ſignificẽtur</reg>. </s> + <s xml:id="s647" xml:space="preserve">Di-<lb/>co nunc quòd ſumma <var>.b.d.</var> cum <var>.q.p.</var> nempe <var>.<lb/>h.d.</var> ſemper duplo <var>.n.u.</var> ſcilicet <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> ſit <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> ſed <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> manifeſtum 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">quorũ</reg> primum <lb/>eſt. </s> + <s xml:id="s650" xml:space="preserve">Datis tribus quantitatibus cognitis, ſi quis quartam inuenire voluerit, <lb/>quæ eiuſmodi ſit reſpectu tertiæ, qualis eſt ſecunda reſpectu primæ, ſecunda cum <lb/>tertia in ſummam colligenda erit, ex qua detracta prima, ſupererit quarta.</s> + </p> + <p> + <s xml:id="s651" xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. ſi quartam inuenire vo <lb/>luerimus eiuſmodi proportionem cum tertia arithmeticè ſeruantem, quam ſecunda <lb/>cum prima, ſecundam cum tertia in ſummam colligemus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> ſumma .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">Idipſum tamen proueniret ſi quis ex tertio termino differentiam primi atque ſe-<lb/>cundi detraheret; </s> + <s xml:id="s653" xml:space="preserve">hæc tamen via non tam vniuerſalis eſtqu àm illa. </s> + <s xml:id="s654" xml:space="preserve">N ſi quartus ter <lb/>minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-<lb/>mino in ſummam colligere oporteret.</s> + </p> + <p> + <s xml:id="s655" xml:space="preserve">Alterum problema eſt, quòd inuentis duobus terminis, ſi tertius requiratur, ſe-<lb/>cundus duplicandus erit, ex qua ſumma detracto primo, ſtatim 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 & ſecun-<lb/>di termini, eam detrahendo aut in ſummam cum ſecunda colligendo, attamen prior <lb/>ratio magis latè patet, ideſt vniuerſalior eſt.</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 ſi quis cupiat ſecundum 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">coniungetq́;</reg> vltimo termino, nempe quarto, ex qua ſum-<lb/>ma tertiam partem deſumet, quæ erit ſecundus terminus quęſitus.</s> + </p> + <p> + <s xml:id="s659" xml:space="preserve">Exempli gratia, ſi 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 ſecundus inueniendus ſit, ſumpto quolibet <lb/>pro primo, ſit 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, ſumma erit .18. cuius eſt tertia pars .6. ſecundus <lb/>numerus ſcilicet ſumpto principio à minimo. </s> + <s xml:id="s662" xml:space="preserve">Idipſum euenit ſumpto principio à <lb/>maximo. </s> + <s xml:id="s663" xml:space="preserve">Nam ſi datur ſecundus à minimo aut à maximo, illico tertius datur diffe-<lb/>rentia inter hunc & primum, ſecundo coniuncta, aut ex eodem detracta.</s> + </p> + <p> + <s xml:id="s664" xml:space="preserve">Cuius ratio ſic demonſtratur, quatuor termini quatuor lineis <var>.m.g</var>: <var>q.p</var>: <var>u.n</var>: <var>c.t.</var> <lb/>ſignificentur, quorum <var>.m.g.</var> et <var>.c.t.</var> tantummodo cognoſcantur. </s> + <s xml:id="s665" xml:space="preserve"><reg norm="ſitque" type="simple">ſitq́;</reg> <var>.m.g.</var> primus ac <lb/>maior terminus: </s> + <s xml:id="s666" xml:space="preserve">k.g. verò ſit 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 ſumma totalis eſt, æqualem eſſe <var>.q.p</var>. </s> + <s xml:id="s667" xml:space="preserve">In primis enim <lb/>certi ſumus <var>.m.f.</var> in <var>.m.g.</var> reperiri æqualem <var>.q.p.</var> <reg norm="ſupereſtque" type="simple">ſupereſtq́;</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>: ſimul 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> ſimul 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">conſiderandum eſt an hætres quantitates <var>.f.g</var>: <var>K.d.</var> et <var>.b.K.</var> ſimul æquales ſint <var>.q.p.</var> <lb/>quod tamen per ſe manifeſtum eſt. </s> + <s xml:id="s670" xml:space="preserve">nam <var>.q.p.</var> ſuperat <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 differentię <var>.f.g.</var> ſuperat <var>.c.t.</var> ita <lb/>que <var>.f.g</var>: <var>k.d.</var> et <var>.K.b.</var> ipſi <var>.q.p.</var> ſunt <reg norm="ae- quales" type="simple">ę-<lb/>quales</reg>, ex quo ſequitur <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, ſumpto or-<lb/>dinis principio à minimo termino <lb/>rum, duplicetur <var>.c.t.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> duplum <lb/>hoc <var>.K.t.</var> cui <var>.k.b.</var> æqualis <var>.m.g.</var> con-<lb/>iungatur, quæſumma ſit <var>.b.t</var>. </s> + <s xml:id="s673" xml:space="preserve">Di-<lb/>co <var>.u.n.</var> tertiam eſſe partem ipſius. <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 ſemel reperitur <var>.u.n.</var> vnà cum <lb/>duabus differentijs, nempe <var>.i.g.</var> in <lb/>ipſa autem <var>.b.t</var>: <var>u.n.</var> ſignificetur pri <lb/>mo loco per <var>.r.K.</var> ex quo ſupererit <var>.b.r.</var> duabus differentijs prædictis æqualis, ſed ex <lb/>præſuppoſito <var>.u.n.</var> componitur ex <var>.o.u.</var> æquali <var>.c.t.</var> et <var>.o.n.</var> ęquali 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 ipſius <var>.b.t.</var> eſſe tertiam partem, quod erat <reg norm="propoſitum" type="context">propoſitũ</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 ſi quis velit ſecundum quinque continuorum proportionalium termi-<lb/>num inuenire, ſolis extremis cognitis. </s> + <s xml:id="s677" xml:space="preserve">Rectè <reg norm="vltimum" type="context">vltimũ</reg> triplo primi coniunget, <lb/>ex qua ſumma quartam partem detraher, quæ erit ſecundus terminus quæſitus. <lb/></s> + <s xml:id="s678" xml:space="preserve">Quod ipſum faciet qui inuenire vult ſecundum terminum ſenarij ſeptenarij, octo-<lb/>narij aut alterius cuiuſcunque, creſcente tamen multiplicatione primi, <reg norm="vltimoque" type="simple">vltimoq́;</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. ſi .18. primus erit, hoc eſt, ſi à genere maioris inæ-<lb/>qualitatis progrediemur, triplicabimus terminum .18. <reg norm="dabunturque" type="simple">dabunturq́;</reg> .54. cui numero <lb/>coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo <lb/>tè ſecundus terminus gratia, aut ſecundi ſex terminorum, quadruplicandus eſſet pri <lb/>mus .18. deinde adiuncto vltimo, quinta pars ſummæ eſſet ſecundus terminus, <reg norm="atque" type="simple">atq;</reg> <lb/>ita deinceps.</s> + </p> + <p> + <s xml:id="s680" xml:space="preserve">Cuius ſpeculationis 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="ſigniſicentur" type="context">ſigniſicẽtur</reg>. <lb/></s> + <s xml:id="s681" xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus <var>.z.h.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> triplum hoc <var>.k.<lb/>h.</var> <reg norm="cuiconiungatur" type="context">cuicõiungatur</reg> <var>.b.k.</var> ęqualis 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 ſum-<lb/>mę <var>.b.h</var>. </s> + <s xml:id="s683" xml:space="preserve">Nam in <var>.k.h.</var> ſecundus 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 eſt tres has differentias <var>.n.h</var>: <var>a.c.</var> et <var>.d.k.</var> ſimul cum <var>.b.<lb/>K.</var> ęquales 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">dubiũ</reg> re <lb/>uocari <reg norm="non" type="context">nõ</reg> poteſt, <lb/>cum <var>.f.s.</var> ſuperet <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/>ſit <var>.x.a.</var> et <var>.a.b.</var> ſit <lb/>æqualis <var>.z.h.</var> & <reg norm="cum" type="context">cũ</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> ſu-<lb/>peret <var>.e.g.</var> quæ in <var>.<lb/>a.b.</var> ſint <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 ſumma <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 eſt vt quæ proportio (licet impropriè dicta) <lb/>eſt ipſius <var>.a.b.</var> ad <var>.c.d.</var> <reg norm="eadem" type="context">eadẽ</reg> ſit ipſius <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 ipſius <var>.a.b.</var> ad <var>.e.f.</var> quæ ipſius <var>.c.d.</var> ad <var>.g.h</var>.</s> + </p> + <p> + <s xml:id="s691" xml:space="preserve">Nam, ex hypotheſi, differentia qua <var>.a.b.</var> ſuperat <var>.c.d.</var> (quæ ſit <var>.m.b.</var>) æqualis eſt <lb/>differentiæ qua <var>.e.f.</var> ſuperat <var>.g.h.</var> (quæ ſit <var>.i.f.</var>) vnde <var>.a.m.</var> reſiduum ex <var>.a.b.</var> æquale erit <lb/><var>c.d.</var> & reſiduum <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 ſcientia. </s> + <s xml:id="s694" xml:space="preserve">Vnde <var>.K.m.</var> æqualis erit <var>.n.d.</var> hoc eſt ipſi <var>.g.h.</var> hoc eſt ipſi <var>e.i</var>. </s> + <s xml:id="s695" xml:space="preserve">Quare ex <lb/>communi conceptu <var>.b.K.</var> æqualis erit ipſi <var>.f.e.</var> ſed <var>.n.d.</var> æqualis eſt <var>.g.h.</var> vt dictum eſt. <lb/></s> + <s xml:id="s696" xml:space="preserve">Cum ergo <var>.b.K.</var> æqualis ſit <var>.e.f.</var> et <var>.d.n.</var> ipſi <var>.g.h.</var> et <var>.a.b.</var> maior ſit ipſa <var>.K.b.</var> per <var>.a.K.</var> æqua-<lb/>lem ipſi <var>.c.n.</var> per quam <var>c.n</var>: <var>d.c.</var> maior eſt ipſa <var>.d.n.</var> ſequitur verum eſſe <reg norm="propoſitum" type="context">propoſitũ</reg> hoc <lb/>eſt, quod eadem proportio ſit ipſius <var>.a.b.</var> ad <var>.e.f.</var> quæ <var>.c.d.</var> ad <var>.g.h.</var> arithmetice ſcilicet.</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 eiuſdem numeri diuiſi-<lb/>bilis, geometricè <reg norm="eandem" type="context">eandẽ</reg> inter ſe <reg norm="proportionem" type="context">proportionẽ</reg> ſeruant, <reg norm="quam" type="context">quã</reg> ipſimet <reg norm="diuidentes" type="context">diuidẽtes</reg>.</s> + </p> + <p> + <s xml:id="s698" xml:space="preserve">Exempli gratia ſi per ſenarium & 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 eſt ratio numerus diuiſibilis ſignificetur rectangulis <var>.u.x.</var> et <var>.n.e.</var> diuidentes <lb/>autem ſint <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/>ſo dabit <var>.x.o.</var> & diuiſo <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 ſub <lb/>ſcriptam figuram conſiderantibus, in qua, <lb/>ex .15. ſexti aut .20. ſeptimi, 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 prorſus <lb/>inter ſe proportionem ſeruabunt, <lb/>quam ipſi 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 aduerſo.</s> + </p> + <p> + <s xml:id="s704" xml:space="preserve">Cuius ratio ex .15. ſexti aut .20. ſeptimi dependet. </s> + <s xml:id="s705" xml:space="preserve">prout in ſubſcripto ordine fa-<lb/>cillimè deprehendi poteſt.</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 ſecta, & per ſingulas <lb/>ipſarum diuiſa, ſumma trium prouenientium quadrato medij prouenientis <lb/>æqualis eſt.</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="ipſeque" type="simple">ipſeq́;</reg> numerus .14. per ſingulas 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 ſecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-<lb/>nient .7. integri, qui in ſummam 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 ſpeculationis gratia, totalis numerus ſignificetur 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 ſint continuæ proportionales, quarum ſingulis, <lb/>numerum <var>.n.c.</var> diuiſum eſſe cogitemus, proueniens autem ex diuiſione <var>.n.c.</var> per <var>.n.<lb/>a.</var> ſit <var>.i.d.</var> quod verò prouenit ex diuiſione <var>.n.c.</var> per <var>.a.e.</var> ſit <var>.d.u.</var> proueniens quoque ex <lb/>diuiſione <var>.n.c.</var> per <var>.e.c.</var> ſit <var>.u.o.</var> quorum ſumma ſit <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 ſit <var>.o.u.</var> <reg norm="erigaturque" type="simple">erigaturq́;</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 ſit æqualis, <reg norm="terminenturque" type="simple">terminenturq́;</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 ſexti aut .20. ſeptimi, quandoquidem tria prouenientia <var>.o.u</var>: <var>u.d.</var> et <var>.d.i.</var> ex <lb/>pręcedenti theoremate ſunt inter ſe continua proportionalia, proportionalitate qua <lb/>partes <var>.n.c</var>. </s> + <s xml:id="s710" xml:space="preserve">Iam verò ſi probauero <var>.q.i.</var> productum, producto <var>.m.p.</var> æquale eſſe, pro-<lb/>poſitum quoque probatum erit. </s> + <s xml:id="s711" xml:space="preserve">Numerus enim producti <var>.q.i.</var> æqualis eſt numero. <lb/></s> + <s xml:id="s712" xml:space="preserve">ſummæ <var>.i.o</var>. </s> + <s xml:id="s713" xml:space="preserve">Habemus autem ex definitione diuiſionis ita ſe habere <var>.n.c.</var> ad <var>.i.d.</var> ſicut <var>.<lb/>n.a.</var> ad <var>.o.q</var>. </s> + <s xml:id="s714" xml:space="preserve">Itaque permutando ſic ſe habebit <var>.n.c.</var> ad <var>.n.a.</var> ſicut <var>.d.i.</var> hoc eſt <var>.m.o.</var> ad <var>.<lb/>o.q.</var> ſed ſicut ſe habet <var>.n.c.</var> ad <var>.n.a.</var> ita pariter ſe habet <var>.i.o.</var> ad <var>.o.u.</var> hoc eſt 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> ſic ſe habebit ſicut <var>.m.o.</var> ad <var>.o.q.</var> ex quo ex .15. ſexti aut .20. ſeptimi <var>.<lb/>q.i.</var> æqualis erit <var>.m.p.</var> & conſequenter quadrato <var>.d.u</var>. </s> + <s xml:id="s716" xml:space="preserve">Vt autem lector minori labo-<lb/>re cognoſcere queat <var>.i.o.</var> ad <var>.o.u.</var> ſic ſe habere, vt <var>.n.c.</var> ad <var>.n.a.</var> ſciendum eſt quòd, ſic <lb/>ſe habet <var>.i.d.</var> ad <var>.d.u.</var> ut <var>.c.e.</var> ad <var>.e.a.</var> ex quo componendo ſic ſe habebit <var>.i.u.</var> ad <var>.d.u.</var> ſi-<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> ſed cum ex <reg norm="præ- cedenti" type="context">præ-<lb/>cedẽti</reg> theoremate ſic ſe habeat <var>.d.u.</var> <lb/>ad <var>.u.o.</var> ſicut <var>.e.a.</var> ad <var>.a.n.</var> permutando <lb/>ſic ſe habebit <var>.d.u.</var> ad <var>.a.e.</var> ſicut <var>.u.o.</var> ad <lb/><var>a.n.</var> ex quo ex .11. quinti ſic ſe habe-<lb/>bit <var>.i.u.</var> ad <var>.c.a.</var> prout <var>.o.u.</var> ad <var>.a.n.</var> per-<lb/>mutandoq́ue <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> ſicut <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 ſecta per-<lb/>q́ue ſingulas diuiſa, ſumma quatuor prouenientium æqualis ſit producto ſe-<lb/>cundi in tertium.</s> + </p> + <p> + <s xml:id="s718" xml:space="preserve">Exempli gratia, ſi triginta in quatuor partes proportionales ſecetur, hoc eſt. <lb/>16. 8. 4. 2. <reg norm="perque" type="simple">perq́;</reg> harum ſingulas idem numerus .30. diuidatur, primum proueniens <lb/>erit .1. cum ſeptem octauis partibus. </s> + <s xml:id="s719" xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7. <lb/>cum dimidio, quartum .15. integri, quorum ſumma erit .28. cum octaua parte, tan <lb/><reg norm="tumque" type="simple">tumq́;</reg> erit productum ſecundi prouenientis in tertium.</s> + </p> + <p> + <s xml:id="s720" xml:space="preserve">Quod vt ſciamus, quantitas <var>.n.c.</var> in partes continuas proportionales quatuor ſe-<lb/>cetur <var>.n.a</var>: <var>a.t</var>: <var>t.e.</var> et <var>.e.c.</var> <reg norm="rurſusque" type="simple">rurſusq́;</reg> per ſingulas partes illa ipſa diuiſa, prouenientia <lb/>ſint <var>.i.d</var>: <var>d.x</var>: <var>x.u</var>: <var>u.o.</var> <reg norm="quorum" type="context">quorũ</reg> ſumma ſit <var>.i.o.</var> hanc <reg norm="ſummam" type="context">ſummã</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">quousq́;</reg> <var>.o.p.</var> æqua <lb/>lis ſit <var>.o.u.</var> <reg norm="erectamque" type="simple">erectamq́;</reg> <var>.m.o.</var> æqualem <var>.i.d.</var> perpendiculariter <var>.o.p.</var> & productam donec <var>.<lb/>o.q.</var> vnitati ſit æqualis. </s> + <s xml:id="s722" xml:space="preserve">Iam terminatis rectangulis <var>.m.p.</var> et <var>.i.q.</var> patebit ex .15. ſexti <lb/>aut .20. ſeptimi, 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 ſi pro-<lb/>bauero productum <var>.i.q.</var> producto <var>.m.p.</var> æquale eſſe, facile patebit propoſitum. </s> + <s xml:id="s724" xml:space="preserve">Cuius <lb/>gratia, ſequuti præcedentis theorematis ordinem, primum ex <reg norm="definitionem" type="context">definitionẽ</reg> diuiſionis, <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> ſic <lb/>ſe habebit vt <var>.i.d.</var> hoc eſt <var>.m.o.</var> ad <var>.o.q.</var> & ſi progrediamur eodem ordine, quo præ-<lb/>cedenti theoremate, ſumpto principio ab <var>.i.d.</var> et <var>.e.c.</var> verſus <var>.d.x.</var> et <var>.e.t.</var> gradatimq́ue <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 ſe 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. ſextiaut .20. ſeptimi <reg norm="produ- ctum" type="context">produ-<lb/>ctũ</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 quantaſuis partes diuiſa ſeruari poterit, cum huiuſmodi <reg norm="ſcientia" type="context">ſciẽtia</reg> in vni <lb/>uerſum 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, ſemper producto <lb/>rectanguli compræhenſi à maximo & medio in minimo termino æqualis ſit.</s> + </p> + <p> + <s xml:id="s728" xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. ſi <lb/>ſumpſerimus productum maximi in medium nempe .54. quod per <reg norm="minimum" type="context">minimũ</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="ſignifi- centur" type="context">ſignifi-<lb/>cẽtur</reg>, cubus autem <var>.e.</var> ſignificetur figura <var>.d.n.</var> <reg norm="productumque" type="simple">productumq́</reg> <var>.a.</var> in <var>.e.</var> ſit <var>.b.n.</var> ipſius <reg norm="au- temmet" type="context">au-<lb/>tẽmet</reg> in <var>.i.</var> ſit <var>.p.o.</var> ita quod <var>.q.p.</var> aut <var>.b.o.</var> cum ſint <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpeciei, 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 abſolui corpus <var>.n.h.</var> ita ut <var>.b.<lb/>o.c.</var>ſit vnica recta linea, ex quo ex .25. <lb/>vndecimi proportio <var>.n.h.</var> ad <var>.n.k.</var> ea-<lb/>dem eſt quæ <var>.o.h.</var> ad <var>o.k.</var> ſed ſic ſe 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 ſexti aut .18. vel .19. ſe-<lb/>ptimi itaque <var>.n.h.</var> ad <var>.n.k.</var> ex .11. <lb/>quinti ſic ſe habebit. vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>ſed <var>.n.h.</var> ad <var>.n.d.</var> ex eiſdem ſic ſe 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 præſuppoſito. </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 propoſitum.</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 ſingulos tres termi <lb/>nos diuiſo, prouenientia, ſingulis dictis terminis ſint 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. diuiſum dabit .4: per .6: 6. per .4: 9.</s> + </p> + <p> + <s xml:id="s735" xml:space="preserve">Cuius gratia, ſint tres termini <reg norm="continui" type="context">cõtinui</reg> <reg norm="proportionales" type="simple">ꝓportionales</reg> <var>.a.o</var>: <var>o.c.</var> et <var>.c.q.</var> <reg norm="quadratum" type="context">quadratũ</reg> <reg norm="autem" type="context">autẽ</reg> <lb/>medij ſit <var>.e.c</var>. </s> + <s xml:id="s736" xml:space="preserve">Iam ſi applicetur <reg norm="rectangulum" type="context">rectangulũ</reg> <var>.a.d.</var> æquale quadrato <var>.e.c.</var> ipſi <var>.a.o.</var> & re-<lb/>ctangulum <var>.q.p.</var> æquale eidem quadrato <var>.e.c.</var> ipſi <var>.c.q.</var> ſi quadratum <var>.e.c.</var> per <var>.a.o.</var> diui <lb/>ſerimus, proueniens erit <var>.o.d.</var> <reg norm="diuiſoque" type="simple">diuiſoq́</reg> per <var>.c.q.</var> proueniens erit <var>.c.p.</var> quod ſi per ſuam <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 ſine dubio æquale eſt <var>.o.c.</var> ſed dico <var>.<lb/>o.d.</var> æqualem eſſe <var>.c.q</var>. </s> + <s xml:id="s737" xml:space="preserve">Nam ex .16. ſexti aut <lb/>20. ſeptimi eadem eſt proportio <var>.a.o.</var> ad <var>.o.<lb/>c.</var> quę <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 eſt <var>.c.q.</var> quandoqui <lb/>dem ex .11. ſic ſe habet <var>.o.c.</var> ad <var>.o.d.</var> ſicut <var>.o.<lb/>c.</var> ad <var>.c.q</var>. </s> + <s xml:id="s738" xml:space="preserve">Applicatis ijſdem rationibus ipſi <var>.<lb/>p.c.</var> probabimus <var>.c.p.</var> æqualem eſſe <var>.a.o.</var> cum <lb/><var>o.c.</var> media ſit 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 eſt <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 propoſitis tribus quantitatibus continuis proportionalibus proportione <lb/>aliarum duarum nobis datarum, multiplicata maiori poſtremarum dua-<lb/>rum in ſummam mediæ cum minima trium primarum, productum æqua-<lb/>le ſit producto minoris duarum in ſummam 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"/> + poſitorum .3. et .2. multiplicato .3. per .10. <reg norm="ſummam" type="context">ſummã</reg> .6. cum .4. dantur .30. quod pro-<lb/>ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</s> + </p> + <p> + <s xml:id="s741" xml:space="preserve">Quod vt cognoſcamus, tres quan <lb/> + <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> + titates continuæ proportionales ſint <lb/><var>b.a.p.</var> proportione <var>.d.q.</var> productum <lb/>autem <var>.d.</var> in ſummam <var>.a.</var> cum <var>.p.</var> ſit <var>.f.t.</var> <lb/>& productum <var>.q.</var> in ſummam <var>.b.a.</var> ſit <var>.<lb/>K.h.</var> et <var>.K.n.</var> ſit 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> ipſi <var>.q.</var> et <var>.f.o.</var> ipſi <var>.d</var>. <lb/></s> + <s xml:id="s742" xml:space="preserve">quare ita ſe habebit <var>.K.n.</var> ad <var>.n.o.</var> ſicut <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 eſt <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 ſicut <var>.k.o.</var> ad <var>.o.t.</var> ex quo ex .15. ſexti aut .20. ſeptimi <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 ſingulis tribus quantitatibus continuis proportionalibus in <lb/>reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum 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 ſimul collecta efficiunt numerum .228 ſed <reg norm="tantum" type="context">tantũ</reg> eſt pro-<lb/>ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</s> + </p> + <p> + <s xml:id="s746" xml:space="preserve">Cuius <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg> cauſa, tres termini <reg norm="continui" type="context">cõtinui</reg> proportionales ſignificentur linea <var>.<lb/>b.e.</var> nempe <var>.b.d</var>: <var>d.c</var>: <var>c.e.</var> cuius duplum ſit <var>.u.e.</var> et <var>.b.f.</var> æqualis ſit <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>ſit <var>.u.s.</var> cui dico æqualem eſſe ſummam productorum <lb/>ſingulorum 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. ſexti aut .20. ſeptimi itemq́ue <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/>prędictas. </s> + <s xml:id="s748" xml:space="preserve">Quare ſex producta æquantur inter ſe, <reg norm="replentque" type="simple">replentq́</reg> productum <var>.u.s.</var> ex quo <lb/>verum eſt propoſitum.</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 cognoſci poſſit <reg norm="verum" type="context">verũ</reg> eſſe proportionem ſummæ quatuor quan-<lb/>titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-<lb/>dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</s> + </p> + <p> + <s xml:id="s750" xml:space="preserve">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-<lb/>es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma + <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 ſicſe habet .20. ad .8. nempe ad ſecundam, vt .30. <lb/>ad .12.</s> + </p> + <p> + <s xml:id="s751" xml:space="preserve">Quod vt ſciamus, quatuor prædictæ quantitates ſignificentur linea <var>.a.e.i.o.</var> pro-<lb/>babo ita ſe 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 ſic ſe 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 ſe habebit <var>.a.</var> ad <var>.i.</var> <lb/>vt <var>.e.</var> ad <var>.o.</var> & è conuerſo 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">permutandoq́</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 propoſitum. <lb/></s> + <s xml:id="s753" xml:space="preserve">Ex quo patet error antiquorum quiidipſum, accidere arbitrati ſunt in quantitatibus <lb/>diſcretæ proportionalitatis, quod tamen falſum eſt.</s> + </p> + <p> + <s xml:id="s754" xml:space="preserve">Exempli gratia, ſi proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eſt quæ .4. <lb/>ad .2. </s> + <s xml:id="s755" xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non ſit eadem quæ .12. ad .6. harum <lb/>autem ſumma erit .24. & ſumma ſecundæ cum tertia .10. ſed primæ cum tertia erit <lb/>16. ex quo .16. ad .6. non ſic ſe habebit vt .24. ad .10. <lb/>At in ſpeculatione quatuor quantitatum <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> + <var>e.i.o.</var> ſi proportio <var>.e.</var> ad <var>.i.</var> non eſſet eadem <lb/>quæ <var>.a.</var> ad <var>.e.</var> minimè licuiſſet dicere ita ſe <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 quibuſlibet, productum duarum in tertiam, vna <lb/>ſemper <reg norm="eademque" type="simple">eademq́;</reg> ſit quantitas.</s> + </p> + <p> + <s xml:id="s757" xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. ſi multiplicauerimus .15. per .8. tum produ <lb/>ctum per .2. tantum erit quantum ſi quis multiplicaret .8. per .2. & hoc per .15. et .15. <lb/>per .2. <reg norm="rurſusque" type="simple">rurſusq́;</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 ſignificentur <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> ſit æquale <lb/><var>m.f.</var> et <var>.u.t</var>: a: et <var>.u.c</var>: o. patebit manifeſtè <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> ſed idipſum 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, & idipſum <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, ſi prima in ſecundam multiplice-<lb/>tur & hoc productum in tertiam, <reg norm="rurſusque" type="simple">rurſusq́</reg> hoc alterum in quartam, vltimum <lb/>productum æquale ſit producto producti ſecundæ 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, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8. <lb/>per .5. & hoc .40. per .3. rurſus 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 ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis <var>.a.e.i.o.</var> <lb/>ſignifi cari, productum autem <var>.e.</var> in <var>.i.</var> eſſe <var>.m.f.</var> et <var>.r.s.</var> ſimiliter & productum <var>.a.</var> in <var>.o.</var> eſ-<lb/>ſe <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/>hocq́ue <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 eſt, in figura præcedentis theore matis ſigni-<lb/>ficetur linea <var>.n.u.</var> & linea <var>.r.u.</var> hu-<lb/>ius, nempe <var>.a.</var> ſignificetur 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/>ſignificabitur producto <var>.n.t.</var> quod <lb/><reg norm="productum" type="simple context">ꝓductũ</reg> <var>.u.s.</var> <reg norm="pręsens" type="context">pręsẽs</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="præsens" type="context">præsẽs</reg> <var>.o.</var> mul<lb/>tiplicatum, quod erat in præceden <lb/>ti <var>.u.c.</var> ſignificabitur 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/>ſentis oritur, ſed 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 eſt propoſitum.</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 quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-<lb/>cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-<lb/>rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</s> + </p> + <p> + <s xml:id="s769" xml:space="preserve">Exempli gratia, ſi horum quatuor diſparium numerorum <reg norm="ſummam" type="context">ſummã</reg>, in dicta pro-<lb/>greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-<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 eſt, vnitatem, tam ſumi pro ſui ipſius radicem, <lb/>quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate. <lb/></s> + <s xml:id="s772" xml:space="preserve">Nunc autem pro quadrato ſumamus per <var>.o.</var> ſignificato, <reg norm="cogitemusque" type="simple">cogitemusq́</reg> quadratum <var>.o.</var> <lb/>includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-<lb/>priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-<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> ſit <var>.o.u.c.e.</var> ita ex communi ſcientia quadrato <var>.o.</var> iun-<lb/>gitur gnomon <var>.e.c.u.</var> conſtans tribus vnitatibus, quare primus gnomon, numero im-<lb/>pari conſtat. </s> + <s xml:id="s774" xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-<lb/>ſparibus ſummam excreſcere, cum propius accedere <reg norm="quam" type="context">quã</reg> binario nequeant, ex quo <lb/>medio binario, ſibi inuicem ſuccedunt. </s> + <s xml:id="s775" xml:space="preserve">Dico igitur quòd quinario ternarium ſub <lb/>ſequente, coniuncto quadrato <var>.o.u.c.e.</var> profertur quadratum, quod in numeris, bi-<lb/>narij quadratum ſequitur, <reg norm="eritque" type="simple">eritq́;</reg> ternarij, <reg norm="quodque" type="simple">quodq́;</reg> ſignificetur 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 eſt <var>.e.c.u</var>. </s> + <s xml:id="s776" xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> ſcimus 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">itemq́;</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 præſuppoſito, maior eſt 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> ſuperabit. </s> + <s xml:id="s777" xml:space="preserve">Qua-<lb/>re <var>.b.f.d.</var> erit impar immediatè ſequens ternarium, qui coniunctus quadrato <var>.o.c.</var> <lb/>quadratum ſubſequens componet. </s> + <s xml:id="s778" xml:space="preserve">Eadem ratione probabitur de quadrato <var>.o.n.</var> ſe <lb/>quenti <var>.o.f.</var> & gnomone <var>.i.n.a.</var> cum hic ordo ſpeculationis ſit vniuerſalis. </s> + <s xml:id="s779" xml:space="preserve">In <lb/>quo cernitur quemlibet gnomonem ſibi <reg norm="contiguum" type="context">contiguũ</reg> inferiorem ſemper duabus vni-<lb/>tat ibus excedere, cumque quadrata non niſi gnomonibus ſibi inuicem ſuccedant. <lb/></s> + <s xml:id="s780" xml:space="preserve">Sed <reg norm="cum" type="context">cũ</reg> primus <var>.e.c.u.</var> diſpar fuerit, <reg norm="proculdubio" type="simple">ꝓculdubio</reg> <reg norm="etiam" type="context">etiã</reg> <reg norm="neceſſarioque" type="simple">neceſſarioq́;</reg> cæteri diſpares <reg norm="erunt" type="context">erũt</reg>. <lb/></s> + <s xml:id="s781" xml:space="preserve">Ex qua ſpeculatione, oritur regula ab antiquis tradita <lb/>inueniendi vltimi numeri diſparis <reg norm="concurrentis" type="context">cõcurrentis</reg> ad <reg norm="compo ſitionem" type="context">cõpo<lb/> + <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> + ſitionem</reg> alicuius quadrati. </s> + <s xml:id="s782" xml:space="preserve">Vt ſi quis ſeire deſideret nu-<lb/>merum vltimum diſparem, quo mediante quadratum <var>.<lb/>o.n.</var> conſtitutum fuit, quod aliud non eſt quam ſcire <lb/>quantus ſit 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 ſciamus hunc gnomonem <var>.i.o.a.</var> <lb/>patet duplicandam eſſe radicem <var>.o.e.b.i.</var> <reg norm="dabiturque" type="simple punctuation">dabiturq́,</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 ſcire gnomonem .i.b.e.o.u.d.a. </s> + <s xml:id="s784" xml:space="preserve">Itaque <lb/>minor eſt vnitate duplo radicis, cum unitas <var>.o.</var> bis repe-<lb/>tatur, quæ tamen in gnomone ſemel tantum ſumebatur.</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 ſumma quadratorum, quorum radices ſunt in proportione ſeſquitertia <lb/>nempe .4. ad .3. quadrata ſit.</s> + </p> + <p> + <s xml:id="s786" xml:space="preserve">Exempli gratia, ſumemus quadratum .3. ſcilicet 9. quod in ſummam cum qua-<lb/>drato .4. colligemus, nempè .16. <reg norm="eritque" type="simple">eritq́;</reg> quadratum .25. & ita quadratum .6. hoc eſt <num value="36">.<lb/>36.</num> collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-<lb/>dratum .9. hoceſt .81. coniunctum quadrato .12. nempè .144. producet quadra-<lb/>tum .225.</s> + </p> + <p> + <s xml:id="s787" xml:space="preserve">In cuius gratiam ſint duo quadrata ſubſcripta <var>.q.o.</var> et <var>.q.a.</var> quorum radices ſint <var>.q.</var> <lb/> + <anchor type="figure" xlink:label="fig-0071-02a" xlink:href="fig-0071-02"/> + g. et <var>.q.p.</var> hoc eſt <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 ſiue la-<lb/>tum <reg norm="quam" type="context">quã</reg> gnomon <var>.b.a.g.</var> nempè vt <var>.h.</var> ſit æqua <lb/>lis .g: g. verò differentia ſit qua <var>.q.g.</var> maior eſt <var>.<lb/>q.p.</var> <reg norm="huncque" type="simple">huncq́;</reg> gnomonem <var>.f.s.h.</var> dico ęqualem eſ <lb/>ſe quadrato <var>.q.o.</var> nam ex preſuppoſito <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 ſe æquales ſunt vnde <lb/>etiam quadratum <var>.q.o.</var> nouem partibus ſuper-<lb/>ficialibus quadratis conſtabit, quarum ſingula <lb/>rum radix æqualis erit <var>.g.</var> cumque præcedenti <lb/>theoremate didicerimus quemlibet gnomo-<lb/>nem quadrati immediatè ſequentis æquę 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 ſuperficialibus creſcere, <reg norm="quarum" type="context">quarũ</reg> <reg norm="ſingularum" type="context">ſingularũ</reg> radix æqualis eſt <var>.g.</var> ne <lb/>ceſſariò ſequitur gnomonem <var>.b.a.g.</var> duabus partibus aut vnitatibus gnomonem <var>.d.<lb/>o.p.</var> ſuperare, ita vt gnomon <var>.b.a.g.</var> ſeptem vnitatibus, aut partibus ſuperficialibus <lb/>quadratis conſtet. </s> + <s xml:id="s789" xml:space="preserve">Quare eadem ratione gnomon <var>.f.s.h.</var> conſtabit nouem ſimilibus. <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 eſt, 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 ſe continuæ proportionales erunt. <lb/></s> + <s xml:id="s792" xml:space="preserve">Idipſum dico ſi <var>.q.p.</var> fuerit .6. et <var>.q.g</var>: 8: </s> + <s xml:id="s793" xml:space="preserve">tunc enim ſingulæ 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 ſummam 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 ſit <var>.g</var>. </s> + <s xml:id="s796" xml:space="preserve">Idem occurret ſi <var>.q.g.</var> erit .12. et <var>.q.p.</var> <lb/>quinque, quod cum fuerit patebitex præce-<lb/>dentis theorematis ſpeculatione, gnomonem <lb/><var>f.s.h</var>: 25. vnitatibus conſtare, cogitatum am-<lb/>plitudinis ſimplicis vnitatis denominatæ in <var>.q.<lb/>p.</var> aut <var>.q.g.</var> non amplitudinis gnomonis <var>.b.a.g.</var> <lb/>qui ſeptem vnitatibus latus eſſet. </s> + <s xml:id="s797" xml:space="preserve">Cum igitur <var>.<lb/>q.p.</var> quinque vnitatibus linearibus conſtet ſcimus <var>.q.o</var>: 25. ſuperficialibus conſtare, <lb/>collecto itaque in ſummam quadrato <var>.q.o.</var> cum quadrato <var>.q.a.</var> cognoſcetur quadra-<lb/>tum <var>.q.s.</var> vnà etiam eius radix. </s> + <s xml:id="s798" xml:space="preserve">Eadem ratione, alia multa quadrata ſimilia 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 propoſito numero pari maiori binario, qui detrahi & in ſummam colli-<lb/>gi debeat ex altero numero quærendo, vt tam reſiduum quam ſumma ſint <lb/>quadrata numerorum integrornm. </s> + <s xml:id="s800" xml:space="preserve">Rectè dimidium propoſiti numeri in ſeipſum <lb/>multiplicamus, & quadrato huic addimus vnitatem, <reg norm="eritque" type="simple">eritq́;</reg> numerus quæfitus.</s> + </p> + <p> + <s xml:id="s801" xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-<lb/>ro inueſtigando, ut reſiduum detractionis, & ſumma ſint quadrati numeri. </s> + <s xml:id="s802" xml:space="preserve">Addi-<lb/>ta vnitate ipſi .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s> + </p> + <p> + <s xml:id="s803" xml:space="preserve">Cuius ſpeculationis gratia, ſubſcripta 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">cogitemusq́;</reg> quadratum <var>.g.p.</var> eſſe quadratum ſummæ, <var>K.n.</var> verò reſidui ſubtractio-<lb/>nis: <var>u.i.</var> <reg norm="autem" type="wordlist">aũt</reg> numerum <reg norm="inueſtigandum" type="context context">inueſtigãdũ</reg>, ex quo gnomon <var>.u.d.i.</var> cognoſcetur ita etiam et <var>.n.<lb/>o.K.</var> qui inter ſe ſunt 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">dimidiũ</reg> eſſe gnomonis <var>.<lb/>n.o.K.</var> ex unitatibus ſuperficialibus quarum una erit <var>.m.a</var>.</s> + </p> + <p> + <s xml:id="s806" xml:space="preserve">Cuius numeri quadratum ſit <var>.t.c.</var> vnde etiam cognitum & cum <var>.K.c.</var> ex communi <lb/>ſcientia ſit vnitas linearis, </s> + <s xml:id="s807" xml:space="preserve">propterea quod <var>.m.a.</var> eſt ſuperficialis hoc eſt quadrata, <lb/>quæ detracta ex <var>.q.c.</var> dimidio gnomonis <var>.n.o.K.</var> (quamuis lineari) ſupererit <var>.K.q.</var> co <lb/>gnita, numerorum integrorum (nota <var>q.K.i.</var> ſemper minor erit duabus vnitatibus li-<lb/>nearibus & maior vna ex dictis vnitatibus, ut ex te ipſo 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> ipſius quadratum numerorum integrorum cognoſcetur, cui addito gnomone <var type="gnomon">.<lb/>n.o.K.</var> cognoſcemus numerum <var>.u.i.</var> quæſitum.</s> + </p> + <p> + <s xml:id="s809" xml:space="preserve">Sed cum nobis hæc via, tenenda propoſitum non fuit, hoc eſt primo loco inue <lb/>niendi quadrati minoris <var>.n.K.</var> ideo ſupereſt probandum gnomonem <var>.t.o.c.</var> vnitati <reg norm="ae- qualem" type="simple">ę-<lb/>qualem</reg> eſſe, nempe quadratulo <var>.m.a.</var> quod patebit, ſi conſideremus nos ſumpſiſſe <lb/>rectangulum <var>.r.c.</var> pro dimidio gnomonis <var>.n.o.K</var>. </s> + <s xml:id="s810" xml:space="preserve">etenim ſi ſupplemento 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 ſupplementa <var>.m.t.</var> et <var>.m.c.</var> inter ſe <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 ſcilicet <var>.t.o.c.</var> ex quo cognoſcitur numerus <var>.u.i.</var> quæſitus. <lb/></s> + <s xml:id="s812" xml:space="preserve">Quod autem quadratum <var>.g.p.</var> numeris integris conſtet, hac ratione probatur viſum <lb/>enim fuit ſupra quadratum <var>.n.K.</var> verè quadratum eſſe, & numeris integris conſtare, <lb/>pariter etiam <var>.t.c.</var> <reg norm="ſeque" type="simple">ſeq́;</reg> mutuo conſequi (nam <var>.K.c.</var> eſt vnitas linearis) ex quo gnomon <lb/><var>n.a.K.</var> numero diſpari conſtabit, 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 eſt gnomonem <var>.t.d.c.</var> etiam numero diſpari conſtare, <lb/>ita vt à numero <var>.n.a.K.</var> non niſi duabus vnitatibus differat, nempe vt <var>.c.p.</var> ſit vnitas li-<lb/>nearis, ſed ita reuera eſt, numerus enim <var>.u.d.i.</var> ex præſuppoſito par eſt, </s> + <s xml:id="s814" xml:space="preserve">quare nume <lb/>rus <var>.t.d.c.</var> diſpar erit, cum alterum vnitate ſuperet, videlicet gnomone <var>.t.o.c.</var> vnita <lb/>ri æquali, tum <var>.n.a.K.</var> minor eſt <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 ſe-<lb/>quitur <var>.g.p.</var> eſſe quadratum <reg norm="integrorum" type="context">integrorũ</reg> ex dicto theoremate ac con ſequens 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 ſe different. </s> + <s xml:id="s817" xml:space="preserve">Vnà etiam ſcienda eſt cauſa, cur numerus propoſitus 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">cũ</reg> ipſe <lb/>ſit futurus gnomon <var>.n.o.K.</var> nec poſſit minor eſſe <lb/>numero ternario, vt patet ex .90. theoremate, <lb/>idcirco ſequitur neceſſariò maiorem eſſe bina-<lb/>rio debere. </s> + <s xml:id="s819" xml:space="preserve">Quòd ſi diſpar numerus propone-<lb/>retur, nec forma operis nec ſpeculationis <reg norm="mutan- da" type="context">mutã-<lb/>da</reg> eſſet. </s> + <s xml:id="s820" xml:space="preserve">Non erit tamen neceſſarium vt ipſa <lb/>quadrata <var>.n.K.</var> et <var>.g.p.</var> numeris integris conſta-<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 ſpeculari nihilo-<lb/>minus fractis integris, <reg norm="ipſisque" type="simple">ipſisq́;</reg> collectis cum ſuis <lb/>fractis ſummæ 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 propoſitis duobus numeris altero pari, altero verò diſpari, duplo primi <lb/>minore per vnitatem, ſi alium inuenire numerum voluerimus, cui alterum iſto <lb/>rum coniunctum proferat quadratum, & altero detracto, quadratum ſuperſit. </s> + <s xml:id="s823" xml:space="preserve">Re-<lb/>ctè datos numeros in ſummam colligemus, quam ſummam in duas quam maximas <lb/>poterimus partes diuidemus, quarum vna pari, altera diſpari conſtet, tum vtran-<lb/>que in ſeipſam multiplicabimus, & quadrato minori, duorum numerorum propo-<lb/>ſitorum quemuis ademus, ex quo cupimus nobis quadratum minus ſupereſſe, & 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 ſit, 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 ſummam collecti dabunt .17. differentiam minoris quadra <lb/>ti & maioris. </s> + <s xml:id="s826" xml:space="preserve">I am ſi ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui <lb/>bus in ſeipſis multiplicatis alterum quadratum erit .64. alterum .81. addito <reg norm="itaque" type="simple">itaq;</reg> ipſi. <lb/>64. 11. aut .6. pro libito, propoſitum numerum conſequemur. </s> + <s xml:id="s827" xml:space="preserve">cui addito .6. vel .11. <lb/>dabit nobis .81. vel ex ipſo 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 ſpeculatio <lb/>ex .90. dependet, quo demonſtratum fuit gnomonem proximè quadratum ſequen <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 ſi quis cupiat ſummam progreſſionis arithmeticæ quam citiſſimè cogno <lb/>ſcere. </s> + <s xml:id="s830" xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius <lb/>poſtea vltimi termini dimidium cum numero terminorum multiplicabit, ex <lb/>quo multiplicationis productum, erit omnium propoſitorum terminorum ſumma, <lb/>aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum <lb/>multiplicabit. </s> + <s xml:id="s831" xml:space="preserve">Nam idipſum eueniet.</s> + </p> + <p> + <s xml:id="s832" xml:space="preserve">Exempli gratia, ſi proponerentur .17. termini in prima progreſſione arithmeti-<lb/>ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino ſumma 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">Idpſum eueniet, multiplicato dimidio numeri <reg norm="terminorum" type="context">terminorũ</reg> <lb/>per vltimum coniunctum vnitati primo termino.</s> + </p> + <p> + <s xml:id="s834" xml:space="preserve">Quod vt ſciamus, cogitemus terminos progreſſionis collocari, vt in figura ſub-<lb/>ſcripta <var>.a.o.n.</var> collocantur, <reg norm="tanquam" type="context">tanquã</reg> per gradus, ſumpto 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 abſoluto parallelogrammo <var>.q.o.</var> ſciemus aper-<lb/>tè ſummam 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 duę <lb/>ambientes diametrum inter ſe ſunt 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 ſumi pluſquam <reg norm="dimidium" type="context">dimidiũ</reg> totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>. </s> + <s xml:id="s838" xml:space="preserve">pro <lb/>tanta portione, quantum eſt dimidiam occupatam à diametro, qui <reg norm="cam" type="context">cã</reg><unsure/> ex diſcretis <lb/>reſpondentibus numero terminorum componatur, conſtat numero æquali eſſe di-<lb/>cto numero terminorum <var>.o.n</var>. </s> + <s xml:id="s839" xml:space="preserve">Iam ſi quis multiplicet <var>.a.o.</var> per dimidium <var>.o.n.</var> procul <lb/>dubio, ex prima ſexti aut .18. ſeptimi, orietur <reg norm="dimidium" type="context">dimidiũ</reg> numeri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg> <var>.q.o.</var> <lb/>quod minus erit ſumma progreſſionis dimidio numeri diametri, aut quod idem eſt <lb/>dimidio <var>.o.n.</var> ſed hoc dimidium <var>.o.n.</var> æquale eſt producto dimidij vnitatis <var>.n.</var> in <var>.o.n.</var> <lb/>ex .20. ſeptimi, cum dimidium <var>.o.n.</var> ſit 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 ſumma quæſita <lb/>propoſitæ progreſſionis. </s> + <s xml:id="s841" xml:space="preserve">Idipſum accidet multiplicata ſumma <var>.o.a.</var> & vnitate <var>.n.</var> <reg norm="per" type="simple">ꝑ</reg> <lb/>dimidium <var>.o.n.</var> ex .20. ſeptimi, cum proportio totius ad totum eadem ſit, quæ dimi <lb/>dijad dimidium, ex cauſa permutationalitatis. </s> + <s xml:id="s842" xml:space="preserve">Patet etiam in progreſſionibus, <lb/>quæ ab vnitate initium ducunt, ſi fiat aſcenſus per binarium ſumma vltimi termini <lb/>cum primo ſemper duplam futuram eſſe numero terminorum, quod ſequentes 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 ſpeculari licebit, Diametros harum figurarum notaui literis ſiue <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, idipſum vt <lb/>monuimus accidit, hoc tamen notato, quòd ex conſequenti 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, ſingulæ partes diametri, vnitati ſui primi termini æ-<lb/>quales ſunt. </s> + <s xml:id="s844" xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eſt 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 eiuſmodi progreſſionibus accidit quoque <reg norm="parallelo- grammum" type="context">parallelo-<lb/>grãmum</reg> à diametro in tres partes diuidi, quarum vnam ipſe occupat, reliquæ ve-<lb/>ro inter ſe æquales ipſum ambiunt. </s> + <s xml:id="s846" xml:space="preserve">Ex quo illud etiam ſequitur, productum <var>.a.o.</var> in <lb/>dimidium <var>.o.n.</var> æquale eſſe dimidio <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quod minus eſt ſumma progreſ-<lb/>ſionis dimidio diametri, quod dimidum ſi inuenire voluerimus, minimum <reg norm="terminum" type="context">terminũ</reg> <var>.<lb/>n.n.n.n.</var> per dimidium <var>.o.n.</var> multiplicabimus, & ex .18. aut .19. ſeptimi ipſum 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 eſt 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. ſeptimi. <lb/></s> + <s xml:id="s849" xml:space="preserve">idem dimidium diametri oriri, ſi 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 ſtatim ſummam propoſitæ progreſſionis <lb/>cognoſcere voluerit, <lb/> + <anchor type="figure" xlink:label="fig-0075-03a" xlink:href="fig-0075-03"/> + ſemper primum termi <lb/>num <var>.n.n.n.n.</var> cum <var>.a.o.</var> <lb/>coniunget, qua ſumma <lb/>per <reg norm="dimidium" type="context">dimidiũ</reg> <var>.o.n.</var> mul-<lb/>tiplicata, aut <var>.o.n.</var> per <lb/>dimidium dictæ ſum-<lb/>mæ, ex prædictis rationibus propofitum conſequemur.</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 ſi 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 eſttamen quoties primus terminus differentia <reg norm="terminorum" type="context">terminorũ</reg> <lb/>eſt, nempe aſcendens ipſorum ter minorum.</s> + </p> + <p> + <s xml:id="s853" xml:space="preserve">Cuius ratio manifeſtè ſpeculari poteſt in figura præcedentis theorematis. </s> + <s xml:id="s854" xml:space="preserve">Nam <lb/>diuiſa <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 diuiſionis. </s> + <s xml:id="s855" xml:space="preserve">At ſuperius dictum fuit ita ſe 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 ſequitur 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 eſt detrahere primum ex vltimo, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> per numerum aſcenden-<lb/>tem differentiam ſcilicet, partiri, <reg norm="proueniensque" type="simple">proueniensq́;</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 ſummam collectis ſimul <lb/>cum vnitatibus primi termini, <reg norm="totque" type="simple">totq́;</reg> funt termini, quot interualla ſimul 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 diuiſo, 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">refiduoq́;</reg> per interuallum, hoc eft numerum dif-<lb/>ferentiæ diuifo, proueniens erit numerus terminorum abſque primo quod vnus eft, <lb/>coni uncto quoque dicto prouenienti propoſitum conſequemur.</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">refiduumq́;</reg> per numerum terminorum excepto vno diuidet.</s> + </p> + <p> + <s xml:id="s864" xml:space="preserve">Huius theorematis ſpeculatio ex .13. theoremate manifeſta crit, nam in præce-<lb/>denti cap. numerus terminorum erat proueniens diuiſionis reſidui ſubtractionis 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 ſi quis maximum omnium terminorum dictæ progreffionis cognofcere <lb/>voluerit, dato primo vnà cum numero aſcendenti, <reg norm="numeroque" type="simple">numeroq́;</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">productoq́;</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 eſt 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 ſummam 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">cũ</reg> toto fequente multiplicabant, <reg norm="productumque" type="simple">productumq́;</reg> ſumma quæſita erat.</s> + </p> + <p> + <s xml:id="s870" xml:space="preserve">Exempli gratia, ſi vltimus terminus eiuſmodi 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 ipſius nempe .3. & dimidio, cum numero ipſum terminum <reg norm="ſequenti" type="context">ſequẽti</reg>, nem <lb/>pè .8. ſumma dictorum terminorum erit .28.</s> + </p> + <p> + <s xml:id="s871" xml:space="preserve">Huius autem ſpeculatio ex .94. theoremate dependet, in quo facilè depræhen-<lb/>dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-<lb/>mo termino ſemper æqualem eſſe; </s> + <s xml:id="s872" xml:space="preserve">ex quo <reg norm="tantum" type="context">tãtum</reg> eſt dimidium numeriterminorum, <lb/>quantum maximi dimidium, <reg norm="tantusque" type="simple">tantusq́;</reg> eſt vltimus terminus vnitati coniunctus, quan <lb/>tus numerus is, qui vltimum terminum conſequitur.</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 diſpar eſt ſcire cupientes, numerum integrorum proximè dimidium <lb/>maximi ſequentem ſumebant, quem per maximum multiplicabant, ex quo <lb/>ſumma quæſita oriebatur.</s> + </p> + <p> + <s xml:id="s874" xml:space="preserve">Exempli gratia, ſi dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor, <lb/>& per maximum .7. multiplicabant, ex quo pariter proferebatur ſumma .28.</s> + </p> + <p> + <s xml:id="s875" xml:space="preserve">Cuius ratio ex .20. ſeptimi Euclidis oritur, cum eadem ſit proportio numeri fe-<lb/>quentis ma ximum ad numerum dimidium maximi ſequentem; </s> + <s xml:id="s876" xml:space="preserve">quæ maximi ad <reg norm="fuum" type="context">fuũ</reg> <lb/>dimidium, eſt 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 eſt à nonnullis, à veteribus obſeruatam fuiſſe hancregulam, qua ſci-<lb/>re poſſent ſummam alicuius progreſſionis arithmeticæ diſcontinuæ 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">dimidiũ</reg> vltimi termini per <lb/>pro ximum numerum dimidio dicto maiorem, ex quo <reg norm="inquiebant" type="context">inquiebãt</reg> ſemper productum <lb/>ſummæ quæſitæ æquale eſſe, <reg norm="ſubijciuntque" type="simple">ſubijciuntq́;</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 eſt, non quia ex ſe vnus ex producentibus numeris dimi-<lb/>dium termini maioris futurus ſit, alter uerò proximè ſequens dimidium, fed quia <lb/>vt dictum eſt .95. theoremate, eadem eſt proportio maximi termini ad numerum <lb/>terminorum, quæ minimi ad vnitatem. </s> + <s xml:id="s881" xml:space="preserve">Cumq́ue in præfenti exemplo minimum <lb/>ſit duplum vnitati in eiuſmodi caſu, numerus terminorum, dimidio maximi termini <lb/>æqualis eſt, qui terminorum numerus ex ſe, vt patet, vnus eſt ex producentibus, al-<lb/>ter verò producens numerus, eſt proximè dimidium ſequens, non exſe, fed quia nu <lb/>merus ſequens, dimidium eſt ſummæ 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 creſcút <lb/>regulafal<unsure/>fa eſt, prout facilè patere poteſt ei, qui ex ſcientiæ legibus ope ſpeculatio-<lb/>nis .95. theorematis ſpeculatus 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 vſos fuiſſe dicunt, quo ſum-<lb/>mam ſcire poſſent progreſſionis diſcontinuæ, quænumero diſpari abſolui-<lb/>tur. </s> + <s xml:id="s884" xml:space="preserve">Ea autem eſt eiuſmodi. </s> + <s xml:id="s885" xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-<lb/>ximas partes diuidebant, quarum vna ſemper altera maior erat, banc autem maio-<lb/>rem in ſeipſam multiplicabant, at que quadratum hoc, ſummam 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 ſanè regula, non ſemper, etſi interdum vera ſit.</s> + </p> + <p> + <s xml:id="s887" xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creſcit per bina <lb/>rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè ſe-<lb/>quens, nempe è duabus partibus vltimi termini maior, æqualis ſit numero termino <lb/>rum, qui per ſe 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 ſe dimidium ſummæ primi & vltimi eſſe debet, per <lb/>accidens pars maior eſt duarum vltimi termini, & alteri producenti æqualis.</s> + </p> + <p> + <s xml:id="s889" xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuſmodi progreſſione dimidium <lb/>ſummæ vltimi termini cum primo, ſemper medium proportionale eſt inter eam <lb/>ſummam & dimidium numeri terminorum, etenim huiuſmodi ſumma numero ter-<lb/>minorum ſemper dupla eſt, prout .94. theoremate tradimus. </s> + <s xml:id="s890" xml:space="preserve">Itaque ex .20. ſeptimi, <lb/>quadratum partis maioris, producto ſummæ dictæ in numerum dimidij <reg norm="terminorum" type="context">terminorũ</reg> <lb/>æquale erit, quod productum per ſe ſummæ progreſſionis eſt æquale. </s> + <s xml:id="s891" xml:space="preserve">At in cæte-<lb/>ris eiuſmodi progreſſionibus fallit regula, vt ex ſupradictis facilè demonſtratur.</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 propoſitis, diſpoſitis nihilominus progreſſio-<lb/>ne, aut proportionalitate geometrica continua, ſi minimus ex maximo & exfe-<lb/>quenti minimum detrahatur, reſiduum maximi, eam proportionem ad fum-<lb/>mam reliquorum omnium terminorum retinebit, quam reſiduum ſecundi 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, ſi primum, hoc eſt minimum, ex ſecundo, & maximo detra <lb/>has, exſecundo ſupererit .9. ex maximo .189. quod ſi minimum per reſiduum maxi <lb/>mi multiplicaueris, hoc eſt .189. orietur .567. tum ſi huiuſmodi productum per .9. <lb/>( refiduum ſecundi ) diuiſeris, proueniet .63. quod proueniens æquale erit ſummæ <lb/>reliquorum omnium terminorum, maximo excepto. </s> + <s xml:id="s894" xml:space="preserve">Ex quo inferre licet ex .20. ſe <lb/>ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut ſi reſiduum ſecundi <lb/>per ſummam dictorum terminorum multiplicaueris produceturidem .567. </s> + <s xml:id="s895" xml:space="preserve">quare <lb/>ex .20. ſeptimi & cætera.</s> + </p> + <p> + <s xml:id="s896" xml:space="preserve">Quod vt <reg norm="ſcientificè" type="context">ſciẽtificè</reg> poſſimus, & in vniuerſum ſpeculari. </s> + <s xml:id="s897" xml:space="preserve">Quatuor termini propo-<lb/>ſiti, quatuor ſubſcriptis lineis <reg norm="ſignificentur" type="context">ſignificẽtur</reg> <var>.b.i</var>: <var>c.a</var>: <var>f.r</var>: <var>m.s.</var> (quod <reg norm="autem" type="wordlist">aũt</reg> de his quatuor di <lb/>co de <reg norm="centummillibus" type="context">centũmillibus</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="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.n.i.</var> <reg norm="idemque" type="simple">idemq́;</reg> <var>.m.s.</var> ex ſecundo termino <var>.f.r.</var> ſubtra-<lb/>hatur, <reg norm="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.o.r</var>. </s> + <s xml:id="s899" xml:space="preserve">Dico proportionem <var>.n.i.</var> ad ſummam 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 ſecundus <var>.f.r.</var> <reg norm="detrahatur" type="simple">detrahat̃</reg>, ex <reg norm="tertioque" type="simple">tertioq́;</reg> ſuperſit <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="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.d.i.</var> ſanè <lb/> + <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> + ſic ſe 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 ſe ſcire poteſt. </s> + <s xml:id="s901" xml:space="preserve">Quare ex <lb/>19. quinti ſic ſe 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> & ſeparando ſic <var>.a.t.</var> ad <var>.<lb/>a.c.</var> (hoc eſt <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 ſic ſe 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 eſt <var>.o.r.</var> ad <var>.m.s.</var> ex .11. quinti. </s> + <s xml:id="s903" xml:space="preserve">Itaque ex communi ſcientia ſic ſe 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 ſit <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 ſit <var>.o.r</var>. </s> + <s xml:id="s905" xml:space="preserve">Iam ſi ſic ſe 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> ſic <lb/>ſe 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 ſic <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 ſic <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 eſt. ut <var>.o.r.</var> ad <var>.m.s.</var> quod erat propoſitum.</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 deſideranti ſummam quorumcunque terminorum progreſſionis conti-<lb/>nuæ geometricæ cognoſcere. </s> + <s xml:id="s907" xml:space="preserve">Rectè minimus terminus ex maximo detrahen <lb/>dus eſt, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> per denominantem progreſſionis dempta vnitate diuidendum, <lb/><reg norm="prouenientique" type="simple">prouenientiq́;</reg> maximum terminum addendum, ex quo oritur ſumma quæſita.</s> + </p> + <p> + <s xml:id="s908" xml:space="preserve">Exempli gratia, ſi darentur quatuor termini continui proportionales .8. 12. 18. <lb/>27. primum hoc eſt minimum .8. ex vltimo .27. detraheremus: </s> + <s xml:id="s909" xml:space="preserve"><reg norm="remaneretque" type="simple">remaneretq́;</reg> .19. qui <lb/>per denominantem progreſſionis, dempta vnitate, diuideretur. </s> + <s xml:id="s910" xml:space="preserve">Quo loco animad <lb/>uertendum eſt, quamlibet <reg norm="denominationem" type="context">denominationẽ</reg> cuiuſcunque proportionis numerorum <lb/>ſupra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eſt, & <lb/>idipſum de ſuperparticularibus, & ſuperpartientibus eſt intelligendum, vt in præ-<lb/>ſenti proportio ſeſquialtera inter duos terminos cogitanda eſt, nempe inter vnum <lb/>& dimidium, atque vnum. </s> + <s xml:id="s911" xml:space="preserve">Seſquitertia autem inter vnum & tertiam partem, <lb/>& vnum. </s> + <s xml:id="s912" xml:space="preserve">Seſquiquinta inter vnum cum quinta parte, & vnum. </s> + <s xml:id="s913" xml:space="preserve">De ſuperpartien <lb/>tibus idem aſſero quod de proportione <reg norm="ſuperbipartiente" type="context">ſuperbipartiẽte</reg> tertias appellata, vt .5. <lb/>ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, ſuperbipartiens quar-<lb/>tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans ſcilicet, ſem <lb/>per ſit 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, ſupererit tantummodo dimi-<lb/>dium, quo diuiſo .19. proueniet .38. qui numerus æqualis erit ſummæ <reg norm="reliquorum" type="context">reliquorũ</reg> <lb/>omnium terminorum, cui coniuncto vltimo termino .27. dabitur ſumma quæſita .65.</s> + </p> + <p> + <s xml:id="s916" xml:space="preserve">Pro cuius ſpeculatione, quatuor termini ſignificentur, 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="reſiduumque" type="simple">reſiduumq́;</reg> ſit <var>.n.i.</var> & ex <lb/>ſecundo <var>.f.r.</var> cuius reſiduum ſit <var>.o.r.</var> proportio verò progreſſionis ea ſit, quæ <var>.g.h.</var> ad <var>.<lb/>y.</var> quo vnitas repræſentatur (ex quo ſic ſe 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> ſuperſit <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> quę diui <lb/>datur in puncto <var>.x.</var> ita vt <var>.n.x.</var> æqualis <lb/>ſit vnitati <var>.y.</var> & in puncto <var>.u.</var> ita. vt <var>.n.<lb/>u.</var> æqualis ſit <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 ſic ſe habeat <var>.<lb/>f.r.</var> ad <var>.m.s.</var> hoc eſt ad <var>.f.o.</var> vt <var>.g.h.</var> ad <var>.y</var> <lb/>hoc eſt ad <var>.g.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> ſic <lb/>ſe 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. eiuſdem <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 eſt ad <var>.m.s.</var> vt <var>.h.</var> ad <var>.g.</var> hoc eſt <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">rectangulũ</reg> <var>.u.z.</var> cogitetur ęquale rectangulo <var>.x.i.</var> cuius <var>.x.i.</var> <lb/>ſuperficialis 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 ſuperficialis <var>.u.z.</var> lineari <var>.n.i.</var> æqualis erit, qui <lb/>quidem numerus in figura rectangu-<lb/>la ſuperficialis cognitandus erit, cum <lb/> + <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> + diuidendus ſit per <var>.h.</var> hoc eſt per <var>.n.u.</var> <lb/>ex quo proueniens ex huiuſmodi di <lb/>uiſione 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. ſepti-<lb/>mi eadem eſt proportio <var>.n.i.</var> ad <var>.n.z.</var> <lb/>quæ <var>.n.u.</var> ad <var>.n.x.</var> hoc eſt <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 ſummæ trium terminorum <var>.c.a</var>: f. num <var>.s.</var> cui coniuncto <lb/>quarto termino <var>.b.i.</var> propoſitum 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">PRopoſuere veteres quæſita nonnulla de itineribus <reg norm="interque" type="simple">interq́;</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 ſin-<lb/>gulis diebus conficiat, alter verò prima die milliare vnum, ſecunda duo, tertia tria, <lb/>atque ita ſingulis diebus milliare addit; </s> + <s xml:id="s928" xml:space="preserve">quærimus quot dierum ſpacio ſocium con <lb/>ſequetur.</s> + </p> + <p> + <s xml:id="s929" xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, ſic ſunt .8. milliaria. <lb/></s> + <s xml:id="s930" xml:space="preserve">ex quo ſemper vnitas detrahicur, quæ in præſenti exemplo erit .7. <reg norm="totque" type="simple">totq́;</reg> dies erunt <lb/>quibus ſocius ſocium conſequetur, & milliarium numerum æqualem abſoluerit.</s> + </p> + <p> + <s xml:id="s931" xml:space="preserve">Cuius rei facilis erit ſpeculatio, ſi ſubſcripta figura diligenter conſideretur, in <lb/>qua primus viator, die prima, quatuor milliaria linea <var>.q.d.</var> ſignificata conficit, at-<lb/>que illa ipſa die alter vnum tantum defignatum per <var>.d.</var> perficit, ita vt primus via-<lb/>tor tribus milliaribus ſocium anteceſſerit, altera verò die fecundus uiator cum duo <lb/>milliaria <reg norm="conficiat" type="context">cõficiat</reg>, excedètur à primo duobus milliaribus tantummodo, quę cum tri-<lb/>bus primæ diei quinque erunt; </s> + <s xml:id="s932" xml:space="preserve">tertia die ijſdem de cauſis primus ſex tantum millia-<lb/>ribus à ſecundo diſtabit, cum verò quarta die tot ſecundus quot primus milliaria <lb/>conficiat, primus à ſecundo amplius quam antea non diſtabit; </s> + <s xml:id="s933" xml:space="preserve">quinta verò cum ſe <lb/>cundus vnum milliare amplius quam primus conficiat. </s> + <s xml:id="s934" xml:space="preserve">propius accedit ad primum <lb/>vno ex ſex milliaribus, quibus anteà diſtabat, tum ſexta cum duobus primum ſupe-<lb/>ret, detrahet ex ſex milliaribus præteritæ diſtantiæ tria, ſeptima tandem illa ſex <lb/>detraxerit. </s> + <s xml:id="s935" xml:space="preserve">In quo conſiderandum eſt ſecundum 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">quarũ</reg> <lb/>duarum figurarum <var>.d.o.p.f.</var> pars <reg norm="communis" type="context">cõmunis</reg> eſſe reperitur, quæ quantitates ſi inuicem <lb/>æquales eſſe debent, neceſſe eſt ſeparatas partes <var>.u.q.n.</var> et <var>.t.i.c.</var> inter ſe æquales eſſe, <lb/>& quoniam quarta die (hoc eſt die ſic diſtante 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 abſoluat vnus <lb/> + <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> + quot alter abſque vlla differentia, quæ ſigni-<lb/>ficetur per <var>.o.s.</var> neceſſe eſtitaque ex communi <lb/>conceptione tot dies eſſe poſt <var>.o.s.</var> quotante-<lb/>ceſſerant, vt exceſſus æqualis ſit defectui, qui <lb/>ſimul collecti, iuncta etiam <var>.o.s.</var> duplum erunt <lb/><var>d.s.</var> dempta vnitate, prout facilè in ſubſcripta <lb/>figura qui ſque per ſe ſcientificè poterit ſpecu <lb/>lari. </s> + <s xml:id="s936" xml:space="preserve">Quamobrem conſultum erit duplicare <lb/>numerum <var>.o.s.</var> & exduplo vnitatem detrahe-<lb/>re, quandoquidem dies ſupra <reg norm="infraque" type="simple">infraq́;</reg> <var>.o.s.</var> cum <lb/>die <var>.o.s.</var> minores ſunt duplo numeri <var>.d.s.</var> aut <var>.o.<lb/>s.</var> (quodidem eſt) 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="ſiſecundus" type="context">ſiſecũdus</reg> viator <reg norm="ordinem" type="context">ordinẽ</reg> <reg norm="ſecundæ" type="context">ſecũdæ</reg> progreſſionis arithmeticæ <reg norm="ſeruansiter" type="context">ſeruãsiter</reg> <lb/>agat, nempe ea quæ ab vno per binarium aſcendit, ſemper numerus dierum <lb/>æqualis erit numero milliarium diurnorum primi viatoris.</s> + </p> + <p> + <s xml:id="s938" xml:space="preserve">In cuius gratiam animaduertendum eſt numerus ne milliarium diurnorum pri-<lb/>mi viatoris par an impar ſit. </s> + <s xml:id="s939" xml:space="preserve">Etenim ſi par eſt, primus viator in fine ſingulorum die-<lb/>rum primæ medietatis numeri omnium dierum ſecundum antecedet numero diſpa <lb/>ri milliarium; </s> + <s xml:id="s940" xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diſpa <lb/>ri præteribitur, vt in ſequenti figura patet. </s> + <s xml:id="s941" xml:space="preserve">Nam prima die, ſecundus ex primo <lb/>milliare vnum ex numero pari, qui à primo conficitur detrahit; </s> + <s xml:id="s942" xml:space="preserve">ſecunda verò die <lb/>idem ſecundus, duo ſubtrahit milliaria exdiſpari, qui primo reliquus fuerat, <reg norm="ſicque" type="simple">ſicq́;</reg> <lb/>perpetuò diſpar remanet vſque ad vnitatem, ad quam cum peruenerint, nempe ad <lb/>illius diei exitum, quo primus ſecundum vnitate tantummodò ſuperat, manifeſtè <lb/>depræhendetur ſubſequente dieſecundum vnitate primum ſuperaturum, altera ve <lb/>rò tribus vnitatibus, prout penultima die ſecundus à primo tribus vnitatibus ſupera <lb/>batur. </s> + <s xml:id="s943" xml:space="preserve">Quare neceſſe erit, tot diebus ſecundum cum primo iter agere, inchoan-<lb/>do ab ea die, qua fecundus primam ſuperabit, quot egerat dum à primo ſuperare-<lb/>tur, vt ex communi conceptione, media figura <var>.A.</var> depræhendi poteſt. </s> + <s xml:id="s944" xml:space="preserve">Quod au-<lb/>tem ſingula dimidia dierum, dimi-<lb/> + <anchor type="figure" xlink:label="fig-0081-02a" xlink:href="fig-0081-02"/> + dia ſint numeri milliarium diurno-<lb/>rum primi; </s> + <s xml:id="s945" xml:space="preserve">patebit exſequenti fi-<lb/>gura, cogitato termino <var>.u.n.</var> vlti-<lb/>mo progreſſionis ſuperatę à primo <lb/>vſque ad vnitatem <var>.e.</var> quiterminus <lb/><var>u.n.</var> coniunctus primo <var>.o.</var> nempe <var>.e.</var> <lb/>ſemper <reg norm="duplum" type="context">duplũ</reg><unsure/> eſt 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 conſtat numero <lb/>milliarium diurnorum primi viatoris, ex quo ſequitur totum numerum dierum, quo <lb/>rum <var>.o.n.</var> dimidium eſt, 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 ſi numerus milliarium primi viatoris diſpar fuerit, ſecundum numero pari <lb/>ſemper ſuperabit, vt facile erit ſequentem figuram conſideranti intelligere, <lb/>ex quo illud ſequetur, futuram quandam diem, qua paria milliaria conficient. </s> + <s xml:id="s948" xml:space="preserve">Sit-<lb/>q́ue illa dies <var>.u.n.</var> ſequitur <reg norm="etiam" type="context">etiã</reg> tranſacta 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 ſuperetur, <reg norm="quanto" type="context">quãto</reg> ſecundum primus <lb/>ſuperauerat, vnde totalis numerus <var>.o.f.</var> mi <lb/>nor erit duplo <var>.o.n.</var> vnitate ex communi <lb/>conceptione, ſed ita etiam ſe habet termi-<lb/>nus <var>.u.n.</var> hoc eſt 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 propoſitum.</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 ſecundi viatoris, non ab vnitate ſed à binario inchoata, <lb/>per binarium quoque aſcenderet, <reg norm="numerusque" type="simple">numerusq́;</reg> milliarium diurnorum primi via <lb/>toris par eſſet, abſque dubio quadam die paria milliaria <reg norm="vterque" type="simple">vterq;</reg> conficeret, quę ſigni <lb/>ficetur <var>.u.n.</var> qua tranſacta, tot diebus vtrique <reg norm="ambulandum" type="context">ambulãdum</reg> erit, quot <reg norm="fuerunt" type="context">fuerũt</reg> <reg norm="dum" type="context">dũ</reg> primus <lb/>ſecundum ſuperaret, vt totidem alijs pri-<lb/>mus à ſecundo ſuperetur, in qua tamen <lb/> + <anchor type="figure" xlink:label="fig-0082-02a" xlink:href="fig-0082-02"/> + progreſſione terminus <var>.u.n.</var> ſemper duplus <lb/>eſt numero terminorum <var>.o.n.</var> ex .95. theo-<lb/>remate, <reg norm="totque" type="simple">totq́;</reg> ſunt infra <var>.u.n.</var> termini vſque <lb/>ad <var>.f.</var> quot ſupra. </s> + <s xml:id="s950" xml:space="preserve">ex quo illud ſequitur 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, ſi 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 ſi in eiuſmodi progreſſione numerus milliarium diurnorum primi viato-<lb/>ris diſpar fuerit, patet quòd primus ſecundum numero diſpari ſuperabit, 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"/> + ciſſimq́;</reg> primum ſecundus, 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, ſit itaque vltima <lb/>dies, qua primus ſecundum vnita <lb/>tem antecedit <var>.u.n.</var> qui terminus <lb/>duplus eſt numero terminorum <lb/><var>o.n.</var> & cum illa die primus ſecun-<lb/>dum milliario antecedat, ſequen + <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 ſcientia neceſſe <lb/>eſt ſecundum tot diebus <reg norm="cum" type="context">cũ</reg> primo iter agere quot ſunt <var>.o.n.</var> qui ſimul æquales erunt <var>.<lb/>u.n.</var> ſed <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/>ſequemur 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 ſecundi viatoris progreſſio per ternarium aſcenderet, ſumpto initio ab <lb/>ipſo ternario, animaduertendum eſt an numerus milliarium diurnorum primi, <lb/>ternario menſuretur necne, etenim ſi menſuretur, tandem aliquando paria millia-<lb/>ria conficient, quæ dies ſit <var>.u.n.</var> </s> + <s xml:id="s955" xml:space="preserve">quare ſub <lb/> + <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> + <var>u.n.</var> totidem quot ſupra termini <reg norm="erunt" type="context">erũt</reg>, <lb/>& <reg norm="cum" type="context">cũ</reg> <var>.o.n.</var> tertia ſit 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 ſumendæ ſint duæ tertiæ partes <var>.u.n.</var> <lb/>ex quibus vnitas detrahatur ſuperſitq́ue <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/>aſcendente ſecundi, patet nullam futuram diem qua pari milliaria conficient, <lb/></s> + <s xml:id="s958" xml:space="preserve">quare illa vltima qua primus ſecundum antecedet, vno aut duobus milliaribus an-<lb/>tecedet in præſenti caſu. </s> + <s xml:id="s959" xml:space="preserve">Antecedat itaque duobus milliaribus, <reg norm="ſitque" type="simple">ſitq́;</reg> dies <var>.u.n.</var> & alte <lb/>ra <var>.t.i.</var> ſecundus primum vno milliari ſuperabit, ita quod ſub <var>.t.i.</var> non poterunt plu-<lb/>res integros dies iter agere, quam ambulauerunt ante diem <var>.u.n.</var> hoc eſt vſquequo <lb/>ſecundtis iunctus ſit primo, qui numerus dierum, tertia parte <var>.o.n.</var> ipſius <var>.u.n.</var> vnitate <lb/>minor erit, cum ex .95. theoremate <var>.o.n.</var> ſit tertia pars <var>.u.n.</var> ex quo numerus <var>.o.f.</var> ter-<lb/>minorum aut dierum intergrorum cognitus erit, qui ſi cum numero alcendente <lb/>cognoſcetur, ſtatim ex .99. theoremate deueniemus in cognitionem vltimi diei in <lb/>tegri <var>.s.f.</var> atque ita etiam totius ſummæ progreſſionis ex .95. theoremate. </s> + <s xml:id="s960" xml:space="preserve">Iam verò <lb/>cognito numero milliarium diurnorum primi, ſimul cum numero terminorum, aut <lb/>dierum conſequenter nouerimus rectanguli ſummam, hoc eſt productum à primo <lb/>viatore formatum, quarum duarum ſummarum in præſenti caſu ſemper ea, quæ <lb/>huiuſmodi producti eſt, maior erit, cum conſtitutum fuerit ſecundum viatorem à <lb/>primo ſuperari ipſa die <var>.u.n.</var> vno milliari amplius quam ſequente die <var>.t.i.</var> primus à ſe <lb/>cundo ſuperatur, tum pari gradu iter egerunt ſub <var>.t.i.</var> quo ſupra <var>.u.n.</var> ambulauerant. <lb/></s> + <s xml:id="s961" xml:space="preserve">Hoc animaduertendo, quòd ſi ſumma progreſſionis maior eſſet rectangulo, ex ea <lb/>ſumma neceſſe eſſet <reg norm="numerum" type="context">numerũ</reg> mil <lb/>liarium vltimi termini in ſumma <lb/> + <anchor type="figure" xlink:label="fig-0083-02a" xlink:href="fig-0083-02"/> + incluſi detrahere, & reſiduo ope-<lb/>rari. </s> + <s xml:id="s962" xml:space="preserve">Nunc verò ſummam pro-<lb/>greſſionis exſumma rectanguli à <lb/>primo viatore facti ſubtrahi de-<lb/>bet, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> ſeruari <reg norm="voceturque" type="simple">voceturq́;</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">primũ</reg> <reg norm="reſiduum" type="context">reſiduũ</reg>. </s> + <s xml:id="s963" xml:space="preserve">Ad hæc <reg norm="numerum" type="context">numerũ</reg> <reg norm="milliariorum" type="context">milliariorũ</reg>, quæ <reg norm="ſecundus" type="context">ſecũdus</reg> viator die <reg norm="ſeqennti" type="context">ſeqẽnti</reg> <var>.s.f.</var> confi <lb/>ciet <reg norm="ſumatur" type="simple">ſumat̃</reg>, ex quo numerus <reg norm="milliarium" type="context">milliariũ</reg> <reg norm="diurnorum" type="context">diurnorũ</reg> primi detrahatur, <reg norm="residuumque" type="context simple">reſiduũq́;</reg> pariter re <lb/>ſeruetur, <reg norm="voceturque" type="simple">voceturq́;</reg> ſecundum reſiduum, poſtmodum numerum milliarium primi <lb/>vnius diei multiplicetur per primum reſiduum ſeruatum, <reg norm="productumque" type="simple">productumq́;</reg> per <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>reſiduum diuidatur <var>.a.c.</var> <reg norm="proueniens" type="context">proueniẽs</reg> (<reg norm="quod" type="simple">ꝙ</reg> erit iter primi in ſequenti die) iungatur reſi-<lb/>ſiduo primo, tot enim erunt milliaria conficienda a ſecundo ſequenti die, vt <lb/>ſeſe conſequantur.</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 ſciamus quantam partem diei <reg norm="ſeqenntis" type="context">ſeqẽntis</reg>, ſingulos itinere agere oporteat, <lb/><reg norm="proueniens" type="context">proueniẽs</reg> per .24. horas multiplicetur (ſuppoſito 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">ꝓductũq́;</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="numerum" type="context">numerũ</reg> milliariorum vuius diei primi viatoris di <lb/>uidatur, ex quo dabitur <reg norm="quantitas" type="context">quãtitas</reg> <reg norm="horarum" type="context">horarũ</reg>, & pars horæ, qua <reg norm="cuique" type="simple">cuiq;</reg> illa die <reg norm="ambulandum" type="context">ambulandũ</reg> <lb/>eſt. </s> + <s xml:id="s965" xml:space="preserve">Idem accideret ſi primum reſiduum reſeruatum cum proueniente in ſummam <lb/>colligeretur, <reg norm="eaque" type="simple">eaq́;</reg> ſumma per .24. horas multiplicaretur, <reg norm="productumque" type="simple">productumq́;</reg> per <reg norm="nume- rum" type="context">nume-<lb/>rũ</reg> milliariorum ſequenti die à ſecundo conficiendorum diuideretur. </s> + <s xml:id="s966" xml:space="preserve">I dipſum quo-<lb/>que eueniret multiplicato primo reſiduo per .24. & producto per ſecundum reſi-<lb/>duum diuiſo.</s> + </p> + <p> + <s xml:id="s967" xml:space="preserve">Exempli gratia, primus viator diurna milliaria vndecim conficit, ſecundus, pri-<lb/>ma die tria, ſecunda .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="ſupereritque" type="simple">ſupereritq́;</reg> .2. </s> + <s xml:id="s968" xml:space="preserve">quare: <var>u.n.</var> ab <var>.e.n.</var> duobus milliaribus ſu-<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">totq́;</reg> diebus ambulandum erit poſt <var>.t.i.</var> hoceſt .6. in vniuerſum inte-<lb/>gris. </s> + <s xml:id="s969" xml:space="preserve">Ad hęc multiplicato <var>.o.f.</var> hoc eſt .6. per <var>.x.o.</var> hoc eſt .3. habebimus <var>.s.f.</var> <reg norm="milliariorum" type="context">milliariorũ</reg> <lb/>18. <reg norm="tum" type="context">tũ</reg> <reg norm="coniuncto" type="context context">cõiũcto</reg> <var>.x.o.</var> primo termino hoc eſt .3. <reg norm="cum" type="context">cũ</reg> <var>.s.f.</var> hoc eſt .18. vltimo termino, habe <lb/>bimus .21. quo multiplicato <reg norm="cum" type="context">cũ</reg> dimidio <var>.o.f.</var> hoc eſt .3. habebimus totam ſummam <lb/>progreſſionis .63. ſex dierum integrorum ex .94. theoremate, tum multiplicato .11. <lb/>nempe numero <reg norm="milliariorum" type="context">milliariorũ</reg> diurnorum primi cum .6. hoc eſt cum <var>.o.f.</var> habebimus pa <lb/>rallelogrammum à primo ſex diebus integris confectum milliariorum .66. ex quo <lb/>detracta .63. ſumma inquam progreſſionis, ſupererit pro primo reſiduo .3. ſumptis <lb/>poſtea milliaribus .21. pro itinere, quod ſecundus die ſequenti <var>.s.f.</var> conficeret, & ex <lb/>ijs detracto numero <reg norm="milliariorum" type="context">milliariorũ</reg> diurnorum primi, nempe .11. ſecundum reſiduum <lb/>erit 10. quod pro diuidenti ſeruabitur. </s> + <s xml:id="s970" xml:space="preserve">Iam multiplicato .11. cum primo reſiduo <num value="3">.<lb/>3.</num> dabitur .33. qui diuiſus per .10. ſecundum reſiduum profert .3. cum tribus decimis, <lb/><reg norm="eritque" type="simple">eritq́</reg> iter à primo viatore ſequenti die conficiendum, hoc etiam ipſum proueniens <lb/>cum primo reſiduo .3. coniunctum, dat .6. cum tribus decimis, quod eſt iter ſecundi <lb/>viatoris illa ſequenti die. </s> + <s xml:id="s971" xml:space="preserve">Ad inueniendam autem quantitatem diei, qua vtrique <lb/>ambulandum eſt, perinde erit multiplicare proueniens .3. & tres decimas per .24. ho <lb/>ras, & productum per .11. dimidium iter primi viatoris partiri, ac multiplicare ſum <lb/>mam .6. & tres decimas cum .24. horis, <reg norm="productumque" type="simple">productumq́</reg> diuidere per .21. hoc eſt periter <lb/>ſecundi viatoris ſequentis diei, vtrinque enim ſemper ſeptem horæcum .12. minu <lb/>tis prouenient. </s> + <s xml:id="s972" xml:space="preserve">Idipſum accidet multiplicato per .24. horas primo reſiduo .3. pro-<lb/><reg norm="ductoque" type="simple">ductoq́;</reg> diuiſo per ſecundum reſiduum .10.</s> + </p> + <p> + <s xml:id="s973" xml:space="preserve">Quarum ſpeculationum gratia, totum iter parallelogrammi primi viatoris die-<lb/>rum integrorum fignificetur linea <var>.n.e.</var> ſumma verò progreſſionis ſecundi linea <var>.f.m.</var> <lb/>parallela <var>.n.e.</var> <reg norm="eritque" type="simple">eritq́;</reg> <var>.f.m.</var> minor <var>.n.e</var>. </s> + <s xml:id="s974" xml:space="preserve">Conſtituamus deinde à termino <var>.f.n.</var> (majoris <lb/><reg norm="intelligentię" type="context">intelligẽtię</reg> gratia) vtranque <reg norm="perpendiculariter" type="context">perpẽdiculariter</reg> duci, <reg norm="producatur" type="simple">ꝓducatur</reg> deinde <var>.n.e.</var> donec <var>.e.<lb/>d.</var> æqualis ſit itineri diurno primi viatoris, item etiam producatur <var>.f.m.</var> donec <var>.m.K.</var> <lb/>æqualis ſit itineri à ſecundo confecto ſequenti 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 poſito. </s> + <s xml:id="s975" xml:space="preserve">Poſtmodum <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 eſt propor-<lb/>tioni <var>.m.e.</var> ad <var>.e.b.</var> ex .2. ſexti, ſed cum <var>.m.k.</var> et <var>.g.k.</var> notæ ſint, pariter cognoſcetur <var>.m.<lb/>g.</var> ſecundum reſiduum, cum etiam notæ ſint <var>.n.e.</var> et <var>.n.a</var>. </s> + <s xml:id="s977" xml:space="preserve">Itaque cognoſcemus <var>.a.e.</var> hoc <lb/>eſt <var>.m.h.</var> cognitis verò <var>.m.g</var>: <var>g.k.</var> et <var>.m.h.</var> ex .15. ſexti aut .20. ſeptimi cognoſcetur <var>.h.<lb/>q.</var> erit igitur <var>.a.e.</var> aut quod idem eſt <var>.m.</var> hprimum reſiduum, et <var>.m.g.</var> ſecundum, et <var>.h.<lb/>q.</var> aut <var>.e.o.</var> proueniens, et <var>.n.o.</var> et <var>.f.q.</var> itinera vtriuſque viatoris inter ſe æqualia. <lb/></s> + <s xml:id="s978" xml:space="preserve">Nec verò prætermittenda eſt ſpeculatio vltimæ rationis inueniendæ quantitatis <lb/>diei, quæ conſtat ope diuiſionis producti <var>.m.h.</var> in .24. per <var>.m.g</var>. </s> + <s xml:id="s979" xml:space="preserve">Ea autem eiuſmodi <lb/>eſt. </s> + <s xml:id="s980" xml:space="preserve">Probatum fuit ſic ſe 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/>ſic ſe 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 ſic ſe habeat <var>.m.h.</var> ad <var>.h.q.</var> vt <var>.m.g.</var> ad <var>.g.k.</var> permutando ſic ſe 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> ſed <lb/><reg norm="cum" type="context">cũ</reg> <var>.m.k.</var> ſit motus toti diei reſpon <lb/>dens, ſecurè dicere poterimus, ſi <lb/><var>m.g.</var> talis eſt reſpectu horarum <num value="24">.<lb/>24.</num> ſignificatarum 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/>ſpondens: </s> + <s xml:id="s983" xml:space="preserve">quæ <lb/><reg norm="poſtmodum" type="context">poſtmodũ</reg> erit <var>.<lb/>m.q.</var> quæ, vt <reg norm="di- ctum" type="context">di-<lb/>ctũ</reg> fuit, talis eſt <lb/>reſpectu <var>.m.k.</var> <lb/>qualis <var>.m.h.</var> re-<lb/>ſpectu <var>.m.g</var>. </s> + <s xml:id="s984" xml:space="preserve">Reli <lb/>quę duæ ſpecula <lb/>tiones priorum <lb/><reg norm="modorum" type="context">modorũ</reg>, vna & <lb/>eadem eſt, <reg norm="facilisque" type="simple">facilisq́;</reg> per ſe mediocriter intelligenti. </s> + <s xml:id="s985" xml:space="preserve">Eodem modo reliquæ omnes <lb/>progreſſiones ſecundi viatoris <reg norm="cum" type="context">cũ</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 ſcilicet viatorem iter incipere diebus aliquot an-<lb/>tè ſecundum, primum tamen lentius, quàm ſecundum ambulare, & utrunque <lb/>eorum certa quædam milliaria conficere. </s> + <s xml:id="s987" xml:space="preserve">Iam ſiſcire voluerimus in quot diebus <lb/>ſeſe conſequentur, uulgaris regula iubet, inſpici quot milliaria primus ſolus 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 ſoli 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, ſi primus octo diebus antequam ſecundus iter arripuiſſet, con-<lb/><reg norm="feciſſetque" type="simple">feciſſetq́;</reg> fingulis diebus .20. milliaria, tum ſecundus .25. quotidie perfeciſſet, mul <lb/>tiplicandus eſſet numerus .8. cum .20. ex quo darentur .160. milliaria à primo ſolo <lb/><reg norm="ambulante" type="context">ambulãte</reg> confecta, quibus diuiſis 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 eſt. </s> + <s xml:id="s990" xml:space="preserve">Sint enim duo rectanguli <var>.a.n.</var> et <var>.u.i.</var> æquales inter <lb/>ſe, quibus motus itinerarium ſignificentur, quorum <var>.a.n.</var> ſit primi, et <var>.u.i.</var> ſecundi, præ <lb/>tereà. <var>a.c.</var> numerum milliarium diurnorum primi .et <var>.u.e.</var> ſecundi, 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/>gnoſcetur. </s> + <s xml:id="s991" xml:space="preserve">Tum <var>.c.o.</var> numerum <reg norm="dierum" type="context">dierũ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> + primi ſoli iter agentis denotet, <reg norm="cumque" type="context simple">cũq;</reg> <lb/>conſtituamus <var>.a.n.</var> æqualem eſſe <var>.u.i.<lb/>o.i.</var> ęqualis erit <var>.o.a.</var> atque <var>.o.a.</var> cogni <lb/>tus ex ſuis producentibus <var>.a.c.</var> et <var>.c.o.</var> <lb/>itaque <var>.o.i.</var> etiam cognitus, qui diui-<lb/>ſus per latus cognitum <var>.o.e.</var> dabit <var>.e.<lb/>i.</var> cognitum numerum ſcilicet dierum, quibus ſecundo ambulandum eſt, vt primum <lb/>conſequatur.</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 ſumpſi ordinem illius operationis, numeris mediantibus, ad <lb/>inueniendam exactam temporis quantitatem, ſeu interuallum, <reg norm="quod" type="simple">ꝙ</reg> tranſit, vel in <lb/>tercedit inter vnam mediocrem coniunctionem & aliam proximam <reg norm="ſequentem" type="context">ſequentẽ</reg> duo <lb/>rum planetarum, vt patet in epiſtola noſtra ad Illuſtrem Bernardum Trottum con-<lb/>tra Benedictum Altauillam repræhenſorem Ephemeridum. </s> + <s xml:id="s993" xml:space="preserve">Verum tamen eſt <reg norm="quod" type="simple">ꝙ</reg> <lb/>cum praxis huiuſmodi theorematis ſit multiplex, viſum fuit vnam proponere, quę <lb/>non ita perſpicua ſit, ſed ſubobſcura, non quòd aliquid voluerim latere illum ami <lb/>cum mihi dilectiſſimum, cui priuatim omnes modos prius oſtenderam, ſed vt cere-<lb/>brum illius mei aduerſarij in laberintum conijcerem inextricabilem vt feci, quam-<lb/>uis modus ille egregius etiam ſit, vt nunc oſtendam.</s> + </p> + <p> + <s xml:id="s994" xml:space="preserve">In dicta epiſtola igitur mente cogitaui medium motum tardioris planetæ, pu-<lb/>ta ſaturni, illius temporis quo velocior planeta, ſcilicet Iupiter, percurrit ſuo medio <lb/>motu totum zodiacum, incipiendo ambo eodem temporis puncto, nec non ab vna <lb/>eorum media coniunctione, hoc eſt ab eodem zodiaci puncto, in quo coniunctę fue <lb/>runt eorum lineæ mediorum motuum, vbi inueni vi regulæ de tribus, quòd Satur <lb/>nus ſpacio dierum vnius mediocris reuolutionis Iouis, qui ſunt .4328. progreditur <lb/>medio motu gra .145. min .4. hoc eſt min .8704. pofito quòd ipſe Saturnus perficiat <lb/>vnam mediam reuolutionem ſpacio 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/>ipſum ſpacio .4328. dierum, certus eram hos dies ſignificatos eſſe à linea <var>.a.u.</var> vel <var>.c.<lb/>o.</var> (æquales enim inuicem ſunt) in figura huiuſmodi theorematis, & quòd rectangu <lb/>lum <var>.a.o.</var> præbebat ſummam graduum .145. min .4. hoc eſt min .8704. et quòd <var>.a.c.</var> <lb/>vel <var>.o.u.</var> ſignificabat iter vnius diei ipſius 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> ſignificari dies .30. & à puncto <var>.x.</var> productam eſſe <var>.x.f.</var> parallelam ipſi <lb/><var>u.o.e.</var> vnde certi erimus rectangulum <var>.e.x.</var> ſignificare iter Ionis ſpacio 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. & ſecundorum .43. et <var>.o.x.</var> minutorum .60. & ſecun. <lb/>20. vt in dicta epiſtola, vnde rectangulum <var>.o.f.</var> erit min .89. & ſecun .23. & quia re-<lb/>ctangulum <var>.o.i.</var> æquale eſt rectangulo <var>.a.o.</var> ergo <var>.o.i.</var> ſimiliter 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. ſecun <var>.o.</var> & tertiarum .40. videlicet tertiarum <num value="7240">.<lb/>7240.</num> ſuppoſito periodo totaliipſius Saturni dierum .10740. et <var>.u.e.</var> erit <reg norm="minutorum" type="context">minutorũ</reg>. <lb/>4. ſecun .59. & ter .27. vel circa hoc eſt tertiarum .17967. vnde <var>.o.e.</var> erit tertiarum <num value="10727">.<lb/>10727.</num> </s> + <s xml:id="s997" xml:space="preserve">Nuncſi dixerimus cum <var>.o.e.</var> tertiarum .10727. dat <var>.o.u.</var> vel <var>.a.c.</var> (nam tam <lb/>vna quam altera eſt tertiarum .7240.) quid dabit <var>.a.u.</var> vel <var>.o.c.</var> (quia tam vna quam <lb/>altera eſt 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/>ſius <var>.a.u.</var> dabunt totam <var>.a.t.</var> <reg norm="partium" type="context">partiũ</reg> .7249. quæ erunt tot dies, hoc eſt periodus quæſita.</s> + </p> + <p> + <s xml:id="s998" xml:space="preserve">Alia methodo ſimiliter poſſumus idem cognoſcere, ſcilicet dicendo ſi rectangu <lb/>lum <var>.f.o.</var> quod eſt minutorum .89. & ſecun .23. hoc eſt ſecundorum .5363. dat rectan <lb/>gulum <var>.o.x.</var> minutorum .60. & ſecun .20. hoc eſt ſecun .3620. quid dabit <var>.a.u.</var> partium <lb/>4328. vnde veniet <var>.u.t.</var> partium .2921. ſimiliter, eo quod eadem proportio eſt rectan <lb/>guli <var>.f.o.</var> ad <var>.o.x.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> ex prima ſexti, vel .18. 19. ſeptimi ſeu .15. quinti.</s> + </p> + <p> + <s xml:id="s999" xml:space="preserve">Poſſet etiam aliquis dicere ſi <var>.f.o.</var> dat <var>.o.x.</var> quid dabit <var>.o.a.</var> vnde veniet <var>.o.t.</var> quo <lb/>diuiſo per <var>.o.u.</var> daret <var>.u.<lb/>t.</var> quia ita ſe 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/>ſupra 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/>ſtola, aliam methodum <lb/>obſeruaui, quæ eſt multi <lb/>plicando minuta .8704. <lb/>per .30. <reg norm="productumque" type="simple">productumq́;</reg> di <lb/>uiſi per min .5363. quaſi <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 ſupradictis propoſitionibus veniet <var>.e.i.</var> & quia permu-<lb/>tando ita ſe habet <var>.o.f.</var> ad <var>.e.f.</var> vt <var>.o.i.</var> ad <var>.e.i.</var> ideo dixi, ſi min .89. cum ſecun .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 ſi duo iter agentes, eodem in-<lb/>ſtanti diuerſis è locis proficiſcantur, ita vt vnus locum vnde alter profectus <lb/>eſt petat, <reg norm="alterque" type="simple">alterq́;</reg> altero velocior ſit, quo loco <reg norm="quanue" type="context">quãue</reg> die ſibi inuicem occurrent.</s> + </p> + <p> + <s xml:id="s1004" xml:space="preserve">Exempli gratia, Patauio profectus quidam Taurinum petit, eodem inſtanti al-<lb/>ter Taurino Patauium, <reg norm="eſtque" type="simple">eſtq́;</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 quiſque confece-<lb/>rit, <reg norm="quotque" type="simple">quotq́;</reg> diebus iter egerit, priuſquam ſibi occurrant.</s> + </p> + <p> + <s xml:id="s1006" xml:space="preserve">Iubent nos veteres dies vtriuſque inuicem inter ſe multiplicare, <reg norm="eritque" type="simple">eritq́;</reg> produ-<lb/>ctum .99. item etiam in ſummam colligere, <reg norm="eritque" type="simple">eritq́;</reg> ſumma .20. per quam <reg norm="productum" type="context">productũ</reg>. <lb/>99.@diuiſerimus dabuntur dies .4. cum .19. vigeſimis vnius diei. </s> + <s xml:id="s1007" xml:space="preserve">At pro milliaribus <lb/>vtriuſque, pro eo qui .11. diebus iter conficit, multiplicatis .400. per .4. et .19. vigeſi <lb/>mis, tum diuiſo 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 quiſque confecerit.</s> + </p> + <p> + <s xml:id="s1008" xml:space="preserve">Dum autem hæc ſpecularer attentius, occurrit alius ſoluendi modus, quamuis pro <lb/>lixior. </s> + <s xml:id="s1009" xml:space="preserve">Is <reg norm="autem" type="wordlist">aũt</reg> eſt eiuſmodi. </s> + <s xml:id="s1010" xml:space="preserve"><reg norm="Accipiatur" type="simple">Accipiat̃</reg> medietas minoris numeri <reg norm="dierum" type="context">dierũ</reg>, <reg norm="nempe" type="context">nẽpe</reg> .4. <reg norm="cum" type="context">cũ</reg> dimi <lb/>dio, & per .400. multiplicetur, <reg norm="productumque" type="context simple">productũq́;</reg> per <reg norm="maiorem" type="context">maiorẽ</reg> numerum diuidemus ſcilicet <lb/>11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio <reg norm="millia- riorum" type="context">millia-<lb/>riorũ</reg> itineris .200. detracto, & <reg norm="preſiduum" type="context">preſiduũ</reg> <reg norm="nempe" type="context">nẽpe</reg> .36. <reg norm="cum" type="context">cũ</reg> .4. vndecimis multiplicato pro <lb/><reg norm="ductoque" type="simple">ductoq́;</reg> diuiſo <reg norm="per" type="simple">ꝑ</reg> <reg norm="ſummam" type="context">ſummã</reg> dimidij itineris .200. <reg norm="cum" type="context">cũ</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">ꝑ</reg> .363. ct .7. vndecimas partes <reg norm="proueniet" type="simple">ꝓueniet</reg> .16. <reg norm="cum" type="context">cũ</reg> .4. vndecimis, quo <reg norm="coniuncto" type="context">cõiuncto</reg> pri <lb/>mo <reg norm="prouenienti" type="simple context">ꝓueniẽti</reg>, primus .180. milliaria <reg norm="confecerit" type="context">cõfecerit</reg>, quæ è .400. detracta ſupererunt .220. <lb/>pro itinere ſecundi, qui .9. diebus iter abſoluit. </s> + <s xml:id="s1011" xml:space="preserve">Ad hæc ſi tempus ſcire velimus <lb/>eius, qui .11. diebus appellit, multiplicabimus .11. cum .180. <reg norm="productumque" type="simple">productumq́;</reg> per .400. <lb/>partiemur, <reg norm="prouenientque" type="simple">prouenientq́;</reg> paulominus, quam quinque dies, nempe .4. cum .22. horis <lb/>et .48. minutis, quod tempus vtrique viatori inſeruiet, quandoquidem idipſum pro <lb/>uenit multiplicato .220. per .9. <reg norm="productoque" type="simple">productoq́;</reg> per .400. diuiſo.</s> + </p> + <p> + <s xml:id="s1012" xml:space="preserve">Huius autem, qui à me pręſcribitur modi, ſpeculatio talis eſt. </s> + <s xml:id="s1013" xml:space="preserve">Duo termini duabus <lb/>rectis lineis æqualibus, & parallelis inter ſe <var>.b.p.</var> et <var>.d.q.</var> ſignificentur, quæ alijs dua-<lb/>bus <var>.b.d.</var> et <var>.q.p.</var> <reg norm="coniungantur" type="simple">coniungant̃</reg>, quę parallelæ & æquales erunt ex .33. primi, quibus ſigni <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/>ſumatur <reg norm="punctum" type="context">punctũ</reg> medium <var>.q.p.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.k.</var> & ab ipſo ad <var>.b.d.</var> ducatur <var>.k.i.</var> parallela <var>.d.q.</var> aut <lb/><var>b.p.</var> quod idem eſt, ex quo <var>.b.i.</var> æqualis erit <var>.p.k.</var> ex .34. primi, hoc eſt <var>.q.k.</var> <reg norm="certique" type="simple">certiq́;</reg> eri-<lb/>mus primum viatorem <var>.q.p.</var> in dimidio itineris <var>.q.k.</var> occurrere non potuiſſe viatori ip <lb/>ſius <var>.b.i.</var> quandoquidem eo tempore, quo is, qui ipſius <var>.q.p.</var> mouetur per <var>.q.k.</var> (cum ſit <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 eſt in <var>.k.</var> ex quo certi erimus eos inter <var>.c.</var> et <var>.i.</var> ſibi in-<lb/>uicem obuiaturos eſſe. </s> + <s xml:id="s1016" xml:space="preserve">Cogito deinde rectam lineam ductam <var>.k.c.</var> & ut ſe habet <var>.i.<lb/>c.</var> ad <var>.c.b.</var> ita cogito ſe 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/>ſtum eſt, lineam <var>.k.c.</var> in puncto <var>.e.</var> interſecabit, à quo cum fuerit ducta <var>.e.o.n.</var> parallela <lb/><var>k.i.</var> habebimus <var>.o.n.</var> ea ſcilicet puncta, quibus occurrunt ſibijpſis, nam cum ſic ſe 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 ſimilitudine manifeſta <lb/>triangulorum, ex æqualitate proportionum ſic ſe 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> ſpatia ſint tempori-<lb/>bus æqualibus confecta, itaque ſpatia <var>.k.n.</var> et <var>.c.o.</var> ex communi ſcientia temporibus <lb/>æqualibus conficientur.</s> + </p> + <p> + <s xml:id="s1017" xml:space="preserve">Quare rectè dicimus, ſi 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> cognoſcitur, quo ex dimidio itineris detracto, remanet <var>.c.i.</var> cogni <lb/>tus, ſed cum probauerimus <var>.q.k.</var> ad <var>.k.n.</var> hoc eſt <var>.i.o.</var> (cum ſint æquales inter ſe, ex .34 <lb/>primi) ita ſe habere. vt <var>.b.c.</var> ad <var>.c.o.</var> permutando ſic ſe 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 ſi ſumma <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/>ſcitur <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 ſi <var>.b.d.</var> tot diebus abſolui <lb/>tur, aut etiam <var>.q.p</var>: quo <var>.b.o.</var> aut <var>.q.n.</var> abſoluetur.</s> + </p> + <p> + <s xml:id="s1022" xml:space="preserve">Vt autem ad ſpeculationem regulæ antiquorum deueniamus, cogitemus pri-<lb/>mum viatorem ipſius <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> abſoluit. </s> + <s xml:id="s1023" xml:space="preserve">Is autem ad <var>.g.</var> pertingat, ex quo eadem pro-<lb/>portio ſpacij <var>.q.g.</var> ad <var>.q.p.</var> hoc eſt <var>.b.d.</var> dabitur, quæ temporis quo <var>.b.d.</var> abſoluitur 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> ſolum, qui per <var>.q.p.</var> mouetur (mo-<lb/>tus enim continui regulares & vniformes conſtituuntur) 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 ſe habere <var>.k.n.</var> ad <var>.c.o.</var> vt <var>.q.k.</var> ad <var>.b.c.</var> itaque ſic ſe 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 ſe habere <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> ex quo <lb/>componendo ſic ſe 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 eſt <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 eſt <var>.b.d.</var> ad <var>.b.o.</var> vt ſumma <reg norm="dierum" type="context">dierũ</reg> vnius & <lb/>alterius viatoris ad <reg norm="minorem" type="context">minorẽ</reg> <reg norm="numerum" type="context">numerũ</reg> <reg norm="dierum" type="context">dierũ</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">tentũ</reg> <reg norm="quando" type="wordlist">qñ</reg> diceremus ſi ſumma dierum, quibus iter agitur à viatoribus talis eſt (20) re-<lb/>ſpectu numeri dierum velocioris(9) qualis & cui <reg norm="reſpondebit" type="context">reſpõdebit</reg> totum ſpacium <var>.b.d</var>? </s> + <s xml:id="s1025" xml:space="preserve">vn-<lb/>de dabitur ſpacium <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 poſtmodum diuidi productum per ſummam omnium <lb/>dierum, rectèid quidem fit. </s> + <s xml:id="s1027" xml:space="preserve">Nam cum ſic ſe habeat <var>.b.d.</var> ad <var>.b.o.</var> vt ſumma omnium <lb/>dierum ad minorem quantitatem dierum velocioris ſcilicet. </s> + <s xml:id="s1028" xml:space="preserve">Ideo temporis propor <lb/>tio à mobili per <var>.b.d.</var> abſumpti ad tempus mobilis per <var>.b.o.</var> eadem erit, quæ ſummæ <lb/>omnium dierum ad namerum dierum velocioris. </s> + <s xml:id="s1029" xml:space="preserve">Quarerectè dicemus, ſi eiuſmodi <lb/>ſumma talem reſpectum habet ad minorem numerum dierum, quem numerum re-<lb/>ſpiciet dies ipſius <var>.b.d</var>? </s> + <s xml:id="s1030" xml:space="preserve">ex quo proferentur dies reſpondentes ipſi <var>.b.o.</var> cætera iam <lb/>dicta fuerunt.</s> + </p> + <p> + <s xml:id="s1031" xml:space="preserve">Huiuſmodi verò ſpeculationis 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> ſit ſubſcripta <lb/>figura pars <reg norm="inquam" type="context">inquã</reg> pręcedentis, in qua <lb/><reg norm="conſtituamus" type="context simple">cõſtituamꝰ</reg> <var>.o.n.</var> <reg norm="locum" type="context">locũ</reg> <reg norm="eum" type="context">eũ</reg> eſſe quo ſibi <lb/>viatores obuient, ex quo ſpacium <var>.q.<lb/>n.</var> à ſuo viatore conficietur, eo ipſo <lb/>tempore, quo à ſuo ſpacium <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. proficiſcitur, & componendo <reg norm="eadem" type="context">eadẽ</reg> <lb/>erit proportio <var>.d.b.</var> ad <var>.b.o.</var> quæ ſum-<lb/>mæ dierum ad minorem numerum ipſorum, & eadem quæ dierum <var>.b.d.</var> ad dies <lb/>ipſius <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 ipſa itinera aliud quæritur peruenuſtè, in quo quæſito illud con<lb/>ſtituitur cognitum eſſe, nempe interuallum inter duo diuerſa loca, è quibus <lb/>duo viatores eodem inſtanti vt ſibi occurrant proficiſcuntur, <reg norm="certaque" type="simple">certaq́;</reg> milliaria ſin-<lb/>gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur <lb/>deinde quoto die ſibi occurrent. </s> + <s xml:id="s1033" xml:space="preserve">Hoc autem fit diuiſo toto interuallo locorum per <lb/>ſummam milliariorum quam vterque quotidie abſoluit.</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, diſtant loca .100. milliaribus à ſe inuicem; </s> + <s xml:id="s1035" xml:space="preserve">vnus autem viator <lb/>ſingulis diebus .15 milliaria, alter .10. conficit ſi ita que .15. cum .10. <reg norm="coniungamus" type="context">coniũgamus</reg>, <lb/>ſumma erit .25. per <reg norm="quam" type="context">quã</reg> diuiſis milliaribus .100. totius interualli proferetur .4. nume <lb/>rus quæſitus dierum quo viatoribus iter agendum erit prius quam ſibi obuient.</s> + </p> + <p> + <s xml:id="s1036" xml:space="preserve">In cuius ſpeculationis gratiam totum iter ſignificetur 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> ſit occurſus ita vt eodem <lb/>tempore, alter ſpacium <var>.a.i.</var> alter <var>.u.i.</var> confecerit, ſpacij autem <var>.a.e.</var> tempus <lb/>per <var>.b.</var> ſignificetur & tempus ſpacij <var>.u.o.</var> per <var>.c.</var> quæ tempora erunt inter ſe <lb/>æqualia, porrò ſpacij <var>.a.i.</var> tempus per <var>.d.</var> & ſpacij <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 ipſius <var>.b.</var> ad iter ipſius <var>.c.</var> quæ <lb/>itineris <var>.d.</var> ad iter ipſius <var>.f.</var> & componendo itinerum ipſius <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"/> + ipſius <var>.f.</var> meritò itaque quęritur ſi 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> ſed <var>.c.</var> <lb/>ſignatum eſt pro vna die, </s> + <s xml:id="s1037" xml:space="preserve">quare in pro <lb/>poſito exemplo <var>.f.</var> ſignificabit 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 ſummæ, rurſus eiuſdem ſummę <lb/>dimidium adiungere, tum quærimus, quoties noueratius totam eam ſummam ingre <lb/>diatur patefactis fractis ſi qui occurrant.</s> + </p> + <p> + <s xml:id="s1039" xml:space="preserve">Exempli gratia, ſi quis cogitaſſet numerum .12. iubebant huic dimidium addi, <lb/>nempe .6. ex quo ſumma erat .18. iubebant, præterea dimidium huius ſummæ nem-<lb/>pe .9. toti ſummæ adiungi, quæ fuiſſet .27. adhæc quærebant ſibi patefieri quoties <num value="9">.<lb/>9.</num> ſummam prædictam ingrederetur, & ſi in prima aut ſecunda diuiſione aut <reg norm="etiam" type="context">etiã</reg> <lb/>vtraque, fracti reperirentur, ac quoties nouem vltimam ſummam ingrediebatur, <lb/>toties .4. multiplicabant. </s> + <s xml:id="s1040" xml:space="preserve">Quod ſi in prima diuiſione fracti erant, vltimo produ-<lb/>cto addebant vnitatem; </s> + <s xml:id="s1041" xml:space="preserve">ſin verò in ſecunda, binarium adiungebant, ex quo exa-<lb/>ctus numerus quæſitus proferebatur.</s> + </p> + <p> + <s xml:id="s1042" xml:space="preserve">Pro cuius rei ratione ſit <var>.a.</var> numerus cogitatione compræhenſus et <var>.e.</var> ipſius <var>.a.</var> cum <lb/>eiuſdem medietate ſumma et <var>.i.</var> ipſius <var>.e.</var> cum eiuſdem medietate itidem ſumma, vn <lb/>de <var>.i.e.a.</var> tres numeri continui proportionales, in ſeſquialtera 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 ęqua <lb/>litatis proportionum ita ſe 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. ipſam <var>.i</var>. </s> + <s xml:id="s1045" xml:space="preserve">Sed <lb/>quia ſępe contingit, vt in ſecunda diuiſione, aut in ambabus etiam diuiſionibus re <lb/>periantur numeri fracti, anima duertendum eſt numerum animo compræhenſum <var>.a.</var> <lb/>ſcilicet aut parem aut imparem ſemper futurum. </s> + <s xml:id="s1046" xml:space="preserve">Si par eſt, aut multiplex erit ad <num value="4">.<lb/>4.</num> aut non. </s> + <s xml:id="s1047" xml:space="preserve">Si priori modo ſe habebit in duabus diuiſionibus, nullus numerus fra-<lb/>ctus admittetur, ſed ſi ad .4. multiplex non erit, à multiplicibus per duo ſemper dif <lb/>feret, & ſi per medium diuidatur, eiuſdem medietas impar ſemper 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 ſumma par nunquam exiſtet, cuius medietatem aliquod medium ſemper <lb/>ingredietur, & hanc ob cauſam poſterior ſumma cum fracto ſemper erit, & nume-<lb/>rum deſumptum maiorem eſſe multiplici ad quatuor per duo ſignificabit.</s> + </p> + <p> + <s xml:id="s1048" xml:space="preserve">At verò ſi inter impares reponatur, aut eorum erit qui ſuperant multiplicem <lb/>ipſius quatuor per vnum, ſeu per tria, quod hinc innoteſcet, nempe, quia ſi eorum <lb/>erit qui dictum multiplicem per vnum tantum vincunt, ſua medietate ipſi numero <lb/>addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior <lb/>ſumma in numerum parem ſemper euadet, vnde in poſteriori ſumma nullus nume-<lb/>rus fractus conſpicietur, & hanc ob <reg norm="causam" type="context">causã</reg> multiplici ipſius .4. vnitas ſemper addetur.</s> + </p> + <p> + <s xml:id="s1049" xml:space="preserve">Sed ſi numerus deſumptus, in ſerie eorum, qui multiplicem ipſius .4. pertria ſu-<lb/>perant, collocabitur, hinc compræhendetur, quia primæ ſummæ numerus cum <lb/>media vnitate ſemper impar erit, vnde ſecunda ſumma præter integras cum me-<lb/>dia vnitate nobis ſemper occur ret.</s> + </p> + <p> + <s xml:id="s1050" xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiſione, & ſecunda numerus <lb/>aliquis fractus conſiſtat, eò tantum nobis inſeruit, quò deueniamus in cognitionem <lb/>an numerus animo conceptus multiplicem ipſius .4. per vnum, per duo, aut tria ſupe <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 poteſt, quin aliquis fra-<lb/>ctus in prima diuiſione, aut in ſecunda ſequatur: </s> + <s xml:id="s1052" xml:space="preserve">vnde à numeris imparibus, qui mul <lb/>tiplicem ipſius .4. unitatis tantum exceſſu <reg norm="ſuperant" type="context">ſuperãt</reg>, poſterior ſumma <reg norm="cum" type="context">cũ</reg> quarta parte <lb/>vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipſius <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 ipſius .4. per duo cum medietate vnitatis præter <lb/>integros ſemper procedit. </s> + <s xml:id="s1054" xml:space="preserve">Ita cum is qui numerum ſecum conſiderat, ſi in nume-<lb/> + <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> + ris fractis verſatus eſſet, qui eum in-<lb/>terrogat prudenter ſe gereret, ſi ſibi <lb/>declarari curaret, quis nam ex fractis <lb/>ſu per integros <reg norm="ſecundæ" type="context">ſecũdæ</reg> <reg norm="summæ" type="context">sũmæ</reg> remane <lb/>ret, quia <reg norm="per" type="simple">ꝑ</reg> quot quarta integros <reg norm="ſecun- dæ" type="context">ſecũ-<lb/>dæ</reg> ſummæ ſuperaret, per <reg norm="totidem" type="context">totidẽ</reg> inte <lb/>gros numerus mente conceptus multiplicem ipſius .4. ſuperaret.</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 ſi ali quis quemuis numerum animo compræhendat, eique <lb/>numero alium etiam quemlibet numerum propoſitum addat, & à tertia par <lb/>te huius ſummæ tertiam partem numeri imaginati detrah et, reſiduum ſecundi nu-<lb/>meri adiuncti, ideſt propoſiti, tertia pars erit.</s> + </p> + <p> + <s xml:id="s1056" xml:space="preserve">Vt exempli gratia, ſi aliquis de numero denario cogitaſſet, <reg norm="huicque" type="simple">huicq́;</reg> .24. adderet, <lb/>vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-<lb/>rij cogitatione concepti, ideſt .3. cum tertia parte vnius, à tertia parte huius ſum mæ <lb/>ideſt ab vndecim & vna tertia parte remanerent .8. ideſt tertia pars numeri additi. <lb/></s> + <s xml:id="s1057" xml:space="preserve">Id quod mihi inter iocos in honeſtorum 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 eſt <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> ſit tertia pars ipſius <var>.a.b.</var> prioris numeri im aginati, et. b <lb/>c. tertia pars ipſius, <var>b.d.</var> ſecundi numeri propoſiti, vnde coniunctum vnius harum ter <lb/>tiarum <reg norm="partium" type="context">partiũ</reg> <reg norm="cum" type="context">cũ</reg> alia ſit <var>.e.c.</var> quod quidem <var>.e.c.</var> eſſe tertiam partem ſummæ duorum <lb/>primorum ideſt <var>.a.d.</var> aſſero. </s> + <s xml:id="s1059" xml:space="preserve">Iam manifeſtum eſt ipſius <var>.d.b.</var> ad <var>.b.c.</var> eſſe quemadmo <lb/>dum ipſius <var>.a.b.</var> ad <var>.b.e.</var> vnde viciſſim ipſius <var>.d.b.</var> ad <var>.b.a.</var> erit quemadmodum ipſius <var>.b.<lb/>c.</var> ad <var>.b.e.</var> & coniunctim ipſius <var>.d.a.</var> ad <var>.a.b.</var> quemadmodum ipſius <var>.c.e.</var> ad <var>.e.b.</var> & viciſ-<lb/>ſim ipſius <var>.d.a.</var> ad <var>.c.e.</var> quemadmodum ipſius <var>.b.a.</var> ad <var>.b.e.</var> ſed proportio ipſius <var>.b.a.</var> ad <var>.<lb/>b.e.</var> eſt tripla, ergo ea quæ eſt ipſius <var>.a.d.</var> ad <var>.e.c.</var> erit quoque tripla; </s> + <s xml:id="s1060" xml:space="preserve">vnde ſumendo <var>.e.<lb/>c.</var> pro tertia parte ipſius <var>.a.d.</var> & ab ipſa <var>.e.c.</var> ſubtrahendo tertiam partem ipſius <var>.a.b.</var> <lb/>tertia pars ipſius <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, ſupponendo <var>.e.c.</var> tertiam partem ipſius <var>.a.d.</var> et <var>.e.b.</var> ipſius <var>.a.<lb/>b.</var> exiſter. </s> + <s xml:id="s1062" xml:space="preserve">Dico <var>.b.c.</var> tertiam partem ipſius <var>.b.d.</var> futuram:</s> + <s xml:id="s1063" xml:space="preserve">quia ſi totius <var>.a.d.</var> ad totum <lb/><var>e.c.</var> ita ſe habet, quemadmodum <var>.a.b.</var> à toto <var>.a.d.</var> diffecti atque diuulſi ad <var>.e.b.</var> à toto <var>.<lb/>e.c.</var> diſractum, 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> reſidui <var>.b.d.</var> totius <var>.a.d.</var> ad reſiduum <var>.b.c.</var> <lb/>totius <var>.e.c.</var> erit, vt totius <var>.a.d.</var> ad <reg norm="totum" type="context">totũ</reg> <var>.e.c.</var> at-<lb/>que hic quidem modus rem <reg norm="propoſitam" type="context">propoſitã</reg> ſpe-<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 reſolu-<lb/>ta, proptereà quòd <reg norm="non" type="context">nõ</reg> ſemper nobis ſuccurrit 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 ſecunda dupla ſit primę, & adhuc eam ſuperet tribus vnitatibus, tertia ve <lb/>rò æqualis ſit aggregato primæ cum ſecunda, & amplius ipſum aggregatum ſuperet <lb/>quinque vnitatibus.</s> + </p> + <p> + <s xml:id="s1066" xml:space="preserve">Ad hoc autem quæſitum ſoluendum antiqui vtebantur regula falſi, quod reuera <lb/>breuiori modo poteſt ſolui, videlicet detra hendo illud ſecundum exceſſum, quin-<lb/>que ſcilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eſt .22. cum dimidia <lb/>vnitate, ſi addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-<lb/>tia parte quæſita ipſius numeri .50. deinde ſi 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 eſt .6. cum dimidia vnitate, prima pars, ex tri-<lb/>bus quæſita erit, quæ quidem ſi detraxerimus ex .19. cum dimidia vnitate, reli-<lb/>quum erit .13. cui <reg norm="cum" type="context">cũ</reg> additus fuerit primus exceſſus ideſt .3. </s> + <s xml:id="s1067" xml:space="preserve">Iam propoſitum re-<lb/>ſultabit nobis .16. pro ſecunda parte quæſita.</s> + </p> + <p> + <s xml:id="s1068" xml:space="preserve">Ratio verò huiuſmodi operationis talis eſt, ſit verbi gratia totalis numerus pro-<lb/>poſitus ſignificatus per lineam <var>.a.b.</var> cuius ſecundæ partis numerus datus ſignificetur <lb/>per lineam <var>.g.</var> & numerus tertiæ partis propoſitus 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 diuiſa in pun <lb/>cto <var>.e.</var> & ipſi <var>.e.f.</var> addita <var>.f.b.</var> tota <var>.e.b.</var> nobis cognita erit, quæ quidem tertia pars <lb/>quæſita ipſius <var>.a.b.</var> erit, proptereà quòd <var>.a.e.</var> (quæ æqualis eſt ipſi <var>.e.f.</var>) erit ſumma <lb/>primæ, & ſecundæ partis. </s> + <s xml:id="s1070" xml:space="preserve">Detrahatur poſteà. g. ex <var>.e.a.</var> & remanebit <var>.d.a.</var> cuius ter <lb/>tia pars ſit <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 ſecundam 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> compoſita eſt ex <var>.<lb/>e.f.</var> æquali <var>.a.e.</var> hoc eſt æquali compoſi-<lb/>to ex prima, & ſe 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 ſint horæ, alius verò reſpondit tot eſſe, quot duæ tertiæ præteriti <lb/>temporis ſimul iuncta cum tribus quintis futuri temporis totius dieri naturalis effi-<lb/>ciunt. </s> + <s xml:id="s1074" xml:space="preserve">Nunc quæritur quot ſint horę.</s> + </p> + <p> + <s xml:id="s1075" xml:space="preserve">Antiqui, hoc etiam problema ſoluebant mediante regula falſi, ſed mihi alio mo <lb/>do ſoluendum 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">cũ</reg> quin-<lb/>que futuri temporis, vnde veniunt .14. vnitates, ex regula poftea de tribus ita dico <lb/>ſi ex .14. mihi prouenit .9. quid reſultabit ex .24. & prouenient mihi horæ .15. cum <lb/>tribus ſeptimis vnius horæ, hoc eſt minuta ferè .26.</s> + </p> + <p> + <s xml:id="s1077" xml:space="preserve">Pro cuius ratione, quinque vnitates, feu partes temporis futuri ſignificentur à <lb/>linea <var>.e.u.</var> quarum trium ſigniſicentur a linea <var>.e.i.</var> ſumpta deinde ſit linea <var>.e.o.</var> æqualis <lb/>lineæ <var>.e.i.</var> et <var>.e.a.</var> tripla ſit ad <var>.o.e.</var> vel ad <var>.e.i.</var> quod idem eſt, vnde <var>.a.e.</var> compoſita erit <lb/>ex <var>.a.o.</var> (hoc eſt ex duabus tertijs ip ſius <var>.a.e.</var>) & ex <var>o.e.</var> (hoc eſt ex. tribus quintis ip-<lb/>ſius <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. ſeptimi 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 ſocios nummos habere, quorum ſumma primi & <lb/>ſecundi cognita ſit, item ſumma primi & tertij cognita & ſumma ſecundi & <lb/>tertij item cognita, at que ex huiuſmodi tribus aggregatis veniunt in cognitionem <lb/>particularem vniuſcuiuſque illorum.</s> + </p> + <p> + <s xml:id="s1080" xml:space="preserve">Gemafriſius ſoluit hoc problema ex regula ſalſi. </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, ſumma primi cum ſecundo .50. & ſecundi cum tertio .70. & primi <lb/>cum tertio .60. harum trium ſummarum accipiantur duæ quæuis, vt puta .50. & .70 <lb/>quæ coniunctæ ſimul dabunt .120. à qua ſumma detrahatur reliqua, ideſt .60. & <lb/>reſtabit nobis .60. cuius medietas erit .30. hoc eſt numerus nummorum ſecundi <lb/>ſocij quo numero detracto à .70. hoc eſt à ſumma ſecundi cum tertio remanebit .40. <lb/>hoc eſt numerus tertij ſocij, & hic numerus deſumptus à .60. reſiduus erit nume-<lb/>rus primi ſocij.</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 conſideremus triangulum hic ſubnotatum <var>.a.b.c.</var> cuius <lb/>unumquodque latus ſignificet ſummam duorum ſociorum, vtputa latus <var>.a.b.</var> ſignifi-<lb/>cet ſummam primi cum ſecundo, latus verò <var>.b.c.</var> ſummam ſecundi cum tertio, la-<lb/>rus autem <var>.a.c.</var> ſummam primi cum tertio, et <var>.a.e.</var> ſeu <var>.a.o.</var> ſit numerus primi ſocij, et <var>.<lb/>e.b.</var> vel <var>.b.u.</var> ſit ſecundi ſocij, et <var>.c.u.</var> ſeu <var>.c.o.</var> ſit tertij, cum autem <var>.a.e.</var> æqualis ſit <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 ſuppoſito ſi <reg norm="dempta" type="context">dẽpta</reg> fuerit ſumma ſeu <lb/>latus <var>.a.c.</var> datum ex aggregato laterum <var>.<lb/>a.b.</var> cum <var>.b.c.</var> reliquarum ſummarum, 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> ſiue <var>.b.u.</var> nobis cognita erit, qua <lb/>detracta exſumma <var>.b.a.</var> relinquetur no <lb/>bis cognitus numerus <var>.a.e.</var> detracto ve-<lb/>ro numero <var>.a.e.</var> hoc eſt <var>.a.o.</var> ex <var>.a.c.</var> ſum-<lb/>ma, ſeu latus, aut <var>.b.u.</var> ex <var>.b.c.</var> remanebit <lb/><var>o.c.</var> ſeu <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="ſolum" type="context">ſolũ</reg> de tribus ſocijs, ſed <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"/> + ſint ſex ſocij <var>.a.b.c.d.e.f.</var> quorum ſumma per binos co-<lb/>gnita, vtputà ſumma numeri <var>.a.</var> cum <var>.b.</var> cognita nobis ſit, <lb/>& ſumma numeri <var>.b.</var> cum <var>.c.</var> & ſumma <var>.c.</var> cum <var>.d.</var> & ſum-<lb/>ma <var>.d.</var> cum <var>.e.</var> & ſumma <var>.e.</var> cum <var>.f.</var> neceſle eft etiam ſcire <lb/>ſummam duorum vno relicto, vtputa ſummam <var>.a.</var> cum <lb/>c. vt poſſimus triangulum <var>.a.b.c.</var> conſtituere. </s> + <s xml:id="s1086" xml:space="preserve">Vnde ex <lb/>præmiffa, cognitus numerus nobis erit vniuſcuiuſque <var>.a.<lb/>b.c</var>. </s> + <s xml:id="s1087" xml:space="preserve">Quapropter dempto numero <var>.c.</var> ex ſumma <var>.c.</var> cum <lb/>d. & numero <var>.d.</var> ex ſumma <var>.d.</var> cum <var>.e.</var> & numero <var>.e.</var> ex ſum <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 ſcribit Vitruuius, proderet, con-<lb/>templarer, mihi etiam viſum eſt, vt aliquem modum ſcientiſicum inueſtigarem, quo <lb/>proportio auri ad argentum, quod in aliquo propoſito corpore exipſis miſto cogni <lb/>ti ponderis cognoſci poſſet. </s> + <s xml:id="s1089" xml:space="preserve">Et cum multos diuerſis temporibus excogitarim offi-<lb/>cio meo deeſſe nolui in ijſdem literarum monumentis mandandis, quorum hic <lb/>vnus erit: </s> + <s xml:id="s1090" xml:space="preserve">propoſita nobis ſint tria corpora <var>.A.M.V.</var> æqualia inter ſe, ſed diuer-<lb/>ſarum ſpecierum materiei, vtputa quod <var>.A.</var> ſit argenteum, & omogeneum <var>.V.</var> ve-<lb/>rò aureum omogeneum, & M. mixtum exauro, & argento, ideſt heterogeneum, <lb/>cupimusergo ſcire <reg norm="iuſtam" type="context">iuſtã</reg> quantitatem auri & argenti, quæ eſt in ipſo corpore <var>.M.</var> <lb/>miſto. </s> + <s xml:id="s1091" xml:space="preserve">Ita igitur faciamus. </s> + <s xml:id="s1092" xml:space="preserve">Videamus primum quantum ſit pondus vniuſcuiuſque <lb/>ipſorum 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> miſti <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 ſeruan-<lb/>da, dematur etiam pondus <var>.M.</var> ex pondere <var>.V.</var> reliquum erit .18. pro ſecunda diffe-<lb/>rentia, etiam ſeruanda, multiplicetur poſteà pondus <var>.A.</var> per ſecundam 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 ipſius corpo-<lb/>ris miſti <var>.M.</var> quo etiam detracto ex pondere totali ipſius <var>.M.</var> reliquum erit quanti-<lb/>tas auri eius corporis, hoc eſt .180.</s> + </p> + <p> + <s xml:id="s1095" xml:space="preserve">In cuius operationis ſpeculatione, aliquid natura ſua prius cognitum præcedere <lb/>oportet hoc eſt, quod omnia corpora omogenea eandem proportionem obtinent <lb/>inter quantitates, quam inter pondera. </s> + <s xml:id="s1096" xml:space="preserve">Quo ſuppoſito 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>: ſed <var>.e.o.</var> ſignificet par-<lb/>tem argenti, et <var>.o.u.</var> partem auri in corpore miſto <var>.M.</var> vnde ex communi conceptu <lb/>habebimus <var>.o.e.</var> æqualem <var>.u.c.</var> cum ex hypotheſi <var>.e.u.</var> æqualis ſit <var>.o.c.</var> et <var>.a.o.</var> ſimiliter. <lb/></s> + <s xml:id="s1097" xml:space="preserve">Significetur poſteà 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> ſit diffe-<lb/>rentia, qua <var>.f.g.</var> maior eſt .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 eſt <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> ſeu <lb/><var>f.g.</var> (quodidem eſt) 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/>ſeparando 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 eſt, vt differentia in <lb/>ter <var>.f.</var> g, et <var>.f</var>: cognita nobis eſt etiam <var>.f</var>: cognoſcimus itidem <var>.d.</var> vt differentiam inter <var>.<lb/>x.b.d.</var> et <var>.b.x.</var> quapropter cognoſcemus <var>.b.</var> ex .20. ſeptimi Eucli. & ſic <var>.x.</var> reſiduum. <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 propoſiti deuenire poſſumus in cognitio-<lb/>nem veræ quantitatis auri, & argenti confuſi in corona Hieronis conſtituen-<lb/>do primum duo corpora ſimplicia æqualia inter ſe, & coronæ hoc modo videlicet, <lb/>immergendo coronam, ſeu corpus miſtum in aliquod vas aqua plenum, & diligen-<lb/>ter colligere aquam, quæ ex eo effundetur, poſteà verò oportet, aliud vas inuenire <lb/>præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-<lb/>ſteà tantum argenti, quantum ſieri poteſt, vnde vnumquodque horum duorum cor <lb/>porum ſimplicium æquale erit mixto, ſeu coronæ, & ſic quod dictum eſt 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 propoſito di-<lb/>ctum eſt, quædam theoremata præmittenda ſunt, videlicet quòd quotíeſcunque <lb/>fuerint tria corpora, quorum duo inuicem æqualia ſint in quantitate, ſed diuerſa- + <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 ſpecierum materiæ, tertium verò corpus maius, vel minus ſit in quantitate vtro-<lb/>que illorum, ſed eiuſdem 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 cuiuſuis æqualium.</s> + </p> + <p> + <s xml:id="s1104" xml:space="preserve">Exempli gratia, ſit <var>.b.</var> corpus aliquod aureum æquale corpori <var>.u.</var> argenteo, ſit <lb/>etiam corpus <var>.a.</var> argenteum maius corpore <var>.b.</var> vel <var>.u.</var> ſed ponderis eiuſdem, 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 eſt 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 ſit, 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. eiuſdem 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 eſt 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">QVotieſcunque nobis propoſita fuerint duo corpora cuiuſuis magnitudinis æ-<lb/>que ponderantia, ſed diuerſarum ſpecierum materiæ, cum ſcire volueri-<lb/>mus proportionem ponderum illarum ſpecierum inter ipſas hoc modo faciemus.</s> + </p> + <p> + <s xml:id="s1108" xml:space="preserve">Sint exempli gratia, duo nobis propoſita corpora <var>.a.</var> et <var>.b.</var> (vt dictum eſt) quæ ſi <lb/>fuerint æqualium magnitudinum inter ſe, clarum erit quod quæritur, ſed inæqua-<lb/>lia erunt, immergatur <reg norm="unumquodque" type="simple">unumquodq;</reg> eorum in vas aqua plenum, & collecta ſit aqua <lb/>effuſa ab vnoquoque illorum, </s> + <s xml:id="s1109" xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iſtarum aquarum æqualis magnitudi-<lb/>nis erit ſui corporis impellentis, & proportio ponderoſitatis illarum eadem erit, <lb/>quæ earum magnitudinum ex omogeneitate, quapropter ſi vnamquamque illarum <lb/>ponderabimus, habebimus propoſitum 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 ſcire voluerimus pondus alicuius magnitudinis aquæ æqualis alicui <lb/>corpori ponderoſo, breuiſſimus modus erit ponderando ipſum corpus tam in ae-<lb/>re, quàm in aqua, & quia ſemper leuius erit in aqua, </s> + <s xml:id="s1111" xml:space="preserve">tunc differentia ponderum ip-<lb/>ſius corporis, erit pondus quæſitum, hoc eſt vnius corporis aquei æqualis magnitu-<lb/>dinis magnitudini corporis propoſiti ex <ref id="ref-0014">.7. propoſitione lib. Archimedis de inſi-<lb/>dentibus aquæ</ref>. </s> + </p> + <p> + <s xml:id="s1112" xml:space="preserve">Quare ex præmiſſis quotieſcunque immerſa fuerint in aquam dicti vaſis duo cor <lb/>pora æquè ponderantia, ſed diuerſarum ſpecierum, vt dictum eſt, proportio pon-<lb/>deris aquæ maioris ad pondus aquæ minoris magnitudinis eadem ſemper erit, quæ <lb/>ponderis minoris corporis ad pondus alicuius corporis eidem æqualis, ſpeciei verò <lb/>maioris, vel eadem proportio ponderis alicuius corporis æqualis maiori, ſpeciei ve <lb/>rò minoris ad pondus ipſius maioris.</s> + </p> + <p> + <s xml:id="s1113" xml:space="preserve">Vt puta ſit 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">aqueũ</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 eſt, 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> eſt vt <lb/>ponderis <var>.a.</var> ad pondus <var>.u.</var> ex corporum omogenei-<lb/>tate, & ex æqualitate magnitudinum corporum antecedentium & conſequentium.</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 quotieſcumque fuerint duo corpora aquea, quo-<lb/>rum vnum æqualis magnitudinis ſit alicui miſto, quod quidem miſtum graue <lb/>ſit tam in aere, quàm in aqua, alterum verò corpus aquem æqualis ſit magnitudi-<lb/>nis alicui corpoli ſimplici, quod quidem corpus ſimplex æqualis ponderis ſit dicto <lb/>corpori miſto. </s> + <s xml:id="s1117" xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni <lb/>tudini corporis miſti, ad pondus corporis aquei, cuius magnitudo æqualis eſt ma-<lb/>gnitudini corporis ſimplicis, eadem erit, quæ proportio ponderis alicuius corpo-<lb/>ris ſimplicis, cuius magnitudo æqualis ſit magnitudini corporis miſti ſuperius dicti, <lb/>ſed ſpeciei corporis ſimplicis iam dicti, ad pondus dicti miſti.</s> + </p> + <p> + <s xml:id="s1118" xml:space="preserve">Exempli gratia, ſit corpus aqueum <var>.e.</var> magnitudinis æqualis corpori <var>.m.</var> mixto, <lb/>corpus verò aqueum <var>.i.</var> æqualis magnitudinis ſit corpori ſimplici <var>.a.</var> quod quidem <lb/>corpus <var>.a.</var> æqualis ponderis ſit cum corpore <var>.m.</var> & corpus <var>.u.</var> ſit æqualis magnitudinis <lb/>cum corpore <var>.m.</var> ſed ſpeciei 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 eſt, quin <lb/>eadem proportio ſit magnitudinis <var>.e.</var> ad magnitudinem <var>.i.</var> quæ magnitudinis <var>.m.</var> ad <lb/>a. ſed <var>.m.</var> ad <var>.a.</var> eſt vt <var>.u.</var> ad <var>.a.</var> ex .7. quinti </s> + <s xml:id="s1120" xml:space="preserve">quare <lb/>ex .11. eiuſdem 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 ipſius magnitudinibus loquendo, ſed propor-<lb/>tio ponderis <var>.u.</var> ad pondus <var>.a.</var> eadem eſt, 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 eſt quæ ponderis <var>.u.</var> ad pondus <var>.a.</var> ex .7. quinti, ergò ex .11. <lb/>eiuſdem 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 eſt propoſitum.</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 cognoſcendam proportionem duarum diuerſarum ſpecierum in <lb/>corpore miſto propoſito, tribus corporibus aqueis mediantibus, quæ <reg norm="quidem" type="context">quidẽ</reg> <lb/>corpora æqualium magnitudinum ſint alijs tribus corporibus vnius & eiuſdem pon <lb/>deris, quorum vnum ſit mixtum, reliqua verò duo ſimplicia, ſed ſpecierum 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/>ſitquinque librarum, quorum vnum ſit aureum, aliud argenteum, reliquum verò <lb/>mixtum ex ijs duobus metallis, vnde corpus aureum ſimplex minus erit, & argen <lb/>teum maius corpore mixto, quod nulli dubium eſt, ſit nunc pondus corporis aquei <lb/>ęqualis corpori aureo, <reg norm="librarum" type="context">librarũ</reg> .3. aquei verò ęqualis miſto, ſit <reg norm="librarum" type="context">librarũ</reg> 3. <reg norm="cum" type="context">cũ</reg> quarta par <lb/>te, aquei demum æqualisargenteo, librarum .4. cum dimidia, vnde exijs, quæ in præ <lb/>cedenti theoremate, & in .126. theoremate diximus, ſi imaginatione concipiemus <lb/>alia duo corpora ſimplicia, auri, & argenti, ſed æqualium magnitudinum mixto, <lb/>habebimus proportionem ponderis aurei ad pondus corporis mixti vt <reg norm="trium" type="context">triũ</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 eſt aurei ad mixtum, vt .13. ad .12. & mixti ad argenteum, vt .18. ad .13. <lb/>& aurei ad argenteum, vt .3. ad .2. ideſt, vt .18. ad .12.</s> + </p> + <p> + <s xml:id="s1126" xml:space="preserve">Nunc inueniantur duo numeri ita inter ſe proportionati, vt .3. ad .2. habentes ta-<lb/>men inter ipſos numerum ita proportionatum ad maximum, vt .12. ſe habet ad <num value="13">.<lb/>13.</num> & ita proportionatum ad minimum, vt ſe habet .18. ad .13. quod hoc modo in-<lb/>ueniemus, multiplicabimus .18. per .12. & proueniet nobis .216. pro numero me-<lb/>dio, poſteà multiplicabimus .18. per .13. & proueniet .234. pro maximo, <reg norm="demum" type="context">demũ</reg> multi <lb/>plicando .12. per .13. proueniet .156. pro minimo, ita quod .234. correſpondebit <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, ſi ordinem theo <lb/>rematis .122. ſequemur, 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. ſed quia ſuppoſitum fuit corpus mixtum eſſe quinque librarum, propte-<lb/>reà dicemus. </s> + <s xml:id="s1129" xml:space="preserve">Si .216. hoc eſt toti corpori mixto correſpondent quinque libræ tunc <lb/>parti .180. hoc eſt auro in ipſo corpore mixto, correſpondent libræ .4. cum duabus <lb/>vncijs, ex regula detribus, reſiduum verò quinque librarum, ideſt vnciæ decem, <lb/>correſpondent parti .36. hoc eſt argento in dicto corpore mixto.</s> + </p> + <p> + <s xml:id="s1130" xml:space="preserve">Sed ſi 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">cũ</reg> pon <lb/>dus mixti fuiſſet .120. librarum, </s> + <s xml:id="s1132" xml:space="preserve">tunc aurum ipſius 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 ſpeculatione nil aliud oportet nunc dicere cum ſatis dictum à no <lb/>bis ſuperius fuerit, vno excepto, hoc eſt rationem reddere, qua motus fui ad inue <lb/>niendos illos .3. numeros ita inter ſe diſpoſitos, vt dictum eſt, quæ quidem ratio fuit, <lb/>vt haberemus .3. numeros ita inter ipſos ordinatè diſpoſitos, vt ſunt 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, reſpectu extremorum, vnde medius numerus in <lb/>ſuo vero ſitu inter .18. et .12. fuiſſent .16. <reg norm="cum" type="context">cũ</reg> .8. tertijs decimis, ſed vt <reg norm="fractorum" type="context">fractorũ</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 eſt .18. ad .12. ex .18. ſepti <lb/>mi, iuſſi etiam multiplicari .18. per .12. vt nobis prodiret .216. ad quem numerum, <lb/>numerus .234. ita ſe haberet, ut .13. ad .12. ex .19. ſeptimi, quod autem ita ſit 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. maniteſtum eſt exijſdem, 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">ſupponitur obſidio alicuius loci, vbi <lb/>alimento ad nutriendos .10000. homines ſufficiunt pro quinque menſibus tan-<lb/>tum, ſed quia eum locum obſidione non liberari putatur niſi .18. menſibus exactis, <lb/>quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eſt .18. <lb/>menſibus.</s> + </p> + <p> + <s xml:id="s1137" xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eſt hominum .10000. cum <lb/>ſecundo, hoc eſt menſium quinque, productum verò diuidatur per .18. hoc eſt men-<lb/>ſium, </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 eſt hæc, ſint exempli gratia duo hic ſubſcripta producta <lb/>ſuperficialia <var>.a.n.</var> et <var>.o.u.</var> inuicem æqualia, ſed tal@ figura delineata, vt proportio <var>.u.<lb/>x.</var> ad <var>.x.o.</var> ſit, vt .10000. ad quinque, & proportio <var>a.x.</var> ad <var>.x.o.</var> ſit vt .18. ad quinque, <lb/>ct <var>.x.n.</var> ſit nobis ignota, quæ quidem eſt illa, quæ indagatur, ita <reg norm="quod" type="simple">ꝙ</reg> vnumquodque <lb/>iſtorum productorum ſignificabit alimentum, et <var>.u.x.</var> ſignificabit numerum homi-<lb/>num .10000. qui quidem homines comederent totum alimentum <var>.u.o.</var> ſpacio tem-<lb/>poris <var>.x.o.</var> quinque menſium, proptereà quòd <var>u.o.</var> ſupponitur productum eſſe ab <var>.<lb/>u.x.</var> in <var>.x.o</var>. </s> + <s xml:id="s1140" xml:space="preserve">Deinde <reg norm="ſupponendo" type="context">ſupponẽdo</reg> <var>.a.x.</var> tem <lb/> + <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> + pus eſſe .18. menſium, ergo <var>.x.n.</var> ſignifi-<lb/>cabit numerum hominum, qui eo tem-<lb/>poris ſpacio ali poſſunt, hoc eſt <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. ſexti, ſeu ex, 20. <lb/>ſeptimi proportio <var>.x.u.</var> ad <var>.x.n.</var> <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quę <var>.a.x.</var> ad <var>.x.o.</var> quapropter rectè factum <lb/>erit accipere <reg norm="productum" type="context">productũ</reg> <var>.u.o.</var> quodidem <lb/>eſt in quantitate, quod productum .2. n. & ipſum 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">QVotieſcunque nobis propoſitum fuerit inuenire tertium terminum, trium ter <lb/>minorum continuè proportionalium armonicæ proportionalitatis, quo-<lb/>tum duo nobis cogniti ſint, 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 ſint 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 ſumma <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 eſt 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. ſeptimi, proptereà quòd proportio <var>.q.o.</var> ad <var>.o.p.</var> hoc eſt ad <var>.<lb/>d.p.</var> eſt vt <var>.a.g.</var> ad <var>.g.n.</var> coniunctim cum diſiunctim it a ſit <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> (ideſt ad <var>.e.c.</var>) ita ſe <reg norm="hent" type="context">hẽt</reg> ut <var>.p.o.</var> (hoc eſt <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 ſi 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 ſi <var>.e.c.</var> tertius terminus nobis propoſitus eſſet ſimul cum <var>.a.g.</var> medio, & volue <lb/>rimus maiorem inuenire <var>.q.p.</var> ſcilicet, oportebit <var>.e.c.</var> ex <var>.a.g.</var> detrahere, differentiam <lb/>verò <var>.n.g.</var> ſimiliter 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 reſiduo <var>.c.t.</var> me-<lb/>diante diuidemus productum, <reg norm="quod" type="simple">ꝙ</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">ꝙ</reg> <lb/><reg norm="productum" type="context">productũ</reg> quod ſit ex <var>.e.t.</var> in <var>.d.p.</var> <lb/>æquale eſt producto quòd fit ex <var>.a.g.</var> in <var>.t.c.</var> per 20. ſeptimi Eucli. eo quòd <var>.a.g.</var> (id-<lb/>eſt <var>.q.d.</var>) ad <var>.d.p.</var> eſt ut <var>.e.t.</var> ad <var>.t.c.</var> diſiunctim, cum coniunctim ita ſit <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> eſt vt <var>.d.p.</var> ad <var>.t.c.</var> hoc eſt 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">Propoſiti enim cum nobis fue <lb/>rint duo termini <var>.c.e.</var> minimus et <var>.g.a.</var> medius, maximus verò quærendus ſit, de <lb/>trahatur differentia <var>.g.n.</var> ex <var>.e.c.</var> & per reſiduum <var>.e.t.</var> diuidatur productum <reg norm="quod" type="simple">ꝙ</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. ſeptimi 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">ꝙ</reg> naſcitur ex <var>.p.d.</var> (hoc eſt <var>.a.g.</var>) in <var>.e.c.</var> æquale eric producto <var>.e.t.</var> in <var>.q.p.</var> qua-<lb/>propter ſi diuiſerimus 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">cũ</reg> nobis propoſiti fuerint duo termini <var>.q.p.</var> maximus, et <var>.a.g.</var> medius, ſi <reg norm="mini- mum" type="context">mini-<lb/>mũ</reg> <var>.e.c.</var> <reg norm="voluerimus" type="simple">voluerimꝰ</reg> inuenire. </s> + <s xml:id="s1152" xml:space="preserve">Termino <var>.q.p.</var> maximo, <reg norm="iungatur" type="context simple">iũgat̃</reg>. <var>p.o.</var> ęqualis, <var>p.d.</var> <reg norm="differentię" type="context">differẽtię</reg> <lb/>propoſitæ, diuidatur poſtea productum <reg norm="quod" type="simple">ꝙ</reg> ex <var>.q.p.</var> in <var>.a.g.</var> generatur per <var>.q.o.</var> prouen <lb/>tus autem ſit <var>.e.c.</var> qui quidem erit terminus quæſitus.</s> + </p> + <p> + <s xml:id="s1153" xml:space="preserve">Cuius operationis ſpeculutio hæc erit, ſupponatur terminum <var>.e.c.</var> inuentum eſſe <lb/>vnde <var>.n.g.</var> differentia ſit 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">productũ</reg> quòd fit ex <var>.a.g.</var> <lb/>in <var>.q.p.</var> (ex .20. ſeptimi) æquale erit <lb/>producto <var>.q.o.</var> in <var>.a.n</var>. </s> + <s xml:id="s1155" xml:space="preserve">Quare ſi diuiſum 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 ſit inuenire tertium <lb/>terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com <lb/>municantibus in tali proportionalitate, hoc eſt harmonica. </s> + <s xml:id="s1157" xml:space="preserve">Vthoc oſtendatur.</s> + </p> + <p> + <s xml:id="s1158" xml:space="preserve">Sint duo termini dati <var>.a.o.</var> et <var>.a.e.</var> inter ſe communicantes, tertius verò inuentus <lb/>ſit <var>.a.c.</var> qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem <lb/>eſſe cum primis datis.</s> + </p> + <p> + <s xml:id="s1159" xml:space="preserve">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-<lb/>dem proportionem eſſe <var>.a.c.</var> ad <var>.a.o.</var> quæ eſt <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. eiuſdem <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/>eſt propoſitum.</s> + </p> + <p> + <s xml:id="s1161" xml:space="preserve">Sed ſi datus fuerit maximus <var>.a.c.</var> cum medio <var>.a.e.</var> interſe 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 ſuppoſito). <var>a.c.</var> commu-<lb/>nicat cum <var>.e.c.</var> hoc eſt 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">erũt</reg> <lb/> + <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> + inter ſe 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> ſimiliter.</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 ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide <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, reſiduum verò <var>.q.x.</var> multiplicetur per <var>.b.r.</var> productum poſteà 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. ſeptimi <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 abſque aliqua <lb/>diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet. <lb/></s> + <s xml:id="s1170" xml:space="preserve">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram <reg norm="proportionem" type="context">proportionẽ</reg> inuenian-<lb/>tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc <lb/>eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-<lb/>dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum + <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 eſt quod fit ex quinque in .3. quod erit .15. </s> + <s xml:id="s1171" xml:space="preserve">Vt <lb/>autem medium terminum harmonicum inter iſtos habeamus, accipiatur <reg norm="duplum" type="context">duplũ</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 ſpeculatio eſt iſta: </s> + <s xml:id="s1173" xml:space="preserve">ſignificentur duo termini datæ proportionis ab <var>.q.b.</var> <lb/>et <var>.b.r.</var> quorum ſumma erit <var>.q.r.</var> cuius quadratum ſit <var>.q.o.</var> ſit 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">Sitq́;</reg> <var>.b.x.</var> æqualis <var>.b.r.</var> et <var>.q.u.</var> ſimiliter, & 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 huiuſmodi proportionalitatis.</s> + </p> + <p> + <s xml:id="s1178" xml:space="preserve">Primum ergo certi ſcimus ex prima ſexti vel .18. ſeptimi eandem exiſtere pro-<lb/>portionem <var>.q.e.</var> ad <var>.b.o.</var> ſeu ad <var>.b.y.</var> quæ <var>.q.b.</var> ad <var>.b.r</var>: ſed <var>.u.y.</var> ad <var>.u.x.</var> eſt vt <var>.y.l.</var> ad <var>.l.x.</var> <lb/>hoc eſt vt <var>.q.b.</var> ad <var>.b.r.</var> ideſt vt <var>.q.e.</var> ad <var>.b.o.</var> & ſumma <var>.u.y.</var> cum <var>.u.x.</var> ideſt <var>.q.y.</var> minor eſt <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 eſt <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 ſpeculatione <reg norm="præcedentis" type="context">præcedẽtis</reg> theo <lb/>rematis, ſequitur <var>.u.y.</var> eſſe differentiam inter <reg norm="maximum" type="context">maximũ</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 eſt <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 ſunt 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">coniunctũ</reg> cum <var>.x.u.</var> medius <lb/>terminus erit, qui quidem (vt dictum eſt) duplus eſt 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 eſt. </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 vnuſquiſque dictorum extremorum, <lb/>deinde multiplicentur ipſi extremi interſe, vnde <lb/>habebimus tria producta eadem proportione inui <lb/>cem exiſtentia, 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 propoſitos 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 ideſt maius eſſet <lb/>7. cum dimidia vnitate, reliquum verò eſſet <lb/>quinque, productum poſteà quod ex ipſis extremis <lb/>prouenit, erit .6. quod quidem eſt harmonicè collo <lb/>catum inter .7. cum dimidia vnitate, & quinque.</s> + </p> + <p> + <s xml:id="s1186" xml:space="preserve">Cuius rei ſpeculatio 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"/> + propoſiti <var>.a.e.</var> maior, et <var>.e.o.</var> minor, <reg norm="Sitque" type="simple">Sitq́;</reg> <var>.o.k.</var> medius arithmeticus inter dictos, vn-<lb/>de clarè patebit <var>.o.k.</var> eſſe dimidium ſummæ 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> ſit <reg norm="productum" type="context">productũ</reg> <lb/>quod fit ex <var>.e.o.</var> in <var>.o.k.</var> et <var>.n.m.</var> ſit productum quod ſit ex <var>.a.e.</var> in <var>.e.o.</var> quorum vnum-<lb/>quodque erit dimid ium vniuſcuiuſque producti præcedentis theorematis, <lb/>ex .18. et .19. ſeptimi Eucli. vnumquodque ſui relatiui. </s> + <s xml:id="s1189" xml:space="preserve">Quare argumentando per <lb/>mutando à concluſionibus 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">harmonicũ</reg> inuenire cum quis voluesit inter duos <lb/>propoſitos terminos, ita faciendum erit, hoc eſt per ſummam datorum ex <lb/>tremorum diuidatur productum quod fit ex minimo termino in <reg norm="differentiam" type="context">differẽtiam</reg> dato-<lb/>rum, prouentus poſtea erit differentia inter maximum & med<unsure/>um quæſitum.</s> + </p> + <p> + <s xml:id="s1191" xml:space="preserve">Vt exempli gratia, ſi nobis propoſiti fuerint hi duo termini .3. et .2. ſumma eo-<lb/>rum erit quinque, per quam cum diuiſerimus productum, quod naſcitur ex mini-<lb/>mo .2. in differentiam eorum, quæ eſt vnum, quod quidem erit .2. </s> + <s xml:id="s1192" xml:space="preserve">tunc duæ quintæ <lb/>partes prouenient, quæ ſi demptæ fuerint ex maximo termino, reliquum erit .2. <reg norm="cum" type="context">cũ</reg> <lb/>3. quintis, hoc eſt 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 propoſitos, in-<lb/>ter quos deſideremus inuenire <var>.o.s.</var> medium ita illis <reg norm="relatum" type="context">relatũ</reg>, vt proportio exceſſus ip-<lb/>ſius ſupra <var>.x.c.</var> (qui ſit <var>.e.n.</var>) ad exceſ-<lb/>ſum <var>.u.d.</var> ſupra <var>.o.s.</var> (qui ſit <var>.n.d.</var>) ea-<lb/> + <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> + dem ſit 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> & hæcſumma 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. ſed quia <var>.d.b</var>: <var>u.b.</var> et <var>.e.d.</var> quantitates no-<lb/>bis cognitę ſunt, ideò <var>.d.n.</var> ex .20. ſeptimi 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 diuerſis vnius <lb/>anni temporibus, </s> + <s xml:id="s1197" xml:space="preserve">tunc in fine anni ſumma torius lucri datur cognita, ſed quæ-<lb/>ritur quantuni<unsure/> vnicuique illorum exipſa ſumma debeatur.</s> + </p> + <p> + <s xml:id="s1198" xml:space="preserve">Exempli gratia, primus in principio anni poſuit .100. aurcos, ſecundus verò .100 <lb/>diebus poſt primum poſuit .50. aureos tertius autem .200. diebus poſt primum po-<lb/>ſuit .25. aureos ſumma lucri poſtea in fine anni fuit aureorum .60.</s> + </p> + <p> + <s xml:id="s1199" xml:space="preserve">Nunc vt ſciamus quantum huius ſummæ vniduique illorum proueniat, præcipit <lb/>regula, vt faciamus tria producta, quorum primum ſit ex numero dierum totius an-<lb/>ni in numerum aureorum primi, vnde tale productum in præſenti caſu erit .36500. <lb/>ſecundum verò ſit ex numero dierum à primo die in quo ipſe ſecundus poſuit uſque <lb/>ad finem anni, in numerum ipſorum nummorum, quod erit .13250. tertium autem <lb/>productum ex diebus tertij in numerum ſuorum aureorum, quod <reg norm="quidem" type="context">quidẽ</reg> erit .4125. <lb/>quæ producta ſimul 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"/> + ipſorum prochictorum per ſummam lucri hoc eſt per .60. vnde multiplicatio primi <lb/>producti erit .2190000. multiplicatio verò ſecundi producti erit .795000. tertij po <lb/>ſtca erit .247500. quarum multiplicationum vnaquæque diuidatur per ſummam <lb/>53875. productori<unsure/>t, & proueniet ex prima diuiſione .40. <reg norm="cum" type="context">cũ</reg> fractis .35000. vnius in-<lb/>tegri diuiſi in partes .53875. quod erit lucrum primi, prouentus autem ſecundæ di-<lb/>uiſionis erit .14. cum fractis .41050. vnius integri diuiſi in partes .53875. lucrum <reg norm="ſecu di." type="context">ſecũ<lb/>di</reg> </s> + <s xml:id="s1200" xml:space="preserve">prouentus verò quartæ diuiſionis erit .4. cum fractis .32000. vnius integri, vt ſu <lb/>pra diuiſi in partes .53875. hoc eſt lucrum tertij.</s> + </p> + <p> + <s xml:id="s1201" xml:space="preserve">Cuius rei ſpeculatio ex ſe in ſub ſcripta figura patet, vbi <var>.a.q.</var> ſignificat numerum <lb/>dierum totius anni pro primo mercatore <var>.q.n.</var> autem ſignificat numerum dierum ſe <lb/>cundi mercatoris <var>.e.q.</var> poſteà ſignificat numerum dierum tertij ſit etiam <var>.s.a.</var> pro nu-<lb/>mero denariorum primi, et <var>.o.n.</var> pro numero ſecundi, et <var>.e.t.</var> pro numero <lb/>tertij, productum autem <var>.q.s.</var> ſignificet valorem primi lucri, et <var>.q.o.</var> ſecundi, <lb/>et <var>.q.t.</var> tertij <var>.x.y.</var> autem ſignificet ſummam lucri omnium, et <var>.x.i.</var> ſignificet <lb/>partem primi, et <var>.i.p.</var> ſecundi, et <var>.p.y.</var> tertij. </s> + <s xml:id="s1202" xml:space="preserve">vnde clarè patebit ex communi <lb/>ſcientia 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 ſi aliquis dice-<lb/>ret, ſi ex dicto aggregato, prouenit <lb/><var>x.y.</var> quid proueniet vnicuique <reg norm="illo- rum" type="context">illo-<lb/>rũ</reg> <reg norm="productorum" type="context">productorũ</reg>. </s> + <s xml:id="s1204" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ſi numerus dena-<lb/>riorum <reg norm="ſecundi" type="context">ſecũdi</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">lucrũ</reg> <lb/>ſignificaretur à rectangulo <var>.q.b.</var> & ita <lb/>de tertio dico <reg norm="quod" type="simple">ꝙ</reg> ſignificaretur à <reg norm="re- ctangulo" type="context">re-<lb/>ctãgulo</reg> <var>.q.c.</var> vel ſi ſi<unsure/>antibus <reg norm="ijſdem" type="context">ijſdẽ</reg> <reg norm="denariorum" type="context">denariorũ</reg> quantitatibus <var>.n.o.</var> et <var>.e.t.</var> omnes ſuas pe-<lb/>cunias eodem tempore poſuiſſent, </s> + <s xml:id="s1206" xml:space="preserve">tunc rectangula ſignificantia eorum lucra eſlent <lb/><var>q.s.q.d.</var> et <var>.q.f.</var> ſed cum nec eodem tempore, nec eandem quantitatem poſueruntr<unsure/>e <lb/>ctè eorum lucra ſignificantur à rectangulis <var>.q.s.q.o.</var> et <var>.q.t.</var> <reg norm="quod" type="simple">ꝙ</reg> ex prima .6. vel .18. aut <num value="19">.<lb/>19.</num> ſeptimi 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 propoſiti trianguli, cuius la-<lb/>r<unsure/>erum proportio nobis data ſit ſimul cum area ſuperſiciali ipſius trianguli, ſed quia <lb/>ipſe Tartalea vtiturregula algebræ, mihi viſum eſt breuiori methodo hoc idein fa <lb/>cere, & etiam vniuerſaliori via.</s> + </p> + <p> + <s xml:id="s1208" xml:space="preserve">Supp onamus igitur duo triangula, quorum vnum <var>.u.n.i.</var> ſit nobis <reg norm="propoſitum" type="context">propoſitũ</reg>, & <lb/>cognitæ ſuperficiei, proportiones ſimiliter laterum <var>.i.n.</var> ad <var>.n.u</var>: et <var>.u.n.</var> ad <var>.u.i.</var> ſint no <lb/>bis datæ, <reg norm="alterum" type="context">alterũ</reg> verò <reg norm="triangulum" type="context">triangulũ</reg> ſit <var>.a.o.u.</var> à nobis tamen ita <reg norm="confectum" type="context">confectũ</reg>, v<unsure/>latera ſint in<lb/>er ſe proportionata eodem modo, quo latera prioris trianguli, ſed hæc nobis <reg norm="etiam" type="context">etiã</reg> <lb/>cognita ſint, <reg norm="quod" type="simple">ꝙ</reg> facillimum eſt. </s> + <s xml:id="s1209" xml:space="preserve">Nunc vero ſi <reg norm="demptum" type="context">demptũ</reg> fuerit <reg norm="quadratum" type="context">quadratũ</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="ſupponendo" type="context">ſupponẽdo</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> conſequenter 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 eſt, ideo cognoſcemus <var>.e.u.</var> ſed <reg norm="cum" type="context">cũ</reg> <var>.e.u.</var> minor ſit <var>.a.u.</var> ex .18. & penultima primi, <lb/>ſi <reg norm="demptum" type="context">demptũ</reg> fuerit quadratum <var>.e.u.</var> ex quadrato <var>.a.u.</var> remanebit nobis <reg norm="cognitum" type="context">cognitũ</reg> <reg norm="quadra- tum" type="context">quadra-<lb/>tũ</reg> <var>.a.e.</var> & ſic nota erit nobis perpendicularis <var>.a.e.</var> ex penultima primi, quæ quidem <var>.<lb/>a.e.</var> ſi multiplicata fuerit in dimidium <var>.o.u.</var> dabit nobis <reg norm="ſuperficiem" type="context">ſuperficiẽ</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 ſimi <lb/>litudinem) eſt vt quadrati <var>.o.u.</var> ad quadratum <var>.n.i.</var> ex communi ſcientia cum vna-<lb/>quæque iſtarum proportionum dupla ſit proportioni <var>.o.u.</var> ad <var>.n.i.</var> ex .17. et .18. ſexti, <lb/>deinde cum nobis cognitæ ſint tres iſtarum quatuor quantitatum hoc eſt ſuperficies <lb/>trianguli <var>.a.o.u.</var> ſuperficies 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/>cognoſcemus etiam quadratum <var>.n.i.</var> & ſic <var>.n.i.</var> latus primi trianguli, vnde reliqua la <lb/>tera illicò nobis innoteſcent exipſa regula de tribus, cum dixerimus, ſi <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/>ſcilicet inuenire <var>.x.</var> quantitatem me-<lb/> + <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> + diam proportionalem inter duas ſu-<lb/>perficies triangulorum, vnde ſuper-<lb/>ficies trianguli <var>.i.a.u.o.</var> ad <var>.x.</var> ſe ha-<lb/>beret ut <var>.o.u.</var> ad <var>.i.n.</var> & ita ex regula <lb/>detribus cognoſcemus <var>.i.n</var>. </s> + <s xml:id="s1214" xml:space="preserve">Multo <reg norm="tem" type="context">tẽ</reg> <lb/>pore poſtquàm hoc theorema conſtruxi, ipſum conſcriptum inueni in decimo <lb/>ſecundi libri Ioannis de monte Regio, ſatis tamen obſcurè 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 algebrę <lb/>quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia, <lb/>ſint tertio longiora per diame-<lb/> + <anchor type="figure" xlink:label="fig-0105-02a" xlink:href="fig-0105-02"/> + trum circuli inſcriptibilis in ip-<lb/>ſo triangulo. </s> + <s xml:id="s1216" xml:space="preserve">ſed hoc breuius <lb/>geometricè poteſt <reg norm="demonſtrari" type="context">demõſtrari</reg>, <lb/>quemadmodum in ſubſcripta <lb/>hic figura videre eſt, proptereà <lb/>quòd cum anguli <var>.A.o.u.</var> et <var>.n.</var> <lb/>omnes ſint recti et <var>.A.u.</var> æqualis <lb/><var>o.n.</var> et <var>.A.n.</var> ęqualis <var>.u.o.</var> ipſæ <var>.A.<lb/>u.</var> et <var>.A.n.</var> æquales erunt diame-<lb/>tro ipſius circuli. </s> + <s xml:id="s1217" xml:space="preserve">Sed eædem <var>.<lb/>A.u.</var> et <var>.A.n.</var> ſunt ſuperfluum, quo <var>.A.B.</var> et <var>.A.C.</var> ſunt maiores <var>.B.C.</var> cum <var>.B.u.</var> et <var>.C.n.</var> <lb/>ſint 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 ſecundi libri nouæ ſcientiæ poterat ipſe Tartalea breuio <lb/>ri methodo abſque vlla operatione ipſius Algebræ inuenire <var>.A.H.</var> reſpectu <var>.A.<lb/>E.</var> eſſe vt .4. <reg norm="cum" type="context">cũ</reg> vno ſeptimo ad <reg norm="vnum" type="context">vnũ</reg>. </s> + <s xml:id="s1219" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ipſe ſupponit <var>.A.E.</var> <reg norm="decimam" type="context">decimã</reg> <reg norm="partem" type="context">partẽ</reg> eſſe ipſius + <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. ideſt .14. cum vno ſepti-<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 ſeptimis. </s> + <s xml:id="s1222" xml:space="preserve">ſed <var>.L.O.</var> ex ſuppoſi-<lb/>to erit .20. eo quòd <var>.L.I.</var> ęquatur ipſi <var>.A.I.</var> ſimiliter et <var>.I.O.</var> vt ipſe 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 ipſi <var>.L.O.</var> ex .<lb/>35. tertij ipſius Eucli. partium .8. <reg norm="cum" type="context">cũ</reg> duabus ſeptimis. cuius <reg norm="dimidium" type="context">dimidiũ</reg> hoc eſt <var>.A.H.</var> erit <lb/>4. cum una ſeptima, quod eſt propoſitum. </s> + <s xml:id="s1224" xml:space="preserve">Reſpice figuram ipſius 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">QVadrageſimum nonum quæſitum ſimiliter poſſumus alio modo ſoluere, vt <lb/>putà cum vnumquodque latus rhombi ſimul cum area cognitum, ſeu datum <lb/>nobis ſit <reg norm="cognitum" type="context">cognitũ</reg> ſimiliter nobis erit quadratum lateris <var>.a.d.</var> hoc eſt ſumma 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/>ſit totalis ſuperficies rhombi, cognita etiam nobis erit eius medietas, hoc eſt produ-<lb/>ctum <var>.o.d.</var> in <var>.o.a.</var> vnde ex methodo .37. Theorematis cognoſcemus .a<unsure/><var>.o.</var> et <var>.o.d.</var> & ſic <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 huiuſmodi eſt. </s> + <s xml:id="s1228" xml:space="preserve">Aliquis habet dolium mero plenum, ex quo <lb/>duas vrnas extrahit ipſius vini, ſed loco ipſius vini infundit duas vrnas aquæ. </s> + <s xml:id="s1229" xml:space="preserve">Dein <lb/>de poſt aliquot dies extrahit iterum alias duas vrnas illius miſti, & iterum infundit <lb/>duas vrnas aquæ, & poſt 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 ipſius Tartaleæ bona eſt, cum ſupponat illas quatuor quantitates vini eſſe <lb/>inuicem continuas proportionales, vt putà primò totum vinum merum, poſteà re-<lb/>ſiduum pro ſecunda quantitate, deinde pro tertia in ſecunda, & pro quarta in ter-<lb/>tia extractione, hoc eſt quòd proportio totius vini meri ad vinum in prima ſit, vt hu <lb/>ius ad vinum in ſecunda, & vt huius ad vinum in tertia miſtione. </s> + <s xml:id="s1232" xml:space="preserve">Sed quia ipſe <lb/>non probat hanc continuam proportionalitatem ex methodo ſcientifica, mihi <reg norm="visum" type="context">visũ</reg> <lb/>eſt hoc loco illam deſcribere.</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 ſupponamus quamlibet partem huius miſti omogeneam eſſe ſuo <lb/>toto, quapropter ſequetur 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 diuiſam, vt proportio <var>.i.e.</var> ad <var>.i.o.</var> eadem ſit 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 miſti hoc eſt primæ miſtionis, vnde cum eadem pro <lb/>portio ſit <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 eſt <var>.i.o.u.</var> (proptereà quòd <var>.i.o.</var> æqualis eſt <var>.a.e.</var> <lb/>vt reſidua 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/>ſto, & poſteà infuſa aqua vſque ad plenitudinem dolij, proportio ingredientium <lb/>huius ſecundi miſti erit ea, quæ eſt inter <var>.o.u.</var> et <var>.o.a.</var> eo quòd in prima miſtione pro-<lb/>proportio ingredientium erat ea, quæ eſt inter <var>.o.u.</var> et <var>.a.e.</var> vel inter <var>.a.e.</var> et <var>.o.u.</var> <lb/>vt demonſtrauimus. </s> + <s xml:id="s1239" xml:space="preserve">Accipiamus ergo <var>.t.m.</var> huiuſmodi ſecundi mifti, magnitudi-<lb/>nis <var>.a.i.</var> vel <var>.e.o.</var> ſignificantis duas vrnas, & permutemus eum in tantam aquam, <lb/><reg norm="ſitque" type="simple">ſitq́;</reg> punctum <var>.o.</var> quod nobis diuidat <var>t.m.</var> in <var>.o.m.</var> et, <var>o.t.</var> partes ſimplices, tali propor <lb/>tione inuicem relatas, vt ſunt <var>.o.u.</var> et <var>.o.a.</var> vnde habebimus ex ſupradictis rationibus <lb/>eandem proportionem ipſius <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 eſt <var>.i.m.u.</var> (eo quod cum <var>.t.m.</var> æqualis ſit <var>.a.i.</var> per conſequens <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> ſed 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 ſupra diximus. </s> + <s xml:id="s1240" xml:space="preserve">Ergo proportio <var>.i.m.u.</var> ad <var>.m.u.</var> erit dupla <lb/>ſimiliter proportioni <var>.i.o.u.</var> ad <var>.o.<lb/>u.</var> quapropter <var>.o.u.</var> erit media pro<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 eſt 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/>eſt vt intellectus acquieſcat, 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 eſt, ſed pulchrum etiam vide-<lb/>tur quærere proportionem ingredientium in ultima miſtione, cum cognita fue <lb/>rit nobis proportio continentiæ dolij ad capacitatis vrnæ ſimul <reg norm="cum" type="context">cũ</reg> numero vitium <lb/>extractionum & impletionum.</s> + </p> + <p> + <s xml:id="s1243" xml:space="preserve">Exempli gratia, ſi proportio <var>.a.u.</var> ad <var>.a.i.</var> cognita nobis fuerit, cognoſcemus etiam <lb/><var>e.i.</var> ex regula de tribus & per conſequens etiam <var>.i.o.</var> reſiduum ex <var>.e.o.</var> & ſimiliter ag-<lb/>gregatum <var>.a.i.</var> cum <var>.i.o.</var> & ſic <var>.o.u.</var> reſiduum totius, et <var>.o.t.</var> ſimiliter, eo quòd <var>.a.u.</var> ad <var>.a.<lb/>o.</var> eſt ut <var>.t.m.</var> ad <var>.o.t.</var> vnde cognoſcemus etiam <var>.o.m.</var> vt reſiduum <var>.t.m.</var> & ſimiliter ag-<lb/>gregatum <var>.a.o.</var> cum <var>.o.m.</var> hoc eſt <var>.a.m.</var> & etiam <var>.m.u.</var> reſiduum totius.</s> + </p> + <p> + <s xml:id="s1244" xml:space="preserve">Cognoſcere autem proportionem totius dolij ad vrnam, vel ècontrà, cum cogni <lb/>ta nobis fuerit proportio ingredientium in vltima miſtione ſimul cum numero vi-<lb/>tium extractionum, & repletionum, quod ſcribit Tartalea, hoc etiam modo <lb/>poſſumus.</s> + </p> + <p> + <s xml:id="s1245" xml:space="preserve">Exempli gratia, ſi proportio <var>.m.u.</var> ad <var>.m.a.</var> cognita nobis fuerit, illicò ſcie-<lb/>mus proportionem <var>.a.u.</var> ad <var>.m.u.</var> & cum ſciuerimus numerum vitium extractionum, <lb/>& impletionum illicò cognoſci-<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 eſt <var>.a.u.</var> ad <var>.i.u.</var> & ſimiliter ea, quæ eſt <var>.a.u.</var> ad <var>.a.i.</var> & è conuerſo ſimiliter.</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 ipſum, vt dictum eſt. </s> + <s xml:id="s1247" xml:space="preserve">Nunc verò velim ſcire proportio-<lb/>nem vini ad a quam hac vltima vice. </s> + <s xml:id="s1248" xml:space="preserve">Nunc igitur ſi procedemus iuxta doctrinam <lb/>primi exempli huius theorematis, obtinebimus quod quærebamus.</s> + </p> + <p> + <s xml:id="s1249" xml:space="preserve">Sed ſi diceret iuxta Tartaleæ quæſitum, hoc eſt 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="ſupradictum" type="context">ſupradictũ</reg> <lb/>eſt, ita vt vltima vice proportio vini ad aquam ſit ſeſquialtera. </s> + <s xml:id="s1250" xml:space="preserve">Tunc ſi iuxta mo-<lb/>dum ſecundi exempli huius theorematis procedemus habebimus quod cupimus.</s> + </p> + <p> + <s xml:id="s1251" xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eſt, habeo <reg norm="dolium" type="context">doliũ</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 ſcire quoties me hoc facere oporteat, ita vt poſtrema vi-<lb/>ce vinum ſe habeat ad aquam in proportione ſeſquialtera, vnde multoties accidet <lb/>vltimam extractionem, & impletionem mutilatam, ſeu imperfectam, euadere.</s> + </p> + <p> + <s xml:id="s1254" xml:space="preserve">Exempli gratia, ſi proportio vini ad aquam in vltima miſtione 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 ſupra dicto <lb/>proportio <var>.a.m.</var> ad <var>.m.u.</var> nobis in-<lb/>noteſceret, & proportio <var>.n.a.</var> ad <var>.<lb/>n.u.</var> nobis data eſt ſimul cum <reg norm="quan titate" type="context">quã<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 ſimiliter, etiam, et <var>.t.n.</var> reſiduum vaſis, quo metimur, vnde <lb/>neceſſe erit, quo<unsure/>d vltima vice vas contineret ſolum <var>.t.n.</var> reliqua uerò per ſe 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. ſuæ arithmeticæ cap .66. quæſtione .56.</ref> quam Car<lb/>danicam vocat, ita inquit.</s> + </p> + <p> + <s xml:id="s1257" xml:space="preserve">Quidam perambulauit prima die certam quantitatem ſpatij, & ſecunda die, <reg norm="tan tò" type="context">tã<lb/>tò</reg> plus proportionaliter, quantò diameter eſt maior coſta, & tertia die tantò plus <lb/>ſecunda, quantò proportionaliter portio lineæ diuiſæ ſecundum proportionem ha <lb/>bentem medium, & duo extrema excedit minorem portionem, & quarta die in <lb/>proportione ad tertiam vt ſecunda ad primam, & quinta die proportionaliter tan-<lb/>tò plus quarta, quantò in tertia plus ſecunda, & 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 eſt, quàm ſi aliquis diceret, propono tibi, exempli gratia, <lb/>lineam <var>.a.l.</var> nouem partibus inuicem non æqualibus ita diuiſam <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> & ſic de cæteris vſque ad poſtremam <var>.k.l.</var> quæ quidem proportiones ſint <lb/>etiam inuicem diſſimiles, ſeu inæquales, do tibi etiam <reg norm="proportionem" type="context">proportionẽ</reg> totius lineæ <var>.a.l.</var> <lb/>ad <var>.a.b.</var> ſuam partem, quæ vt in propoſito exemplo nonupla eſt.</s> + </p> + <p> + <s xml:id="s1260" xml:space="preserve">Quæro nunc quam proportionem habebit <var>.a.c.</var> ad <var>.a.b.</var> & ſic de cæteris partibus <lb/>eiuſdem ad eandem <var>.a.b</var>.</s> + </p> + <p> + <s xml:id="s1261" xml:space="preserve">Quod quidem facillimum erit ſpeculari, nec non operari vnicuique, qui omnino <lb/>practicæ numerorum ignarus non fuerit, dum ab ordine ſcientifico non diſcedat.</s> + </p> + <p> + <s xml:id="s1262" xml:space="preserve">Cum enim cognoſcimus proportionem <var>.a.c.</var> ad <var>.c.d.</var> conſequenter cognoſcemus <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> ſi <var>.c.d.</var> accipiemus, vt medium inter <var>.a.d.</var> et <var>.d.e.</var> cognoſcemus 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 poſteà. <lb/><var>d.e.</var> inter <var>.e.f.</var> et <var>.a.e.</var> innoteſcet ea, quæ eſt <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/>cognoſcemus proportionem totius <var>.a.l.</var> ad <var>.i.k.</var> & hac mediante, cam cognoſcemus, <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> & ſic gradatim, co <lb/>gnita nobis erit proportio totius <lb/>lineæ <var>.a.l.</var> ad ſuam partem <var>.a.c.</var> be-<lb/> + <anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/> + neficio poſteà totius lineæ <var>.a.l.</var> co <lb/>gnoſcemus proportionem <var>a.c.</var> ad <lb/><var>a.b.</var> & ſic aliarum reſpectu lineæ <var>.a.b.</var> vt quærebatur, quæ quidem propoſitio, etſi car <lb/>danica uocetur leuiſſima tamen eſt.</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 ſumma colligendi, ſubtrahendi, <reg norm="multiplicandi" type="context">multiplicãdi</reg>, & di <lb/>uidendi proportiones ſcripſerint, nullus tamen (quod ſciam) perfectè, ac <lb/>ſcientificè ſpeculatus eſt has operationes, quapropter hanc rem cum ſilentio tranſi <lb/>re nolui, quin aliquid de ipſa conſcribam à ſumma dictarum proportionum in-<lb/>cohando.</s> + </p> + <p> + <s xml:id="s1266" xml:space="preserve">Quotieſcunque igitur volunt duas proportiones inuicem aggregare, ſimul ea-<lb/>rum antecedentia multiplicant, & ſimiliter earum conſequentia. </s> + <s xml:id="s1267" xml:space="preserve">Tunc proportio <lb/>terminata ab illis productis euadit in ſummam illarum duarum propoſitarum <lb/>proportionum.</s> + </p> + <p> + <s xml:id="s1268" xml:space="preserve">Vt exempli gratia, ſi voluerimus colligere proportionem ſeſquialteram cum ſeſ-<lb/>quitertia, multiplicando .3. cum .4. antecedentia ſcilicet, pro ductum erit .12. poſteà <lb/>multiplicando .2. cum .3. conſequentia, tunc productum erit .6. </s> + <s xml:id="s1269" xml:space="preserve">Proportio igitur, <lb/>quæ inter .12. et .6. reperitur. (quæ dupla eſt) eſt ſumma propoſitarum <reg norm="proportionum" type="context">proportionũ</reg>.</s> + </p> + <p> + <s xml:id="s1270" xml:space="preserve">Cuius rei ſpeculatio erit huiuſmodi ſint <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 ſint eorum conſequentia, productum <lb/>autem antecedentium ſit <var>.a.g.</var> illud verò quod <reg norm="con" type="context">cõ</reg> <lb/>ſequentium ſit <var>.d.a.</var> vnde proportio <var>.a.g.</var> ad <var>.a.d.</var> <lb/>compoſita erit ex proportione <var>.x.</var> ad <var>.t.</var> & ex ea, <lb/>quæ eſt <var>.u.</var> ad <var>.n.</var> per .24. ſexti vel quintam octaui. <lb/></s> + <s xml:id="s1271" xml:space="preserve">Patet igitur ratio rectè faciendi, vt ſuprà dictum <lb/>eſt.</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">QVotieſcunque deinde detrahere volunt vnam proportionem ex altera mul-<lb/>tiplicant antecedens vnius cum conſequenti alterius. </s> + <s xml:id="s1273" xml:space="preserve">Tunc proportio, quę <lb/>inter talia duo producta incluſa reperitur, eſt reſiduum, ſeu differentia illarum dua-<lb/>rum proportionum datarum.</s> + </p> + <p> + <s xml:id="s1274" xml:space="preserve">Vt exempli gratia, ſi aliquis vellet ex proportione dupla detrahere ſeſquialte-<lb/>ram, multiplicaret .2. antecedens duplæ cum .2. conſequenti ſeſquialteræ, quorum <lb/>productum eſſet .4. pro antecedenti reſiduę proportionis. </s> + <s xml:id="s1275" xml:space="preserve">Deinde multiplicaret .3 <lb/>antecedens ſeſquialteræ cum .1. conſequenti duplæ, & productum eſſet .3. pro <reg norm="con- ſequenti" 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"/> + ſequenti</reg> reſiduę proportionis; </s> + <s xml:id="s1276" xml:space="preserve">quæ quidem reſidua proportio eſſet vt .4. ad .3. hoc <lb/>eſt ſeſquitertia, & ſic de cæteris.</s> + </p> + <p> + <s xml:id="s1277" xml:space="preserve">Pro cuius ratione, ſit proportio <var>.x.</var> ad <var>.n.</var> ea quæ (exempli gratia) maior ſit, à <lb/>qua volumus demere proportionem <var>.t.</var> ad <var>.u.</var> minorem ſcilicet. </s> + <s xml:id="s1278" xml:space="preserve">Nunc autem <lb/>productum <var>.x.</var> in <var>.u.</var> ſit <var>.a.g.</var> illud verò <var>.t.</var> in <var>.<lb/>n.</var> ſit <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 reſiduam 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> quę <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 ſexti, ſeu .18. vel .19. ſe-<lb/>ptimi, ſed proportio <var>.a.g.</var> ad <var>.a.b.</var> hoc eſt <var>.x.</var> ad <var>.<lb/>n.</var> componitur ex ea, quæ eſt <var>.a.g.</var> ad <var>.a.d.</var> & ea, <lb/>quæ eſt <var>.a.d.</var> ad <var>.a.b.</var> hoc eſt <var>.t.</var> ad <var>.u.</var> ergò ea, quę <lb/>eſt <var>.a.g.</var> ad <var>.a.d.</var> erit quàm 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, quotieſcunque aliquam proportionem dupli-<lb/>care volentes, quadramus terminos ipſius proportionis, vel ſi eam triplicare <lb/>voluerimus, cubamus ipſos terminos, vel ſi eam quadruplicare voluerimus <lb/>inuenimus cenſicos cenſicos terminorum ipſius proportionis, & ſic de ſingulis, in <ref id="ref-0016">.17 <lb/>Theo. huiuſmodi tractatus</ref> manifeſta eſt.</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">QVotieſcunque nobis propoſiti fuerint duo numeri ad libitum, deſideraremus<lb/>q́ue duas proportiones tali relatione inuicem refertas, quali ſunt hi duo pro <lb/>poſiti numeri inter ſe, ita faciendum erit.</s> + </p> + <p> + <s xml:id="s1283" xml:space="preserve">Sciendum primo eſt proportionem maioris numeri propoſiti ad minorem ſem-<lb/>per eſſe alicuius ex quinque generum, hoc eſt aut erit generis multiplicis, aut ſu-<lb/>perparticularis, aut multiplicis ſuperparticularis, aut ſuper partientis, aut multi-<lb/>plicis ſuperpartientis.</s> + </p> + <p> + <s xml:id="s1284" xml:space="preserve">Nunc autem ſi erit ex genere multiplici, iam ab antiquis traditus eſt modus, <reg norm="quem" type="context">quẽ</reg> <lb/>ſequi debemus. </s> + <s xml:id="s1285" xml:space="preserve">Cuius ſpeculatio 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 ſi talis proportio datorum numerorum erit alicuius aliorum generum, ita <lb/>agemus, ſi fuerit ſuperparticularis.</s> + </p> + <p> + <s xml:id="s1287" xml:space="preserve">Sit exempli gratia, ſeſquialtera, tunc ſumantur duo numeri inuicem inæquales, <lb/>quos à caſu volueris <var>.o.</var> et <var>.c.</var> qui quidem cubentur, & eorum cubi ſint <var>.a.</var> et <var>.e</var>. </s> + <s xml:id="s1288" xml:space="preserve">Inuenia <lb/>tur poſteà. u. ita proportionatus ad <var>.o.</var> vt <var>.o.</var> eſt ad <var>.c.</var> ex regula de tribus, hoc eſt diui-<lb/>dendo quadratum ipſius <var>.o.</var> per <var>.c.</var> vnde nobis proueniat <var>.u.</var> & quia proportio <var>.a.</var> ad <var>.e.</var> <lb/>tripla eſt proportioni <var>.o.</var> ad <var>.c.</var> & proportio <var>.u.</var> ad <var>.c.</var> dupla eſt <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> ſeſquialtera erit proportioni <var>.u.</var> ad <var>.c</var>.</s> + </p> + <p> + <s xml:id="s1289" xml:space="preserve">Sed ſi proportio numerorum propoſitorum fuerit ſeſquitertia, faciemus <var>.a.</var> et <var>.e.</var> <lb/>eſſe cenſica cenſica ipſius <var>.o.</var> et <var>.c</var>. </s> + <s xml:id="s1290" xml:space="preserve">tunc ſumemus <var>.u.</var> conſequentem ad <var>.o.</var> vt dictum eſt, <lb/>deinde inueniremus <var>.i.</var> conſequens ad <var>.u.</var> ita ut <var>.u.</var> conſequens ipſius <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 eſt <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 ſuperparticularibus.</s> + </p> + <p> + <s xml:id="s1293" xml:space="preserve">Sed ſi data proportio numerorum fuerit ex ſuper partientibus, vt exempli gra-<lb/>tia de quinque ad tria, efficiemus, vt <var>.a.</var> et <var>.e.</var> ſint prima relata ipſius <var>.o.</var> et <var>.c.</var> vnde <lb/>proportio <var>.a.</var> ad <var>.e.</var> ita ſe 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">vnũ</reg> & propor-<lb/>tio <var>.i.</var> ad <var>.c.</var> ut tria ad <reg norm="vnum" type="context">vnũ</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> ſe habebit, <lb/>vt quinque ad tria, & ſic 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 ſeruando, 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. ſexti et .20. ſeptimi manifeſtè pateat ratio, quare rectè fiatac <lb/>cipiendam radicem quadratam illius producti, quod fit ex duobus datis <lb/>terminis, vt medium proportionale geometricè inter ipſos habeamus: </s> + <s xml:id="s1297" xml:space="preserve">nihilomi-<lb/>nus, quia per aliam methodum hoc idem ſcire 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 ſint, 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 eſt <var>.h.b.</var> <lb/>ad <var>.b.g.</var> ex prima .6. aut .18. vel .19. ſepti-<lb/>mi, ſed per .11. octaui ita eſt 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/>eſt 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 eſt diuidere vnam datam <lb/><reg norm="proportionem" type="context">proportionẽ</reg> per æqualia, hoc eſt in duas <lb/>æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poſteriori-<lb/>bus Theorematibus ad demonſtrandum priora illis, ſed hoc .148. dictum ſit 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">ꝙ</reg> ſi quis inuenire voluerit ſecundum terminum ex quatuor nume <lb/>ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-<lb/>modo nobis cogniti ſint, rectè factum ſit quadrare primum eorum, & hoc quadra-<lb/>tum poſteà per alium terminum cognitum multiplicare, cuius producti demum ac-<lb/>cipere radicem cubam pro ſecundo termino quæſito, hocloco videbimus.</s> + </p> + <p> + <s xml:id="s1301" xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eſt, 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 ſint, <reg norm="ſitque" type="simple">ſitq́</reg> imagina <lb/>tione deſcriptus cubus <var>.a.q.</var> primi termini, <reg norm="cubusque" type="simple">cubusq́</reg> <var>.d.k.</var> ſecundi rermini, conſidere-<lb/>mus etiam baſim <var>.a.i.</var> quadratam ipſius cubi <var>.a.q.</var> hoc eſt præcedentem dignitatem ip <lb/>ſius cubi eiuſdem radicis, quæ quidem baſis <var>.a.i.</var> multiplicetur per quartum <reg norm="terminum" type="context">terminũ</reg> <lb/><var>g.h.</var> productum autem ſit <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, ſed per primam ſexti, vel .18. aut .19. ſeptimi ita eſt <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> ideſt <lb/> + <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> + vt <var>.a.b.</var> ad <var>.g.h.</var> ſed vt eſt <var>.a.b.</var> ad <var>.g.h.</var> <lb/>ſic eſt <var>.a.q.</var> ad <var>.k.d.</var> per .36. vndecimi, <lb/>ſeu per .11. octaui, vnde per .11. quin <lb/>ti ſic 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. eiuſdem <var>.a.g.</var> ęqualis 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="ſecundo" type="context">ſecũdo</reg> termino <var>.c.d.</var> <lb/>id, quod nobis inſeruit ad inueniendam tertiam partem vnius propoſitæ 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 ſpeculatio iſta ita vniuerſalis 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, ſed eiuſdem <lb/>ſpeciei, 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 ipſius <var>.a.i.</var> multiplicatione in <var>.g.h.</var> reſultet <var>.a.<lb/>g.</var> vnde ex .18. vel .19. ſeptimi eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.a.b.</var> ad <var>.g.h.</var> ſed <lb/>eadem etiam eſt <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 huiuſmodi dignitatis ex quantitate <var>.a.g.</var> habebimus <var>.c.d.</var> ſecundum 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, ſi accipiat radices quadratas illorum datorum rer-<lb/>minorum, etſi voluerit tertiam partem, accipiat radices cubas: </s> + <s xml:id="s1310" xml:space="preserve">ſi autem quartam, <lb/>accipereradices cenſicas cenſicas ipſorum, & ſic de ſingulis 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 diuiſerit per denominationem ipſorum fractorum.</s> + </p> + <p> + <s xml:id="s1313" xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem ſeſquiquar-<lb/>tam per duo tertia, multiplicabit prius ipſam proportionem per numeratorem .2. <lb/>& productum, erit proportio .25. ad .16. qua poſtea diuiſa 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 ſunt, cu <lb/>tam vna, quàm alia, ſit 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 eſt, quàm accipere duas tertias partes vnius <lb/>totius ſuperficialis, imaginemur igitur hoc totum <var>.a.b.</var> lineare diuiſum eſſe in tertias <lb/>partes mediantibus <var>.e.</var> et <var>.d</var>. </s> + <s xml:id="s1316" xml:space="preserve">& tunc multiplicando ipſum per 2. tertias lineares produ-<lb/>ctum erit <var>.a.c.</var> ſex vnitatum ſuperficialium, quod quidem productum poſteà diuiſum <lb/>per .3. dabit <var>.d.c.</var> hoc eſt duas tertias ſuperficiales (quæ eſt tertia pars ipſius <var>.a.c.</var>) & <lb/>ęquales numero <var>.c.b.</var> duabus vnitatibus linearibus, ideſt duabus tertijs ipſius <var>.a.b</var>. </s> + <s xml:id="s1317" xml:space="preserve">No <lb/>tandum etiam eſt, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà <lb/>etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eſt quàm <lb/>quædam operatio ipſius regulæ de tribus, vt eyempli gratia volo multiplicare .25. <lb/>per 20. hoc nihil aliud eſt niſi quærere alium numerum ita proportionatum ad .25. <lb/>vt 20. ſe habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo <lb/>per vnum exregula de tribus, prouentus eſt idem numerus ipſius producti, & propte <lb/>rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eſt <lb/>quàm quærere aliquem numerum ita proportionatum ad ipſum numerum datum, <lb/>vt ſe habet numerator ad denominatorem, exempli gratia ſi .24. aliquis voluerit mul <lb/>tiplicare per duo tertia hoc idem eſt vt ſi quæreret numerum ad quem .24. ita ſe <lb/>habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eſt quod aliud non eſt mulri-<lb/>plicare aliquam proportionem per fractos, quàm aliam proportionem quærere ad <lb/><reg norm="quam" type="context">quã</reg> data ſe habeat, vt denominator ſe <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="conſtituendo" type="context context">cõſtituẽdo</reg> <reg norm="denominatorem" type="context">denominatorẽ</reg> in primo loco, quilocus eſt diuiſoris, numerato <lb/><reg norm="rem" type="context">rẽ</reg> verò in <reg norm="ſecundo" type="context">ſecũdo</reg> loco, <reg norm="multiplicando" type="context">multiplicãdo</reg> poſteà pro <lb/>portionem per <reg norm="numeratorem" type="context">numeratorẽ</reg>, & <reg norm="productum" type="context">productũ</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 ſe habebit, vt denomi-<lb/>nator ſe <reg norm="hent" type="context">hẽt</reg> ad numeratorem ex ratione ipſius re <lb/>gulę 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 ſe ſatis patet, <lb/>cum idem ſit modus diuidendi quemhbet nume <lb/>rum integrum per fractos. </s> + <s xml:id="s1320" xml:space="preserve">Quare, quæ vnius, <lb/>& alterius eſt 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 ſoluit .24. quæſitum ſi-<lb/>bi propoſitum à Hieronymo Cardano, via particulari & non generali. </s> + <s xml:id="s1322" xml:space="preserve">Quæ-<lb/>ſitum autem tale eſt quamlibet propoſitam rectam lineam in duas partes ita diuide <lb/>re via Euclidis, ut cubus totius lineæ ad cubos partium ſe habeat in proportione <lb/>tripla.</s> + </p> + <p> + <s xml:id="s1323" xml:space="preserve">Tartalea igitur inquit quòd vt ſatisfiat ſpeculatiuis ingenijs ſoluendum ſit huiuſ-<lb/>modi quæſitum, ſecando lineam propoſitam <var>.a.b.</var> in tres æquales partes, quarum vna <lb/>fit <var>.c.b.</var> vnde problema ſolutum erit.</s> + </p> + <p> + <s xml:id="s1324" xml:space="preserve">Verum dicit, ſed hæc non eſt methodus generalis, proptereà, quod cum tale <lb/>problema alterius fuiſlet 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> diuiſibilis in puncto <var>.c.</var> ita vt cubum totius dictæ <var>.a.b.</var> lineæ ad <lb/>ſummam cuborum <reg norm="ſuarum" type="context">ſuarũ</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 ſe contemplari po-<lb/>teſt, conſtituendo punctum <var>.c.</var> in medio loco inter <var>.a.</var> et <var>.b.</var> vnde proportio totalis <lb/>cubi ad ſummam 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 propoſitum reuertamur, conſiderabimus cubum totalem ipſius <var>.a.b.</var> <lb/>eſſe vt .125. & ſummam partialium vt .65. quam detrahemus ex cubo totali & nobis <lb/>remanebit .60. pro ſumma trium ſolidorum inuicem æqualium, quorum longitu-<lb/>do vniuſcuiuſque erit tota linea <var>.a.b.</var> nobis cognita vt radix dati cubi totalis, quæ erit <lb/>in hoc exemplo quinque partium, latitudo verò vniuſcuiuſque dictorum <reg norm="ſolidorum" type="context">ſolidorũ</reg> <lb/>erit <var>.a.c.</var> pars maior ipſius <var>.a.b.</var> quæ quidem <var>.a.c.</var> adhuc nobis ignota eſt, profunditas <lb/>ſeu altitudo vniuſcuiuſque illorum ſolidorum, erit <var>.c.b.</var> pars reliqua ipſius <var>.a.b.</var> & <reg norm="etiam" type="context">etiã</reg> <lb/>nobis incognita, ſed quia ſumma horum trium ſolidorum nobis manifeſta ſuperius <lb/>fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuſcuiuſque illorum ſoli-<lb/>dorum, vt tertia pars totius ſummæ ipſorum quæ erit .20. in propoſito <reg norm="exemplo" type="context">exẽplo</reg>, dein <lb/>de cum vnumquodque illorum ſolidorum producatur à ſuperficie contenta ſeu pro<lb/> ducta ab <var>.c.a.</var> in <var>.c.b.</var> in tota linea <var>.a.b.</var> ſequitur quòd ſi diuiſerimus hoc ſolidum .20. <lb/>per lineam <var>.a.b.</var> quinque partium proueniet nobis cognita ſuperficies producta ab <var>.<lb/>a.c.</var> in <var>.c.b.</var> quatuor partium, ſed cum quadratum totius <var>.a.b.</var> nobis cognitum ſit, eo <lb/>quod <var>.a.b.</var> vt eius latus etiam cognitum eſt. </s> + <s xml:id="s1328" xml:space="preserve">Tunc dictum quadratum erit .25. quod <lb/>quidem æquale eſt quadruplo illius quod fit ex <var>.a.c.</var> in <var>.c.b.</var> ſimul cum quadrato diffe <lb/>rentiæ inter <var>.a.c.</var> et <var>.c.b.</var> per .8. ſecundi 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 eſt, vt <lb/>16. eo quod ſimplum quod eſt .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 ſi hoc quadru-<lb/>plum .16. demptum fuerit ex totali <lb/>quadrato .25. reliquum erit .9. qua <lb/><reg norm="dratum" type="context">dratũ</reg> ſcilicet vnius partis <var>.a.c.</var> ipſius <lb/>hoc eſt illius partis, quæ differentia <lb/>eſt inter <var>a.c.</var> et <var>.c.b.</var> quæ quidem erit <num value="3">.<lb/>3.</num> partium quæ differentia cum ſub-<lb/>tracta fuerit ex <var>.a.b.</var> reliquum erit du <lb/>plum ipſius <var>.c.b.</var> duo ſcilicet. </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 ſuperficialium. <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 ſuam Arithmeticam re <lb/>cepiſſet, mox poſteà per literas petijt <reg norm="explicationem" type="context">explicationẽ</reg> regulæ falſi.</s> + </p> + <p> + <s xml:id="s1333" xml:space="preserve">Similiter poſt inciſas omnes ſuperiorum Theorematum figu-<lb/>ras, <reg norm="opereque" type="simple">opereq́;</reg> Typographo commiſſo, amicus quidam omnium <lb/>ſcientiarum ornatiſſimus maxima neceſſitudine mecum coniun-<lb/>ctus monuit me, vt aliquid de regula falſi ſcribere vellem, cuius <lb/>ſuaſu hæc, quæ ſequuntur appendicis vice ponere libuit, nelector, quidpiam quod <lb/>ad hancrem pertinet iure merito à nobis deſiderare poſſet; </s> + <s xml:id="s1334" xml:space="preserve">vt autem ad ipſam <reg norm="re- gulam" type="context">re-<lb/>gulã</reg> accedamus Ego ſicut, & in alijs multis, ita & in huiuſcæ regulę inuentione cum <lb/>ipſo Stifelio maximè conuenio, putans regulam falſi, ſeu falſarum poſitionum in-<lb/>uentam fuiſſe per paruos numeros in quæſtionibus facillimis & cognitis, eodem fer <lb/>mè modo, quo ipſe monſtrat illis duobus exemplis, quæ quamuis ipſe appellet theo <lb/>remata, nihilominus the oremata ego illa non vocarem, niſi adiuncta fuerit ſpecu-<lb/>latio ab ipſo præterita, & non experientia tantummodo, vt ipſe fecit. </s> + <s xml:id="s1335" xml:space="preserve">Primum eius <lb/>exemplum eſt, quòd.</s> + </p> + <p> + <s xml:id="s1336" xml:space="preserve">Quorumcumque <reg norm="duorum" type="context">duorũ</reg> numerorum differentia, ſi fuerit multiplicata in aggre <lb/>gatum eorum, producit ipſam differentiam, quæ eſt inter quadrata eorum.</s> + </p> + <p> + <s xml:id="s1337" xml:space="preserve">Secundum verò exemplum eſt, quod.</s> + </p> + <p> + <s xml:id="s1338" xml:space="preserve">Datis tribus numeris ſecundum progreſſionem arithmeticam diſpoſitis, facit mul <lb/>tiplicatio medij in ſe, <reg norm="quantum" type="context">quãtum</reg> multiplicatio extremorum inter ſe cum multiplicatio <lb/>ne differentiarum inter ſe.</s> + </p> + <p> + <s xml:id="s1339" xml:space="preserve">Talia enim exempla ipſe aliter non probat niſi experientia in aliquibus numeris, <lb/>arbitratus ex eo inuentam eſſe regulam falſi, experientia tantummodo confirma-<lb/>tam, quod quidem etiam & ego credo. </s> + <s xml:id="s1340" xml:space="preserve">At experientia in philoſophia mathema-<lb/>tica, aut <reg norm="nullam" type="context">nullã</reg> prorſus facit <reg norm="ſcientiam" type="context">ſcientiã</reg>, aut omnino ſuperfluus fuit Euclides in multis <lb/>ſuis propoſitionibus, & præcipuè in eius ſecundo libro, ſi ſufficeret experientia. </s> + <s xml:id="s1341" xml:space="preserve">Id-<lb/>circo quo magis ad euidentiam ipſius veritatis, quam profiteor, deuenire poſſim, <lb/><reg norm="accipiam" type="context">accipiã</reg> primò primum exemplum <lb/>ipſius Stifelij hic ſuperius citatum, <lb/>& pro numero maiori, in prima hic <lb/> + <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> + ſubſcripta figura .AE. accipio <var>.a.i.</var> <lb/>cuius quadratum ſit <var>.a.c</var>: pro minori <lb/>vero numero capio <var>.a.e.</var> <reg norm="partem" type="context">partẽ</reg> ipſius <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 ipſius <var>.a.i</var>: & differen <lb/>tia ipſorum quadratorum erit gno-<lb/>mon <var>.e.c.o</var>: Nunc autem protraho <var>.<lb/>i.c.</var> latus quadrati maioris quouſque <lb/><var>c.n.</var> æqualis ſit <var>.a.e.</var> numero minori, <lb/><reg norm="perficioque" type="simple">perficioq́;</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, ſed 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 ipſi <var>.u.o.</var> reliquæ ſcilicet parti ipſius 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> ſed <var>.e.t.</var> ęquatur <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 ſit ipſi <var>.a.e.</var> erit etiam æqualis ipſi <var>.<lb/>o.t</var>. </s> + <s xml:id="s1343" xml:space="preserve">quare <var>.u.n.</var> æqualis erit ipſi <var>.u.o.</var> <lb/>& tunc intellectus quieſcit, & <reg norm="abſque" type="simple">abſq;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> + aliqua alia experientia verè ſcientifi <lb/><reg norm="ceque" type="simple">ceq́;</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/>ipſam <reg norm="differentiam" type="context">differentiã</reg>, quæ eftinter qua-<lb/>drata eorum.</s> + </p> + <p> + <s xml:id="s1345" xml:space="preserve">Hæcautem propoſitio à me ipſo <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 ſpeculatione autem, etſcientia ſecundi exempli, in ſecunda hic ſubſcripta <lb/>figura <var>.ω.</var> cogitemus lineam <var>.u.a.</var> tribusin partibus arithmeticè diuiſam, qua <lb/>rum maxima ſit <var>.u.o.</var> media. ſit <var>.o.e.</var> minima verò ſit <var>.e.a.</var> multiplicatio autem mediæ <var>.<lb/>o.e.</var> in ſe ſit quadratum <var>.o.t.</var> abſcindatur 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 ſe ducta procreabit quadratum <var>.o.c.</var> quod erit productum ex differen <lb/>tijs ipſarum partium, & erit pars quadrati <var>.o.t.</var> ſuperius dicti, vt exſe 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> quouſque <var>.t.r.</var> æqualis ſit ipſi <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/>eſt. </s> + <s xml:id="s1352" xml:space="preserve">Claudatur ergo rectangulum <var>.i.r.</var> quod erit æquale producto ipſius <var>.e.a.</var> in <var>.o.u.</var> <lb/>Nam <var>.e.i.</var> ſumpta fuit <lb/>æqualis <var>.e.a.</var> ſed 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">ꝓductum</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, ſcientifice-<lb/>q́ue poſſumus affirma <lb/>re, quòd. </s> + <s xml:id="s1354" xml:space="preserve">Datis tribus <lb/>numeris <reg norm="ſecundum" type="context">ſecundũ</reg> pro <lb/>greffionem arithme-<lb/>ticam diſpofitis, fa-<lb/>cit multiplicatio me-<lb/>dij in ſe quantum mul <lb/>tiplicatio extremorum inter ſe, cum multiplicatione differentiarum inter ſe.</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 ſic de alijs huiuſmodi inuentionibus infero.</s> + </p> + <p> + <s xml:id="s1356" xml:space="preserve">DIcturus igitur aliquid circa <reg norm="regulam" type="context">regulã</reg> falſi, videtur mihi nullam oportere facere <lb/>mentionem de origine huiuſcæ regulæ, cum in hoc Stifelius ſatisfecerit, ſed + <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">propriaq́;</reg> fundamenta huiuſmodi operationis oftendere, fu-<lb/>mendo eadem exempla propoſita abipſis practicis, & maximè à Nicolao Tartalea <lb/>viro accuratiffimo, qui vbicunque potuit ſpeculatus eſt cauiſas <reg norm="ipſarum" type="context">ipſarũ</reg> operationum, <lb/>etſi de huiuſmodi falſi regula circa finem cap .8. lib. 17. promittat poſtea loqui, nub-<lb/>libi tamen loquutus eft. </s> + <s xml:id="s1357" xml:space="preserve">Monendum etiam cenſeo, me nihil de rationibus regulæ <lb/>falſi ſimplicis dicturum, cum ex ſeipſis ſatis appareant, quod non ita eſt de poſitio-<lb/>nibus duplis. </s> + <s xml:id="s1358" xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipſius Tartaleæ, quo <reg norm="etiam" type="context">etiã</reg> <lb/>ipſe vtitur pro exemplo docendi gratia, ipſam regulam duplæ poſitionis, quod qui <lb/>dem problema aliter à me <reg norm="ſolutum" type="context">ſolutũ</reg> fuit in .118. </s> + <s xml:id="s1359" xml:space="preserve">Theoremate huius mei lib. quod ſimi <lb/>liter ob hanc demum occaſionem mihi oblatam, alia etiam via, ſpeculatus ſumidem <lb/>poſſe fieri, quæ quidem via ſeu methodus generalis erit, & ita ſe habet.</s> + </p> + <p> + <s xml:id="s1360" xml:space="preserve">Accipio enim propoſitum numerum diuiſibilem, à quo detraho ſummam <lb/>datorum numerorum, primo duplicato, eo quòd tam in ſecunda quam in <lb/>tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à <lb/>quo detraho ſummam dictorum numerorum, quæ eſt .11. nam tres, & tres, & <lb/>quinque ſunt vndecim, eo quòd primus ingreditur in ſecunda, & in tertia parte, <lb/>dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-<lb/>dus eſt pro ſumma trium partium ſimplicium adhuc incognitarum, à quo extrahen <lb/>da eſt prima, eo modo quo nunc proponam exregula de tribus, hoc eſt aggregan <lb/>do dictas partes ſimplices ſine aliqua additione vtcunque volueris (ſed commodius <lb/>erit in minimis numeris) iuxta propoſitum, quod quidem propoſitum eſt, vt ſecun <lb/>da pars dupla ſit primæ, tertia verò æqualis fit primæ & ſecundæ, quæ partes in di-<lb/>ctis minimis numeris, ita diſpoſitæ erunt .1. 2. 3. quarum ſumma erit .6. </s> + <s xml:id="s1361" xml:space="preserve">Nunc ſi ex <lb/>regula de tribus dixerimus, cum hæc ſumma proueniat nobis ab vno, à quo proue-<lb/>niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoſito nume-<lb/>ro .39. cum ergo habuerimus primam <reg norm="partem" type="context">partẽ</reg>, reliquas poſteà illicò cognoſcemus.</s> + </p> + <p> + <s xml:id="s1362" xml:space="preserve">Huiuſmodi verò operationis ratio ex ſe manifeſta patet, eo quòd proportio ſum <lb/>mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ <lb/>ipſius .39. ad primam partem quæſitam huiuſmodi aggregati partium <reg norm="ſimplicium" type="context">ſimpliciũ</reg>, ſed <lb/>quia nemo adhuc, quod ſciam, ſatis animaduertit rationem modorum, qui ab anti-<lb/>quis obſeruati ſunt, qui quidem modi duo ſunt circa hoc Helcataym duplæ falſæ <lb/>pofitionis, igitur non prætermittam aliquid de hacreſpeculari, & primo de pri-<lb/>mo modo.</s> + </p> + <p> + <s xml:id="s1363" xml:space="preserve">In primis igitur <reg norm="ſciendum" type="context">ſciendũ</reg> eft, <reg norm="quod" type="simple">ꝙ</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 ſimpli<lb/>cibus partibus, vt <reg norm="etiam" type="context">etiã</reg> <reg norm="median- tibus" type="simple">median-<lb/>tibꝰ</reg> <reg norm="compoſitis" type="context">cõpoſitis</reg>, ut in pręſenti <reg norm="exem" type="context">exẽ</reg> <lb/>plo pro primis pofitionibus ac-<lb/>ceperunt .10. et .8. pro ſecundis <lb/>verò compoſitis <reg norm="cum" type="context">cũ</reg> numero .3. <lb/><reg norm="inuenerunt" type="context">inuenerũt</reg> .23. et .19. pro tertijs <lb/><reg norm="autem" type="wordlist">aũt</reg> <reg norm="compoſitis" type="context">cõpoſitis</reg> <reg norm="cum" type="context">cũ</reg> <reg norm="quinque" type="simple">quinq;</reg>, notaue <lb/>runt .38. et .32. vnde prima ſum <lb/>marefultauit .71. ſecunda verò <lb/>59. ita <reg norm="quod" type="simple">ꝙ</reg> <reg norm="primus" type="simple">primꝰ</reg> error remanebat <lb/>21. <reg norm="ſecundus" type="context simple">ſecũdꝰ</reg> <reg norm="autem" type="wordlist">aũt</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 exſummis partium ſimplicium.</s> + </p> + <p> + <s xml:id="s1365" xml:space="preserve">Vtexempli gratia, in figura <var>.B.</var> ſumma propoſita partium ſimplicium eſt .39. <lb/>vt diximus, eo quòd ab ipſo .50. detraxerimus .11. ſumma ſcilicet numerorum adij <lb/>ciendorum ad efficiendas partes compofitas, ſumma poſteà fimplicium partium <lb/>primæ poſitionis, erit .60. eo quòd prima pars erat .10. ſecunda autem ſimplex 20. <lb/>tertia verò fimplex .30. iuxta ordinem propoſiti. </s> + <s xml:id="s1366" xml:space="preserve">Summa deinde ſimplicium <reg norm="partium" type="context">partiũ</reg> <lb/>fecundæ poſitionis effet .48. quia prima eius pars erat .8. ſecunda verò ſimplex .16. <lb/>tertia autem ſimplex .24. vnde prima ſumma excederet datam .39. per .21. differen-<lb/>tiæ, ſecunda verò per .9. vt ſupra vidimus de ſummis compoſitis à dato .50. compo-<lb/>fito, & hoc quidem mirandum non eft, quod ſcilicet tres ſummæ fimplicium par-<lb/>tium ſintinuicem inæqua-<lb/>les, ijſdem differentijs me-<lb/> + <anchor type="figure" xlink:label="fig-0118-01a" xlink:href="fig-0118-01"/> + diantibus, quibus <reg norm="differunt" type="context">differũt</reg> <lb/>dictæ tres ſummæ compofi <lb/>tæ, cum ab vnaquaque <reg norm="con" type="context">cõ</reg> <lb/><reg norm="poſitarum" type="context">poſitarũ</reg> ablatus fit nume-<lb/>rus .11. æqualiter, vnde ex <lb/>neceſſitate, permutando, <lb/><reg norm="earum" type="context">earũ</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 noſtri lib.</ref> ſummæ 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, ſed à fimplici-<lb/>bus per ſe, & 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 ſcilicet ſecundum 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 ſe-<lb/>cundæ poſitionis. <reg norm="&c." type="unresolved">&c.</reg> quòd benè ſe habet. </s> + <s xml:id="s1368" xml:space="preserve">Vbi animaduertendum eſt, quod ille <lb/>numerus .12. non eft accipiendus per ſe vt differentia errorum hoc eft .21. et .9. nifi <lb/>peràccidens, fed benè perfe, vt <reg norm="differentia" type="context">differẽtia</reg> inter .60. er .48. ſimplices ſummas, quem <lb/>admodum .9. in hoc propoſito eft differentia per ſe inter .48. et .39 per accidens ve-<lb/>ro inter .59. et .50.</s> + </p> + <p> + <s xml:id="s1369" xml:space="preserve">Cognoſcendum igitur eft mediante .24. quinti Eucli. quod eadem proportio <lb/>eft primæ ſummæ (ſimplicium dico) ad ſuam primam partem, quæ ſecundæ ſum-<lb/>mæ ad ſuam, & tertiæ ſummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-<lb/>co antiqui ſi multiplicauiffent tertiam ſummam fim plicem cum prima parte prioris <lb/>fummæ fimplicis, & productum diuififfent per primam ſummam, vnde prima pars <lb/>quæſita tertiæ ſummæ orta fuiffet, abſque ullo negotio ipfius plus velminus) </s> + <s xml:id="s1370" xml:space="preserve">Quare <lb/>habebimus tres terminos antecedentes ab vna parte, & tres terminos conſequen-<lb/>tesab alia parte continentes vnam <reg norm="eandemque" type="simple">eandemq́;</reg> proportionem, vnde ex .19. quinti, <lb/>vel .12. ſeptimi eorum differentiæ proportionales erunt, hoc eft, <reg norm="quod" type="simple">ꝙ</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 eſt inter primam & fecundam ſummam, ad differen-<lb/>tiam quæ eſt inter primas earum partes, quæ illius differentiæ, quæ eſt inter ſecun-<lb/>dam & tertiam ſummam, ad differentiam, quæ eft inter primas illarum partes, ſed <lb/>harum .4. differentiarum, tres nobis cognitæ ſunt, ideft .12. 2. et .9. ergo ex regula de <lb/>tribus ab Eucli. in .20. ſeptì<unsure/>mi ſpeculata inueniebatur quarta differentia, quæ eft .1. <lb/>cum dimidio.</s> + </p> + <p> + <s xml:id="s1371" xml:space="preserve">A compofitis ſummis idem etiam proueniet, ſed non vt ex proprijs caufis, & per <lb/>ſe, ſedper 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) ipſius .71. ad .59. quæ ipſius .60. ad .48. nec ea quæ ipſius .59. ad <num value="50">.<lb/>50.</num> eft quæ ipſius .48. ad .39: </s> + <s xml:id="s1374" xml:space="preserve">Vnde non erit eadem proportio ipſius .71. ad .59. quæ <lb/>ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipſius .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 inęqualitatis.</s> + </p> + <p> + <s xml:id="s1377" xml:space="preserve">A fimplicibus igitur ſummis pendet ratio huiuſmodi effectus.</s> + </p> + <p> + <s xml:id="s1378" xml:space="preserve">Si vero prima pars fecundæ poſitionis effet .4. tunc ſecunda eius pars effet .8. & ter-<lb/>tia .12. quarum ſumma effet .24. (harum fimplicium partium ſeilicet) & minor vera <lb/>(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum <lb/>partium effet .6. differentia vero primæpartis ſecundæ poſitionis, a prima parte quę <lb/>fita effet .2. cum dimidio. </s> + <s xml:id="s1379" xml:space="preserve">Vnde in huiuſmodi 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 cognoſcatur differentia .36. <lb/>differentia ſcilicet ſimplicium ſummarum ipſarum poſitionum.</s> + </p> + <p> + <s xml:id="s1381" xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eſt, quod multiplicabant <lb/>diametraliter errores cum primis partibus, hoc eſt primum errorem cum prima par <lb/>te, hoc eſt cum numero ſecundæ poſitionis, ſecundum vero errorem cum prima <lb/>parte, hoc eſt cum numero primæ poſitionis, differentiam poſteà vel aggregatum <lb/>horum duorum productorum diuidebant per differentiam vel aggregatum dicto-<lb/>rum errorum, proueniens poſteà erat prima pars quæſita numeri propoſiti. </s> + <s xml:id="s1382" xml:space="preserve">Vn-<lb/>de oriebantur tria producta, quorum <reg norm="tertium" type="context">tertiũ</reg>, hoc eſt differentia, ſeu aggregatum il-<lb/>lorum conſtituebatur ex differentia feuaggregato errorum, & ex numero quæ-<lb/>fito.</s> + </p> + <p> + <s xml:id="s1383" xml:space="preserve">Vtin præfenti exemplo, primus error eſt .21. qui multiplicatus cum prima par-<lb/>te ſecundæ poſitionis, quæ eſt .8. producit .168. <reg norm="ſecundus" type="context">ſecũdus</reg> verò error eſt .9. qui multi-<lb/>plicatus cum prima parte primę poſitionis producit .90. differentia autem horum <lb/>productorum eſt .78. quæ diuifa per differentiam errorum, quæ eſt 12. dabit .6. <reg norm="cum" type="context">cũ</reg> di <lb/>midio, pro prima parte quæſita dati numeri diuiſibilis, qui erat .50.</s> + </p> + <p> + <s xml:id="s1384" xml:space="preserve">Hæc omnia rectè ſe habent. </s> + <s xml:id="s1385" xml:space="preserve">Sed, vt ſupra dixi diuiſor non eft per ſe differentia <lb/>errorum, neque etiam differentia per ſe ſummarum compoſitarum, fed bene fim-<lb/>plicium.</s> + </p> + <p> + <s xml:id="s1386" xml:space="preserve">Pro cuius rei ſpeculatione, accipiendæ ſunt ſummæ ſimplices, quarum differen-<lb/>tiæ per ſe vtiles ſunt in huiuſmodi operatione; </s> + <s xml:id="s1387" xml:space="preserve">& quia etiam rationes veritatis ex <lb/>iſtis, & non ex illis fluunt; </s> + <s xml:id="s1388" xml:space="preserve">quamuis tam vnæ, quam aliæ ſint eædem in quantitate, <lb/>ideſt 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">Diſponantur igitur huiuſmo-<lb/> + <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> + di numeri tali ordine, vt fim-<lb/>plex ſumma, quæ ab vna reli-<lb/>quarum ſuperatur, & aliam ſupe-<lb/>rat, medium locum teneat; </s> + <s xml:id="s1390" xml:space="preserve">@t <lb/>in propoſito exemplo ſumma <lb/>mediocris eft .48. quę à ſumma <num value="60">.<lb/>60.</num> ſuperatur, & ſuperat ſum-<lb/>mam .39. locata igitur fit hęc .48. <lb/>inter illas, ſuæ verò primæ partes <lb/>fimiliter conftitutæ ſint ſupra di-<lb/>ctas ſummas, cum ſuis <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 ipſius plus ſimpli-<lb/>citer: </s> + <s xml:id="s1392" xml:space="preserve">figura verò <var>.D.</var> pro exem-<lb/>plo ipſius 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 ſolum pro hypoteſi, <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">Demonſtrandum nobis nunc relinquetur, maximum productum æquale effere-<lb/>liquis duobus; </s> + <s xml:id="s1397" xml:space="preserve">hoc eſt productum .168. æquale effe productis .90. et .78. quorum <lb/>duorum productorum alterum .90. ſcilicet, generatur à differentia .9. quæ eft ſe-<lb/>cundę, & tertię ſummæ, in primum numerum antecedentem, qui eſt .10. alterum vc-<lb/>ro productum .78. ſcilicet, generatur à differentia .12. quę eſt primę, & ſecundę, ſum <lb/>mę in tertium numerum antecedentem, qui eſt .6. cum dimidio, maximum vero <lb/>productum .168. ſcilicet generatur à differentia maxima .21. quę eft primę, & tertię <lb/>ſummę (& ſemper ęqualis prioribus duabus differentijs .12. et .9.) in ſecundum nu-<lb/>merum antecedentem, qui eſt .8.</s> + </p> + <p> + <s xml:id="s1398" xml:space="preserve">Conſtituantur igitur duo producta fimul iuncta ęqualia duobus .90. et .78. <lb/>lateralibus ſupra vnam aliquam rectam lineam <var>.q.p.</var> <reg norm="fitque" type="simple">fitq́;</reg> productum <var>.f.g.</var> ęquale <num value="90">.<lb/>90.</num> productum verò <var>.g.n.</var> ęquale .78. fit etiam baſis <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 ſubſcripta videre licet, et <var>.q.p.</var> erit .21. <lb/>Cogitemus nunc differentiam <var>.d.i.</var> diuiſam eſſe in puncto <var>.e.</var> ita vt eadem proportio <lb/>ſit ipſius <var>.d.e.</var> ad <var>.e.i.</var> quæ ipſius <var>.q.g.</var> ad <var>.g.p.</var> hoc eſt 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 ipſorum antecedentium numerorum, deinde à puncto <var>.e.</var> ducatur imagina-<lb/>tione <var>.u.e.o.</var> æ quidiſtans ipſi <var>.q.p.</var> & producatur <var>.q.n.</var> vſque ad <var>.u.</var> vnde ita ſe 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. ſexti vel .20. ſeptimi <var>.n.e.</var> <lb/>rectangulum æquale crit ipſi <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>: ſed cum <var>.g.i.</var> ſit 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 eſt igitur quod dictum fuit, hoc eſt <reg norm="quod" type="simple">ꝙ</reg> maximum productum ęquale ſit 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 poſteà quo volueris horum altero productorum ex maximo, <reg norm="diuiſoque" type="simple">diuiſoq́;</reg> <lb/>reliquo per differentiam conſequentium, ipſi diametraliter oppoſitam, pro <lb/>ueniet tibi numerus antecedens <reg norm="correſpondensque" type="simple">correſpondensq́;</reg> illi.</s> + </p> + <p> + <s xml:id="s1401" xml:space="preserve">Animaduertendum tamen eſt, quòd ſi in figura à me ita ordinata, ſumma ſim-<lb/>plex propoſita medium locum occuparet, vt in figura <var>.D.</var> arithmetica videri poteſt; <lb/></s> + <s xml:id="s1402" xml:space="preserve">tunc vt habeatur eius productum, addenda ſimul erunt circunſtantia producta .eo <lb/><reg norm="quod" type="simple">ꝙ</reg> eius ſecundum latus eſſet antecedens medio loco conſtitutum, & prima pars <reg norm="quae- ſita" type="simple">quę-<lb/>ſita</reg> numeri propoſiti: </s> + <s xml:id="s1403" xml:space="preserve">in qua figura <var>.D.</var> manifeſtè patet ratio, quare colligendi ſint <lb/>tam errores, quam producta, dum eorum alterum eſt plus, reliquum verò minus.</s> + </p> + <p> + <s xml:id="s1404" xml:space="preserve">Speculatio figurę <var>.D.</var> arithmeticę videbitur in figura <var>.D.</var> geometrica, eodem fe <lb/>rè modo quo fecimus in figuris <var>.C.</var> mutatis mutandis, reſpectu ipſius plus, & minus.</s> + </p> + <p> + <s xml:id="s1405" xml:space="preserve">Collectio namque <reg norm="errorum" type="context">errorũ</reg> ſimiliter accidentalis eſt, eo quod eſſentialis numerus <lb/>diuiſor per ſe, eſt maxima differentia ſummarum ſimplicium, vt in dicta figura <var>.D.</var> <lb/>cerni poteſt.</s> + </p> + <p> + <s xml:id="s1406" xml:space="preserve">Sed vt ſuperius dixi, nunc etiam repeto, quòd rectè hoc loco multiplicabatur <lb/>ſumma ſimplex propoſita, cum prima par <lb/>te primę poſitionis, vt productum diuide <lb/>retur per primam ſimplicem ſummam, <lb/> + <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> + vnde proueniret nobis pars prima <reg norm="quaeſi- ta" type="simple">quęſi-<lb/>ta</reg> noſtri numeri propoſiti, ex regula de <lb/>tribus, vnica poſitione.</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 ſit .100. in quinque partes, tales <lb/>verò, <reg norm="quod" type="simple">ꝙ</reg> ſecunda duplo maior ſit prima <lb/>cum .2. ſimul, tertia autem æqualis ſit pri-<lb/>mæ & ſecundæ cum .3. vnitatibus iunctis, <lb/>quarta poſteà maior ſit prima ſecunda, & <lb/>tertia per .4. vnitates, quinta demum ſu-<lb/>peret reliquas omnes per quinque vnita <lb/>tes, vt in figura <var>.E.</var> videre eſt, quæ quidem <lb/>partes compoſitæ (ſumpta vnitate pro <lb/>prima) ita diſpoſitæ erunt .1. 4. 8. 17. 35. <lb/>quarum ſumma erit .65. ſimplices autem <lb/>cum diſpoſitæ fuerint erunt .1. 2. 3. 6. 12. <lb/>quarum ſumma erit .24. dempta igitur <lb/>cum fuerit hæc ſimplex ſumma .24. à com <lb/>poſita .65. reſiduum erit .41. hoc eſt ſum-<lb/>ma numerorum propoſitorum cum ſuis <lb/>iterationibus in ipſis partibus, quod cum <lb/>per ſe clariſſimum ſit, ſuperſluum eſt <reg norm="ipsam" type="context">ipsã</reg> <lb/>ſummam annatomizare per ſingulas par-<lb/>tes, niſi quis habuerit eius cerebrum à fi-<lb/>gura Omega <reg norm="terminatum" type="context">terminatũ</reg>, cui tamen poſ-<lb/>ſemus dicere dictam ſummam .41. in .4. <lb/>partes diuidi, cuius prima eſſet .2. pro ad <lb/>ditione ad <reg norm="ſecundam" type="context">ſecũdam</reg> partem ſimplicium, + <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"/> + ſecunda verò eſſet .5. pro additione ad tertiam partem ſimplicium, tertia autem eſ-<lb/>ſet .11. pro additione ad quartam partem ſimplicium, quarta demum eſſet .23. pro <lb/>additione quintæ partis ſimplicium, quarum partium .2. 5. 11. 23. ſumma eſt .41. vt <lb/>diximus. </s> + <s xml:id="s1408" xml:space="preserve">Hæc igitur ſumma .41. ſubducenda eſt à numero .100. propoſito, vnde re-<lb/>linquetur .59. pro ſumma partium ſimplicium numeri propoſiti, quarum prima erit <lb/>2. cum vndecim vigeſimisquartis ex diuiſione huiuſmodi .59. per .24. ſummam par-<lb/>tium ſimplicium ex viregulæ de tribus, dicendo ſi .24. prouenit nobis ab .1. prima <lb/>partium ſimplicium, à quo proueniet nobis .59? </s> + <s xml:id="s1409" xml:space="preserve">vnde proueniet à .2. cum vndecim <lb/>vigeſimisquartis pro prima parte quæſita, ſecunda verò iuxta propoſitum, erit .6. <lb/>cum .22. vigeſimisquartis, tertia autem .12. cum nouem vigeſimisquartis, quarta po<lb/>ſteà .25. cum .18. vigeſimisquartis, quinta demum erit .52. cum .12. vigeſimisquartis, <lb/>quarum omnium ſumma erit .100.</s> + </p> + <p> + <s xml:id="s1410" xml:space="preserve">STifelius in primo exemplo regulæ falſi, ita inquit.</s> + </p> + <p> + <s xml:id="s1411" xml:space="preserve">Quæratur numerus, à cuius dimidio ſubtractæ partes tertia, & quarta relin-<lb/>quatur .300.</s> + </p> + <p> + <s xml:id="s1412" xml:space="preserve">Ipſe enim ſupponit .300. pro reſiduo cognito alterius numeri incogniti, deinde <lb/>accipit .24. pro prima poſitione numeri cogniti, à cuius medietate abſcindit tertiam <lb/>& quartam partem ipſius medietatis, vnde remanet .5. qui quidem numerus .5. ex <num value="22">.<lb/>22.</num> quinti vel .15. ſeptimiſe ha-<lb/>bebit ad .24. vt .300. ad <reg norm="numerum" type="context">numerũ</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 diuiſerit 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">Conſideremus igitur <reg norm="meam" type="context">meã</reg> di-<lb/>ſpoſitionem numerorum huiuſ-<lb/>modi exempli, in figura hic ſup-<lb/>poſita <var>.F.</var> in qua videre licebit <lb/>quo pacto ipſe etiam Stifelius ac <lb/>cipiat diuiſorem .5. vt <reg norm="differentiam" type="context">differentiã</reg> <lb/>errorum & non ut differentiam <lb/>duorum conſequentium .5. et .10 <lb/>ſicuti eſt re uera, ut diuiſor dico, <lb/>ex rationibus à me hic ſupra ad-<lb/>ductis, quamuis vna & eadem ſit <lb/>quantitas neceſſariò ut patet.</s> + </p> + <p> + <s xml:id="s1415" xml:space="preserve">ACcipiamus adhuc aliud exemplum à Tartalea propoſitione .9. <reg norm="datum" type="context">datũ</reg>, & <reg norm="oppoſitum" type="context">oppoſitũ</reg> <lb/>priori; </s> + <s xml:id="s1416" xml:space="preserve">nam ſicut in illo numerus ſimplex habebatur per ſubtractionem ſum-<lb/>mæ numerorum adijciendorum, in hoc fitèconuerſo, hoc eſt per additionem nu-<lb/>merorum ſubtrahendorum.</s> + </p> + <p> + <s xml:id="s1417" xml:space="preserve">Problema igitur ita ſe habet. </s> + <s xml:id="s1418" xml:space="preserve">Fuit quidam mercator qui habebat aliquot au-<lb/>reos, cuius quantitas poſteà quærenda erit, hic enim fecit duo itinera, ut aliquod <lb/>dictis aureis mediantibus lucrum faceret, in primo autem itinere duplicauit nume-<lb/>rum ſuorum aureorum, ex quibus poſteà conſumpſit .4. pro aliquibus expenſis, 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"/> + ſecundo itinere iterum duplicauit ſuos aureos, ex quibus etiam poſtea conſumpſit <num value="8">.<lb/>8.</num> numeratis poſteà pecunijs reperit tantummodo .24. aureos in eius marſupio, <reg norm="quae ritur" type="simple">quę<lb/>ritur</reg> nunc quot habebat aureos in principio primi itineris.</s> + </p> + <p> + <s xml:id="s1419" xml:space="preserve">Intali caſu, cum ipſe quolibet itinere duplicabat eius pecuniam, nulli dubium <lb/>eſt quòd in fine ſecundi itineris ipſe habuiſſet pecuniam ſuam quadruplicatam, ſi <lb/>ex ipſa nihil detractum fuiſſet, ſed quia in fine primi itineris conſumpſit .4. aureos, <lb/>quibus alios .4. lucratus eſſet in ſecundo itinere, poſteà conſumpſit iterum <num value="8">.<lb/>8.</num> aureos, ita <reg norm="quod" type="simple">ꝙ</reg> ex quadruplo ſuæ primæ pecuniæ, rectè dici poteſt, quod conſum-<lb/>pſerit .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 ſi addiderimus .16. ipſi .24. habebimus .40. pro quadruplo eius <lb/>prioris pecuniæ. </s> + <s xml:id="s1422" xml:space="preserve">Rectè igitur dici poteſt, ſi .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 reſpondeat cum methodo antiquorum. <lb/></s> + <s xml:id="s1424" xml:space="preserve">Ego enim inueni duas poſitiones ſcriptas à Tartalea pro prima pecunia hoc eſt .12. <lb/>et .14. ſed à .12. pro primo errore reperi .8. more antiquo à .14. verò pro ſecundo er-<lb/>rore proueniebat .16. producta autem horum numerorum diametraliter, ſunt .112. <lb/>et .192. quorum differentia eſt .80. pro tertio producto, quo diuiſo per differen-<lb/>tiam <reg norm="errorum" type="context">errorũ</reg> .8. ſcilicet, præbetnobis .10. pro pecunia quæſita, vt etiam ego inueni.</s> + </p> + <p> + <s xml:id="s1425" xml:space="preserve">Sed hoc mihi viſum eſt ſubtilius examinare mea methodo mediante, vtin figu-<lb/>ra <var>.G.</var> videre eſt, prius enim ſuo loco poſuitria producta dicta, deinde duas poſitio <lb/>nes .12. et .14. & quia ſciebam productum .112. oriri à multiplicatione .14. cum .8. <lb/>ideo poſui talem numerum .8. ſuo loco diametraliter oppoſito ei producto .112. <lb/>& quia ſciebam etiam productum .192. naſci ex .12. et .16. ideo ſuo loco poſui hunc <lb/>numerum .16. qui eſt maxima differentia inter duos conſequentes ( ita à me ſupra <lb/>nominatos) à qua differentia dempta priori .8. iam inuenta, reliqua .8. mihi daba-<lb/>tur, <reg norm="quam" type="context">quã</reg> ſuo loco notaui, ſuo <reg norm="etiam" type="context">etiã</reg> <lb/>loco ſcripſi .2. differentiam inter <lb/>12. et .14. antecedentium. </s> + <s xml:id="s1426" xml:space="preserve">ſed <lb/> + <anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/> + quia ſciebam eandem proportio <lb/>nem eſſe inter hanc differentiam <lb/>& <reg norm="differentiam" type="context">differentiã</reg> .8. huic <reg norm="ſuppoſitam" type="context">ſuppoſitã</reg>, <lb/>quæ reperitur inter .12. <reg norm="antecedem" type="context">antecedẽ</reg> <lb/>tem, & <reg norm="ſuum" type="context">ſuũ</reg> conſequentem; </s> + <s xml:id="s1427" xml:space="preserve">ideo <lb/>poſui .48. pro dicto conſequenti, <lb/>diuiſi poſtea productum .80. per <num value="8">.<lb/>8.</num> differentiam ei diametraliter <lb/>oppoſitam, vnde prouenit mihi <lb/>10. cui ita proportionatus eſt <reg norm="ſuus" type="simple">ſuꝰ</reg> <lb/>numerus conſequens .40. vt .48. <lb/>ad .12. et .56. ad .14. exijſdem ra-<lb/>tionibus à meſupra dictis. <lb/></s> + <s xml:id="s1428" xml:space="preserve">In tali igitur figura videntur nu-<lb/>merinaturaliter <reg norm="correſpondentes" type="context">correſpondẽtes</reg> <lb/>ipſis poſitionibus, & hac metho-<lb/>do poſſumus inuenire tales numeros conſequentes in omnibus alijs exemplis à no-<lb/>ſtris maioribus ſcriptis.</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 ſit quarta pars totius vaſis cum oper <lb/>culo, pars autem media ſine operculo, ſit quinta pars ipſius 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 eſt pes .4. cum .14. decimisnonis ta-<lb/>lium partium, ſeu librarum, qualium operculus eſt .18. </s> + <s xml:id="s1432" xml:space="preserve">Videamus igitur & nos ex <lb/>noſtra figura, quo pacto hoc reſpondeat veritati.</s> + </p> + <p> + <s xml:id="s1433" xml:space="preserve">Inuenta enim ſunt tria producta, <reg norm="iam" type="context">iã</reg> orta ex dicta methodo .10. 100. 90. quæ ſuis <lb/>locis notaui, vt in ſigura <var>.H.</var> ſubſcripſi etiam duas illorum poſitiones .5. et .10. cum <lb/>ſua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat <num value="1">.<lb/>1.</num> quod ſuo loco notaui, ſimiliter quia .100. productum, pro vno eius laterum erat <num value="5">.<lb/>5.</num> reliquum autem .20. ſuo loco poſui, & quia differentia inter .20. et .1. duo latera, <lb/>quę eſt .19. æqualis eſt ei, quæ inter duo conſequentia duarum poſitionum, etiam <lb/>ſuo loco ipſam conſtitui, ſed quia hæc differentia eſt vnum laterum producti .90. er <lb/>go reliquum latus <reg norm="quæſitum" type="context">quæſitũ</reg> erit .4. cum .14. decimisnonis, rectè igitur operatur. <lb/></s> + <s xml:id="s1434" xml:space="preserve">ſed cum eadem proportio ſit inter differentiam .5. ſuperiorem, et .19. inferiorem, <lb/>quæ eſt vnius <reg norm="antecedentis" type="context">antecedẽtis</reg> ad ſuum <lb/>conſequens, </s> + <s xml:id="s1435" xml:space="preserve">quare .10. <reg norm="antecedens" type="context">antecedẽs</reg> <lb/>habebit pro ſuo conſequenti .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 ſi .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">Huiuſmodi autem rei ratio ita <lb/>ſe <reg norm="hent" type="context">hẽt</reg>, eſto linea <var>.a.e.u.</var> cuius pars <lb/>a. ſit quarta reliquarum <var>.e.u.</var> iuncta <lb/>rum, ſed <var>.e.</var> ſit quinta ipſius <var>.a</var>. </s> + <s xml:id="s1437" xml:space="preserve">Tunc <lb/>clarum erit quod <var>.e.</var> erit vigeſima <lb/>dictarum <var>.e.u.</var> </s> + <s xml:id="s1438" xml:space="preserve">quare erit decima-<lb/>nona ipſius <var>.u.</var> ſed <reg norm="cum" type="context">cũ</reg> u. <reg norm="sumpta" type="context">sũpta</reg> ſit vt <num value="18">.<lb/>18.</num> rectè igitur dici poteſt, ſi .u: ut <num value="19">.<lb/>19.</num> prouenit ab <var>.a.</var> ut quinque, à <lb/>quot ipſius <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/>ſa cum fuerit <var>.u.</var> in partes .19. quod <lb/>quinque illarum æquabuntur ipſi <var>.<lb/>a.</var> cum quælibet fuerit æqualis <var>.e.</var> <lb/>quintæ parti ipſius <var>.a</var>.</s> + </p> + <p> + <s xml:id="s1440" xml:space="preserve">HAc igitur mea numerorum diſpoſitione mediante reperiuntur ipſi numeri in <lb/>feriores naturaliter conſequentes, correſpondentesq́ue ipſis ſuperioribus an <lb/>tecedentibus; </s> + <s xml:id="s1441" xml:space="preserve">quamuis multoties <reg norm="contingere" type="context">cõtingere</reg> poſſit, ut generationes ſeu com-<lb/>poſitiones ipſorum ignorentur: </s> + <s xml:id="s1442" xml:space="preserve">& quia tam à differentijs errorum, quam ab illis, <lb/>quę ſunt inter ueros conſequentes numeros ( propter eorum æqualitatem ) elicitur <lb/>ipſa ueritas, proptereà rectè antiqui illis vſi ſunt, quamuis ſint potius ſenſum <lb/>ſequuti, uel experientiam, quam rationem: </s> + <s xml:id="s1443" xml:space="preserve">quæ quidem ratio pendet ab ipſis na-<lb/>turalibus numeris conſequentibus ( ut ſupra uidimus ) etſi incognitis ut plurimum, <lb/>quod ſi ipſos inuenire primò nobis datum fuiſſet, unica <reg norm="tantummodo" type="context">tantũmodo</reg> poſitio ſuffice- + <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 ipſa regula de tribus, vt <reg norm="iam" type="context">iã</reg> ſępius <reg norm="dictum" type="context">dictũ</reg> eſt, quod <reg norm="etiam" type="context">etiã</reg> clarè patet ex di-<lb/>uerſis problematibus .17. lib. ipſius Tartaleæ, vt ex primo, quod aſſumpſimus pro <lb/>noſtro etiam primo exemplo, ex .9. 15. 16. 17. 18. 19. 20. 27. 28. 29. 30. 33. & ex <lb/>alijs multis, vbi facillimè inue nitur conſequens ipſius poſitionis, qui quidem nume-<lb/>rus eſt diuiſor producti ipſius numeri propoſiti in numerum poſitionis, vnde poſteà <lb/>prouenit <reg norm="ſecundum" type="context">ſecundũ</reg> latus huiuſmodi producti, hoc eſt 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 manifeſtè pateant <reg norm="eorum" type="context">eorũ</reg> rationes abſque regula falſi, cuius <lb/>regulæ rationes non ita promptè ipſi intellectui ſe offerunt, vt ſupra vidimus.</s> + </p> + <p> + <s xml:id="s1445" xml:space="preserve">Accipiamus pro exemplo .21. problema ipſius Tartalæ in dicto .17. libr. vbi ſup-<lb/>ponit vnum hædum diuiſum in .4. partes, quarum quælibet vendebatur eodem pre <lb/>cio, interiora vero .6. denarijs minus quam quælibet dictarum partium, ſumma <lb/>autem omnium iſtorum denariorum fuit .127. quæritur nunc precium cuiuſque <lb/>partis.</s> + </p> + <p> + <s xml:id="s1446" xml:space="preserve">Tale enim problema hoc etiam alio breuiori modo poteſt ſolui, vt rationes ma-<lb/>gis pateant, quam ex regula falſi.</s> + </p> + <p> + <s xml:id="s1447" xml:space="preserve">Nam ſi illi numero .127. denariorum, additus fuerit numerus .6. ſumma erit .133. <lb/>qua diuiſa per quinque, illico proueniet .26. cum tribusquintis pro precio vniuſcu-<lb/>iuſque 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">preciũ</reg> <reg norm="vniuſcuiuſque" type="simple">vniuſcuiuſq;</reg> illorum.</s> + </p> + <p> + <s xml:id="s1450" xml:space="preserve">Hoc etiam problema, hac alia methodo ſolui poteſt, dicendo exregula de tribus, <lb/>ſi ex .20. vtrorunque qui ea vendit, vult .44. quid volet ex .12? </s> + <s xml:id="s1451" xml:space="preserve"><reg norm="manifeſtum" type="context">manifeſtũ</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, ſed 12. cum .28. pomis valebant .36. ergo .16. poma ſola valebunt .9. <lb/>cum tribus quintis, hoc enim clarè ex ſe patet; </s> + <s xml:id="s1453" xml:space="preserve">quare cum dixerimus, ſi .16. poma ſo <lb/>la valent .9. cum tribusquintis, vnum valebit <var>.o.</var> cum tribusquintis, ſed 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/>ſcilicet 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 ſupponit quod quidam comparaſſet <lb/>quatuor petias, vt vulgo dicitur, panni pro ducatis .96. quarum primæ precium ob-<lb/>litus ſit, ſed memoria tenet pro ſecunda ſoluiſſe .6. plusquam pro prima, & pro ter-<lb/>tia ſoluiſſe .8. plus quam pro ſecunda, & pro quarta ſoluiſſe .10. plus quam pro ter-<lb/>tia, quæritur nunc quantum fuerit precium vniuſcuiuſque 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 ſolui, vt in <lb/>ſubſcripta figura <var>.I.</var> videri <lb/>poteſt, <reg norm="addendo" type="context">addẽdo</reg> ſimul omnes <lb/>exceſſus. </s> + <s xml:id="s1456" xml:space="preserve">Nam exceſſus <reg norm="ſecum" type="context">ſecũ</reg> <lb/>dæ ſupra primam eſt .6. ſed <lb/>cum exceſſus tertiæ ſupra ſe <lb/>cundam ſit .8. ergo exceſſus <lb/>tertiæ ſupra 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"/> + ſed exceſſus quartæ ſupra tertiam eſt .10. vnde ſupra ſecundam erit .18. & ſupra pri-<lb/>mam erit .24. quæ omnia ſimul addita erunt .44. & in qualibet harum trium remane-<lb/>bit una pars æqualis primæ quantitati, </s> + <s xml:id="s1457" xml:space="preserve">quare ſi 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. ſecunda .19. ter-<lb/>tia .27. & quarta .37. quarum omnium ſumma eſt .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 poſitionibus autem Tartaleæ in noſtra figura <var>.K.</var> digeſtis, videre poſſumus <lb/>quo pacto <reg norm="colligantur" type="context">colligãtur</reg> huiuſ <lb/>modi <reg norm="conſequentes" type="context">conſequẽtes</reg> numeri ſimpli-<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 compoſi <lb/>tæ .9. 15. 23. 33. ex quarum ſumma <lb/>80. ſubtrahitur .36. ſumma ſim-<lb/>plex ex ſimplicibus partibus .9. 9. <lb/>9. 9. & <reg norm="reſiduum" type="context">reſiduũ</reg> quod eſt .44. ſubdu <lb/>citur ex .96. ſumma compoſita & <lb/>propoſita, vnde remanet .52. pro <lb/>ſumma ſimplici, ex numero dato, <lb/>cuius proportio ad .13. eadem eſt <lb/>quæ .36 ad .9. & proptereà ſuper-<lb/>flua eſt ſecunda poſitio, <reg norm="quando" type="context">quãdo</reg> ſci <lb/>mus inuenire tales duos numeros <lb/>conſequentes, vt in hoc exemplo <lb/>ſunt .36. et .52. quia ex regula de <lb/>tribus poſteà elicitur veritas quæ-<lb/>ſita. </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 poſitiones primi <reg norm="exem" type="context">exẽ</reg> <lb/>pli Tonſtalli hoc eſt .33. et .31. vt in figuris hic ſubiectis <var>.P.Q.</var> facile quis poteſt vi-<lb/>dere, vbi in figura P. videbit nume-<lb/>ros compoſitos, in figura verò <var>.Q.</var> cer <lb/> + <anchor type="figure" xlink:label="fig-0127-02a" xlink:href="fig-0127-02"/> + net numeros ſimplices, à quibus pro <lb/>ueniunt rationes per ſe huiaſmodi <lb/>operationis, in figura autem <var>.R.</var> vide <lb/>bitur meus ordo, & iſtæ tres figuræ ſi <lb/>miles <reg norm="erunt" type="context">erũt</reg> tribus illis primis <var>.A.B.C.</var> <lb/>ita quòd cum quis illas intellexerit, il <lb/>lico etiam iſtas cognoſcet, 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/>ſimplicium <reg norm="numerorum" type="context">numerorũ</reg>, à quibus fluit <lb/>tota ratio (vt ſupra dixi) huiuſcemo <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 poteſt dici de .15. problemate (ſicut de alijs multis) vbi ponit tres <lb/>homines habentes .40. aureos quorum primus habet duas quintas partes ſecun-<lb/>di, <reg norm="ſecundus" type="context">ſecũdus</reg> verò <reg norm="quinque" type="simple">quinq;</reg> octauas tertij, <reg norm="quæritur" type="simple">quærit̃</reg> <reg norm="nunc" type="context">nũc</reg> quot ducatos habeat vnuſquiſque.</s> + </p> + <p> + <s xml:id="s1463" xml:space="preserve">Quis non videt quæſo, <reg norm="quod" type="simple">ꝙ</reg> omnes partes erunt .15. quare cum dixerimus ſi .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. ſimiliter aſſero, vbi ponit <reg norm="alium" type="context">aliũ</reg> emiſſe tria fruſta panni pro ducatis .48. <lb/>quarum ſecundam habuit pro dimidio precio primæ, tertiam autem pro quarta <lb/>parte ipſius ſecundæ, </s> + <s xml:id="s1466" xml:space="preserve">quare omnes partes erunt .13. quapropter precium tertiæ pe-<lb/>tiæ erit tertiadecima pars ipſius .48. hoc eſt .3. cum .9. tertijs decimis.</s> + </p> + <p> + <s xml:id="s1467" xml:space="preserve">Adhuc duo exempla videtur mihi proponere, quorum primum eſt .38. eiuſdem <lb/>lib. vbi ſupponitur operarium quendam velle perficere opus quoddam ſpacio die-<lb/>rum .36. tali pacto, quod qualibet die, in qua ipſe operaturus ſit lucretur ſolidos <num value="16">.<lb/>16.</num> qualibet verò die, in qua nihil agat perdat ſolidos .24. </s> + <s xml:id="s1468" xml:space="preserve">Tunc accidit, vt exacto <lb/>termino <reg norm="perfectoque" type="simple">perfectoq́;</reg> opere, tantum lucratus ſit, quantum perdiderit. </s> + <s xml:id="s1469" xml:space="preserve">Quæritur <reg norm="nunc" type="context">nũc</reg> <lb/>quot fuerint dies lucri, quotúe perditionis.</s> + </p> + <p> + <s xml:id="s1470" xml:space="preserve">Huiuſmodi problematis operatio breuiſſima abſque vlla falſa poſitione ita erit, <lb/>hoc eſt diuidendo productum .36. in .24. per .40. ideſt per aggregatum ipſius .24. <reg norm="cum" type="context">cũ</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 ſe ſatis 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/>& vigeſimaſepti <lb/>mi ex regula de <lb/>tribus reperiun <lb/>tur partes ipſius <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 ſe inuicem habentes, vt .24. et .16. quæ ſunt .21. cum tribus quintis, et .14. <reg norm="cum" type="context">cũ</reg> dua <lb/>bus quintis, ex quo ſequitur, vt quod fit ex .21. cum tribus quintis, in 16. ęquale ſit 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 eſt .39. quod quidem à ſuperiori non differt, niſi quod <lb/>in fine operationis, operarius dictus lucratus eſt ſolidos .60: </s> + <s xml:id="s1473" xml:space="preserve">quęritur <reg norm="nunc" type="context">nũc</reg> vt ſu-<lb/>pra, quot fuerunt dies lucri, & quot perditionis.</s> + </p> + <p> + <s xml:id="s1474" xml:space="preserve">Hoc etiam abſque vlla falſa poſitione dicto citius poteſt ſolui, hoc modo, <reg norm="diuidem" type="context">diuidẽ</reg> <lb/>do ſcilicet illos .60 ſolidos per .40. ideſt per aggregatum .24. cum .16. proueniens <lb/>autem, quod erit .1. cum dimidio, adde ad latus ſuperius inuentum, hoc eſt .21. cum <lb/>tribus quintis, & ſunima erit .23. cum decima parte pro numero dierum lucri, dein-<lb/>de idem prouentum deme ex alio latere ſuperius 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 ſpeculatione cogitemus in figura <var>.N.</var> duo dicta producta inuicem <lb/>æqualia <var>.o.b.</var> et <var>.n.c.</var> exiſtente 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 ſi mente concepta fuerit recta <var>.e.<lb/>a.t.</var> æquidiſtans <var>.o.c.</var> ita vt rectangulum <var>.o.e.</var> ſit .60. </s> + <s xml:id="s1478" xml:space="preserve">tunc rectangulum, ſeu productum <lb/><var>b.t.</var> ſuperabit rectangulum ſeu productum <var>.n.e.</var> per idem .60. ex communi conceptu, <lb/>eo quòd ex producto <var>.n.c.</var> ſublatum eſt productum <var>.a.c.</var> 24. & producto <var>.o.b.</var> additum <lb/>eſt productum <var>e.a.</var> 16. rectè igitur feci cum diuiſerim .60 per .40. vnde prouenit mi <lb/>hi <var>.u.a.</var> ideſt .1. cum dimidio, quod additum ipſi <var>.b.u.</var> compoſuit <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 ſi idem operator perdidiſſet .60. tunc cogitaremus parallelam dictam <var>.e.a.</var> t <lb/>ſuperius ductam eſſe ita vt ſecaret <var>.b.u.</var> & non <var>.u.n.</var> vnde adderet .24. ipſi 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 ſi aliquid notatu di-<lb/>gnum aduerterim, hoc eſt quod ſæpe accidere poterit ut caſus impoſſibiles <lb/>proponantur. </s> + <s xml:id="s1481" xml:space="preserve">Quemadmodum ſi aliquis diceret, cupio mihi ueſtimentum 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 ſit albus, rubeus uerò alter, deinde albus ſit pretij .40. ſolidorum uniuſcuiuſ-<lb/>que cubiti, rubeusuerò precij .50. uellemq́ue omnes cubitos eſſe .8. nec plus <lb/>nec minus. </s> + <s xml:id="s1482" xml:space="preserve">Vellem etiam ſoluere ſolidos 450. neque minus.</s> + </p> + <p> + <s xml:id="s1483" xml:space="preserve">Hic igitur caſus impoſſibilis eſt, eo quòd .8. cubiti totius rubei eſſent precij ſo-<lb/>lidorum .400. tantummodo, unde ex alio panno albo minoris precij ſumere ali-<lb/>quid non poſſumus.</s> + </p> + <p> + <s xml:id="s1484" xml:space="preserve">Idem etiam eueniret ſi uoluiſſet ſoluere ſolidos .320. neque plus, eo quòd .8. cu-<lb/>biti illius minoris precij, hoc eſt .40. ſolidorum, eſsent ualoris .320. ſolidorum tan <lb/>tummodo, quare pro alio panno nullus eſset locus. </s> + <s xml:id="s1485" xml:space="preserve">Animaduertendum igitur erit <lb/>quod numerus poſſibilis ad ſoluendum tale quæſitum erit inter .400. et .320. & non <lb/>extra iſtos terminos, vt vnicuique patere poteſt.</s> + </p> + <p> + <s xml:id="s1486" xml:space="preserve">Similiter idem in hoc alio caſu accidere poterit, ut ſi quis diceret.</s> + </p> + <p> + <s xml:id="s1487" xml:space="preserve">Emi quinque petias panni pro aureis .55. pretium tamen primæ oblitus ſum, ſed <lb/>memoria teneo, quòd ſecunda altioris pretij erat quam ipſa prima per .4. & ter-<lb/>tia precioſior ſecunda per .7. et quarta carior tertia per .9. quinta verò ſuperabat <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 ſolutum eſſet, ſuperat aureos .55 quòd <lb/>quidem nullo pacto fieri poteſt, vt veri ſint ſupra dicti exceſsus, ſi verus eſt 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. ſed .22. cum .20. et .11. et .4. dant .57. qui numerus maior <lb/>eſt 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 ſciam) veras <reg norm="internasque" type="simple">internasq́</reg> cauſas <lb/>operationis perſpectiuæ perſectè docuerit, operæpre-<lb/>cium exiſtimaui <reg norm="aliquam" type="context">aliquã</reg> de ijs diſputationem ſuſcipere.</s> + </p> + <p> + <s xml:id="s1491" xml:space="preserve">Multi enim <reg norm="eorum" type="context">eorũ</reg>, qui huiuſmodi operationis regulas <lb/>præſcribunt, cum eius effectuum veras cauſas igno-<lb/>rent, varios <reg norm="diuerſosque" type="simple">diuerſosq́;</reg> errores committunt, vt exempli <lb/>gratia in ſubſcripta figura ſuperficiali <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> ipſi <var>.q.d.</var> à puncto <var>.B.</var> interſecationis lineæ-<lb/>o.l. cum latere <var>.i.d.</var> trianguli, & (idem) indifferenter, ean-<lb/>dem quoque à puncto <var>.Z.</var> interſecationis ipſius <var>.o.l.</var> cum perpendiculari <var>.x.i.</var> ducunt. <lb/></s> + <s xml:id="s1492" xml:space="preserve">neſcientes hunc ſolum eſſe verum modum, n onitem alium, quia ſi alius, talis eſſet, <lb/>hic, verus non exiſteret, nam ſi vellent ſeſe excuſ are, quòd ducendo dictam paralle-<lb/>lam à puncto <var>.B.</var> hoc fiat præſupponendo planum ipſius <var>.i.d.q.</var> verſus rectangulum <var>.<lb/>q.a.</var> orizontale inclinatum, ſecundum angulum <var>.i.d.q.</var> hæc excuſatio accipien-<lb/>da non eſſet, quia horum conſenſu, præſupponendo planum <var>.i.d.q.</var> inclinatum, <lb/>anguli inferiores rectanguli degradati, non tam acuti, quam ſunt duo <var>.i.d.q.</var> et <var>.i.q.<lb/>d.</var> eſſe deberent, quod facilè eorum ratione innoteſcet, quæ de figura corporea <var>.<lb/>A.</var> hîc ſubſcripta mox proponam, præter id, quòd volentes deinde aſpicere qua-<lb/>dratum degradatum, oporteret huiuſmodi planum reſpectu oculi ita collocare, <lb/>quemadmodum ſe habet linea <var>.i.d.</var> reſpectu <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/>ſis <var>.o.l.</var> et <var>.x.i.</var> perpendiculari ipſi <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 ſpeculatione imaginemur in figura corporea .A: <var>q.a.</var> eſſe figuram re-<lb/>ctangulam <reg norm="orizontalemque" type="simple">orizontalemq́;</reg> ad degradandam ſuper 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, ſit autem oculus reſpicientis <var>.o.</var> cuius altitudo <var>.o.p.</var> ab orizonte, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>conſpicit 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> ſit verò primum ita <lb/>collocatus pes <var>.p.</var> eius qui reſpicit, vt linea <var>.p.l.</var> perpendicularis ipſi <var>.u.a.</var> lateri re-<lb/>ctanguli, medio loco poſita ſit, inter <var>.a.n.</var> et <var>.u.s</var>. </s> + <s xml:id="s1495" xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</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 ſuperficie-<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> poſitæ erunt. </s> + <s xml:id="s1497" xml:space="preserve">Imaginemur etiam lineam <var>.p.s.n.</var> perpendicula-<lb/>rem ipſi <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 conſtituet, cum ex .28. duæ <var>.u.q.s.</var> et <var>.a.d.n.</var> ſint parallelæ ipſi <var>.p.l.</var> et <var>.s.n.</var> ipſi <var>.u.<lb/>a.</var> & quia ſupponitur <var>.o.p.</var> perpendicularis plano orizontali, Angulus ergò <var>.o.p.l.</var> re-<lb/>ctus erit ex ſecunda 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 ſuperficiei, in qua ſunt duæ lineæ <var>.o.p.</var> et <var>.s.n.</var> ex .4. <lb/>11. perpendicularis erit, & ſuperficies orizontalis <var>.a.s.</var> perpendicularis erit cum dicta <lb/><var>o.s.n.</var> ex .18. eiuſdem 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 ſcientia, in ſuperficiebus duorum triangulorum <var>.o.u.q.</var> et <var>.<lb/>o.a.d.</var> erunt, ſi noluerimus cogere aduerſarium ad confitendum duas lineas rectas in-<lb/>cludere ſuperficiem, quemadmodum cogere-<lb/> + <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> + tur facere, ſi opinaretur duas alias rectas per <lb/>eadem puncta <var>.o.s.n.</var> tranſire, quæſunt in di-<lb/>ctis ſuperficiebus. </s> + <s xml:id="s1499" xml:space="preserve">Vnde <var>.o.s.</var> et <var>.o.n.</var> communes <lb/>erunt ſectiones duarum dictarum <reg norm="ſuperficierum" type="context">ſuperficierũ</reg> <lb/>cum ſuperficie <var>.o.s.n</var>. </s> + <s xml:id="s1500" xml:space="preserve">Imaginemur nunc has <lb/>duas ſuperficies <var>.o.u.</var> et <var>.o.a.</var> quarum commu-<lb/>nis ſectio ſit <var>.o.t.</var> (quæ erit linea recta ex .3. lib. <lb/>II.) quæ erunt perpendiculares ſuperficiei <var>.o.s.<lb/>n.</var> ex .4. et .14. iam dictis. </s> + <s xml:id="s1501" xml:space="preserve">& ex .19. eiuſdem <lb/><var>o.t.</var> perpendicularis eidem ſuperficiei <var>.o.s.n.</var> <lb/>erit, & ex .6. eiuſdem hæc linea <var>.o.t.</var> duabus <var>.u.<lb/>q.s.</var> et <var>.a.d.n.</var> parallela exiſter, & ex .9. eiuſdem <lb/>hæc linea <var>.o.t.</var> duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> parallela <lb/>exiſtet, & ex eadem .9. erit parallela ipſi <var>.p.l.</var> <lb/>Imaginemur nunc planum, ſuper quod deſide <lb/>remus videre quadrangulum orizontale, quod <lb/>planum, exempli gratia, ſit primo, vt iam dixi-<lb/>mus, locatum in linea <var>.q.d.</var> ad angulos rectos <lb/>cum plano orizontali, cuius communes ſectio <lb/>nes cum ſuperficiebus <var>.s.t.</var> et <var>.n.t.</var> viſionis la-<lb/>terum <var>.u.q.</var> et <var>.a.d.</var> ſint <var>.i.q.</var> et <var>.i.d.</var> & com-<lb/>munis ſectio trianguli <var>.o.u.a.</var> ideſt viſionis <lb/>lateris <var>.a.u.</var> cum dicto plano, ſit <var>.r.e</var>. </s> + <s xml:id="s1502" xml:space="preserve">Vnde ex <lb/>communi ſcientia rectangulum orizontale, <lb/>oculo <var>.o.</var> ſeipſum patefaciet in plano <var>.i.q.d.</var> ſe- + <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 ſectio ſuperficiei <var>.p.t.</var> <lb/>cum dicto plano, ſit <var>.i.x.</var> quæ <var>.i.x.</var> perpendicularis erit <var>.s.a.</var> ſuperficiei orizontali ex <lb/>19. lib. 11. quia <var>.p.t.</var> eſt etiam orizonti perpendicularis ex .18. eiuſdem, cum <var>.o.p.</var> ei-<lb/>dem perpendicularis exiſtat. </s> + <s xml:id="s1504" xml:space="preserve">Vnde <var>.i.x.</var> erit altitudo trianguli <var>.i.q.d.</var> & æqualis ipſi <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> ſectio ſuperficiei triangularis <var>.o.a.u.</var> <reg norm="cum" type="context">cũ</reg> <lb/>ſuperficie <var>.p.t.</var> quæ <var>.o.l.</var> ſecando lineam <var>.e.r.</var> in puncto <var>.Z.</var> nobis oſtendet quantum di-<lb/>ſtare ſeu eminens eſſe debeat latus <var>.e.r.</var> in plano ab <var>.q.d.</var> medio ipſius <var>.z.x</var>. </s> + <s xml:id="s1506" xml:space="preserve">Et quia <lb/>præſuppoſuimus <var>.p.l.</var> in eodem medio, inter <var>.u.s.</var> et <var>.a.n.</var> ideo <var>.x.q.</var> ęqualis erit <var>.x.d.</var> <lb/>& ex .4. lib. primi <var>.i.q.</var> ipſi <var>.i.d.</var> et <var>.e.r.</var> parallela ipſi <var>.q.d.</var> ex .6. lib. 11. cum ipſa quoque <lb/>ſit perpendicularis ſuperficiei <var>.p.t.</var> ex <ref id="ref-0020">.19. eiuſdem</ref>. </s> + <s xml:id="s1507" xml:space="preserve">Hucuſque igitur in figura cor-<lb/>porea <var>.A.</var> prodeunt in lucem omnes cauſæ effectuum figuræ ſuperficialis <var>.A.</var> ideſt vn <lb/>de fiat, vt in ipſa figura ſuperficiali, triangulum <var>.o.p.l.</var> tale conſurgat, & quid ſignifi-<lb/>cet <var>.o.</var> et <var>.o.p.</var> et <var>.p.l.</var> et <var>.o.l.</var> & quam ob cauſam tale quoque formetur triangulum <var>.i.<lb/>q.d.</var> atque in tantam altitudinem, quantam obtinet <var>.o.p.</var> & quid ſint latera <var>.i.q.</var> et <var>.i.<lb/>d.</var> & quare erigatur <var>.x.i.</var> parallela ipſi <var>.p.o.</var> ab eadem <var>.p.o.</var> tanto ſpatio diſtans, & qua <lb/>ratione producatur à puncto <var>.Z.</var> ipſa <var>.Z.r.e.</var> parallela ipſi <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 obſeruandum eſt, quòd ſi planum ipſius <var>.i.q.d.</var> in figura corporea aliquan-<lb/>tulum inclinatum eſſet orizontem verſus, anguli <var>.i.q.d.</var> et <var>.i.d.q.</var> maiores exiſterent, <lb/>quàm cum idem eſt ipſi orizonti perpendiculare, quemadmodum clarè demonſtra-<lb/>tum fuit in .39. primi Vitelionis.</s> + </p> + <p> + <s xml:id="s1509" xml:space="preserve">Non igitur rectè fit ſi in figura ſuperficiali ducatur à puncto <var>.B.</var> parallela ipſi <var>.q.d.</var> <lb/>abſque 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 ſubſequentes <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 ſumere ſemper <var>.p.x.</var> figuræ ſuperfi-<lb/>cialis æqualem ei, quæ eſt corporeæ, & eidem ſuperficiali, adiungere <var>.x.d.</var> æqualem <lb/>ei, quæ eſt corporeæ, & compoſito <var>.p.d.</var> ex dictis duabus lineis, in figura ſuperficiali, <lb/>addere <var>.d.q.</var> æqualem ei, quæ eſt 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 diſtans 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"/> + ſteà. <var>x.i.</var> erigetur æqualis lineæ <var>.p.o.</var> & ſuis terminis <lb/>concludetur triangulum <var>.i.q.d.</var> & id quod remanet. <lb/></s> + <s xml:id="s1512" xml:space="preserve">Vnde ſi longius diſerendo progrediare, patebit ex <num value="4">.<lb/>4.</num> primi <var>.i.d.</var> ſuperficialem, 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 eſt, & vniuerſalis, licet in figura ſuperficiali <var>.A.</var> ſupe-<lb/>rius poſita, ſecundum communem antiquorum conſuetudinem exemplum <lb/>dederim, effectus enim idem eſt. </s> + <s xml:id="s1515" xml:space="preserve">Sed ſi quis vellet conſiderare dictam figuram ſu-<lb/>perficialem <var>.A.</var> ſecundum eum modum, quem de figura <var>.B.</var> ſuperficiali pręſcripſi, id <lb/>poterit in ſubſcripta figura <var>.C.</var> ſpeculari in hunc modum. </s> + <s xml:id="s1516" xml:space="preserve">Accipiet enim <var>.p.x.</var> ſuper <lb/>ficialem, æqualem corporeæ, & quia in ipſa corporea <var>.A.</var> ſuppoſita fuit linea <var>.p.l.</var> <lb/>ideſt punctum <var>.x.</var> inter duas <var>.u.s.</var> et <var>.a.n.</var> ſecabimus <var>.p.x.</var> ſuperficialem in puncto <var>.d.</var> ita, <lb/>vt <var>.d.x.</var> ſuperficialis, æqualis ſit corporeæ, & ipſi ſuperficiali <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> ſuperfi-<lb/>cialis æqualis erit corporeę,<lb/> + <anchor type="figure" xlink:label="fig-0134-03a" xlink:href="fig-0134-03"/> + et <var>.p.x.</var> ſuperficiali addetur <var>.<lb/>x.l.</var> æqualis corporeæ. </s> + <s xml:id="s1518" xml:space="preserve">De <lb/>ijs poſtea quæ dicenda ſuper-<lb/>ſunt, iam ſatis ſuperq́ue 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 ſe-<lb/>quentes, in errorem non in-<lb/>cidemus, imò efficietur qua -<lb/>drilaterum <var>.q.r.</var> ſuperficiale, <lb/>ſimile, & 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 eſt punctum perſpectiuæ, vt practici dicere ſo <lb/>lent) quid ſit, hac via & ratione ſub noſtram 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 eſt, quàm punctum, varijs ſectionibus commune, & huiuſmodi punctum, ocu <lb/>lus non eſt, quemadmodum multi Pictores, Sculptores, Architecti, & Perſpectiui <lb/>ignari, ipſum punctum, oculum appellando, falsò crediderunt, quaſi <reg norm="punctum" type="context">punctũ</reg> <var>.i.</var> per-<lb/>ſpectiuæ oculus eſſet.</s> + </p> + <p> + <s xml:id="s1522" xml:space="preserve">In ſupradictis igitur figuris manifeſte eluceſcit cauſa diminutionis obiectorum, <lb/>& altitudinis trianguli æqualis ei, quæ eſt oculi à plano orizontali, vt etiam diſtantię <var>.<lb/>p.l.p.x.</var> & cuiuſuis tandem rei. </s> + <s xml:id="s1523" xml:space="preserve">Sed vt huius effectus ſcientia magis in vniuerſum pa-<lb/>retur. </s> + <s xml:id="s1524" xml:space="preserve">Volo duas hic ſubſcriptas figuras <var>.D.</var> corpoream, & <var>.D.</var> ſuperficialem à vo-<lb/>bis conſiderari, in quarum corporea, linea <var>.p.l.</var> ſit 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> diſiunctum ſit à rectangulo ſuper-<lb/>ficiali, ideſt, vt ſeparatum exiſtat à linea <var>.q.d.</var> latere ipſius rectanguli, & ſit etiam obli <lb/>quum, reſpectu ipſius rectanguli, ideſt vt communis ſectio dicti plani cum ſuperficie <lb/><var>a.s.</var> orizontalis ipſi <var>.u.a.</var> parallela non ſit, ſed ſit obliqua, ſi tamen idem planuni per-<lb/>pendiculare dictæ ſuperficiei orizontali <var>.a.s.</var> erit: </s> + <s xml:id="s1525" xml:space="preserve">& dicta communis ſectio exprima<lb/>tur characteribus .q.ω.α.d.x. nunc in figura corporea habebimus figuram <var>.e.r.c.m.</var> <lb/>in plano, quod viſualem pyramidem ſecat, medio cuius figuræ <var>.e.r.c.m.</var> oculus po-<lb/>ſitus in <var>.o.</var> rectangulum orizontale conſpicit. </s> + <s xml:id="s1526" xml:space="preserve">Volentes vero nunc in figura <var>.D.</var> ſuper-<lb/>ficiali eam deſcribere, faciem us <var>.p.x.</var> ſuperficialem, æqualem corporeæ, eiq́ue <lb/>addemus <var>.x.l.</var> æqualem corporeæ, aut ſumemus <var>.p.l.</var> eidem corporeæ <reg norm="aequa- lem" type="simple">ęqua-<lb/>lem</reg>, quam ſecabimus in puncto <var>.x.</var> eodem planè modo, quo corporea reperi-<lb/>tur diuiſa; </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> ſimiles & 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 ſecta fuit corporea, & ducemus lineam <var>.d.i.</var> vnde habebi-<lb/>mus triangulos <var>.i.d.q.</var> et <var>.i.d.x.</var> ſimiles 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"/> + cuſque habebimus ſitus duorum laterum figurę <lb/>rectanguli degradati, ideſt ſitus ipſius <var>.e.m.</var> et <var>.r.<lb/>c.</var> etiam ſi adhuc neſciatur in qua parte ipſius <var>.i.<lb/>q.</var> & ipſius <var>.i.d.</var> eſſe <reg norm="debeant" type="context">debeãt</reg>. </s> + <s xml:id="s1531" xml:space="preserve">Quod ſi ſcire volue <lb/>rimus <reg norm="ſecabitur" type="simple">ſecabit̃</reg>. <var>p.l.</var> in <reg norm="puncto" type="context">pũcto</reg> <var>.g.</var> ſimilis corporeæ, <lb/>ſi in ipſa tamen corporea prius protraxerimus <lb/>lineam <var>.q.d.</var> latus rectanguli vſque 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> ſuperficialis, <lb/>quæ ſecabit lineam <var>.i.x.</var> in puncto <var>.f.</var> linea vero <var>.<lb/>o.l.</var> in puncto <var>.z.</var> punctis ſitis in <var>.i.x.</var> ſuperficiali, <lb/>pręcisè vt in corporea, <reg norm="quemadmodum" type="context">quemadmodũ</reg> quilibet <lb/>ex ſe facilè cognoſcere poteſt. </s> + <s xml:id="s1533" xml:space="preserve">Deinde in cor <lb/>porea, in ſuperficie 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 ſuperficie <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 ſectiones dictorum <reg norm="duorum" type="context">duorũ</reg> <lb/>triangulorum cum plano trianguli <var>.i.q.x.</var> ipſi quoque plano ex .19. eiuſdem lib. erunt <lb/>perpendiculares. </s> + <s xml:id="s1534" xml:space="preserve">Nuncautem ſecetur <var>.q.x.</var> ſuperficialis in punctis <var>.ω.</var> et .α. eadem ra-<lb/>tione; </s> + <s xml:id="s1535" xml:space="preserve">qua corporea ſecta ſuit à duabus <var>.p.q.</var> et <var>.p.u.</var> à quibus punctis <var>.ω.</var> et. α. ſuperficia <lb/>libus ductæ ſint duæ <var>ω.m.</var> et .α.e. perpendiculares vſque ad latus <var>.i.q.</var> in punctis <var>.m.</var> et <var>.<lb/>e.</var> quę ſitum habebunt in <var>.i.q.</var> ſuperficiali pręcisè, vt in corporea, ex .26. primi, du-<lb/>cendo deinde in ſuperficiali duas <var>.m.f.</var> et <var>.e.Z.</var> eæ æquales erunt corporeis ex .4. pri-<lb/>mi, & ſic anguli <var>.i.e.z.</var> et <var>.i.m.f.</var> & eę duę lineę <var>.e.z.</var> et <var>.m.f.</var> fectę 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, ſed ita quo-<lb/>que ſe habent duę <var>.e.m.</var> et <var>.r.c.</var> ſi verum eſt <reg norm="quod" type="simple">ꝙ</reg> dif <lb/>ferentię rerum æqualium ſint 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> ſuperficialem om <lb/>ninò ſinlilem, & ęqualem corporeæ. </s> + <s xml:id="s1537" xml:space="preserve">Is tamen <lb/>modus prolixus eſt, & arduus, quam ob cau-<lb/>ſam neque ego vnquam <reg norm="eum" type="context">eũ</reg> viui accommo-<lb/>darem, neque alijs, vt eodem vterentur ſua-<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 ſciendum, quòd qui ſciuerit vnum ſolum punctum locare in perſpe <lb/>ctiua, eo modo quem nunc proponam, facilè quoque ſciet ſupra quoduis <reg norm="planum" type="context">planũ</reg> <lb/>(quod tamen ſit perpendiculare orizonti) quamlibet rem locare. </s> + <s xml:id="s1539" xml:space="preserve">Quam ob cauſam <lb/>imaginemur hic ſubſcriptas duas figuras <var>.E.</var> <reg norm="corpoream" type="context">corporeã</reg>, & E. ſuperficialem, & 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 ſuppo-<lb/>nebatur in figura <var>.A.</var> corporea. </s> + <s xml:id="s1540" xml:space="preserve">Imaginemur ergo in ipſa figura <var>.E.</var> corporea radi-<lb/>um viſualem <var>.o.b.</var> qui ſectus ſit à noſtro plano in <var>.k.</var> quod quidem <var>.k.</var> quærendum eſt <lb/>in triangulo <var>.i.q.d.</var> ipſius 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/>ſuperficiali ducatur <var>.b.c.</var> ad rectos <reg norm="cum" type="context">cũ</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 ipſi <var>.q.d.</var> quę ab ipſa <var>.x.l.</var> in puncto <var>.m.</var> erit diuiſa, & hęc <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> ſecta 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 diuiſa. </s> + <s xml:id="s1542" xml:space="preserve">Atque id erit quod nobis <lb/>inquirendum propoſueramus.</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="ſpeculationem" type="context">ſpeculationẽ</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 eſ <lb/>ſe vſque ad <var>.y.</var> lineæ <var>.s.n.</var> & imaginatione ſit com <lb/><reg norm="præhenſa" 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> ſimul cum linea <var>.o.b.</var> erunt <lb/>in vna eademq́ue ſuperficie plana, quam cha-<lb/>racteribus <var>.y.</var> R<unsure/>. notemus .et <var>.i.c.</var> eius erit ſe-<lb/>ctio communis cum plano, in quo quæritur <reg norm="pun- ctum" type="context">pũ-<lb/>ctum</reg>, et <var>.f.k.</var> ipſius plani cum triangulo <var>.o.b.m.</var> <lb/>erit ſectio communis, & parallela ipſi <var>.q.d.</var> ex <ref id="ref-0022">.<lb/>6. lib. 11.</ref> quia <var>.k.f.</var> perpendicularis eſt ſuperfi-<lb/>ciei <var>.p.t.</var> ex .19. eiuſdem cum triangulus <var>.o.<lb/>b.m.</var> eidem ſuperficiei <var>.p.t.</var> ex .18. eiuſdem <lb/>perpendicularis exiſtat. </s> + <s xml:id="s1545" xml:space="preserve">Vnde perſpicuè pa-<lb/>tet ratio quare protracta<unsure/> ſit parallela <var>.b.c.</var> et <lb/>quare ducta ſit <var>.i.c.</var> et coniuncta <var>.x.m.</var> cum <var>.x.<lb/>p.</var> directè, & quare ducta ſit <var>.o.m.</var> et <var>.f.k</var>. </s> + <s xml:id="s1546" xml:space="preserve">Lau-<lb/>do igitur vt ſemper præſupponatur <var>.p.x.</var> perpen <lb/>dicularis baſi ipſius plani & præſupponatur, (vt <lb/>rem totam vnò verbo complectar) ſuperficies <var>.<lb/>p.t.</var> perpendicularis plano, & orizonti. </s> + <s xml:id="s1547" xml:space="preserve">Quod <lb/>reliquum eſt, neceſſariv<unsure/>m non eſt, niſi ad ſpe-<lb/>culandum. </s> + <s xml:id="s1548" xml:space="preserve">Neceſſariæ ergo non ſunt aliæli-<lb/>neæ, quàm.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 ſit ipſi <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 cauſas, 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 noſtro plano perpendiculari orizonti, & ita <reg norm="locatum" type="context">locatũ</reg><unsure/>, <lb/>vt poſtremo diximus, inueniamus; </s> + <s xml:id="s1551" xml:space="preserve">duas hîc ſubſcriptas figuras conſiderabimus <var>.<lb/>G.</var> corpoream, & G. ſuperficialem, ſimiles duabus <var>.E.E.</var> proximè præcedentibus, <lb/>in quarum corporea ſit linea <var>.b.M.</var> altitudinis perpendicularis orizonti. </s> + <s xml:id="s1552" xml:space="preserve">Quare ſi <lb/>deſiderabis inuenire in noſtro plano ſitum puncti <var>.M.</var> ideſt punctum radij <var>.o.M.</var> vi-<lb/>ſualis in quo ipſe radius à plano eſt diuiſus, quod ſit <var>.R.</var> quamuis extra <reg norm="triangulum" type="context">triangulũ</reg> <lb/><var>i.q.d.</var> tibi imaginatione confige ductam eſſe lineam <var>.p.b.</var> quæ erit ſectio commu-<lb/>nis orizontis cum ſuperficie <var>.o.p.b.M.</var> quæ ſuperficies erit perpendicularis ipſi ori-<lb/>zonti ex .18. lib 11. </s> + <s xml:id="s1553" xml:space="preserve">Quòd autemnon minus <var>.o.p.</var> quàm.M.b. ſit in vna eademq́ue <lb/>ſuperficie dubitandum non eſt, quia ſi 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/>demq́ue ſuperficie conſtantem, 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 ſuperficie ſit ex .2. lib. 11. ſe-<lb/>quetur ſuperficiem <var>.o.p.b.M.</var> planam eſſe, & vnicam, cuius communis ſectio cum no-<lb/>ſtro plano ſit. θ.K.R. quæ perpendicularis orizonti exiſtet ex .19. lib. 11. eritq́ue pa-<lb/>rallela ipſi <var>.i.x.</var> ex .6. eiuſdem. </s> + <s xml:id="s1555" xml:space="preserve">Imaginare nunc erectam eſſe <var>.m.T.</var> æqualem ipſi <var>.<lb/>b.M.</var> orizonti perpendicularem, quæ extenſa erit in ſuperficie <var>.p.t.</var> quod ex ſe ad <lb/>conſiderandum admodum facilè, clarumq́ue exiſtit, 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> ęqualis <var>.b.M.</var> parallela eſt <lb/>ipſi <var>.b.M.</var> ex .6. lib. 11. præter hæc <var>.b.m.</var> parallela eſt ipſi <var>.q.d.</var> quia ſic fuit ducta <lb/>ſuperius, vnde <var>.M.T.</var> parallela erit ipſi <var>.q.d.</var> ex .9. vndecimi, & obid perpendi-<lb/>cularis erit ſuperficiei <var>.b.t.</var> ex .8. eiuſdem. </s> + <s xml:id="s1557" xml:space="preserve">Nunc ſit <var>.R.V.</var> communis ſectio trian-<lb/>guli <var>.o.M.T.</var> cum noſtro plano, vnde <var>.R.V.</var> perpendicularis erit ſuperficiei <var>.p.t.</var> <lb/>ex .19. lib. 11. quam ob cauſam parallela erit ipſi <var>.q.d.</var> ex .6. aut ex .9. eiuſdem <lb/>quia ex .6. dicta, parallela eſt ipſi <var>.M.T</var>. </s> + <s xml:id="s1558" xml:space="preserve">Atſi <var>.R.V.</var> parallela eſt ipſi <var>.q.d.</var> <lb/>etiam <var>.f.K.</var> probatum iam fuit parallelam eſſe eidem, ergo <var>.R.V.</var> parallela erit <lb/>ipſi <var>.K.f.</var> ex .30. primi, </s> + <s xml:id="s1559" xml:space="preserve">Vnde ex .34. æqualis erit ipſi <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> ſupra figuram <var>.E.</var> ſuperficialem, & erigamus <var>.m.T.</var> perpendi-<lb/>cularem ipſi <var>.m.p.</var> ſed æqualem perfectæ altitudini, & ducamus <var>.T.o.</var> vt ſecet li-<lb/>neam <var>.i.x.</var> in puncto <var>.V.</var> ab ipſo ducentes <var>.V.R.</var> parallelam ipſi <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 ipſi <var>.i.a.</var> habebimus altitudinem <var>.k.R.</var> quam quærebamus in <lb/>noſtro plano. </s> + <s xml:id="s1561" xml:space="preserve">Quod cum ſui 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 ſcien-<lb/>tia erudito relinquo. </s> + <s xml:id="s1562" xml:space="preserve">ideſt, vt probetur <var>.k.R.</var> <reg norm="ſu- perficialem" type="context">ſu-<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="ſuper- fluitatem" type="context">ſuper-<lb/>fluitatẽ</reg> linearum, & hoc <reg norm="accommodantes" type="context">accõmodantes</reg> vt in figura <var>.<lb/>F.</var> diligenter conſideretur 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 ſit 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 abſcindatur in puncto <var>.s.</var> ita quòd <var>.c.s.</var> æqualis ſit alti <lb/>tudini perfectæ, deinde coniungatur rectà. s. cum <var>.i</var>. </s> + <s xml:id="s1566" xml:space="preserve">Tunc ſi ab <var>.k.</var> vſque 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"/> + neę <var>.c.s.</var> hæc <var>.R.k.</var> erit altitudo <lb/>quæſita ſeu 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 eſt quin <var>.f.V.</var> ſit æqualis alti-<lb/>tudini quęſitæ ſeu degradatę, quo <lb/><reg norm="tieſcunque" type="simple">tieſcunq;</reg> ergo <reg norm="probauerimus" type="simple">ꝓbauerimus</reg> <var>.k.R.</var> <lb/>æqualem eſſe lineæ <var>.f.V.</var> habebi-<lb/>mus propoſitum. </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 ſimilitudine <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/>ſed <var>.c.s.</var> ſumpta 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">ꝗnti</reg>, & ex .11. eiuſ-<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. eiuſdem <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 philoſophis obſeruatus, eſt etiam vtilis, <reg norm="compendioſaque" type="simple">compendioſaq́;</reg> via <lb/>progreditur, cuius ſpeculationem, in ſubſcripta figura, quadam ex parte <reg norm="ſecun- dum" type="context">ſecũ-<lb/>dum</reg> morem antiquum, quadam etiam ex parte ſecundum ingenij mei vires <reg norm="conſtru- cta" type="context">cõſtru-<lb/>cta</reg>, cognoſcemus. </s> + <s xml:id="s1573" xml:space="preserve">In qua ego diuiſi <var>.x.i.</var> in puncto <var>.s.</var> ab <var>.x.</var> ita eleuato, quanta eſt <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 ipſius <var>.M.T.</var> et <var>I.s.</var> duxi ſupponendo eſſe <var>.I.</var> <reg norm="punctum" type="context">punctũ</reg> pſpectiuæ <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>antiquos, ideſt angulum ſupremum trianguli antiquorum à punctoq́ue <var>.k.</var> meo duxi <lb/><var>k.f.</var> parallelam ipſi <var>.c.m.p.</var> vſque ad <var>.i.x.</var> in puncto <var>.f.</var> & à puncto à communi ipſis <var>.k.f.</var> <lb/>et <var>.i.x.</var> vſque ad <var>.I.s.</var> duxi quoque <var>.A.B.</var> parallelam ipſi <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 conſiderans modum, quem ego de figuris <var>.G.H.</var> antecedentibus <lb/>præſcripſi, videndum eſt, an punctum <var>.B.</var> tribus lineis <var>.A.B.I.s.</var> et <var>.R.V.</var> quarum hęc vl <lb/>tima à me iam ducta fuit, commune exiſtat, ideſt vtrum <var>.A.B.</var> æqualis exiſtat ipſi <var>.K.<lb/>R.</var> quam ſecundum modum à me adinuentum, reuera ſcimus eſſe deſideratam altitu <lb/>dinem in perſpectiua. </s> + <s xml:id="s1575" xml:space="preserve">Quod tunc à nobis probatum erit, quando rationibus clarè <lb/>patebit ipſam <var>.A.B.</var> æqualem eſſe ipſi <var>.f.V</var>. </s> + <s xml:id="s1576" xml:space="preserve">Quamobrem ducamus <var>.I.f.</var> vſque ad <var>.ω.</var> <lb/>lineæ <var>.c.p.</var> vnde ratione ſimilitudinis triangulorum manifeſtè intelligemus, eandem <lb/>proportionem eſſe ipſius <var>.m.T.</var> ad <var>.f.V.</var> quæ eſt <var>.m.o.</var> ad <var>.f.o.</var> & eius, quæ eſt <var>.m.o.</var> ad <var>.f.<lb/>o.</var> quæ eſt <var>.ω.I.</var> ad <var>.f.I.</var> & eius, quæ eſt <var>.ω.I.</var> ad <var>.f.I.</var> quæ eſt <var>.x.I.</var> ad <var>.A.I.</var> & eius, quæ eſt <var>.x.<lb/>I.</var> ad <var>.A.I.</var> quæ eſt <var>.x.s.</var> ad <var>.A.B.</var> ideſt vt eius, quæ eſt <var>.m.T.</var> ad <var>.A.B.</var> ſed 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 ſequitur <var>.A.B.</var> æqualem eſſe <var>.f.V.</var> ex .9. quinti Eucli. <reg norm="atque" type="simple">atq;</reg> etiam <lb/>ipſi <var>.k.R.</var> quod à nobis propoſitum eſt 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 ſermonem de figuris ſu-<lb/>perficialibus orizontalibus, ſeu de plantis, <lb/>pulcherrimum quendam modum, quem ego ad <lb/>locandum quodlibet punctum in perſpectiua, <lb/>(degradatum cum fuerit <reg norm="parallelogrammum" type="context">parallelogrãmum</reg> quod <lb/>dam rectangulum, in noſtro plano perpendicula <lb/>ri orizonti, quemadmodum in ſuperioribus figu-<lb/>ris <var>.A.</var> demonſtrauimus) conſideraui, ſilentio <lb/>haud prætereundum eſſe.</s> + </p> + <p> + <s xml:id="s1579" xml:space="preserve">Sit igitur in ſubſcripta hîc figura <var>.K.</var> in paralle <lb/><reg norm="logrammo" type="context">logrãmo</reg> perfecto <reg norm="punctum" type="context">pũctum</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> vſque 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> ipſorum <lb/>laterum, ideſt vt non ſecent 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="correſpondet" type="context">correſpõdet</reg> <var>.<lb/>u.</var> perfecti, mediante linea <var>.e.f.</var> quæ erit <var>.u.f.</var> degra <lb/>dita in noſtro plano. </s> + <s xml:id="s1582" xml:space="preserve">Deinde ſumatur punctum <var>.<lb/>n.</var> in linea <var>.q.d.</var> tam diſtans à <var>.q.</var> quàm.g. diſtat ab <var>.<lb/>u.</var> ducaturq́ue linea <var>.i.n.</var> quæ lineam <var>.e.r.</var> in puncto <lb/>c. diuidet, quod exijs, quæ ſuperius iam diximus <lb/>ad ipſum <var>.g.</var> referetur. </s> + <s xml:id="s1583" xml:space="preserve">Ducendo poſtea lineam oc <lb/>cultam <var>.q.c.</var> patebit eam correſpondere lineæ <var>.q.g.</var> quæ ſecans lineam <var>.e.f.</var> in puncto <var>.<lb/>t.</var> hoc, communi ſcientiæ ratione, reſpondebit ipſi <var>.b.</var> vt omnes cognoſcent.</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 ſi 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 ſit vna diagonalis quam <lb/>volueris puta <var>.q.a.</var> deinde à puncto <var>.b.</var> ad reli-<lb/>quum angulum oppoſiti lateris ducta ſit recta <var>.<lb/>b.d.</var> ita quod à diagonali ſecetur in puncto <var>.ω.</var> <lb/>per quod punctum demum à reliquo angulo la <lb/>teris <var>.q.u.</var> ducta ſit <var>.u.ω.</var> vſque ad latus <var>.q.d.</var> in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.f.</var> quo facto, ita faciendum erit in rectangu <lb/>lo degradato, hoc eſt ducenda erit diagonalis <var>.<lb/>q.r.</var> quę correſpondet diagonali <var>.q.a.</var> perfecti <lb/>deinde <var>.f.e.</var> quæ correſpondet rectæ <var>.f.u.</var> perfe-<lb/>cti, quæ etiam interſecabitur à diagonali <var>.q.r.</var> <lb/>in puncto <var>.o.</var> correſpondens <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> vſque ad <var>.t.</var> <lb/>in latere <var>.q.e.</var> hoc <reg norm="punctum" type="context">punctũ</reg> <var>.t.</var> correſpondebit 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 ſi loco diametri <var>.q.a.</var> ſumpta <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 correſpondens ipſi <var>.f.</var> in figura de-<lb/>gradata erit in latere ſupremo <var>.e.r.</var> correſpon-<lb/>dens lateri <var>.u.a.</var> & ita ducenda erit diameter <var>.d.<lb/>e.</var> correſpondens diametro <var>.d.u.</var> et <var>.q.f.</var> ſurſum <lb/>verſus correſpondens <var>.q.f.</var> imum verſus deinde <var>.<lb/>r.o.</var> reſpondens <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> ſuperficiali perſpectiuæ facillimum modum locandi quoduis <lb/>punctum in perſpectiua 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, quàm latitudinis protenſa, quæ quidem latitudo erat ad duos circiter pedes <lb/>deſignata 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 ſuperficiem eiuſdem tabulæ excederent, vt <reg norm="vnum" type="context">vnũ</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 poſtea, 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 poſteà. <var>o.i.</var> parallela ipſi <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 extenſam, quanta conſtat <var>.A.B.</var> <reg norm="conſtitui" type="context">cõſtitui</reg> etiam, vt <reg norm="quoddam" type="context">quoddã</reg> angulum re <lb/>ctum materiale tantæ magnitudinis, quanta nobis vſui eſſe poterat ſuper eadem ta-<lb/>bula; </s> + <s xml:id="s1588" xml:space="preserve">necnon regula quædam materialis neceſſariæ longitudinis <reg norm="ſtatuerentur" type="simple">ſtatuerent̃</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/>ideſt <var>.p.E.</var> ſuper eadem tabula adeò diſtantem 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, ſignaui. </s> + <s xml:id="s1590" xml:space="preserve">Hæc autem diſtantia, 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 ſignificabat. </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/>quotieſcunque volebam in perſpectiua aliquid delineare. </s> + <s xml:id="s1592" xml:space="preserve">Quod cum facere deſi-<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 conſtitaere <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.b.</var> <reg norm="ſumebam" type="context">ſumebã</reg> <reg norm="angulum" type="context">angulũ</reg> <reg norm="rectum" type="context">rectũ</reg> <reg norm="materialem" type="context">materialẽ</reg>, ſeu <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> tranſire faciebam, & vbi hoc latus <reg norm="lineam" type="context">lineã</reg> <lb/><var>E.p.</var> diuidebat, punctum .c ſignabam per quod efficiebatur, vt regula <var>.i.M.</var> tranſiret, <lb/>quieſceretq́ue aliquantulum aliquo modo in huiuſmodi ſitu, opera deinde circini <lb/>interuallum <var>.b.c.</var> ſumebam, & in <var>.p.E.</var> à puncto <var>.x.</var> verſus <var>.E.</var> punctum <var>.m.</var> <reg norm="ſignabam" type="context">ſignabã</reg>: </s> + <s xml:id="s1594" xml:space="preserve">per <lb/>quod faciebam, vt tranſiret 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> tranſibat, quod quidem latus regulam <var>.i.M.</var> in puncto <var>.k.</var> <lb/>(quod ſtatim ſuper folio papyri ſignabatur) interſecabat, atque hoc erat <reg norm="punctum" type="context">punctũ</reg>, <lb/>quod quærebam, puncto <var>.b.</var> correſpondens. </s> + <s xml:id="s1596" xml:space="preserve">Huiuſmodi effectus rationes ab ijs, quæ <lb/>ſuperius dixi eliciuntur. </s> + <s xml:id="s1597" xml:space="preserve">Atque hæc ad baſes rerum, vt in ſubſcripta figura eluceſcit, <lb/>ſpectabant.</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, vſus ſum 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 ſeſe offerebat occaſio. </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, ſeu normæ regulæ <var>.B.C.</var> anniteretur, aliud verò <lb/>per <var>.m.</var> in rectitudine cuius ſignabam <var>m.T.</var> interuallum æquale altitudini perfecti, <lb/>ideſt punctum <var>.T.</var> æqualiter diſtans ab <var>.m.</var> tranſire faciebam, deinde regulam <var>.o.Q.</var> <lb/>per punctum <var>.T.</var> tranſire quoque faciebam: </s> + <s xml:id="s1600" xml:space="preserve">& notabam interſectionem ipſius 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, aliudq́ue per punctum <var>.k.</var> tranſire faciebam, & in huiuſmodi <lb/>rectitudine à puncto k. ſignabam quandam menſuram æ qualem lineæ <var>.f.V.</var> quę 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, ideſt ba <lb/>ſis eiuſdem perfecti tranſire faciebam. </s> + <s xml:id="s1603" xml:space="preserve">Et in huiuſmodi rectitudine ſignabam <var>.c.s.</var> <lb/>æquale interuallum altitudini perfecti, ideſt punctum <var>.s.</var> ita diſtans à <var>.c.</var> efficiendo de <lb/>inde, vt latus anguli recti, lateri <var>.B.C.</var> tabulæ anniteretur, aliudq́ue per punctum <var>.k.</var> <lb/>tranſire faciens ſignabam <var>.k.R.</var> indeterminatam. </s> + <s xml:id="s1604" xml:space="preserve">Faciens deinde tranſire regulam <lb/><var>i.M.</var> per punctum <var>.s.</var> notabam punctum <var>.R.</var> interſectionis eiuſdem 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, expeditiorq́ue 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 poſtquam de perſpectiuis inter nos ſermonem <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 ſatis laudatus Daniel Barba-<lb/>rus ſe accepiſſe profitetur à Ioanne Zamberto patritio Veneto, qui ad verbum om <lb/>nia deſumpſerat a Ioanne Cuſino Pariſienſe. </s> + <s xml:id="s1609" xml:space="preserve">Nec parum mirabar peritiſſimum il-<lb/>lum Cuſinum, quod in capite quarto ſecundæ partis perſpectiuæ, vt quodpiam pla-<lb/>num quadrilaterum in quadratam figuram redigeret, ſuper vnam datam <reg norm="lineam" type="context">lineã</reg> qua-<lb/>dratam compoſuiſſe. </s> + <s xml:id="s1610" xml:space="preserve">Non animaduertens diſtantiam aut interuallum <var>.b.c.</var> degra-<lb/>datum ín linea <var>.b.f.</var> (quod eſt <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 ſubſcriptæ hic figuræ facilè depræhendi poteſt. </s> + <s xml:id="s1611" xml:space="preserve">Vbi <var>.b.c.</var> latitudo eſſe po-<lb/>teſt, 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">latũ</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> ſcientificè intelligi poteſt. </s> + <s xml:id="s1613" xml:space="preserve">Hinc, ad inueniendum perfectum alicuius pla-<lb/>ni degradati, non ſufficere degradationem ſolum interualli inter duos terminos <lb/>ſolos, ideſt <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">ꝙ</reg> non ſunt omnia quadrata. </s> + <s xml:id="s1614" xml:space="preserve">Ad <reg norm="inquirendum" type="context">inquirendũ</reg> <lb/>igitur perfectum alicuius plani <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, alicuius propoſiti degradati, oportet <lb/>vniuerſam degradationem tam latitudinis, <reg norm="quam" type="simple">ꝗ̃</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 ſubſcripta hic figura, <reg norm="volendo" type="context">volẽdo</reg> inuenire perfectum paral <lb/>lelogrammum degradati <var>.b.h.l.m.</var> dando diſtantiam 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 vſque ad planiſitum in quo degradatio facta ſit: </s> + <s xml:id="s1616" xml:space="preserve">ſtatim altitudo <var>.A.</var> oculi à pe <lb/>de, quæ tanta ſemper eſſe debet quanta eſt altitudo trianguli <var>.b.n.m.</var> qui clauditur, <lb/>protrahendo <var>.m.l.</var> et <var>.b.h.</var> vſque ad concurſum 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> & vſque 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> vſque ad <var>.c.</var> ipſius <var>.d.b.m.</var> produ <lb/>ctæ ducere <var>.A.E.c.</var> atque deinde protrahere lineam <var>.o.b.</var> vſque ad <var>.T.</var> ita vt <var>.b.T.</var> ęqua <lb/>lis ſit ipſi <var>.b.c.</var> & ad ipſam à puncto <var>.m.</var> ducere parallelam <var>.m.R.</var> & à puncto <var>.T.</var> ducere <var>.<lb/>T.R.</var> parallelam ipſi <var>.b.m</var>. </s> + <s xml:id="s1618" xml:space="preserve">Vnde ex .34. primi Eucli <var>.m.R.</var> æqualis erit ipſi <var>.b.T.</var> et <var>.R.<lb/>T.</var> ipſi <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 eſt hic quod magis nos commoueat, quia cum ex linea <var>.b.c.</var> quadratum <var>.b.g.</var> pro <lb/>duxerit, vult eum poſtea degradare. </s> + <s xml:id="s1620" xml:space="preserve">Quod vt faciat (hanc figuram videbis in cap <num value="4">.<lb/>4.</num> ſecundæ partis Danielis Barbari) oculum <var>.A.</var> in eadem ſuperficie extenſa, <lb/>quadrati <var>.b.g.</var> collocat. </s> + <s xml:id="s1621" xml:space="preserve">Quod rectè fieti non poteſt, quia oculum hoc mo-<lb/>do locantes, viſualesq́ue radios beneficio vnius plani ſituati in <var>.b.f.</var> ſecantes in ipſo <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, & ſi <lb/>natura ſua ſit omnibus notum, ponit <reg norm="tantum" type="wordlist/context">tñ</reg> id ipſum Vitelio pro quinta propoſitione <lb/>quarti libri de perſpectiua. </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 ſemper videri mediantibus angulis <var>.b.A.c.</var> et <var>.f.A.g.</var> quod fieri <reg norm="non" type="context">nõ</reg> poteſt, quem-<lb/>a dmodum ex mea figura corporea <var>.A.</var> facilè cognoſcere 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 ſuperfi-<lb/>ciem <var>.s.a.</var> exiſtunt videntur, vnde ſi quis imaginaretur in puncto <var>.p.</var> oculum eſſe, & <lb/>ab ipſo 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 diuerſis igitur an-<lb/>gulis, pro maiori parte, deteguntur latera, à partibus quadrati tam degradati, quàm <lb/>perfecti, quæ non ſunt anguli <var>.b.A.c.</var> et <var>.f.A.G</var>. </s> + <s xml:id="s1626" xml:space="preserve">Quod vero idem poſtea dicat eam <lb/>proportionem eſſe ab <var>.b.E.</var> ad <var>.f.h.</var> & ſimul ad <var>.c.g.</var> quæ eſt 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 conſiderandum relinquo; </s> + <s xml:id="s1628" xml:space="preserve">cum rationabilis degrada-<lb/>tio eſſe debeat, qua ratione neceſſarium ſit, vt diſtantiæ reſq́ue, in vna & <reg norm="eadem" type="context">eadẽ</reg> pro-<lb/>portione cum altitudine oculi ad rem degradatam exiſtant? </s> + <s xml:id="s1629" xml:space="preserve">Cum poſtea degrada-<lb/>uerit <reg norm="quadratum" type="context">quadratũ</reg>, is ſcriptor, in figura <var>.d.b.c.e.</var> eum bene & ex perſpectiuæ optimis <lb/>legibus degradatum fuiſſe probare nititur; </s> + <s xml:id="s1630" xml:space="preserve">ſolum probans <var>.d.e.</var> æqualem eſſe ipſi <var>.<lb/>E.h.</var> <reg norm="qua" type="wordlist">q̃</reg> <var>.E.h.</var> <reg norm="ſecundum" type="context">ſecundũ</reg> ipſum eſt degra datio lateris <var>.c.g.</var> & <reg norm="cum" type="context">cũ</reg> ſuperius dixerit, ſetria <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 ſi <var>.c.g.</var> <lb/>perfecti, degradatum eſt in <var>.E.h</var>: et <var>.d.e.</var> rectè protracta exiſtit, cum ſit æqua-<lb/>lis ipſi <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/>ſa ipſis <var>.b.E.</var> et <var>.f.h.</var> quæ, ex ipſo, ſunt 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 reſpondere, <reg norm="quod" type="simple">ꝙ</reg> hoc pacto nulla ſuperficies 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 ſunt degradata, <reg norm="eiusque" type="simple">eiusq́;</reg> <reg norm="proportionalitates" type="simple">ꝓportionalitates</reg> ma <lb/>lè intellectæ nil probant. </s> + <s xml:id="s1634" xml:space="preserve">quia ſi 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 exiſterent, quarum quælibet maior eſt <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 ſcribit eam eſſe rationem, aut comparationem ab <var>.A.d.</var> ad <var>.b.<lb/>E.</var> quæ eſt ab <var>.d.c.</var> ad <var>.b.c.</var> eandemq́ue eſſe ab <var>.E.h.</var> ad <var>.c.g.</var> quæ eſt ab <var>.A.E.</var> ad <var>.A.c.</var> nil <lb/>probet; </s> + <s xml:id="s1636" xml:space="preserve">nec ſimilitudinem triangulorum, nec aliquam propoſitionem Eucli. citans. <lb/></s> + <s xml:id="s1637" xml:space="preserve">In quo excuſari non poteſt, quòd non ſoleat Euclidem, aut alium quemuis autorem <lb/>citare, cum vel in ipſo operis principio capite .3. primæ partis, A pollonium <reg norm="Pergeum" type="context">Pergeũ</reg> <lb/><reg norm="Euclidemque" type="simple">Euclidemq́;</reg>, & ſi etiam præter rem, citet. </s> + <s xml:id="s1638" xml:space="preserve">Deinde <reg norm="quum" type="context">quũ</reg> idem probare vult <var>.d.e.</var> æqua <lb/>lem eſſe ipſi <var>.E.h.</var> eandem inquit eſſe proportionem <var>.a.b.</var> ad <var>.a.d.</var> quæ eſt ipſius <var>.A.c.</var> <lb/>ad <var>.A.E.</var> quod & ſi verum ſit, hic tamen modus ratiocinandi nullo ordine nititur, <lb/>quia rectius dixiſſet pro clariori intelligentia ipſius <var>.a.c.</var> ad <var>.a.e.</var> eandem proportio-<lb/>nem eſſe, quæ eſt <var>.A.c.</var> ad <var>.A.E.</var> propter ſimilitudinem, quæ inter duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> intercedit, cum <var>.E.e.</var> ſupponatur parallela ipſi <var>.A.a.</var> quod etiam vt de-<lb/>monſtraretur longiori oratione ei opus fuiſſet ſi voluiſſet intellectum eorum, qui pa <lb/>rum ſunt exercitati, perduci ad <reg norm="cognoſcendum" type="context">cognoſcendũ</reg> idem planè futurum de <var>.a.c.</var> ad <var>.a.e.</var> vt <lb/>eſt ipſius <var>.A.c.</var> ad <var>.A.E.</var> in hunc modum, ideſt 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/>ſint parallelæ. </s> + <s xml:id="s1639" xml:space="preserve">Vnde ex .4. ſexti. 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 ipſius <var>.E.c.</var> ad <var>.e.c.</var> & ex .19. eiuſdem de <var>.A.E.</var> ad <var>.a.<lb/>e.</var> vt ipſius <var>.A.c.</var> ad <var>.a.c.</var> & ex .16. iam dicta de <var>.A.E.</var> ad <var>.A.c.</var> vt ipſius <var>.a.e.</var> ad <var>.a.c.</var> ideſt <lb/>ipſius <var>.A.c.</var> ad <var>.A.E.</var> vt eſt ipſius <var>.a.c.</var> ad <var>.a.e</var>: Aut hoc alio modo, qui breuior eſt pro-<lb/>cedendum, incipiendo ſcilicet à ſecunda ſexti Eucli. dicendo <reg norm="quod" type="simple">ꝙ</reg> exiſtente <var>.E.e.</var> paral <lb/>lela ipſi <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 ſtatim 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 ſupradictis duabus propoſitionibus ideſt .29. primi, & 4. ſexti, cogno-<lb/>ſcitur idem planè eſſe de <var>.b.c.</var> ad <var>.d.e.</var> quod ipſius <var>.a.c.</var> ad <var>.a.e.</var> & ex eiſdem idem eſſe <lb/>de <var>.c.g.</var> ad <var>.E.h.</var> quod ipſius <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> ſed cum ex ſuppoſito <var>.c.g.</var> ſit æqualis ipſi <var>.c.b.</var> idem <lb/>erit de <var>.c.g.</var> ad <var>.e.d.</var> quod ipſius <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 eiuſdem <var>.c.g.</var> ad <var>.e.d.</var> ex .9. igitur eiuſdem <var>.d.e.</var> æqualis erit ipſi <var>.E.<lb/>h.</var> atque hic verus eſt modus ducendi intellectum parum exercitatum in cognicio-<lb/>nis campum. </s> + <s xml:id="s1641" xml:space="preserve">quem quidem mihi obſeruandum proponerem ſi onus ſcribendi ſu-<lb/>ſciperem ijs, qui in ſcientijs parum verſati ſunt, 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 ipſo adducta propter quam <var>.E.</var> repreſentatur oculo al-<lb/>tius quam <var>.b.</var> nempe eo quod <var>.A.</var> ſuperſtet ipſi <var>.E.</var> nihil valet, quia ſi inferius eſſet, <lb/>idem contingeret, ſed hoc euenit eo quod <var>.E.</var> altius eſt ipſo <var>.b</var>. </s> + <s xml:id="s1643" xml:space="preserve">Idem dico de <var>.h.</var> <lb/>vbi ſimiliter decipitur. </s> + <s xml:id="s1644" xml:space="preserve">Idem etiam in .7. cap. fallitur in ſecundo modo, quem oſten <lb/>dit pro ſecundo quadrato aliquo degradato à parallelogrammo degradato magis <lb/>longo quàm lato, cum ducat parallelam <var>.l.m.</var> ad <var>.b.c.</var> à puncto <var>.l.</var> interſection is ipſius <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 innoteſcit.</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 proceſsit, quia <reg norm="cum" type="context">cũ</reg> <lb/>angulus .2. trianguli perfecti magis diſtet à plano ſuper quod degradari debet <lb/>triangulum, quàm latus .1. 3. oppoſitum dicto angulo .2. & per confequens longère <lb/>motior ſit ab oculo, ipſe in degradato, <reg norm="eum" type="context">eũ</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 eſt, <reg norm="admittendum" type="context">admittendũ</reg>. <lb/></s> + <s xml:id="s1647" xml:space="preserve">Quod verò in ſecunda parte ab eo traditur, ideſt alius quidam modus quem de <reg norm="tranſ" type="context">trãſ</reg> <lb/>ferendis punctis à perfecto in degradato proponit, non eſt modus vniuerſalis; </s> + <s xml:id="s1648" xml:space="preserve">quia <lb/>ſi 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 ſit in ſubſcripta 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 ſit parallela duobus lateribus <var>.A.B.</var> et <var>.C.D.</var> in cuius quolibet puncto <var>.<lb/>Q.</var> ſit infimus terminus altitudinis oculi, & in <var>.<lb/> + <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> + T.</var> ad perpendiculum ipſius <var>.Q.</var> ſit verus ſitus <lb/>eiuſdem, tantum eleuatus à <var>.Q.</var> quanta eſt <lb/>medietas ipſius <var>.D.B.</var> ſitq́ue figura corpo-<lb/>rea finita ſimilis meæ <var>.A.</var> vnde <var>.Q.T.</var> æqualis <lb/>erit ipſi <var>.Q.æ.</var> & planum perpendiculare <reg norm="orizon- ti" type="context">orizõ-<lb/>ti</reg>, ſuper quod punctum <var>.k.</var> perfecti duci debet <lb/>ſit <var>.R.D.B.</var> ſintq́ue ductæ per imaginationem <lb/>lineæ <var>.T.K</var>: <var>Q.K.</var> et ſit <var>.K.N.</var> perpendicularis la-<lb/>teri <var>.C.D.</var> à quo puncto <var>.N.</var> imaginatione ſit <reg norm="con" type="context">cõ</reg> <lb/>præhenſa linea <var>.N.Q.</var> at que hæ tres lineæ ſectæ <lb/>ſint à plano in punctis <var>.c.i.</var> et .2. quorum <reg norm="punctum" type="context">punctũ</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 ſecti <reg norm="erunt" type="context">erũt</reg> <lb/>à plano <var>.R.B.D.</var> quorum communes ſectiones <lb/>erunt .1. 2. et <var>.D.c.</var> & quia <var>.N.K.D.i.</var> et <var>.æ.Q.</var> <lb/>inuicem ſunt parallelæ, ſequitur eandem pro-<lb/>portionem futuram ipſius <var>.Q.K.</var> ad <var>.K.i.</var> quæ eſt <lb/>ipſius <var>.æ.N.</var> ad <var>.N.D.</var> imaginatione concipien <lb/>do a puncto <var>.K.</var> vſque ad <var>.æ.Q.</var> quandam paral-<lb/>lelam ipſi <var>.N.æ.</var> quemadmo dum ex te ipſo intel <lb/>ligere potes. </s> + <s xml:id="s1651" xml:space="preserve">Sed ratione ſimilitudinis trian-<lb/>gulorum ita ſe 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 ipſius <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. eiuſdem <lb/>de <var>.Q.T.</var> ad <var>.Q.æ.</var> quod de .1. 2. ad <var>.c.D.</var> & exiſtente <var>.æ.Q.</var> ex ſuppoſito æquali ipſi.</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 huiuſmodi regula tunc bona redditur, quando <var>T.Q.</var> æqualis eſt ipſi .œ.<lb/>Q. ideſt medietati ipſius <var>.D.B.</var> at verò ſi æqualis non eſſet hoc minime ſequeretur, <lb/>vt facilè patet. </s> + <s xml:id="s1654" xml:space="preserve">Quòd verò .2. <var>R.Z.</var> &. ſint benè diſpoſita, dubitandum non eſt, quia <lb/>punctum <var>.i.</var> meæ hic ſubſcriptæ figuræ, quod coreſpondet K. eius ſiguræ adeò diſtat <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 manifeſtè cognoſci poteſt, ab eo, quod in <lb/>ſuperius poſitis figuris corporeis dixi. </s> + <s xml:id="s1657" xml:space="preserve">Huiuſmodi modus ducendi res in perſpectiua, <lb/>non ſolum à Gallis, ſed à Germanis etiam in vſum reducitur. </s> + <s xml:id="s1658" xml:space="preserve">Sed quia ad hæc <reg norm="vſque" type="simple">vſq;</reg> <lb/>tempora eiuſdem perfectionis ratio, quam ego ſuperius propoſui, <reg norm="nondum" type="context">nõdum</reg> in lucem <lb/>emerſit, factum fuit, vt <reg norm="errorum" type="context">errorũ</reg> laqueis irretirentur, ſumentes <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/>vniuerſalem eſſe putet, labitur in errorem, cum debuiſſet longitudinem ipſius <var>.T.Q.</var> <lb/>debere eſſe æqualem medietati ipſius <var>.D.B.</var> proferre. </s> + <s xml:id="s1660" xml:space="preserve">Aſſerit deinde diſtantiam ip-<lb/>ſius <var>.T.Q.</var> à latere <var>.B.D.</var> æ qualem eſſe debere lateri <var>.C.D.</var> quod neceſſarium non eſt, <lb/>quia in quibuslibet diſtantijs, iuſta operatio fieri poteſt, quemadmodum in ſubſcri-<lb/>pta hîc figura facile patet, ideſt, quòd quibuſcunque modis <var>.c.D.</var> æqualis remaneat <lb/>ipſi .1. 2. & ſic interualla, quæ <reg norm="per" type="simple">ꝑ</reg> tranſuerſum aguntur <reg norm="vſque" type="simple">vſq;</reg> ad <reg norm="medium" type="context">mediũ</reg> trianguli <var>.D.R.B.</var> <lb/>Neque etiam probandus eſt auctor ille, cum pro oculo, ſuum <var>.T.</var> loco <var>.Q.</var> à me poſi-<unsure/><lb/>ti, ponit, cum is locus ſit verus ſitus pedis eius quireſpicit, & non oculi. </s> + <s xml:id="s1661" xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> <lb/>Auctor iſte, modo vniuerſali intelligat, vt iam diximus, <reg norm="conſideretur" type="context">cõſideretur</reg> figura tertij mo <lb/>di primi cap. tertiæ partis, in qua ſuum oculum (vt ita dicam) ponit in <var>.o.</var> altius ſeu <lb/>diſtans à rectitudine lateris <var>.c.d.</var> plus quam ſit 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/>ſam, quæ me in alijs meis literis impulit ad <reg norm="dicendum" type="context">dicendũ</reg>, <reg norm="angulum" type="context">angulũ</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 cauſam imagineris in ſubſcripta hîc figura duo triangula <var>.q.o.u.</var> et <var>.q.p.u.</var> <lb/>quorum <var>.q.p.u.</var> perpendiculariter ſit ſuper ſuperficie trianguli <var>.q.o.p.</var> collocatum, <lb/>præcisè vt in mea figura corporea <var>.A.</var> ſuperficies verò trianguli <var>.q.o.p.</var> ſit exempli-<lb/>gratia <var>.V.M.</var> & trian-<lb/>guli <var>.u.o.p.</var> ſit <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> ſe-<lb/>ctio ſit <var>.V.p.o.x.</var> non <lb/>eſt enim <reg norm="dubitandum" type="context">dubitãdum</reg> <lb/>quin triangulum <var>.q.<lb/>p.u.</var> ſit perpendicula-<lb/>re triangulo <var>.q.o.p.</var> <lb/><reg norm="cum" type="context">cũ</reg> hoc ex .18. lib. 11. <lb/>Eucli. perpendicula-<lb/>re ſit ſuperficiei <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 ſuperficiei <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"/> + eſt angulum <var>.p.q.u.</var> obtuſum eſſe; </s> + <s xml:id="s1665" xml:space="preserve">Imaginemur ergo circa triang ulum <var>.p.q.u.</var> circun-<lb/>ſcriptum eſſe circulum, cuius portio <var>.p.q.u.</var> minor erit medietate eiuſdem medij cir-<lb/>culi, vt iam ex <ref id="ref-0023">30. Eucli. lib. tertij</ref> nouiſti. </s> + <s xml:id="s1666" xml:space="preserve">nunc imaginemur dictum circulum circum <lb/>lineam <var>.q.u.</var> loco axis verſus <var>.x.</var> moueri, vnde girus eiuſdem, per quem tranſibat linea <lb/><var>V.x.</var> remouebitur ab eadem linea non nihil cum motus erit à primo ſitu vſquequò <lb/>ad ſecandam 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 ſi <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 ſi <lb/>idem <reg norm="punctum" type="context">punctũ</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">ꝗ-<lb/>dẽ</reg> re tu ipſe 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">ꝙ</reg> <lb/>hæc Ioannis Cuſini di <lb/>cta ad hæc vſque tempora tanto in prætio ſint habita, vt ab excellentibus ſcriptori-<lb/>bus quaſi ſi proprij eorum ingenij partus eſſent, de verboad verbum vt theſauros, in <lb/>fuis <reg norm="ipſorummet" type="context">ipſorũmet</reg> libris reſcripta fuerint, quemadmodum iam omnes admonui in mea <lb/>gnomonica Orontium, Munſterum, <reg norm="aliosque" type="simple">aliosq́;</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 perſpectiua tertíum corpus regu <lb/>lare, <reg norm="quod" type="simple">ꝙ</reg> octo triangulis æquilateribus eſt term inatum, ſcire deſideras, hoc <reg norm="vnum" type="context">vnũ</reg> <lb/>eſt caput: </s> + <s xml:id="s1671" xml:space="preserve">vnde fiat, aut quomodo probetur quaſlibet duas facies oppoſitas eiuſ-<lb/>dem corporis octoaedri <reg norm="inuicem" type="context">inuicẽ</reg> æquidiſtantes eſſe. </s> + <s xml:id="s1672" xml:space="preserve">Quamobrem ſit hîc <reg norm="ſubſcriptum" type="context">ſubſcriptũ</reg> <lb/><reg norm="octoaedrum" type="context">octoaedrũ</reg>, cuius diameter vna ſit <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> æquidiſtantes eſſe contendo <lb/>ſint 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 ſint 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 ęquales erunt <reg norm="ſumendo" type="context">ſumẽdo</reg> eas vt <lb/>baſes <reg norm="triangulorum" type="context">triangulorũ</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="æquidiſtabunt" type="simple context">æꝗdiſtabũt</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">ꝗa</reg> ſi imaginabimur dia <lb/>metrum <var>.b.q.</var> tunc ex .4. aut ex .8. eiuſdem 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">ſed ob eaſdem rationes <var>.p.l.</var> paralle-<lb/>la eſt ipſi <var>.d.k.</var> vnde ex 15. lib. 11. facies <var>.b.p.l.</var> <lb/>parallela fit, aut æquidiſtans ipſi <var>.q.d.k.</var> ideſt <lb/>primum propoſitum.</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 diſtantiæ, aut interualli ſimul cum ſitu, in fa-<lb/>cie <var>.q.d.k.</var> quem latus <var>.p.l.</var> perpendiculariter reſpicit. </s> + <s xml:id="s1676" xml:space="preserve">Imaginemur à puncto <var>.u.</var> ſuper <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 conſtet, <reg norm="quodquidem" type="context">quodquidẽ</reg> triangulum, medietas eſt qua-<lb/>drilateri, ſeu. rumbi <var>.q.a.b.u.</var> cui vnaquæque dictarum quatuor facierum perpendi-<lb/>cularis exiſtit ex .4. ct .18. lib. 11. & ob id linea <var>.u.o.</var> extenſa in ſuperficie 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. eiuſdem lib <var>.o.u.</var> perpendicularis erit faciei <var>.b.p.l</var>. </s> + <s xml:id="s1677" xml:space="preserve">Ha bebimus ergo ſitum in fa-<lb/>cie <var>.q.d.k.</var> qui reſpicietur 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> ſit latus exagoni æquilateris circumſcrip tibilis ab eodem circu <lb/>lo, qui vnam ex faciebus triangularibus æquilateribus propoſiti corporis circunſcri-<lb/>bere pot eſt, ita oſtenditur. ſit <reg norm="comprehenſum" type="context">cõprehenſum</reg> imaginatione, triangulum <var>.a.q.u.</var> ſepara <lb/>tim, cuius latus <var>.a.u.</var> æquale eſt vni ex lateribus <reg norm="triangulorum" type="context">triangulorũ</reg> eiuſdem 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 ſit perpendicularis <var>.u.o.</var> ab vna <reg norm="extremitatum" type="context">extremitatũ</reg> <lb/>lateris maioris, ad vnum ex minoribus lateribus, quę perpendicularis intra triangu-<lb/>lum cadet, quia dictum triangulum oxigonium eſt. </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 ſint 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 an<lb/>gulus <var>.q.</var> ſit <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">ꝗa</reg><var>.a.u.</var> minor eſt diametro <reg norm="ſphae ræ" type="simple">ſphę<lb/> + <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> + ræ</reg> datum corpus circumſcribentis, cum <var>.q.</var> <lb/>dictæ ſphęrę ſuperficiem 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> ęqualem eſſe lateri <lb/>exagoni dicti, ſatis erit probare <var>.a.q.</var> ſeſqui <lb/>alteram eſſe ad <var>.a.o.</var> quia ſi in ſubſcripto <lb/>hîc circulo ducemus duas ſemidiametros <var>.<lb/>n.p.</var> et <var>.n.l.</var> ad. angulos <reg norm="trianguli" type="context">triãguli</reg> ęquilateri <var>.p.</var> <lb/>et <var>.l.</var> & cum quodlibet laterum ipſius exago <lb/>ni, ęquale ſit ſemidiametro 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. eiuſ <lb/>dem <var>.o.n.</var> ęqualiserit ipſi <var>.o.q.</var> ideſt <var>.q.a.</var> ſeſ <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> ſeſquialteram eſſe ad <var>.a.o.</var> eſt <reg norm="quoque" type="simple">quoq;</reg> <lb/>ſciendum primò omne latus trianguli ęquilateri in potentia ſeſquitertium eſſe ad <lb/>perpendicularem eiuſdem trianguli, quod vndecima lib. 14. Eucli. breuiter demon <lb/>ſtratum eſt.</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 poſteriore loco dixi <lb/>mus, & quia quadratum ipſius <var>.q.a.</var> eſt tantò minus aliorum duorum quadratorum <lb/>ſumma, quantum eſt duplum producti ipſius <var>.q.a.</var> in <var>.a.o.</var> ex .13. ſecundi, ſed alia duo <lb/>quadrata ſimul collecta faciunt .21. à quo numero ſubtrahendo quadratum ipſius <var>.a.<lb/>q.</var> ideſt nouem, remanebit numerus .12. pro duplo producti ipſius <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/>eſt ſimplex <lb/>productum <lb/>ipſius <var>.q.a.</var> <lb/><reg norm="in" type="wordlist">ĩ</reg> <var>a.o.</var> erit 6. <lb/>Sed <reg norm="quia" type="simple">ꝗa</reg> qua <lb/>dratum ip-<lb/>ſius <var>.q.a.</var> eſt <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/>ipſius <var>.q.a.</var> <lb/>in <var>.a.o.</var> pro <lb/>latere <var>.a.o.</var> <lb/>conſurgent <lb/>duo, cum er <lb/>go <var>.a.o.</var> ſint <lb/>duo tertia <lb/>ipſius <var>.a.q.</var> <lb/>certi <reg norm="erimus" type="simple">erimꝰ</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="DEſiderantes" type="context">DEſiderãtes</reg> ſcire 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 perſpectiuę Danielis <lb/>Barbari, ſeu Zamberti, eſſe veram <reg norm="altitudinem" type="context">altitudinẽ</reg> cor-<lb/>poris octoaedri, <reg norm="primum" type="context">primũ</reg> ſcire debemus <reg norm="quod" type="simple">ꝙ</reg> <reg norm="exiſtente" type="context">exiſtẽte</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, quę ſuperius <reg norm="iam" type="context">iã</reg> <lb/>diximus, facile percipi poteſt; </s> + <s xml:id="s1688" xml:space="preserve">ex penultima primi <var>.<lb/>b.l.</var> in potentia, ſeſquioctaua erit ad <var>.k.l.</var> ipſa 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">cũ</reg> eſſet <var>.h.<lb/>l.</var> exiſteret in potentia tripla ad <var>.h.k.</var> & ſeſquialtera <lb/>ad <var>.l.k.</var> & ſeſquitertia ad <var>.l.b.</var> & ſic ad <var>.h.b.</var> vnde <var>.l.h.</var> <lb/>æqualis eſſet vni ex lateribus <reg norm="trianguli" type="context">triãguli</reg> ęquilateri di-<lb/>cti corporis. </s> + <s xml:id="s1689" xml:space="preserve">Ex rationibus igitur ſuperius hîc poſi-<lb/>tis <var>.l.k.</var> erit altitudo dicta, id eſt diſtantia inter duas <lb/>facies inuicem oppoſitas, 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">aliũ</reg> <reg norm="non" type="context">nõ</reg> <reg norm="paruum" type="context">paruũ</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">cũ</reg> ſint acuti, <reg norm="nam" type="context">nã</reg> <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> minor eſt angulo cubi ſolido.</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 ſcitißimè, de mechn<unsure/>-<lb/>nicis, at cum natura <reg norm="vſusque" type="simple">vſusq;</reg> aliquid ſemper vel nouum, vel <lb/>Latens in apertum emittere ſoleant, nec ingenui aut grati ſit <lb/>animi, posteris inuidere, ſi quid ei contigerit comperuiße prius <lb/>tenebris inuolutum: </s> + <s xml:id="s1693" xml:space="preserve">cum tam multa ipſe ex aliorum diligentia <lb/>ſit conſequut 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 biſce mechanicis verſantur, nuſquam ante bac tentata, <lb/>aut ſatis exastè explicata in medium proferre volui: </s> + <s xml:id="s1695" xml:space="preserve">quo vel iuuandi deſiderium, vel <lb/>ſaltem non ocioſi 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 ſitus 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 poſitum 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 diuerſa ratione ſitus ipſius brachij. </s> + <s xml:id="s1698" xml:space="preserve">ſit 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> ſit pondus, & <var>.C.O.</var> linea inclinationis, ſeuicineris<unsure/> <var>.C.</var> verſus cen-<lb/>trum mundi, cum qua <var>.B.C.</var> angulum rectum conſtituat in puncto <var>.C</var>. </s> + <s xml:id="s1699" xml:space="preserve">Exiſtente <lb/>igitur in huiuſmodi ſitu brachio <var>.B.C.</var> dico pondus <var>.C.</var> grauius futurum, quam <lb/>in alio quolibet ſitu. </s> + <s xml:id="s1700" xml:space="preserve">quia ſupra centrum <var>.B.</var> omninò non quieſcet, quemadmodum <lb/>in quouis alio ſitu faceret. </s> + <s xml:id="s1701" xml:space="preserve">Ad quod intelligendum, ſit dictum brachium, in ſitu <var>.B.<lb/>F.</var> cum eodem pondere in puncto <var>.F.</var> & linea itineris ſeu inclinationis dicti ponderis <lb/>ſit <var>.F.u.M.</var> per quam lineam dictum pondus progredi non poteſt, niſi 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 ſupra cen <lb/>trum <var>.B.</var> mediante brachio <var>.B.F.</var> nititur. <lb/></s> + <s xml:id="s1703" xml:space="preserve">Eſt quidem verum, quòd pondus <var>.C.</var> nec <lb/>ipſum etiam per lineam <var>.C.O.</var> proficiſce-<lb/>tur, quia iter extremitatis brachij eſt cir-<lb/>cularis, & <var>.C.O.</var> in vno <reg norm="quodam" type="context">quodã</reg> puncto eſt <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 ſuæ inclinatio-<lb/>nis (ponamus <var>.F.u.M.</var>) propinqua erit di <lb/>cto centro <var>.B.</var> quod ſequenti cap. proba-<lb/>bo, vt exempli gratia, ſit <var>.F.</var> ſuper <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> vndeſe- + <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 eſt <var>.A.F.</var> & <lb/>minus ſupra 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-ſupra dictum <lb/>centrum <var>.B.</var> quieſcet, & hac ratione grauius quoque erit, & quanto magis vicinum <lb/>erit ipſi <var>.A.</var> à dicto <var>.F.</var> tantò magis ſuper centrum <var>.B.</var> quoque quieſcet, vnde <reg norm="tantò" type="context">tãtò</reg> quo-<lb/>que leuius exiſtet. </s> + <s xml:id="s1707" xml:space="preserve">Idem dico de omni ſitu brachij per girum inferiorem <var>.C.Q.</var> vbi <lb/>pondus pendebit à centro <var>.B.</var> dictum centrum attrahendo, quemadmodum ſuperius <lb/>illud impellebat. </s> + <s xml:id="s1708" xml:space="preserve">Hæc verò omnia cap. ſequenti 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 diuerſo ſitu 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> poſitam inter centrum & lineam <var>.F.u.M.</var> inclinatio-<lb/>nis, quam pondus ab extremitate <var>.F.</var> liberum verſus mundi <reg norm="centrum" type="context">centrũ</reg> conficeret. </s> + <s xml:id="s1710" xml:space="preserve">Quod <lb/>vt facilius intelligamus imaginemur <reg norm="alterum" type="context">alterũ</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 eſt <var>.B.D.</var> cla-<lb/>rè cognoſcetur ex .6. lib. primi de ponderibus Archimedis, quòd ſi in puncto <var>.u.</var> col-<lb/>locatum erit pondus ipſius <var>.C.</var> libra nihil penitus à ſitu orizontali dimouebitur. </s> + <s xml:id="s1711" xml:space="preserve">Sed <lb/>perinde eſt quòd pondus <var>.F.</var> æquale <var>.C.</var> ſit in extremo <var>.F.</var> in ſitu brachij <var>.B.F.</var> <reg norm="quam" type="context">quã</reg> vt ſit <lb/>in puncto <var>.u.</var> in ſitu ipſius <var>.B.u.</var> orizontali. </s> + <s xml:id="s1712" xml:space="preserve">Ad cuius rei euidentiam imaginemur <reg norm="filum" type="context">filũ</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 ſi 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 eſt ſitu ipſius <var>.C.</var> quantò <var>.u.<lb/>B.</var> minus eſt ipſo <var>.B.C</var>. </s> + <s xml:id="s1713" xml:space="preserve">Idem aſſero ſi brachium eſſet in ſitu <var>.e.B.</var> quod facilè cogno-<lb/>ſcere poterimus, ſi imaginemur filum appenſum ipſi <var>.u.</var> brachij <var>.B.C.</var> & vſque ad <var>.e.</var> <lb/><reg norm="perpendicularem" type="context">perpendicularẽ</reg>, in quo extremo <reg norm="appensum" type="context">appensũ</reg> eſſet pondus æquale ponderi <var>.C.</var> & <reg norm="liberum" type="context">liberũ</reg> <lb/>ab <var>.e.</var> brachij <var>.B.e.</var> vnde libra orizontalis manebit. </s> + <s xml:id="s1714" xml:space="preserve">Sed ſi brachium <var>.B.e.</var> conſolida-<lb/>tum fuiſſet in tali ſitu cum orizontali <var>.B.D.</var> <lb/> + <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> + & <reg norm="appenſo" 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="aſcenderet" type="context">aſcẽderet</reg>, <reg norm="neque" type="simple">neq;</reg> deſcenderet. </s> + <s xml:id="s1715" xml:space="preserve">quia tantum <lb/>eſt quod ipſum ſit appenſum filo, <reg norm="quod" type="simple">ꝙ</reg> pendet <lb/>ab <var>.u.</var> quantum quòd ab ipſo liberum <reg norm="appem" type="context">appẽ</reg> <lb/>nſum fuiſſet <var>.e.</var> brachij <var>.B.e.</var> & hoc procede <lb/>ret ab eo quòd partim pendereta centro <var>.<lb/>B.</var> & ſi <reg norm="brachium" type="context">brachiũ</reg> eſſet in ſitu <var>.B.Q.</var> totum <reg norm="pon" type="context">põ</reg> <lb/>dus centro <var>.B.</var> remaneret appenſum, <reg norm="quem- admodum" type="context">quem-<lb/>admodũ</reg> in ſitu <var>.B.A.</var> <reg norm="totum" type="context">totũ</reg> dicto centro an-<lb/>niteretur. </s> + <s xml:id="s1716" xml:space="preserve">vnde fit vt hoc modo pondus <lb/>magis aut minus ſit 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 eſt cauſa proxima, & per ſe, <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">demq́;</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"/> + ſtat. </s> + <s xml:id="s1718" xml:space="preserve">Et quamuis appellem latus <var>.B.C.</var> orizontale, ſupponens illud angulum rectum <lb/>cum <var>.C.O.</var> facere, vnde angulus <var>.C.B.Q.</var> fit vt minor ſit recto, ob quantitatem vnius <lb/>anguli ęqualis ei, quem duæ <var>.C.O.</var> et <var>.B.Q.</var> in centro regionis <reg norm="elementaris" type="context">elemẽtaris</reg> <reg norm="conſtituunt" type="context">conſtituũt</reg>, <lb/>hoc tamen nihil refert, cum dictus angulus inſenſibilis ſit magnitudinis. </s> + <s xml:id="s1719" xml:space="preserve">Ab iſtis au-<lb/>tem rationibus elicere poſſumus, quod ſi 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> & ſi dictum <var>.u.</var> erit propius <var>.B.</var> quam puncto <var>.C.</var> pendebit ab ipſo, aut nitetur <lb/>ipſi amplius <reg norm="quam" type="context">quã</reg> exmedietate, & ſi magis verſus <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 cuiuſlibet ponderis, aut uirtus mouens re-<lb/>ſpectu alterius quantitatis cognoſcatur 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 hucuſque ſunt dicta, facilè intelligi poteſt, <reg norm="quod" type="simple">ꝙ</reg> quantitas <var>.B.u.</var> <lb/>quæ ferè perpendicularis eſt à centro <var>.B.</var> ad lineam <var>.F.u.</var> inclinationis, ea eſt, <lb/> + <anchor type="handwritten" xlink:label="hd-0155-01a" xlink:href="hd-0155-01"/> + quæ nos ducit in cognitionem quantitatis virtutis ipſius <var>.F.</var> in huiuſmodi ſitu, conſti <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/>ſint appenſa duo pondera, aut duæ virtutes mouentes <var>.e.</var> et <var>.c.</var> ita tamen <reg norm="quod" type="simple">ꝙ</reg> linea incli-<lb/>nationis <var>.e.</var> ideſt <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> ideſt <var>.a.c.</var> faciat angulum acutum, aut obtuſum 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. ſecetur 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">ſit <var>.o.t.</var> & puncto <var>.i.</var> appenſum ſit pondus æquale ipſi <var>.c.</var> cuius inclinationis <lb/>linea parallela ſit lineæ inclinationis ponderis <var>.e.</var> ſupponendo tamen pondus aut vir <lb/>tutem <var>.c.</var> ea ratione maiorem eſſe ea, quæ eſt <var>.e.</var> qua <var>.b.o.</var> maior eſt <var>.o.t.</var> abſque dubio <lb/>ex .6. lib. primi Archi. de ponderibus <var>.b.o.i.</var> non mouebitur ſitu, ſed ſi loco <var>.o.i.</var> imagi <lb/>nabimur <var>.o.t.</var> conſolidatam cum <var>.o.b.</var> & per lineam <var>.t.c.</var> attractam virtute <var>.c.</var> ſimiliter <lb/>quoque continget ut <var>b.o.</var> t; </s> + <s xml:id="s1724" xml:space="preserve">communi quadam ſcientia, non moueatur ſi tu. </s> + <s xml:id="s1725" xml:space="preserve">Eſt ergo <lb/> + <anchor type="handwritten" xlink:label="hd-0155-02a" xlink:href="hd-0155-02"/> + quod propoſuimus verum quantitatem alicuius ponderis reſpectu ad eam, quæ eſt <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 innoteſcit 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 ſequetur, 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 exſupradictis cauſis 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 ſtateræ, aut vectis, maior breuioris, ab ijs, quæ in ſu <lb/>perioribus capitibus diximus, ideſt <reg norm="quod" type="simple">ꝙ</reg> nitatur <reg norm="pendeatuem" type="context">pendeatuẽ</reg> magis aut minus à <lb/>centro pondus in extremitate brachij maioris poſitum, oboritur. </s> + <s xml:id="s1729" xml:space="preserve">Quamobrem illud <lb/>à nobis primò eſt cognoſcendum, ſtateras, aut vectes, puras mathematicas li-<lb/>neas non eſſe, ſed naturales, hincque exiſtere corpora cum materia coniuncta. </s> + <s xml:id="s1730" xml:space="preserve">Nunc <lb/>igitur imaginemur <var>.n.s.</var> eam ſuperficiem eſſe, quæ ſecundum longitudinem axem ſta <lb/>teræ ſcindit. </s> + <s xml:id="s1731" xml:space="preserve">& ſupponamus ipſius 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 ſit, quanta eſt ſpiſſitu-<lb/>do, aut craſſities ipſius ſtateræ à ſuperiori latere ad inferius, ad faciliorem intelligen-<lb/>tiam, ſupponendo <var>.n.s.</var> <reg norm="parallelogrammam" type="context">parallelogrãmam</reg>. </s> + <s xml:id="s1732" xml:space="preserve">Poſitis igitur duobus ponderibus æquali-<lb/> + <anchor type="handwritten" xlink:label="hd-0156-01a" xlink:href="hd-0156-01"/> + bus in extremitatibus brachiorum, experientia innoteſcit, <reg norm="quod" type="simple">ꝙ</reg> pondus ad <var>.u.s.</var> appen-<lb/>ſum, viol entiam faciet ponderi appenſo ad <var>.n.x.</var> ſed nos volumus inueſtigare <reg norm="causam" type="context">causã</reg> <lb/>huius effectus, quæ à nemine vnquam literarum monumentis, <reg norm="quod" type="simple">ꝙ</reg> ſciam, conſignata <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 ſtateram, aut vectem materialem eſſe & <var>.n.s.</var> eius ſuperficiem me-<lb/>diam, ſupponendo <var>.i.</var> eſſe centrum quo nititur dicta ſtatera aut vectis; </s> + <s xml:id="s1734" xml:space="preserve">Cum hocer-<lb/>go ita ſe habeat, ſint <var>.u.s.</var> et <var>.n.x.</var> lineæ inclinationum ponderum, & imaginemur, <reg norm="quod" type="simple">ꝙ</reg> <lb/>dicta pondera pendeant à punctis <var>.u.</var> et <var>.n.</var> vt reuera pendent, etiam ſi appenſa eſſent <lb/>ſub <var>.s.</var> et <var>.x.</var> quia punctum <var>.u.</var> & punctum <var>.n.</var> ita coniuncta ſunt cum <var>.s.</var> et <var>.x.</var> ut qui <reg norm="vnum" type="context">vnũ</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> quę <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, ſi quis ipſi <lb/>e. pondus ipſius <var>.u.</var> (<reg norm="quod" type="simple">ꝙ</reg> æquale eſt ponderi <var>.n.</var>) appenderet, id eandem planè vim habe <lb/>ret, quam pondus ipſius <var>.n.</var> habet, & ſtateram neque ſurſum, neque deorſum moue-<lb/>ret, quia ambo pondera ad centrum <var>.i.</var> mediantibus lineis <var>.e.i.</var> et <var>.n.i.</var> exęquo annite-<lb/>rentur, ſed dicto pondere poſito 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 diſtans à centro <var>.i.</var> <lb/> + <anchor type="handwritten" xlink:label="hd-0156-03a" xlink:href="hd-0156-03"/> + quàm linea <var>.e.t.</var> vnde huiuſmodi pondus magis quoque liberum à centro <var>.i.</var> reſultat. <lb/></s> + <s xml:id="s1737" xml:space="preserve">magisque ponderoſum, quam cum erat in <var>.e.</var> ratione eorum, quæ primo & ſecundo <lb/>capitibus diximus, & ob hanc cauſam ſuperat pondus poſitum in <var>.n</var>. </s> + <s xml:id="s1738" xml:space="preserve">Sed ſi centrum <lb/>fuerit .in <var>.o.</var> imaginabimur duas lineas <var>.o.s.</var> et <var>.o.x.</var> & ſupponemus quòd pondera po-<lb/>ſita ſint in <var>.s.</var> et <var>.x.</var> vnde exiſtente magis orizontali linea <var>.o.s.</var> quam erit <var>.o.x.</var> & linea <lb/><var>u.s.</var> inclinationis longius diſtante à centro <var>.o.</var> quàm 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 ſuperius dixi-<lb/>mus, inueniemus eundem effectum verum eſſe. </s> + <s xml:id="s1739" xml:space="preserve">In ſtateris, rectè & propriè appella <lb/>ri poteſt <var>.x.i.s.</var> aut <var>.n.o.u.</var> orizontalis, ſed in omni vectium ſpecie, hoc <reg norm="tantum" type="context">tãtum</reg> per quan <lb/>dam ſimilitudinem dicetur. </s> + <s xml:id="s1740" xml:space="preserve">Idem contemplari licet ſupponendo centrum in medio <lb/>inter <var>.o.</var> et <var>.i.</var> quod vnuſquiſque ex ſe abſque 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 quibuſdam rebus animaduerſione 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/>ſunt neceſſaria. </s> + <s xml:id="s1742" xml:space="preserve">Quod autem quærimus, in eo conſiſtit, quòd aliqui vectes <lb/>adhibeantur ad opus, quorum centrum, quod Græci <reg norm="hypomochlion" type="context">hypomochliõ</reg> appellant vnum <lb/>eſt ex extremis ipſius vectis, & pondus, quod ſurſum eleuari debet, inter ipſa-<lb/>met extrema iacet, propinquum tamen hypomochlio, vt exempli gratia, ſi vectis <lb/>eſſet infraſcripta 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">ꝙ</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 conſiderandum eſt quomodo pondus <var>.n.</var> annitatur ad <var>.u</var>. </s> + <s xml:id="s1744" xml:space="preserve">Hanc ob cau <lb/>ſam 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> verſus mundi cen <lb/>trum ſit poſita, 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 ſuperius conſtante, vt <var>.n.</var> ad inferius (re-<lb/>mota tamen grauitate materiæ vectis) huiuſmodi virtus, totum pondus ipſius <var>.n.</var> com <lb/>muni quadam ſcientiæ notione ſuſtinebit. </s> + <s xml:id="s1746" xml:space="preserve">& ſi <reg norm="pondus" type="context">põdus</reg> ipſius <var>.n.</var> eſſet in <var>.x.</var> è directo ſu-<lb/>per <var>.o.</var> totum pondus ſuper hypomochlio ſe haberet, & tanta virtus ipſius hypomo-<lb/>chlij ſufficeret ad reſiſtendum pro ſuſtinendo, quanta eſt grauitas ipſius ponderis, <lb/>ſed ipſum iterum ponamus in <var>.n.</var> ibi clarum erit, quòd ſi alia virtus à parte inſeriori <lb/>ad ſuperiorem vectis non opponitur, excepto tamen hypomochlio, oportebit virtu <lb/>te cuiuſdam partis ponderis <var>.n.</var> (abſque conſideratione tamen, vt iam dixi, ponderis <lb/>materiæ vectis) vt vectis à parte <var>.s.u.</var> deprimatur, & dixi vnius cuiuſdam partis pon-<lb/>deris <var>.n.</var> quia alia <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ponderis pars annititur ipſi 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 ſeſe huiuſ-<lb/>modi reſiſtentia, vt vectis non deprimatur, clarum erit communi ſcientia, <reg norm="quod" type="simple">ꝙ</reg> virtus <lb/>ponderis <var>.n.</var> diuiſa erit per medium æqualiter, cuius vna medietas ſuper <var>.o.</var> quieſcet, <lb/>& alia ſuper <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 reſiſtentiam <lb/>t. ablatam eſſe, <reg norm="poſitamque" type="simple">poſitamq;</reg> in <var>.e.</var> clarum quoque erit, <reg norm="quod" type="simple">ꝙ</reg> maior pars ponderis <var>.n.</var> ipſi <var>.e.</var> <lb/>annitetur beneficio lineæ <var>.n.e.</var> quàm ipſi <var>.o.</var> cum linea <var>.n.i.</var> inclinationis ipſi <var>.e.</var> ſit pro <lb/>pinquior quam <var>.o.</var> quia omnis reſiſtentia aut in <var>.i.</var> aut in <var>.e.</var> aut in <var>.t.</var> aut in <var>.u.</var> eſt loco <lb/>centri, quemadmodum eſt <var>.o.</var> & alter alterius opera iuuatur. </s> + <s xml:id="s1749" xml:space="preserve">Si verò eadem reſiſten <lb/>tia poſita erit in <var>.u.</var> clarum quoque erit, <reg norm="quod" type="simple">ꝙ</reg> minor pars ponderis <var>.n.</var> annitetur ipſi <var>.u.</var> <reg norm="quam" type="context">quã</reg> <lb/>ipſi <var>.o.</var> cum dicta <var>.n.i.</var> à centro <var>.u.</var> longius quam à centro <var>.o.</var> diſter, & 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="ſecundum" type="context">ſecũdum</reg> propor <lb/>tionem angulorum <var>.u.n.i.</var> et <lb/><var>o.n.i.</var> ſed ſecundum propor <lb/>tionem <var>.u.i.</var> ad <var>.i.o.</var> quod cla <lb/>rè compræhendi poteſt 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 conuerſo, ideſt, vt quemadmodum nunc ſupponuntur <var>.o.</var> et <var>.u.</var> eſſe duo <lb/>centra quibus <reg norm="ſuſtinetur" type="simple">ſuſtinet̃</reg> pondus <var>.e.</var> ipſius <var>.n.</var> imaginemur <var>.n.</var> eſſe quoddam centrum à <lb/>quo pendeant duo pondera <var>.o.</var> et <var>.u.</var> ſic inuicem proportionata, ut ſunt <var>.u.i.</var> et <var>.i.o.</var> <lb/>certe horum ponderum cauſa ſtatera <var>.o.s.</var> quam vectem appellabamus à nulla parte <lb/>inclinabitur. </s> + <s xml:id="s1750" xml:space="preserve">Redeuntes nunc ad propoſitum, dicemus <reg norm="quod" type="simple">ꝙ</reg> annitente pondere ipſius <var>.<lb/>n.</var> minus ad <var>.u.</var> quam ad <var>.o.</var> ideſt ad <var>.t.</var> minori vi opus erit in <var>.u.</var> quàm in <var>.t.</var> ad attollen-<lb/>dum pondus ipſius <var>.n.</var> & ſic per conſequens quantò longius erit punctum <var>.u.</var> ab <var>.t.</var> tan <lb/>tò minori quoque vi egebit, & conſequenter quando vis, aut reſiſtentia in <var>.u.</var> ita pro <lb/>portionata erit illi, quæ eſt ipſius <var>.o.</var> vt eſt <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, reſiſtentiæ ipſius <var>.u.</var> ad eam, quæ eſt ipſius <var>.o.</var> ea, quæ eſt <var>.o.<lb/>i.</var> ad <var>.i.u.</var> </s> + <s xml:id="s1752" xml:space="preserve">tunc vectis à par-<lb/>teipſius <var>.u.s.</var> eleuabitur, ſi <lb/> + <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> + vero proportio minor eſſet <lb/>quàm.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 cuiuſdam uis adauctæ.</head> + <head xml:id="head206" xml:space="preserve">CAP. VI.</head> + <p> + <s xml:id="s1754" xml:space="preserve">QVibuſdam in locis vtuntur <reg norm="quidam" type="context">quidã</reg> <reg norm="quodam" type="context">quodã</reg> <reg norm="inſtrumento" type="context">inſtrumẽto</reg> piſtorio ad <reg norm="ſubigendam" type="context context">ſubigẽdã</reg> pa-<lb/>ſtam, 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 deſcriptione imaginemur planum, in quo ſedet ille, qui voluit paſtam, & in quo <lb/>ipſa paſta eſt repoſita <var>.T.S.D.</var> & triangulum <var>.T.A.S.</var> immobile perpendiculare-<lb/>q́ue ſuperficiei dicti plani, angulo autem <var>.A.</var> coniunctum lignum <var>.A.E.</var> vt ſemidiame <lb/>trum mobilem, & æqualem perpendiculari ipſius trianguli, </s> + <s xml:id="s1755" xml:space="preserve">unde <var>.A.</var> loco centri erit <lb/>et <var>.D.O.</var> ſit ſemidiameter, qui paſtam contundit, & ab eius extremo <var>.O.</var> (quod <var>.O.</var> <lb/>quando <var>.D.O.</var> orizontalis eſt, in baſi dicti trianguli reperitur) veniat lignum <var>.O.V.</var> <lb/>quod cum <var>.A.V.</var> ſit æquale perpendiculari imaginatæ ab angulo <var>.A.</var> baſi <var>.T.S.</var> <reg norm="deno- datum" type="context">deno-<lb/>datũ</reg> <reg norm="tantum" type="wordlist/context">tñ</reg> utvulgo <reg norm="dicitur" type="simple">dicit̃</reg> ſeu flexile in <var>.O.</var> & in <var>.V.</var> vt elleuare <reg norm="atque" type="simple">atq;</reg> deprimere ſemidiame <lb/>trum <var>.D.O.</var> poſſit, et <var>.V.O.</var> ſit æqualis <var>.A.V.</var> et <var>.V.</var> medium ſit 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> ſunt deinde duo ligna <reg norm="perpendicularia" type="context">perpẽdicularia</reg> ab <var>.A.</var> ad baſim <lb/>fixa, & immobilia inter ſe adeò diſtantia, vt inter ipſa <reg norm="pertranſeant" type="context context">pertrãſeãt</reg> <var>.O.V.</var> et <var>.D.O.</var> ſupra <lb/>& infra, ne deuiet ſemidiametrum <var>.D.O</var>. </s> + <s xml:id="s1756" xml:space="preserve">In extremitate deinde ipſius <var>.E.</var> ſit lignum <lb/>quoddam tenue, vt digitus polex, ad angulos rectos cum <var>.A.E.</var> quod ab aliquo, qui <lb/>antedictam machinam ſtet, manibus teneatur, qui quidem homo idipſum lignum, <lb/>ideſt ſemidiametrum <var>.A.E.</var> à ſuperficie trianguli dicti, ad ſe trahendo, & deinde ver <lb/>ſus eundem triangulum impellendo, vim quandam maximam mediante ſemidia <lb/>metro <var>.D.O.</var> ſuper paſtam excitat.</s> + </p> + <p> + <s xml:id="s1757" xml:space="preserve">Pro cuius rei contemplatione volo vt ſecundam hanc ſubſcriptam 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> baſem trianguli <var>.a.u.o.</var> cui <var>.o.t.</var> perpendicularis dictæ baſi <var>.<lb/>u.a.</var> addatur. </s> + <s xml:id="s1758" xml:space="preserve"><reg norm="Hucuſque" type="simple">Hucuſq;</reg> igitur <var>.u.o.</var> æqualis erit <var>.o.x.</var> & ipſi <var>.o.a.</var> imaginemur etiam <var>.a.o.</var> <lb/>vſque ad <var>.b.</var> ita productam vt <var>.o.b.</var> æqualis ſit <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 verſus <var>.u.</var> vnde linea eius inclinationis ſit ſemper <var>.a.u.</var> ſupponamus etiam <var>.a.o.b.</var> <lb/>eſſe <reg norm="libram" type="context">librã</reg>, aut ſtateram, aut vectem, & <var>.o.</var> eius centrum, vnde vis, aut virtus ipſius <var>.a.</var> <lb/>proportionalis erit ipſi <var>.o.t.</var> reſpectu virtutis, aut vis imaginatæ in <var>.b.</var> inclinationis <lb/>perpendicularis ipſi <var>.b.a.</var> quæ quidem virtus, aut vis in <var>.b.</var> proportionalis erit ipſi <var>.b.<lb/>o.</var> ex tertio capite huius tractatus; </s> + <s xml:id="s1759" xml:space="preserve">Si ergo fuiſſet poſita in <var>.b.</var> virtus quædarn ad an-<lb/>gulum rectum, trahens lineam <var>.b.o.</var> tam proportionatam virtuti perpendiculari ip-<lb/>ſius <var>.a.</var> quam eſt <var>.o.t.</var> proportionata ipſi <var>.o.b.</var> ſtatera <var>.b.o.a.</var> non moueretur, ſed quæuis <lb/>portio maior in <var>.b.</var> ſuperaret <var>.a.</var> cum autem fuerit <var>.o.x.</var> æqualis ipſi <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 ſcientia, 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 ſuperare debet reſiſtentiam <lb/>in <var>.a.</var> quæ ipſi <var>.u.</var> contraponitur, debet ha-<lb/>bere aliquantulum maioris proportionis <lb/>ad reſiſtentiam, quæ in <var>.a.</var> angulum re-<lb/>ctum efficeret cum <var>.a.o.</var> ea, quæ eſt <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 quibuſdam erroribus Nicolai Tartaleæ circa pondera <lb/>corporum & eorum motus, quorum aliqui deſumpti <lb/>fuerunt à fordano ſcriptore 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 quàm cuiuſquam hominis, quemad-<lb/>modum Ariſto. ſcribit, detegam hoc loco quoſdam errores Nicolai Tartaleę <lb/>de ponderibus corporum, & velocitatibus motuum localium. </s> + <s xml:id="s1762" xml:space="preserve">Et primum decipitur <lb/>is in .8. lib. ſuarum diuerſarum inuentionum in ſecunda propoſitione, cum non ani-<lb/>maduerterit quanti momenti ſint extrinſecæ reſiſtentiæ.</s> + </p> + <p> + <s xml:id="s1763" xml:space="preserve">Subiectum quoque tertiæ propoſitionis eſt malè demonſtratum, quia idem pla-<lb/>nè ex eius demonſtratione iam dicta corporibus hætereogeneis, aut figura diuerſis <lb/>contingeret, quod ad velocitates attinet.</s> + </p> + <p> + <s xml:id="s1764" xml:space="preserve">In quarta propoſitione, quod ad <reg norm="diſputandum" type="context context">diſputãdũ</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="ſequitur" type="simple">ſequit̃</reg>, quod Archimedes in .6. propoſitione lib. primi de <reg norm="ponderibus" type="context">põderibus</reg> <reg norm="probauit" type="simple">ꝓbauit</reg>.</s> + </p> + <handwritten/> + <p> + <s xml:id="s1766" xml:space="preserve">Sed in ſecunda parte quintę propoſitionis non uidet <reg norm="quod" type="simple">ꝙ</reg> uigore ſitus eo modo, quo <lb/>ipſe diſputat, nulla elicitur ponderis differentia. </s> + <s xml:id="s1767" xml:space="preserve">quia ſi corpus <var>.B.</var> deſcendere debet <lb/>per arcum <var>.i.l.</var> corpus <var>.A.</var> aſcendere debet per arcum <var>.u.s.</var> æqualem, & ſimilem. eadem <lb/>quoque ratione ſituatum, vt eſt arcus <var>.i.l.</var> vnde vt eſt facilè corpori <var>.B.</var> deſcendere <lb/>per arcum <var>.i.l.</var> difficile ita erit corpori <var>.A.</var> aſcendere per arcum <var>.u.s</var>. </s> + <s xml:id="s1768" xml:space="preserve">Hęc autem qnin <lb/>ta propoſitio Tartaleæ eſt ſecuuda quæſtio à Iordano propoſita.</s> + </p> + <p> + <s xml:id="s1769" xml:space="preserve">Quòd autem ad primum corollarium dictæ propoſitionis attinet, verum ille qui <lb/>dem ſcribit, eius tamen effectus cauſa & à Iordano prius, & ab ipſo poſtea citata, na-<lb/>tura ſua vera non eſt. </s> + <s xml:id="s1770" xml:space="preserve">quia vera cauſa per ſe ab eo oritur, <reg norm="quod" type="simple">ꝙ</reg> à centro libræ dependeat <lb/>vt primo cap. huius tractatus oſtendi. </s> + <s xml:id="s1771" xml:space="preserve">Secundum verò corollarium falſum eſſe, ijs ra <lb/>tionibus quas nunc ſubiungam, 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 reſpectu ad <var>.u.</var> & brachijs æqualibus <reg norm="conſtantem" type="context">conſtãtem</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 ſint <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 ipſius <var>.a.</var> in huiuſmodi ſitu 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">ꝙ</reg> tertio cap. hu-<lb/>iustractatus probaui, ſed linea <var>.o.t.</var> maior eſt linea <var>.o.e.</var> quod ſic probo. </s> + <s xml:id="s1775" xml:space="preserve">Imaginemur <lb/>triangulum <var>.u.a.b.</var> circunſcriptum eſſe à circulo <var>.u.a.n.b.</var> cuius <var>.c.</var> ſit centrum, <reg norm="quod" type="simple">ꝙ</reg> erit <lb/>extra lineam <var>.u.o.</var> cum ſupponatur <var>.a.o.b.</var> obliquam eſſe reſpectu 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> vſque ad circunferentiam, quæ perpendicula-<lb/>ris erit ipſi <var>.a.b.</var> ex tertia lib. 3. Eucli. </s> + <s xml:id="s1777" xml:space="preserve">ſi poſteà 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> ſed ſi imaginabimur <var>.u.o.</var> ad circunferentiam vſque <lb/>productam, clarum erit <reg norm="quod" type="simple">ꝙ</reg> arcum <var>.s.b.</var> ſecaret in puncto <var>.n.</var> vnde arcus <var>.n.b.</var> minor erit <lb/>arcu <var>.n.a.</var> & ſic 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> ſit diameter, cuius circunferentia <lb/>per duo puncta <var>.e.</var> et <var>.t.</var> <reg norm="prætergradiatur" type="simple">prætergradiat̃</reg>, cum in ipſis ſint angulirecti, quod quilibet ex <lb/>ſeratio cinando colligere poteſt, ſi .30. lib. 3. in mentem reuocauerit. </s> + <s xml:id="s1780" xml:space="preserve">Sed cum angu-<lb/>lus <var>.o.u.t.</var> ſit 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 ipſius <var>.o.e.</var> ex conuerſo .27. lib. 3. quod eſt propoſitum. </s> + <s xml:id="s1781" xml:space="preserve">Pon-<lb/> + <anchor type="handwritten" xlink:label="hd-0160-01a" xlink:href="hd-0160-01"/> + dusigitur ipſius <var>.a.</var> in huiuſmodi ſitu, pondere ipſius <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 propoſitionis ediſerit, & per conſequens <lb/>2. corollarij falſitatem oſtendit, vt eam quoque, quæ in 6. propoſitione latet. </s> + <s xml:id="s1783" xml:space="preserve">quia <reg norm="cum" type="context">cũ</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/>dusipſius <var>.b.</var> eadem ſit cum <lb/>ea quę eſt <var>.o.t.</var> ad <var>.o.e.</var> ſub co <lb/><reg norm="gnitionem" type="context">gnitionẽ</reg> noſtram cadere po <lb/>teſt, primum cognoſcendo <lb/>angulos obliquitatis librę, <lb/>ideſt angulos <var>.b.o.u.</var> et <var>.a.o.<lb/>u.</var> quia oportet ſemper ſup-<lb/> + <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> + ponere ſitum aliquem no-<lb/>tum. </s> + <s xml:id="s1784" xml:space="preserve">Si nobis deinde co-<lb/>gnita erit proportio ipſius <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 conſe-<lb/>quens ipſius <var>.o.a.t.</var> eius reſi-<lb/>dui, vnde poſtea 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. propoſitione, & Iordanus in ſecunda parte. <lb/></s> + <s xml:id="s1786" xml:space="preserve">ſecundæ propoſitionis ſcribunt, maximum quoque errorem inſe continet. <lb/></s> + <s xml:id="s1787" xml:space="preserve">Dicunt enim <reg norm="angulum" type="context">angulũ</reg> <lb/><var>h.a.f.</var> differentem ab <lb/>angulo <var>.d.b.f.</var> alia ra-<lb/>tione non eſſe quàm <lb/>per angulum conta-<lb/>ctus <reg norm="duorum" type="context">duorũ</reg> <reg norm="circulorum" type="context">circulorũ</reg>, <lb/>vt in ſua figura ſcribit <lb/>Tartalea; </s> + <s xml:id="s1788" xml:space="preserve">id quod fal-<lb/>ſiſſimum eſt. </s> + <s xml:id="s1789" xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> ob <lb/>cauſam in ſubſcripta <lb/>figura ſit 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">centrũ</reg> regionis ele <lb/>mentaris, et <var>.A.u.</var> et <var>.B.<lb/>u.</var> lineæ <reg norm="inclinationum" type="context">inclinationũ</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/>ipſi <var>.A.u.</var> quæ gyrum <var>.<lb/>B.F.A.</var> in puncto <var>.K.</var> <lb/>communi ſcientiæ <reg norm="prae cepto" type="simple">prę<lb/>cepto</reg> ſcindet, & habe <lb/>bimus angulum <var>.K.B.<lb/>Z.</var> æqualem angulo <var>.<lb/>H.A.F.</var> ideſt <var>.u.A.F.</var> <lb/>(quia <var>.H.u.</var> et <var>.D.</var> <reg norm="unum" type="context">unũ</reg> <lb/>ſunt) 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 ſit <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">nũc</reg> <lb/>comparatio eſt inter <lb/>angulum <var>.D.B.F.</var> & an <lb/>gulum <var>.K.B.Z.</var> miſtili-<lb/>neos, qui quidem duo <lb/>anguli, <reg norm="communem" type="context">cõmunem</reg> ha-<lb/>bent angulum miſtili <lb/>neum <var>.K.B.F.</var> quapro-<lb/> + <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> + pter ſi angulus <var>.K.B.Z.</var> <lb/>miſtilineus maior eſt <lb/>angulo <var>.D.B.F.</var> miſti-<lb/>lineo per angulum <var>.<lb/>K.B.Z.</var> contingentiæ, <lb/>circulorum ergo angu <lb/>lus miſtilineus com-<lb/>munis <var>.K.B.F.</var> æqualis <lb/>erit miſtilineo, angu-<lb/>lo <var>.D.B.F.</var> pars vide-<lb/>licet ſui 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/>lapſi fuerunt ab eo, <reg norm="quod" type="simple">ꝙ</reg> <lb/> + <anchor type="handwritten" xlink:label="hd-0162-0a" xlink:href="hd-0162-01"/> + lineas inclinationum <lb/>pro parallelis viciſſim <lb/>ſumpſerunt, 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 propoſitio Tartaleæ, quæ eſt <reg norm="quinta" type="simple">ꝗnta</reg> quæſtio Iordani mihi <reg norm="videtur" type="simple">videt̃</reg> excipien-<lb/>da riſu, cum pondus ipſius <var>.A.</var> ponderi ipſius <var>.B.</var> exiſtens æquale, grauius ſit pondere <lb/>eiuſdem <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 ſcilicet ipſe putet <lb/>lineas <var>.A.E.</var> et <var>.B.D.</var> figuræ ab eo confictæ ſibi inuicem eſſe parallelas, quæ etiam ſi <lb/>æquidiſtantes eſſent (vnde angulus <var>.E.A.G.</var> minor eſſet angulo <var>.D.B.F.</var>) non eam ta <lb/>men ob cauſam huiuſmodiangulorum differentia cauſa eſſet differentiæ <reg norm="grauitatum" type="context">grauitatũ</reg> <lb/>ipſorum <var>.A.</var> et <var>.B.</var> ob ea quæ cap .4. huius tractatus poſui.</s> + </p> + <p> + <s xml:id="s1794" xml:space="preserve">Octaua autem propoſitio, quæ eſt .6. quæſtio Iordani Iongè melius demonſtratur <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">Tartalęa citat, qui <lb/>neque etiam probat nonam .10. 11. 12. et .13. cum ad pręcedentes probandas mini <lb/>mè acceſſerit.</s> + </p> + <p> + <s xml:id="s1796" xml:space="preserve">Quartadecima verò, quæ eſt .10. quęſtio Iordani, duas ob cauſas eſt falſa, quarum <lb/>vna eſt, <reg norm="quod" type="simple">ꝙ</reg> (ſupponendo <var>.A.D.E.G.B.</var> eſſe vnum brachium librę et <var>.A.</var> punctum <reg norm="centri" type="context">cẽtri</reg> <lb/>eiuſdem, et <var>.D.</var> pondus ęquale 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> ſibi <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="non" type="context">nõ</reg> ſunt ęquales; </s> + <s xml:id="s1797" xml:space="preserve"><reg norm="cum" type="context">cũ</reg> ille angulus ſit intrinſecus, hic <lb/>verò extrinſecus & oppoſitus dicto intrinſeco <reg norm="vnius" type="simple">vniꝰ</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 vſque ad centrum regionis elementaris, vnde dictus angu-<lb/>lus <var>.M.E.G.</var> maior eſt alio, ex .16 lib. primi Eucli. </s> + <s xml:id="s1798" xml:space="preserve">Qua ratione fit, vt hanc ob cauſam <lb/>E. grauius ſit ipſo <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 eſtratio, qua dictum <var>.E.</var> grauius fit ipſo <var>.D.</var> quę quidem eſt <lb/>maior diſtantia à centro <var>.A.</var> libræ, per ſimiles 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, quę eſt .11. quęſtio Iordani, cuius Autho-<lb/>ris opuſculum opera Traiani Bibliopolę Venetijs è tenebris in lucem emerſit.</s> + </p> + </div> + <div type="section" level="3" n="9"> + <head xml:id="head210" style="it" xml:space="preserve">Quòdſummaratione ſtateræper æqualia interualla <lb/>ſint 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">diuidũtur</reg> ſtateræ per interualla ęqualia, in libras, aut in <lb/>vncias, aut quoquo alio modo. </s> + <s xml:id="s1802" xml:space="preserve">Nam ſit ſtatera exempli gratia <var>.a.b.</var> <lb/>& punctum, <reg norm="quod" type="simple">ꝙ</reg> eam ſuſtinet ſit <var>.c.</var> & vas illud, <reg norm="quod" type="simple">ꝙ</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">cũ</reg> <lb/>eo, <reg norm="quod" type="simple">ꝙ</reg> eſt dicti vaſis <var>.f.</var> ab altera parte, ſint cauſę, quibus ſtatera <var>.a.b.c.</var> ſtet orizonta-<lb/>lis. cui ſic orizontali manenti imaginemur ad punctum <var>.a.</var> adiunctum eſſe pondus, <lb/>veluti vnius librę. & ad punctum <var>.d.</var> tam diſtanti à <var>.c.</var> ut eſt <var>.a.</var> ab ipſo <var>.c.</var> aliud quoque <lb/>pondus vnius libræ <reg norm="additum" type="context">additũ</reg> eſſe, vnde <reg norm="coni" type="context">cõi</reg> <reg norm="quadam" type="context">quadã</reg> ſcientia ſtatera, non mouebitur ſitu. <lb/></s> + <s xml:id="s1804" xml:space="preserve"><reg norm="quia" type="simple">ꝗa</reg> exiſtentibus duobus hiſce 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> abſque dubio <var>.a.d.</var> non mutaret ſitum, ſed <var>.d.b.</var> et, f. in ſitu, in <lb/>quo reperiuntur, à centro paribus viribus prędita ſunt. </s> + <s xml:id="s1805" xml:space="preserve">Addendo igitur <var>.d.b.</var> ipſi <var>.d.</var> <lb/>et <var>.f.</var> ipſi .a: ſumma earum, æqualibus quoque viribus conſtabunt. ex communi ſen-<lb/>tentia, quæ habet ſi ęqualibus addas ęqualia, tota quoque fient ęqualia. </s> + <s xml:id="s1806" xml:space="preserve">Si verò <lb/>ponderi ipſius <var>.a.</var> aliud adderetur eidem ęquale, haberemus in <var>.a.</var> duplum pon-<lb/>dus ei <reg norm="quod" type="simple">ꝙ</reg> eſt ipſius <var>.d.</var> ſed volentes vt ſolum cum pondere ipſius <var>.d.</var> ſtatera ſtet orizon <lb/>talis, ſi dictum pondus ipſius <var>.d.</var> longè diſtabit à centro <var>.c.</var> per duplum ipſius <var>.c.a.</var> ideſt <lb/>ipſius <var>.c.d.</var> id <reg norm="quod" type="simple">ꝙ</reg> volumus aſſeque-<lb/>mur, beneficio ſupradictarum ra <lb/> + <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> + tionum, adiuti opera ſextę lib. pri<lb/>mi de <reg norm="ponderibus" type="context">põderibus</reg> Archimedis. </s> + <s xml:id="s1807" xml:space="preserve">Et <lb/>ſi quis aliud <reg norm="quoque" type="simple">quoq;</reg> pondus adiun <lb/>geret ipſi <var>.a.</var> æquale illi priori, ad <lb/><reg norm="efficiendum" type="context">efficiẽdum</reg>, vt ſtatera ſemper ori <lb/>zontalis maneret, oporteret, vt <reg norm="pondus" type="context">põdus</reg> ipſius <var>.d.</var> ab <var>.c.</var> longè diſtaret, ita vt huiuſmodi <lb/>diſtantia tripla eſſet primæ, & ſic per quoſdam quaſi 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">ARiſtoteles 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 falſum <lb/>eſt: </s> + <s xml:id="s1809" xml:space="preserve">quia huiuſmodi linea partes nullas ſecundum latitudinem habet, (vt ipſe etiam <lb/>confirmat) ſed eſt idem conuexum circuli: </s> + <s xml:id="s1810" xml:space="preserve">linea verò quæ terminus eſt ſuperficiei <lb/>ambientis, & amplectentis circulum eſt eadem concauitas dictæ ſuperficiei eun-<lb/>dem circulum ambientis, quæ nullam conuexitatem habet. </s> + <s xml:id="s1811" xml:space="preserve">& hæ duæ ſunt lineæ, <lb/>quarum vna diuerſa eſt 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 Ariſtoteles ſcribit de duplici reſpectu motus vnius puncti ſecun <lb/>dum vnam datam pro portionem, non ſufficit, ille enim ſic ait.</s> + </p> + <p> + <s xml:id="s1813" xml:space="preserve">Sit proportio ſecundum quam latum fertur, quam habet <var>.A.B.</var> ad <var>.A.C.</var> et <var>.A.</var> qui <lb/>dem feratur verſus .B: <var>A.B.</var> verò ſubterferatur verſus <var>.M.C.</var> latum autem ſit <var>.A.</var> <reg norm="quidem" type="context">quidẽ</reg> <lb/>ad <var>.D.</var> vbi autem eſt <var>.A.B.</var> verſus <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 eſt & <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 eſt 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 reſpondeo, punctum <var>.A.</var> quod mouetur in linea <var>.A.M.</var> ab <var>.A.</var> verſus <var>.M.</var> vſque <lb/>ad <var>.F.</var> non moueriab aliqua proportione determinata magis quàm ab alia: </s> + <s xml:id="s1818" xml:space="preserve">vnde <reg norm="non" type="context">nõ</reg> <lb/>ſolum poſſumus imaginari dictum punctum <var>.A.</var> moueri ab <var>.A.</var> vſque ad <var>.F.</var> eiuſdem <lb/>velocitatis ſub alia quadam proportione, ſed etiam ſub alia, quæ iam datæ contraria <lb/>ſit, vt eſt proportio ipſius <var>.A.C.</var> ad <var>.A.B.</var> <reg norm="imaginantes" type="context">imaginãtes</reg> moueri <var>.A.</var> verſus <var>.C.</var> et <var>.A.C.</var> ver <lb/>ſus <var>.B.M.</var> delatam. </s> + <s xml:id="s1819" xml:space="preserve">Dico etiam idem <var>.A.</var> moueri vſque ad <var>.F.</var> ſecundum proportio-<lb/>nem ipſius <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">cũ</reg> linea <var>.A.F.</var> face-<lb/>re <reg norm="angulum" type="context">angulũ</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">triangulũ</reg> <var>.A.H.F.</var> ęqui <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="ipſius" type="simple">ipſiꝰ</reg> <lb/><var>A.H.</var> ad <var>.F.H.</var> quę <reg norm="enipſius" type="context">ẽipſius</reg> <lb/><var>A.O.</var> ad <var>.O.P.</var> punctum <lb/>igitur <var>.A.</var> vſque ad <var>.F.</var> mouetur ſecundum proportionem etiam ipſius <var>.A.O.</var> ad <var>.O.P.</var> <lb/>Huiuſmodi igitur conſideratio, ab Ariſtotele facta, nullius eſt 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 cauſam non attulerit.</head> + <head xml:id="head215" xml:space="preserve">CAP. XI.</head> + <p> + <s xml:id="s1823" xml:space="preserve">QVærens Ariſtoteles vnde fiat, vt eæ libræ, quæ brachia habent alijs longiora, <lb/>ſint exactiores cæteris, ait hoc euenire ratione maioris velocitatis extremo <lb/>rum earundem. </s> + <s xml:id="s1824" xml:space="preserve">Quod verum non eſt; </s> + <s xml:id="s1825" xml:space="preserve">quia hîc effectus nil aliud eſt, quam clarius pro <lb/>ponere ob omnium oculos obliquitatem brachiorum à linea orizontali, & oſtende-<lb/>re etiam facilius à dicto orizontali ſitu exire brachia iam dicta. </s> + <s xml:id="s1826" xml:space="preserve">Quæ quidem per ſe <lb/>neque à velocitate, neque à tarditate motus, ſed à ratione vectis, & à ma-<lb/>iori interuallo inter ſecundum ſitum extremorum à primo proficiſcuntur. </s> + <s xml:id="s1827" xml:space="preserve">Vt exem-<lb/>pli gratia, imaginemur magnam libram <var>.A.B.</var> orizontalem, cuius centrum ſit <var>.E.</var> et <lb/>pondus <var>.B.</var> maius ſit pondere ipſius <var>.A.</var> vnde conceditur, quòd ob hanc rationem di-<lb/>cta libra ſitum mutabit, qui ſecundus ſitus ſit 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 & ſecundus ſitus ſit in <var>.h.f.</var> ita tamen vt anguli circa <var>.e.</var> æquales <lb/>ſint ijs, qui ſunt circa <var>.E.</var> ideſt <var>.b.e.f.</var> ſit ęqualis <var>.B.E.F</var>. </s> + <s xml:id="s1829" xml:space="preserve">Nunc dico ſitum <var>.H.F.</var> <reg norm="exa- ctiorem" type="context">exa-<lb/>ctiorẽ</reg> futurum & clariorem ſitu <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 eſt ipſo <var>.b.f.</var> in qua <var>.B.E.</var> maius eſt <var>.b.e.</var> <lb/>quod autem interuallum <var>.B.F.</var> breuiori, aut longiori temporis ſpacio quam <var>.b.f.</var> ſit fa <lb/>ctum, nil planè refert. </s> + <s xml:id="s1830" xml:space="preserve">Ratione vectis deinde, dico <reg norm="quod" type="simple">ꝙ</reg> ſi ſupponemus duas libras pa-<lb/>res <reg norm="æqualesque" type="simple">æqualesq́;</reg> in omni alio reſpectu, præter quàm in brachiorum longitudine, pon-<lb/>dus <var>.B.</var> maiorem vim habebit ad deprimendum brachium <var>.E.B.</var> quàm pondus <var>.b.</var> quia <lb/>libræ materiales, cum ſuſtineantur ab <var>.E.e.</var> & non à puncto mathematico, ſed <lb/>à linea, aut ſuperficie naturali in materia exiſtente. </s> + <s xml:id="s1831" xml:space="preserve">vnde aliqua reſiſtentia ipſi mo-<lb/>tui brachiorum oritur, & hanc ob cauſam, ſupponendo hanc reſiſtentiam æqualem <lb/>tam in <var>.E.</var> quàm in <var>.e.</var> clarum erit ob ea, quæ in cap .4. huius tractatus oſtendi <var>.B.</var> cum <lb/>minus dependeat ab <var>.E.</var> aut minus quoque eidem <var>.E.</var> annitatur, ponderoſum magis <lb/>futurum, quam <var>.b.</var> & hac de cauſa mouebit ad partem inferiorem, maiori cum agilita <lb/>te, brachium <var>.E.B.</var> multo magis etiam illud ipſum deprimet, ideſt maiorem etiam an <lb/>gulum <var>.B.E.F.</var> quàm 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 cauſa ſecundæ, & tertiæ quæstionis mechanicæ <lb/>ab Ariſtotele nonperſpecta.</head> + <head xml:id="head217" xml:space="preserve">CAP. XII.</head> + <p> + <s xml:id="s1832" xml:space="preserve">ARiſtoteles in ſecunda quæſtionum mechanicarum quærens illius rationem ſic <lb/>ſcribit.</s> + </p> + <p> + <s xml:id="s1833" xml:space="preserve">Cur ſiquidem ſurſum fuerit ſpartum quando deorſum lato pondere quiſpiam id <lb/>admouet rurſus aſcendit libra: </s> + <s xml:id="s1834" xml:space="preserve">ſi autem deorſum conſtitutum fuerit non aſcendit, <lb/>ſed manet? </s> + <s xml:id="s1835" xml:space="preserve">an quia ſurſum quidem ſparto exiſtente plus libræ extra perpendiculum <lb/>ſi (ſpartum enim eſt perpendiculum) quare neceſſe eſt deorſum ferriid, quod plus <lb/>eſt, quare & cætera.</s> + </p> + <p> + <s xml:id="s1836" xml:space="preserve">Sed vera cauſa, vnde fiat, vt ſi ſpartum fuerit ſurſum, & brachium vnum <lb/>ipfius libræ deprimendo, & idem liberum deinde permittendo, ad ſitum ori-<lb/>zontalem redeat, non ſolum eſt maior quantitas ponderis brachiorum quæ iam præ <lb/>tergreſſa eſt vltra verticalem lineam, ſed etiam eſt 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, ſit <var>.A.B.</var> libra in ſitu orizontali, cuius <lb/>ſpartum ſit <var>.E.</var> ſuper ipſam. </s> + <s xml:id="s1837" xml:space="preserve">& deprimentes brachium ipſius <var>.A.</var> vſque ad <var>.F.</var> eius ſitus <lb/>ſit in <var>.F.H.</var> vnde medium <reg norm="punctum" type="context">pũctum</reg> <var>.G.</var> prætergreſſum erit lineam verticalem <var>.V.Z.</var> ver <lb/>ſus <var>.B.</var> quæ <var>.V.Z.</var> ſecabit brachium <var>.F.G.</var> in puncto <var>.D.</var> vnde <var>.D.H.</var> longius erit ipſo <var>.<lb/>F.D</var>. </s> + <s xml:id="s1838" xml:space="preserve">Nunc nobis ſupponendum eſt id, <lb/>quod veriſſimum exiſtit, dictam ſcilicet li <lb/> + <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> + bram in ſitu <var>.F.H.</var> <reg norm="etiam" type="context">etiã</reg> ſi ſuſtineatur à pun-<lb/>cto <var>.E.</var> idem tamen futurum ac ſi ſuſtenta-<lb/>retur in puncto <var>.D.</var> vnde ſequitur, quod <lb/>pondus appenſum ex ipſa <var>.H.</var> ita grauius <lb/>reddatur, ipſo <var>.F.</var> in eadem propor-<lb/>tione, quæ maioreſt <var>.D.H.</var> ipſo <var>.D.F.</var> ob <lb/>rationes quas in primis huius tractatus ca-<lb/>pitibus poſui, vt etiam ſi <var>.D.H.</var> quodmate <lb/>riale eſſe ſupponitur, nullam planſe4; </s> + <s xml:id="s1839" xml:space="preserve">graui-<lb/>tatem haberet, <reg norm="ſolustamem" type="context">ſolustamẽ</reg> exceſſus vis pon <lb/>deris in <var>.H.</var> poſiti, longè maior pondere in <lb/>F. collocato pro maiorilongitudine ipſius <lb/><var>D.H.</var> ſufficiat. ad præſtandum vt libra ad <lb/>ſitum 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 ſecunda deinde huius quęſtionis par <lb/> + <anchor type="handwritten" xlink:label="hd-0166-01a" xlink:href="hd-0166-01"/> + te, in qua ſcribit libram in ſitu, in quo poſi <lb/>ta eſt, firmam manere, toto cęlo aberrat, quia <reg norm="neceſſarium" type="context">neceſſariũ</reg> eſt, vt omninò cadat, eò <reg norm="uſque" type="simple">uſq;</reg> <lb/>quò ſpartum ſurſum remaneat: </s> + <s xml:id="s1841" xml:space="preserve">ablato tamen omni impedimento, quod nulla eget <lb/>probatione, cum natura ſua 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">Cauſa, deinde, vera tertiæ quæſtionis non eſt ea, quam Ariſtoteles ponit, ſed hu-<lb/>iuſmodi effectus ab eo, quod capitibus .4. et .5. huius tractatus propoſui 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 Ariſtotelisratio in 6. quæſtione poſit a non ſit admittenda.</head> + <head xml:id="head219" xml:space="preserve">CAP. XIII.</head> + <p> + <s xml:id="s1843" xml:space="preserve">VOlens Ariſtoteles rationem proponere, vnde fiat, vt nauis velocius moueatur <lb/>cum antennam altiorem quàm cum depræſſiorem habet, id ad vectis ratio-<lb/>nem refert, quod verum <reg norm="non" type="context">nõ</reg> eſt. </s> + <s xml:id="s1844" xml:space="preserve">Huiuſmodi enim ratione nauis tardius potius, quàm <lb/>velocius ferri deberet, quia quantò altius eſt velum, vi venti impulſum, <reg norm="tantò" type="context">tãtò</reg> magis <lb/>proram ipſius nauis in aquam demergit. </s> + <s xml:id="s1845" xml:space="preserve">Sed huiuſmodi effectus à maiori potius <lb/>quantitate venti quam recipit, quàm ab alia aliqua cauſa oritur, quia ventus liberius <lb/><reg norm="vehementiusque" type="simple">vehementiusq́;</reg> in altiore parte, quàm 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 Ariſtotele de octaua quæstione confictæ <lb/>ſufficient es non ſint.</head> + <head xml:id="head221" xml:space="preserve">CAP. XIIII.</head> + <p> + <s xml:id="s1846" xml:space="preserve">RAtiones etiam ab Ariſtotele propoſitæ pro indaganda octauæ quæſtionis ve-<lb/>ritate, in qua quærit vnde fiat, vt corpora rotundæ figuræ, ad <reg norm="voluendum" type="context">voluendũ</reg> ſint <lb/>faciliora reliquis, quarum reuolutionum corporum tres ſpecies aſſignat, <reg norm="quarum" type="context">quarũ</reg> vna <lb/>eſt, vt rotarum <reg norm="curruum" type="context">curruũ</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 vaſorum a figulis fabricatorum, <reg norm="ſufficientes" type="context">ſufficiẽtes</reg> <reg norm="non" type="context">nõ</reg> ſunt.</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 ſolo quodam puncto contingat, quemadmodum <lb/>probat Theodoſius in .3. lib. primi & Vitellio in .71. lib. primi, & <reg norm="ducendo" type="context">ducẽdo</reg> per <reg norm="centrum" type="context">centrũ</reg> <lb/>ſphæræ lineam vſque ad punctum contactus, ipſa erit perpendicularis plano contin-<lb/>genti ſphęram dictam, vt probat <reg norm="idem" type="context">idẽ</reg> Thęodoſius 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 eſt omnem inclinationem ponderoſam huiuſ <lb/>modi corporis homogęnei totam hanc lineam æqualiter omni ex parte circundare; <lb/></s> + <s xml:id="s1851" xml:space="preserve">cuius quidem rei exemplum in carta deſcribere poſſumus mediante figura circulari <lb/>hîc ſubſcripta <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 ipſi <var>.b.d.</var> ex .17. lib. 3. Eucli. & <reg norm="tantum" type="context">tantũ</reg> ponderis habebimus à parte <var>.a.u.e.</var> quan <lb/>tum ab ipſa <var>.a.n.e</var>. </s> + <s xml:id="s1852" xml:space="preserve">Nuncigitur ſi imaginabimur ductum eſſe centrum verſus <var>.u.</var> per <lb/>lineam <var>.o.u.</var> parallelam ipſi <var>.a.d.</var> clarum nobis <lb/>erit, <reg norm="quod" type="simple">ꝙ</reg> <reg norm="abſque" type="simple">abſq;</reg> vlla difficultate aut reſiſtentia <reg norm="idem" type="context">idẽ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> + ducemus, quia huiuſmodi centrum ab inferiori <lb/>parte ad ſuperiorem, nunquam mutabit ſitum <lb/>reſpectu <reg norm="diſtantiæ" type="context">diſtãtiæ</reg> ſeu interualli, quę inter ipſum <lb/>lineamq́ue <var>.a.d.</var> intercedit, <reg norm="quod" type="simple">ꝙ</reg> quidem centrum <lb/>in ſe colligittotum pondus figurę <var>.a.n.e.u.</var> & be <lb/>neficio lineæ <var>.e.o.a.</var> illud ipſum 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 verſus ipſum <var>.d.</var> aut verſus <lb/>b. <reg norm="dirigatur" type="simple">dirigat̃</reg>; </s> + <s xml:id="s1853" xml:space="preserve">ita vt <reg norm="cum" type="context">cũ</reg> non oporteat vt huius figuræ <lb/><reg norm="pondus" type="context">põdus</reg>, vna vice, magis eleuetur, quàm alia, ſed <lb/>ſemper ęqualiter ſuper lineam <var>.b.a.d.</var> quieſcat.</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> ſemper diuiſum à linea <var>.a.o.e.</var> per medium, ſequitur 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, ſi imaginabimur pentagonum <var>.K.i.h.f.l.</var> quie <lb/>ſcere <reg norm="ſuper" type="simple">ſuꝑ</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">primũ</reg> <reg norm="totum" type="context">totũ</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> poſteà centrum <var>.o.</var> (ponamus.) verſus <var>.l.</var> dubium non eſt, 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 diſtet, voluens ſe per <reg norm="arcum" type="context">arcũ</reg> vnum <lb/>circuli, <reg norm="qui" type="simple">ꝗ</reg> <reg norm="pro" type="simple">ꝓ</reg> ſuo ſemidiametro habeat <var>.o.K.</var> quę maior eſt ipſa <var>.o.a.</var> ex .18. li. primi Eu <lb/>cli. vnde ſi à puncto <var>.K.</var> imaginabimur lineam <var>.K.c.</var> reſpicientem centrum regionis <lb/>elementaris, dubium non eſt, quin ſi velimus transferre <reg norm="centrum" type="context">cẽtrum</reg> hoc à priori ſitu <reg norm="vſque" type="simple">vſq;</reg> <lb/>ad dictam lineam, oporteat addere pondus parti ipſius <var>.l.</var> quæ à linea <var>.K.c.</var> fuit ſecta, <lb/>aut aliquid de ipſo pondere partis centri detrahere. </s> + <s xml:id="s1856" xml:space="preserve">quod quibuſuis modis fiat, ar-<lb/>duum certè eſt 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">ꝙ</reg> perfectè in medio ipſius ponderis eſt, reperiatur ſemper in linea per-<lb/>pendiculari ipſi plano, in quo animaduertendum eſt, <reg norm="quod" type="simple">ꝙ</reg> etiam ſi ipſum planum ap-<lb/>pellem; </s> + <s xml:id="s1858" xml:space="preserve">pro plano tamen perfecto intelligi nolo, ſed pro ſuperficie perfectè <reg norm="ſphaeri- ca" type="simple">ſphęri-<lb/>ca</reg> circa centrum à corporibus grauibus expetitum; </s> + <s xml:id="s1859" xml:space="preserve">nam ratione magnæ amplitudi-<lb/>nis huiuſmodi ſuperficiei, nullam differentiam notatu dignam à perfecto aliquo pla <lb/>no exigui interualli ad curuitatem eiuſdem ſuperficiei imaginari poterimus. </s> + <s xml:id="s1860" xml:space="preserve">Sed ut <lb/>redeamus ad ſermonem de reuolutione figuræ rotundæ ſuſceptum, <reg norm="clarum" type="context">clarũ</reg> igitur erit <lb/>quamlibet minimam vim (vt ita dicam) quę trahat, aut impellat centrum <var>.o.</var> verſus <var>.u.</var> <lb/>huiuſmodi figuram reuoluturam, cuius media pars ad trahendum, aut impellendum <lb/>punctum <var>.e.</var> ſufficiere; </s> + <s xml:id="s1861" xml:space="preserve">Imaginemur autem <reg norm="quod" type="simple">ꝙ</reg> li <lb/>nea <var>.n.o.u.</var> eſſet libra <reg norm="quędam" type="context">quędã</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> poſita, & vis, quę trahere cen <lb/>trum deberet, diuiſa eſſet per medium, cuius <lb/>medietas appenſa eſſet extremitati <var>.u.</var> diame-<lb/>tri <var>.n.o.u.</var> <reg norm="clarum" type="context">clarũ</reg> erit, <reg norm="quod" type="simple">ꝙ</reg> abſque vlla difficultate <lb/>reuolueret figuram ſuper lineam <var>.b.a.d.</var> verſus <var>.<lb/>d.</var> quia huius vis, aut pondus <reg norm="nullum" type="context">nullũ</reg> contra pon <lb/>dus haberet vltra centrum <var>.o.</var> uerſus <var>.n.</var> <reg norm="quod" type="simple">ꝙ</reg> cen-<lb/>trum <var>.o.</var> perpetuo quieſcit <reg norm="ſuper" type="simple">ſuꝑ</reg>. a. in linea <var>.e.o.<lb/>a.</var> per medium diuidente ſemper totum pon-<lb/>dus figurę ſuppoſitę. </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">ipsũ</reg> ver <lb/>ſus <var>.u.</var> <reg norm="trahens" type="context">trahẽs</reg> <lb/>per lineam <lb/><reg norm="parallelam" type="context">parallelã</reg> ip <lb/>ſi <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/>ſi linea qua <lb/>dictum cen <lb/>trum trahi-<lb/>tur ab ipſo + <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 æquediſtaret, ſed ſurſum traheret ſuper <var>.u.</var> aut ſubter, aliquid de ſua vi vir <lb/><reg norm="tuteque" type="simple">tuteq́;</reg> amitteret, & tantò plus, quantò inclinata magis eſſet verſus <var>.a.o.e.</var> & tandem <lb/>cum eſſet vnita cum <var>.a.o.e.</var> aut ad ſuperius, aut ad inferius quantalibet ui, etiam ſi in-<lb/>finita, figuram extra ſitum primæ lineæ <var>.a.o.e.</var> non moueret, ſed ſi ſurſum traheret ſe <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 ſpecies, tribus reuolutionum modis, abſque axis mutatione conſta <lb/>re poteſt, ideſt 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 ſi priorem <lb/>modum conſiderabimus, vt mediante fune quælibet figura, quæ circularis non ſit, <lb/>voluatur, ſupponamus 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">laterumq́;</reg> diſparium) plures inæqualitates, quæ reuolutionem eiuſdem fi-<lb/>guræ irregularem efficient; </s> + <s xml:id="s1868" xml:space="preserve">quarum vna erit, quod duæ partes funis, ideſt <var>.u.q.</var> et <var>.i.p.</var> <lb/>non erunt in vna <reg norm="eademque" type="simple">eademq́;</reg> inter ſe diſtantia ſemper, quod facile intellectu erit, ſi ima <lb/>ginabimur ductas eſſe lineas <var>.a.i</var>: <var>u.i</var>: et <var>.i.c.t.</var> ſi funis duo pondera habebit alterum <lb/>altero maius, ſuis extremis appenſa, 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> eiuſdem funis, <reg norm="mundi" type="context">mũdi</reg> centrum, dum firmæ ma <lb/>nebunt, reſpicient; </s> + <s xml:id="s1870" xml:space="preserve">ſed 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, eiuſdem figuræ mundi <reg norm="quoque" type="simple">quoq;</reg> cen <lb/>trum reſpiciet, vt in <lb/> + <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> + figura <var>.A.</var> <reg norm="ſicque" type="simple">ſicq́;</reg> etiam <lb/>linea <var>.i.c.t.</var> (pro <reg norm="exem- plo" type="context">exẽ-<lb/>plo</reg>) erit menſura di-<lb/>ſtantiæ funium inter <lb/>ipſas, & deinde <reg norm="circum" type="context">circũ</reg> <lb/>uoluendo etiam di-<lb/>ſtabuntinter ſe 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">ĩnotuit</reg> <reg norm="exem" type="context">exẽ</reg> <lb/>plo, & ſic etiam ali-<lb/>quando erunt magis <lb/><reg norm="diſtantes" type="context">diſtãtes</reg>, quàm linea <lb/><var>t.i.</var> & minus quàm.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 ſunt 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, ſi <lb/>extrema <var>.q.</var> et <var>.p.</var> eſſent quoque ſemper in vna <reg norm="eademque" type="context simple">eadẽq́;</reg> <lb/>diſtantia; </s> + <s xml:id="s1876" xml:space="preserve">neque à corpore <reg norm="ponderoſo" type="context">põderoſo</reg> eſſent attracta, <lb/>quia aliæ partes ipſius <var>.u.q.</var> et <var>.i.p.</var> ex ſupradictis ratio-<lb/>nibus vnam <reg norm="eademque" type="simple">eademq́;</reg> diſtantiam <reg norm="non" type="context">nõ</reg> ſemper <reg norm="ſeruarent" type="context">ſeruarẽt</reg>. <lb/></s> + <s xml:id="s1877" xml:space="preserve">vnde fieret vt cum diuerſis 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> ſemidiametros <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/>ſemper traherent ope ſeu virtute anguli æqualis ipſi <var>.<lb/>c.i.p</var>. </s> + <s xml:id="s1878" xml:space="preserve">Hæc autem inę qualitas communis eſt 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, quàm diſparis numeri. </s> + <s xml:id="s1879" xml:space="preserve">Sed aliam quandam maiorem <lb/>inęqualitatem habent hæ figuræ numeri diſparis, quæ eſt, quòd <reg norm="quando" type="context">quãdo</reg> linea <var>.t.i.</var> tam <lb/> <var>.u.q.</var> quàm ipſi <var>.i.p.</var> <lb/> + <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> + <reg norm="perpendicularis" type="simple context">ꝑpẽdicularis</reg> fuerit, <lb/>ideſt <reg norm="quando" type="context">quãdo</reg> <var>.t.i.</var> cum <lb/>dictis partibus funis <lb/>angulos rectos con-<lb/>ſtituerit, <reg norm="tunc" type="context">tũc</reg> ratione <lb/><reg norm="longitudinis" type="context">lõgitudinis</reg> ipſius <var>.c.<lb/>i.</var> maioris quam <var>.t.<lb/>c.</var> (quia cum ſit <var>.c.i.</var> <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg> ipſi <var>.c.a.</var> et <var>.c.a.</var> <lb/>maior ipſa <var>.c.t</var>: <var>c.i.</var> <lb/>etiam maior ſit ipſa <var>.<lb/>c.t.</var>) pondus aut vis <lb/>ipſius <var>.p.</var> ſuperabit <reg norm="eam" type="context">eã</reg> <lb/>quæ eſt ipſius <var>.q.</var> ſed <lb/>quando <var>.t.</var> erit in oppoſita parte, et <var>.i.</var> in ea, quæ eſt <lb/>ipſius .t: q. <reg norm="eandem" type="context">eãdem</reg> ob cauſam ſuperabit <var>.p.</var> & ſic 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="aſcendentem" type="context">aſcendẽtem</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 inęqualitatem habent figuræ pares, quæ <lb/>etiam in imparibus cernitur, etſi aliquantulum di-<lb/>uerſa; </s> + <s xml:id="s1882" xml:space="preserve">quæ ab eo oritur, quod funes ſit modò ma-<lb/>gis, modo minus propinquę centro; </s> + <s xml:id="s1883" xml:space="preserve">quæ inæqualis <lb/>diſtantia, maiorem <reg norm="minoremque" type="simple">minoremq́;</reg> vim ſuper dictum <lb/>centrum ob rationes in ſecunda parte cap. decimi <lb/>huius tractatus propoſitas, 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 diſparibus accommodari poteſt.</s> + </p> + <p> + <s xml:id="s1886" xml:space="preserve">Secundus modus eſt earum rotarum, in quibus aliquod animal incedit, quæ ſi cir-<lb/>culares non eſſent, tantò difficilius voluerentur, quantò pauciores angulos haberent. <lb/></s> + <s xml:id="s1887" xml:space="preserve">quod cum per ſe pateat, non demonſtrabo. </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 eſt 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, quàm <reg norm="violentię" type="context">violẽtię</reg>, quam anguli aeri faciunt, eum <lb/>expellendo, vt ipſi occupent locum, quem ipſe <reg norm="aer" type="conjecture">aér</reg> implebat. </s> + <s xml:id="s1892" xml:space="preserve">Quod nullo modo po <lb/>teſt euenire circulari figuræ.</s> + </p> + <p> + <s xml:id="s1893" xml:space="preserve">Nunc nobis ad dicendum reſtat de ſpecie reuolutionis rotarum, quæ parallelæ <lb/>ſunt orizonri, quibus accidit poſſe volui primo <reg norm="tertioque" type="simple">tertioq́;</reg> modo ſecundę ſpeciei, & ob <lb/>id ſi circulares non erunt, eadem ſubibunt incommoda, de quibus in ſecunda illa ſpe <lb/>cie loquuti ſumus. </s> + <s xml:id="s1894" xml:space="preserve">ſed circulares rotæ huius tertiæ ſpeciei ad reuoluendum erunt re-<lb/>liquis eò faciliores, <reg norm="quod" type="simple">ꝙ</reg> vno <reg norm="ſolum" type="context">ſolũ</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 ſpecie formari poteſt problema, vnde fiat, vt quieſcens huiuſ-<lb/>modi rota parallela orizonti ſuper vnum punctum, & quantò fieri poteſt exiſtens <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg>, ſi eam circunuoluamus maiore qua poterimus ui, & <reg norm="eandem" type="context">eãdem</reg> poſtea dimitten-<lb/>tes non perpetuò circunuoluatur.</s> + </p> + <p> + <s xml:id="s1897" xml:space="preserve">Hoc quidem, quatuor fit ob cauſas. quarum prima eſt, quia huiuſmodi motus, eius <lb/>rotæ non ſit naturalis. </s> + <s xml:id="s1898" xml:space="preserve">ſecunda eſt, quia etiamſi rota ſuper punctum mathematicum <lb/>quieſceret, oporteret tamen vt ſuperius <reg norm="alterum" type="context">alterũ</reg> haberet polum, qui ipſam <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 quędam <lb/>conſequeretur, ex qua reſiſtentia prodiret.</s> + </p> + <p> + <s xml:id="s1900" xml:space="preserve">Tertia eſt, quia aer contiguus eam perpetuò aſtringit, <reg norm="hocque" type="simple">hocq́;</reg> modo eius motui <lb/>reſiſtit.</s> + </p> + <p> + <s xml:id="s1901" xml:space="preserve">Quarta eſt, quia quęlibet pars corporea, quę à ſe mouetur, impetu eidem à quali-<lb/>bet extrinſeca virtute mouente impręſſo, habet naturalem inclinationem ad rectum <lb/>iter, non autem curuum, vnde ſi à dicta rota particula aliqua ſuę circunferentiæ <reg norm="diſium" type="context">diſiũ</reg> <lb/>geretur, abſque dubio per aliquod temporis ſpatium pars ſeparata recto itinere fer <lb/>retur per aerem, vt exemplo à fundis, quibus iaciuntur lapides, ſumpto, cognoſce <lb/>re poſsumus, in quibus, impetus motus impręſſus naturali quadam propenſione <lb/>rectum iter peragit, cum euibratus lapis, per lineam rectam contiguam giro, quem <lb/>primo faciebat, in puncto, in quo dimiſſus fuit, rectum iter inſtituat, vt rationi con-<lb/>ſentaneum eſt.</s> + </p> + <p> + <s xml:id="s1902" xml:space="preserve">Eadem, quoque ratione fit, vt quantò maior eſt aliqua rota, tantò maiorem quo <lb/>que impetum, & impreſſionem motus eius circunferentiæ partesrecipiant, vnde <reg norm="ſae pe" type="simple">ſę<lb/>pe</reg> euenit, vt dum eam ſiſtere volumus, id <reg norm="cum" type="context">cũ</reg> labore & cum diſſicultate agamus ; </s> + <s xml:id="s1903" xml:space="preserve">quia <lb/>quantò maior eſt diameter vnius circuli, tantò minus curua eſt eiuſdem circunferen <lb/>tia, & tantò propius accedit angulum eiuſdem circunferentiæ ad quantitatem duo-<lb/>rum angulorum rectorum rectilineorum, ideſt circunferentia ad rectitudinem linea <lb/>rem. </s> + <s xml:id="s1904" xml:space="preserve">Vnde earundem partium dictæ circunferentiæ motus ad inclinationem ſibi à <lb/>natura tributam, quæ eſt 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 queſtionis <lb/>admittendanon ſit.</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, accerſiri 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 queſtione <lb/>ſint reijciende.</head> + <head xml:id="head225" xml:space="preserve">CAP. XVI.</head> + <p> + <s xml:id="s1906" xml:space="preserve">ARiſtotelis rationes, vnde fiat, vt facilius moueantur libræ vacuæ, quàm plenè <lb/>ad propoſitam diſputationem non pertinent; </s> + <s xml:id="s1907" xml:space="preserve">quia ſemper ineunda eſt ratio <lb/>proportionis virtutis mouentis ſuper mobile; </s> + <s xml:id="s1908" xml:space="preserve">quod ipſe 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 ſolum <lb/>ponderis obtineat, & ſit libra <var>.n.i.u.</var> æqualis priori, quæ pro ſingula 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">Ariſtoteles admiratur, quòd addendo ipſi <var>.e.</var> mediam pon <lb/>deris vnciam, brachium <var>.i.e.</var> velocius cadat, quàm <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> ipſi <var>.u.</var> <lb/>brachij <var>.i.u</var>. </s> + <s xml:id="s1911" xml:space="preserve">Quod à duabus cauſis proficiſcitur, quarum prior eſt, magna differentia <lb/>proportionis vnius libræ ad medietatem vnius vnciæ, ad proportionem vnius vnciæ <lb/>ad ipſam medietatem, quia ſi pondus adiectum extremo <var>.u.</var> dimidiæ eſſet libræ , & <lb/>cum eadem tarditate brachium moueret, optimo iure in admirationem poſſet Ari-<lb/>ſtoteles duci. </s> + <s xml:id="s1912" xml:space="preserve">Sed hoc fieri non poſſet, quia ipſum deprimeret cum eadem quaſi ve <lb/>locitate, qua media vncia brachium <var>.i.e</var>. </s> + <s xml:id="s1913" xml:space="preserve">Dixi autem quaſi, quia nonnihil diſcrimi-<lb/>nis intercederet, quod proficiſcitur à ſecunda ratione. </s> + <s xml:id="s1914" xml:space="preserve">Et hæc, reſiſtentia eſt , quæ <lb/>oritur à ſparto, quia quantò maius pondus continet libra, tantò magis præmit ſpar <lb/>tum in loco, in quo ſuſtinetur; </s> + <s xml:id="s1915" xml:space="preserve">vnde maior reſiſtentia in circunuolutione <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpar <lb/>ti, in loco, in quo quieſcit, exoritur, quia ipſum eſt 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> deſcenderet, oporteret, vt propor-<lb/>tio dimidiæ librę adiectæ ponderi ipſius <var>.u.</var> <lb/> + <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> + quod eſt vnius libræ, vim ſuam haberet, <lb/>quæ excederet reſiſtentiam ſui ſparti (me-<lb/>dio brachiorum maiorum ijs qui ſunt <var>.a.i.<lb/>e.</var>) ita proportionatam, vt proportionata <lb/>eſt vis dimidiæ vnciæ ipſi e. iunctæ, reſiſten <lb/>tiæ ſui ſparti. </s> + <s xml:id="s1917" xml:space="preserve">Huiuſmodi rationes cum ro-<lb/>tis grauioribus <reg norm="leuioribusque" type="simple">leuioribusq;</reg>, & ijs, quę à cor <lb/>poribus quibuſlibet 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 cauſa .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, quàm fieret manu, quod corpus liberatum deinde cum <lb/>fuerit à funda, natura duce, iter <reg norm="fuum" type="context">fuũ</reg> à puncto, à quo proſilijt, per lineam contiguam <lb/>giro, quem poſtremò faciebat, ſuſcipit. </s> + <s xml:id="s1919" xml:space="preserve"><reg norm="Dubitandumque" type="simple">Dubitandumq́</reg> non eſt, quin dicta funda <lb/>maior impetus motus dicto corpori imprimi poſſit, <reg norm="cum" type="context">cũ</reg> ex multis circumactibus, ma-<lb/>ior ſemper impetus dicto corpori accedat. </s> + <s xml:id="s1920" xml:space="preserve">Manus autem eiuſdem corporis motus, <lb/>dum illud ipſum circunuoluitur (pace Ariſtotelis dixerim) centrum non eſt, neque <lb/>funis eſt ſemidiameter. </s> + <s xml:id="s1921" xml:space="preserve">Immo manus quam maximè fieri poteſt in orbem cietur; <lb/></s> + <s xml:id="s1922" xml:space="preserve">qui quidem motus in orbem, vt circumagatur etiam ipſum corpus, cogit, quod qui-<lb/>dem corpus, naturali quadam inclinatione, exiguo quodam impetu iam incępto, <lb/>vellet recta iter peragere, vt in ſubſcripta figura patet, in qua <var>.e.</var> ſignificat 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 eſt, impręſſum illum impetum, continuò paulatim decreſcere <lb/>vnde ſtatim inclinatio grauitatis eiuſdem corporis ſubingreditur, quæ ſeſe miſcens <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, ſed citò fiat curua, cum dictum corpus <var>.a.</var> duabus virtutibus moueatur, qua-<lb/>rum vna eſt, 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 ſimul & ſemel moueri <lb/>poſſe. </s> + <s xml:id="s1924" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> eſt <reg norm="ſilentio" type="context">ſilẽtio</reg> <reg norm="prætereundus" type="context">prætereũdus</reg> <lb/>hac in re <reg norm="quidam" type="simple context">ꝗdã</reg> notatu <reg norm="dignus" type="simple">dignꝰ</reg> effectus, <lb/>qui eiuſmodi eſt, <reg norm="quod" type="simple">ꝙ</reg> quanto magis <lb/>creſcit impetus in corpore <var>.a.</var> cauſa <lb/>tus ab augumento velocitatis giri <lb/>ipſius <var>.e.</var> <reg norm="tantò" type="context">tãtò</reg> magis oportet, vt ſen-<lb/>tiat ſe trahi manus à dicto corpore <lb/>a. mediante fune, quia quantò ma-<lb/>ior impetus motus ipſi <var>.a.</var> eſt impręſ <lb/>ſus, 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 eſt 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 reſtat, illud ipſum totum de <lb/>pendet à .4. <reg norm="quintoque" type="simple">quintoq́;</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">vectemq́;</reg> reperitur, quæ eſt, <reg norm="quod" type="simple">ꝙ</reg> iugum aliquam reſiſtentiam pro coniunctione <lb/>calcata in loco, in quo voluitur, magis quàm 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 queſtione.</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 ſunt vectis, vt <lb/>exempli gratia ſit 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">ꝙ</reg> tenentes marlus longè à medio, in locis <lb/>a. et <var>.c.</var> facilius <reg norm="minorique" type="simple">minoriq́;</reg> cum labore illum frangemus, quàm ſi eaſdem vicinas me-<lb/>dio eiuſdem 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> ſunt <reg norm="cum" type="context">cũ</reg> ijs, quæ <lb/>primis huius tractatus capitibus propoſitæ 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 manifeſtè perſpiciemus eorum, quæ iam diximus <lb/>ratione, <reg norm="quod" type="simple">ꝙ</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, quàm loco <var>.a.</var> et <var>.c.</var> <reg norm="duarum" type="context">duarũ</reg> <reg norm="linearum" type="context">linearũ</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">habebũt</reg> <lb/> + <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> + in <var>.a.</var> et <var>.c.</var> <lb/>quàm 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. queſtionis.</head> + <head xml:id="head233" xml:space="preserve">CAP. XX.</head> + <p> + <s xml:id="s1933" xml:space="preserve">DEcimaſeptima quæſtio ab Ariſtotele haud benè percepta fuit, quia is non ac-<lb/>commodat partes vectis ſuis 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 ſint <var>.o.</var> & pondera, <lb/>quæ ſunt attollenda ſint <var>.a.</var> et <var>.e.</var> inter ſe æqualia, & diſtantię ſint <var>.a.o.</var> et <var>.e.o.</var> ſibi <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>æquales, ſed <var>.o.n.</var> æqualis ſit ipſi <var>.o.u</var>: clarum erit, <reg norm="quod" type="simple">ꝙ</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 ſupponuntur æ qua <lb/> + <anchor type="handwritten" xlink:label="hd-0174-01a" xlink:href="hd-0174-01"/> + lia, clarum quoque erit, commu-<lb/>ni ſcientia, tantam virtutem in <lb/> + <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> + n. quanta ſufficiet ad <reg norm="attollendum" type="context">attollendũ</reg> <lb/>a. in <var>.u.</var> <reg norm="quoque" type="simple">quoq;</reg> ſuffecturam ad ele-<lb/>uandum <var>.e.</var> quia <reg norm="cum" type="context">cũ</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 ſuo 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">quintoq́;</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 ſatis ſu <lb/><reg norm="perque" type="simple">perq́;</reg> dixi in dicto capit .5. conuenire poſ-<lb/>ſunt.</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 ſit aliqua pars ligni cindenda ſe-<lb/>cundum venulas ſuas <var>.d.e.f.g.</var> & ſit cuneus <lb/><var>a.b.c.</var> qui vi mallei <var>.P.</var> vſque 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, poſt quam infigitur cuneus ſe <lb/>cundum venas, longíor erit parte <var>.x.b.t.</var> cu <lb/>nei, quæ ingreſſa eſt. </s> + <s xml:id="s1939" xml:space="preserve">Oportet nunc ima-<lb/>ginari duos vectes ſimiles ſupradictæ <var>.u.e.<lb/>o.</var> in hunc modum, vt puncta <var>i.r.</var> lìni ſint <lb/>loco <var>.u.</var> extremi <reg norm="ipſius" type="simple">ipſiꝰ</reg> vectis, et <var>.t.x.</var> loco vir <lb/>tutis applicatæ ipſi <var>.u.</var> & reſiſtentia circa <lb/>punctum <var>.m.</var> loco ponderis <var>.e.</var> vectis <var>.o.e.u.</var> <lb/>dicti, & pars <var>.K.</var> quaſi immediata poſt <var>.m.</var> <lb/>verſus extremitatem <var>.f.e.</var> ligni, ſit 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 & intrinſeca cauſa trocble arum.</head> + <head xml:id="head235" xml:space="preserve">CAP. XXI.</head> + <p> + <s xml:id="s1941" xml:space="preserve">PRo intelligenda vera, & intrinſeca ratione, vnde fiat ut multitudo rotularum in <lb/>trochleis cauſa ſit, ut exigua vis ſurſum moueat, aut attollat <reg norm="pondera" type="context">põdera</reg> magna. </s> + <s xml:id="s1942" xml:space="preserve">Ima <lb/>ginemur duas hîc ſubſcriptas trochlæas explicatas tranſuerſaliter in hunc modum, <lb/>ideſt ſit <reg norm="paruum" type="context">paruũ</reg> <reg norm="tignum" type="context">tignũ</reg> <var>.a.b.</var> fixum & <reg norm="parallelum" type="context">parallelũ</reg> orizonti. cui ſint rotulæ appenſe ab infe <lb/>riori parte ad ſuperiorem <reg norm="huicque" type="simple">huicq́;</reg> è regione <reg norm="oppoſitus" type="simple">oppoſitꝰ</reg> ſit aliud <reg norm="tignum" type="context">tignũ</reg> <var>.c.d.</var> quod moueri <lb/>poſſit ab imo ad ſumum, ſuper quod totidem ſint rotulæ aut radij, <reg norm="cum" type="context">cũ</reg> annexa poſtea <lb/>fuerit funis puncto <var>.b.</var> fixo, eam faciendo pertranſire per rotulas tam à parte ſupe-<lb/>riore, quam ab inferiore; </s> + <s xml:id="s1943" xml:space="preserve">& appenſum deinde cum erit paruo illi tigno <var>.c.d.</var> mobili <lb/>pondus <var>.E.</var> ducendo poſtmodum extremum <var>.f.</var> funis tranſeuntis per rotulas, idem pla <lb/>nè fiet quod à trochlęis ſimul unitis fieri ſolet. </s> + <s xml:id="s1944" xml:space="preserve">Cuius quidem effectus ratio ſub no-<lb/>ſtram cognitionem cadet facilius in huiuſmodi figura. </s> + <s xml:id="s1945" xml:space="preserve">Imaginemur ſeparatim ſta-<lb/>teram <var>.g.h.</var> cuius <reg norm="centrum" type="context">cẽtrum</reg> ſit <var>.K.</var> ita ſitum, ut brachium <var>.g.k.</var> ſit duplum ad brachium <var>.K.<lb/>h.</var> ſupponendo igitur in puncto <var>.g.</var> pondus, aut virtutem mouentem unius libræ, & in <lb/>h. duarum librarum, <reg norm="abſque" type="simple">abſq;</reg> dubio hæ duæ uirtutes in huiuſmodi diſtantijs à centro <lb/> + <anchor type="handwritten" xlink:label="hd-0175-01a" xlink:href="hd-0175-01"/> + ęquales <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="erunt" type="context">erũt</reg>, ob rationes prioribus capitibus iam allatas, & ſtatera orizontalis <lb/>manebit. </s> + <s xml:id="s1946" xml:space="preserve">Vnde clarum erit, <reg norm="quod" type="simple">ꝙ</reg> quæuis etiam exigua virtus adiuncta ipſi <var>.g.</var> mouebit <lb/>ſtateram extra orizontalem ſitum. </s> + <s xml:id="s1947" xml:space="preserve">Nunc ſi puncto <var>.i.</var> ex æquo medio inter <var>.g.</var> et <var>.K.</var> <lb/>applicata erit virtus ipſius <var>.h.</var> non amplius conſiderato brachio <var>.K.h.</var> inclinante uirtu-<lb/>te ipſius <var>.i.</var> eandem partem verſus, in quam inclinabat, quando erat in <var>.h.</var> ſed uirtus ip <lb/>ſius <var>.g.</var> inclinet contrario modo, <reg norm="diuerſoque" type="simple">diuerſoq́;</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 ſunt dicta <var>.g.h.</var> ſemper <lb/>in eodem ſitu abſque motu manſuram, <reg norm="hancque" type="simple">hancq́;</reg> ſtateram appellabimus mobilem, & <lb/>primam. </s> + <s xml:id="s1949" xml:space="preserve">Imaginemur nunc à puncto <var>.e.</var> fixo deſcendere 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">& ſit <var>.n.l.m.</var> diameter vnius ex rotulis ſuperioribus alterius parui <lb/>tigni defixi à parte inclinationis ipſius <var>.g.</var> & parallela diametro <var>.g.K.</var> cuius diametri <lb/>centrum fixum ſit <var>.l.</var> & ſit coniunctum <var>.g.</var> punctum, à fune cum puncto <var>.m.</var> quæ <reg norm="tam" type="context">tã</reg> per-<lb/>pendicularis ſit primo diametro <var>.g.i.K.</var> quàm ſecundo <var>.n.m.</var> ideſt 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> ſint recti. </s> + <s xml:id="s1951" xml:space="preserve">Imaginemur <reg norm="quoque" type="simple">quoq;</reg> virtutem ipſius <var>.g.</var> applicatam eſſe extremo <var>.<lb/>n.</var> cum inclinatione tamen contraria, ideſt ad inferiorem partem, quæ quidem virtus <lb/>communi quodam conceptu eandem poſſidebit vim ſuſtentandi 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 ſuperiorem partem, <lb/>& ſic etiam diameter <var>.n.l.m.</var> non magis ab una, quàm ab alia parte declinabit, quia <lb/>cum quædam virtus in <var>.n.</var> reperiatur æqualis medietati uirtutis ipſius <var>.i.</var> quæ uirtus ip <lb/>ſius <var>.i.</var> uim habet deprimendi ipſum <var>.g.</var> ideſt <var>.m.</var> pro dimidia ſui ipſius parte, ſequitur <var>.<lb/>n.m.</var> debere immobilem permanere. </s> + <s xml:id="s1952" xml:space="preserve">Nunc ſi alia diameter rotulæ mobilis erit de-<lb/>ſumpta, quæ ſit <var>.p.q.o.</var> cuius centrum ſit <var>.q.</var> in ſitu parallelo ipſi <var>.n.l.m.</var> & ſic collocata, <lb/>vt coniungendo <var>.o.</var> cum <var>.n.</var> anguli <var>.m.n.o.</var> et <var>.n.o.p.</var> ſint recti: </s> + <s xml:id="s1953" xml:space="preserve">ſi imaginati fuerimus <reg norm="tranſ" type="context">trãſ</reg> <lb/>latum eſſe <reg norm="pondusipſius" type="simple">pondusipſiꝰ</reg> <var>.n.</var> in <var>.o.</var> <reg norm="cum" type="context">cũ</reg> <reg norm="eadem" type="context">eadẽ</reg> inclinatione ad depræſſiorem partem, illud ip <lb/>ſum, ac ſi eſſet in <var>.n.</var> communi conceptu, ſine alicuius diametri mutatione præſtabit. <lb/></s> + <s xml:id="s1954" xml:space="preserve">Et ſi centrum <var>.q.</var> fixum eſſet, & extremo <var>.p.</var> appoſitum fuiſſet pondus ipſius <var>.o.</var> cum in <lb/>clinatione ad ſuperiorem partem, idem etiam planè pręſtaret, etiam ſi nullum ullius <lb/>diametri ſitum, communi ſcientia, mutaret, cum extremum <var>.m.</var> deorſum ſit ductum <lb/>à. g. uirtute dimidiæ partis ipſius <var>.i.</var> & ab alia huic ſimili <var>.m.</var> quoque deorſum ſit tra-<lb/>ctum ab .o: quod quidem <var>.o.</var> deorſum eſt alteratum, ob inclinationem ad ſuperius <lb/>à uirtute poſita in <var>.p.</var> ſupponendo centrum <var>.q.</var> fixum. </s> + <s xml:id="s1955" xml:space="preserve">Sed ſi loco centri fixi, imagina <lb/>bimur in <var>.q.</var> pondus aliquod æquale ipſi <var>.i.</var> quod duplum erit in uirtute ad eam, quæ <lb/>eſt ipſius <var>.p.</var> & ipſius quoque .g: ſequetur <reg norm="etiam" type="context">etiã</reg> eadem immobilitas horum trium dia-<lb/>metrorum. </s> + <s xml:id="s1956" xml:space="preserve">Quia cum ſit huiuſmodi pondus ſeu virtus in <var>.q.</var> cum inclinatione con-<lb/>traria virtuti in <var>.p.</var> quæ æquipollet dimidiæ parti ipſius <var>.q.</var> & ſic ei quæ eſt ipſius <var>.o.</var> ſi-<lb/>militer quia <var>.o.</var> tractum eſt ſupra ab <var>.n.</var> virtute ipſius <var>.g.</var> quod <var>.m.</var> deorſum trudit; </s> + <s xml:id="s1957" xml:space="preserve">idcir <lb/>co quanta erit vis quam habebit virtus in <var>.q.</var> ferendi deorſum diametrum <var>.p.o.</var> tanta <lb/>quoque virtutes ipſorum <var>.p.</var> et <var>.o.</var> æquales, & æqualiter diſtantes à <var>.q.</var> ipſum ad ſupe-<lb/>riorem partem inclinabunt. </s> + <s xml:id="s1958" xml:space="preserve">Quamobrem nec aſcender, nec deſcendet, nec locum <lb/>mutabit. </s> + <s xml:id="s1959" xml:space="preserve">Supponamus nunc quartum diametrum rotulæ <var>.s.t.r.</var> quæ ſit ſecunda rotu <lb/>larum fixarum, parallela ipſi <var>.p.o.</var> & in eo ſitu, quo coniungendo extrema <var>.r.p.</var> anguli <lb/><var>o.p.r.</var> et <var>.p.r.s.</var> ſint recti, & imaginemur virtutem ipſius <var>.p.</var> reperiri in <var>.s.</var> cum inclinatio <lb/>ne tamen contraria, ideſt deorſum verſus, ex his <reg norm="idem" type="context">idẽ</reg> quoque planè ſequetur, ideſt <reg norm="quod" type="simple">ꝙ</reg> <lb/>nulla <reg norm="harum" type="context">harũ</reg> quatuor diametrorum mouebitur. </s> + <s xml:id="s1960" xml:space="preserve">quia eundem <reg norm="effectum" type="context">effectũ</reg> <reg norm="cum" type="context">cũ</reg> inclinatione <lb/>deorſum verſus efficeret dicta virtus in <var>.s.</var> quem in <var>.p.</var> cum inclinatione ſurſum verſus. <lb/></s> + <s xml:id="s1961" xml:space="preserve">et iam dictum eſt virtutem ipſius <var>.g.</var> dimidium virtutis ipſius <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 ipſius <var>.r</var>. </s> + <s xml:id="s1962" xml:space="preserve">Hucuſque <reg norm="ſcien- tificè" type="context">ſciẽ-<lb/>tificè</reg> nouimus pondus, aut virtutem ipſius <var>.s.</var> quæ eſt dimidium <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.i.</var> ſuſtinere uim <lb/>ipſorum <var>.i.</var> et <var>.q.</var> nam quater tantum, quanta ipſamet virtus ipſius <var>.s.</var> eſſe conſpicitur. <lb/></s> + <s xml:id="s1963" xml:space="preserve">Et ſi adiunctę nobis eſſent duæ aliæ diametri cum ijſdem planè conditionibus <reg norm="ijſdem" type="context">ijſdẽ</reg> <lb/>rationibus vtentes, cognoſceremus quod eadem medietas ipſius <var>.i.</var> ſexies tantum <reg norm="pon" type="context">põ</reg> <lb/>deris, quanta ipſa exiſteret, ſeſtineret. </s> + <s xml:id="s1964" xml:space="preserve">Vnde <reg norm="manifeſtum" type="context">manifeſtũ</reg> euadit, <reg norm="quod" type="simple">ꝙ</reg> eidem medietati <lb/>ipſius <var>.i.</var> in <var>.s.</var> nonnihil virtutis addendo, dictæ diametri, illicò <reg norm="mouerentur" type="context">mouerẽtur</reg> ſitu. </s> + <s xml:id="s1965" xml:space="preserve">Et quia <lb/>rotulæ in quolibet puncto, aliquam diametrum habent, neceſſariò ſequitur <reg norm="quod" type="simple">ꝙ</reg> infe-<lb/>riores ad ſuperiores accedere debeant. </s> + <s xml:id="s1966" xml:space="preserve">Attamen ſi forte extremum immobile ip-<lb/>ſius funis non pendet à puncto <var>.e.</var> trochleæ ſuperioris, ſed alligatum fuerit ad <reg norm="medium" type="context">mediũ</reg> <lb/>inferioris trochleæ ut ad punctum <var>.i.</var> ope unius trochleę ſuperioris immobilis vt in fi <lb/>gura <var>.A.</var> videre licet, clarè patebit <reg norm="quod" type="simple">ꝙ</reg> à tribus virtutibus æqualibus pondus in <var>.i.</var> <reg norm="poſitum" type="context">poſitũ</reg> <lb/>ſuſtinebitur: </s> + <s xml:id="s1967" xml:space="preserve">hoc eſt à <var>.g.</var> ab <var>.i.</var> & ab <var>.k.</var> <reg norm="quarum" type="context">quarũ</reg> vnaquęque tertia pars erit ipſius <var>.i.</var> in con <lb/>c<unsure/>ontrariam <reg norm="partem" type="context">partẽ</reg>, hoc eſt tertia pars reſiſtentiæ. </s> + <s xml:id="s1968" xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</reg> ex æquo inter ſe <reg norm="diſtant" type="context">diſtãt</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 reſiſtentię, 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="ſuffieiant" type="context">ſuffieiãt</reg> reſiſtentiæ in <var>.o.</var> et <var>.p.</var> <lb/>ipſum ſuſtinere, idcirco ipſum pondus <var>.q.</var> ſubſeſquialter erit <reg norm="ponderi" type="context">põderi</reg> in <var>.i.</var> poſiti. </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 ſi adhuc. duo diametri vnus in-<lb/>ferior, alter verò ſuperior 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 ſeptima pars trium ponderum <var>.i.q.</var> & tertij additi, ex <lb/> + <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> + ſupradictis rationibus. </s> + <s xml:id="s1976" xml:space="preserve">Et quia virtus ſuſti <lb/> + <anchor type="figure" xlink:label="fig-0177-02a" xlink:href="fig-0177-02"/> + nens totale pondus trochleæ inferiori ap-<lb/>penſum in tot diuiditur partes æquales, <lb/>quot ſunt diametri orbiculorum trochleæ <lb/>inferioris, quando extremum immobile fu <lb/>nis alligatum fuerit trochleę ſuperiori, 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 ſemper ſeſqui <lb/>altera erit vnicuique aliorum <reg norm="diametrorum" type="context">diametrorũ</reg> <lb/>ideò virtus reſiſtentię alterius extremi mo <lb/>bilis funis, puta <var>.s.</var> ſubmultiplex erit totalis <lb/>ponderis, eo modo quo diximus, cuius vir <lb/>tus, ſeu grauitas diuiditur ſeu diſtrubuitur <lb/>diametris inferioris trochleæ vt dictum eſt.</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 cauſa .24. quæſtionis.</head> + <head xml:id="head237" xml:space="preserve">CAP. XXII.</head> + <p> + <s xml:id="s1977" xml:space="preserve">VEra cauſa effectus, qui vigeſimaquarta quæſtione exprimitur, adhuc à nemine <lb/>(quod ſciam) animaduerſa fuit, licet non ſit admodum ardua vel obſcura. </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> ſimul coniunctos, vt ſi ip <lb/>ſorum 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 ſuper lineam <var>.f.i.</var> reuolui maiorem, & quando <lb/>idem circulus erit in <var>.l.</var> dictam lineam <var>.f.i.</var> tangere circunferentiam eiuſdem 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 ipſi <var>.f.l.</var> quia ex .17. tertii, anguli <var>.f.</var> et <var>.g.</var> <lb/>ſunt æquales, vnde ex .28. primi <var>.f.l.</var> et <var>.g.K.</var> ſunt parallelæ. </s> + <s xml:id="s1980" xml:space="preserve">& ſic erunt <var>.k.l.</var> cum <var>.f.g.</var> ex <lb/>eadem ſupradicta. </s> + <s xml:id="s1981" xml:space="preserve">Ratio autem, qua arcus <var>.g.b.</var> tranſierit lineam <var>.g.K.</var> maiorem ipſa, <lb/>eſt, quia dum mouetur, quodlibet punctum ipſius <var>.g.b.</var> virtute reuolutionis ipſius <var>.f.c.</var> <lb/>omne punctum eiuſdem arcus <var>.g.b.</var> vlterius verſus <var>.K.</var> quam ſi moueretur virtute re-<lb/>uolutionis ipſius <var>.g.b.</var> ſuper 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 ſitu 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 ſint 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/>ſam 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 ipſo <var>.b.</var> et <var>.g.</var> fa-<lb/>ciet iter <var>.g.n.</var> quę itinera alijs multò breuiora ſunt, quia breuioribus cruribus reuolu-<lb/>ta ſunt dicta puncta; </s> + <s xml:id="s1985" xml:space="preserve">& ſic dico de reliquis omnibus punctis inter <var>.g.</var> et <var>.b.</var> & in hoc ca <lb/>ſu 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 ſolum non erunt delata anteà, ſed potius à primo ſitu retrorſum <lb/>erunt repulſa. </s> + <s xml:id="s1987" xml:space="preserve">Vnde non eſt, quòd in tantam admirationem ducamur ſi dum reuol <lb/>uitur circulus maior, arcus <var>.g.b.</var> circuli minoris, totam lineam <var>.g.K.</var> tranſire 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 ſeſe 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 ſolum mouetur ver <lb/>ſus <var>.m.</var> quòd circa centrum <var>.a.</var> feratur, cum ipſum etiam centrum moueatur verſus <var>.m.</var> <lb/>ſed quia pręter hoc deferantur quoque à circulo maiori verſus <var>.m.</var> vſque 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 verſus <var>.i.</var> virtute reuolutionis circuli minoris, <lb/>& alter ex eo, <reg norm="quod" type="simple">ꝙ</reg> dictus circulus maior circa centrum <var>.a.</var> voluatur, vnde omne <reg norm="punctum" type="context">punctũ</reg> <lb/>contactus circuli maioris cum recta <var>.f.i.</var> tetrorſum pellitur verſus <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 cauſa .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 ſedet ( non tamen Turcarum more ) ſi velit <lb/>ſeſe in pedes erigere, calcaneos retrahit, vt efficiat angulum acutum, cum <reg norm="fae- moribus" type="simple">fę-<lb/>moribus</reg> coxis à parte inferiori, & ventrem inclinat, ad conſtituendum etiam angu <lb/>lum acutum in ſuperiori parte, ea eſt; </s> + <s xml:id="s1991" xml:space="preserve">vt totius corporis pondus, ex ęquo, ideſt ab <lb/>oppoſitis partibus circundet lineam rectam, quæ tranſit per locum, in quo conquie <lb/>ſcunt pedes verſus mundi centrum. </s> + <s xml:id="s1992" xml:space="preserve">ideſt, ut edatur ęquilibrium ponderis ipſius cor-<lb/>poris circum lineam illam, quę ſub pedibus inſeruit pro ſparto. </s> + <s xml:id="s1993" xml:space="preserve">Vnde aperiendo, <lb/>deinde dictos duos angulos circa dictam <reg norm="lineam" type="context">lineã</reg>, abſque vlla difficultate erigitur cor-<lb/>pus, & abſque 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, quę reperiuntur in vorticibus aquarum, ſemper verſus <lb/>medium ipſarum vertiginum vniuntur, inde promanat, quod media <lb/>vertiginum ſemper depreſſiora ſunt. </s> + <s xml:id="s1995" xml:space="preserve">vnde quòd dicta corpora ad medium acce-<lb/>dant, nihil aliud eſt, quàm ipſa corpora ſuo pondere grauitateq́ue deſcendere, figu <lb/>ra enim vorticibus eſt quaſi conica, & concaua cum angulo deorſum, & gyro baſis <lb/>ſurſum. </s> + <s xml:id="s1996" xml:space="preserve">Atque hæc vera eſt huius effectus cauſa, & non ea quam Ariſtoteles ponit, <lb/>à<unsure/> quo aliarum omnium quæſtionum, quas ego omiſi rationes ſunt benè propoſitæ.</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 eſt certè Ariſtotelis amplitudo at que authoritas, vt dif-<lb/>ficillimum ac periculoſum ſit quidpiam ſcribere contra quam <lb/>ipſe docuerit, & mihi præſertim, cui ſemper viſa est viri <lb/>illius ſapientia admirabilis. </s> + <s xml:id="s1998" xml:space="preserve">Veruntamen studio veritatis im-<lb/>pulſus, cuius ipſe amore in ſeipſum ſiviueret excitaretur, in me <lb/>dium <reg norm="quædam" type="context">quædã</reg> proferre non dubitaui, in quibus me inconcußa mathematicæ philoſophiæ <lb/>baſis, cui ſemper inſiſto, ab eo dißentire coegit.</s> + </p> + <div type="section" level="3" n="1"> + <head xml:id="head243" style="it" xml:space="preserve">Qualiter & ubi Ariſtoteles de uelocitate motuum natura-<lb/>lium localium aliter tractauerit quam nos <lb/>ſentiamus.</head> + <head xml:id="head244" xml:space="preserve">CAP.I.</head> + <p> + <s xml:id="s1999" xml:space="preserve">VOlens Ariſtoreles probare vacuum non eſſe in rerum natura .8. cap. lib. 4. phy<lb/>ſicorum ait, idem corpus per varia <reg norm="diuerſaque" type="simple">diuerſaq́;</reg> media, vt per <reg norm="aerem" type="context">aerẽ</reg>, & per <reg norm="aquam" type="context">aquã</reg> <lb/>ſi moueretur, proportionem velocitatis eiuſdem corporis per aerem, ei, quæ per <lb/>aquam fit, vnam <reg norm="eandemque" type="simple">eandemq́;</reg> futuram cum ea, quæ eſt ſubtilitatis aereę ad ſubtilitatem <lb/>aquæ. </s> + <s xml:id="s2000" xml:space="preserve">In poſtrema autem parte eiuſdem capitis ſic ſcribit: </s> + <s xml:id="s2001" xml:space="preserve">Nam cum ea quę ma-<lb/>iorem vel ponderis velleuitatis pręſtantiam habent, ſi ſimili ſigura ſint, <reg norm="ſpacium" type="context">ſpaciũ</reg> par, <lb/>& æquale, maiore celeritate conficere cernamus, ea quam magnitudines inter ſe 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 phyloſophus .2. cap. ſexti phyſicorum ſcribens eademmet proportione, <lb/>qua tempus diuiditur, magnitudinem etiam diuidi. </s> + <s xml:id="s2004" xml:space="preserve">Sexto autem cap. primi de cœ-<lb/>lo ſcribit, tempora eandem proportionem habere, quam habentè conuerſo ponde-<lb/>ra; </s> + <s xml:id="s2005" xml:space="preserve">vt ſi media pars vnius ponderis, vnius horæ ſpatio moueretur, vniuerſum pondus <lb/>in media hora moueretur. </s> + <s xml:id="s2006" xml:space="preserve">Secundo cap. lib. 3. de cœlo duobus in locis apertè com<lb/>monſtrat velocitatem corporis minoris, maiori corpori comparatam, in eadem exi-<lb/>ſtere proportione, in qua dicta corpora adinuicem relata exiſtunt. </s> + <s xml:id="s2007" xml:space="preserve">Quinto cap. eiuſ<lb/>dem lib. idem affirmat, exemplo ab igne deſumpto. </s> + <s xml:id="s2008" xml:space="preserve">Ex alijs etiam plurimis locis <lb/>cognoſci poteſt, ſenſiſſe Ariſtotelem duo corpora eadem ſpecie, & figura prædita <lb/>eandem planè proportionem in ſuorum motuum velocitatibus, quam in ſuis 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">omniũ</reg> poſtremus Nicolaus Tartalea, ſecunda propoſitione vigeſiminoni <lb/>quæſiti octaui libri, vbi profitetur ſe demonſtratiuè probare hanc propoſitio-<lb/>nem veram exiſtere; </s> + <s xml:id="s2010" xml:space="preserve"><reg norm="neque" type="simple">neq;</reg> videt quàm magna reſiſtentiarum ſit differentia, quætam <lb/>ex diuerſitate figurarum, quàm ex magnitudinum varietate exoriri poteſt; </s> + <s xml:id="s2011" xml:space="preserve">quas qui <lb/>dem diuerſitates ne conſiderat 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 ſupponenda ut conſtet cur circa uelocit atem motuum <lb/>natur alium localium ab Ariſtotelis placitis <lb/>recedamus.</head> + <head xml:id="head246" xml:space="preserve">CAP. II.</head> + <p> + <s xml:id="s2012" xml:space="preserve">CVM ſuſceperimus prouinciam probandi quod Ariſtoteles circa motus <lb/>locales naturales deceptus fuerit, ſunt quædam primo veriſſima & obie-<lb/>cta intellectus perſe cognita pręſupponenda, ac primum quælibet duo corpora, <lb/>grauia, aut leuia, area æquali, <reg norm="ſimilique" type="simple">ſimiliq́</reg> figura, ſed ex materia diuerſa conſtantia, <reg norm="eodem q́ue" type="context">eodẽ<lb/>q́ue</reg> modo ſitum habentia, eandem proportionem velocitatis inter ſuos motus loca <lb/>les naturales, ut inter ſuamet pondera aut leuitates in vno <reg norm="eodemque" type="simple">eodemq́;</reg> medio, ſeruatu-<lb/>ra. </s> + <s xml:id="s2013" xml:space="preserve">Quod quidem natura ſua notiſſimum eſt ſi conſiderabimus non aliunde maio-<lb/>rem tarditatem, aut velocitatem gigni, quàm à .4. cauſis (dummodo medium vnifor <lb/>mè ſit & quietum) ideſt à maiori aut minori pondere aut leuitate; </s> + <s xml:id="s2014" xml:space="preserve">à diuerſa figura; <lb/></s> + <s xml:id="s2015" xml:space="preserve">à ſitu ciuſdem figuræ diuerſo, reſpectu lineę 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 eiuſdem figu-<lb/>ræ ſitum, 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 & ſitu eiuſdem figuræ dico, reſpectu reſi-<lb/>ftentiæ ipſius medii dico: </s> + <s xml:id="s2019" xml:space="preserve">Quia diffimilitudo aut inęqualitas figurarum, aut ſitus di-<lb/>uerſus non <reg norm="parum" type="context">parũ</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 cęlerius idem facit acuta, quàm ob <lb/>tufa; </s> + <s xml:id="s2021" xml:space="preserve">& illa quæ cum angulo, qui antecedat mouebitur velocius quàm illa, quæ ſecus. <lb/></s> + <s xml:id="s2022" xml:space="preserve">Quotieſcunque igitur duo corpora vnam <reg norm="eandemque" type="simple">eandemq́;</reg> reſiſtentiam ipſorum ſuperfi-<lb/>ciebus, aut habebunt aut recipient, eorum motus inter ſeipſos eodem planè modo <lb/>proportionati conſurgent, quo erunt ipſorum virtutes mouentes: </s> + <s xml:id="s2023" xml:space="preserve">& è conuerſo, quo <lb/>tieſcunque duo corpora vnam <reg norm="eandemque" type="simple">eandemq́;</reg> grauitatem, aut leuitatem, & diuerſas reſi <lb/>ſtentias habebunt, eorum motus inter ſeipſos eandem <reg norm="proportionem" type="context">proportionẽ</reg> ſortientur, <reg norm="quam" type="context">quã</reg> <lb/>habebunteorum reſiſtentiæ conuerſo modo; </s> + <s xml:id="s2024" xml:space="preserve">quæ quidem reſiſtentiæ inter ſeipſas, <lb/>eandem proportionem quàm ipſarum ſuperficies habebunt, aut in qualitate ſola fi <lb/>guræ, aut in quantitate ſola, aut in ſitu, aut in aliquibus ex dictis rebus, eo tamen mo <lb/>do, quiſuperius poſitus fuit, vt ſcilicet corpus illud quod alteri comparatum, æqua-<lb/>lis erit ponderis, aut leuitatis, ſed minoris reſiſtentiæ, exiſtet velocius altero, in <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportione, cuius ſuperficies reſiſtentiam ſuſcipit minorem ea quæ alterius eſt cor-<lb/>poris, ratione facilioris diuiſionis continuitatis aeris, aut aquæ; </s> + <s xml:id="s2025" xml:space="preserve">Vt exempli gratia, <lb/>ſi proportio ſuperficiei corporis maioris ſuperficiei minoris ſeſquitertia eſſet, pro-<lb/>portio velocitatis dicti corporis maioris, velocitati corporis minoris, eſſet ſubſeſqui <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 exiſtit.</s> + </p> + <p> + <s xml:id="s2027" xml:space="preserve">Aliud quoque ſupponendum eſt, velocitatem ſcilicet motus naturalis alicuius <lb/>corporis grauis, in diuerſis medijs, propor-<lb/>tionatam eſſe ponderi eiuſdem corporis in <lb/>ijſdem medijs; </s> + <s xml:id="s2028" xml:space="preserve">Vt exempli gratia, ſi pondus <lb/> + <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> + totale alicuius corporis grauis ſignificatum <lb/>crit ab <var>.a.i.</var> quo corpore poſito 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 denſo, quàm ipſum ſit, (quia in medio ſe denſiore ſi poner etur, non graue <lb/>e<gap extent="2"/>et, ſed leue, quemadmodum Archimedes oſtendit) illud medium ſubtrahat par-<lb/>tem <var>.e.i.</var> vnde pars <var>.a.e.</var> eiuſdem ponderis libera manear; </s> + <s xml:id="s2029" xml:space="preserve">& poſito deinde eodem cor <lb/>pore in aliquo alio medio denſiore, minus tamen denſo quam ipſum ſit corpus, <lb/>hoc medium ſubtrahat partem <var>.u.i.</var> dicti ponderis, vnde pars <var>.a.u.</var> ei uſdem <lb/>ponderis remanebit. </s> + <s xml:id="s2030" xml:space="preserve">Dico proportionem velocitatis eiuſdem corporis per <reg norm="medium" type="context">mediũ</reg> <lb/>minus denſum, ad velocitatem eiuſdem per medium magis denſum futuram vt <var>.a.e.</var> <lb/>ad <var>.a.u.</var> vt eſt etiam rationi conſonum, magis quàm ſi dicamus huiuſmodi velocitates <lb/>eſſe, vt <var>.u.i.</var> ad <var>.e.i.</var> cum velocitates à virtutibus mouentibus ſolum (cum figura vna, <lb/><reg norm="eademque" type="simple">eademq́;</reg> in qualitate, quantitate <reg norm="ſituque" type="simple">ſituq́</reg> erit) proportionentur. </s> + <s xml:id="s2031" xml:space="preserve">Quæne<unsure/>nc diximus, <lb/>planè ſimilia ſuntijs, quæ ſupra ſcripſimus, quia idem eſt dicere, proportionem velo <lb/>citatum, duorum corporum hetereogeneorum, ſed ſimilium figura, & magnitudine <lb/>æ qualium, in vno ſolo medio, æqualem eſſe <lb/>proportioni ponderum ipſorum, vt ſi dicam?</s> + <s xml:id="s2032" xml:space="preserve"><unsure/> <lb/>proportionem velocitatum vnius ſolum cor-<lb/> + <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> + poris per diuerſa media eandem eſſe cum ea. <lb/>quæ eſt <reg norm="ponderum" type="context">ponderũ</reg> dicti corporis in iſidem 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 diuerſis medijs habere cum denſitate eorum.</head> + <head xml:id="head248" xml:space="preserve">CAP. III.</head> + <p> + <s xml:id="s2033" xml:space="preserve">POſſibile eſt in rerum natura corpus aliquod huiuſmodi denſitate præditum re-<lb/>periri, vt velocitas eius motus naturalis per aerem, velocitati per aquamita pro <lb/>portionata exiſtat, vt eſt <reg norm="denſitas" type="context">dẽſitas</reg> aquæ denſitati aeris. </s> + <s xml:id="s2034" xml:space="preserve">Denſitas aquæ notetur (exem-<lb/>pli gratia) per <var>.u.i.</var> & ea, quæaeris eſt per <var>.e.i.</var> & pondus alicuius corporis in aere per <lb/><var>e.a.</var> & pondus eiuſdem corporis in aqua per <var>.u.a.</var> ita tamen, quod eadem proportio <lb/>ſit <var>.e.a.</var> ad <var>.u.a.</var> vt <var>.u.i.</var> ad <var>.e.i.</var> vnde per vltimam ſuppoſitionem <reg norm="præcedentis" type="context">præcedẽtis</reg> capitis, pro <lb/>portio velocitatis prædicti corporis per aerem, <lb/>proportioni eiuſdem 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">Oſcitanter ab Ariſtotele nonnibil prolatum cap 8. <lb/>lib. 4 Phyſicorum.</head> + <head xml:id="head250" xml:space="preserve">CAP. IIII.</head> + <p> + <s xml:id="s2035" xml:space="preserve">EX ſupradictis patet in vniuerſum non eſſe verum quod Ariſto .8. cap .4. lib. phy<lb/>ſicorum ſcribit, velocitates ſcilicet motuum alicuius corporis per diuerſa me-<lb/>dia, proportionatas eſſe denſitatibus eorundem mediorum. </s> + <s xml:id="s2036" xml:space="preserve">Quocirca, ſit propor-<lb/>tio <var>.u.i.</var> ad <var>.e.i.</var> vt <reg norm="denſitatis" type="context">dẽſitatis</reg> aquę ad <reg norm="aeream" type="context">aereã</reg> <reg norm="denſitatem" type="context">dẽſitatem</reg> .et <var>.e.a.</var> ad <var>.u.a.</var> vt ponderis alicuius <lb/>corporis in aere ad pondus eiuſdem in aqua, ita tamen vt maior aut minor propor-<lb/>tio ſit <var>.e.a.</var> ad <var>.u.a.</var> quam <var>.u.i.</var> ad <var>.e.i.</var> vnde exiſtente 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/>cauſam nimis diſſentaneum eſt rationi, opi-<lb/>nari proportionem velocitatis omnium cor <lb/>porum grauium per aerem vnam <reg norm="eandemque" type="simple">eandemq́;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> + eſſe cum velocitate eorundem per aquam, <lb/>quemadmodum Ariſtoteles ſenſit.</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 denſitate dupla ad aerem, & aliquod <lb/>graue corpus in aqua duplum ad denſitatem ipſius aquæ, vnde dictum corpus in <lb/>denſitate ad aerem quadruplum erit; </s> + <s xml:id="s2039" xml:space="preserve">quam ob cauſam, mediam ſui ponderis tota-<lb/>lis partem in aqua, & in aere quartam partem, ex .7. lib. de inſidentibus aquæ ab Ar-<lb/>chimede conſcripto, 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> ſui, in aere <reg norm="autem" type="wordlist">aũt</reg> uirtute <reg norm="trium" type="context">triũ</reg> <reg norm="quartarum" type="context">quartarũ</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 eiuſde m in aqua ſeſquialtera erit. </s> + <s xml:id="s2042" xml:space="preserve"><reg norm="hocque" type="simple">hocq́;</reg> <lb/>corpus appelletur <var>.A</var>. </s> + <s xml:id="s2043" xml:space="preserve">Sit aliud quoque corpus, quod <var>.B.</var> nominetur, ſimile figura, & <lb/>magnitudine corporea corpori <var>.A.</var> ſed <reg norm="denſitate" type="context">dẽſitate</reg>, in proportione ſeſquialtera ad <reg norm="aquam" type="context">aquã</reg>, <lb/>& denſius 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 ſeſquialtera, vnde etiam velocius erit ipſo <var>.B.</var> in aere <lb/>in eadem proportione, ſed 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 Ariſtoteli concedam corpus <var>.B.</var> in aere, quam in aqua velocius futurum in ea-<lb/>dem proportione, in qua, aqua eſt <reg norm="denſior" type="context">dẽſior</reg> aere, ex Euclidis vndecima propoſitione <lb/>lib. quinti. </s> + <s xml:id="s2045" xml:space="preserve">Sed præter hæc omnia, ſi corpus <var>.A.</var> eſſet etiam velocius in aere, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, in eadem proportione, ſequeretur ex .16. dicti lib. quinti proportionem velo-<lb/>citatis <var>.A.</var> in aqua ad <reg norm="velocitatem" type="context">velocitatẽ</reg> ipſius <var>.B.</var> in aqua etiam ſeſquialteram eſſe. </s> + <s xml:id="s2046" xml:space="preserve">Sed cum <lb/>corpus <var>.A.</var> in denſitate ad aquam <reg norm="duplum" type="context">duplũ</reg> ſit, & corpus <var>.B.</var> <reg norm="ſeſquialterum" type="context">ſeſquialterũ</reg> ad ipſam <reg norm="aquam" type="context">aquã</reg>, <lb/>ſequetur <reg norm="proportionem" type="context">proportionẽ</reg> ponderis ipſius <var>.A.</var> ad <reg norm="pondus" type="context">põdus</reg> ipſius <var>.B.</var> in aqua eſſe in propor-<lb/>tione dupla; </s> + <s xml:id="s2047" xml:space="preserve">Vnde ex primo ſuppoſito capitis ſecundi proportio velocitatis <var>.A.</var> ad <lb/>velocitatem <var>.B.</var> in aqua dupla erit, non ſeſquialtera. </s> + <s xml:id="s2048" xml:space="preserve">Si ergo proportio velocitatis <var>.<lb/>A.</var> ad eam quæ eſt <var>.B.</var> in aqua dupla eſt, & ea, quæeſt <var>.B.</var> in aere, ad eam, quæ eſt ipſius <lb/>per aquam eſt etiam dupla (vnde ea quę eſt <var>.A.</var> per aquam ęqualis erit ei, quæ eſt <var>.B.</var> <lb/>peraerem, ex .9. lib. quinti) & cum ea, quæ eſt <var>.A.</var> ſit ei, quæ eſt <var>.B.</var> per aerem ſeſqui-<lb/>altera, erit ergo ea, quæ eſt <var>.A.</var> per aerem, ei, quæ eſt ſuimet ipſius per aquam ſeſqui <lb/>altera, non autem dupla, ex .7. eiuſdem libr. quinti. </s> + <s xml:id="s2049" xml:space="preserve">Hiſce rationibus accedimus ad <lb/>confirmandam veritatem vltimi ſuppoſiti cap .2. proportionem videlicet velocita <lb/>tis <reg norm="motus" type="simple">motꝰ</reg> naturalis in diuerſis medijs <reg norm="alicuius" type="simple">alicuiꝰ</reg> corporis <reg norm="ponderoſi" type="context">põderoſi</reg> in ipſis medijs eſſe ean <lb/>dem cum ea, quæeſt 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/>ctuoſa, aut pinguia non ſunt, ut ſunt <lb/>oleum, lac, aut huiuſmodialia, 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 eiuſdem corporis in diuerſis medijs pro <lb/>portiones eorum mediorum denſit atum non ſeruant. 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 diuerſa media, vnde <lb/>proportiones velocitatum inæquales exiſtunt. </s> + <s xml:id="s2051" xml:space="preserve">Vt exempli gratia, ſi fue-<lb/>rit corpus <var>.A.</var> cuius pondus totale ſit <var>.o.a.</var> quod in aqua diminutum ſit ratione partis <var>.<lb/>e.o.</var> ita vt ei ſolum relinquatur pondus <var>.a.e.</var> & in aeie adempta ſit ei pars <var>.i.o.</var> vnde ſo <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 denſum, quàm aer, quemadmodum aer minus denſus eſt, aqua, in quo, cor <lb/>pus <var>.A.</var> ammittat partem <var>.t.o.</var> ponderis ſui, vnde ex .7. lib. de inſidentibus aquæ Ar-<lb/>chimedis, eadem proportio erit <var>.e.o.</var> ad <var>.i.o.</var> quæ eſt <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> eſt <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 eſt <var>.i.o.</var> ad <var>.t.o</var>. </s> + <s xml:id="s2055" xml:space="preserve">Cum ſit ergo proportio <var>.a.i.</var> <lb/>ad <var>.a.e.</var> ut <var>.e.o.</var> ad <var>.i.o.</var> erit diſiunctim <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. quin<lb/>ti erit <var>.a.e.</var> æqualis <var>.i.o.</var> ſed cum ita ſehabeat <var>.e.o.</var> ad <var>.i.o.</var> vt <var>.i.o.</var> ad <var>.t.o.</var> ita quoque <lb/>ſe 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/>ęqualis ſit ipſi <var>.i.o.</var> non poterit eſſe proportio <var>.t.a.</var> ad <var>.i.a.</var> vt eſt <var>.o.i.</var> ad <var>.t.o.</var> quia ſi <lb/>hoc eſſet, eſſet etiam diſiunctim proportio <var>.i.t.</var> ad <var>.i.a.</var> vt eſt <var>.i.t.</var> ad <var>.t.o.</var> & ex ſupradicta <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> ideſt minorem <var>.a.e.</var> æqualem eſſe <var>.a.i.</var> quæ maior eſt <var>.a.e</var>. </s> + <s xml:id="s2059" xml:space="preserve">Oſtenſiuè tamen idem <lb/>hoc modo probari poteſt, vt exiſtente <var>.i.o.</var> ęquali ipſi <var>.a.e.</var> per conſequens <reg norm="quoque" type="simple">quoq;</reg> erit <lb/>minor ipſa <var>.a.i.</var> cum <var>.a.e.</var> pars ſit ipſius <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> minoreſt <var>.<lb/>o.i</var>. </s> + <s xml:id="s2061" xml:space="preserve">Tanto magis igitur minor erit <var>.t.o.</var> ipſa <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="eiuſdem" type="context">eiuſdẽ</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, quàm.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> quàm.t.a. ad <var>.i.a.</var> ita ergo ſe habebunt ipſorum 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 eiuſdem figur æ et materiæ ſed inæqualis <lb/>magnitudinis, in ſuis motibus natur alibus uelocit atis, in eo <lb/>dem medio, proportionem longè diuerſam ſeruatura <lb/>eße quam Aristoteliuiſum 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">ꝙ</reg> in uno <reg norm="eodemque" type="simple">eodemq́;</reg> mcdio duo corpora inæqualia, ſed <lb/>ſimili figura & materia, mouebuntur naturali motu, diuerſa 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, quàm Ariſtoteles præſcripſit.</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 ſit, & per conſequens in eadem quoque proportione grauius ipſo <var>.o.</var> in <lb/>qua eſt maius, communi omnium ſententia.</s> + </p> + <p> + <s xml:id="s2065" xml:space="preserve">Scribit ergo Ariſtoteles proportionem velocitatis corporis <var>.a.</var> ad eam, quæ eſt <lb/>corporis <var>.o.</var> (naturaliterſe vnoquoque mouente) eandem futuram, quæ eſt 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 eſt, <lb/>ſed eandem grauitatem obtinere, quæ communicata eſt corpori <var>.o.</var> quod ex quauis <lb/>materia conſter. </s> + <s xml:id="s2067" xml:space="preserve">Hinc ex primo ſuppoſito ſecundi capitis certi erimus proportio-<lb/>nem velocitatis corporis <var>.a.</var> ſi comparetur cum velocitate corporis <var>.u.</var> futuram, vt <reg norm="eam" type="context">eã</reg>, <lb/>quæ eſt ponderis corporis <var>.a.</var> ad pondus ipſius corporis <var>.u</var>. </s> + <s xml:id="s2068" xml:space="preserve">Ex .9. igitur lib. quinti Eu-<lb/>cli. cogitur fateri Ariſtoteles velocitatem corporis <var>.o.</var> eſſe vnam <reg norm="eandemque" type="simple">eandemq́;</reg> in ſpe-<lb/>cie, quæ eſt corporis <var>.u</var>. </s> + <s xml:id="s2069" xml:space="preserve">Quod primo ſuppoſito cap. ſecundi huius lib. planè repugna<lb/>ret. </s> + <s xml:id="s2070" xml:space="preserve">Igitur hæc Ariſtotelis opinio falſa eſt. </s> + <s xml:id="s2071" xml:space="preserve">Idem quoque probaretur mediante cor <lb/>pore <var>.i.</var> æquali magnitudine, <reg norm="ſimilique" type="simple">ſimiliq́;</reg> figura cum corpore <var>.o.</var> prædito, ſed, quòd ad <lb/>quantitatem attinet, æquali corpori <var>.a.</var> vnde ex primo ſuppoſito cap. ſecundi huius li <lb/>bri in eadem pro <lb/>portione <reg norm="velocius" type="simple">velociꝰ</reg> <lb/>eſſet corpore <var>.o.</var> <lb/> + <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> + in qua grauius eſt. <lb/>ex .9. igitur quin-<lb/>ti cogitur Ariſto-<lb/>teles affirmare <reg norm="tam" type="context">tã</reg> <lb/>velox eſſe corpus <lb/>a. <reg norm="quam" type="context">quã</reg> eſt corpus <lb/>i. vnde idem pla-<lb/>nè inconueniens emergit ex ſecundo ſuppoſito cap. ſecundi 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 eiuſdem materia in diuerſis <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 ſimili tamen figura & eadem materia con-<lb/>ſtantia, naturaliter ſe per diuerſa media mouentia, vnam <reg norm="eandemque" type="simple">eandemq́;</reg> ſem-<lb/>per proportionem velocitatum ſeruant.</s> + </p> + <p> + <s xml:id="s2073" xml:space="preserve">Sint duo corpora <var>.A.</var> et <var>.B.</var> ſibi inuicem inæqualia quorum <var>.A.</var> ſit maius, ſed ſimile <lb/>figura & idem materia, <lb/>cuius pondus totaleſit <var>.<lb/> + <anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/> + x.o.</var> & pondus totaleip <lb/>ſius <var>.B.</var> ſit <var>.u.s</var>. </s> + <s xml:id="s2074" xml:space="preserve">Imagine-<lb/>mur quoque corpus <var>.A.</var> <lb/>poſitum 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 correſpondet <var>.e.o.</var> æquale eſt ipſi <var>.A.</var> & corpus <lb/>aqueum, cui correſpondet <var>.c.s.</var> æquale eſt i pſi <var>.B.</var> vt eſt ab Archimede <reg norm="probatum" type="context">probatũ</reg>: </s> + <s xml:id="s2076" xml:space="preserve">com <lb/>muni quadam ſcientiæ ratione, ſequitur eandem proportionem futuram <var>.o.x.</var> ad <var>.e.o.</var> <lb/>quæ eſt <var>.u.s.</var> ad <var>.c.s.</var> ob <reg norm="eaſdemque" type="simple">eaſdemq́;</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/>quintí 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, ſi ſubtra <lb/> + <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> + hantur proportiones @reſi <lb/><reg norm="ſtentiarum" type="context">ſtẽtiarum</reg> extrinſecus <reg norm="ad- uenentium" type="context">ad-<lb/>uenẽtium</reg>, proportiones <lb/>quæ remanebunt, exter-<lb/>tio communi axiomate <lb/>ab Eucli. in principio pri<lb/>mi lib. poſito, ad inuicem <lb/>erunt æquales, ſecundum quas eorundem corporum ſunt 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 diſeruerit 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 Ariſtoteles circa finem cap .8. lib. 4. phyſicorum ſubiungit quod ea-<lb/>dem proportione dicta corpora mouerentur in vacuo, vt in pleno, id pace <reg norm="eius" type="simple">eiꝰ</reg> <lb/><reg norm="dictum" type="context">dictũ</reg> ſit planè <reg norm="erroneum" type="context">erroneũ</reg> eſt. </s> + <s xml:id="s2080" xml:space="preserve">quia in pleno dictis corporibus ſubtrahitur proportio reſi <lb/>ſtentiarum extrinſecarum à proportione ponderum, vt velocitatum proportio re-<lb/>maneat, quę nulla eſſet, ſi dictarum reſiſtentiarum proportio, ponderum propor-<lb/>tioni æqualis eſſet, & hanc ob cauſam diuerſam velocitatum proportionem in va-<lb/>cuo haberent ab ea, quæ eſt 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 eiuſdem 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 ſupradicta 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> ſit dimidia pars ipſius <var>.o.</var> ſint alia <lb/>quoque duo corpora <var>.a.</var> et <var>.e.</var> omogenea primis, quorum quodlibet æquale ſit ipſi <var>.g.</var> <lb/>& imaginatione compręhendamus ambo poſita in extremitatibus alicuius lineæ, cu <lb/>ius medium ſit <var>.i.</var> clarum erit, tantum pondus habiturum, punctum <var>.i.</var> quantum <reg norm="centrum" type="context">centrũ</reg> <lb/>ipſius <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 ipſius .<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 ſuam 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 eſt .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 eiuſdem materiæ & figuræ, ſireſiſten-<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 præſcripſimus, poſſet oſtendi, ſi duo cor-<lb/>pora <var>.o.</var> et <var>.g.</var> ſuas reſiſtentias, ita ad inuicem proportionatas haberent, utſunt <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 ipſius <var>.o.</var> cum à tanto pondere <lb/>i. motum eſſet; </s> + <s xml:id="s2085" xml:space="preserve">quanto centrum ipſius <var>.o.</var> atquetan <lb/> + <anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/> + tam reſiſtentiam duo corpora <var>.a.</var> et <var>.e.</var> <reg norm="quanta" type="context">quãta</reg> ipſum <lb/>o. ſolum haberet ex hypotheſi, dicta tamen corpo <lb/>ra <var>.a.</var> et <var>.e.</var> tam ſeparata, 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 demonſir atur eſſe proportio ponder is corpor is den <lb/>ſioris ad pondus minus denſi in <reg norm="medijs" type="lig">medijs</reg> <reg norm="denſioribus" type="context">dẽſioribus</reg>, quam <lb/>ſit eorundem corporum in medio minus denſo, nec <lb/>corporum ponder a ſeruare proportionem <lb/>denſitatis mediorum.</head> + <head xml:id="head266" xml:space="preserve">CAP. XII.</head> + <p> + <s xml:id="s2086" xml:space="preserve">PRopoſita nobis cum fuerint duo corpora <var>.A.</var> et <var>.B.</var> area corporea æqualia, quo-<lb/>rum <var>.A.</var> denſius ſit ipſo <var>.B.</var> probabo in medio magis denſo, maiorem proportio <lb/>nem futuram ponderis ipſius <var>.A.</var> ad pondus <var>.B.</var> quàm in medio minus denſo.</s> + </p> + <p> + <s xml:id="s2087" xml:space="preserve">Sit igitur <var>.p.g.</var> pondus totale ipſius corporis <var>.A.</var> et <var>.q.k.</var> ipſius corporis <var>.B.</var> vnde <var>.p.g.</var> <lb/>maius erit ipſo <var>.q.k</var>. </s> + <s xml:id="s2088" xml:space="preserve">Sit quoque <var>.o.g.</var> pondus, quod medium magis denſum ſubtra-<lb/>hit à pondere <var>.p.g.</var> et <var>.n.k.</var> ſit pondus, quod idem medium ſubtrahit à pondere <var>.q.k.</var> et <lb/><var>f.g.</var> ſit pondus, quod medium minus denſum ſubtrahit à <var>.p.g.</var> et <var>.i.k.</var> illud, quodid@m <lb/><reg norm="medium" type="context">mediũ</reg> ſubtrahit ab <var>.q.k.</var> vnde <var>.o.g.</var> æquale erit <var>.n.k.</var> et <var>.f.g.</var> ipſi <var>.i.k.</var> quia quod ad <reg norm="aream" type="context">areã</reg> <lb/>attinet, corpora ſupponuntur æqualia, vnde proportio <var>.p.f.</var> ad <var>.q.i.</var> maior erit ea, quæ <lb/>eſt <var>.o.f.</var> ad <var>.n.i.</var> communi <lb/> + <anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/> + ſcientiæ notione, quia ſi <lb/>ſcinderet <reg norm="aliquis" type="simple">aliꝗs</reg>.p.f. in pun <lb/>cto <var>.c.</var> ita. vt <var>.c.f.</var> æquale eſ-<lb/>ſet ipſi <var>.q.i.</var> proportio <var>.c.f.</var> <lb/>ad <var>.q.i.</var> eſſet vt ea, quæ eſt <var>.<lb/>o.f.</var> ad <var>.n.i.</var> (hoc eſt 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"/> + ſed proportio <var>.p.f.</var> ad <var>.q.i.</var> maior eſſet ea, quæ eſt <var>.c.f.</var> ad <var>.q.i.</var> ex. octaua lib. quinti, vn-<lb/>de ex .12. eiuſdem lib. maior eſſet <var>.p.f.</var> ad <var>.q.i.</var> quàm.o.f. ad <var>.n.i.</var> ex .33. igitur eiuſdem, <lb/>maior erit proportio <var>.p.o.</var> ad <var>.q.n.</var> quàm.p.f. ad <var>.q.i</var>. </s> + <s xml:id="s2089" xml:space="preserve">Sic quoque ſe habebunt ad inui <lb/>cem velocitates, quod eſt propoſitum. </s> + <s xml:id="s2090" xml:space="preserve">Cum autem proportio <var>.p.o.</var> ad <var>.q.n.</var> maior ſit, <lb/>quàm.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 euerſim maior erit proportio <var>.q.i.</var> ad <var>.q.n.</var> quàm.p.f. ad <var>.p.o.</var> vnde ſi proportio <lb/><var>p.f.</var> ad <var>.p.o.</var> eſſet ac ea, quæ eſt <var>.o.g.</var> ad <var>.f.g.</var> non eſſet <var>.q.i.</var> ad <var>.q.n.</var> ut eſt <var>.o.g.</var> ad <var>.f.g.</var> aut <lb/>vt <var>.n.k.</var> ad <var>.i.k.</var> quodidem <lb/>eſt, de quibus quidem re-<lb/> + <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> + bus, exemplis propoſitis <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 ſe-<lb/>quentibus pondera, ſequi <lb/>tur proportionem veloci-<lb/>citatum duorum corporum hetereogeneorum eandem non eſſe per diuerſa media, <lb/>contra id, quod ſequeretur ſi Ariſtotelis opinionem .8. cap. lib. 4. phyſicorum 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 ſe habere quam Aristoteles <lb/>doceat in fine libri ſeptimi phyſicorum.</head> + <head xml:id="head268" xml:space="preserve">CAP. XIII.</head> + <p> + <s xml:id="s2092" xml:space="preserve">NOn tam facile eſt aſſignare proportionem velocitatum duorum corporum na <lb/>turalium, quam Ariſtoteles vltimo cap. lib. 7. phyſicorum putauit.</s> + </p> + <p> + <s xml:id="s2093" xml:space="preserve">Quamobrem ſint duo corpora <var>.B.</var> et <var>.D.</var> materia <reg norm="magnitudineque" type="simple">magnitudineq́;</reg> diuerſa, pondere <lb/>tamen, & figura ſimilia, & proportio reſiſtentiarum, quas recipiunt à medio <reg norm="dum" type="context">dũ</reg> mo-<lb/>uentur, ſit. ut <var>.o.i.</var> ad <var>.a.e.</var> denotentur deinde velocitates totales abſque vlla reſiſten-<lb/>tia ab <var>.a.u.</var> et <var>.o.c.</var> quæ æquales erunt ad inuicem per communem ſcientiam ex ſup-<lb/>poſito, ſint alia deinde duo corpora <var>.V.</var> et <var>.M.</var> eodem modo ſe habentia ut prima <var>.B.</var> <lb/>et <var>.D.</var> in eodem medio, ſed ex diuerſa materia ab ea, quæ eſt illorum duorum corpo <lb/>rum, magnitudine tamen & figura ijſdem ſimilia: </s> + <s xml:id="s2094" xml:space="preserve">ſignificentur quoque eo-<lb/>rundem reſiſtentiæ per <var>.t.s.</var> et <var>.n.r.</var> & eorundem velocitates à nulla ex reſiſtentijs 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> ipſi <var>.o.i.</var> et <var>.n.x.</var> ipſi <var>.t.g</var>: <var>n.x.</var> ta-<lb/>men et <var>.t.g.</var> non erunt ęqualia <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 ipſi <var>.B.</var> cum applicata <lb/>erit reſiſtentia <var>.a.e.</var> dicto corpori <var>.B.</var> quæ diminutam facit totam <var>.a.u.</var> per <var>.a.e.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.i.c.</var> <lb/>ea, quę remanet ipſi <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 eſt ex <var>.t.g.</var> corporis <var>.M</var>. </s> + <s xml:id="s2097" xml:space="preserve">Vnde communi omnium <reg norm="conſenſu" type="context">cõſenſu</reg> aſſequemur <var>.e.u.</var> ma <lb/>iorem futuram <var>.r.x.</var> et <var>.i.c.</var> ipſa <var>.s.g</var>. </s> + <s xml:id="s2098" xml:space="preserve">Scindatur deinde <var>.a.m.</var> ad ęqualitatem <var>.n.x.</var> et <var>.o.z.</var> <lb/>ipſius <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 ipſa <var>.z.i.</var> maior igitur erit proportio <var>.z.c.</var> <lb/>ad <var>.z.i.</var> quàm.m.u. ad <var>.m.e.</var> (quia <var>.z.c.</var> ad <var>.z.i.</var> ita ſe habet vt <var>.m.u.</var> ad <var>.i.z.</var> ex .7. lib. quin-<lb/>ti, ſed <var>.m.u.</var> ad <var>.i.z.</var> maior eſt quam ad <var>.m.e.</var> ex .8. dicti lib. vnde ex .12. eiuſdem <var>.z.c.</var> ad <lb/>ad <var>.z.i.</var> maior erit, quàm.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> ideſt <var>.s.</var> g <lb/>ad <var>.r.x.</var> quod Ariſtoteli in mentem non venerat. </s> + <s xml:id="s2100" xml:space="preserve">Alijs quoque modis idem proba-<lb/>ri poteſt, vt ſi 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 eſt ratio vt ad <var>.z.c.</var> ex .7. quinti, ſed proportio <var>.e.m.</var> ad <var>.<lb/>z.c.</var> maior eſt quam <var>.i.z.</var> ad <var>.z.c.</var> ex .8. eiuſdem, ergo ea, quæ eſt <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 eſt <var>.e.u.</var> ad <var>.m.u.</var> maior erit illa, <lb/>quæ eſt <var>.i.c.</var> ad <var>.z.c.</var> & <reg norm="permutando" type="context">permutãdo</reg>, quam ea, quæ eſt <var>.e.u.</var> ad <var>.i.c.</var> ea, quæ eſt <var>.m.u.</var> ad <var>.z.c.</var> <lb/>& ex .33. quinti, ea, quæ eſt <var>.e.m.</var> ad <var>.i.z.</var> maior erit ea, quæ eſt <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 ſequatur ex ſupradistis.</head> + <head xml:id="head270" xml:space="preserve">CAP. XIIII.</head> + <p> + <s xml:id="s2101" xml:space="preserve">EX præcedenti capite manifeſtè depræhenditur, in vniuerſum Ariſtotelis opi-<lb/>nionem veram non eſſe in prima parte vltimi capitis. lib. 7. phyſicorum; </s> + <s xml:id="s2102" xml:space="preserve">quia <lb/>in eo loco ſupponens ipſe corpus <var>.B.</var> pręcedentis capitis eſſe dimidiam partem ipſius <lb/>D. quantum ad aream corpoream ſpectat (ſunt tamen pondere ad inuicem æqualia) <lb/>ait <var>.B.</var> futurum duplo velocius ipſo <var>.D</var>. </s> + <s xml:id="s2103" xml:space="preserve">Ego verò præcedenti capite accepi <var>.e.u.</var> pro <lb/>velocitate reſidua corporis <var>.B.</var> (ſubtracta ea tamen parte, quam ei reſiſtentia adimit, <lb/>quæ erat <var>.e.a.</var>) et <var>.i.c.</var> pro ea, quæ eſt corporis <var>.D.</var> et <var>.r.x.</var> pro ea, quæ eſt corporis <var>.V.</var> et <var>.<lb/>s.g.</var> pro ea, quæ eſt corporis <var>.M</var>. </s> + <s xml:id="s2104" xml:space="preserve">Dicat nunc Ariſtoteles, quę nam harum duarum pro <lb/>portionum dupla erit? </s> + <s xml:id="s2105" xml:space="preserve">quia ſi earum aliqua talis erit, alia nullo modo eſſe poterit, <lb/>vt iam oſtendi, etiamſi duo corpora <var>.V.</var> et <var>.M.</var> eaſdem conditiones habeant, quas <var>.B.</var> <lb/>et <var>.D</var>. </s> + <s xml:id="s2106" xml:space="preserve">Ratio autem, quæ Ariſtotelem induxerit ad illud credendum, nulla alia eſſe <lb/>potuit, quàm quod putarit reſiſtentias proportionatas eſſe magnitudinibus corpo-<lb/>reis, ideſt quemadmodum <var>.B.</var> erat corporaliter dimidia pars ipſius <var>.D.</var> ſic etiam habe <lb/>ret medietatem eius reſiſtentiæ, quam habuiſſet corpus <var>.D</var>. </s> + <s xml:id="s2107" xml:space="preserve">Quod etſi verum eſſet, <lb/>non tamen ſequeretur neceſſariò in quibuſlibet corporibus futuram velocitatum <lb/>proportionem eandem, quæ reſiſtentiarum eſt, vt ſuperiore capite oſtendimus.</s> + </p> + </div> + <div type="section" level="3" n="15"> + <head xml:id="head271" style="it" xml:space="preserve">Numrestè ſenſerit Philoſophus reſistentias 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 Ariſtoteles crediderit reſiſtentias proportionatas eſſe corporibus, erra-<lb/>uit. </s> + <s xml:id="s2109" xml:space="preserve">Si ſuperficies ijſdem proportionatæ eſſent, dubium non eſt, quin <lb/>reſiſtentiæ quoque ipſæ, ijſdem proportionatæ exiſterent, ſupponendo eas ſimiles <lb/>ſitu, dum eadem corpora mouerentur. </s> + <s xml:id="s2110" xml:space="preserve">Sed eadem proportio non eſt inter ſuperfi- + <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">Ariſtoteles igitur in eo defecit. </s> + <s xml:id="s2112" xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> inter <lb/>ſuperficies non eadem ſit proportio, quæ inter corpora extat, ſi primo ad ſphęricas <lb/>mentem verterimus, intelligemus proportionem eam, quæ inter duas ſphæras repe <lb/>ritur triplam ſemper exiſtere ei, quæ eſt inter ipſarum diametros ex vltima .12. libr. <lb/>Euclid. </s> + <s xml:id="s2113" xml:space="preserve">Eſt autem proportio, quæ eſt inter ſuperficies ſphęricas ęqualis ei, quæ eſt <lb/>ipſorum circulorum maiorum ex .16. lib. quinti, cum ex .31. primi de ſphæra & cy-<lb/>lindro Archimedis, omnis ſphærica ſuperficies quadrupla, ſit maiori circulo ipſius <lb/>ſphęræ, ſed proportio, quæ eſt inter dictos circulos, eſt dupla ei, quæ eſt inter <reg norm="eorun- dem" type="context context">eorũ-<lb/>dẽ</reg> diametros ex .2. lib. 12. Euc. </s> + <s xml:id="s2114" xml:space="preserve">ergo <reg norm="proportio" type="simple">ꝓportio</reg>, quæ eſt inter corpora, ſeſquialtera erit <lb/>ei, quæ eſt ſuperficierum, & non æqualis, ut Ariſtoteles putauit. </s> + <s xml:id="s2115" xml:space="preserve">Idem de corporibus <lb/>ſimilibus à ſuperficiebus planis terminatis dico, ratiocinando mediante .36. lib. 11. <lb/>et .18. ſexti, vnde cognoſcemus proportionem corporum, proportioni laterum, tri-<lb/>plam futuram, & ſuperficierum proportionem, laterum proportioni duplam. </s> + <s xml:id="s2116" xml:space="preserve">Quare <lb/>corporum proportio, ei, quæ ſuperficierum eſt, ſeſquialtera erit, ita ut ſi velocitates <lb/>extitiſſent ad inuicem proportionatæ, vt ſuperficies, proportio velocitatis corporis <var>.<lb/>B.</var> ei, quæ eſt corporis <var>.C.</var> fuiſſet ſubſeſquialtera 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">Fdipſum aliter demonſtr atur.</head> + <head xml:id="head274" xml:space="preserve">CAP. XVI.</head> + <p> + <s xml:id="s2117" xml:space="preserve">ALio quoque modo probari poteſt non eſſe in vniuerſum verum id, quod Ari-<lb/>ſtoteles in prima parte capitis vltimi lib. 7. phyſicorum ait, ſic ſcribens.</s> + </p> + <p> + <s xml:id="s2118" xml:space="preserve">Si <var>.A.</var> quidem ſit id quod mouet <var>.B.</var> verò id quod mouetur, et <var>.C.</var> ſit longitudo per <lb/>quam, et <var>.D.</var> tempus in quo eſt motum, in tempore nimirum ęquali, potentia æqua-<lb/>lis <var>.A.</var> dimidium ipſius <var>.B.</var> per duplum mouebit ipſius <var>.C.</var> per ipſum autem <var>.C.</var> in dimi <lb/>dio temporis <var>.D.</var> ſic enim erit rationis ſimilitudo.</s> + </p> + <p> + <s xml:id="s2119" xml:space="preserve">Sit ergo corpus <var>.o.</var> ſeptimi capitis pondere æquali corpori <var>.u.</var> eiuſdem capitis, ſed <lb/>area corporea minusipſo <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">nũc</reg> <lb/>tertium aliud corpus omogeneum ipſi <var>.u.</var> quod ſit <var>.i.</var> magnitudine & figura ſimile ipſi <lb/>o. vnde minor erit ipſo <var>.u.</var> pro media parte, & hanc ob cauſam ipſum <var>.u.</var> erit duplo ma <lb/>gis graue, quàm ipſum <var>.i.</var> & per conſequens ipſum quoque <var>.o.</var> duplo grauius erit <reg norm="quam" type="context">quã</reg> <lb/>ſit ipſum <var>.i.</var> ex .7. libr. quinti Euclidis. </s> + <s xml:id="s2122" xml:space="preserve">Ipſum ergo corpus <var>.o.</var> duplo velocius erit, <lb/>quàm ipſum <var>.i.</var> ex primo ſuppoſito cap .2. huius lib. </s> + <s xml:id="s2123" xml:space="preserve">Vnde ex .9. quinti, velocitas ipſius <lb/>i. æqualis eſſet ei, quæ eſt ipſius u. cum Ariſtoteles ſcribat <var>.o.</var> quoque futurum duplo <lb/>velocius ipſo <var>.u.</var> <reg norm="quod" type="simple">ꝙ</reg> cap .7. huius lib. falſum eſſe demonſtraui.</s> + </p> + </div> + <div type="section" level="3" n="17"> + <head xml:id="head275" style="it" xml:space="preserve">De alio Aristo. lapſu.</head> + <head xml:id="head276" xml:space="preserve">CAP. XVII.</head> + <p> + <s xml:id="s2124" xml:space="preserve">SCribit Ariſtoteles in ultimo cap. lib. 7. phyſicorum in hunc modum. <lb/></s> + <s xml:id="s2125" xml:space="preserve">Si duo quædam ſeorſum per tantum ſpatium tanto tempore duo ſeorſum pon <lb/>dera mouent, & compoſita per longitudinem æqualem, <reg norm="ęqualiuem" type="context">ęqualiuẽ</reg> in tempore, com-<lb/>poſitum ex ponderibus <reg norm="vtriſque" type="simple">vtriſq;</reg> mouebunt, eſt 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 vniuerſum nec etiam poteſt eſſe verum in pleno, quia cap .14. iam pro-<lb/>baui, non eandem proportionem eſſe inter ſuperſicies corporum, & ipſa corpora.</s> + </p> + </div> + <div type="section" level="3" n="18"> + <head xml:id="head277" style="it" xml:space="preserve">Quomodo dignoſcatur proportio uelocitatis duorum ſimilium <lb/>corporum omogeniorum inaqualium.</head> + <head xml:id="head278" xml:space="preserve">CAP. XVIII.</head> + <p> + <s xml:id="s2127" xml:space="preserve">ETiam ſi reperire in qua proportione motus naturaliter moueantur duo corpo-<lb/>ra, figura & materia ſimilia, inęqualia tamen ad inuicem, non facile ſit, oſten-<lb/>dam tamen qua ratione id conſequi poſſimus.</s> + </p> + <p> + <s xml:id="s2128" xml:space="preserve">Proponantur nobis, exempli gratia, duo corpora <var>.a.</var> et <var>.o.</var> ſphęrica, inęqualia inui-<lb/>cem, omogenea tamen materia, quorum <var>.a.</var> maius ſit; </s> + <s xml:id="s2129" xml:space="preserve">ſi 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/>ſphęricum, alia tamen & diuerſa materia conſtans, ſed pondere ęquale corpori <var>.o.</var> & <lb/>ſuperſicie tam proportionata ſuperficiei corp oris <var>.a.</var> quàm eſt ea, quæ eſt ſui ponde-<lb/>ris ad pondus ipſius <var>.a</var>. </s> + <s xml:id="s2131" xml:space="preserve">Hoc facto, indagetur, quænam erit proportio inter ſu-<lb/>perficies corporum <var>.i.</var> et <var>.o.</var> quę ſemper dupla eſt, vel ſubdupla ei quæ eſt diametro-<lb/>rum; </s> + <s xml:id="s2132" xml:space="preserve">ut iam cap .15. dixi, & hęc proportio ſuperficierum ſphęricarum <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.o.</var> et <var>.i.</var> ſub <lb/>trahatur ab æqualitate, quod igitur remanebit, erit proportio <reg norm="velocitatum" type="context">velocitatũ</reg> inter duo <lb/>corpora <var>.o.</var> et <var>.i.</var> ideſt inter <var>.o.</var> et <var>.a.</var> vt exempli gratia, ſi proportio ſuperficiei <var>.o.</var> ſuperfi <lb/>ciei ipſius <var>.i.</var> ſeſquitertiα<unsure/> eſſet, ſub <lb/>trahendo eam ab ęqualitate, rema-<lb/> + <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> + neret <reg norm="proportio" type="simple">ꝓportio</reg> ſubſeſquitertia, vnde <lb/>velocitas corporis maioris ( quod in <lb/>pręſenti loco ſupponitur eſſe <var>.o.</var>) ei, <lb/>quę eſt corporis minoris, quale eſt <lb/>corpus <var>.i.</var> ſubſeſquitertia eſſet; </s> + <s xml:id="s2133" xml:space="preserve">aut <lb/>dicamus quòd <var>.i.</var> eſſet velocius ipſo <lb/>o. in proportione ſeſquitertia ex ſe <lb/>cundo ſuppoſito ſecundi capitis huius libri. </s> + <s xml:id="s2134" xml:space="preserve">Sed <var>.i.</var> tam velox eſt quam ipſum <var>.a.</var> ex <num value="11">.<lb/>11.</num> cap. ergo proportio velocitatis ipſius <var>.a.</var> ſeſquitertia erit ei. quæ eſt ipſius <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 ſit inanis ab Ariſtotele ſuſcepta demonſtratio 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 ſuperius <reg norm="demonſtrauimus" type="context">demõſtrauimus</reg> facilè cognoſci poteſt irritam eſſc eam ratio <lb/>nem, quam Ariſtoteles .8. cap. lib. 4. phyſicorum ad deſtruendum vacuum, <reg norm="con" type="context">cõ</reg> <lb/>finxit. </s> + <s xml:id="s2136" xml:space="preserve">Vtigitur idem facilius oſtendamus, compræhendamus imaginatione infini-<lb/>ta media corporea, quorum vnum altero rarius ſit, in qua placuerit nobis ex propor <lb/>tionibus, incipiendo ab uno, imaginemur etiam corpus <var>.Q.</var> denſius primo medio, cu-<lb/>ius corporis, totalis grauitas ſit <var>.a.b.</var> & poſitum in ipſo medio, amittat partem <var>.e.b.</var> ip-<lb/>ſius grauitatis, & in ſecundo medio amittat <var>.i.b.</var> & ſic 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 remanſuram ſuam totalem grauitatem <var>.a.b.</var> in quolibet <lb/>ex-dictis medijs. </s> + <s xml:id="s2138" xml:space="preserve">Nunc ſi quærat à me Ariſtoteles proportionem velocitatis corpo-<lb/>ris <var>.Q.</var> per vacuum ad velocitatem dicti corporis per plenum, ego ei proponam pro-<lb/>portionem ipſius <var>.a.b.</var> ad <var>.a.e.</var> exempli gratia, dicens, <reg norm="quod" type="simple">ꝙ</reg> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> <var>.a.b.</var> maius eſt <lb/>ip ſo <var>.a.e.</var> ſic etiam corpus <var>.Q.</var> velocius erit in vacuo, quàm in pleno, dicti autem ple-<lb/>ni denſitatem appellabimus <var>.e.b</var>. </s> + <s xml:id="s2139" xml:space="preserve">Ariſtoteles dicet nunc, <reg norm="quod" type="simple">ꝙ</reg> aliud quoddam medium <lb/>in eadem proportione ſubtilius ipſo <var>.e.b.</var> deſumatur; </s> + <s xml:id="s2140" xml:space="preserve">quemadmodum <var>.a.e.</var> minus eſt <lb/>ipſo <var>.a.b.</var> ſit ergo iſtud <var>.i.b.</var> in quo Ariſtoteles credit corpus Q. futurum tam velox ut <lb/>in vacuo, in quo aberrat, <reg norm="quia" type="simple">ꝗa</reg> proportio velocitatis corporis <var>.Q.</var> in medio <var>.i.b.</var> ad velo <lb/>citatem eiuſdem in medio <lb/><var>e.b.</var> ita ſe hàbebit, 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 ſuppoſito ca <lb/>pit .2. huius libr. quæ minor <lb/>eſſet ea, quæ eſt <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 ſatis dilucidè Ariſtotelem de loco ratiocinatum fuiße.</head> + <head xml:id="head282" xml:space="preserve">CAP. XX.</head> + <p> + <s xml:id="s2141" xml:space="preserve">QVæ Ariſtoteles de loco ſcribit multas in ſe continent difficultates. </s> + <s xml:id="s2142" xml:space="preserve">Primum, <lb/>cap .4. lib. 4. phyſicorum ait, omne corpus eſſe in ſuo proprio loco, ſupponen <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 poteſt eſſe locus ipſius corporis, cum omni dimen <lb/>ſione <reg norm="capacitateque" type="simple">capacitateq́;</reg> ſit denudatum? </s> + <s xml:id="s2144" xml:space="preserve">vnde ſi <reg norm="centrum" type="context">centrũ</reg> locus eſſet corporum grauium, om <lb/>nia dicta corpora grauia, extra proprium locum exiſterent, quia nullum ex iis eſt, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſit in centro. </s> + <s xml:id="s2145" xml:space="preserve">Adde quod neque hoc cum loci definitione ab ipſo poſita conſentiret <lb/>cum ipſe dicat in eodem cap. locum eſſe ſuperſiciem quandam, & non interuallum, <lb/>licet huiuſmodi definitio falſa appareat primo ex <reg norm="inconuenienti" type="context">incõuenienti</reg> falſo, quod ipſe hinc <lb/>ſequuturum dicit, ideſt, quod ſi locus interuallum eſſet, infinita loca exiſterent, quod <lb/>reuera nec ob hanc cauſam inconueniens exiſtit, quia eodem planè modo quo ali-<lb/>quod corpus poteſt eſſe infinita corpora, (quod ipſe diceret in potentia) ſic etiam in <lb/>teruallum aliquod poſſet eſſe infinita interualla. </s> + <s xml:id="s2146" xml:space="preserve">Cum autem dicat ſuperficies cor-<lb/>poris ambientis eſſe locum eius corporis, quod continetur, cogitur dicere lineam, <lb/>quæ circundat ſuperficiem, ſuperficiei locum eſſe, & puncta ipſius lineæ, quod reue <lb/>ra abſurdum eſt. </s> + <s xml:id="s2147" xml:space="preserve">Locus corporis eſt interuallum illud eadem magnitudine & figu-<lb/>ra, qua corpus ipſum pręditum eſt, quod ſi non eſſet, ſed eſſet ſuperficies, quemad-<lb/>modum Ariſtoteles voluit, maximum inconueniens ſequeretur, ſcilicet æquales lo-<lb/>cos capere inęqualia corpora, aut corpora æqualia, locos inęquales occupare, quod <lb/>ſcitu facillimum eſt, cum Theon ſuper Ptolomęi Almageſtum iam probarit ſphæ-<lb/>ricam ſuperficiem maius interuallum corporeum continere, quàm aliam <reg norm="quanuis" type="context">quãuis</reg> ſu-<lb/>perficiem dictæ ſphęricæ 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, quę tamen corpora apta eſſent <lb/>ad occupandos locos ęquales, quamuis Ariſtoteles dicat, locum, neque maiorem ne <lb/>que minorem eſſe debere locato. </s> + <s xml:id="s2148" xml:space="preserve">Sed interualla corporea ęqualia à quauis figura <lb/>terminata, continebunt ſemper corpora ęqualia. </s> + <s xml:id="s2149" xml:space="preserve">Corporeum igitur interuallum eſt + <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 adęquatus, cum corpus in interuallum ſuperſiciale non intret, <lb/>quam @is interuallum corporeum ingrediatur. </s> + <s xml:id="s2150" xml:space="preserve">Et hoc modo <reg norm="nullum" type="context">nullũ</reg> eſt corpus, quod <lb/>in m@ do aut extra mundum ( dicat autem Ariſtoteles quicquid voluerit ) locum <lb/>ſuum 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 ſenſerit de infinito.</head> + <head xml:id="head284" xml:space="preserve">CAP. XXI.</head> + <p> + <s xml:id="s2151" xml:space="preserve">TRactans Ariſtoteles in fine quinti cap. lib. 3. phyſicorum de infinito ait, impoſ<lb/>ſibile cum ſit inuenire locum infinitum, & omne corpus in loco cum ſit, impoſ <lb/>ſibile 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 Ariſtoteles debuerit beneficio loci deſtruere infinitum, ordine peruerſo <lb/>de infinito prius, quàm de loco diſputationem inſtituat; </s> + <s xml:id="s2154" xml:space="preserve">ſed dicamus ipſum intelli-<lb/>gere de infinito corporeo, & cum probauerimus corporis locum eſſe corporeum in <lb/>teruallum, non autem ſuperficiem, neque opus ſit in definitione interualli mentio <lb/>nem aliquam facere terminorum, vnde ipſum infinitum eſſe poteſt, neque aliqua ra <lb/>tione de hac re dubitari poteſt; </s> + <s xml:id="s2155" xml:space="preserve">hoc modo nullum inconueniens ſequeretur, quòd <lb/>extra cęlum reperiri poſſit corpus aliquod infinitum, quamuis, id ipſe nulla euiden-<lb/>ti ratione inductus perneget. </s> + <s xml:id="s2156" xml:space="preserve">Senſit quoque, abſque eo, <reg norm="quod" type="simple">ꝙ</reg> aliquam rationem propo <lb/>nat, aliquid extra cœlum reperiri quemadmodum apparet ex fine cap .9. lib. primi <lb/>de cœlo, cum etiam ait cap .8. lib. 8. phyſicorum, infinitas partes alicuius continui eſ-<lb/>ſe ſolum in potentia, non item in actu, hoc non eſt illico concedendum, quia ſi omne <lb/>totum continuum, & re ipſa exiſtens, in actu eſt, omnis quoque eius pars erit in actu, <lb/>quia ſtultum eſſet credere, ea quæ actu ſunt, ex ijs, quæ potentia exiſtunt, componi. <lb/></s> + <s xml:id="s2157" xml:space="preserve">Neque etiam dicendum eſt continuationem earundem partium efficere, vt poten-<lb/>tia ſint 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 eſt, quin ante <reg norm="diuiſionem" type="context">diuiſionẽ</reg>, <lb/>medietas <var>.a.e.</var> tam in actu (licet coniuncta cum alia <var>.e.u.</var>) reperiretur, quàm totum .2. <lb/>u. licet à ſenſu diſtincta non eſſet. </s> + <s xml:id="s2159" xml:space="preserve">Idem affirmo de medietate <var>.a.e.</var> ideſt de quarta <lb/>parte totius <var>.a.u.</var> & pariter de octaua, de milleſima, & de quauis, ita vt eſſentia actua <lb/>lis infiniti hoc modo tutò concedi poſſit, <reg norm="cum" type="context">cũ</reg> ita ſit in natura. </s> + <s xml:id="s2160" xml:space="preserve">Sed peius etiam ſenſit <lb/>Ariſtoteles eodem loco capitis quinti lib. 3. phyſicorum, 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 conſiderat eadem etiam ratione intelligi poſſe <lb/>aliquam quantitatem <reg norm="infinitorum" type="context">infinitorũ</reg> cubitorum, & in quantitatis definitione nullam eſ-<lb/>ſe neceſſitatem terminorum, vt exempli gratia in definitione numeri, non eſt neceſ <lb/>ſitas alicuius determinati numeri, quia multitudo, non minus infinita, quàm finita, <lb/>intelligi poteſt. </s> + <s xml:id="s2162" xml:space="preserve">Vbi poſteà cap .8. libr .4. phyſicorum ait nullam eſſe differentiam <lb/>inter infinitum, & vacuum, reuera nihil abſurdius 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 Ariſtotele adductatemporis definitio.</head> + <head xml:id="head286" xml:space="preserve">CAP. XXII.</head> + <p> + <s xml:id="s2163" xml:space="preserve">CVM ſenſerit Ariſtoteles <reg norm="tempus" type="context">tẽpus</reg> abſque motu eſſe <reg norm="non" type="context">nõ</reg> poſſe, ea tamen ab <reg norm="inui- cem" type="context">inui-<lb/>cẽ</reg> ſeparans, <reg norm="volens" type="context">volẽs</reg> definire <reg norm="tempus" type="context">tẽpus</reg> ait, <reg norm="ipsum" type="context">ipsũ</reg> eſſe <reg norm="motus" type="simple">motꝰ</reg> <reg norm="menſuram" type="context">menſurã</reg> <reg norm="numerumque" type="context simple">numerũq́;</reg>. </s> + <s xml:id="s2164" xml:space="preserve">Quæ <lb/>quidem definitio, natura ſua non eſt bona, quia tempus, neque numerus eſt, neque <lb/><reg norm="etiam" type="context">etiã</reg> eſt <reg norm="menſura" type="context">mẽſura</reg> motus <reg norm="per" type="simple">ꝑ</reg> ſe, ſed <reg norm="tantum" type="wordlist">tm̃</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="accidens" type="context">accidẽs</reg>, quia nihil eſt, <reg norm="quod" type="wordlist">qđ</reg> numeret aut menſuret <lb/>aliud, quod non ſit <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpeciei <reg norm="cum" type="context">cũ</reg> illo quod <reg norm="menſuratur" type="context">mẽſuratur</reg>, aut numero <reg norm="circunſcribitur" type="simple">circunſcribit̃</reg>, <lb/>vt <reg norm="exempli" type="context">exẽpli</reg> gratia, nulla <reg norm="vnquam" type="context">vnquã</reg> ſuperficies <reg norm="per" type="simple">ꝑ</reg> ſe numerabit aut <reg norm="menſurabit" type="context">mẽſurabit</reg> <reg norm="lineam" type="context">lineã</reg>, aut cor-<lb/>pus; </s> + <s xml:id="s2165" xml:space="preserve">neclinea ſuperficiem <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 ſuperfi-<lb/>ciem; </s> + <s xml:id="s2167" xml:space="preserve">Sed linea lineam menſurabit; </s> + <s xml:id="s2168" xml:space="preserve">ſuperficies ſuperficiem; </s> + <s xml:id="s2169" xml:space="preserve">& corpus corpus; </s> + <s xml:id="s2170" xml:space="preserve"><reg norm="etianſi" type="context">etiãſi</reg> <lb/>tam vna ex iis quantitatibus quàm altera ſit continua. </s> + <s xml:id="s2171" xml:space="preserve">Cum verò motus non ſit tem <lb/>pus, neque tempus ſit motus, ſed inter ſe maximè differant, ſequetur ex iis, alterum <lb/>nullo modo per ſe eſſe menſuram alterius, niſi per accidens. </s> + <s xml:id="s2172" xml:space="preserve">Et ſi alicui videtur, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ad ſignificandam aliquam quantitatem motus, dicere huiuſmodi operationem dua-<lb/>rum horarum, aut duorum dierum, aut duorum annorum ſpatio completam eſſe, ſit <lb/>ponere tantum tempus: </s> + <s xml:id="s2173" xml:space="preserve">animaduertere debet hoc ſimpliciter non eſſe verum, quia <lb/>horarum, dierum. </s> + <s xml:id="s2174" xml:space="preserve">& annorum interualla, imaginatione <reg norm="concipiuntur" type="context">concipiũtur</reg> vt motus corpo-<lb/>rum cęleſtium, ſine quibus, neque anni, neque dies, neque horę exiſterent, <reg norm="etiam" type="context">etiã</reg> ſi om <lb/>nis motus ſit (vt ita dicam) locatus in tempore, ut corpus in loco, vnde motus motu, <lb/>& tempus tempore, non autem aliud ab alio menſuratur. </s> + <s xml:id="s2175" xml:space="preserve">Tempus ex neceſſita-<lb/>te (phyloſophicè tamen loquendo) res eſt æterna, motus non item, quia diuerſis mo <lb/>dis terminari poteſt & ceſſare, & interim dum ceſſabit quieſcet corpus, quod <lb/>primo mouebatur. </s> + <s xml:id="s2176" xml:space="preserve">nihilominus tamen, tempus continuabit curſum ſuum. </s> + <s xml:id="s2177" xml:space="preserve">Tempus <lb/>igitur potius locus motus erit dicendum, quàm numerus aut menſura eius, & tale eſt, <lb/>vt conſumatum uideatur à continuò quodam fluxu vnius inſtantis, quemadmodum <lb/>iam dixi in .38. capite meę gnomonicæ, & cum dico ab vno inſtanti, vnum in ſpecie, <lb/>& non in numero intelligo, quod à ſenſibus noſtris percipi non poteſt, <reg norm="neque" type="simple">neq;</reg> etiam <lb/>notari, quia nouum ſemper inſtans nobis occurrit. </s> + <s xml:id="s2178" xml:space="preserve">& ſi aliquis aliquod <reg norm="exemplum" type="context">exemplũ</reg> (lar <lb/>go modo) incompræhenſibilitatis ipſius inſtantis deſideraret, imaginetur rotam ali <lb/>quam albam, in qua ſit nigrum aliquod punctum ſenſibile, aut è contra rotam <reg norm="nigram" type="context">nigrã</reg> <lb/>imaginetur, in qua ſit punctum album, quæ rota velociſſimè moueatur; </s> + <s xml:id="s2179" xml:space="preserve">huiuſmodi <lb/>punctum, nullo modo aſſignari poterit, magis ab una parte quàm ab altera; </s> + <s xml:id="s2180" xml:space="preserve">immo ſe <lb/>ſe nobis offeret ſemper in forma lineæ circularis. </s> + <s xml:id="s2181" xml:space="preserve">poſſumus aliquo modo etiam ſu-<lb/>mere exemplum à ſono, quia omnis chorda cuiuſlibet inſtrumenti muſici, dum ſo-<lb/>nus editur, tremit, unde huiuſmodi ſonus, appellari poteſt aggregatum aliquod ex <lb/>innumerabilibus ſonis. </s> + <s xml:id="s2182" xml:space="preserve">eodem modo ſe habet ſonus, quem ędunt campanę, & omnia <lb/>inſtrumenta tam naturalia, quàm artificialia, quæ quantò velocius <reg norm="tremunt" type="context">tremũt</reg>, tanto acu <lb/><reg norm="tiorem" type="context">tiorẽ</reg> generant ſonum, & quantò tardius, tantò grauiorem. </s> + <s xml:id="s2183" xml:space="preserve">Neque eſt quòd in ad-<lb/>mirationem ducamur, quòd ſenſui unum aliquod continuum appareat id, quod di-<lb/>ſcretorum eſt multitudo ( non putet tamen aliquis me negare continuitatem ſucceſ <lb/>ſiuam ipſius temporis) quod clare cognoſci poteſt à niue, aut à chryſtallo, 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 inſpiciendum offerunt, quod nihil aliud eſt, quàm innumerabilis quædam <lb/>multitudo minutorum reflexorum. </s> + <s xml:id="s2184" xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de ſputo, & qualibet ſpuma, & 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 ſunt corpuſcula à quibus vt à ſpeculis reflectitur lumen, tantò magis ag <lb/><reg norm="gregatum" type="context">gregatũ</reg> illud <reg norm="album" type="context">albũ</reg> apparet. </s> + <s xml:id="s2185" xml:space="preserve">Hæc <reg norm="autem" type="context">autẽ</reg> exempla <reg norm="cum" type="context">cũ</reg> ſint, nec non largo modo ſumpta, <lb/><reg norm="mirum" type="context">mirũ</reg> non erit ſi claudicare <reg norm="videbuntur" type="context simple">videbũt̃</reg>. </s> + <s xml:id="s2186" xml:space="preserve">Sed ut ad <reg norm="motum" type="context">motũ</reg>, & <reg norm="tempus" type="context">tẽpus</reg> reuertamur ( quæ ſunt <lb/><reg norm="continua" type="context">cõtinua</reg> ſucceſſiua) Ariſtoteles in definiendo tempore, non reduxit in mentem, quod <lb/>ſcribit decimo metaphyſicę et .4. cap. ſecundo. libr. de cęlo, omnia videlicet, ab eo, <lb/>quod minimum eſt in ſuo genere, menſurari, & ex ſeipſo in phyſicorum libris, tem-<lb/>pus non eſt de genere motus; </s> + <s xml:id="s2187" xml:space="preserve">ergo eius ipſius rationum ui, tempus non erit menſura <lb/>motus, ſed motus quidem poteſt menſurare motum, videlicet velocior minus velo-<lb/>cem, & breuior longiorem; </s> + <s xml:id="s2188" xml:space="preserve">& <reg norm="numerus" type="simple">numerꝰ</reg> <reg norm="menſuratur" type="simple">menſurat̃</reg> numero, & tempus tempore in quan <lb/>tum longum eſt, aut breue, non in quantum velox, aut tardum; </s> + <s xml:id="s2189" xml:space="preserve">Nullum autem in-<lb/>conueniens ſequetur ſumendo tempus tam ptoportionale motui, quam locus cor-<lb/>pori, quia motus decem milliarium, quæ aliquis vnius horæ ſpatio conficiat, erit pro <lb/>portionalis corpori denſo, & motus vnius milliaris eadem hora peracti, proportiona <lb/>lis erit corpori raro; </s> + <s xml:id="s2190" xml:space="preserve">& quemadmodum corpus denſum occupat minus interuallum <lb/>loci, contra quam fiat in corpore raro: </s> + <s xml:id="s2191" xml:space="preserve">ſic etiam motus velox breuiori temporis ſpa-<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/>Ariſtotele.</head> + <head xml:id="head288" xml:space="preserve">CAP. XXIII.</head> + <p> + <s xml:id="s2192" xml:space="preserve">ARiſtoteles .8. capi .8. phyſicorum ait impoſſibile eſſe aliquid per <reg norm="lineam" type="context">lineã</reg> rectam <lb/>nunc vno modo, nunc altero, ideſt eundo, & redeundo per dictam lineam in <lb/>extremis abſque 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 ſpecula-<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="ſiniſtram" type="context">ſiniſtrã</reg> ferri; </s> + <s xml:id="s2195" xml:space="preserve">& imaginemur <reg norm="punctum" type="context">pũctum</reg> <var>.b.</var> extra ipſum, ubi <lb/>magis nobis videbitur, à quo ducantur duæ lineæ recte <var>.b.u.</var> et <var>.b.n.</var> contiguæ ipſi 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 ſit <var>.<lb/>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> conſtituamus in quali <lb/> + <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> + bet parte, ſumemus etiam punctum <var>.a.</var> circun-<lb/>ferentiæ dicti circuli, à quo vſque ad <var>.b.</var> lineam <var>.<lb/>b.a.</var> imaginemur <reg norm="fixam" type="context">fixã</reg> in <var>.b.</var> ſed quod remanat mo <lb/>bile, ſecundum 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> verſus <var>.b.<lb/>n.</var> proficiſcetur, & aliquando ab <var>.b.n.</var> verſus <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 quieſcet, quia in inſtanti reuertetur, quia <lb/><var>b.u.</var> et <var>.b.n.</var> in puncto, <reg norm="dictum" type="context">dictũ</reg> circulum tangunt, & <lb/>dicta <var>.b.a.</var> interſecabit ſemper aliquam ex dictis <lb/><var>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> quod interſectionis <lb/>punctum ſit <var>.t</var>. </s> + <s xml:id="s2198" xml:space="preserve">Imaginemur nunc quod <reg norm="ſecundum" type="context context">ſecũdũ</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 quieſcet, etiam ſi ſit in quouis ex <lb/>tremo. </s> + <s xml:id="s2199" xml:space="preserve">Ariſtotelis igitur opinio, tuta non eſt.</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 grauisſimus an bene ſenſerit 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">ARiſtoteles in fine .8. phyſicorum ſentit corpus per vim motum, & ſeparatum à <lb/>primo mouente, moueri, aut motum eſſe per aliquod tempus ab aere, aut ab <lb/>aqua, quæ ipſum <reg norm="ſequuntur" type="context">ſequũtur</reg>. quod fieri non poteſt; </s> + <s xml:id="s2201" xml:space="preserve">quia imo aer, qui in locum defer-<lb/>tum à corpore ſubintrat ad fugandum vacuum, non ſolum hoc corpus non impellit, <lb/>ſed potius id cohibet à motu, quia aer per vim à corpore ducitur retrò, & diuiſus à <lb/>parte anteriori à dicto corpore, reſiſtit ſimiliter, & quantum dictus aer in dicta parte <lb/>condenſatur, tantum in poſteriori rarefit, vnde per vim ſeſe 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/>ipſum ab aere rapitur. </s> + <s xml:id="s2203" xml:space="preserve">Huiuſmodi autem rarefactio aeris, naturalis non eſt, ſed vio <lb/>lenta; </s> + <s xml:id="s2204" xml:space="preserve">& hanc ob cauſam reſiſtit, & ad ſe trahit, ſed non ſufferente natura, vt inter <reg norm="vnum" type="context">vnũ</reg> <lb/>& aliud ex dictis corporibus reperiatur vacuum; </s> + <s xml:id="s2205" xml:space="preserve">iccirco funt hæc ſemper contigua, <lb/>& mobile corpus aerem deſerere cum nequeat, eius velocitas impeditur. </s> + <s xml:id="s2206" xml:space="preserve">Huiuſmo <lb/>di igitur corporis ſeparatim à primo mouente velo citas oritur à quadam naturali im <lb/>pręſſione, ex impetuofitate recepta à dicto mobili, quæ impręsſio & impetuoſitas, <lb/>in motibus rectis naturalibus continuò creſcit, cum perpetuò inſe cauſam <reg norm="mouentem" type="context context">mouẽtẽ</reg>, <lb/>ideſt propenſionem eundi ad locum ei à natura asſignatum habeat. </s> + <s xml:id="s2207" xml:space="preserve">Ariſto .8. cap. <lb/>primi lib. de cœlo, dicere non deberet <reg norm="qui" type="simple">ꝗ</reg> quantò propius accedit corpus ad <reg norm="terminum" type="context">terminũ</reg> <lb/>ad quem, tantò magis ſit velox; </s> + <s xml:id="s2208" xml:space="preserve">ſed potius, <reg norm="qui" type="simple">ꝗ</reg> quantò longius diſtat à termino à quò <lb/>tantò velocius exiſtit. </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 eſt inclinatio ad locum ſuum eundi, extra quem per vim confiftit. <lb/></s> + <s xml:id="s2210" xml:space="preserve">Neque etiam rectè ſeripſit Ariſto .9. cap. lib. 8. phyficorum et .2. lib. primi de cœlo <lb/>eſſe aliquem motum ex recto & circulari mixtum, <reg norm="qui" type="simple">ꝗ</reg> omninò imposſibile eſt.</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 ſe <lb/>quicquid Ariſtoteli uiſum ſit.</head> + <head xml:id="head292" xml:space="preserve">CAP. XXV.</head> + <p> + <s xml:id="s2211" xml:space="preserve">MOtus rectus corporum naturalium ſurſum, aut dcorfum, non eft naturalis pri <lb/>mò & per ſe, quia motus naturalis perpetuus eſt, aut vt melius dicam, inceſ-<lb/>fabilis, & alius eſſe non poteft quàm circularis, <reg norm="nullaque" type="simple">nullaq́;</reg> pars cum fuo toto coniun-<lb/>cta, alium motum naturalem habere poteft, quàm 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">libereq́;</reg> vagetur, ſpontè, & quàm breuisſima poteft <lb/>via, ad locum, <reg norm="ſuitotius" type="simple">ſuitotiꝰ</reg> à natura ſtatutum proficiſcitur. </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 ſuum locum, vbi contra naturam ſuam reperitur. </s> + <s xml:id="s2214" xml:space="preserve">Vnde hu-<lb/>iuſmodi motus, partim & non omninò, naturalis eft. </s> + <s xml:id="s2215" xml:space="preserve">Is autem proprius eſt & natura <lb/>lis motus, qui dicti corporis eſſentiam conſeruat. </s> + <s xml:id="s2216" xml:space="preserve">hoc autem non præſtat hic rectus, <lb/>cum deſtruat, ergò hic motus primò & per ſe 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 cęlo fic ſcribit.</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 ſe caufam nec grauitatis, nec leuitatis corporum <lb/>naturalium nofce, quæ eft denfitas auto raritas corporis grauis, aut leuis, maior denſi-<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">ꝗ</reg> ipſe de vtre inflato proponit, debuiſſet ſaltem ei oculos ad verita-<lb/>tem, quæ clarisſimè fulget, inſpiciendum aperire. </s> + <s xml:id="s2221" xml:space="preserve">Verisſimum eſt, vtrem inflatum <lb/>plus ponderis habere quàm 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, quàm ſi eidem liberè vagari permit-<lb/>teretur, vnde violenter, quodam modo, con denfata eft, & quia corpus denfum in <lb/>minus denfo, femper deſcendit, & minus denſum in magis denſo aſcendit. </s> + <s xml:id="s2223" xml:space="preserve">Hanc ob <lb/>caufam vter inflatus plenus corpore magis denſo, quàm eft medium quod eum cir-<lb/>cundat, deſcendit, 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, & ſpbæra.</head> + <head xml:id="head296" xml:space="preserve">CAP. XXVII.</head> + <p> + <s xml:id="s2224" xml:space="preserve">CVm Ariftoteles fenſerit circulum eſſe figurarum ſuperficialium <reg norm="primam" type="context">primã</reg>, & <reg norm="ſphae ram" type="simple context">ſphę<lb/>rã</reg> eſſe <reg norm="primam" type="context">primã</reg> <reg norm="corporearum" type="context">corporearũ</reg> <reg norm="prope" type="wordlist">ꝑꝑ</reg> <reg norm="earum" type="context">earũ</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 ſentit) perfectè, licet rationem huius rei non <lb/>nouerit. </s> + <s xml:id="s2227" xml:space="preserve">Nam <reg norm="centrum" type="context">centrũ</reg> cuiuſlibet rei, eiuſdem rei <reg norm="principium" type="context">principiũ</reg> eft, & eę figurę, quæ ipſum <lb/>æqualiter circundant, poſſunt appellari perſectæ, ſiue ſint ſuperficiales, ſiue corpo-<lb/>reæ, & ècontrà illæ, quæ contrario modo ſe habent, imperfectæ. </s> + <s xml:id="s2228" xml:space="preserve">Quòd autem per-<lb/>ſectum eſt, licet natura fit primum, eſt tamen vltimum generatione. </s> + <s xml:id="s2229" xml:space="preserve">Sed quando <lb/>Ariftoteles duas dictas figuras pronuntiauit primas, vt perfectas, prioritate ſcilicet <lb/>ea, quæ oritur à perfectione, verum dixit; </s> + <s xml:id="s2230" xml:space="preserve">fed quando de figuris ſuperficialibus 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 ſeu elipſi, 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/>ſtrat, quia aliæ figuræ, à lineis curuis terminatę, eandem conditionem fortiuntur. <lb/></s> + <s xml:id="s2233" xml:space="preserve">Circulus ſphę<reg norm="raque" type="simple">raq́;</reg>, non ex vno ſolo angulo recto conſtant, 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 cęlo, etiam fi triangulus ex duobus angulis rectis conſurgat, ſed ſunt <lb/>figurę infinitorum angulorum rectorum, & hanc ob cauſam à me dicuntur vltimæ & <lb/>perfectę, quia infinito nihil addi poteſt. </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">folidorũ</reg> <lb/>ſphęræ, minor eft quadruplo infinito per .4. infinita angulorum ſolidorum <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg>, quæ .4. infinita, minora ſunt quouis angulo ſolido acuto terminato à tribus pla-<lb/>nis. </s> + <s xml:id="s2235" xml:space="preserve">Triangulus inter figuras planas ſuperſiciales eft primus, & circulus vltimus; </s> + <s xml:id="s2236" xml:space="preserve">& <lb/>pyramis quadrilatera, inter corpora eft prima, & ſphęra vltima.</s> + </p> + </div> + <div type="section" level="3" n="28"> + <head xml:id="head297" style="it" xml:space="preserve">Occultam fuiße grauisſimo Stagirit & canſam ſcintilla-<lb/>tionis ſtellarum.</head> + <head xml:id="head298" xml:space="preserve">CAP. XXVIII.</head> + <p> + <s xml:id="s2237" xml:space="preserve">VBi Ariſtoteles ait ſcintillationem ſtellarum ſieriratione aſpectus @oſtri ob, ma <lb/>ximam diſtantiam, maximum errorem committit, vt etiam facid quum putat <lb/>vifionem fieri extramittendo, contra id, quod alio loco, immo contra veritatem ip <lb/>ſam afferuit. </s> + <s xml:id="s2238" xml:space="preserve">Scintillatio ergo ſtellarum, neque aſpectus noſtri ratione, neque ali-<lb/>cuius mutationis earundem ſtellarum, ſed ab inæqualitate motus corporum diapha <lb/>norum mediorum naſcitur, <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> clarè cernitur, quòd fi inter aliquod obie <lb/>ctum, & nos, aliquis ſumus, qui aſcendat, intercefferit, videbimus obiectum illud qua <lb/>ſi tremere. </s> + <s xml:id="s2239" xml:space="preserve">Hoc autem tantò magis fiet, quantò magis diſtabit obiectum ab ipſo fu <lb/>mo; </s> + <s xml:id="s2240" xml:space="preserve">vnde admirationi locus non erit, fi ftellas fixas magis ſcintillare, quam errantes <lb/>cernamus. </s> + <s xml:id="s2241" xml:space="preserve">Lumen ſtellæ ad oculum noſtrum 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, quàm in propinquis ſtel <lb/>lis apparet, quemadmodum ab exemplo de fumo allato, & etiam ab aliquibus vi-<lb/>tris ex ſuperficie non plana, ſed irregulari conſtantibus, quilibet cognoſcere poteft.</s> + </p> + </div> + <div type="section" level="3" n="29"> + <head xml:id="head299" style="it" xml:space="preserve">Daricontinuum infinitum motum ſuper 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 ſenſerunt imposfibile eſſe dari per <reg norm="imaginationem" type="context">imaginationẽ</reg> motum con-<lb/>tinuum & <reg norm="perpetuum" type="context">perpetuũ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> + ſuper 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">quarũ</reg> <lb/><var>b.a.</var> fit <reg norm="infinita" type="context">ĩfinita</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 ſolis calorem à motu localι ipſius corporis ſolaris, <lb/>ut Ariſtoteli placuit.</head> + <head xml:id="head302" xml:space="preserve">CAP. XXX.</head> + <p> + <s xml:id="s2245" xml:space="preserve">JD nullo planè modo eſt admittendum quod Ariftoteles credidit calorem folis à <lb/>motu locali ipſiuſmet 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, ſatis eſt eſſicere, & c. ſed <lb/>cap .7. lib. 2. de cælo fic ſeribit, Caliditas autem ab ipſis, <reg norm="lumenque" type="simple">lumenq́;</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 cœleſtium fit caufa ca <lb/>loris, ſed eriam luminis, paulò autem poſt dicit, ſuperiorum autem corporum vnum <lb/>quodque fertur in ſphæra, vt ipſa 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 ſolate fixum fit in ſpisfitudine ſui orbis deferentis, fe-<lb/>cundum communem opinionem, non mouetur per fe, ſed accidentaliter, cum ſei-<lb/>licet fertur à dicto ſuo orbe, vnde fieri poteſt, 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/>ſam, 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 posſibile eſſet, quid ergo fricatio ſuperficierum orbis ſui, 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 ſcilicet 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">Mercuriusq́;</reg>, magis calefacere quam fol de <reg norm="berent" type="context">berẽt</reg>, <lb/>cum ita ſint veloces vt ipſe Sol, & eodem magis propinqua terræ. </s> + <s xml:id="s2254" xml:space="preserve">Verum Ari-<lb/>ſtotelis <reg norm="temporibus" type="context">tẽporibus</reg>, <reg norm="nullum" type="context">nullũ</reg> <reg norm="alium" type="context">aliũ</reg> <reg norm="planetam" type="context">planetã</reg> quam folem <reg norm="putabant" type="context">putabãt</reg> philofophi ſupra <reg norm="Lunam" type="context">Lunã</reg> eſ-<lb/>ſe. </s> + <s xml:id="s2255" xml:space="preserve">Atque etiam <reg norm="contigeret" type="context">cõtigeret</reg> menfe Decembri, quam Iunio, magis inualeſceret calor, <lb/>cum huiuſmodi menſe ſolad nos propius accedat, quam menfe Iunii. per differen-<lb/>tiam maiorem diametro regionis elementaris, (nam folaris eccentricitas maior eft <lb/>ſemidiametro <reg norm="elementaris" type="context">elemẽtaris</reg> regionis) non confiderans Ariftoteles differentiam ca-<lb/>loris, quæ naſcitur ex eo, <reg norm="qui" type="simple">ꝗ</reg> Sol aut altius ſupra orizontem excurrat, aut infra <reg norm="eundem" type="context">eundẽ</reg> <lb/>feratur; </s> + <s xml:id="s2256" xml:space="preserve">neque eam, quę à longitudine, aut breuitate diei proficiſcitur. </s> + <s xml:id="s2257" xml:space="preserve">Sed quia Ari <lb/>ſtoteles eodem cap tertio Metheororum intelligit de motu rapto, ideſt diurno, ſiue <lb/>dicamus vniuerfali, hinc ſequi deberet, <reg norm="qui" type="simple">ꝗ</reg> Sol maiorem caloris uim menſe Martij & <lb/>Septembris, quàm aliis menfibus, profunderet, quia in iiſdem temporibus, ſol virtu <lb/>te huiuſmodi motus velocior exiftat, quàm alio quolibet tempore anni, cum tunc <lb/>per æquatorem circuũoluatur. </s> + <s xml:id="s2258" xml:space="preserve">Multa quoque alia incommoda ſequerentur ſi Ari <lb/>ſtorelis 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">ꝗ</reg> vniente ſeſe 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 ipſius ſolis, & ita in lumine ſedem habet, ut fi ſol <lb/>quieſceret, neque in orbe ſuo circumager etur, infęliciſſima eſſet ea regio, in cuiu; <lb/></s> + <s xml:id="s2261" xml:space="preserve">Zenith ipſe reperiretur.</s> + </p> + </div> + <div type="section" level="3" n="31"> + <head xml:id="head303" style="it" xml:space="preserve">Vnde caloris ſolis 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 ſuperiore oſtenderim calorem ſolis non aliunde, quàm à lumine <lb/>prouenire, oſtendam nunc ex ordine, ex quot, <reg norm="quibusque" type="simple">quibusq́;</reg> cauſis oriatur magna <lb/>differentia eius caloris æſtatis ad hyemem, quarum nonnullæ ab antiquis obſerua-<lb/>tæ fuerunt, aliæ autem à nemine, quod ſciam. </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 ſupra orizontem, quæ cau-<lb/>ſa ab antiquis pofita, & citata fuit. </s> + <s xml:id="s2264" xml:space="preserve">Aliam quoque huius rei cauſam iidem antiqui <lb/>dicebant eſſe propinquitatem ſolis noftro Zenith, ſed hæc cauſa immediata non <lb/>eſt, quia ab ea tres caufæ immediatæ naſcuntur; </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">ſecunda maior quantitas luminis in ſuperficie terrę; <lb/></s> + <s xml:id="s2267" xml:space="preserve">tertia, minor <reg norm="reſiſtentia" type="context">reſiſtẽtia</reg> vaporum ipſi radio luminoſo 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 reſlexi cum incidente, quiſque, uel ſaltem mediocriter <lb/>in cathoptricę cognitione verſatus hoc verum eſſe cognoſcet. </s> + <s xml:id="s2270" xml:space="preserve">Vt hoc tamen in-<lb/>noteſcat facilius. </s> + <s xml:id="s2271" xml:space="preserve">Imaginemur <var>.q.p.</var> et <var>.b.d.</var> eſſe duas particulas ęquales ſuperficiei <lb/>ipfius terræ, ſuper quas cadant duo radii luminofi ſolis <var>.e.q.</var> et <var>.n.d.</var> quorum <var>.e.q.</var> fit ad <lb/>modum obliquus, et <var>.n.d.</var> quaſi perpendicularis, vnde radii reſlexi <var>.p.a.</var> et <var>.b.u.</var> aſcen <lb/>dent cum angulis æqualibus eis, qui funt ſuorum cadentium, cum omnis angulus re-<lb/>flexionis femper æqualis ſit angulo ſuæ incidentię, 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 ſint angulis <var>.b.</var> et <var>.d.</var> trianguli <var>.b.i.d.</var> ex ſuppoſito. </s> + <s xml:id="s2273" xml:space="preserve">Quòd uero attinet ad ma <lb/>iorem quantitatem luminis ſuper terræ ſuperſiciem; </s> + <s xml:id="s2274" xml:space="preserve">Imaginemur radium <var>.a.q.</var> cuius <lb/>reſpectu etiam imagine mur duos ſuperficiei terræ ſitus, 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 eſt ex ſuppoſito, 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 ſuper <var>.q.o.</var> cadat vniuerſum lumen, <lb/>quod ſuper <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 ſuperficies <var>.q.u.</var> ipſius terræ, quàm.q.o. quia maius lumen in ſe maio-<lb/>rem calorem includit: </s> + <s xml:id="s2278" xml:space="preserve">quod manifeſtè apparet in radiorum vnione mediante refle-<lb/>xione, aut refractione. </s> + <s xml:id="s2279" xml:space="preserve">Sed quod attinet ad minorem refiſtentiam vaporum ad ip-<lb/>ſum radium luminoſum, etfi primo capite meæ Gnomonicæ leuiter id attigerim, ni <lb/>hilominus tamen, & idem ipſum hoc loco proponam. </s> + <s xml:id="s2280" xml:space="preserve">Denotetur, exempli gratia, <lb/>ſuperficies terræ ab <var>.o.g.</var> et ea, quæ eft vaporum ab <var>.n.a.</var> ſupponatur etiam ſol 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 ſit Zenith <reg norm="puncti" type="context">pũcti</reg> <var>.o.</var> & <reg norm="etiam" type="context">etiã</reg> in <var>.p.</var> ipſi <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 ipſis vaporibus offuſcatæ, ſed <var>.o.n.</var> breuior eſt <var>.o.a.</var> ex .7. lib. 3. Eucli. </s> + <s xml:id="s2283" xml:space="preserve">mino-<lb/>rem ergo reſiſtentiam habebit à vaporibus ſol in Zenith, quàm extra cundem com -<lb/>morans, & quantò longius erit idem ab ipſo Zenith, tanto maiorem reſiſtentiam 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 ſenſus 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 ſonum corpus aliquod, quod ſenſu ſit deſtitutum, vt Ariſtoteles .9. cap. li-<lb/>br .2. de cælo putauit, offendere, eſt falſum.</s> + </p> + <p> + <s xml:id="s2285" xml:space="preserve">Corpus enim non niſi à corpore poteſt lædi, non ergo à ſono, cum ſonus corpus <lb/>non ſit. </s> + <s xml:id="s2286" xml:space="preserve">Sed aer, & ignis, cum è contra ſint corpora, hoc facilè præſtare poſſunt im-<lb/>plendo aliquem locum velociter ad excludendum vacuum; </s> + <s xml:id="s2287" xml:space="preserve">vnde generatur ſonus. <lb/></s> + <s xml:id="s2288" xml:space="preserve">Quod hucuſque à nemine animaduerſum 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 ſonitu corporum cælestium non <lb/>fuiſſe ab Aristotele ſublatam.</head> + <head xml:id="head308" xml:space="preserve">CAP. XXXIII.</head> + <p> + <s xml:id="s2289" xml:space="preserve">SEnſerunt Pythagorici orbes cæleſtes dum circunuoluuntur, non autem corpora <lb/>ſtellarum ſolum, æd ere ſonu. </s> + <s xml:id="s2290" xml:space="preserve">Quibus dum Ariſtoteles contradicere cogitat, <lb/>maximè fauet. </s> + <s xml:id="s2291" xml:space="preserve">Eatamen opinio è phyloſophorum ſcholis eſt explodenda, quia aut <lb/>orbes ſunt ſibi ipſis contigui, aut inuicem diſtantes: </s> + <s xml:id="s2292" xml:space="preserve">ſi ab inuicem diſtant (quod <lb/>nemo adhuc conceſſit, quia hac ratione vacuum introduceretur) clarum eſt, quod <lb/>cum ſe minime tangant, ſonum edere nequeunt: </s> + <s xml:id="s2293" xml:space="preserve">Si verò eorum vnus alteri ſit conti <lb/>guus, <reg norm="neque" type="simple">neq;</reg> etiam ab ipſis ſonus reſultare poterit, quia pro certo putandum eſt, <reg norm="ipſorum" type="context">ipſorũ</reg> <lb/>ſuperficies tam politas eſſe, ac lenas, vt nihil omnino aſperitatis, aut inæqualitatis <lb/>contineant. </s> + <s xml:id="s2294" xml:space="preserve">Vt exempli gratia, ſi aliquis duo ſpecula plana inuicem confricaret, nul <lb/>lum planè ſonum audiret, ſed ſi hoc faceret cum duabus ſuperficiebus a ſperis, <reg norm="ſonum" type="context">ſonũ</reg> <lb/>perſentiret, & tanto maiorem; </s> + <s xml:id="s2295" xml:space="preserve">quantò aſperiores eſſent dictæ ſuperficies, & qui vult <lb/>vtarcus lirę, ex corda ſonum eliciat, colophonia dictum arcum illinet, vt aſperiorem <lb/>reddat. </s> + <s xml:id="s2296" xml:space="preserve">Neceſſarium quoque eſt vt tremat ſiue trepidet corpus, quod <reg norm="ſonum" type="context">ſonũ</reg> edere <lb/>debet; </s> + <s xml:id="s2297" xml:space="preserve">Neque etiam abſque aere ſonus efficipotelt, quia aer ſonat ingrediendo <lb/>velociter ad implendum locum, vt non remaneat vacuus. </s> + <s xml:id="s2298" xml:space="preserve">Sed ſupponendo in æche <lb/>rea regione neque aerem eſſe, neque corpus aliquod fluidum, clarè patebit orbes <lb/>cœleſtes ex ſeſe nullum emittere ſonum. </s> + <s xml:id="s2299" xml:space="preserve">Idem affirmo de fricatione ſuperficiei con <lb/>cauæ infimi orbis lunaris cum conuexa materiæ à dicto orbe contentæ, ſuperioribus <lb/>rationibus fultus, vt etiam experientia à corpore aliquo fluido, quod in alio velociſ <lb/>ſimè moueretur deſumpta fretus, cuius corporis ſuperficies tamen lenis eſſet, à quo <lb/>ſonus 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 ſuum axem velociſſimè moueatur, nullum ſo-<lb/>num efficiet, quia nullam aeris partem extra ſuum <reg norm="locum" type="context">locũ</reg> impellit dum mouetur non <lb/>ſecundum totum, ſed ſecundum ſuas partes, quarum quælibet abſque reſiſtentia im-<lb/>mediatè ſubintrat locum alterius, abſque temporis interpoſitione. </s> + <s xml:id="s2302" xml:space="preserve">nec huiuſmodi <lb/>locum aliquo modo eadem materia dicti corporis, quod circunuoluitur: </s> + <s xml:id="s2303" xml:space="preserve">deſtitutum <lb/>dimittat. </s> + <s xml:id="s2304" xml:space="preserve">Sed ſi Pythagorici de alia quadam harmoniæ ſpecie ab ea, quæ eſt ſono-<lb/>rum, vt à diuerſis velocitatibus motuum, aut à diuerſis magnitudinibus aut diſtantiis, <lb/>aut ſtellarum influxibus intellexiſſent, rectè ſenſiſſent exparte, non autem omnino, <lb/>quia ea harmoniam efficere nequeunt, quæ ad <reg norm="inuicem" type="context">inuicẽ</reg> ſecundum interualla harmoni-<lb/>ca proportionata non ſunt, vt ſunt dupla, ſeſquial tera, ſeſquitertia, ſeſquiquarta, ſeſ-<lb/>quiquinta, ſupertripartientia quintas, <reg norm="ſuperbipartientia" type="context">ſuperbipartiẽtia</reg> tertias, & quę ab ijs <reg norm="dependent" type="context">dependẽt</reg> <lb/>ideſt coniuncta ſunt cum duplis; </s> + <s xml:id="s2305" xml:space="preserve">de conſonantijs loquendo. de diſſonantiis idem di <lb/>co, quæ harmonicis inſeruiunt modulationibus, vt <reg norm="ſeſquioctauum" type="context">ſeſquioctauũ</reg>, <reg norm="ſeſquinonum" type="context">ſeſquinonũ</reg>, ſeſqui <lb/><reg norm="quintundecimum" type="context context">quintũdecimũ</reg>, <reg norm="ſequiuigeſimunquartum" type="context context">ſequiuigeſimũquartũ</reg>, <reg norm="ſeſquioctuogeſimum" type="context">ſeſquioctuogeſimũ</reg>, & ſuperbipartiens vigeſi <lb/>masquintas. </s> + <s xml:id="s2306" xml:space="preserve"><reg norm="Verum" type="context">Verũ</reg> quidem eſt nonnulla harmonica interualla in aſpectibus <reg norm="comperta" type="context">cõperta</reg> <lb/>fuiſſe, vt Prolomeus oſtendit, & alii quoque aſſerunt. </s> + <s xml:id="s2307" xml:space="preserve">ineſt tamen huic rei nonnihil <lb/>difficultatis. </s> + <s xml:id="s2308" xml:space="preserve">vt exempli gratia, ſi ſubtrahamus diateſſaron extra diapaſon, remanet <lb/>diapente, & ſi à diapente ſubtrahamus ſemiditonum, 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"/> + conſonantiæ, eum habent reſpectum ad inuicem, quem habent diateſſaron, & dia <lb/>pente, quia quemadmodum ſemiditonum, & ditonum ſimul coniuncta, compo <lb/>nunt diapente, ſic diateſſaron, & diapente ſimul vnita componunt diapaſon; </s> + <s xml:id="s2309" xml:space="preserve">& <reg norm="quem" type="context">quẽ</reg> <lb/>admodum terminus, qui diuidit diapaſon in <reg norm="diateſſaron" type="context">diateſſarõ</reg>, & diapente, eſt mediator ha@ <lb/>monicus inter extrema diapaſon diuiſi, ſic etiam terminus, qui diuidit <reg norm="diapente" type="context">diapẽte</reg> in ſe-<lb/>miditonum, & ditonum, mediator eſt harmonicus inter extrema ipſius diapente diui <lb/>ſi) ſubtrahendo deinde à diapaſon ſemiditonum remanet exachordum maius, & ab <lb/>eodem diapaſon ſubtrahendo ditonum remanet exachordum minus, quę <reg norm="quidem" type="context">quidẽ</reg> <reg norm="non" type="context">nõ</reg> <lb/>accidunt aſpectuum circulo, quia ſubtrahendo aſpectum quadratum ab oppoſito, <lb/>remanet aliud quadratum, & ſubtrahendo ſextilem à trino remanet quoque alius <lb/>ſextilis. </s> + <s xml:id="s2310" xml:space="preserve">Quòd autem attinet ad motus, ad magnitudines, ad diſtantias, & ad influ-<lb/>xus, nihil eſt, quod hiſce proportionibus conueniat, ſed quia hæc omnia <reg norm="dependent" type="context">depẽdent</reg> <lb/>ab <reg norm="infinita" type="context">ĩfinita</reg>, & diuina <reg norm="prouidentia" type="simple context">ꝓuidẽtia</reg> Dei, neceſſariò fit vt iſtæ velocitates, eæ magnitudines, <lb/>diſtantiæ, & influxus, talem ordinem, & reſpectum inter ſeipſa, & vniuerſum <reg norm="habeant" type="context">habeãt</reg>, <lb/>qualis perfectiſſimus ſit.</s> + </p> + </div> + <div type="section" level="3" n="34"> + <head xml:id="head309" style="it" xml:space="preserve">Deraro et denſo nonnulla, minus diligenter à Peripateticis <lb/>perpenſa.</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 diſputauerunt, quia hoc naſciturà condenſatione hali <lb/>tus, quę ab ambiente frigore fit. </s> + <s xml:id="s2313" xml:space="preserve">quia halitus is abore, aut naſo animalis <reg norm="exiens" type="context">exiẽs</reg> <lb/>non eſt purus aer attractus primò, ſed mixtus eſt cum quodam vapore excrementi-<lb/>tio, & ſubtili, quo ſemper ab ea parte <reg norm="euacuatur" type="simple">euacuat̃</reg> corpus, qui ſtatim ab aere frigido cir-<lb/>cundatur, & denſatur, quam ob cauſam ab ipſo ea luminis pars reflectitur, quæ eum <lb/>penetrare non poteſt, quod in hypocauſtis, <reg norm="huiuſmodique" type="simple">huiuſmodiq́;</reg> calidis locis non fit. </s> + <s xml:id="s2314" xml:space="preserve">Idem <lb/>exemplo ab aqua ſtatim à ciſternis, aut profundis puteis in hyeme extracta compro <lb/>bari poteſt, quia tunc temporis, huiuſmodi aqua, cum magis calida ſit, quàm fri-<lb/>gida, emittit vaporem, qui facillimè videtur, ob rationem iam dictam, quod <lb/>in æſtate non cernitur in aqua, etſi 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 antiperiſtaſi deſumptam citarunt iidem ad inquirendum, <lb/>cur aqua ſubterranea magis calida, aut minus frigida, hyberno tempore, quàm ea, <lb/>quæ eſt ſupra terram ſit, vana eſt, quia hoc non aliunde fit, quàm ab eo, <reg norm="quod" type="simple">ꝙ</reg> terræ por-<lb/>ri à frigoris ſiccitate ſint clauſi, vnde vapores & exalationes non tam facilè exire poſ <lb/>ſunt. </s> + <s xml:id="s2316" xml:space="preserve">quamobrem calefiunt ſubterraneæ partes. </s> + <s xml:id="s2317" xml:space="preserve">Fimum, fœnum, frumentum hac in <lb/>re ſunt nobis exemplo, in quibus ſępiſſimè viſum eſt ignem accendi.</s> + </p> + <p> + <s xml:id="s2318" xml:space="preserve">Priore illa quoque ratione de antiperiſtaſi dicta, volunt philoſophi maiorem ca-<lb/>liditatem hyęme, quàm ęſtate in animalium ſtomacho contineri, non animaduerten <lb/>tes ſiccitatem, frigiditatis partes ſuperficiales corporis, <reg norm="reſtringentem" type="context">reſtringẽtem</reg>, ſanguinem ver <lb/>ſus originem ſuam impellere, qui in eo loco copioſior cum ſit, eas partes tunc tem-<lb/>poris calefacit magis.</s> + </p> + <p> + <s xml:id="s2319" xml:space="preserve">Neque etiam ijdem nouerunt cauſam, vnde fiat, ut in æſtate impleto vaſe vitreo, <lb/>aut argenteo, aut ex materia non poroſa conſtante, aqua frigida, vas ſudet, 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 niſi in calidis locis euenit, quem ſudorem, dicebantipſi, eſſe <lb/>eandem aquam, quæ per porros vaſis exiret, quod falſiſſimum eſt, quia ſi per porros <lb/>aqua frigida exiret multò magis exiret calida, cum ſubtilior ſit, & ad penetrandum <lb/>aptior. </s> + <s xml:id="s2320" xml:space="preserve">Sed hoc non aliunde oritur, quàm à condenſatione aeris vas circundantis, <lb/>cauſata à frigiditate vaſis refrigerati ab aqua, quemadmodum tempore hyberno <lb/>clarè videmus mane ſuperficies interiores vitri feneſtrarum ſudare, quia <reg norm="extrinſecum" type="context">extrinſecũ</reg> <lb/>frigus refrigerando vitrum, intrinſecum aerem ſibi contiguum congelat.</s> + </p> + <p> + <s xml:id="s2321" xml:space="preserve">Neque ſilentio inuoluendum eſt, nec Ariſtotelem, neque alium ex ſuis fautoribus <lb/>animaduertiſſe denſum, & rarum eſſe cauſam ventorum. </s> + <s xml:id="s2322" xml:space="preserve"><reg norm="Rarum" type="context">Rarũ</reg> autem & denſum, me <lb/>diante calore & frigore fit, & ſi à partibus, in omogeneis, licet <reg norm="argumentari" type="context">argumẽtari</reg>, de toto <lb/>deducat conſequentiam qui velit, obſeruans in calidis æſtatis diebus, dum aliqua nu <lb/>becula ad Solem cooperiendum incedit, ibi ſtatim agitationem aeris ſentiri; </s> + <s xml:id="s2323" xml:space="preserve">ea verò <lb/>nubecula prætergreſſa cum fuerit, & in ea parte, aer ad priſtinam raritatem cauſa-<lb/>tam à calore Solis redierit, quieſcit; </s> + <s xml:id="s2324" xml:space="preserve">huiuſmodi autem aeris agitatio, à nulla certè ex <lb/>halatione proficiſcitur, ſed à motu ſolum locali, quem dum condenſatur, facit. </s> + <s xml:id="s2325" xml:space="preserve">Om <lb/>ne denſum natura ſua frigidum eſt; </s> + <s xml:id="s2326" xml:space="preserve">omne rarum calidum, & è conuerſo. </s> + <s xml:id="s2327" xml:space="preserve">Et frigida <lb/>aura, quæ à flabellis cauſatur, non ſolum à nouo aere qui nos tangit, ſed etiam à den-<lb/>ſo, quod in agitatione eiuſdem aeris fit, naſcitur.</s> + </p> + <p> + <s xml:id="s2328" xml:space="preserve">Cum autem de raritate & denſitate diſputationem ſuſceperim, non ſine ratione <lb/>mihi <reg norm="videtur" type="simple">videt̃</reg> <reg norm="illorum" type="context">illorũ</reg> <reg norm="opinionem" type="context">opinionẽ</reg> <reg norm="explodendam" type="context context">explodẽdã</reg> eſſe, qui Lunę maculas <reg norm="non" type="context">nõ</reg> aliud eſſe dixerunt, <lb/>quàm aliquas partes rariores aliis eiuſdem Lunæ partibus, non obſeruantes rarum, & <lb/>denſum, proportionabilia lumini, quod ab huiuſmodi corporibus reflectitur, non eſ<lb/>ſe. </s> + <s xml:id="s2329" xml:space="preserve">quia corpus aliquod rarum aliquando aptum erit ad reflectendum maius lumen, <lb/>quàm corpus minus rarum ut manifeſtè 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, ideſt, eas Lunę partes, den <lb/>ſiores eſſe; </s> + <s xml:id="s2332" xml:space="preserve">neque unquam aliquis fuit qui de diaphano, aut opaco mentionem fece <lb/>rit, quia melius eſt credere, eas partes diaphanas, ſiue perſpicuas magis eſſe, quàm a-<lb/>lias, quę per aliquod ſpatium, ſolis radio ingreſſum permittant, & alię partes <reg norm="cum" type="context">cũ</reg> ſint <lb/>opacæ ipſum à ſuperficie reflectant. </s> + <s xml:id="s2333" xml:space="preserve">diuerſa tamen ratione à ſpeculo, cum in pleni-<lb/>lunio tota ferè Lunę pars illuminata cernatur, quamuis dictum lumen extenſiuè & in <lb/>tenſiuè ſit minus eo, quod ipſa in nouilunio recipit. </s> + <s xml:id="s2334" xml:space="preserve">Indignum autem mihi videtur <lb/>ijs reſpondere, qui dixerunt huiuſmodi maculas, terræ vmbras exiſtere, cum craſſiſſi-<lb/>mæ ignorantiæ tenebris ſint circunfuſi, vt <reg norm="etiam" type="context">etiã</reg> fuit Cornelius Agrippa, qui primo de <lb/>occulta philoſophia dicens ſe noſſe modum <reg norm="quendam" type="context">quendã</reg> naturalem à Pythagora inuen-<lb/>tum, quo in Luna id totum, quod ipſe ſuper ſpeculum ſcripſiſſet, videretur. </s> + <s xml:id="s2335" xml:space="preserve">oſtendit <lb/>manifeſtè ſe ignorare luminum <reg norm="vmbrarumque" type="simple">vmbrarumq́;</reg> naturam. </s> + <s xml:id="s2336" xml:space="preserve">quia nulla vnquam vmbra ge <lb/>nerari poteſt à corpore, quod aut opacum non ſit, 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">adũ-<lb/>brare</reg> poteſt, niſi opacum illud in linea recta ſitum obtineat, quæ inter lucidum & il <lb/>luminatum extenditur. </s> + <s xml:id="s2338" xml:space="preserve">Neque etiam reſpondebimus ijs, qui ſentiunt quotieſcun-<lb/>que nulla eſſet terra, ſed totus hic globus eſſet aqua, toties non futuram eclipſim lu-<lb/>narem, ratione diaphaneitatis aquæ. </s> + <s xml:id="s2339" xml:space="preserve">Quod <reg norm="falſiſſimum" type="context">falſiſſimũ</reg> eſt, quia omne corpus <reg norm="ſphae- ricum" type="simple">ſphę-<lb/>ricum</reg> quantumuis diaphanum ſit, dummodo ſit denſius aere, luminoſos radios re-<lb/>frangit, & eos ad inuicem interſecare facit, qui deinde vltra interſectionis <reg norm="punctum" type="context">punctũ</reg> di-<lb/>ſgregantur, ita vt amittant illuminationis actum. </s> + <s xml:id="s2340" xml:space="preserve">Adde <reg norm="quod" type="simple">ꝙ</reg> etſi huiuſmodi corpus <lb/>aqueum, ſphęricum non eſſet, ſed cubicum, illud ſuper <reg norm="aliquam" type="context">aliquã</reg> ex eius ſuperficiebus <lb/>ad angulos rectos radius ſolaris 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 ſit denſius aere acquirit, nec totus radius vnquam dictum corpus <lb/>ingr editur, cum ab eius ſuperficie magna pars reflectatur. </s> + <s xml:id="s2341" xml:space="preserve">Reſiſtit ergo huiuſmo-<lb/>di corpus lumini, & quantò magis ſpiſſum aut profundum exiſtit, tantò, validius reſi <lb/>ſtit. </s> + <s xml:id="s2342" xml:space="preserve">Habemus huius rei teſtes, piſcatores vnionum, in ijs mundi partibus, quæ pau-<lb/>cis ab hinc annis Hiſpanorum opera nobis innotuerunt, qui aſfirmanu ad maris <reg norm="vſque" type="simple">vſq;</reg> <lb/>fundum lumen Solis non peruenire.</s> + </p> + <p> + <s xml:id="s2343" xml:space="preserve">Immediata ratio, cur nebulę in ijs locis in quibus <reg norm="conſpiciuntur" type="context">cõſpiciuntur</reg> permaneant, & <reg norm="uunc" type="context">uũc</reg> <lb/>altiores, nunc vero depręſſiores cernantur, non ea eſt, quam Ariſtoteles cap .3. lib. 1. <lb/>metheororum proponit, ſed inde oritur, quòd ſint eædem denſiores ea parte aeris, <lb/>quæ ipſis ſupereminet, & rariores e a, quæ ijſdem ſubiacet. </s> + <s xml:id="s2344" xml:space="preserve">Quòd autem alicuius cor <lb/>poris denſitas maior ea, quæ eſt medij, in quo reperitur, cauſa ſit, vt ipſum corpus de <lb/>ſcendat, & maior raritas eiuſdem corporis, ea, quæ eſt quoque me dij, efficiat, vt di-<lb/>ctum corpus aſcendat, iam Archimedes in lib. de inſidentibus aquæ docuit.</s> + </p> + <p> + <s xml:id="s2345" xml:space="preserve"><reg norm="Rectiſſimem" type="context">Rectiſſimẽ</reg> inſtituit natura, vt corpora denſiora verſus loca anguſtiora, & minora <lb/>(intelligendo ea loca orbicularis figuræ) quæ ad centrum propius accedunt, & rario <lb/>ra ad ampliora loca, & maius ſpatium occupantia, ſeſe reciperent. </s> + <s xml:id="s2346" xml:space="preserve">tum quia eadem <lb/>quantitas materiæ condenſatæ, eget minori loco quam rarefacta, <reg norm="cum" type="context">cũ</reg> etiani, quia <reg norm="cum" type="context">cũ</reg> <lb/>corpus denſum non ita ad velocitatem motus localis, vt rarum, idoneum ſit, ad eas <lb/>partes accedat, quæ motibus tardioribus magis ſunt aptæ, corpora <reg norm="autem" type="context">autẽ</reg> rara ad eas, <lb/>quæ velocioribus motibus ſunt aptiores ſeſe transferant. </s> + <s xml:id="s2347" xml:space="preserve">præterquam <reg norm="quod" type="simple">ꝙ</reg> reuera ap-<lb/>pareat pro maiori parte, corpus magis denſum, minus diaphanum; </s> + <s xml:id="s2348" xml:space="preserve">aut magis <reg norm="opacum" type="context">opacũ</reg> <lb/>futurum, quàm rarum, licet ſæpiſſimè videamus contrarium, vt ſuperius innuimus. <lb/></s> + <s xml:id="s2349" xml:space="preserve">eſt tamen naturale <reg norm="proportionatumque" type="simple">proportionatumq́;</reg> magis opacum denſo, & diaphanum raro, <lb/>quàm è contra. </s> + <s xml:id="s2350" xml:space="preserve">Quamobrem ſumma ratione inducta natura voluit, vt corpora ma <lb/>gis opaca, aut minus diaphana, magis vicina centro colligantur, vt ſpatium, quòd re <lb/>manet, abſque vllo impedimento à radijs ſolaribus penetrari poſſit. </s> + <s xml:id="s2351" xml:space="preserve">Tres autem eæ <lb/>cauſæ, quas hoc loco poſui, propriæ ſunt, immediatæ, & per ſe, ex quibus fit, vt corpo <lb/>ra denſiora deſcendant, & rariora aſcendant in mediis minus denſis, aut minus raris <lb/>dictorum corporum, quæ à nemine, <reg norm="quod" type="simple">ꝙ</reg> ſciam, hucuſque propoſitæ fuerunt.</s> + </p> + <p> + <s xml:id="s2352" xml:space="preserve">Qui autem aſſerunt cucurbitæ, quam apponunt chirurgi, effectum ex eo naſci, <reg norm="quod" type="simple">ꝙ</reg> <lb/>calidi ſit attrahere, valdè aberrant à vero quia hoc, non niſi à raro, & à denſo imme-<lb/>diatè, à calido & frigido cauſatis efficitur, quia aer in cucurbita rarefactus à calore <lb/>& per conſequens dilatatus, ſtatim vt à dicto calore deſeritur, iterum condenſatur & <lb/>tantò citius, quantò aer ambiens frigidior exiſtet, & quia eadem materia cum con-<lb/>denſata fuerit minorem ſemper occupat locum, reſtringens igitur ſeſe in cucurbi-<lb/>ta aer dum condenſatur, neceſſariò fit, ne ulla, ſcilicet pars vacua remaneat, <reg norm="quod" type="simple">ꝙ</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 ſit corpus experiri poſſumus, ſi <reg norm="eam" type="context">eã</reg> ad ignem pri-<lb/>mò calefactam, deinde <reg norm="cum" type="context">cũ</reg> ore in amplo aliquo cyatho, aut alio vaſe. </s> + <s xml:id="s2354" xml:space="preserve">vino, aut aqua <lb/>pleno vbi videbimus huiuſmodi liquorem ſtatim ſurſum ferri, quia dum calefit am-<lb/>phora, rarefit quoque aer qui in ea continetur, & quia rateſcit 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 remanſerit, iterum condenſatur ob defectum caloris, reſtringitur, <lb/><reg norm="minorique" type="simple">minoriq́;</reg> indiget loco; </s> + <s xml:id="s2357" xml:space="preserve">Quod cum ita ſe habeat, neceſſarium eſt, 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 ſi corpus admodum non erit fluxile, aut humidum, ita vt ingre <lb/>di amphoram poſſit ita amphorę 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">cũ</reg> admiratione <reg norm="videmus" type="simple">videmꝰ</reg> fragile vas <reg norm="vitreum" type="context">vitreũ</reg> <reg norm="magnum" type="context">magnũ</reg>, & graue <reg norm="lapideum" type="context">lapideũ</reg> corpus eleuare. <lb/></s> + <s xml:id="s2359" xml:space="preserve">Sed vt ad denſum & ad rarum redeamus, mihi videtur frigidum eſſe conſequentem <lb/>qualitatem denſi, & calidum rari, quia quæuis res dum calefit, rarefit, & quælibet <lb/>materia dum refrigeratur, ſimul condenſatur. </s> + <s xml:id="s2360" xml:space="preserve">Qua ratione fit, vt terra frigidior <lb/>ſit aqua, & ignis calidior ſit aere.</s> + </p> + <p> + <s xml:id="s2361" xml:space="preserve">Nec propriè locutus eſt Ariſtoteles .9. & .10. capite primi lib. & ſecundo ſecundi <lb/>metheororum cum dixerit <reg norm="calorem" type="context">calorẽ</reg> Solis eum eſſe, qui ſurſum humores, <reg norm="vaporesque" type="simple">vaporesq́;</reg> eue <lb/>hat, quia Sol nil aliud facit, quam calefacere, cuius caloris ratione, ea materia rarefit, <lb/>& ob rarefactionem leuior facta aſcendit, non quia ſurſum à Sole feratur.</s> + </p> + <p> + <s xml:id="s2362" xml:space="preserve">Quę ſubſequuntur, cum raro ac denſo ſimbolum habere videntur. </s> + <s xml:id="s2363" xml:space="preserve">cum raro, ſcili-<lb/>cet calidum, humidum, leue, ſublime, diaphanum, lumen, clarum, lux, <reg norm="album" type="context">albũ</reg>, dies, mo-<lb/>tus, velox, ſimplex, diſgregatum, molle, lene, acutum, ſubtile, coctum, <reg norm="ſpaciosum" type="context">ſpaciosũ</reg>, <lb/>dulce, voluptas, audacia, lætitia, liberalitas, veritas, induſtria, amor, miſericordia, hu-<lb/>manitas, ſanitas, vita, & iis ſimilia. </s> + <s xml:id="s2364" xml:space="preserve">Cum denſo verò frigidum, ſiccum, graue, imum, <lb/>opacum, vmbra, obſcurum, tenebræ, nigrum, nox, quies, tardum, mixtum, congrega <lb/>tum, durum, aſperum, ob tuſum, craſſum, crudum, anguſtum, amarum, dolor, cimor, <lb/><reg norm="melancholia" type="context">melãcholia</reg>, auaritia, mendacium, inertia, odium, crudelitas, feritas, infirmitas, mors, <lb/>& ijs ſimilia.</s> + </p> + <p> + <s xml:id="s2365" xml:space="preserve">Verum eſt quod ea ratio, qua Ariſtoteles ait aerem humidum eſſe, parui eſt mo-<lb/>menti, quia ſimiliter deigne inferri poſſet, qui facilius à termino alieno, <reg norm="quam" type="context">quã</reg> aer, aut <lb/>aqua terminari poteſt.</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 diſentiente <lb/>Ariſtotele.</head> + <head xml:id="head312" xml:space="preserve">CAP. XXXV.</head> + <p> + <s xml:id="s2366" xml:space="preserve">SEd vt ad <reg norm="Ariſtotelem" type="context">Ariſtotelẽ</reg> redeamus, rectè dicere non poteſt motum rectum ad <reg norm="curuum" type="context">curuũ</reg> <lb/>comparabilem non eſſe .4. cap. lib. 7. phyſicorum, 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 ſemi-<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 ſuis lateribus circundantibus angulum rectum æquale ſemidiametro dicti <lb/>circuli, aliud quoque latus ipſum angulum rectum circundans, ex neceſſitate, <reg norm="circun- ferentiæ" type="context">circũ-<lb/>ferentiæ</reg> dicti circuli habebit æquale. </s> + <s xml:id="s2369" xml:space="preserve">Poteſt igitur dari vna quædam recta linea <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg> circulari contra Ariſtotelis opinionem, qui non benè reuocauit in mentem, <lb/>quod ſcripſit de relatiuis, cum dixit quadraturam circuli poſſe quidem dari, etſi <reg norm="tunc" type="context">tũc</reg> <lb/><reg norm="temporis" type="wordlist">tꝑis</reg> de ea <reg norm="non" type="context">nõ</reg> <reg norm="haberetur" type="simple">haberet̃</reg> ſcientia. </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>, poteſt <reg norm="etiam" type="context">etiã</reg> dari vna <lb/>recta linea ęqualis circunferentiæ <reg norm="eiuſdem" type="context">eiuſdẽ</reg> circuli, ob rationes <reg norm="iam" type="context">iã</reg> dictas. </s> + <s xml:id="s2371" xml:space="preserve">Sed ſi Ariſt. <lb/>dixiſſet, circularem corporum cęleſtium motum, comparabilem non eſſerecto cor-<lb/>porum elementarium, verum dixiſſet, non quia eorum alter circularis, alter ve-<lb/>rò ſit rectus, ſed quia cœleſtis regularis ſit, neque modò tardus, modò velox, <lb/>ſed vnam ſemper & eandem velocitatem retinens, <reg norm="motus" type="simple">motꝰ</reg> <reg norm="autem" type="wordlist">aũt</reg>, qui eſt <reg norm="corporum" type="context">corporũ</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">ꝙ</reg> nunquam fuit neque ſit futurus aliquis <reg norm="horum" type="context">horũ</reg> <lb/>rectorum, qui naturales dicuntur, qui tam velociter moueatur, ut motus cœli, quia ſi <lb/>voluerimus conſiderare motum diurnum .24. horarum, ſecundum opinionem com-<lb/>munem, reperiemus calculando, Lunam in quadraturis cum Sole, dum inuenitur in <lb/>æquatore, ſingulis horarum minutis moueri per .500. milliaria Italica vel circa, & in <lb/>coniunctionibus, & oppoſitionibus ipſius Solis .1000. vel circa, & Solem tempore <reg norm="ae- quinoctiorum" type="simple context">ę-<lb/>quinoctiorũ</reg> .18000. & <reg norm="Saturnum" type="context">Saturnũ</reg> circa æquatoris <reg norm="ſitum" type="context">ſitũ</reg> .260000. & <reg norm="amplius" type="simple">ampliꝰ</reg> de ſtellis <reg norm="autem" type="wordlist">aũt</reg> <lb/>fixis circa æquatorem poſitis quiuis cogitet; </s> + <s xml:id="s2372" xml:space="preserve">quod reuera diffi cillimum quibuſdam <lb/>videbitur, quod quidem non occurrit <reg norm="ſecundum" type="context">ſecũdum</reg> pulcherrimam Ariſtarchi ſamij opi-<lb/>nionem, diuinitus à Nicolao Copernico expreſſam, contra quam nil planè valent <lb/>rationes ab Ariſtotele; </s> + <s xml:id="s2373" xml:space="preserve">neque etiam à Ptolomeo propoſitę. </s> + <s xml:id="s2374" xml:space="preserve">Motu verò proprio, quo <lb/>libet horę 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> eſt, aut op <lb/>poſita 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, & ſi <reg norm="moueretur" type="simple">moueret̃</reg> motu recto hoc <reg norm="mon" type="context">mõ</reg>, <lb/>& velocius etiam corpore cęleſti, non obſeruans <reg norm="tamen" type="wordlist">tamẽ</reg> uniformitatem, ut dictum cœ <lb/>leſte facit, cum eodem nullo modo comparari poſſet, quia rectus dictus naturalis, <lb/>ſuam ſemper velocitatem adauget, ob continuam impreſſionem, quam recipit à cau <lb/>ſa perpetuò coniuncta cum ipſo corpore, quę eſt propenſio illa naturalis eundi bre-<lb/>uiori quadam via ad locum ſuum, ita vt etiam ſi dictum corpus elementare à motu <lb/>tardiore ad velociorem, ſuperare poſſet <reg norm="motum" type="context">motũ</reg> alicuius corporis cęleſtis, ij duo motus <lb/>interſecarent ſeſe in vno ſolo <reg norm="puncto" type="context">pũcto</reg>, quod diuidi <reg norm="diſtribuique" type="simple">diſtribuiq́;</reg> in partes nequiret, ideſt <lb/>non niſi in vno ſolo temporis inſtanti redderentur æquales, vt ita dicam. </s> + <s xml:id="s2377" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> <reg norm="ſolum" type="context">ſolũ</reg> <lb/>loquor de circulari cœleſti cum recto elementari, ſed de qualibet alia motuum ſpe-<lb/>cie, ſiue ſint ambo recti, ſiue 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 ſufficienter exploſam fuiſſe ab Ariſtotele opinionem cre-<lb/>dentium plures mundos exiſtere.</head> + <head xml:id="head314" xml:space="preserve">CAP. XXXVI.</head> + <p> + <s xml:id="s2378" xml:space="preserve">MAior ratio, qua Ariſtoteles eorum opinionem, qui plures eſſe mundos dixe <lb/>runt, refutare nititur, in eo conſiſtit, quod is credat partes terræ, quæ alijs <lb/>mundis aſſignarentur, ad huius mundi centrum inclinationem habere, & ſic ignem <lb/>illorum, propenſionem habiturum ad circunferentiam huius.</s> + </p> + <p> + <s xml:id="s2379" xml:space="preserve">Quæ certè ratio tam debilis eſt, vt per ſe cadat, non conſideransipſe, quòd ſi <lb/>eſſent dicti mundi, eorum quilibet ſuum proprium centrum, <reg norm="ſuamque" type="simple">ſuamq́;</reg> propriam cir-<lb/>cunferentiam haberet, <reg norm="terrasque" type="simple">terrasq́;</reg> & ignes haberent inclinationem ad centra circunfe-<lb/><reg norm="rentiasque" type="simple">rentiasq́;</reg> ſuorum mundorum, abſque eo, <reg norm="quod" type="simple">ꝙ</reg> vna terra, alterius centrum appeteret; </s> + <s xml:id="s2380" xml:space="preserve">vt <lb/>exempli gratia, ſi doctiſſimi Ariſtarchi opinio eſt vera, rationi <reg norm="quoque" type="simple">quoq;</reg> conſentaneum <lb/>erit maximè, vt quod Lunæ contingit, cuilibet <reg norm="etiam" type="context">etiã</reg> ex aliis quinque planetis eue-<lb/>niat, ideſt, vt quemadmodum Luna ſuorum epicyclorum ope <reg norm="circum" type="context">circũ</reg> terram voluitur, <lb/>quaſi per circunferentiam alterius cuiuſdam epicycli, in quo terra ſit inſtar centri <lb/>naturalis (ideſt ſit 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 ſui 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, ſitum habens, voluantur; </s> + <s xml:id="s2382" xml:space="preserve">quod quidem corpus, & aliquem quoque ha-<lb/>beat motum circa ſuum axem, ſit opacum, ijs conditionibus, quæ terræ ſunt ſimi-<lb/>les, præditum exiſtat, & in dicto epyciclo ſint res ſimiles iſtis lunaribus.</s> + </p> + </div> + <div type="section" level="3" n="37"> + <head xml:id="head315" style="it" xml:space="preserve">Anrectè loquutus ſit Phyloſopbus de extenſione luminis <lb/>per uacuum.</head> + <head xml:id="head316" xml:space="preserve">CAP. XXXVII.</head> + <p> + <s xml:id="s2383" xml:space="preserve">ARriſtoteles ſecundo lib. de anima ſentit <reg norm="quod" type="simple">ꝙ</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 veriſimile <reg norm="non" type="context">nõ</reg> eſt; </s> + <s xml:id="s2385" xml:space="preserve"><reg norm="quia" type="simple">ꝗa</reg> <reg norm="quemadmo dum" type="wordlist">quẽadmo<lb/>dum</reg> quantò rarius eſt aliquod corpus, tanto aptius eſt vt diaphanum exiſtat; </s> + <s xml:id="s2386" xml:space="preserve">& <reg norm="quan- tò" type="context">quã-<lb/>tò</reg> rarius eſt dictum corpus, tantò minorem quantitatem materiæ contineat; </s> + <s xml:id="s2387" xml:space="preserve">ſic <reg norm="quam" type="context">quã</reg> <lb/>tò magis diaphanum eſt, cum ex perexigua materia conſtet, tantò magis liber tran-<lb/>ſitus luminis patet; </s> + <s xml:id="s2388" xml:space="preserve">Vnde quantò minor quantitas materiæ erit in dicto ſpatio, tan <lb/>tò nitidius pertranſibit lumen. </s> + <s xml:id="s2389" xml:space="preserve">Sequitur ergo, quòd vbi nulla eſſet materia, totum <lb/>lumen libere tranſiret. </s> + <s xml:id="s2390" xml:space="preserve">Color cęruleus quem videmus in profunditate aquæ, & ae-<lb/>ris, color eſt a quæ & aeris, qui denotat reſiſtentiam factam ab aere & ab aqua ipſi lu <lb/>mini; </s> + <s xml:id="s2391" xml:space="preserve">Quod quidem lumen ubi corpus aliquod non eſſet, minime reflecteretur, ſed <lb/>abſque vllo impedimento rectà tranſiret.</s> + </p> + </div> + <div type="section" level="3" n="38"> + <head xml:id="head317" style="it" xml:space="preserve">An rectè phyloſophiœ<unsure/> penus Ariſtoteles ſenſerit 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">ARiſtoteles .8. cap. lib. 3. de cœlo, diſputans contra antiquos de elementorum <lb/>figuris, ait pyramidem implere poſſe locum corporeum. </s> + <s xml:id="s2393" xml:space="preserve">quod verum non <lb/>eſt. </s> + <s xml:id="s2394" xml:space="preserve">Cubus quidem id facit ab .8. enim cubis perfectè impletur locus, ſed non <lb/>item .12. pyramides, ut Ariſtoteles ſenſit (ideſt ſex ſuper aliquam exagonam figu-<lb/>ram ſuperficialem & ſexſub eadem) id præſtant, cum potius maius vacuum rema-<lb/>neatad quamlibet partium ſupra, & infra, quam plenum. </s> + <s xml:id="s2395" xml:space="preserve">Rectius Ariſtoteles <lb/>egiſſet, ſi probaſſet ratione immobilitatis conuenire pyramidem terræ, quam cu-<lb/>bum. </s> + <s xml:id="s2396" xml:space="preserve">quamuis, de horum corporum altero, ſit ſtultum 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 eſt illud, quod ita ſe habet, vt multis rationibus probari poteſt, quarum vna <lb/>hæc nobis ſufficiet. </s> + <s xml:id="s2398" xml:space="preserve">Scimus iam ex communi conceptu corpus ſphęricum eſſe ma-<lb/>gis volubile, <reg norm="inſtabileque" type="simple">inſtabileq́;</reg>, quàm alia ſint. </s> + <s xml:id="s2399" xml:space="preserve">Illud ergo corpus, cuius figura ad ſphæri-<lb/>cam magis accedet, ad uoluendum, & ad mouendum facilius erit quouis alio, quod <lb/>æqualis ſit quantitatis, & ſibi omogeneum materia, vt exempli gratia corpus .20. ba <lb/>ſium ad voluendum, & ad mouendum promptius erit eo, quod ex .12. conſtat, & id, <lb/>quod eſt .12. eo, quod eſt .8. & id, quod eſt .8. eo, quod eſt .6. & id, quod eſt .6. vt <lb/>cubus eſt, eo, quod eſt .4. cuiuſmodi eſt pyramidale. </s> + <s xml:id="s2400" xml:space="preserve">Huc accedit, quòd pyrami-<lb/>dale corpus aliam conditionem habet, quàm 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 ſit, cubicum autem econtrà ſit alterabile vndequaque, <reg norm="ſuaque" type="simple">ſuaq́;</reg> quadrata in <reg norm="rhum- bos" type="context">rhũ-<lb/>bos</reg> mutare poſſit, iiſdem exiſtentibus lateribus.</s> + </p> + </div> + <div type="section" level="3" n="39"> + <head xml:id="head319" style="it" xml:space="preserve">Examinatur quam ualida ſit ratio Aristotelis de <lb/>inalterabilitate Cœli.</head> + <head xml:id="head320" xml:space="preserve">CAP. XXXIX.</head> + <p> + <s xml:id="s2401" xml:space="preserve">ARriſtoteles textu .22. primi lib. de Cœlo ita inquit.</s> + </p> + <p> + <s xml:id="s2402" xml:space="preserve">Accidit autem, & hoc per ſenſum ſufficienter, quo ad humanam dixiſſe fi-<lb/>dem, & omni pręterito tempore ſecundum traditam inuicem memoriam, nihil vi-<lb/>detur tranſmutatum neque ſecundum totum vltimum cęlum, neque ſecundum par-<lb/>tem ipſius propriam vllam.</s> + </p> + <p> + <s xml:id="s2403" xml:space="preserve">Hoc autem in loco Ariſto. non conſiderauit, <reg norm="quod" type="simple">ꝙ</reg> ſimiliter de terra dici poſſet, quan <lb/>do ipſa ita eminus proſpiceretur, imo abſque dubio putandum eſt, <reg norm="quod" type="simple">ꝙ</reg> ſi terra luce So <lb/>lis prædita eſſet, & aliquis ipſam ab octauo orbe vellet videre, nullo pacto cerne-<lb/>ret, cum ſidera illa quæ primæ magnitudinis vocantur, & quæ pluſquam centies ma <lb/>iora ip<unsure/>ſa terra putantur non niſi 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 ſint. <lb/></s> + <s xml:id="s2405" xml:space="preserve">Animaduertimus tamen permultos ſumma <reg norm="cum" type="context">cũ</reg> <lb/>difficultate <reg norm="eorum" type="context">eorũ</reg> demonstr ationes percipere. </s> + <s xml:id="s2406" xml:space="preserve">Prœ<unsure/>-<lb/>cipuè ubi quint a, aut ſeptima deffinitiones eiuſ-<lb/>dem libri neceſſariœ<unsure/> ſunt. </s> + <s xml:id="s2407" xml:space="preserve">Illœ enim adeo obſcurœ <lb/>uidentur, ut longè facilius admißuri ſint hœc no-<lb/>ſtra poſtulat at anquam clarior a. </s> + <s xml:id="s2408" xml:space="preserve">At que etiam tanquam intellectui <lb/>commodiora, quam ſit illud quintum <reg norm="idemque" type="simple">idemq́ꝫ</reg> ultimum postulatum <lb/>eiuſdem in primo libro poſitum, de line a duas alias ſecante. </s> + <s xml:id="s2409" xml:space="preserve">Quan-<lb/>doquidem <reg norm="ijs" type="lig">ijs</reg> noſtris postulatis admißis, ſequentia 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 compoſita ex æquali numero partium æqualium, ſunt inuicem <lb/>æqualia.</s> + </p> + <p> + <s xml:id="s2411" xml:space="preserve">Vtſi quis diceret omnes proportiones quæ <reg norm="compoſitæ" type="context">cõpoſitæ</reg> ſunt ex æquali numero alia-<lb/>rum proportionum inuicem æqualium, ſunt etiam inuicem æquales, quod Eucli-<lb/>des conatur demonſtrare 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> ſi 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 conuerſo ſi æqualibus æqualia addas compoſita erunt inuicem æqualia.</s> + </p> + <p> + <s xml:id="s2414" xml:space="preserve">Quod in ipſis proportionibus hoc loco ſemper intelligendum eſt.</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">Quę est εuclidis ſeptima propoſitio.</head> + <p> + <s xml:id="s2415" xml:space="preserve"><emph style="sc">Qvod</emph> ſi fuerint plures termini æquales inuicem, ratio ſeu proportio vnius ip-<lb/>ſorum ad alium tertium terminum maiorem, minoremúe, ſed eiuſdem generis, erit <lb/>cadem quæ cuiuſuis alterius termini ad eundem tertium. </s> + <s xml:id="s2416" xml:space="preserve">Et è conuerſo, quæ fuerit <lb/>proportio tertij termini ad vnum prædictorum æqualium, eadem erit, ſpecie, 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 propoſitio.</head> + <p> + <s xml:id="s2417" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius plurium terminorum collatorum cum ali <lb/>quo tertio eiuſdem generis, eadem fuerit cum ea quæ eſt cuiuſuis alterius dictorum <lb/>terminorum cum eodem tertio, aut proportio dicti tertij, cum aliquo dictorum, ea-<lb/>dem fuerit cum ea quæ ipſius eſt ad aliquem alium eorundem <reg norm="terminorum" type="context">terminorũ</reg>, tunc eiuſ-<lb/>modi termini, æquales erunt inter ſe.</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 propoſitio.</head> + <p> + <s xml:id="s2418" xml:space="preserve"><emph style="sc">Qvoties</emph> plures erunt termini, quorum vnus fuerit maior altero, ſi compa-<lb/>rentur alicui tertio eiuſdem generis, proportio maioris adtertium illum, maior erit <lb/>ea, quæ eſt minoris ad prædictum tertium, & proportio illius tertij ad maiorem, mi-<lb/>nor erit ea quæ eiuſdem 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 propoſitio.</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, eiuſmodi 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 propoſitio.</head> + <p> + <s xml:id="s2421" xml:space="preserve"><emph style="sc">Proportiones</emph>, quarum vnaquęque cum aliqua tertia æqualis eſt, ipſæ quo-<lb/>que inter ſe ſunt æquales. </s> + <s xml:id="s2422" xml:space="preserve">Vtillud, Quæ vni & eidem ſunt æqualia, ſibi inuicem <lb/>ſunt 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 propoſitio.</head> + <p> + <s xml:id="s2423" xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius ex pluribus antecedentibus cum ſuo ex <lb/>pluribus conſequentibus, æqualis fuerit ei cuiuſuis alterius dictorum <reg norm="antecedentium" type="context">antecedentiũ</reg>, <lb/>cum ſuo plurium <reg norm="conſequentium" type="context">cõſequentium</reg>, proportio totius aggregati antecedentium cum to-<lb/>to aggregato conſequentium, dictæ primę proportioni ęqualis erit, nempe illius an <lb/>tecedentis ad ſuum conſequens.</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 propoſitio.</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 ſe, 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 (ſiue coniuncti ſiue di-<lb/>ſiuncti ſint) æquales ſinguli 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="ſuum" type="context">ſuũ</reg> ter-<lb/>tium, æqualis erit proportioni aggregati reliquorum terminorum ad ſuum tertium, <lb/>& è conuerſo, ita ſe habebit primus tertius terminus ad ſuos multos terminos, ſicut <lb/>ſe habet ſecundus tertius terminus ad ſuos ſimul ſumptos.</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, ſem <lb/>per multiplex eſt ad ſingulas partiales proportiones, ex quibus ipſum 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 diuiſa fuerit, ex iis partibus componitur, in <lb/>quas diuiditur.</s> + </p> + <p style="it"> + <s xml:id="s2429" xml:space="preserve">Cum enim bæ præpoſitiones ſint ita conſpicuæ ipſi intellectui, ut <reg norm="abſque" type="simple">abſq;</reg> dubio inter obie <lb/>ct a ipſius intellectus connumerari poſſint, nullus ſanæ 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, ſecundum, & tertium theorema quinti Euclidis ab ipſo ſatis exactè de <lb/>monſtratur, ſtudioſus itaque autorem conſulat.</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 ſic <reg norm="demonſtrarem" type="context">demonſtrarẽ</reg>. <lb/></s> + <s xml:id="s2432" xml:space="preserve">ſit, verbi gratia, proportio <var>.a.</var> ad <var>.b.</var> <lb/>quæ eſt <var>.c.</var> ad <var>.d.</var> ſumptis 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 eſt <var>.f.</var> ad <var>.h</var>. </s> + <s xml:id="s2433" xml:space="preserve">Habemus enim ex .10 <lb/>poſtulato præmiſſo, eandem futuram <lb/>proportionem <var>.e.</var> ad <var>.a.</var> quæ eſt <var>.f.</var> ad <var>.c.</var> <lb/>& ita <var>.b.</var> ad <var>.g.</var> quæ eſt <var>.d.</var> ad <var>.h.</var> ex præ-<lb/>ſuppoſito verò <reg norm="cum" type="context">cũ</reg> ſic ſe habeat <var>.a.</var> ad <lb/>b. ſicut <var>.c.</var> ad <var>.d.</var> erit ex primo poſtula-<lb/>to <reg norm="eadem" type="context">eadẽ</reg> proportio <var>.e.</var> ad <var>.g.</var> quæ eſt <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 ſunt <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> & ſimiliter 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 ſint de obiectis intelligibilibus, ſine vllo medio, <lb/>ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti <lb/>Euclid. fecimus tertium Poſtulatum, .8. quintum, .9. quartum, .10. ſextum, .11. ſepti<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 demonſtrabitur, mutatis tantum <lb/>theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur <lb/>tertium .5. 4. et .6. poſtulatum à me propoſitum.</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 ſic demonſtrabo; </s> + <s xml:id="s2439" xml:space="preserve">Sit, exempli gratia, a. termi-<lb/>nus antecedens. et <var>.b.</var> conſequens, qui-<lb/>bus duo multiplices ſumantur <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/>ſtè patet quamlibet partem ipſius <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/>ſtulato propoſitum eluceſcet.</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 ſic demonſtrabitur. </s> + <s xml:id="s2443" xml:space="preserve">Sit, exempli cauſa, eadem pro <lb/>portio <var>.a.</var> ad <var>.b.</var> quæ eſt <var>.c.</var> ad <var>.d</var>. </s> + <s xml:id="s2444" xml:space="preserve">Dico <reg norm="quod" type="simple">ꝙ</reg> ita ſe habebit <var>.a.</var> ad <var>.c.</var> ſicut <var>.b.</var> ad <var>.d</var>. </s> + <s xml:id="s2445" xml:space="preserve">Cogi-<lb/>temus itaque alterum iſtorum 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 ipſius <var>.a.</var> ad <var>.d.</var> componetur ex ea quę <lb/>eſt <var>.a.</var> ad <var>.b.</var> & ea quæ eſt <var>.b.</var> ad <var>.d.</var> ex .12. poſtulato. </s> + <s xml:id="s2447" xml:space="preserve">Et ex eodem, illa ipſa proportio <var>.<lb/>a.</var> ad <var>.d.</var> pariter componetur ex ea quæ eſt <var>.a.</var> ad <var>.c.</var> & ea quæ eſt <var>.c.</var> ad <var>.d.</var> ſumpto <var>.c.</var> pro <lb/>medio termino. </s> + <s xml:id="s2448" xml:space="preserve">Ex quo ſequitur, 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 ſi duas proportiones æquales ſubtraxerimus, eam videlicet quæ eſt <var>.a.</var> ad <var>.b.</var> & il <lb/>lam quæ eſt <var>.c.</var> ad <var>.d.</var> ſupererunt duæ proportiones <lb/>inter ſe 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 ſecundi po <lb/>ſtulati diuiſim.</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 idipſum <reg norm="demonſtrari" type="context">demõſtrari</reg> poteſt, <lb/>ſumpto <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 eſt <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"/> + poſtulato. </s> + <s xml:id="s2452" xml:space="preserve">Sed cum proportio <var>.a.</var> ad <var>.b.</var> ęqualis ſit <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 ſecunda <lb/>parte .2. poſtulati compoſitè, & ſic habebimus pro <lb/>poſitum, ita quòd quotieſcunque <reg norm="dabuntur" type="context">dabũtur</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">DEcimiſeptimi theorematis hæc eſt demonſtratio. </s> + <s xml:id="s2455" xml:space="preserve">Ita ſe ha beat <var>a.c.b.</var> ad <var>.c.<lb/>b.</var> ſicut ſe habet <var>.d.f.e.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2456" xml:space="preserve">Probo ita ſe habere <var>.a.c.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.d.<lb/>f.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2457" xml:space="preserve">Cogitemus itaque alterum terminum ſcilicet <var>.n.f.</var> qui ſic ſe habeat. ad <var>.f.e.</var> <lb/>ſicut ſe habet <var>.a.c.</var> ad <var>.c.b</var>. </s> + <s xml:id="s2458" xml:space="preserve">Quare ex præcedenti theoremate ita ſe habebit <var>.a.c.</var> ad <var>.n.<lb/>f.</var> ſicut ſe habet <var>.c.b.</var> ad <var>.f.e.</var> & ex .8 poſtulato ita ſe habebit <var>.a.c.b.</var> ad <var>.n.f.e.</var> ſicut ſe ha-<lb/>bet <var>.c.b.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2459" xml:space="preserve">Sed cum ex præſuppoſito ita ſe habeat <var>.a.c.b.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.<lb/>d.f.e.</var> ad <var>.f.e.</var> ideo ex præcedenti theoremate ita ſe habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> ſicut ſe ha <lb/>bet <var>.c.b.</var> ad <var>.f.e.</var> demonſtratum autem eſt ita ſe habere <var>.c.b.</var> ad <var>.f.e.</var> ſicut ſe habet <var>.a.c.b.</var> <lb/>ad <var>.n.f.e</var>. </s> + <s xml:id="s2460" xml:space="preserve">Quare ex .7. poſtulato 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. poſtulato <var>.d.f.e.</var> æqualis erit <var>.n.f.e</var>. </s> + <s xml:id="s2461" xml:space="preserve">Itaque ex 3. poſtulato 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> ęqualis 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 ſecunda par-<lb/>te tertij axiomatis præmiſſi. </s> + <s xml:id="s2463" xml:space="preserve">Igitur ita <lb/>ſe habebit <var>.a.c.</var> ad <var>.d.f.</var> ſicut <var>.c.b.</var> ad <var>.f.e.</var> ex <lb/>7. poſtulato. </s> + <s xml:id="s2464" xml:space="preserve">& ſic ex præcedenti theo-<lb/>remate ita ſe habebit <var>.a.c.</var> ad <var>.c.b.</var> ſicut <var>.d.f.</var> ad <var>.f.e.</var> quod erat propoſitum: </s> + <s xml:id="s2465" xml:space="preserve">Quotieſ-<lb/>cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiſim 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 demonſtrari poteſt. </s> + <s xml:id="s2467" xml:space="preserve">Detur proportio <var>.a.c.</var> ad <var>.c.b.</var> ſi-<lb/>milis ei quæ eſt <var>.d.f.</var> ad <var>.f.e.</var> probo ita ſe habere <var>.a.c.b.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.d.f.<lb/>e.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2468" xml:space="preserve">In primis notum eſt ex .16. theoremate ita ſe habiturum, <var>a.c.</var> ad <var>.d.f.</var> ſi <lb/>cut <var>.c.b.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2469" xml:space="preserve">Quare ex .8. poſtulato ita <lb/>ſe habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> ſicut <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 ſe habebit <var>.<lb/>a.c.b.</var> ad <var>.c.b.</var> ſicut <var>.d.f.e.</var> ad <var>.f.e</var>. </s> + <s xml:id="s2471" xml:space="preserve">Quod erat <lb/>propoſitum. </s> + <s xml:id="s2472" xml:space="preserve">Quotieſcunque igitur .4. <lb/>quantitates dabuntur vnius <reg norm="eiuſdemque" type="simple">eiuſdemq́;</reg> generis diſiunctim 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. ſatis quidem apud Euclidem demonſtratur: </s> + <s xml:id="s2474" xml:space="preserve">eius tamentertia <lb/>pars commodius hac ratione demonſtrari poterit (nempe) quod cum ſit 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 eſt <var>.c.</var> ad <var>.d.</var> probabo ita ſe habituram proportionem <var>.b.</var> ad <var>.a.</var> ſicut <lb/>ſe habet <var>.d.</var> ad <var>.c.</var> hoc argumento: </s> + <s xml:id="s2475" xml:space="preserve">ſi <var>.a.</var> ad <var>.b.</var> ita ſe <lb/>habet ſicut <var>.c.</var> ad <var>.d.</var> ex .16. theoremate ita ſe ha <lb/> + <anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/> + bebit <var>.a.</var> ad <var>.c</var>, ſicut <var>.b.</var> ad <var>.d</var>. </s> + <s xml:id="s2476" xml:space="preserve">Quare ſic ſe habebit <lb/>b. ad <var>.d.</var> ſicut <var>.a.</var> ad <var>.c</var>. </s> + <s xml:id="s2477" xml:space="preserve">Itaque ex eodem .16. ita ſe <lb/>ſe habebit <var>.b.</var> ad <var>.a.</var> ſicut <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 demonſtratum fuerit, poteſt ni-<lb/>hilominus & hac via demonſtrari. </s> + <s xml:id="s2479" xml:space="preserve">Sic ſe habeat proportio <var>.a.</var> ad <var>.b.</var> ſicut ſe <lb/>habet <var>.c.</var> ad <var>.d.</var> & proportio <var>.b.</var> ad <var>.e.</var> ſicut <var>.d.</var> ad <var>.<lb/>f</var>. </s> + <s xml:id="s2480" xml:space="preserve">Dico <reg norm="quod" type="simple">ꝙ</reg> ſi <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> & ſi <var>.a.</var> minus fuerit .e: c. <reg norm="quoque" type="simple">quoq;</reg> minus erit <lb/>f. ſin verò ęquale, <reg norm="ent" type="context">ẽt</reg> æquale erit. </s> + <s xml:id="s2481" xml:space="preserve">Nam ex pri <lb/>mo poſtulato certi ſumus ita ſe habere pro <lb/><reg norm="portionem" type="context">portionẽ</reg> <var>.a.</var> ad <var>.e.</var> ſicut ſe habet proportio <var>.c.</var> ad <lb/>p. </s> + <s xml:id="s2482" xml:space="preserve">Quare ex .12. theor <reg norm="propoſitum" type="simple context">ꝓpoſitũ</reg> <reg norm="manifeſtum" type="context">manifeſtũ</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">VIgeſimum primum theorema, ſatis apud Eucli. probatum, nihilominus præ-<lb/>ſcripto nunc modo demonſtrari 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 poſtulatum collegimus. <lb/></s> + <s xml:id="s2485" xml:space="preserve">Sequentia verò cum exactè apud Eucli. demonſtrentur non eſt 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, Philoſophi 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 eſt proprium, quàm <lb/>agitari, & inceßabili motu prodeße. </s> + <s xml:id="s2487" xml:space="preserve">Ac velu <lb/>ti fulgidum ſydus ante oculos <reg norm="ſpectantium" type="context">ſpectantiũ</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 philoſophicis ſpeculationibus dedito, ſapiſ-<lb/>ſimè, ut in principium ſummorum aulis, & <lb/>amplißimis ciuitatibus degenti, ubi multa ſem <lb/>per Nobilium mir a curioſitate, ſciendi deſiderio, & conferendicu <lb/>piditate referta, uerſantur ingenia, contigit, modo ab his, modo ab <lb/>illis, aut uerbis tentari, aut literis prouocari ad diſſerendum, de <lb/>his, in quorum ſtudijs uerſamur. </s> + <s xml:id="s2489" xml:space="preserve">Quarum concertationum & re <lb/>ſponſionum, quoniam non omnino indigna exiſtimaui, quæmemoriæ<unsure/> <lb/>comendarentur, partem aliquam apud me conſeruaui. </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 ſcientia ipſa quo magis diffundetur, creſcat; </s> + <s xml:id="s2492" xml:space="preserve">& quicquid <lb/>ualeo, ſine inuidia in communem utilitatem conferam. </s> + <s xml:id="s2493" xml:space="preserve">Tum ut ui-<lb/>rorum præctantiβimorum, qui me ſuis interrogationibus excitaue <lb/>runt, quantum in me erit, gratitudinis ergo, nomina reddam im-<lb/>mortalia, & eorum exemplo alios, ocio ſordidiore abiecto, quod ſolet <lb/>ourialium præcipuè excelſa ingenia corrumpere, ad ſciſcit andum <lb/>conferendum, & diſſerendum, derebus <reg norm="ſerijs" type="lig">ſerijs</reg>, & quæuſui 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/>conſule, & 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, Philoſophi <lb/>Mathematici.</head> + <head xml:id="head358" style="it" xml:space="preserve">AD SERENISS. CMANVELEM PHILIB. <lb/>Allobrogum & ſubalpinarum 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>, Quàm lectione epiſtolæſeu (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 Celſitu <lb/>do miſit ex Nicea, vt meam de ea re ſententiam proferrem, <lb/>delectatus ſim; </s> + <s xml:id="s2497" xml:space="preserve">ex quo, non tantum recta illius mens ac verè <lb/>ſancta cogitatio, ſed etiam aperta <reg norm="maximaque" type="simple">maximaq́;</reg>, ſi ad exitum per <lb/><reg norm="ducatur" type="simple">ducat̃</reg>, imo ſummè neceſſaria vniuerſo orbi vtilitas percipi <lb/>poteſt; </s> + <s xml:id="s2498" xml:space="preserve">qua de re memini cum Celſitudine tua aliquando ſer-<lb/>monem habuiſſe. </s> + <s xml:id="s2499" xml:space="preserve">Vidi præterea cum ipſo breui tranſ-<lb/>miſſum compendium Domini Aloiſij Lilij: </s> + <s xml:id="s2500" xml:space="preserve">cuius mihi ſententia perplacet, de corre <lb/>ctione eius diei, qui 134. quoque anno præter, neceſſitatem, gignitur. </s> + <s xml:id="s2501" xml:space="preserve">qui ſanè dies <lb/>perpetuæ retrogradationis ingreſſus Solis in Zodiaci ſigna, cauſa fuit. </s> + <s xml:id="s2502" xml:space="preserve">quod ita per-<lb/>ſpicuè patebit. </s> + <s xml:id="s2503" xml:space="preserve">Cum Numa Pompilius anni curſum correxit <reg norm="emendauitque" type="simple">emendauitq́;</reg>, ea ſanè <lb/>mente id videtur præſtitiſſe, vt principium Ianuarij primi menſis anni, præcisè in ip <lb/>ſo hyemalis ſoltitij puncto collocaretur. </s> + <s xml:id="s2504" xml:space="preserve">quod hac tempeſtate, dictam ob cauſam <lb/>adeò retroceſſit, vt circa vndecimam diem Decembris eſſe reperiatur. </s> + <s xml:id="s2505" xml:space="preserve">quod ſi cen-<lb/>teſimo trigeſimo quarto quoque anno detractus dies vnus fuiſſet, nihil erroris pror-<lb/>ſus accidiſſet. </s> + <s xml:id="s2506" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dies hic (vt alias Celſit. tuæ ſignificaui) 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. ſecundorum ferè .16. (anni æqualis <lb/>ſiue medij) fallimur quarto quoque anno in minutis .42. <reg norm="ſecundis" type="context">ſecũdis</reg> propè .56. amplius <lb/>quàm par ſit minutis ſcilicet .10. ſecundis ferè .44. ſingulis annis; </s> + <s xml:id="s2507" xml:space="preserve">qui numerus .134. <lb/>multiplicatus, diem penè horarum .24. conſtituit; </s> + <s xml:id="s2508" xml:space="preserve">penè inquam, quia minutum <reg norm="vnum" type="context">vnũ</reg> <lb/>deeſſet <reg norm="tantummodo" type="wordlist">tm̃modo</reg>, & ſecunda .44. ſi <reg norm="decem" type="context">decẽ</reg> illa minuta, & .44. ſe cunda annua, exquiſita eſ <lb/>ſent 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/>eiuſmodi eſſet emendatio, <reg norm="aptaque" type="simple">aptaq́;</reg> eius ratio à D. </s> + <s xml:id="s2511" xml:space="preserve">Lilio oſtenditur, prout etiam Pe-<lb/>trus Pitatus Veronenſis tradidit, in eo, quem de vera anni quantitate tractatu con-<lb/>ſcripſit, nempe vt tribus primis centeſimis annis, centeſimus quiſque annus commu-<lb/>nis ſit, quartus ſubſequens centeſimus intercalaris: </s> + <s xml:id="s2512" xml:space="preserve">quod ſanè fierineceſſe eſt. </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 centeſimis <reg norm="communibus" type="context">cõmunibus</reg>, tres quartas diei partes plus æquo detraxerimus, <lb/>non enim centeſimo quoque anno, ſed centeſimo trigeſimo quarto, dictus dies de-<lb/>trahi debet, poſtquam tres integros dies, qui quadringentis detrahendi erant, tre-<lb/>centorum annorum ſpacio detraxerimus; </s> + <s xml:id="s2514" xml:space="preserve">ſitq́ue 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 ſuo loco reſtituta erunt. </s> + <s xml:id="s2516" xml:space="preserve">Idcirco dictus iam <lb/>quadringenteſimus annus intercalaris & non communis conſtituendus erit, non alia <lb/>de cauſa, quam vt biſſexti ordinem ſequamur.</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 eſt, de ratione inueniendi ſingulis menſibus <lb/>Nouilunij diem, interdum fallere nos poſſet vno die; </s> + <s xml:id="s2519" xml:space="preserve">prout Ianuario proximè <lb/>lapſo accidit; </s> + <s xml:id="s2520" xml:space="preserve">quo ex præſcripto modo nouilunij, dies nonus illius menſis eſſe debuiſ <lb/>ſet, qui fuit tamen dies ſeptimus, ſexta decima hora cum dimidia poſt meridiem. </s> + <s xml:id="s2521" xml:space="preserve">Ne <lb/>que etiam tutum eſt, via integrorum dierum, nulla habita horarum aut minutorum <lb/>ratione, nec minus ea, quæ à Pitato tradita eſt, mediorum ſeu æqualium <reg norm="motuum" type="context">motuũ</reg> pro <lb/>gredi: </s> + <s xml:id="s2522" xml:space="preserve">At cenſerem potius veros motus ſequendos eſſe ex calculis exactarum tabu-<lb/>larum, quales Prutenicas eſſe iudico; </s> + <s xml:id="s2523" xml:space="preserve">Et cum ſolius Paſchæ cauſa laboremus hac in <lb/>re, pleniluniorum verorum, in multos annos tabulas formarem, quæ æquinoctia ver <lb/>nalia ſequuntur, cum aſſignatione diei Paſchatis præcisè, prout fecit Pitatus; </s> + <s xml:id="s2524" xml:space="preserve">non <lb/>via tamen æqualium pleniluniorum ſed verorum. </s> + <s xml:id="s2525" xml:space="preserve">Porrò quod ad Paſchatis cele-<lb/>brationem attinet, rationi conſentaneum eſt, concilij Niceni decretum ea de re ſer <lb/>uari, prima ſcilicet dominica die poſt primum plenilunium, quod æquinoctium ver-<lb/>nale ſequitur; </s> + <s xml:id="s2526" xml:space="preserve">hoc tamen animaduerſo, ſi dictum plenilunium primum poſt æquino-<lb/>ctium contingens, <reg norm="diem" type="context">diẽ</reg> dominicum ſortiretur; </s> + <s xml:id="s2527" xml:space="preserve">nulla ratione tali die Paſcha celebran-<lb/>dum eſſe; </s> + <s xml:id="s2528" xml:space="preserve">verum ſubſequenti, ne cum Hębreis conſentiat Eccleſia Chriſti: </s> + <s xml:id="s2529" xml:space="preserve">quæ fuit <lb/>cauſa, vt in decreto concilij Niceni ſtatutum ſit, à quartadecima, in vigeſimam 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">eiꝰ</reg> <reg norm="ſententia" type="context">ſentẽtia</reg> in ſuis tabulis die Paſchate declarata, huiuſce anni Paſca <reg norm="celebrandum" type="context">celebrandũ</reg> <lb/>fuerit .23. </s> + <s xml:id="s2531" xml:space="preserve">Martij, ipſomet de plenilunij non tantum æqualis, ſed veri.</s> + </p> + <p> + <s xml:id="s2532" xml:space="preserve">Dies autem Paſchatum elapſorum, 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 poſt nouilunium Februarij, carnis priuij diem eſſe; </s> + <s xml:id="s2534" xml:space="preserve">non autem <reg norm="cum" type="context">cũ</reg> <lb/><reg norm="sanctione" type="context">sãctione</reg> Patrum concilij Niceni, qua ſtatuerunt à vigeſima prima Martii dirigen-<lb/>dum eſſe Paſchatis diem, vt potè qui ſibi perſuaſerunt, circa eum diem <reg norm="æquinoctium" type="context">æquinoctiũ</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/>ſcha ex huiuſmodi ſuppoſitione concilij, poſt vigeſimam primam lunę celebretur, <lb/>cum ſeruata regula concilij non fuerit. </s> + <s xml:id="s2537" xml:space="preserve">Prout <reg norm="manifeſtum" type="context">manifeſtũ</reg> eſt de Paſchate anni .1566. <lb/>celebrato .14. </s> + <s xml:id="s2538" xml:space="preserve">Aprilis (quę fuit .24. lunę) quod .7. dicti menſis <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 ſolenne fuit Paſca, quod tertia eiuſdem eſſe debuerat. </s> + <s xml:id="s2541" xml:space="preserve">Anno <lb/>deinde 1572. 6. Aprilis, dies fuit Paſchatis, quæ .30. Martij futura erat, anno vero <lb/>1575. in tertiam Aprilis Paſcha incidit, caſurum in .27. Martii.</s> + </p> + <p> + <s xml:id="s2542" xml:space="preserve">Cum igirur (vt ex diplomate ad Celſit. tuam miſſo patet) <var>S.D.N.</var> mens ſit <reg norm="atque" type="simple">atq;</reg> vo <lb/>luntas, ut quiſque liberè in medium proferat quid hac dere ſentiat: </s> + <s xml:id="s2543" xml:space="preserve">quædam mihi <lb/>non omnino præmittenda occurrunt, quæ tantis cœptis 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 Paſcha cætera-<lb/>q́ue feſta mobilia ab illo manantia, vt decreto concilij Niceni ſancitum eſt, ſcilicet <lb/>vt ipſum Paſcha celebretur prima dominica poſt 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 ſequitur; </s> + <s xml:id="s2545" xml:space="preserve">verum etiam quò anni principium emendetur, <lb/>ſcilicet vt ad ſuum verum principium reuocetur annus. </s> + <s xml:id="s2546" xml:space="preserve">Nempè ad diem hyemalis <lb/>ſolſtitij, quæ prima Ianuarij dies eſſe debet.</s> + </p> + <p> + <s xml:id="s2547" xml:space="preserve">Deinde, tot dierum menſes conſtituantur, quot hac noſtra tempeſtate, ſol in ipſis <lb/>Zodiaci ſignis verſatur. </s> + <s xml:id="s2548" xml:space="preserve">Poſtremò, quædam feſta immobilia in alios dies <reg norm="transferam" type="context">transferã</reg> <lb/>tur, <reg norm="celebrenturque" type="simple">celebrenturq́;</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 feſtis mobilibus quam immobilibus agat, imo etiam planè <lb/>æquum ſit, vt habeatur vtrorunque ratio, quò ſtatutis 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">cenſerem <lb/>ex eo anno, quem corrigere voluerimus, non modò dies .10. eſſe detrahendos, <reg norm="verum" type="context">verũ</reg> <lb/>etiam vnum & uiginti, illo ipſo anno; </s> + <s xml:id="s2553" xml:space="preserve"><reg norm="idque" type="simple">idq́;</reg> duplici via; </s> + <s xml:id="s2554" xml:space="preserve">aut partiendo menſes, 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 menſes ſint die. <lb/></s> + <s xml:id="s2555" xml:space="preserve">rum duorum ſpatio ſolito breuiores, alter menſis vno deficiat: </s> + <s xml:id="s2556" xml:space="preserve">aut conſtituto <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> ſequitur, ſit & <lb/>primus Ianuarij, & dies ſolſtitij ob quam cauſam exiſtimarem conſultiſſimum eiuſ <lb/>modi annum eſſe mileſimum quingenteſimum ſeptuageſimum 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. ſuis temporibus huius correctionis manifeſtos effe-<lb/>ctus experiri & perpendere, atque diſpoſitionem anni non ſolum principio, ſed <reg norm="cae teris" type="simple">cę<lb/>teris</reg> partibus ſuis in vniuerſum tam concinnè <reg norm="apteque" type="simple">apteq́;</reg> reſpondere, & aſtrorum moti-<lb/>bus, & Eccleſiæ ſacroſanctæ ſanctionibus, ſe 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 elapſi erroris: </s> + <s xml:id="s2560" xml:space="preserve">non de <lb/>cem tantum, quo hyemmalis conuerſio ad initium Ianuarij reuocetur; </s> + <s xml:id="s2561" xml:space="preserve">idq́ue ne à <lb/>communi opinione de ipſo anni principio veritas diſcrepet, 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">annũ</reg> à Ianuario inchoa-<lb/>ri, veritas autem ipſa ſic ſe habeat, vt nobis ſeptentrionalibus tunc inchoet annus, <lb/>cum ad nos Sol accedere incipit, aut dies augetur; </s> + <s xml:id="s2563" xml:space="preserve">non conuenit principia eiuſmo-<lb/>di ſeparata & diſcrepantia eſſe. </s> + <s xml:id="s2564" xml:space="preserve">Et hanc fuiſſe Numæ Pompilio mentem credibile <lb/>eſt, qui ad annum Romuli decem menſium, Ianuarium & Februarium addidit, vt <lb/>principium Ianuarij principium eſſet anni: </s> + <s xml:id="s2565" xml:space="preserve">cuius rei argumentum eſſe poteſt, quod <lb/>C. </s> + <s xml:id="s2566" xml:space="preserve">Iulij Cæſaris temporibus (qui multis annis poſt 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 feſtorum curam ſuſcepit hyemale ſolſtitium per aliquot dies retro-<lb/>ceſſerat; </s> + <s xml:id="s2569" xml:space="preserve">nec mirum tamen eſſet, ſi Numæ temporibus, exactè prima Ianuarij die non <lb/>fuiſſet hyemale ſolſtitium, adhuc pubeſcente in Italia Aſtronomia.</s> + </p> + <p> + <s xml:id="s2570" xml:space="preserve">Huiuſmodi autem correctio dierum .21. poſt .2300. annos à Numa, quæ ſit per-<lb/>petuo ſeruitura, media emendatione ea, quæ de tribus centeſimis annis communi-<lb/>bus, & quarto intercalari, ſuperius propoſita fuit, non repudianda ei videatur, qui <lb/>ſciet, 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 eſt hyemale ſolſtitium, reduceretur.</s> + </p> + <p> + <s xml:id="s2571" xml:space="preserve">Alio item argumento cuique patere poteſt, priſcos Romanos ſtatuiſſe annum ab <lb/>hyemali ſolſtitio initium ſumere, vt inquit Ouidius primo Faſtorum.</s> + </p> + <p> + <s xml:id="s2572" xml:space="preserve">Bruma noui prima eſt, <reg norm="veterisque" type="simple">veterisq́;</reg> nouiſſima Solis.</s> + </p> + <p> + <s xml:id="s2573" xml:space="preserve">Principium capiunt Phębus, & annus idem.</s> + </p> + <p> + <s xml:id="s2574" xml:space="preserve">co quod diem naturalem à medio noctis inchoarent, ab eo puncto ſcilicet, quo Sol <lb/>ad noſtrum hemiſpherium 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, reſpondeat puncto meridiani ſub terra, in quo Sol ſemel in die reperitur: <lb/></s> + <s xml:id="s2577" xml:space="preserve">Quòd apertè norunt hi, qui ſub polo boreali conſtituti ſunt. </s> + <s xml:id="s2578" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> facilè diſcerne <lb/>re poſſumus, diem ſcilicet & annum, quaſi ſibi ad inuicem medio ſuarum partium <lb/>reſpondere; </s> + <s xml:id="s2579" xml:space="preserve">ſolſtitium inquam hyemale, mediæ nocti, æſtiuum meridiei, æquino-<lb/>ctium vernale ortui Solis, autumnale occaſui. </s> + <s xml:id="s2580" xml:space="preserve">Quam tamen ſimilitudinem, multò <lb/>quam nos manifeſtius deprehendunt, hi qui (ut diximus) ſub polo borcali verſantur.</s> + </p> + <p> + <s xml:id="s2581" xml:space="preserve">Quod ſi quis dubitet hac ratione correcto anno, quo nam pacto ad calculos coe-<lb/>leſtes motus medijs tabulis aſtronomicis hactenus in lucem æditis redigi poſſint, id <lb/>facilimum ſanè erit, exempli gratia; </s> + <s xml:id="s2582" xml:space="preserve">aliquis planetę ſitum, aut alicuius ſtellę fixæ, quo <lb/>cunque die menſis anni correcti inuenire cupit, detrahat ex huiuſmodi <reg norm="tempore" type="context">tẽpore</reg> dies <num value="21">.<lb/>21.</num> ab Aera Chriſti, cum reſiduo ſupputet ſtellam, cuius ſitum ſcire deſiderat; </s> + <s xml:id="s2583" xml:space="preserve">ſum-<lb/>pta quacunque tabula, ſupputatio erit exacta: </s> + <s xml:id="s2584" xml:space="preserve">Cuius ratio cuilibet manifeſta erit, <lb/>qui ſciet annum vt potè .1579. dierum .344. tantummodo conſtitutum fuiſſe. </s> + <s xml:id="s2585" xml:space="preserve">Nam <lb/>in ijſdem locis cœli prima die Ianuarij correcti, erunt ſtellæ quibus eſſe ſolebant .11 <lb/>Decembris præcedentis anni ex ſupputatione 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 cęleſtium ipſo princi <lb/>pio anni .1580. correcti: </s> + <s xml:id="s2588" xml:space="preserve">hoc ſtatuto, vt hi motus radices eſſent Aeræ <var>S.D.N.</var> Grego <lb/>rij XIII. quod ſi alio <reg norm="tempore" type="context">tẽpore</reg> quiſpiam motus cęleſtes ad calculos redigere voluerit, <lb/>ſupputabit ab Aera huiuſmodi, quæ anno .1580. principium habuerit: </s> + <s xml:id="s2589" xml:space="preserve">Quæ vt nobi <lb/>lius nomen ſortiatur, <reg norm="idque" type="simple">idq́;</reg> merito ex nomine Gregorij. XIII. Pont. Max. appelletur; <lb/></s> + <s xml:id="s2590" xml:space="preserve">exemplo antiquarum, quæ ex Principum nominibus ſunt 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 ſummæ, quæ ex huiuſmodi Aera Gregoriana ex tabu-<lb/>lis colligetur, ipſiuſmet Aeræ radices addantur, vt exactus calculus habeatur. </s> + <s xml:id="s2593" xml:space="preserve">Et <lb/>hæc ſit primæ ſententiæ noſtræ explicatio.</s> + </p> + <p> + <s xml:id="s2594" xml:space="preserve">Altera erit numerum dierum menſium anni alia ratione quam nunc ſe 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. ſinguli con <lb/>tineant, Februarius, Martius, & October .30. </s> + <s xml:id="s2596" xml:space="preserve">Aprilis, Maius, Auguſtus, & September <lb/>dies .31. Iunius, ac Iulius .32. atque id hac potiſſimum de cauſa, vt Sol unum <reg norm="quodque" type="simple">quodq́;</reg> <lb/>ſignum calendis menſrum ingredi poſſit. </s> + <s xml:id="s2597" xml:space="preserve">Nam detractis (ut dictum eſt) diebus .21. <lb/>& reuocato ingreſſu Solis in principium Capricorni ad principium Ianuarij, in quo <lb/>ſigno hac noſtra tempeſtate, Sol, dies propè .29. & quartam vnam verſatur: </s> + <s xml:id="s2598" xml:space="preserve">ſi Ianua <lb/>rius .29. dies continebit, exactis hiſce diebus, ingredietur Aquarium circa princi-<lb/>pium Februarii; </s> + <s xml:id="s2599" xml:space="preserve">hæret autem hoc noſtro ſæculo in Aquario Sol dies propè .29. cum <lb/>dimidio; </s> + <s xml:id="s2600" xml:space="preserve">quare ſi Februarius erit .30. dierum, elapſis ipſis diebus, Sol ingredietur pi-<lb/>ſces circa principium Martii: </s> + <s xml:id="s2601" xml:space="preserve">& ſic de cæteris.</s> + </p> + <p> + <s xml:id="s2602" xml:space="preserve">Quamobrem ſi generali correctione annus emendandus erit, pulcherrimè acci-<lb/>det, ſi menſes anni cum duodecim partibus cœleſtibus, itineris annui Solis, concor-<lb/>dauerint; </s> + <s xml:id="s2603" xml:space="preserve"><reg norm="eiſque" type="simple">eiſq́;</reg> aptè reſponderint. </s> + <s xml:id="s2604" xml:space="preserve">Qua ex re, varię vtilitates promanabunt, pręſertim <lb/>Nautis, Agricolis, Medicis, & alijs qui vera principia, & interualla temporum per <lb/>ſpecta 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 cauſa, ſcient 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 huiuſmodi conuerſione Solis ad nos dependent. </s> + <s xml:id="s2609" xml:space="preserve">Pariter <lb/>ſcient omnes primam diem Iulij, non tantum æqualiter annum diuidere, ſed 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, quę Solis conuerſionem <lb/>ad auſtrales ſequuntur.</s> + </p> + <p> + <s xml:id="s2611" xml:space="preserve">Neconon intelliget vnuſquiſque primam diem Aprilis, <reg norm="primamque" type="simple">primamq́;</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 ſuperat, & alia huiuſmodi, quæ ab æquinoctijs <reg norm="dependent" type="context context">depẽdẽt</reg>.</s> + </p> + <p> + <s xml:id="s2616" xml:space="preserve">Si vero quiſpiam obijciat, modum hunc noſtrum & 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">tũc</reg> <lb/>Sol hærebit in ſigno Sagittarij .32. diebus, totidem in Capricorno, in Geminis vero <lb/>29. totidem in Cancro; </s> + <s xml:id="s2618" xml:space="preserve">ex quo ſequetur prioribus <reg norm="contrarius" type="context">cõtrarius</reg> effectus; </s> + <s xml:id="s2619" xml:space="preserve">huic ego reſpon <lb/>debo, tale <reg norm="quidpiam" type="context">quidpiã</reg> non euenturum, niſi exactis ab hoc anno annis .240@0. quod ſi <lb/>mundus poſthac totidem annis, quot fuit antehac, perdurauerit, punctus augis non <lb/>amplius à ſitu præſenti, quàm .45. gradibus diſtabit. </s> + <s xml:id="s2620" xml:space="preserve">Verum demus <reg norm="modum" type="context">modũ</reg> noſtrum <lb/>& regulam in annos ter, aut quater mille ſubſeruire poſſe, nec amplius, certè hoc <lb/>toto tempore nullius momenti penè erit, quæ accidere poterit mutatio, tametſi ela-<lb/>pſis quatuor millibus annorum Februarius eſſe debebit .29. dierum. Aprilis & No-<lb/>uember .30. Iunius & October .31. Auguſtus .32. in aliis verò menſibus nihil mutan-<lb/>dum erit. </s> + <s xml:id="s2621" xml:space="preserve">Ecce quam ſit nullius momenti mutatio.</s> + </p> + <p> + <s xml:id="s2622" xml:space="preserve">Quæ ſi Iulij Cæſaris temporibus fuiſſent animaduerſa nunquam omiſſa fuiſſent, <lb/>ſed ſcientiæ Aſtronomicæ nondum (vt ita dicam) confirmata ætas, cum alibi, <reg norm="tum" type="context">tũ</reg> maxi <lb/>mè in Italia, quo minus hæc aut ſcirentur aut ſtatuerentur impediebat.</s> + </p> + <p> + <s xml:id="s2623" xml:space="preserve">Tertia ratio eſt, vt non <reg norm="ſolum" type="context">ſolũ</reg> feſta mobilia, verum <reg norm="ent" type="context">ẽt</reg> immobilia ad meliorem regu <lb/>lam (ut dictum eſt) reuocentur, ſi ſuis temporibus celebranda erunt. </s> + <s xml:id="s2624" xml:space="preserve">Quorum <reg norm="primum" type="context">primũ</reg> <lb/>eſt Natiuitas Domini, & quæ ab ea pendent; </s> + <s xml:id="s2625" xml:space="preserve">nempe Circuncifio, Epiphania, Purifi-<lb/>catio, Annunciatio, & Natiuitas Io. Baptiſtæ. </s> + <s xml:id="s2626" xml:space="preserve">ita vt dies Natalis Domini celebretur <lb/>prima die anni, cum Dei filius naſci voluerit circa verum principium anni, quod à <lb/>ſolſtitio hyemali initium ducit, & in ipſo principio diei naturalis ex <reg norm="Romanorum" type="context">Romanorũ</reg> ſen <lb/>tentia, media ſcilicet nocte, tanquam qui ſummæ lætitiæ principium, poſt longos & <lb/>graues filiorum Adæ mærores, eſſet allaturus. </s> + <s xml:id="s2627" xml:space="preserve">Nec forſan Ianuarij nomini, à vete-<lb/>ribus Iano bifronti dicati hæc mutatio non conueniret, cum in ip ſo ſeruatore, duæ <lb/>veluti frontes & formæ vnitæ ſint, duæ ſcilicet naturæ diuina & humana. </s> + <s xml:id="s2628" xml:space="preserve">Hac ratio <lb/>ne abuſus tolletur, natus ex diuerſis moribus Tabulariorum, quorum alij monumen-<lb/>ta, ſeu quæ uocant Inſtrumenta, à die Natiuitatis Domini incohant, alij à Circunci <lb/>ſione, alij à Calendis Martij, nonnulli à Paſchate; </s> + <s xml:id="s2629" xml:space="preserve">quæ varietas innumerabiles lites <lb/>affert & abuſus propè infinitos, ob dubiam & ancipitem ſcripturam. </s> + <s xml:id="s2630" xml:space="preserve">Indictionum <lb/>præterea ordini, hic noſter modus nihil officiet; </s> + <s xml:id="s2631" xml:space="preserve">celebrato Natali celebrabitur Cir-<lb/>cunciſio octaua Ianuarij. </s> + <s xml:id="s2632" xml:space="preserve">Epiphania .13. eiuſdem. </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 ſeruatoris. </s> + <s xml:id="s2635" xml:space="preserve">Prima Aprilis Annunciatio Virginis ſolennis erit, <lb/>ipſo nempè die æquinoctij, natiuitas Diui Io: </s> + <s xml:id="s2636" xml:space="preserve">Baptiſtę celebrabitur Prima Iulij die <lb/>quæ erit ſolſtitij æſtiui, cum illa diminutionem capit. vtrectè Diuus Auguſtinus il-<lb/>la verba Io: </s> + <s xml:id="s2637" xml:space="preserve">Baptiſtę interpretatus fuerit. </s> + <s xml:id="s2638" xml:space="preserve">Illum opportet creſcere, me autem minui: <lb/></s> + <s xml:id="s2639" xml:space="preserve">in quibus ſic tantus Doctor philoſophatur, vt tempus etiam natiuitatis ſerui & do-<lb/>mini præclare notet dicens, natus eſt ſeruus cum decreſcunt dies, natus eſt Dominus <lb/>cùm creſcere incipiunt.</s> + </p> + <p> + <s xml:id="s2640" xml:space="preserve">Inſignes etiam Theologi admonuerunt habendam rationem eſſe nonnullorum <lb/>feſtorum, vt Diui Antonij, diuorum Fabiani & Sebaſtiani, & aliorum ſanctorum, + <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 hęc <var>.S.D.N.</var> curæ erunt, ut in aptiſ <lb/>ſima tempora transferantur.</s> + </p> + <p> + <s xml:id="s2642" xml:space="preserve">Admonuerunt præterea transferendos eſſe dies feſtos Beati Stephani, Ioannis, <lb/>& Innocentium, vt quemadmodum factum eſt hactenus, diem natalis proximè ſe <lb/>quantur, ob multorum Doctorum, non recentium modo, ſed etiam antiquorum ob <lb/>ſeruantiam; </s> + <s xml:id="s2643" xml:space="preserve">qui ſuis omelijs & concionibus multa piè, de myſteriis ſucceſſionis Feſto <lb/>rum huiuſimodi 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/>ſe in ſigno Virginis ſol verſabitur, ita Eccleſia Der in cęlebrandi tantæ Virginis ma <lb/>tris Deilaudibus occupetur.</s> + </p> + <p> + <s xml:id="s2645" xml:space="preserve">Atque hęc ſunt Serenisſime Princeps, quę longa & attenta cogitatione à me exa <lb/>minata, atque perpenſa fuerunt; </s> + <s xml:id="s2646" xml:space="preserve">quæſitam diligenter & accuratè expendentur ab <lb/>his, quorum intereſt, quam mihi apta & rationi conſentanea, ac vera penitus, imo <lb/>(quod me magis afficit) etiam tibi viſa fuerunt; </s> + <s xml:id="s2647" xml:space="preserve">non dubito quin placitura ſint; </s> + <s xml:id="s2648" xml:space="preserve">& vo <lb/>tis ſummi Pont. aliqua ex parte ſatisfactura. </s> + <s xml:id="s2649" xml:space="preserve">eò magis quòd te iubente, & cogitata à <lb/>me, & ſcripta fuerint. </s> + <s xml:id="s2650" xml:space="preserve">Vale Princeps Sereniſſime, & qua ſoles hylaritate cętera no <lb/>ſtra, etiam has breues vigilias ſuſcipe & foue. </s> + <s xml:id="s2651" xml:space="preserve">Dat. Auguſtæ 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. Celſitudinis.</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 celſitudine tua nobis proponitur non ſolum poſſibile eſt, ſed <lb/>facile etiam ad ſoluendum, hoc eſt quod circulus talis inueniatur, qui poſſit cir <lb/>cunſcribere, ſeu capere quadrilaterum ex quatuor datis rectis lineis terminatum, vel <lb/>ſic, datis quatuor rectis lineis ex quibus quadrilaterum poſſit eftici, tale efficiatur vt <lb/>circa ipſum, circulus poſſit circunſcribi.</s> + </p> + <p> + <s xml:id="s2657" xml:space="preserve">Sint igitur .4. lineæ propoſitæ <var>.b.d</var>: <var>q.b</var>: <var>a.q</var>: et <var>.a.d.</var> ex <reg norm="quibus" type="simple">quibꝰ</reg> poſſibile ſit quadrilate <lb/>rum conſtitui, tale vero conſtituatur, vt aliquis circulus poſſit ipſum circunſcribere. <lb/></s> + <s xml:id="s2658" xml:space="preserve">imaginemur autem hoc factum eſſe, quod quidem quadrilaterum ſit <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 ſint <var>.q.d.</var> et <var>.a.b.</var> quæ ſe inuicem interſecent in puncto <var>.o.</var> vnde cum anguli <lb/>contra ſe poſiti circa <var>.o.</var> æquales inuicem ſint 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> ſimilis erit triangulo <var>.d.o.b.</var> et <var>.q.o.b.</var> ſimilis trian-<lb/>gulo <var>.a.o.d.</var> ex definitione. </s> + <s xml:id="s2660" xml:space="preserve">Vnde eadem proportio erit ipſius <var>.q.o.</var> ad <var>.b.o.</var> quæ ipſius <lb/><var>q.a.</var> ad <var>.b.d.</var> & ipſius <var>.b.o.</var> ad <var>.o.d.</var> eadem quæ <var>.q.b.</var> ad <var>.a.d.</var> & ipſius <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 compoſita ex <lb/>ea quæ eſt <var>.q.o.</var> ad <var>.o.b.</var> ex <var>.o.b.</var> ad <var>.o.d.</var> quæ nobis cognitę ſunt, mediante <lb/>proportione ipſius <var>.q.a.</var> ad <var>.b.d.</var> & ipſius <var>.q.b.</var> ad <var>.a.d.</var> proportio ſimiliter ipſius <var>.b.o.</var> <lb/>ad <var>.o.a.</var> nobis cognita erit, vt compoſita ex proportione ipſius <var>.b.o.</var> ad <var>.o.q.</var> & <lb/>ipſius <var>.o.q.</var> ad <var>.o.a.</var> cognitis, mediante proportione ipſius <var>.b.d.</var> ad <var>.q.a.</var> & ipſius <var>.q.b.</var> ad <lb/><var>a.d.</var> cum <reg norm="autem" type="context">autẽ</reg> proportio ipſius <var>.q.o.</var> ad <var>.o.b.</var> nobis cognita ſit, </s> + <s xml:id="s2661" xml:space="preserve">tunc nobis cognita erit <lb/>proportio ipſius <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> eſt vt <var>.a.o.</var> ad <var>.o.d.</var> ex ſimilitudine, <lb/></s> + <s xml:id="s2663" xml:space="preserve">quare proportio compoſiti ex primo, & quarto terminorum ad compoſitum ex .2. & <lb/>tertio, cognita erit. </s> + <s xml:id="s2664" xml:space="preserve">ſed quod fit ex <var>.q.d.</var> in <var>.a.b.</var> cognitum nobis eſt, vt æquale duobus <lb/>productis, hoc eſt ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex ſecunda primi Almageſti. </s> + <s xml:id="s2665" xml:space="preserve">quæ <lb/>producta nobis cognita ſunt, cum nobis data ſint eorum latera. </s> + <s xml:id="s2666" xml:space="preserve">Quapropter facta <lb/>cum fuerit figura quadrilatera rectangula ſimilis alicui alterirectangulæ figuræ pro <lb/>ductæ à duobus lateribus inuicem ita proportionatis, vt ſe habet <var>.q.d.</var> ad <var>.a.b.</var> æqua-<lb/>lis tamen duobus productis, hoc eſt 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. ſexti Eucli quæ quidem figura, exempli gratia, ſit <var>.u.t.</var> eius verò latera <lb/>ſint <var>.u.n.</var> et <var>.n.t.</var> Hæc enim dico æqualia eſſe <var>.q.d.</var> et <var>.b.a.</var> hoc eſt <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> ſed, <var>s.n.</var> æqualis ſit <var>.q.d.</var> et <var>.n.r.</var> æqua-<lb/>lis <var>.a.b.</var> <reg norm="ſintque" type="simple">ſintq́;</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">eademq́</reg> proportio erit <var>.u.n.</var> ad <var>.n.t.</var> quę <var>.s.n.</var> ad <var>.n.r.</var> eo <reg norm="quod" type="simple">ꝙ</reg> ita fa-<lb/>ctum fuit, cum autem ita ſit <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> ſed quia ita eſt <var>.u.n.</var> ad <var>.n.r.</var> vt <var>.s.n.</var> ad <var>.n.t.</var> ex 15. ſexti, </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 ſequitur <var>.n.t.</var> et <var>.n.r.</var> inuicem æquales eſſe, et <var>.u.n.</var> ſimiliter <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> ipſius quadrilateri, difficile <lb/>non erit eius angulos inuenire, eo <reg norm="quod" type="simple">ꝙ</reg> mediante <var>.a.b.</var> cognita, ſimul 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 circunſcriptibilis cuiuſuis di-<lb/>ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo <lb/>dem circulo circunſcriptum erit, ex communi ſcientia.</s> + </p> + <p> + <s xml:id="s2674" xml:space="preserve">SEd vt ipſa operatio facilior fiat, Sint eędem 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">quadrilaterũ</reg> effici. </s> + <s xml:id="s2675" xml:space="preserve">Videatur deinde primò quas volumus oppoſitas ſibi <lb/>inuicem eſſe, ponatur ergò ut <var>.q.a.</var> et <var>.b.d.</var> velimus oppoſitas inuicem facere, et <var>.q.b.</var> <lb/>cum <var>.a.d.</var> ſimiliter, accipiemus nunc <var>.K.</var> cuiuſuis magnitudinis, cui comparetur <var>.e.</var> <lb/>ita proportionata, vt <var>.q.b.</var> eſt ipſi <var>.a.d.</var> ex doctrina .10. ſexti 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 ſimiliter <var>.h.</var> ita pro-<lb/>portionata ad <var>.e.</var> et <var>.g.</var> ad <var>.k.</var> vt <var>.b.d.</var> eſt ad <var>.q.a.</var> vel <var>.g.</var> ad <var>.h.</var> vt <var>.a.d.</var> ipſi <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 compoſitum ſit <var>.g.e.</var> & <lb/>ita duæ <var>.K.</var> et <var>.h.</var> ex quibus ſit <var>.K.h</var>. </s> + <s xml:id="s2677" xml:space="preserve">Nunc ex iſtis 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/>ſit ex <var>.q.a.</var> in <var>.b.d.</var> reliquum verò ſit ex <var>.q.b.</var> in <var>.a.d.</var> quæ quidem ſint <var>.f.m</var>.</s> + </p> + <p> + <s xml:id="s2678" xml:space="preserve">Quo facto deſignetur rectangulum <var>.u.t.</var> ex .25. ſexti, quod æquale ſit duobus re-<lb/>ctangulis <var>.f.</var> et <var>.m.</var> ſimile tamen <var>.Z.</var> cuius rectanguli vnum latus correſpondet <var>.e.g.</var> reli-<lb/>quum verò <var>.K.h.</var> in proportione, ſed in æqualitate, vnum correſpondet <var>.q.d.</var> <reg norm="reliquum" type="context">reliquũ</reg> <lb/>vero <var>.a.b.</var> diametris ipſius quadrilateri.</s> + </p> + <p> + <s xml:id="s2679" xml:space="preserve">Accipiatur nunc latus illud quod correſpondet <var>.K.h.</var> hoc eſt ipſi <var>.a.b.</var> maius ſcili-<lb/>cet, & ſimul cum <var>.b.d.</var> et <var>.a.d.</var> formetur <reg norm="triangulum" type="context context">triãgulũ</reg> <var>.a.b.d.</var> ex .22. primi Eucli. circa quod <lb/>circunſcribatur circulus ex .5. quarti. </s> + <s xml:id="s2680" xml:space="preserve">& inuentum erit quod quęrebamus.</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/>abſque correctione ſemper 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> lapſus in correctione erroris nauis. <lb/>Et alij Petri Medinæ errores.</head> + <head xml:id="head364" xml:space="preserve">ILLVSTRISSIMO ANDREAE PROVANAE <lb/>Leinici Domino, Fruzaſci comiti, Aequiti Torquato, inthimo Sere-<lb/>niſsimi Sabaudiæ Ducis Conſiliario, <reg norm="eiuſque" type="simple">eiuſq́;</reg>, & ſacræ religio-<lb/>nis ſanctorum Mauritij, & Lazari Claſsi Præfecto.</head> + <p> + <s xml:id="s2681" xml:space="preserve"><emph style="sc">INter</emph> Eximiastuas virtutes, reinauticæ peritia Illuſtris emicat <lb/>merito ad te ſcribendum duxi, quod ad eam facultatem perti-<lb/>nens excogitaui, ſimul cum quibuſdam alijs inſtrumentis, vt non-<lb/>nihil commodi attuliſſe videar maritimis negotijs, & aliqua ex <lb/>parte animi mei erga te propenſionem indicauiſſe.</s> + </p> + <p> + <s xml:id="s2682" xml:space="preserve">PEr vnum <reg norm="eum" type="context">eũ</reg><reg norm="demque" type="simple">demq́;</reg> parallelum in primis <reg norm="abſque" type="simple">abſq;</reg> aliqua corre <lb/>ctione ſemper nauigari poſſe, omninò nego. </s> + <s xml:id="s2683" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, <reg norm="verum" type="context">verũ</reg> eſtid quod Petrus Nonius in <lb/>initio ſui operis oſtendit, ideſt nauim verſus æquatorem ſemper declinare: </s> + <s xml:id="s2684" xml:space="preserve">qui <reg norm="cum" type="context">cũ</reg> <lb/>corrigit errorem, fallitur, cum ipſe, eandem nauim, parallelam æquatori in vno ver <lb/>ticali ipſi æquatori propinquiori, & non in primo parallelo dirigit, itaque exiſtimat <lb/>in fine itineris, vbi deſcribit punctum <var>.o.</var> eam in eodem parallelo priori repe-<lb/>riri debere, quod <reg norm="verum" type="context">verũ</reg> <reg norm="non" type="context">nõ</reg> eſt, quia ea correctio efficit, vt motus nauis <reg norm="effectum" type="context">effectũ</reg> <reg norm="cuiuſdam" type="context">cuiuſdã</reg> <lb/><reg norm="deſcenſus" type="context">deſcẽſus</reg> <reg norm="per" type="simple">ꝑ</reg> ſcaligradum pręſtet, in quo à gradu in <reg norm="gradum" type="context">gradũ</reg> fiat deſcenſus, ſed ſi per gra <lb/>dus <reg norm="tantum" type="wordlist">tm̃</reg> aſcenderet <reg norm="quantum" type="context context">quãtũ</reg> <reg norm="deſcendit" type="context">deſcẽdit</reg>, <reg norm="dubium" type="context">dubiũ</reg> <reg norm="non" type="context">nõ</reg> eſt quin in fine ita eſſet ſe habitura <reg norm="quem" type="context">quẽ</reg> <lb/>ad modum in medio & in principio, cum verò ſemper deſcendat, abſque vlla aſcen-<lb/>ſione, neceſſariò ſic ſemper procedens, remota cum eſſent impedimenta terræ, ſub <lb/>æquatore reperiretur, ſub quo perpetuò circuiret globum.</s> + </p> + <p> + <s xml:id="s2685" xml:space="preserve">Idem ſub quolibet meridiano præſtare poteſt, ideſt vno <reg norm="eodemque" type="simple">eodemq́;</reg> vento circun-<lb/>uerti: </s> + <s xml:id="s2686" xml:space="preserve">ſed per alios circulos quam per hos duos (ſiue circulus magnus ſiue paruus) id <lb/>nunquam perfectè efficere poteſt, de parallelis iam manifeſtum eſt, cum impetus na <lb/>turalis corporum, quæ mota ſunt ſint ſemper in ſuperficiebus circulorum maiorum, <lb/>quorum circunferentię cum circunferentijs minorum, præter quam per vnum quod <lb/>dam punctum quando adinuicem contiguæ ſunt, aut per duo ideſt cum ſe ſe interſe-<lb/>cant non communicant, ita quod ad efficiendum, vt triremis aliqua, aut nauis, per <lb/>aliquem ex parallelis ad æquatorem moueatur, neceſſario ſit 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 conſtet virtus mouens remis, aut velis. </s> + <s xml:id="s2688" xml:space="preserve">Sed per quemlibet <reg norm="alium" type="context">aliũ</reg> <lb/>circulum maiorem, qui non ſit aut æquator aut aliquis ex meridianis, eſt penitus im <lb/>poſſibile. </s> + <s xml:id="s2689" xml:space="preserve">ideſt vt vnius venti vi nauis impellatur. </s> + <s xml:id="s2690" xml:space="preserve">Quod vt clarè pateat, ſit orizon <var>.<lb/>a.c.b.d.</var> & æquator <var>.c.q.t.d.</var> & vnus meridianorum ſit <var>.a.r.n.t.b.</var> in quo <var>.n.</var> ſit Zenit ſub <lb/>quo primum nauis reperiatur et <var>.r.</var> ſit polus ſeptentrionalis. </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> conſtituat angulum <var>.a.n.p.</var> ſeu <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 ipſius orizontis cum azimut, erit medio in loco inter <var>.b.</var> et <var>.c.</var> & ideo quarta <var>.n.f.</var> ip <lb/>ſius azimut ſecabit quartam <var>.c.t.</var> ipſius 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 ſimul 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 ſi voluerimus.</s> + </p> + <p> + <s xml:id="s2692" xml:space="preserve">Ponamus nunc nauem à puncto <var>.n.</var> diſcedere ſeu iter facere verſus <var>.u.</var> punctum, & <lb/> + <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> + in ipſo <var>.u.</var> reperiri, iam in hoc ſitu ha-<lb/>bebimus angulum huius ſecundi me-<lb/>ridiani <var>.r.u.p.</var> qui quidem in hoc caſu <lb/>minor eſſet angulo <var>.r.n.p.</var> extrinſeco <lb/>trianguli <var>.r.u.n.</var> ex conuerſo ſecundæ <lb/>partis .48. <reg norm="propoſitionis" type="simple">ꝓpoſitionis</reg> tertij lib. de <reg norm="trian gulis" type="context">triã<lb/>gulis</reg> Monteregij, ſeu ex .13. primi Me <lb/>nelai, cuius anguli <var>.u.</var> arcus orizontalis <lb/>ſit <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. verſus <var>.q.</var> diuerſus ab illo qui <lb/>prius ab <var>.n.</var> verſus <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 eſt quod aliquo modo fieri <lb/>non poteſt, vt nauis ab aliquo loco ad <lb/>alium, breuiſſimo interuallo ire poſſit <lb/>ideſt per gyrum circuli maioris ſphæræ vno tantummodo <reg norm="vento" type="context">vẽto</reg> eam impellente, præ <lb/>ter quam in ęquatore, ſeu in aliquo quouis meridianorum, nos autem ire per gyrum <lb/>alicuius paralleli dementia eſſet, niſi neceſſitas cogeret.</s> + </p> + <p> + <s xml:id="s2694" xml:space="preserve">Huiuſmodi demonſtrationis ope, quantum decipiatur Petrus Medina cap .6. lib. <lb/>3. cognoſcitur, vbi ſic ſcribit; </s> + <s xml:id="s2695" xml:space="preserve">Vbicunque locorum reperiatur homo, aliquem circu-<lb/>lum qui vniuerſum ambiat imaginatione ſibi confingens, per totum eum circulum <lb/>vno <reg norm="eodemque" type="simple">eodemq́;</reg> vento nauigatio ſuſcipitur. </s> + <s xml:id="s2696" xml:space="preserve">Ex hac etiam demonſtratione, quàm fal <lb/>ſa ſit charta maritima patet, cuius beneficio exiſtimant nautę ſe per breuiſſimum iter <lb/>a loco ad locum vehi etiamſi dicti loci non ſint ambo in æquatore, aut in aliquo me <lb/>ridiano, ſed extra dictos circulos vnico tantum vento impellente & ſi 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 ſuperius igitur dicta demonſtratione iam oſtendi, quod ſi velimus vehi ab vno lo <lb/>co ad alium beneficio alicuius circuli maioris, præter duos iam dictos, hoc fieri non <lb/>poteſt vno <reg norm="eodemque" type="simple">eodemq́;</reg> <reg norm="vento" type="context">vẽto</reg> impellente. </s> + <s xml:id="s2698" xml:space="preserve">Vnde ſequitur, omnia ea interualla quæ vno <lb/><reg norm="eodemque" type="simple">eodemq́</reg> vento tranſibimus 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 eſ-<lb/>ſe, planiſphęrium Prolomei & Iordani citat, non animaduertit quam diuerſo mo-<lb/>do a charta maritima huiuſmodi inſtrumentum ſit fabricatum, cum exceptis orizon <lb/>te recto, & meridiano in dicto inſtrumento quilibet alius circulus ſit circulus, ſiue ſit <lb/>almicantarat, ſiue azimur, ſiue æquator, ſiue tropicus, ſiue zodiacus, ſiue alius quiuis <lb/>circulus, eum in charta maritima ne vna quidem ſit linea, quę non ſit 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 poſitæ meæ demonſtrationis ope, deuenimus in cognitionem magnitu <lb/>dinis arcus <var>.n.q.</var> cognoſcimus etiam angulum <var>.n.q.t.</var> vnde nobis <reg norm="manifeſtum" type="context">manifeſtũ</reg> eſſet quo <lb/>vento oporteret iter facere. cum à puncto <var>.q.</var> nauis aliqua diſceſſura eſſet, in eodem <lb/>azimut propoſito. </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 ſupponitur, ſimul 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 cognoſcendam diſtantiam <lb/>vnius meridiani ab alio præſcribit, in genere eſt falſus, etiam ſi is ab antiquis eum de <lb/>ſumat, qui, hic non viderunt quam magna inter meridianos differentia ſit interuallo <lb/>rum eorum quæ ſunt vicina polis & eorum quæ ſunt circa æquatorem.</s> + </p> + <p> + <s xml:id="s2703" xml:space="preserve">Falſus eſt etiam modus ab eo traditus ad cognoſcendos gradus longitudinis per <lb/>medium itineris cogniti in quouis parallelo extra æquatorem facti, & hoc cap .14. li <lb/>bri tertij eiuſdem, & primo cap. lib. 4. <reg norm="continetur" type="context">cõtinetur</reg>, vbi .17. leucas cum dimidia cuilibet <lb/>gradui tam paralleli quàm meridiani aſſignat.</s> + </p> + <p> + <s xml:id="s2704" xml:space="preserve">Falſum eſt etiam quod ab eo aſſeritur, Solem, cum reperiretur in æquatore, circa <lb/>eos qui ſub ipſo æquatore habitant, vnius diei <reg norm="noctisque" type="simple">noctisq́;</reg> ſpatio per omnes uentos cir-<lb/>cunuolui. </s> + <s xml:id="s2705" xml:space="preserve">quia illis æquator idem eſt cum verticali, qui duos tantum rhumbos pro-<lb/>ducit, ideſt orientis, & occidentis: </s> + <s xml:id="s2706" xml:space="preserve">hic verò error, in ſecundo cap. lib. 6. habetur.</s> + </p> + <p> + <s xml:id="s2707" xml:space="preserve">Falſum eſt etiam quod profert Solem ijs qui habitant ſphæram obliquam, qua-<lb/>libet hora tertia, regulariter ab vno rhumbo ad alium ex præcipuis ideſt ab vno azi <lb/>mut ad alium progredi, quemadmodum eadem cap .2. lib. 6. et .7. cap. ſeptimi libr. <lb/>ſcribit. </s> + <s xml:id="s2708" xml:space="preserve">Huius autem rei falſitas ita facile depræhendetur, ponamus hemiſphęrium <lb/>orientale, verbi gratia, cuius meridianus ſit <var>.p.z.b.</var> æquator <reg norm="autem" type="wordlist">aũt</reg> <var>.e.m.</var> vnus verò paral <lb/>lelorum ſeptentrionalium ſit <var>.c.a.</var> in quo Solem exiſtere ponamus, orizon autem ſit <var>.<lb/>b.m.</var> zenit vero <var>.z.</var> polus arcticus <var>.p.</var> ſit poſtea azimut <var>.z.q.</var> à meridiano diſtans per gra <lb/>dus .45. qui quidem azimut in hoc hemiſphęrio erit rhumbus illius venti, quem uul-<lb/>go Itali Sirocum dicunt, et <var>.z.m.</var> ſit azimut verticalis qui in hoc hemiſphærio erit <reg norm="ron" type="context">rõ</reg> <lb/>bus venti orientalis, ita <reg norm="quod" type="simple">ꝙ</reg> ſecundum Medinam à rhumbo <var>.z.m.</var> ad <var>.z.q</var>. </s> + <s xml:id="s2709" xml:space="preserve">Solabſoluet <lb/>ſpatium <reg norm="temporis" type="context">tẽporis</reg> trium horarum, & aliud <lb/>æquale temporis ſpatium abſoluet à <reg norm="rhum" type="context">rhũ</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 ipſo 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">nũc</reg> Solem reperiri in ęqua <lb/>tore, vbi per ipſum Medinam arcus <var>.u.m.</var> <lb/>ſimiliter eſſet graduum .45. & ſic <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. ſed cum <lb/>arcus <var>.e.u.</var> ſit graduum .45. ex ſuppoſito <lb/>ipſius Medinæ, ſequeretur arcum <var>.e.f.</var> <lb/>æqualem eſſe arcui <var>.e.u.</var> pars igitur æqua <lb/>lis erit ſuo 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 diſtantijs Solis à meridiano ſcribit, hanc opinionem Petrià Medina & <lb/>corum qui idem ei perſuaſerunt falſam eſſe demonſtrat.</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">Falſum eſt 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> inſularum quæ azore dicuntur, pyxidem <lb/>verſus eum ventum qui vulgò Græcus dicitur, & ab occidentali verſus eum qui Ma <lb/>giſter dicitur, vergere, huius rei nulla eſt ratio.</s> + </p> + <p> + <s xml:id="s2714" xml:space="preserve">Ego enim huiuſmodi rationem reperiri poſſe contendo, quæ talis eſt, quia pars <lb/>roſæ (ut vocant) à magnete tacta, ad aliquod punctum, aut ſitum globi terrę, in eo-<lb/>dem meridiano inſularum, quæ Azore dicuntur, vltra ſitum poli arctici in terra diri-<lb/>geretur, ita vt ſitus dicti poli in terra eſſet in dicto meridiano, inter locum qui ab in <lb/>dice roſæ aut pyxidis reſpiceretur, & dictas inſulas, id quod ſuperius ſcripto meridia <lb/>no facile cognoſci poteſt, ſumendo pro inſulis ſitum <var>.e.</var> in meridiano et <var>.z.</var> pro polo, <lb/>et <var>.p.</var> pro loco qui à pyxide ſit viſus, <reg norm="imaginando" type="context">imaginãdo</reg> deinde pyxidem in <var>.f.</var> magis orienta <lb/>li quam eſt <var>.e.</var> clarum eſt lineam quæ reſpicit (ponamus) <var>f.p.</var> verſus Græcum & ab alia <lb/>parte verſus Magiſtrum 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, ſeu nauigatoriæ chartę, quemadmodum aliquoties inter nos ſermonem <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">efficeretq́;</reg> vt nauis ſuper aquę globum, beneficio circulorum maiorum, quam <lb/>optimè poſſet, ideſt 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">ꝙ</reg> mihi ex <lb/>animi voto ſucceſſurum putaui, beneficio <reg norm="quinque" type="simple">quinq;</reg> circulorum circundantium <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>globum <reg norm="terreſtrem" type="context">terreſtrẽ</reg> & maritimum, quales ij ſunt qui in inferiori Germania à Gerardo <lb/>Mercatore ſtruuntur, qui vno pede cum dimidio diametri conſtet, ideſtſeſquipede.</s> + </p> + <p> + <s xml:id="s2718" xml:space="preserve">Sit ergo, exempli gratia, huiuſmodi globus <var>.a.b.d.</var> circa quem duo circuli, aut cir<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 ſe globi polos mediantibus extremitatibus axis mun-<lb/>di contineat, qui quidem poli à punctis ſuarum interſectionum per <reg norm="quartam" type="context">quartã</reg> ex æquo <lb/>in punctis <var>.f.</var> et <var>.g.</var> ita diſtent, vt globus circa eoſdem, in ſitu longitudinis mundi vol-<lb/>ui poſſit. </s> + <s xml:id="s2719" xml:space="preserve">Huiuſmodi 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 ſit <var>.h.e.K.</var> cum primo ad angulos rectos in puncto <var>.e.</var> & <lb/>in ſuo oppoſito connexus, & is appellabitur æquator, & poli <var>.f.g.</var> primi poli dicentur.</s> + </p> + <p> + <s xml:id="s2721" xml:space="preserve">Circa huiuſmodi duos circulos, alios <reg norm="etiam" type="context">etiã</reg> duos exiſtere <reg norm="vellem" type="context">vellẽ</reg> ſimul <reg norm="coniunctos" type="context context">cõiũctos</reg> medio <lb/>ad angulos rectos. </s> + <s xml:id="s2722" xml:space="preserve">In quibus <reg norm="quidem" type="context">quidẽ</reg> interſectionis punctis ſint duo poli, qui hos duos <lb/>circulos cum ſecundo priorum ideſt cum æquatore in duobus punctis inuicem op-<lb/>poſitis connectant; </s> + <s xml:id="s2723" xml:space="preserve">quæ æquatoris puncta à punctis interſectionis eiuſdem cum ſuo <lb/>deferente, ratione vna quarta diſtent, quorum duorum circulorum primus ſit <var>.n.i.<lb/>m.</var> quem deferentem azimut appellabimus; </s> + <s xml:id="s2724" xml:space="preserve">ſecundus <var>.r.n.s.m.</var> azimut locorum no<lb/>minabimus. </s> + <s xml:id="s2725" xml:space="preserve">eorundem interſectionis rectæ, puncta ſint <var>.n.</var> et <var>.m.</var> à quibus duo poli ex <lb/>aurichalco confecti ſimiles primis <var>.n.h.</var> et <var>.m.K.</var> vſque ad puncta <var>.h.</var> et <var>.K.</var> æquatoris <lb/>perueniant, qui ſpisſitudinem æquatoris diſtantem 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 ſitu latitudinis mundi verti queat. </s> + <s xml:id="s2726" xml:space="preserve">Et <lb/>hos, ſecundos polos nominabimus.</s> + </p> + <p> + <s xml:id="s2727" xml:space="preserve">Alius deinde circulus <var>.q.i.p.</var> duos poſteriores circulos ambiat, cum deferente ta-<lb/>men azimut mediantibus duobus polis in puncto <var>.i.</var> & in ſuo oppoſito exęquo diſtan <lb/>tibus à ſecundis polis vnius quartæ ſpatio 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 vniuerſa <lb/>lem vocabimus. </s> + <s xml:id="s2729" xml:space="preserve">Hic vero orizon ſuper quatuor quartas circuli, aut ſuper quatuor <lb/>paruis columnis, ut fieri ſolet innixis ſuæ baſi, 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 conſtet, incipiendo ab interſectionibus <var>.e.</var> & eius op <lb/>poſito æquatoris, & numeri in polis <var>.f.</var> et <var>.g.</var> globi finem ſortiantur. </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> verſus <var>.K.</var> deferens autem azi <lb/>mut <var>.n.i.m.</var> ab omni diuiſione liber maneat, ſed azimut <var>.n.s.m.r.</var> in .360. gradus inci <lb/>piendo à puncto <var>.n.</var> verſus <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 ſit nonaginta <reg norm="graduum" type="context">graduũ</reg> <lb/>incipiendo à puncto <var>.i.</var> & eius oppoſito ideſt à polis poſtremis & terminando in pun-<lb/>ctis <var>.q.</var> et <var>.p.</var> in medio ipſorum polorum, & quarta <var>.i.p.</var> orientalis ſeptentrionalis, et <var>.i.<lb/>q.</var> orientalis meridiana appellentur. </s> + <s xml:id="s2733" xml:space="preserve">& ſic ordine ſeruato occidentales.</s> + </p> + <p> + <s xml:id="s2734" xml:space="preserve">Præterea pręparata ſit quædam quarta, ex aurichalco, circuli æqualis ipſi orizon-<lb/>ti, & in .90. gradus diſtincta quæ cum quauis ſuarum extremitatum ipſi zenit, in azi-<lb/>mut applicari poſſit, quemadmodum circa globos cęleſtes fieri ſolet; </s> + <s xml:id="s2735" xml:space="preserve">quę quidem ad <lb/>cognoſcendam altitudinem poli ipſius globi ab orizonte nobis inſeruiet.</s> + </p> + <p> + <s xml:id="s2736" xml:space="preserve">Atque hac ratione hanc noſtram machinam perfectè abſoluemus <reg norm="quam" type="context">quã</reg> appellan-<lb/>dam eſſe Armillam nauticam ſentio. </s> + <s xml:id="s2737" xml:space="preserve">Hic autem illud non omittam, concauum <lb/>duorum priorum circulorum à ſuperficie globi non nimis diſtare debere & con-<lb/>cauum aliorum à ſuperficie conuexa priorum longe poſitos eſſe <reg norm="non" type="context">nõ</reg> debere, & con <lb/>cauam orizontis à conuexa ſecundorum procul abeſſe non debere.</s> + </p> + <p> + <s xml:id="s2738" xml:space="preserve">Neque illud etiam prætermittendum eſt, opere pretium fore ſi in interſectione <lb/>e. priorum, erit foramen elicum, vt clauo elico ex aurichalco confecto, poſſimus <lb/>ſiſtere globum, quando oportuerit, ne amplius circa primos ſuos polos <var>.f.g.</var> circun-<lb/>uoluatur, cum <reg norm="ſitus" type="simple">ſitꝰ</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, ideſt <var>.f.e.g.</var> aliud quoddam foramen huiuſmodi ſit poſitum, 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 vſque ad circulum <var>.f.e.g.</var> perueniens, æquatorem ſiſtere poſſit, ne circa <lb/>ſecundos polos <var>.h.</var> et <var>.K.</var> amplius moueatur quum noluerimus eum mutare ſitum.</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">Deuſu Armillæ nauticæ.</head> + <p> + <s xml:id="s2740" xml:space="preserve">V Tautem noſtra Armilla nautica vti poſſimus pyxidem nos prius oportebit <lb/>habere, diuerſam tamen ab ijs, quibus nautæ hactenus vſi 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, ſed ratione graduum orizontis in .360. partes diſtincti, atque ob hanc cauſam <lb/>ſentio, vt ima pars pyxidis penitus detecta videatur, & in .360. partes dinidatur, <reg norm="nilque" type="simple">nilq́</reg> <lb/>aliud quam quandam lanceo lam ſupra eius acum eſſe volo, quæ dum mouebitur na <lb/>uis, per gradus quamlibet orizontis partem oſtendet; </s> + <s xml:id="s2742" xml:space="preserve">hos autem .360. gradus, ita ſe <lb/>habere volo, vt quęlibet quarta .90. contineat, <reg norm="ſupputatioque" type="simple">ſupputatioq́;</reg> à linea meridiana inci-<lb/>piat, & in verticali deſinat, vt huiuſmodi diuiſio cum ea, quæ eſt orizontis Armillæ <lb/>eadem ſit.</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 diuerſis meridia-<lb/>nis quomodolibet a dinuicem diſtantes, à quorum vno ad alium ſit nauigandum iti-<lb/>nere quo ad fieri poterit breuiori, ideſt per gyrum circuli maioris, dixi autem extra <lb/>æquatorem, ideſt vt ambo, nec in æquatore, nec in uno <reg norm="eodemque" type="simple">eodemq́;</reg> meridiano <reg norm="exiſtant" type="context">exiſtãt</reg>, <lb/>quia vt aliàs dixi in huiuſmodi 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">circũuolutione</reg> globi circa primos polos <var>.f.g.</var> & æqua <lb/>toris circa ſecundos <var>.h.K.</var> hoc eſt per longitudinem, & latitudinem, hi duo loci in <lb/>globo propoſiti ſub azimut <var>.r.n.s.m.</var> ſecundorum circulorum <reg norm="ſitum" type="context">ſitũ</reg> ſortiantur, qui azi-<lb/>mut orizontem in punctis <var>.q.</var> et <var>.p.</var> ſemper ad angulos rectos diſpeſcit <reg norm="ibique" type="simple">ibiq;</reg> globum <lb/>ita quieſcere vt circa polos <var>.f.g.</var> non voluatur, & æquatorem etiam ſic ſirmare, vt cir-<lb/>ca ſecundos polos <var>.h.K.</var> non vertatur faciamus.</s> + </p> + <p> + <s xml:id="s2745" xml:space="preserve">Quod cum factum fuerit, ſecundorum circulorum primus, qui eſt <var>.n.i.m.</var> deferens <lb/>azimut, circa tertios polos <var>.i.</var> & eius oppoſitum, eo uſque voluatur quouſque prior <lb/>globi locus, ideſt is a quo iter eſt incohandum per .90. gradus azimut diſtet ab ori-<lb/>zonte, ideſt ſub zenit orizontis <var>.q.i.p.</var> ſit poſitus, quemadmodum, exempli gratia, ſi <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 diſtans ab <var>.q.</var> et <var>.p.</var> <lb/>ideſt per .90. gradus poneretur ſub <var>.r</var>.</s> + </p> + <p> + <s xml:id="s2746" xml:space="preserve">Conſideretur deinde vbi æquator <var>.h.e.K.</var> ſecundus circulus duorum primorum, <lb/>ab orizonte <var>.q.i.p.</var> ſecabitur, exempli gratia, in puncto <var>.c.</var> quartę orientalis ſepten-<lb/>trionalis eiuſdem orizontis. </s> + <s xml:id="s2747" xml:space="preserve">Videatur deinde quot nam gradibus conſtabit ar-<lb/>cus <var>.i.c.</var> & per totidem gradus conſtituatur extremitas ſeptentrionalis lineæ meridia-<lb/>nę pyxidis nauticę, diſtantis à cuſpide ſeptentrionali ipſius lanceolæ orientem ver-<lb/>ſus, mediante nauis circunuolutione. </s> + <s xml:id="s2748" xml:space="preserve">vnde ipſamet nauis in huiuſmodi ſitu azimut, <lb/>qui per duos hos loc<unsure/>os tranſit, dirigetur, eſſiciendo vt eius prora verſus <reg norm="locum" type="context">locũ</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/>ſeu leucarum iter conficiemus, quot quarta pars vnius gradus requirit. </s> + <s xml:id="s2750" xml:space="preserve">& dum <lb/>hociter abſoluitur, ille qui pręeſt naui, defferentem azimut <var>.n.i.m.</var> circa ſuos polos <var>.i.</var> <lb/>& eius oppoſitum, ſic circunuoluat, vt interſe ctio azimut <var>.r.n.s.m.</var> cum orizonte <var>.q.i.<lb/>p.</var> diſtet à prima ratione dictæ quartę partis vnius gradus, conſtituendo ſecundum lo <lb/>cum, proximiorem zenit, ratione dictæ quartæ partis gradus azimut. </s> + <s xml:id="s2751" xml:space="preserve">Hiſce ita pera <lb/>ctis, obſeruetur deinde vbiæquator <var>.h.e.K.</var> hac ſecunda vice interſecabit orizontem <lb/><var>q.i.p.</var> quod quidem interſectionis punctum ſemper appelletur <var>.c.</var> quod dico non am <lb/>plius in eadem diſtantia manſurum, ut prius à puncto <var>.i.</var> ſed aut longius diſtabit, aut <lb/>propius accedet, vt in præſenti exemplo. </s> + <s xml:id="s2752" xml:space="preserve">quemadmodum ex ſe manifeſtum eſt, cú <lb/>poli globi, ideſt ęquatoris ſint extra azimut, vt præſupponitur, quia loci ſunt in diuer <lb/>ſis meridianis.</s> + </p> + <p> + <s xml:id="s2753" xml:space="preserve">Pro huiuſmodi autem diſtantiæ ratione denuo dirigatur nauis prout æquator <var>.h.<lb/>e.K.</var> in orizonte <var>.q.i.p.</var> nobis oſtendet, atque hoc modo omnium iter quaſi breuiſſi-<lb/>mum fiet. </s> + <s xml:id="s2754" xml:space="preserve">dico autem, quaſi, quia omnibus modis neceſſariò conficitur iter contor-<lb/>tum & in formam ſerpigineæ lineæ. </s> + <s xml:id="s2755" xml:space="preserve">Applicantes deinde per vices extremitatem <lb/>quartæ appoſitæ (de qua ſuperius mentionem fecimus) ipſi zenit <var>.r.</var> efficientes ut per <lb/>ſitum poli globi pertranſeat, deueniemus in cognitionem altitudinis eiuſdem ab <lb/>orizonte, & per conſequens quantum itineris per latitudinem eiuſdem globi pere-<lb/>gerit. </s> + <s xml:id="s2756" xml:space="preserve">mediante deinde interſectione orizontis <var>.q.i.p.</var> cum æquatore, cognoſcemus <lb/><reg norm="quantum" type="context">quãtum</reg> itineris per longitudinem eiuſdem globi, in ipſo ęquatore 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, & occaſum Lunacognoſcendum <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 Illuſtriſs. <reg norm="modum" type="context">modũ</reg> <reg norm="conficiendi" type="context">conficiẽdi</reg> inſtrumenti nuper à me <reg norm="inuenti" type="context">inuẽti</reg>, vt tibi ſi <lb/>gnificaui, quo ſcire poſſis fermè in dies, qua hora (de aſtronomicis loquor) ad <lb/>determinatum parallelum & abſque multa ſupputatione, etiam abſque Aſtrolabio <lb/>Luna oriatur <reg norm="occidatque" type="simple">occidatq́;</reg>. </s> + <s xml:id="s2758" xml:space="preserve">In quo inſtrumento poteris etiam videre quo in ſigno Sol, <lb/>& ſæpius itidem Luna permeat, & huiuſce aſpectus cum Sole, atque longitudinem <lb/>diei <reg norm="noctisque" type="simple">noctisq́;</reg> toto anni tempore exactè diſcernere.</s> + </p> + <p> + <s xml:id="s2759" xml:space="preserve">Circularis lamina ex argento, aut ære, aliaúe materia paranda eſt, in cu-<lb/>ius ſuperficie ambarum facierum Zodiacus delineabitur, modo inferius depicto, <lb/>deinde pro anno quinque circuli ſibi inuicem <reg norm="concentrici" type="context">cõcentrici</reg>, at reſpectu Zodiaci excen <lb/>trici cęlabuntur in ea, adeo vt vtriuſque centri diſtantia ſit pro .32. parte ſemidia-<lb/>metri concauitatis Zodiaci è regione locis augis, temporis qui noſtra ætate circa ſi-<lb/>nem ſecundi gradus cancri inuenitur, eandem viam, in hoc, ſequuti, quam Stofle-<lb/>rus in dorſo Aftrolabij docet. </s> + <s xml:id="s2760" xml:space="preserve">At nomina menſium media ponantur inter duos <lb/>maiores circulos, poſtea inter ſecundum, & tertium ab vna facierum laminæ, ar-<lb/>cus ſemidiurni, ab altera vero arcus ſeminocturni, per quinos quoſque dies collo-<lb/>centur, ita exactè, vt hic ſubtus videbis. </s> + <s xml:id="s2761" xml:space="preserve">adeo vt numeri dierum & ipſorum dierum <lb/>ſigna ſint in interuallis vicinioribus centro communi dictorum quinque circulorum.</s> + </p> + <p> + <s xml:id="s2762" xml:space="preserve">Poſteaquam ab vna & altera <reg norm="facierum" type="context">facierũ</reg> laminæ hæc inſculpra fuerint, aliæ duæ circu <lb/>lares laminæ, magnitudinis ſemidiametri minimi quinque circulorum accipiantur: <lb/></s> + <s xml:id="s2763" xml:space="preserve">quarum vna pro ortu, & altera pro occaſu Lunæ deſeruiet. </s> + <s xml:id="s2764" xml:space="preserve">In qualibet ipſarum <lb/><reg norm="conſtituentur" type="context">conſtituẽtur</reg> circuli quatuor, eo modo qui paulo inferius cernitur, quos omnes diui-<lb/>demus in triginta ſpacia æqualia: </s> + <s xml:id="s2765" xml:space="preserve">& in interuallo <reg norm="quod" type="wordlist">qđ</reg> inter duos primos circulos poſi <lb/>tum eſt, triginta dies <reg norm="annotabimus" type="simple">annotabimꝰ</reg> qui ipſos Lunætriginta dies <reg norm="pręſcribent" type="context">pręſcribẽt</reg>, vt in figara.</s> + </p> + <p> + <s xml:id="s2766" xml:space="preserve">Poſtmodum in lamina quæ ortus Lunæ indicabit, ac duorum maiorum <reg norm="circulo- rum" type="context">circulo-<lb/>rũ</reg> interuallo è regione numeri .1. videlicet primi diei, ponemus horas .12. & minuta <num value="48">.<lb/>48.</num> ex aduerſo diei ſecundi ho .13. et min .36. ex oppoſito tertij ho 14. min .24. & ſic <lb/>ſucceſſiuè augendo per min .48. & indicem è diuerſo diei .30. <reg norm="ſtatuendo" type="context">ſtatuẽdo</reg>, qui coitus <lb/>Lunæ cum Sole ſigniſicabit: </s> + <s xml:id="s2767" xml:space="preserve">atque lineas aſpectuum, vt inferius videre eſt facilè in <lb/>ueniemus.</s> + </p> + <p> + <s xml:id="s2768" xml:space="preserve">Altera in lamina quæ occaſum Lunæ indicabit, poſtquam diſtincta fuerit, vt alte-<lb/>ra .30. dies ac cęteræ lineæ, eo modo quo in ſuperiori collocabuntur, at numeri in-<lb/>terualli maioris, aliter diſponentur, vt potè ex aduerſo diei primi ſolum .48. minu-<lb/>ta deſcribi debent, è directo ſecundi diei ponenda erit hora vna cum minutis .36. & <lb/>è regione tertij inſcribentur .2. horæ, & min .24. & ſic ex ordine per .48. minuta au-<lb/>gendo.</s> + </p> + <p> + <s xml:id="s2769" xml:space="preserve">Nunc lamina ortus Lunę, cum anno arcuum ſeminocturnorum, & illa occaſus cum <lb/>anno arcuum ſemidiurnorum <reg norm="concentrari" type="context">concẽtrari</reg> debet, & ita noſtrum inſtrumentum perfe-<lb/>ctum erit & abſolutum.</s> + </p> + <p> + <s xml:id="s2770" xml:space="preserve">Quoties igitur voluerimus medio inſtrumento dignoſcere fermè in tali orizonte <lb/>qua hora Luna oriatur, ita neceſſe erit volubilem rotam ortus flectere, ut index ve <lb/>niat è regione diei menſis in quo talis operatio fit & talirota ſirma manente perſpi- + <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 aduerſo diei Lunæ, numerum horarum & minutorum in maiori interuallo <lb/>ipſius rotæ notatorum, qui cum arcu ſeminocturno anni, quo cum in ipſa rectitudi-<lb/>ne centri conueniet colligetur, & ſumma quæ ex tali ſupputatione proueniet aper-<lb/>tas faciet horas aſtronomicas, quibus ferè etſi non exactè in die propoſito Luna <lb/>orietur. </s> + <s xml:id="s2771" xml:space="preserve">Idipſum fiet pro occaſu Lunæ.</s> + </p> + <p> + <s xml:id="s2772" xml:space="preserve"><emph style="sc">DIem</emph> ætatis Lunæ iam totus orbis ſcit inuenire, media ſupputatione nu-<lb/>meri Epactę currentis cum numero menſium, ſumpto principio à Martio, <lb/>adiunctis diebus menſis currentis, & detracto numero .30. à ſumma prędicta, ſi ab ip <lb/>ſa dictus numerns .30. ſuperatur.</s> + </p> + <p> + <s xml:id="s2773" xml:space="preserve">Sed ne aliquis putet ſufficere tantummodo additionem quatuor quintarum ho-<lb/>rę qualibet die. à nouilunio inchoando, ſciendum eſt huiuſmodi receſſum Lunæ <lb/>(quamuis non ita exactæ fiat) non <reg norm="computandum" type="context">computandũ</reg> eſſe ab orizonte aliquo, ſed à recto, <lb/>ſeu à meridiano quod idem eſt, quemadmodum <reg norm="vnicuique" type="simple">vnicuiq;</reg> mediocriter erudito pa-<lb/>tere poteſt. </s> + <s xml:id="s2774" xml:space="preserve">At propoſitum nobis non eſt ſcire qua hora Luna in meridiano repe-<lb/>riatur, ſed in noſtro obliquo orizonte, in parte orientali ſeu occidentali, </s> + <s xml:id="s2775" xml:space="preserve">propterea <lb/>igitur addendus eſt, ei ſummæ temporis, qua Luna diſtat à meridiano, arcus ſemi-<lb/>diurnus, vel ſeminocturnus illius loci Zodiaci, in quo Luna reperitur illa die in pro <lb/>poſito parallelo, vt ſciatur proximę, qua hora (ex aſtronomicis) 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">ſupra dicta enim additio quatuor quin <lb/>tarum horæ tantummodo, ſufficiens erit temporibus æquinoctij, ſed 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 occaſu. <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 collendiſs. anno .1570. conſtruxi.</head> + <head xml:id="head371" style="it" xml:space="preserve">CLARISS. FRANCISCO BARBARO VENETORVM <lb/>apud Sereniſſimum Sabaudiæ Ducem Oratori Illuſtriſſimo.</head> + <p> + <s xml:id="s2777" xml:space="preserve"><emph style="sc">HEron</emph> varias ac diuerſas hydraulicas, & ſpiritales machinas propoſuit, in <lb/>ter quastamen nullam ſimilem, ei <reg norm="quam" type="context">quã</reg> ego Sereniſſimo Sabaudiæ Duci <lb/>conſtruxi, deſcribit, quæ quidem fuit Lucerna, & erat huiuſmodi, vt à <lb/>magno aliquo vaſe oleo pleno ſupra alicuius triclinij tabulatum poſito, <lb/>ſubtilis 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 vſque ad <lb/>medium deſcenderet, ita tamcn <lb/>vt hic ſolus 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, ſimile co <lb/>operculo alicuius pyxidis, è cuius <lb/>ambitu prope baſim multi diuer-<lb/>ſi quæ tubi æquales & orizonta-<lb/>les, cuiuſuis longitudinis proſili-<lb/>rent, quorum quilibet in extremi <lb/>tate ſua, 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">chnũ</reg> eſſet <reg norm="cum" type="context">cũ</reg> mixo. </s> + <s xml:id="s2778" xml:space="preserve"><reg norm="oleum" type="context">oleũ</reg> deinde <lb/>medio <reg norm="perpendicularis" type="simple context">ꝑpẽdicularis</reg> tubi ad rece <lb/><reg norm="ptaculum" type="context">ptaculũ</reg> extrinſece deſcendebat, <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">cũ</reg> eædem erant extinctæ <lb/>ne minima quidem olei gutta de <lb/>ſcendebat: </s> + <s xml:id="s2780" xml:space="preserve">id quod eos qui aſta-<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 ſic erat <reg norm="con" type="context">cõ</reg> <lb/>ſtructa. </s> + <s xml:id="s2782" xml:space="preserve">Vas <reg norm="olearium" type="context">oleariũ</reg> cylindricum <lb/>vt in ſubſcripta figura patet, cuiuſ <lb/>uis magnitudinis, omni ex parte <lb/>clauſum faciendum curaui, ita ta <lb/>men vt eius coopereulum <reg norm="aliquam" type="context">aliquã</reg> <lb/><reg norm="tulum" type="context">tulũ</reg> concauum eſſet, in cuius me <lb/>dio erat foramen <var>.e.</var> quod erat os <lb/>tubi <var>.e.g.</var> qui ſub <reg norm="eiuſdem" type="context">eiuſdẽ</reg> vaſis fun-<lb/>do vſque ad <var>.g.</var> tranſibat, ſed po-<lb/>ſtea ſurſum, quaſi <reg norm="vſque" type="simple">vſq;</reg> ad cooper <lb/>culum in ſitu <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 poſtea 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">contignationemq́;</reg> vſque <lb/>ad centrum ſupradictireceptaculi (circa quod <reg norm="tuborum" type="context">tuborũ</reg> ope appenfæ erant ellychnio-<lb/>rum pyramides) tranſibat, atque huiuiuſmodi tubi <var>.n.u.</var> extremitates tam ſuperius <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 vaſis fundo feceram, cui <lb/>paruum <reg norm="tubum" type="context">tubũ</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">ꝑ</reg> hunc reflexum tu <lb/>bum <var>.a.o.t.</var> oleum vaſis exibat, <reg norm="quod" type="simple">ꝙ</reg> <lb/>per oſculum <var>.t.</var> in quendam cana-<lb/>lem tubo <var>.n.u.</var> inſertum, ab extra <lb/>oleum effundebat, & ab exteriori <lb/>parte arundinis <var>.n.u.</var> ingredieba-<lb/>tur receptaculum appenſum ex-<lb/>tremo dicti tubi <var>.n.u.</var> quod extre-<lb/>mum <reg norm="apertum" type="context">apertũ</reg> erat, ut dixi, & à fun-<lb/>do receptaculi tantum <reg norm="diſtans" type="context">diſtãs</reg>, <reg norm="quam" type="context">quã</reg> <lb/>tum volebamus oleum ab ipſius <lb/>receptaculi fundo altum exiſtere <lb/>quod quidem oleum ſtatim vt ad <lb/>oſculum 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 conſequens, neque <lb/>amplius <reg norm="oleum" type="context">oleũ</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="tubum" type="context">tubũ</reg> <var>.a.o.t.</var> eflue-<lb/>bat, nec etiam per <reg norm="tubum" type="context">tubũ</reg> <var>.e.g.c.</var> aer <lb/>ingredi poterat, cum <var>.c.g.</var> ſemper <lb/>oleo exiſteret <reg norm="plenum" type="context">plenũ</reg> <reg norm="quum" type="context">quũ</reg> <var>.g.</var> magis <lb/><reg norm="quam" type="context">quã</reg> <var>.t.</var> <reg norm="adimum" type="context">adimũ</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">cũ</reg> fumitate digiti claudere oſcu-<lb/>lum <var>.t.</var> exiguitubi <var>.a.o.t.</var> vnde aer <lb/><reg norm="impulsus" type="simple">impulsꝰ</reg> ab oleo tubi <var>.e.g.c.</var> extra <lb/>per tubum <var>n.u.</var> quouſque oleum <lb/>vaſis ad ęquilibrium ipſius <var>.n.</var> per-<lb/>ueniebat, <reg norm="pertubum" type="context">pertubũ</reg> <var>.n.u.</var> ingredie-<lb/>batu. </s> + <s xml:id="s2788" xml:space="preserve">& quando dictum oleum <reg norm="qui" type="simple">ꝗ</reg> <lb/>dictum tubum <var>.n.u.</var> extrinſecè 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">ꝑ</reg> ipſum <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> & peripſum <var>.t.</var> denuo tantum olei exire permitte-<lb/>bamus, quantum in receptaculo ad claudendum foramen <var>.u.</var> idoneum exiſteret. </s> + <s xml:id="s2789" xml:space="preserve">Ra-<lb/>tio vero, quę me mouit, ut punctum <var>.g.</var> inferius ipſius <var>.t.</var> conſtituam, eſt, quia <reg norm="cum" type="context">cũ</reg> clau-<lb/>ſum 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 vſque 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> ſurſum verſusreflexum conſtitui, vt aerem ab ingreſſu per foramen <var>.t.</var> <reg norm="arcerem" type="context">arcerẽ</reg>, <lb/>quia huiuſmodi aer nunquam deſcendit ſi corpus magis denſum non deſcendat.</s> + </p> + <p> + <s xml:id="s2790" xml:space="preserve">Verum eſt, <reg norm="quod" type="simple">ꝙ</reg> melius erit, vt maiores difficultates euitemus, ſtatuere dictum <reg norm="tubum" type="context">tubũ</reg> <lb/><var>a.o.t.</var> ita curuum vt eſt .ω qui cum ſuo extremo inferiori ipſi <var>.n.u.</var> ſit contiguus ita ta-<lb/>men ut dictum extremum inferius <reg norm="non" type="context">nõ</reg> ſit inferius quam <var>.o.</var> quia totum oleum exiret.</s> + </p> + <p> + <s xml:id="s2791" xml:space="preserve">Volui etiam vt ſuperior extremitas <var>.n.</var> tubi <var>.n.u.</var> ſit in aere vaſis & non in oleo, ne <lb/>per eam oleum exeat, quia cum extremitas <var>.u.</var> inferior ſit <var>.g.</var> totum oleum quod ſu-<lb/>peraret oſculum <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 philoſophicis rebus verſato innoteſcet.</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 ſcripta quædam, quorum titulus <emph style="sc">animadver-<lb/>siones in ephemeridas</emph>. </s> + <s xml:id="s2793" xml:space="preserve">& breuis alia diſputatio de er <lb/>roribus calculorum Aſtronomicorum. ac <reg norm="demum" type="context">demũ</reg> Theſes quæ-<lb/>dam typis datæ .11. Auguſti .1581. quę omnia cum ad manus <lb/>meas perueniſſent, non potui, non eis animum admouere, <reg norm="cum" type="context">cũ</reg> <lb/>ibi de his ſtudijs <reg norm="ageretur" type="simple">ageret̃</reg>, in quibus partem non exiguam an-<lb/>norum meorum conſumpſi. </s> + <s xml:id="s2794" xml:space="preserve">nec tamen ſcribere aliquid ſta-<lb/>tueram; </s> + <s xml:id="s2795" xml:space="preserve">tum quod exiſtimarem viros Aſtronomiæ 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 ſi ingenue meam ſententiam profer <lb/>re voluiſſem<unsure/>, non poteram, ſine maxima authoris moleſtia ferè omnia reprobare. <lb/></s> + <s xml:id="s2797" xml:space="preserve">Quandoquidem ſolet vnuſquiſque indignationem concipere ex his, quæ ſuæ opi-<lb/>nioni repugnant, id omne maleuolentiæ potius quam veritatis ſtudio tribuens. </s> + <s xml:id="s2798" xml:space="preserve">Qui <lb/>nimo cum nec deeſſent qui dicerent in meipſum directa ea tela fuiſſe, nullam fidem <lb/>eis adhibendam duxi, nec enim qui in ephemeri das inuehitur, me arguere poteſt, <lb/>qui nullas ephemeridas ſcripſi, nec tabulas compoſui. </s> + <s xml:id="s2799" xml:space="preserve">Nec ſi author quidpiam ex <lb/>his quæ à nobis edita fuiſſent impugnare voluiſſet, ægrè ferre debuiſſem, modo à ve <lb/>ritate nuſquam deuiaſſet. </s> + <s xml:id="s2800" xml:space="preserve">Liberum enim eſt cuique ſcribere quodlibet. </s> + <s xml:id="s2801" xml:space="preserve">nec Ari-<lb/>ſtotelem afficit iniuria, quicunque illi fidem ſuam non accommodat, & ſi valdè ini-<lb/>quus ſit, quiſquis 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 ſatis <reg norm="oimbus" type="context">oĩbus</reg> feciſſe videbar, cum his qui de ſcriptis illis me <reg norm="inter- rogauerant" type="context">inter-<lb/>rogauerãt</reg> reſpondiſſem, ea non ſatis 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 deſideres tibi clarius, quæ nam de his mea ſit ſententia explicari, non tam tuis <lb/>precibus deuictus quam mea ipſius cupiditate de te benemerendi impulſus, non ſu-<lb/>ſtineo diutius animum tuum hęſitantem relinquere. </s> + <s xml:id="s2804" xml:space="preserve">Atque vt tibi adeo honeſta cu <lb/>pienti <reg norm="morem" type="context">morẽ</reg> geram paucis hiſce ſcriptis incultis quidem, vt ab homine omni pror-<lb/>ſus facundia deſtituto exaratis, ſed ex quibus nihilominus facile, atque perſpicue, vt <lb/>ſpero, conceptum animi noſtri percipere poſſis, ſi tamen eam præſtantis ingenij tui <lb/>aciem adhibueris, qua ſoles intima quæque ſcientiarum penetrare, noſtræ opinio-<lb/>nis ſummam perſtrinxi, quę ad te mittere decreui. </s> + <s xml:id="s2805" xml:space="preserve">& quamuis ipſa res de qua agitur, <lb/>quæ exactiorem deſiderat expoſitionem, prolixiorem me eſſe cogerit quam voluiſ-<lb/>ſem. </s> + <s xml:id="s2806" xml:space="preserve">multa tamen me obmiſiſſe intelliges, non admodum neceſſaria his quibus <lb/>Aſtrologiæ noti ſunt termini, vt tuarum occupationum rationem me etiam habere <lb/>intelligeres, at que vt ſummam oblectationem <reg norm="concipiam" type="context">cõcipiam</reg> animo ſi me tibi aliqua ex <lb/>parte ſatisfeciſſe intellexero ita humanitati tuæ gratiam habebo, quę mihi occaſio <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 ſimilibus, <lb/>ideſt <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">eorũ</reg> animis falſam 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, ſi quam fortaſſe ex <reg norm="illorum" type="context">illorũ</reg> ſcriptorum lectione <reg norm="conceperunt" type="context">cõceperunt</reg> circa ea, de qui-<lb/>bus nunc ſum acturus.</s> + </p> + <p> + <s xml:id="s2808" xml:space="preserve"><reg norm="Quemadmodum" type="context">Quemadmodũ</reg> igitur ab hoc authore ter ſcriptum fuit de <reg norm="contradictionibus" type="context">cõtradictionibus</reg>, ſiue <lb/>erroribus Ephemeridum, & earum calculos ſequentium, & de ratione qua cognoſci <lb/>poteſt ſitus & locus alicuius ſuperioris planetæ, diuerſus ab eo, qui ab ipſis Epheme <lb/>ridibus aſſignatus eſt, ita diſputationem hanc meam diuidam in tres partes, quo ſci <lb/>licet minus confusè, & magis diſtinctè à me ſcribatur, <reg norm="praeſupponendo" type="simple">p̃ſupponendo</reg>, vt animaduer-<lb/>tere potes, huius ſcriptoris intentionem, aliam non fuiſſe, <reg norm="quam" type="context">quã</reg> <reg norm="oſtendere" type="context">oſtẽdere</reg>, quod ſcripto <lb/>res Ephemeridum diuerſimode eiuſdem temporis locum planetæ aſſignauere, & <lb/>quod eum faciant modo nimium velociter currere, modo nimium in vno ſigno mo-<lb/>rari, vt (exempli gratia) Martem interdum faciunt morari ſex, aut ſeptem menſibus <lb/>in vno ſigno. </s> + <s xml:id="s2809" xml:space="preserve"><reg norm="Idque" type="simple">Idq;</reg> poſtea in cauſa eſſe ait, vt Aſtrologi indiciarij <reg norm="fallantur" type="context">fallãtur</reg>, & ſimul <lb/>careant certis fundamentis rationum quibus futura indicent, & prædicant. </s> + <s xml:id="s2810" xml:space="preserve">Primum <lb/>ergo videndum eſt, quam rectè hic vſus ſit arte, & ſcientia, vt aliorum opiniones, & <lb/>ſcripta redarguere poſſet. </s> + <s xml:id="s2811" xml:space="preserve">Deinde videbimus quomodo verum ſit, & poſſibile id <lb/>quod ab Aſtrologis hactenus creditum, atque traditum eſt, & qua ratione poſſint ſie <lb/>ri veri calculi à peritis regularum ſcientiæ.</s> + </p> + <p> + <s xml:id="s2812" xml:space="preserve">In primo igitur tractatu inſcripto Animadnerſiones, præſupponit Author pro-<lb/>feſſores huius ſcientiæ neſcire inuenire vera loca planetarum, quia vtuntur Ephe-<lb/>meridibus, in quibus eorum loca non rectè ſunt notata. </s> + <s xml:id="s2813" xml:space="preserve">Quod ſecundum ipſum ori <lb/>tur, ex errore calculatorum, ſeu computiſtarum, potius quam ex varietate tabula-<lb/>rum, à quibus Ephemerides ſumptæ ſunt, hoc tamen verum non eſt, Ephemeridas, <lb/>ſcilicet, ita inter ſe differre, ratione errorum computiſtarum tantummodo, ſed po-<lb/>tius ratione ipſarum tabularum, & ſi interdum contingere poſſit error aliquorum mi <lb/>nutorum, nec non graduum, non propterea Ephemerides ita ſpernendæ ſunt. </s> + <s xml:id="s2814" xml:space="preserve">In <lb/>multis enim calculis, tales errores excuſabiles ſunt, 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 quinquageſimum enuntiatum centiloquij Ptole-<lb/>męi, ſatis mendoſe. </s> + <s xml:id="s2816" xml:space="preserve">Ptolemęus enim ibi ſic ait.</s> + </p> + <quote> + <s xml:id="s2817" xml:space="preserve">Non obliuiſcaris eſſe centum viginti coniunctiones, quæ ſunt in ſtellis erraticis, <lb/>in illis enim eſt maior ſcientia eorum quæfiunt in hunc <reg norm="mundum" type="context">mũdum</reg> ſuſcipiendi incre-<lb/>mentum, & decrementum.</s> + </quote> + <p> + <s xml:id="s2818" xml:space="preserve">Nam, neque eo in loco, neque alibi, Ptolemęus quidquam eius dicere voluit <lb/>quod ab hoc profertur.</s> + </p> + <p> + <s xml:id="s2819" xml:space="preserve">Pergatur poſtea in pag .2. & videbitur hunc exiſtimare abſurdum quod Saturni, <lb/>& Iouis coniunctio vera anni .1563. potuerit eſſe in Leone ſigno igneæ triplicitatis <lb/>cum eorum coniunctio vera anni .1544. fuerit in Scorpione, ſigno triplicitatis <lb/>aqueæ, & cum coitus eorum anni .1583. futurus ſit in Piſcibus, ſigno 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 poſtquam duæ ſtellæ coiuerint, non prius ſub alio alterius triplicitatis ſigno <lb/>inter ſe ſunt conuenturæ, quam per omnia ſigna quæ eiuſdem ternarij cum primo ex <lb/>titerint prius <reg norm="coniungantur" type="context">coniungãtur</reg>. </s> + <s xml:id="s2822" xml:space="preserve">Ita ſentit Ptolemęus, <reg norm="cæterique" type="simple">cæteriq́;</reg> non aſpernendi nominis <lb/>Aftronomi.</s> + </quote> + <p> + <s xml:id="s2823" xml:space="preserve">Et <reg norm="tamen" type="wordlist">tamẽ</reg> Ptolemęus nunquam <reg norm="quidquam" type="context">quidquã</reg> huius rei attigit, & quamuis Albumaſar <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, ſed de mediocribus aut æqualibus, & quidem re-<lb/>ctè dicunt, quia lineæ eorum mediorum motuum non coeunt in aliquo ſigno alte-<lb/>rius triplicitatis, prius quam pertranſiuerint omnia ſigna illius, in qua incęperunt. <lb/></s> + <s xml:id="s2824" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> nullum inconueniens ſequitur, ſi 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 poteſt, vt lineæ mediorum motuum coniungantur in vno ſigno, cor <lb/>pora verò eorum planetarum coeant in alio, cum rarò eueniat, vt linea medij mo-<lb/>tus, eadem ſit 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">vanũ</reg> eſſe cre <lb/>dere id eueniſſe propter ſimilem naturam, aut qualitatem ſignorum. </s> + <s xml:id="s2828" xml:space="preserve">Hunc enim <lb/>reſpectum non habent illi planetæ in verisſuis coniunctionibus. </s> + <s xml:id="s2829" xml:space="preserve">Exempli autem <lb/>cauſa ponamus, quod rectè ſupputatæ fuerint coniunctiones annorum .1484. 1504. <lb/>& 1524. quod attinet ad differentiam duodecatemorij, ſcilicet prima in .24. gradu <lb/>Scorpij, ſecunda in .20. Cancri, tertia in .10. Piſcium. </s> + <s xml:id="s2830" xml:space="preserve">Cum ſecunda anticipauerit <lb/>trigonum perfectum cum prima, gradibus .4. & tertia anticipauerit trigonum perfe-<lb/>ctum <reg norm="cum" type="context">cũ</reg> ſecunda gradibus .10. ſi forte prima vt facta fuit in .24. gradu Scorpij facta <lb/>fuiſſet in .2. gradu eiuſdem, <reg norm="planum" type="context">planũ</reg> eſt <reg norm="quod" type="simple">ꝙ</reg> <reg norm="ſecunda" type="context">ſecũda</reg> facta fuiſſet in .28. gradu geminorum <lb/>& tertia in .18. Aquarij, quæ ſigna ſunt diuerſæ triplicitatis ab illa Cancri. </s> + <s xml:id="s2831" xml:space="preserve">Inſuper <lb/>ſi coniunctio anni .1544. quæ fuit in .28. gradu Scorpij fuerit recta <reg norm="correſpondens" type="context">correſpondẽs</reg> <reg norm="prae cedenti" type="simple">prę<lb/>cedenti</reg>, anni .1524. per gra .18. ſine dubio ſi coniunctio anni .1524. facta fuiſſet in <lb/>18. gradu Aquarij, illa anni .1544. fuiſſet in .6. Scorpij ſigni alterius triplicitatis <reg norm="quam" type="context">quã</reg> <lb/>ſint Gemini. </s> + <s xml:id="s2832" xml:space="preserve">Præterea, vt anno .1544. <reg norm="coniunctio" type="context">cõiunctio</reg> facta eſt in .28. gra. Scorpij, & 1563 <lb/>in .29. Cancri, ponendo eas eſſe rectas, quod attinet ad ſuperandum trigonum vno <lb/>gradu, ſi anno .1544. facta fuiſſet in .30. Scorpij, anno .1563. proculdubio facta fuiſ-<lb/>ſet in primo gradu Leonis. </s> + <s xml:id="s2833" xml:space="preserve">Et ſuppoſitis ijs interuallis, quæ ſuperſunt, aut deſunt per <lb/>fectis trigonis, ſi coniunctio anni .1524. fuiſſet in .20. gradu Piſcium, anno .1544. <lb/>fuiſſet in .8. Sagittarij. </s> + <s xml:id="s2834" xml:space="preserve">Quæ quidem omnia aduerſantur opinioni huius ſcriptoris.</s> + </p> + <p> + <s xml:id="s2835" xml:space="preserve">Quodautem opinatur coniunctionem anni .1583. fore in Ariete, ſic dicens pagi <lb/>na ſecunda.</s> + </p> + <quote> + <s xml:id="s2836" xml:space="preserve">Non erit ab re ſi & eandem Saturni, & Iouis coniunctionem in primo igneæ tri-<lb/>plicitatis ſigno, quod eſt Aries futuram afferamus anno .1583. ſi ab accidentibus no <lb/>bis licet, vt ab omnibus paſſim conceditur, planetarum loca diſcernere.</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">ꝑ</reg> <lb/>4. <reg norm="quidem" type="context">quidẽ</reg> in <reg norm="qui" type="simple">ꝗ</reg> buſuis Ephemeridibus aut tabulis. </s> + <s xml:id="s2838" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> videbit eiuſmodi <reg norm="coniunctionem" type="context context">cõiunctionẽ</reg> <lb/>contra <reg norm="ſententiam" type="context">ſententiã</reg> <reg norm="ſuam" type="context">ſuã</reg> fieri in Piſcibus, <reg norm="non" type="context">nõ</reg> <reg norm="autem" type="wordlist">aũt</reg> in Ariete. (vt poſtea res ipſa nos docuit <lb/>ſub <reg norm="menſe" type="context">mẽſe</reg> Aprili. poſt <reg norm="quidem" type="simple context">ꝗdẽ</reg> <reg norm="ſcriptam" type="context">ſcriptã</reg> <reg norm="hanc" type="context">hãc</reg> <reg norm="epiſtolam" type="context">epiſtolã</reg>, <reg norm="vulgarique" type="simple">vulgariq́;</reg> ſermone <reg norm="tranſmiſsam" type="context context">trãſmiſsã</reg>, ſed <reg norm="an- tequam" type="context">an-<lb/>tequã</reg> in <reg norm="latinum" type="context">latinũ</reg> <reg norm="tranſlata" type="context">trãſlata</reg>, & huic volumini inſerta <reg norm="cum" type="context">cũ</reg> alijs Typographo <reg norm="committeretur" type="context">cõmitteretur</reg>.)</s> + </p> + <p> + <s xml:id="s2839" xml:space="preserve">Vbi poſtea 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">antiquorũ</reg>, vt videri <reg norm="pont" type="context">põt</reg> apud <reg norm="Albumaſarem" type="context">Albumaſarẽ</reg> & <reg norm="Alchibicium" type="context">Alchibiciũ</reg>, <lb/>ſed etiam perpendendum an eſſet vera, <reg norm="priuſquam" type="context">priuſquã</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 ſcientiæ. </s> + <s xml:id="s2841" xml:space="preserve">Sunt <lb/>tamen illi antiqui excuſatione aliqua digni. </s> + <s xml:id="s2842" xml:space="preserve">Ponebant enim vigeſimo quoque an-<lb/>no præcisè fieri mediam coniunctionem Saturni <reg norm="cum" type="context">cũ</reg> Ioue, & in quolibet ſigno eiuſ-<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 ſecundum primum ſuppoſitum finiebatur ſpacio annorum .240. qui nume <lb/>rus fit. ex .20. duodecies multiplicatis. </s> + <s xml:id="s2845" xml:space="preserve">Et quia triplicitates ſunt .4. ideo credebant <lb/>in ſpacio annorum .960. qui numerus fit ex .240. quater multiplicato, perfici .48. <reg norm="con- iunctiones" type="context">cõ-<lb/>iunctiones</reg>, priuſquam redirent ad ſe coniungendos in eodem loco, ubi prius iun-<lb/>cti fuiſſent. </s> + <s xml:id="s2846" xml:space="preserve">Primum autem ſuppoſitum, quod vigeſimo quoque anno <reg norm="iungerentur" type="context">iũgerentur</reg>, <lb/>colligebant ſic ratiocinantes. </s> + <s xml:id="s2847" xml:space="preserve">Si Saturnus annis .30. peragit ſuum curſum per om-<lb/>nia ſigna Zodiaci, Iupiter autem peragit eum annis .12. Saturnus ambulauerit .4. ſi <lb/>gna, et .4. quintas partes ſigni, ſiue gra .24. dum Iupiter peragit integrum ambitum <lb/>ideſt annis .12. </s> + <s xml:id="s2848" xml:space="preserve">Itaque deſunt ei anni .8. ad perueniendum ad .20. quibus .8. annis Sa-<lb/>turnus <reg norm="perambulat" type="simple">ꝑambulat</reg> ſigna tria & <reg norm="quintam" type="context">quintã</reg> <reg norm="partem" type="context">partẽ</reg> unius ſigni <var>.i.</var> gradus .6. qui iuncti dictis ſi <lb/>gnis .4. & gra .24. faciunt ſigna .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. ſigna <reg norm="antequam" type="context">antequã</reg> perueniat ad <reg norm="Saturnum" type="context">Saturnũ</reg>, cum Saturnus eo <lb/>dem tempore perfecerit curſum ſignorum .8. </s> + <s xml:id="s2850" xml:space="preserve">Eandem concluſionem etiam fortaſ <lb/>ſe collegerant ex dictis ſuppoſitis, dicentes, ſi Saturnus annis .30. ambulat .12. ſigna <lb/>proculdubio annis .20. ambulat .8. ſigna, quo tempore Iupiter perambulat .20. ad ra <lb/>tionem .12. ſignorum in annis .12.</s> + </p> + <p> + <s xml:id="s2851" xml:space="preserve">Verum hoc ſuppoſitum non eſt bonum, quoniam, ſi ita eſſet, coniunctiones <reg norm="horum" type="context">horũ</reg> <lb/>duorum planetarum nunquam exirent ex vna triplicitate, & non modo .960. <reg norm="quoque" type="simple">quoq;</reg> <lb/>anno, ſed etiam ſexageſimo rurſus coniungerentur in eodem puncto. </s> + <s xml:id="s2852" xml:space="preserve">nec coniunctio <lb/>nes eorum (ſemper autem intelligo de medijs) unquam egrederentur ex illis tribus <lb/>ſignis Zodiaci.</s> + </p> + <p> + <s xml:id="s2853" xml:space="preserve">Sed periodus æqualis Saturni, eſt dierum circiter .10740. atque ita minor an .30. <lb/>atque etiam .29. cum dimidio periodus autem æqualis Iouis, eſt circiter .4328. vt <lb/>ego eam comperio, quidquid alij <reg norm="dicant" type="context">dicãt</reg>, <reg norm="vtque" type="simple">vtq́;</reg> planius alias oſtendam. </s> + <s xml:id="s2854" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> hæc per <lb/>iodus Iouis, etiam minor eſt ann .12. prætermittendo in ſupputatione <reg norm="tam" type="context">tã</reg> Saturni <reg norm="quam" type="context">quã</reg> <lb/>Iouis quaſdam 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 cognoſcere 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 ſunt periodus <lb/>Iouis, conficiet gradus .145. & min .4. ideſt min .8704. & eadem regula inueniemus <lb/><reg norm="quod" type="simple">ꝙ</reg> Saturnus .30. <reg norm="quibuſque" type="simple">quibuſq́;</reg> diebus, conſiciet min .60. & ſecunda .20. </s> + <s xml:id="s2857" xml:space="preserve">Iupiter autem ſin-<lb/>gulis .30. diebus, conficiet min .149. & ſecunda .43. vnde ſubtrahendo minuta Satur-<lb/>ni à minutis Iouis, ſupererunt min .89. cum ſecun .23. </s> + <s xml:id="s2858" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> Iupiter .30. <reg norm="quibuſque" type="simple">quibuſq́;</reg> die-<lb/>bus velocitate curſus, ſuperabit Saturnum minutis .89. cum ſecundis .23. </s> + <s xml:id="s2859" xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dicen-<lb/>do, ſi minuta .89. cum ſecundis .23 dant nobis dies .30. ſupradicta, minuta .8704. da-<lb/>bunt nobis dies .2921. quibus iunctis cum diebus .4328. periodi Iouis, efficientur dies <lb/>7249. ideſt 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 oſtendatur hanc operationem rectam eſſe (nam demonſtrationem <reg norm="ſpeculatiuam" type="context">ſpeculatiuã</reg> <lb/>huius operationis in .113. Theoremate noſtræ Arithmeticę <reg norm="cuique" type="simple">cuiq;</reg> videre licet) fieri <lb/>poteſt his alijs calculis.</s> + </p> + <p> + <s xml:id="s2861" xml:space="preserve">Si Saturnus diebus .10740. tranſit per gra .360. in ſpacio dierum .2921. tranſibit <lb/>per gradus .97. min .54. quibus iunctis cum gra .145. min .4. ſupra notatis, efticientur <lb/>gra .242. min .58. </s> + <s xml:id="s2862" xml:space="preserve">Deinde, ſi Iupiter ſpatio dierum .4328. tranſit per gra .360. igitur <lb/>ſpatio .2921. per eandem regulam inueniemus eum tranſire gradus .242. mi .58. qui <lb/>numerus par eſt illi Saturni. </s> + <s xml:id="s2863" xml:space="preserve">Cum ergo Iupiter confecerit vnum ambitum poſt 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 rurſus aſſequatur Saturnum, tranſeundum ei erit <reg norm="per" type="simple">ꝑ</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/>termiſimus quaſdam exiguas particulas periodorum perfectorum cuiuſque planetę <lb/>in ſuperioribus ſupputationibus.) </s> + <s xml:id="s2864" xml:space="preserve">Illos verò gradus .43. Iupiter conſiciet 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 Alfonſi conueniant cum hoc <reg norm="numero" type="wordlist">nr̃o</reg> calculo, <reg norm="con- ſiderabimus" type="context">cõ-<lb/>ſiderabimus</reg>, <reg norm="quod" type="simple">ꝙ</reg> Era (vt vocant) dicti temporis annorum .19. cum diebus .314. eſt dua-<lb/>rum tertiarum ſexagenarum, ſecundæ nullius, & 53. <reg norm="primarum" type="context">primarũ</reg> ſiue 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">ꝙ</reg> <reg norm="vtriuſque" type="simple">vtriuſq;</reg> planetæ lineæ eiuſmodi motus tranſi <lb/>bunt per min .56. tertij gradus Sagittarij, ideſt coniunctæ erunt.</s> + </p> + <p> + <s xml:id="s2869" xml:space="preserve">In fine poſtea ſecundæ periodi, cuius era erit .4. tertiarum, ſecundæ .1. et .47. pri-<lb/>marum ſexagenarum, locus mediocris <reg norm="vtriuſque" type="simple">vtriuſq;</reg> erit in min .56. gra. ſexti Leonis. </s> + <s xml:id="s2870" xml:space="preserve">In <lb/>fine verò tertiæ periodi, cuius era erit .6. tertiarum .2. ſecundarum, 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">cuiusq́;</reg> periodi, locus <reg norm="eorum" type="context">eorũ</reg> mediocris coniunctim ſemper diſtabit à loco me <lb/>diocri præcedentis coniunctionis gradibus .117. ideſt in trigono antecedenti, minus <lb/>gra .3. </s> + <s xml:id="s2872" xml:space="preserve">Vnde apparet has coniunctiones procedere in contrariam partem reſpectu or <lb/>dinis <reg norm="ſignorum" type="context">ſignorũ</reg> Zodiaci, ſed reſpectu ordinis graduum <reg norm="ſignorum" type="context">ſignorũ</reg>, ſemper <reg norm="progrediuntur" type="simple">progrediunt̃</reg> or <lb/>dine per ternos gradus nunquam retrogradientes. </s> + <s xml:id="s2873" xml:space="preserve">Hinc ſe quitur, vt non duodecies <lb/>in omni triplicitate coniungantur hi duo planetę, vt antiqui putauerunt, ſed decies <lb/>tantum. </s> + <s xml:id="s2874" xml:space="preserve">& ad ſummum ter in ſingulo ſigno, ſpatio 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 ſigno ſequenti alterius triplicitatis. </s> + <s xml:id="s2876" xml:space="preserve">Exempli gratia, po-<lb/>namus <reg norm="quod" type="simple">ꝙ</reg> prima vice <reg norm="coniungantur" type="context simple">cõiungant̃</reg> in gra .2. Arietis, ſecunda 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/>ſeptima in .20. Arietis, octaua in .23. Sagittarij, nona in .26. Leonis, decima in .29. <lb/>Arietis, et vndecima erit in gra .2. Capricorni ſigni <reg norm="ſequentis" type="context">ſequẽtis</reg> triplicitatis. </s> + <s xml:id="s2878" xml:space="preserve">Decem igi<lb/>tur interualla ſingula annorum .19. & dierum .314. faciunt annos .198. & dies .220. <lb/>Immo pertabulas Alfonſi, eiuſmodi periodus non modo non reperitur <reg norm="annorum" type="context">annorũ</reg> .242 <lb/>nec .240. vt antiqui credidere, ſed tribus diebus minor annis .198. & diebus .220. id-<lb/>eſt 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 ſe 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">aggregatũ</reg> <lb/>annorum .794. & dierum .138. cum annis .198. & diebus .217. quorum ſumma eſt <num value="992">.<lb/>992.</num> & dies .355. inuenietur locus mediocris ipſorum <reg norm="planetarum" type="context">planetarũ</reg> in dicto vltimo gra <lb/>du Sagittarij. </s> + <s xml:id="s2881" xml:space="preserve">Sed ſi quęſiuerimus eorum locum mediocrem per aggregatum anno <lb/>rum .198. & dierum .217. cum annis .960. quod erit ſumma annorum .1158. & <reg norm="dierum" type="context">dierũ</reg> <lb/>217. reperiemus Iouem in gradu .18. </s> + <s xml:id="s2882" xml:space="preserve">Sagittarij & Saturnum in .16. Leonis diſtanti-<lb/>bus inter ſe 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">ꝙ</reg> coniunctio eorum mediocris fuerit multis annis ante <lb/>omittendo (vt dixi) radices, quia ſatis eſt 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 animaduerſionum non quaſi cæcus eæcos ſequi, ſed 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 eſt, <reg norm="quod" type="simple">ꝙ</reg> idem author paulo inferius citat coniunctiones horum duorum <lb/>planetarum anni .1493. et .1512. quas neſcio vnde ſumpſerit.</s> + </p> + <p> + <s xml:id="s2887" xml:space="preserve">Nam, etſi inter hos annos eſt interuallum annorum .19. tamen tantum abeſt, vt <lb/>coiuerint dictis annis, vt Saturnus anno .1493. ante finem Auguſti fuerit in .28. gra-<lb/>du Aquarij, Iupiter verò in .28. Leonis ex diametro oppoſiti. </s> + <s xml:id="s2888" xml:space="preserve">Et anno .1512. per to <lb/>tum menſem Iunium & Auguſtum, Saturnus fuerit in Libra, Iupiter verò in Ariete, <lb/>itaque inter ſe ſimiliter oppoſiti, & ſi perfecta oppoſitio non fuit poſtea niſi 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, eſt mendoſus. </s> + <s xml:id="s2890" xml:space="preserve">Nam ipſe Montere gius po <lb/>nit dictam coniunctionem in mi .42. gra .24. non autem in min .4. ipſius gradus. </s> + <s xml:id="s2891" xml:space="preserve">Sed <lb/>hic error nullius eſt momenti, fortaſſe qui impræſſorum incuria irrepſit.</s> + </p> + <p> + <s xml:id="s2892" xml:space="preserve">Pergatur poſtea obſecro ad paginam .3. ipſarum Animaduerſionum, vbi hic co-<lb/>natur oſtendere calculatores non obſeruaſſe verum modum, ſic dicens.</s> + </p> + <quote> + <s xml:id="s2893" xml:space="preserve">Anno .1484. Nouembris .25. Saturno locum conſtituit Monteregius in grad .23. <lb/>min .4. Scorpij. </s> + <s xml:id="s2894" xml:space="preserve">Anno poſtmodum ſubſequenti qui eſt .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">Interq́;</reg> tempora duo interſunt menſes dies .26. <lb/></s> + <s xml:id="s2896" xml:space="preserve">At cum ex motus ſui natura Saturnus hoctemporis ſpacio gradus .4. non debeat <reg norm="tran- ſcendere" type="context">trã-<lb/>ſcendere</reg>, ſit tamen inter <reg norm="vtrunque" type="simple">vtrunq;</reg> tempus differentia graduum .7. minutorum .3. quæ <lb/>ratione ſui motus requirunt menſes .6. vt eos perficiat, conſtat pluſquam tribus men <lb/>ſibus fallere nos Saturnum.</s> + </quote> + <p> + <s xml:id="s2897" xml:space="preserve">Hic videre licet quam veram viam hic ſecutus ſit ad aperiendos errores Epheme <lb/>ridum, & miſeri Monteregij, qui Saturnum claudum facit tantum itineris conficere <lb/>tribus <reg norm="menſibus" type="context">mẽſibus</reg>, <reg norm="quantum" type="context context">quãtũ</reg> vix confeciſſet <reg norm="menſib" type="context">mẽſib</reg>. ſex. </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 cęlum annis .30. igitur menſi-<lb/>bus .30. conficiet duodecimam partem circuitus, cum menſes .30. ſint duodecima <lb/>pars annorum .30. & quia duodecima pars circuitus cęli intelligitur conſtare ex .30. <lb/>gradibus, igitur quilibet menſis poſtulabit gradum vnum. </s> + <s xml:id="s2900" xml:space="preserve">Ideo illi .6. aut .7. gradus <lb/>poſtulant tempus, amplius menſium ſex.</s> + </p> + <p> + <s xml:id="s2901" xml:space="preserve">Atque eiuſmodi mira ratiocinatio poteſt in .2. exemplo eius, inſcripto.</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">ꝙ</reg> Monteregius faciat Saturnum ambulare gra .9. min .10. in menſi-<lb/>bus .7. & diebus .6. </s> + <s xml:id="s2904" xml:space="preserve">Ad quod iter Saturnc ſeni opus eſſet ſaltem menſibus .9. eius <lb/>iudicio.</s> + </p> + <p> + <s xml:id="s2905" xml:space="preserve">Sed ſi hoc miratur, quid dicturus fuiſſet, ſi animaduertiſſet, quod idem calculator <lb/>Monteregius facit Saturnum ambulare immo volare gra .9. min .48. non in 7. ſed in <lb/>2. menſibus cum dimidio, videlicet à .10. die Iunij vſque ad .26. Auguſti <reg norm="eiuſdem" type="context">eiuſdẽ</reg> an-<lb/>ni .1504.</s> + </p> + <p> + <s xml:id="s2906" xml:space="preserve">Quid ſi etiam animaduertiſſet <reg norm="quod" type="simple">ꝙ</reg> à .10. die Iunij ſupradicti vſque ad .16. Ianuarij <lb/>anni ſequentis, faciunt Saturnum, ſurſum, deorſum curſitare amplius gra .17. mi .54. <lb/>Imino ſi animaduertiſſet, quod anno .1524. </s> + <s xml:id="s2907" xml:space="preserve">Stoflerinus ab initio anni, <reg norm="vſquem" type="context">vſquẽ</reg> ad <lb/>medium Maium, ideſt menſib .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 abſurdas, obſtupuiſſet.</s> + </p> + <p> + <s xml:id="s2909" xml:space="preserve">Vbi autem in tergo eiuſdem paginę ait, quod gradibus .13. min .42. reſpondent <lb/>menſes .19. errauit in calculo, nam ex eiuſmodi tempore ſecundum 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 ſinguli 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">ꝙ</reg> in primo exemplo circa <reg norm="annum" type="context">annũ</reg>. <lb/>1484. repręhendit Monteregium, quia facit Iouem ambulare gradus .14. cum min <num value="6">.<lb/>6.</num> in menſibus .2. diebus .4. ad quod iter, vt ipſe ait, opus eſſet ſaltem <reg norm="menſibus" type="context">mẽſibus</reg> .11. <reg norm="atque" type="simple">atq;</reg> <lb/>ita ſecundum ipſum, Ioui opus eſſet anno vno pro ſingulo medio ſigno. </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 ſecundo exemplo ſumpto à Stoflerino ait, <reg norm="quod" type="simple">ꝙ</reg> Ioui ad curſum vnius gra-<lb/>dus, & min .5. opus eſt diebus .30. non autem menſibus .7. & diebus .28. vbi oſtendit, <lb/>ſe paruum diſcrimen facere inter Iouem, & Saturnum.</s> + </p> + <p> + <s xml:id="s2913" xml:space="preserve">Miratur poſtea <reg norm="quod" type="simple">ꝙ</reg> Stoflerinus faciat laborare generoſum Iouem ferè menſibus <lb/>ſex in vno gradu. </s> + <s xml:id="s2914" xml:space="preserve">Sed multo magis, vt puto, miratus eſſet, ſi vidiſſet, quod idem <lb/>Stoflerus in eodem anno facit, quod Iupiter die .4. </s> + <s xml:id="s2915" xml:space="preserve">Ianuarij ſit in eodem puncto, in <lb/>quo poſtea reperitur die vltima Auguſti. </s> + <s xml:id="s2916" xml:space="preserve">At fortaſſe dici poſſet, quod Iupiter pro-<lb/>pter prudentiam, & bonitatem ſuam factus eſt R ex omnium Deorum, vt ait Home-<lb/>rus, & ideo expulit è ſede Saturnum, & aſcendit in altiori cœlo. </s> + <s xml:id="s2917" xml:space="preserve">Vnde euenit vt fa-<lb/>ctus fuerit lentior in curſu, 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 ſenem, & pariter tardio-<lb/>rem in ſe <reg norm="mouendo" type="context">mouẽdo</reg>. </s> + <s xml:id="s2919" xml:space="preserve">aut <reg norm="tunc" type="context">tũc</reg> temporis illum detentum fuiſſe in ſibi dilecta Arcadia <reg norm="cum" type="context">cũ</reg> <lb/>Caliſto. </s> + <s xml:id="s2920" xml:space="preserve">Aut fortaſſe erat in alta ſpecula intentus audiendo ingenti certami-<lb/>ni Timoclis & Damidis, vnde pendebat exitium aut gloria familiæ ſuæ, nam alio-<lb/>quin Stoflerus non depræhendiſſet eum tam otioſum & 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 ſemper bella & ignes ſpirare ſolet, etiam, & ipſe ab <lb/>Aſtrologis factus eſt piger, & languidus, vt velint eum nonnunquam commorari <lb/>in vno ſigno ſex aut ſeptem menſibus; </s> + <s xml:id="s2923" xml:space="preserve">quod nullo pacto placet authori Animaduer-<lb/>ſionum, cum pag .4. ita ſcribat.</s> + </p> + <quote> + <s xml:id="s2924" xml:space="preserve">Quod citra notam, ab omnibus creditur poſſe obſeruari, 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 ſuum 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 cœleſtium obſeruatores, ipſis oculis certi fa <lb/>cti ſunt tam de his effectibus Martis, quam aliorum, vnde coacti ſunt fingere tantam <lb/>magnitudinem eius epicycli, cum ipſe nunquam obſeruauerit motum, nec huius nec <lb/>alterius planetæ, ſed tantum viderit eius moram in Ephemeride ſcriptam. </s> + <s xml:id="s2928" xml:space="preserve">Si enim <lb/>ſaltem diceret, ſe aliquo tempore obſeruaſſe iter Martis, & comperuiſſe aliorum opi <lb/>nionem falſam, attuliſſet aliquem colorem ſententiæ ſuæ. </s> + <s xml:id="s2929" xml:space="preserve">Sed ſi obſeruaſſet, non <lb/>ſcripſiſſet poſtea contra, vt puto. </s> + <s xml:id="s2930" xml:space="preserve">Res enim ita ſe habet, quod Mars in omni circui <lb/>tu ſui epicycli tranſiens per inferiorem partem ipſius epicycli, ſemper commoratur <lb/>multis menſibus in vno duodecatemorio Zodiaci, ſcilicet .6. et .7. menſibus, atque <lb/>etiam amplius, quod quidem ego ſæpe obſeruaui, præſertim anno .1565. et .1566. <lb/>hoc ordine. </s> + <s xml:id="s2931" xml:space="preserve">Primum inſpiciens Ephemeridas ſtadij, reperi <reg norm="quod" type="simple">ꝙ</reg> Mars ſecundum eum <lb/>egrediebatur retrogradationem circa diem .12. Ianuarij anni 1566. in .16. grad. <lb/>Geminorum. </s> + <s xml:id="s2932" xml:space="preserve">Et ſimiliter quod anno .1565. die vltima Auguſti Mars futurus erat in <lb/>eodem ſupradicto loco, priuſquam retrogradi inciperet. </s> + <s xml:id="s2933" xml:space="preserve">Poſtea inueni, quod poſt <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 conſumebatur ſpatium <lb/>menſium .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 ſupputato, ſumpſi inſtrumenta, & ad experimentum me paraui, & vltima <lb/>nocte menſis Auguſti anni .1565. reperi Martem eſſe in dicto gradu geminorum vt <lb/>ſcribebat Stadius. </s> + <s xml:id="s2936" xml:space="preserve">Deinde ſingulis ebdomadibus obſeruans retrogradationem; </s> + <s xml:id="s2937" xml:space="preserve">vidi <lb/>circa finem Octobris quod retrogradi incipiebat, & ea retrogradatio perſeuerauit <lb/>vſque ad medium menſem Ianuarium, aut circiter, anni .1566. obſeruaui poſtea <lb/>etiam ſitum eiuſdem planetæ die .11. </s> + <s xml:id="s2938" xml:space="preserve">Aprilis ſequentis <reg norm="eumque" type="simple">eumq́;</reg><unsure/> inueni in gradu .16. <lb/>Cancri, vti eum poſuerat Stadius. </s> + <s xml:id="s2939" xml:space="preserve">Atque ita experimentum meum conuenit cum <lb/>calculo Stadij, <reg norm="comperique" type="simple">comperiq́;</reg> eum non erraſſe: </s> + <s xml:id="s2940" xml:space="preserve">Et ſic quiſque binis <reg norm="quibusque" type="simple">quibusq́;</reg> annis pote-<lb/>rit certior fieri de veritate. </s> + <s xml:id="s2941" xml:space="preserve">Si autem delectationis cauſa id experiri volueris, expe-<lb/>ctato primam retrogradationem Martis, cuius initium ſecundum Stadium futurum <lb/>eſt 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 aſpice quum erit in grad .9. Leonis, quod erit circa diem .7. Mai .1583. & vide-<lb/>bis <reg norm="quod" type="simple">ꝙ</reg> ipſe Mars in his gra .30. <reg norm="morabitur" type="simple">morabit̃</reg> <reg norm="per" type="simple">ꝑ</reg> menſes .7. & dies .18. <reg norm="atque" type="simple">atq;</reg> vt eius rei pericu-<lb/>lum facias, obſerua noctem præcedentem diei .19. </s> + <s xml:id="s2944" xml:space="preserve">Septem .1582. locum <reg norm="longitudinis" type="context">lõgitudinis</reg> <lb/>eius ſtellæ, & idem poſtea obſerua nocte præcedente diei .7. Mai, aut nocte ſequen-<lb/>ti .1583. & inter duos hoſce terminos obſerua aliqua alia nocte ſtatum eius. </s> + <s xml:id="s2945" xml:space="preserve">Mani-<lb/><reg norm="feſtoque" type="simple">feſtoq́;</reg> videbis Martem conſumere 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 Ptolomęi <lb/>Almageſtum, minori labore poſſet per calculos ſcientificos colligere verita-<lb/>tem, ſuppoſitis tamen terminis ſcriptis in theoricis planetarum. </s> + <s xml:id="s2947" xml:space="preserve">Qui enim vidit <lb/>Almageſtum vel reuolutiones orbium cœleſtium Nicolai Copernici, non poteſt de <lb/>hoc vllo pacto dubitare. </s> + <s xml:id="s2948" xml:space="preserve">Sed qui nondum tantopere progreſſus eſt, <reg norm="ſaltem" type="context">ſaltẽ</reg> capiat <reg norm="huius" type="simple">huiꝰ</reg> <lb/>rei notitiam vniuerſalem, 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">taliũ</reg>, quales ſunt ſexageſimæ ſemidiametri ipſius <lb/>deferentis, & ſemidiametrum epicycli eſſe, partium <reg norm="ſupradictarum" type="context">ſupradictarũ</reg> .39. cum dimidia, <lb/>& quod argumenta vera, in temporibus primarum ſtationum ( cum epicyclus eſt in <lb/>auge, aut in eius oppoſito, aut in <reg norm="lungitudinib" type="context">lũgitudinib</reg>. me dio cribus ) <reg norm="iam" type="context">iã</reg> ab antiquis rectè ſuppu <lb/>tata ſint, ſicuti ſunt. </s> + <s xml:id="s2950" xml:space="preserve">Et præſupponat motum diurnum centri epicycli. min .31. cum di <lb/>midio, quamuis reuera ſit min .31. & ſecundorum .27. aut circiter, nunc <reg norm="quidem" type="context">quidẽ</reg> præter <lb/>mittens, quod vnus habeat reſpectum ad augem mediam epicycli, & alter ad cen-<lb/>trum æquantis. </s> + <s xml:id="s2951" xml:space="preserve">Atque his præſuppoſitis fingat ( exempli gratia ) quod centrum <lb/>epicycli ſit in quauis longitudinum mediarum, & Mars in prima maxima æqua-<lb/>tione argumenti, ſcilicet 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="proficiſcens" type="context">proficiſcẽs</reg> <lb/>Mars perget ad lineam primæ ſtationis, vt poſtea retrogradiatur, veluti ſi in infrapo <lb/>ſita figura maiori, <reg norm="centrum" type="context context">cẽtrũ</reg> <reg norm="mundi" type="context">mũdi</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> ſint notatæ <var>.o.i.</var> et <var>.o.t.</var> & li-<lb/>nea primæ ſtationis <var>.o.n.b.</var> & linea ſecundæ <var>.o.u.d.</var> ſi 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 Alfonſi ponatur eſſe gra .41. <lb/>minut .10. quod euenit quia calculatores ipſarum tabularum interuallum <var>.o.c.</var> quod <lb/>in eo ſitu epicycli interponitur inter centrum mundi, & centrum dicti epicycli, ac-<lb/>ceperunt partium ſexaginta præcisè, nihili facientes minuta illa .18. aut circiter, quę <lb/>verè ſunt 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 ſitu epicycli ſit baſis vnius trianguli orthogonij, cuius vnum ex illis duobus late-<lb/>ribus eſt ſemidiameter eccentrici partium .60. pręcisè, aliud eſt interuallum eccen-<lb/>tricitatis partium .6. eiuſmodi. </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 ſecundum ſitum epicycli. </s> + <s xml:id="s2953" xml:space="preserve">& cum centrum <lb/>eius eſt in auge eccentrici. eſt minimus <reg norm="quam" type="context">quã</reg> eſſe poſſit. </s> + <s xml:id="s2954" xml:space="preserve"><reg norm="eſtque" type="simple">eſtq́;</reg> tantum grad .36. min <num value="46">.<lb/>46.</num> & in oppoſito ipſius augis eſt grad .47. min .1. maximus quam alibi vnquam ſit, <lb/>& ſic continuò variatur, ſecundum ſitum, 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> eſt ſemper rectangulus in puncto <var>.i.</var> & latus <var>.c.i.</var> reſpectu <lb/>ſemidiametri eſt 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> ſit linea veri motus epicycli, & latus ſimiliter <lb/>vnius trian guli, cuius duo latera ſunt ſupradicta, ſcilicet ſemidiameter eccentrici, & <lb/>eccentricitas, inter ſe compræhendentes angulum datum. </s> + <s xml:id="s2958" xml:space="preserve">Nam ſemper præſuppo <lb/>nitur datus locus centri ipſius epicycli, cum ipſe eſt extra augem aut oppoſitum eius <lb/>quia in auge linea <var>.o.c.</var> conſtat ex ſemidiametro eccentrici & interualli eccentricita-<lb/>tis. </s> + <s xml:id="s2959" xml:space="preserve">& in eius oppoſito, ipſa linea <var>.o.c.</var> eſt minor dicto ſemidiametro eccentrici per in <lb/>teruallum dictæ eccentricitatis. </s> + <s xml:id="s2960" xml:space="preserve">Vnde etiam poſſumus extra augem, vel oppoſitum <lb/>eius cognoſcere <var>.o.c.</var> tanquam latus dicti trianguli duorum laterum <reg norm="cum" type="context">cũ</reg> angulo cogni <lb/>torum. </s> + <s xml:id="s2961" xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</reg> per .49. propoſitionem libri primi <reg norm="eiuſdem" type="context">eiuſdẽ</reg> Monteregij cum ſcilicet dictus <lb/>angulus <reg norm="non" type="context">nõ</reg> fuerit rectus. </s> + <s xml:id="s2962" xml:space="preserve">Nam ſi fuerit rectus videbitur per .27. et .28. ſupra 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 reſpondet arcus <var>.i.g.</var> epicycli confectus à Marte <lb/>in diebus circiter .105. ad rationem min .28. aut circiter in ſingulos dies, prætermiſ-<lb/>ſis nunc quidem minutijs cum exigui momenti ſit error .15. aut .20. dierum ad verifi <lb/>cationem longæ morę 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">cũ</reg> <lb/>dimidio in ſingulos dies. qui numerus graduum .55. min .7: differt à numero <reg norm="graduum" type="context">graduũ</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 reſpectum ad centrum æquantis, magis quam ad centrum <reg norm="mundi" type="context">mũdi</reg>, <lb/>quia differentia non eſt tanta, vt poſſit inducere errorem menſium. </s> + <s xml:id="s2964" xml:space="preserve">Hinc ſequitur <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, ſed <lb/>gradibus .14. min .12. vlterius quam in primo loco, in quo erat in Zodiaco, & erit in <lb/>medio ſuæ retrogradationis. </s> + <s xml:id="s2966" xml:space="preserve">Sed quoniam Mars manifeſtè retrogradi non incipit <lb/>in puncto <var>.i.</var> conting entiæ, imo ab illo puncto vſque ad terminum primæ ſtationis li <lb/>neæ <var>.o.n.</var> interponitur arcus <var>.i.n.</var> epicycli, qui eſt graduum .32. minu .14. </s> + <s xml:id="s2967" xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</reg> cogno-<lb/>ſcitur ſubtrahendo arcum <var>.f.i.n.</var> graduum .163. mi .9. qui eſt inter augem, & primam <lb/>ſtationem, à gradibus .180. ( qui arcus <var>.f.i.n.</var> erit verum argumentum, quod ſi-<lb/>militer variatur ſecundum ſitum epicycli, etſi eiuſmodi varietas, nobis <reg norm="non" type="context">nõ</reg> eſt magni <lb/>momenti, vnde poſſumus præſupponere, quod <var>.c.</var> centrum epicycli non alteret <reg norm="in- teruallum" type="context">in-<lb/>teruallũ</reg> <var>.c.o.</var> à centro <reg norm="mundi" type="context">mũdi</reg>, <reg norm="cum" type="context">cũ</reg> non posſit intercedere, error <reg norm="menſium" type="context context">mẽſiũ</reg> reliquum verò <var>.g.<lb/>n.</var> graduum .16. min .51. ſubtrahendo 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 eiuſmodi tamen ſitu 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 ſupradictam, omittendo quod ipſa ſtella habeat reſpectum ad <lb/>augem mediocrem epicycli, & quod dicta aux mediocris mutet diſtantiam 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 ſupradictam. </s> + <s xml:id="s2969" xml:space="preserve">Reſtat nunc no-<lb/>bis inuenire angulum <var>.b.o.c.</var> in centro <reg norm="mundi" type="context">mũdi</reg> inter duas lineas, <var>b.o.</var> et <var>.c.o.</var> <reg norm="quarum" type="context">quarũ</reg> prior <lb/>eſt primæ ſtationis, altera eſt 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> ſint 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 ſi-<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 ſinus anguli <var>.c.n.h.</var> reſtantis ex vno recto, erit partium .95706. dicendo <lb/>poſtea ſi <var>.n.e.</var> tanquam ſinus 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 ſubtractis extota <var>.c.o.</var> quę eſt <reg norm="partium" type="context">partiũ</reg> .60. mi .18. reliqua <lb/>erit nobis <var>.h.o.</var> partium .22. min .30. capiendo poſtea radicem <reg norm="quadratam" type="context">quadratã</reg> ſummæ 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> eſt partium .11. min .27. ſi igitur ad <var>.o.n.</var> tanquam partium .25. min .12. reſpondet <var>.<lb/>n.h.</var> partium .11. minuti .27. linea <var>.n.h.</var> ad <var>.o.n.</var> tanquam partium 100000. reſpon <lb/>debit part .45436. tanquam ſinus anguli <var>.n.o.h.</var> qui angulus erit gra .27. minut .1. ſub <lb/>tracto poſtea 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/>ſtationis in eiuſmodi ſitu. </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. ( præſuppoſita ſemper eadem di-<lb/>ſtantia <var>.o.c.</var> quamuis nonnulla ſit 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. reſtabunt gra .22. min .19. itaque in diebus .69. </s> + <s xml:id="s2973" xml:space="preserve">Mars promo-<lb/>tus fuerit a primo ſitu 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/>ſtatione vſque ad <lb/><reg norm="lineam" type="context">lineã</reg> veri motus <lb/>epicycli ſunt gra <num value="16">.<lb/>16.</num> min .51. ipſius <lb/>epicycli, vt ſupra <lb/><reg norm="vidimus" type="simple">vidimꝰ</reg> quos Mars <lb/>tranſit in diebus <lb/>36. aut circiter ad <lb/>rationem min .28. <lb/>in ſingulos dies, <lb/>quo tempore cen <lb/>trum epicycli, in <lb/>tali diſtantia 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 ſingulos dies, <lb/><reg norm="quibus" type="simple">quibꝰ</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. ſubtractis à <lb/>grad .22. minu .19. per quos Mars progreſſus erat, ſupererunt grad .14. minut .12. <lb/>quibus ipſe in media retrogradatione exiſtens in linea <var>.o.c.</var> veri motus epicycli pro-<lb/>motus erit à principio primi ſitus. </s> + <s xml:id="s2976" xml:space="preserve">Quod cum eo concordat quod ſupra diximus cir <lb/>ca hos gradus .14. min .12. vltra primum ſitum in ſpatio dierum .105. vt ſupra, ad <reg norm="tam" type="context">tã</reg> <lb/>tum enim aſcendunt .69. et .36. nunc fingendo <reg norm="quod" type="simple">ꝙ</reg> Mars pergat in ſuo motu compo-<lb/>ſito qui conſtat ex his duobus circulis, vi eccentrici, & epicycli ( quanquam <lb/>vt dixi omittimus illam ſummam ſubtilitatem ſeu ſcrupuloſi<unsure/> atem <reg norm="continuæ" type="context">cõtinuæ</reg> inæqua-<lb/>litatis diſtantiæ 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 ſuo pergat, vſque ad punctum ſecundæ ſta-<lb/>tionis, præteribunt alij dies .36. vt prius, quibus iunctis cum .105. fiet ſum-<lb/>ma dierum .141. & Mars retrogreſſus erit per alios grad .8. minut .7. quibus ſub-<lb/>tractis a gradibus .14. m in .12. per quos progreſſus erat ſupererunt. gra .6. min .5. qui <lb/>bus ipſe Mars in fine ſuæ retrogradationis promotus erit à primo loco vnde moueri <lb/>cœpit. </s> + <s xml:id="s2977" xml:space="preserve">Inter hanc igitur ſecundam ſtationem lineæ <var>.o.u.d.</var> & lineam <var>.o.t.</var> ſecundæ con <lb/>tingentiæ, Mars diebus .69. vt prius, confecerit gradus .32. min .14. ſui epicycli, & eo-<lb/>dem tempore linea contingens <var>.o.t.</var> ambulauerit gra .36. min .13. vt prius, à quo itine <lb/>re ſubtracto angulo ſecundo <var>.d.o.t.</var> graduum .13. min .54. ſupererunt 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. ideſt in .141. et .36. qui ſunt ferè m. n <lb/>ſes .6. </s> + <s xml:id="s2978" xml:space="preserve">Itaque Mars partim ſurſum partim, deorſum ambulans detentus erit <reg norm="menſibus" 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> ſi 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/>ſus oppoſitum au-<lb/>gis, <reg norm="longior" type="context">lõgior</reg> erit mo <lb/>ra planetæ in eiuſ <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 eſt poſſibi-<lb/>litas huius moræ <lb/>Mattis. </s> + <s xml:id="s2981" xml:space="preserve">Quod qui <lb/>dem mihi ſuffice-<lb/>re videtur <reg norm="non" type="context">nõ</reg> mo-<lb/>do cibi, ſed etiam <lb/>cuiuis, qui harum <lb/>ſcientiarum prin-<lb/>cipia teneat. </s> + <s xml:id="s2982" xml:space="preserve">Ne-<lb/>que enim <reg norm="nunc" type="context">nũc</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 ſunt conſumati, nec curam mihi ſuſcipere erudiendi imperitos. </s> + <s xml:id="s2983" xml:space="preserve">Satis igitur ſit <lb/>oſtendiſſe, quod qui ſcripſit Martem commorari poſſe tam multos menſes in vno ſi <lb/>gno, non impoſſibilem rem tradidit. </s> + <s xml:id="s2984" xml:space="preserve">Immo per obſeruationes huius veritatis mil-<lb/>lies factas, Aſtrologi fecere ſupradictas ſuppoſitiones neceſſarias ad <reg norm="reducendum" type="context">reducẽdum</reg> in <lb/>ſuas cauſas, & ad regulam, eiuſmodi 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 ſcriptorem harum animaduerſionum tantopere eiuſmodi <lb/>mora commoueri, ſed cogitare <reg norm="quod" type="simple">ꝙ</reg> fortaſſe calculi facti fuerunt eo tempore quo mi-<lb/>ſer Mars à Vulcano rete vinctus erat. </s> + <s xml:id="s2986" xml:space="preserve">Vnde cum nonita celeriter ſe expedire poſſet <lb/>iter eius ſegnius peractum fuit. </s> + <s xml:id="s2987" xml:space="preserve">Aut <reg norm="quod" type="simple">ꝙ</reg> <reg norm="quum" type="context">quũ</reg> vulneratus fuit in bello Troiano, vis eius <lb/>& agilitas per aliquantulum temporis imminuta fuit. </s> + <s xml:id="s2988" xml:space="preserve">Atque ſi hic etiam intellexiſ <lb/>ſet eum <reg norm="aliquando" type="context">aliquãdo</reg> fuiſſe in poteſtate Othi, & Ephialtis <reg norm="vinctum" type="context">vinctũ</reg> & carceri incluſum <reg norm="men- ſes" type="context">mẽ-<lb/>ſes</reg> tredecim, dum ab Eribea ſolutus fuit, vt tu, antiquos ſequens, eleganter ſcribis <lb/>in illis tuis pulcherimis dialogis. non exiſtimaſſet, credo, tam abſurdum quod alius <lb/><reg norm="eum" type="context">eũ</reg> detinuiſſet ſex aut ſeptem menſibus, ſed operam dediſſet vt a te intelligeret quid <lb/>ſibi 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 diſtat à me-<lb/>dio circiter dies .8. ſupponens medium motum eſſe dierum .683. <reg norm="quod" type="simple">ꝙ</reg> etiam falſum eſt. <lb/></s> + <s xml:id="s2991" xml:space="preserve">Sed vtcunque ſit, 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 poteſt etiam <lb/>eſſe multo breuior, ſicuti erit à die 3. </s> + <s xml:id="s2993" xml:space="preserve">Decembris anni .1593. <reg norm="vſque" type="simple">vſq;</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 ſine ratione, nam dicto <lb/>initio Decembris Mars paulo ante cæperiteſſe directus, cum centrum epicycli erit <lb/>circa medium Tauri, & eius ſtella in principio Arietis & initio Iunij .1595. </s> + <s xml:id="s2995" xml:space="preserve">Mars pa <lb/>rum diſtabit ab initio retrogradationis, regreſſus tamen ad initium ipſius Arietis, & <lb/>centrum epicycli erit circa medium Aquarij, in cuius ſigni medio, hac ætate repe-<lb/>ritur oppoſitum augis, & in quo ſitu, æquationes <reg norm="argumenti" type="context">argumẽti</reg> ſunt, quam maximę eſſe <lb/>poſſint, quum centrum epicycli circuiuerit ſolum circiter tres quartas totius ambi-<lb/>tus, & Mars circuiuerit per partem ſuperiorem epicycli circiter gradus .252. </s> + <s xml:id="s2996" xml:space="preserve">Hoc au <lb/>tem dico, vt oſtendam 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 poſtea idem author miratur interualla, quæ <reg norm="ponuntur" type="context">ponũtur</reg> inter coniunctio <lb/>nes Iouis, & Martis in eodem ſigno, <reg norm="eaque" type="simple">eaq́;</reg> vocat errores maximos, oſtendit ſe non re <lb/>ctè conſideraſſe motus eorum. </s> + <s xml:id="s2999" xml:space="preserve">Et præcipuè primum miratur <reg norm="quod" type="simple">ꝙ</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 ſe diſtent ann .25. afferens pro ratione, quod hæc duo ſydera, altero quoque anno <lb/>coniunguntur, ſic dicens.</s> + </p> + <quote> + <s xml:id="s3001" xml:space="preserve">Qui ſciet has duas ſtellas ſecundo quoque anno inter ſe coniungendas, mirabitur <lb/>quomodo non poterunt numeratores, huiuſmodi 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> conſiderauit, <reg norm="quod" type="simple">ꝙ</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">aũt</reg> in eo ſæpe fuit. </s> + <s xml:id="s3004" xml:space="preserve">Vnde im <lb/>poſſibile eilvidetur eos non conueniſſe in dicto ſigno. </s> + <s xml:id="s3005" xml:space="preserve"><reg norm="Idemque" type="simple">Idemq́;</reg> dici poteſt de alijs <lb/>coniunctionibus eorundem planetarum, atque has differentias temporum inter di-<lb/>ctas coniunctiones ipſe tribuit erroribus calculorum Ephemeridum, non autem ta-<lb/>bularum, vt ſupra dixit. </s> + <s xml:id="s3006" xml:space="preserve">ſed neſcio quare vellet dictos planetas coire in Leone, ſi <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 ſignis <reg norm="vſque" type="simple">vſq;</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">Idq́;</reg> fortaſſe, Iupiter data opera fecit, vt huiuſ-<lb/>modi Aſtrologos 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 poſtea ait, eos ſecundo quoque anno coniungi, animaduertendum eſt, <reg norm="quia" type="simple">ꝗa</reg> <lb/>(vt <reg norm="iam" type="context">iã</reg> dixi) duæ ſunt ſpecies coniunctionum, quarum vna eſt linearum eorum me-<lb/>diorum motuum, altera corporum eorum, ſaltem in longitudine, cum ambo inue-<lb/>niuntur in eodem circulo, qui tranſit per polos ecclipticæ, nam eos inueniri in <reg norm="eadem" type="context">eadẽ</reg> <lb/>linea recta <reg norm="tranſeunte" type="context context">trãſeũte</reg> per centrum mundi, rarisſimum eſt. </s> + <s xml:id="s3010" xml:space="preserve">Atque coniunctio ſupra di-<lb/>ctarum linearum vocatur media, & inter Iouem & Martem fieri ſolet ſpatio <reg norm="dierum" type="context">dierũ</reg>. <lb/>816. cum dimidio, aut circiter. </s> + <s xml:id="s3011" xml:space="preserve">Altera <reg norm="dicitur" type="simple">dicit̃</reg> vera, ſiue apparens, & irregulatiſſima, <lb/>quæ quidem non ſeruat 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 tranſeat per totum Zodiacum, non ideo <lb/>neceſſe eſt, vt in ſpatio .24. annorum coniungantur in ſingulis ſignis, nunquam in eo <lb/>deſicientes, vtipſe credit loquens de veris coniunctionibus apparentibus, eo quod <lb/>ſint irregulatiſſimæ, vt dixi.</s> + </p> + <p> + <s xml:id="s3014" xml:space="preserve">Atque ſi 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, quę <lb/>eſt dierum .4328. & Martis, quæ eſt dierum .687. in quo tempore Martis, Iupiter am <lb/>bulat gra .57. min .8. & diebus .30. conficit. grad .2. minut .29. & ſecun .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">graduũ</reg> .360. in diebus .687. <lb/>ſingulis .30. diebus conficit. gra .15. mi .43. <reg norm="ſecum" type="context">ſecũ</reg> .14. vnde differentia inter eos eſt gra <lb/>duum .13. mi .15. <reg norm="ſecum" type="context">ſecũ</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 hęc eſt periodus infallibilis mediarum con <lb/>iunctionum Iouis cum Marte.</s> + </p> + <p> + <s xml:id="s3019" xml:space="preserve">Nunc venientes ad tabulas Animaduerſionum, videbimus hæc mirabilia eius, in <lb/>quo conſiſtant & vbi ſint tam multi inſignes 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, & ſimiliter Iouem. </s> + <s xml:id="s3021" xml:space="preserve">ſed 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">mirũ</reg> videtur, quod vnus ex his calculatoribus detineat Saturnum plu-<lb/>ribus menſibus in vno ſigno, & alter in alio, non animaduertens dictam <reg norm="differentiam" type="context">differentiã</reg> <lb/>eſſe eius rei cauſam. </s> + <s xml:id="s3023" xml:space="preserve">Miratur item, quod vnus ex is faciat Saturnum morari paucis <lb/>menſibus in vno ſigno, alter vero eum ibi detineat integris annis. </s> + <s xml:id="s3024" xml:space="preserve">Vt exempli gra-<lb/>tia, verſus finem ſuæ tabulæ Saturni, dicit quod Leouitius eum carceri includit in <lb/>geminis annis .2. menſe vno, & diebus .9. </s> + <s xml:id="s3025" xml:space="preserve">Stadius vero clementior eum liberat intra <lb/>menſes .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 vſque ad diem .6. Septembris, <lb/>eiuſdem anni gra .6. min .34. <reg norm="eumque" type="simple">eumq́;</reg> poſtea retrogradum inde exire die .22. Decem. <lb/>eiuſdem anni, cum ingreditur in Taurum, vbi partim retrogradus, & partim dire-<lb/>ctus manet vſque ad diem .20. Februa .1560. rediens poſtea in geminos, in quibus <lb/>manet vſque ad diem Iunii .1561. & inde ingreditur in Cancrum, <reg norm="ambulatque" type="simple">ambulatq́;</reg> dire-<lb/>ctus. gra .4. min .59. vſque ad diem .4. Octob. </s> + <s xml:id="s3027" xml:space="preserve">Vnde retrogradiens rurſum intratin Ge <lb/>minos die .28. Decemb. eiuſdem anni, at que ibi partim retrogradus partim directus <lb/>manet vſque ad diem .12. Apr .1562. itaque in pluribus vicibus facit eum morari in <lb/>Geminis dies circiter .816. ideſt circiter menſes .27. ſumpſit autem hic ſcriptor 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 cauſa 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">ꝙ</reg> planeta non perambulet totum ſignum, in quod eſt ingreſſus <lb/>vel directus vel totum retrogradus. </s> + <s xml:id="s3030" xml:space="preserve">Atque hæc opinio ſimilis eſt ſuperiori de con <lb/>iunctionibus veris Saturni, & Iouis, vbi dicit quod nunquam coniunguntur in vno ſi <lb/>gno alterius triplicitatis, niſi perfecerit coniunctionem in omnibus ſignis primæ tri <lb/>plicitatis. </s> + <s xml:id="s3031" xml:space="preserve">Verum vt ſuperſedeam vlterius diſputare, mihi videtur, quod hactenus <lb/>dixi, poſſe tibi ſatisfacere, quod attinet ad ſciendam ſententiam mean ſuper dictis <lb/>Animaduerſionibus latinè ſcriptis. </s> + <s xml:id="s3032" xml:space="preserve">Hoc tamen non prætermittam, <reg norm="quod" type="simple">ꝙ</reg> hic non ani-<lb/>maduertit, <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quod" type="simple">ꝙ</reg> <reg norm="differentiæ" type="context">differẽtiæ</reg> locorum <reg norm="planetarum" type="context">planetarũ</reg> quæ <reg norm="suntinter" type="context">sũtinter</reg> ephemeridas Leouitij <lb/>& Stadij, euenere, quia vnus ſupputat <reg norm="cum" type="context">cũ</reg> radicibus, & <reg norm="fundamentis" type="context">fundamẽtis</reg> Alfonſi alter verò <lb/>Reinoldi ex Copernico recentius obſeruatis, ita idem euenire poterit futuris tem-<lb/>poribus, ſi ſupputati fuerint dicti motus, & loci cum recentioribus obſeruationibus <lb/>cum impoſſibile ſit tam ſubtiliter, tanq́ue perfectè ſupputare loca & <reg norm="motus" type="simple">motꝰ</reg> eorum, <lb/>vtlungo interuallo temporis non comperiantur in eis aliquæ differentię, cuius rei re <lb/>medium eſt ſemper ſequi recentiores obſeruationes & tabulas.</s> + </p> + <p> + <s xml:id="s3033" xml:space="preserve">Atque vt tibi ſatisfaciam etiam circa alia ſcripta vulgari lingua edita menſibus .4 <lb/>poſt latina, etſi intelligere potes, qualia poſſint eſſe alia eius ſcripta, ex ijs quæ ſupra <lb/>dicta ſunt, atque etiam ex eo, quod dicit ſemiſiſſe multa exempla ſuarum Animad <lb/>uerſionum in varias terras, illis qui profitentur has ſcientias, aut earum ſtudioſi ſunt, <lb/>nec <reg norm="quenquam" type="context">quenquã</reg> inueniſſe qui ad <reg norm="tam" type="context">tã</reg> laudabilem prouinciam motus ſit, nec vidiſſe, <reg norm="quod" type="simple">ꝙ</reg> ali <lb/>quis reſponderit eius rationibus; </s> + <s xml:id="s3034" xml:space="preserve">laudabilem prouinciam, autem puto, <reg norm="quod" type="simple">ꝙ</reg> intelligat <lb/>correctionem ephemeridum, verens, ne culpa calculatorum, qui eas ſumpſere e ta-<lb/>bulis, tam differentes ſint, vt <reg norm="quibuſdam" type="context">quibuſdã</reg> locis cap .1. </s> + <s xml:id="s3035" xml:space="preserve">Videtur, & præcipuè vbi ſic ait.</s> + </p> + <quote xml:lang="it"> + <s xml:id="s3036" xml:space="preserve">Perche eſſendo impoſſibile alli ſtudioſi di dette ſcientiæ di non ſeruirſi delle <lb/>ephemeridi, maggiormente a quelli che non ſanno ſeruirſi delle tauole, e cono-<lb/>ſcendo d'incorrere in errori ſenza hauerui altro rimedio, ſarebbono forzati di ab <lb/>bandonare i ſtudij loro.</s> + </quote> + <p> + <s xml:id="s3037" xml:space="preserve">Quanquam circa finem dicti capitis redeat in meliorem viam & aduerſetur ſi-<lb/>bijpſi vbi ſic ait.</s> + </p> + <quote xml:lang="it"> + <s xml:id="s3038" xml:space="preserve">Che poi eſſi poſſeſſori della ſcienza, & c.</s> + </quote> + <p> + <s xml:id="s3039" xml:space="preserve">Etiam aperiam tibi, quæ mea ſit de ijs ſententia.</s> + </p> + <p> + <s xml:id="s3040" xml:space="preserve">Hicigitur in ſcriptis Italicis, vt morderet aliquem ex ijs, qui eius ſuperiora ſcripta <lb/>non laudauerant, occaſionem capit aperiendi aliquos illius errores, per editionem <lb/>collationis quorundam calculorum a ſe collectorum illius, atque etiam aliorum, cu <lb/>ius calculi ſunt in ſecunda, & ſeptima figura. </s> + <s xml:id="s3041" xml:space="preserve">Sed prius quam veniamus ad defenſio <lb/>nem harum duarum figurarum vide obſecro quam alienum ei videatur, quod alij <lb/>dixerint differentiam ephemeridum non eſſe magni momenti, non afferens reſpe-<lb/>ctum vllum, qui enim dixerunt eiuſmodi differentiam non eſſe magni momen-<lb/>ti id dixerunt habito reſpectu ad ſignum in quo eſt 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 ſigno, tuncid nullius momenti eſt, <lb/>& ſic in coniunctionibus aut alijs aſpectibus duo, aut .3. gradus non faciunt alteratio <lb/>nem ſenſibilem, cum virtus coniunctionum, & aſpectuum inſit, & duret per mul-<lb/>tos gradus ante aut poſt ipſum punctum. </s> + <s xml:id="s3042" xml:space="preserve">Nec quicquam <reg norm="tamen" type="wordlist">tamẽ</reg> eſt qui dubitet, quin <lb/>præſtaret ſcire ſubtiliter ipſum 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 correſpondeant gradibus æquatoris. </s> + <s xml:id="s3044" xml:space="preserve">Et præterea in ephe <lb/>meridibus videntur certè motus & aſpectus luminarium, quamuis inſit <reg norm="differentia" type="context">differẽtia</reg> mi <lb/>nutorum. </s> + <s xml:id="s3045" xml:space="preserve">Nam non differunt gradibus, præter ſitum parum diſtantem à vero om-<lb/>nium planetarum, quorum cognitio in cœlo, quamuis circa eorum locum error eſ-<lb/>ſet gra .10. tamen in hoc prodeſſet, & tempus aſpectus eorum, etſinon diei præcisè, <lb/>quia influentia eiuſmodi a ſpectuum, præterquam Lunæ durat multis diebus, & non <lb/>vno tantum. </s> + <s xml:id="s3046" xml:space="preserve">præterquam quod ipſæ ephemerides <reg norm="oſtendunt" type="context">oſtendũt</reg> nobis tempus <reg norm="ecclipſium" type="context">ecclipſiũ</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 ſunt 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 ſignis alterius triplicitatis prius quam peręgerit <lb/>præcedentem, quam de faciendo currere <reg norm="Saturnum" type="context">Saturnũ</reg>, & de retinendo Ioue, de <reg norm="detinem" type="context">detinẽ</reg> <lb/>do Marte .6. aut .7. menſibus in vno ſigno, de Marte, & Ioue non coeuntibus ſingu-<lb/>lis .24. annis in quolibet ſigno, & cius generis alia, minime verum eſt quod ſint er-<lb/>rores, quamuis huic præbuerint occaſionem toties errandi.</s> + </p> + <p> + <s xml:id="s3049" xml:space="preserve">Comparatio poſtea inter eius calculos ſumptos partim ex tabulis Iunctini, & par <lb/>tim ex ephemeridibus Stadij tan quam calculis Copernici, & calculos figurarum ſu-<lb/>per eis poſitarum ſupputatarum à diuerſis per ephemeridas Alfonſinas, etiam pro-<lb/>poſita ab eo eſt ad oſtendendum magnam & monſtruoſam differentiam, vt ait cap <num value="2">.<lb/>2.</num> vbi miratur, quod cum ex communi ſententia calculi Copernici meliores ſint, cal <lb/>culatores dictarum figurarum potius eos ſumpſerint à tabulis Alfonſi, quam Coper <lb/>nici. </s> + <s xml:id="s3050" xml:space="preserve">Quæ admiratio quam aliena ſit, conſiderandum permittam cuiuis intelligen-<lb/>ti harum ſacultatum, cum ſæpe accidere poſſit. </s> + <s xml:id="s3051" xml:space="preserve">vt cum aliquis velit ſcire ſolum vni <lb/>uerſalia alicuius geneſis, ſiue natiuitatis, cum non inueniantur ephemerides Coper-<lb/>nici, ſed tantum Alfonſi, calculator vtatur tantum ephemeridibus, quas inuenit, <reg norm="tum" type="context">tũ</reg> <lb/>cauſa vitandi tædij calculi tabularum, qui magni laboris eſt, pręcipuè in tabulis Pru <lb/>tenicis Reinoldi. </s> + <s xml:id="s3052" xml:space="preserve">tum quia ſuperflua ei eſt ſumma ſubtilitas, cum non curet laborare <lb/>circa directiones vt <reg norm="factum" type="context">factũ</reg> eſt pro <reg norm="ſecunda" type="context">ſecũda</reg> figura ab hoc propoſita, quæ erat anni .1551 <lb/>quo non <reg norm="inueniebantur" type="context">inueniebãtur</reg> ephemerides Copernicæ, quæ non editæ ſunt ante annum <lb/>1554. præter quam quod ille nobilis vir pro quo ſupputata fuit dicta ſecunda natiui <lb/>tas dubitabat de anno, vt hic ſimiliter ſcit. </s> + <s xml:id="s3053" xml:space="preserve">quare potuiſſet perdi tempus, & labor, ſi <lb/>ſupputata fuiſſet per tabulas Reinoldi, nam Iunctini tabulæ nondum editæ fuerant. <lb/></s> + <s xml:id="s3054" xml:space="preserve">Calculus poſtea ſeptimæ figuræ, qui erat reuolutio dictæ ſecundæ natiuitatis, <reg norm="duabus" type="simple">duabꝰ</reg> <lb/>de cauſis non factus eſt 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="deſinunt" type="context">deſinũt</reg> ann .1576. & <reg norm="continuatæ" type="context">cõtinuatæ</reg> poſtea quæ perueniunt vſque <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 eſt, 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 ſolum non curant eas ſupputare per <lb/>tabulas, ſed nec exquiſitè quidem per ephemeridas. </s> + <s xml:id="s3058" xml:space="preserve">Calculi poſtea ab hoc ſumpti <lb/>ex tabulis Iunctini, & poſiti ſub dicta ſecunda figura, adeò rectè facti ſunt, vt cum ſe <lb/>cundum ip ſas tabulas oporteat Saturnum eſſe circa .32. minutum gradus .23. Aqua-<lb/>rij, ipſe eum ſcribat in gra .11. mi .3. dicti ſigni. Iupiter ſimiliter qui ſecundum dictas <lb/>tabulas inuenitur circa finem gradus .5. Cancri, ab eo ponitur in min .28. gra .19. eiuſ <lb/>dem. ex quibus planetis Saturnus in figura poſitus eſt 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 ſecun 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"/> + ſi & Iunctini in Saturno non erat differentia plus quam minu .5. & in Ioue min .4. ſed <lb/>ſecundum 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 ſunt quidem differentiæ magnæ, & monſtruoſæ vtipſe <lb/>eas vocat, vt etiam eſtilla Veneris, & Mercurij inter tertiam figuram, & eius calcu-<lb/>lum ſumptum, non quidem à tabulis laborioſis, ſed à ſimplicibus ephemeridibus Sta <lb/>dij, quæ differentia <reg norm="non" type="context">nõ</reg> eſt quidem paucorum graduum, cum ſit tertiæ partis cœli in <lb/>quolibet dictotum planetarum. </s> + <s xml:id="s3061" xml:space="preserve"><reg norm="Huiuſmodique" type="simple">Huiuſmodiq́;</reg> monſtra certè non ſunt orta à ta-<lb/>bulis ſiue ephemeridibus diuerſis, ſed ſunt partus huius authoris.</s> + </p> + <p> + <s xml:id="s3062" xml:space="preserve">Pergens poſtea aſſiduè bonus hic vir hominibus dare ſpecimen doctrinæ ſuæ ape <lb/>riendo (vt conatur) aliorum errores, proponit duas differentias inter primam fi-<lb/>guram, & ſuum calculum ſuppoſitum Saturni, & Iouis. </s> + <s xml:id="s3063" xml:space="preserve">Primum de Saturno ait, <reg norm="quod" type="simple">ꝙ</reg> <lb/>cum differentia ſit gra .1. min .30. oſtendit in directione, <reg norm="quod" type="simple">ꝙ</reg> accidens ſit euenturum <lb/>anno vno, & menſibus ſex ante, aut poſt, quaſi eiuſmodi differentia eſſet partium æ-<lb/>quatoris, ſicuti eſt 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/>manifeſtum eſt inditium ſcientiæ ſuæ, & quantum ea intelligat de quibus loquitur.</s> + </p> + <p> + <s xml:id="s3066" xml:space="preserve">Quod poſtea attinet ad differentiam inter Copernicum & Alfonſum, circa <reg norm="Solem" type="context">Solẽ</reg>, <lb/>nullus eſt harum ſcientiarum peritus, qui id neſciat, & ſimiliter de differentia ſitus <reg norm="cae li" type="simple">cę<lb/>li</reg> in reuolutionibus annuis.</s> + </p> + <p> + <s xml:id="s3067" xml:space="preserve">Quod vero ait ſeptimam ſiguram malè <reg norm="ſupputatam" type="context">ſupputatã</reg> fuiſſe, ſi non eſt maximus cer-<lb/>tè non eſt minimus monſtruoſorum eius errorum. </s> + <s xml:id="s3068" xml:space="preserve">Vbi itidem videri poteſt, quam <lb/>alienus hic ſit ab hac ſcientia. </s> + <s xml:id="s3069" xml:space="preserve">Nam ſi ſaltem curaſſet ſibi ab aliquo ſupputandum <lb/>locum Solis per tabulas Alfonſi in inſtanti 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/>ideſt præterierat gra .10. cum min .54. vel ſi curaſſet ſibi inueniendum tempus, per <lb/>dictas tabulas cum grad .10. min .54. </s> + <s xml:id="s3071" xml:space="preserve">Geminorum vt faciendum eſt, ſequendo <reg norm="tamen" type="wordlist">tamẽ</reg> <lb/>Alfonſum, & non per calculum Solis poſitum in ephemeridibus, vt parum periti fa <lb/>cere ſolent, vidiſſet <reg norm="quod" type="simple">ꝙ</reg> inuenta eſſent min .36. pomeridiana. </s> + <s xml:id="s3072" xml:space="preserve">Leuis tamen occaſio hu <lb/>ic fuit ſuſpicandi eiuſmodi tempus eſſe falſum, quod viderit in illa figura <reg norm="Solem" type="context">Solẽ</reg> po <lb/>ſitum eſſe cum gra .11. & non cum gra .10. min .54. non animaduertens ita notatum <lb/>fuiſſe Solem vt omnes alios planetas, ſcilicet ſine minutis, quum, vt dixi, in reuolu <lb/>tionibus non adhibeatur tanta ſcruploſitas.</s> + </p> + <p> + <s xml:id="s3073" xml:space="preserve">Quod deinde ait, in illa figura Solem poſitum eſſe in decima domo, & non in .9. <lb/>id relinquam iudicio eorum qui ſciunt numerare domos, ſaltem poſuiſſet authori-<lb/>tate ſua Solem in dicta decima diuersè ab exemplo ei dato ab amico, vt oſtenderet <lb/>ſe dicere verum, vt in ſecunda figura diſcrepat ab ipſo exemplo in collocando Leo-<lb/>ne, Virgine, & Libra, & Scorpio, quos malè locauit, & ſi alii bene ſe habent.</s> + </p> + <p> + <s xml:id="s3074" xml:space="preserve">Atque quod hactenus à me dictum eſt, ſatis ſit ad <reg norm="intelligendum" type="context">intelligẽdum</reg> quale ſit <reg norm="reliquum" type="context">reliquũ</reg> <lb/>dictæ eius diſputationis. </s> + <s xml:id="s3075" xml:space="preserve">Sienim velim pergere notare omnia eius errorum loca, eſ <lb/>ſet mihi inanis labor, & tibi nimia moleſtia. </s> + <s xml:id="s3076" xml:space="preserve">Et quamuis non defuerint præſtantiſſi-<lb/>mi viri, qui viſis eius ſcriptis familiariter eum monuere, & tu ipſe, vt audiui, cum <reg norm="in- ſtrumento" type="context">in-<lb/>ſtrumẽto</reg> theoricę in <reg norm="manibus" type="simple">manibꝰ</reg> ei <reg norm="oſtenderis" type="context">oſtẽderis</reg> quo <reg norm="mon" type="context">mõ</reg> Mars poſſit morari amplius ſex <reg norm="mem" type="context">mẽ</reg> <lb/>ſibus in vno ſigno. </s> + <s xml:id="s3077" xml:space="preserve">& præterea cum iam ab initio Taurinum aduenit, mecum com-<lb/>municauerit illa ſua prima ſcripta, <reg norm="egoque" type="simple">egoq́;</reg> eum monuerim, quod in varijs <reg norm="rebus" type="simple">rebꝰ</reg> falleba <lb/>tur, <reg norm="diſſuaſerimque" type="simple">diſſuaſerimq́;</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/>uerſiones differentiarum ephemeridum attinet, quod id iam <reg norm="oens" type="context">oẽs</reg> animaduerterant. <lb/></s> + <s xml:id="s3078" xml:space="preserve">Mihi reſpondit ſe decreuiſſe illa edere, vt poſtea 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/>ieſcens, die .11. </s> + <s xml:id="s3079" xml:space="preserve">Auguſti edidit chartam illam impreſſam inuitans ad diſputa-<lb/>tionem quotquot adhęrerent contrariæ ſententiæ, volens ſuſtinere <reg norm="Martem" type="context">Martẽ</reg> non poſ <lb/>ſe commorari in vno ſigno amplius duobus menſibus, ſupponens partem princi-<lb/>piorum ab omnibus admiſſorum, & in fine paginæ exponens modum, quo vtitur ad <lb/>probationem ſuæ intentionis. </s> + <s xml:id="s3080" xml:space="preserve">Puto autem quodſecum ratiocinabatur de Marte, <lb/>vt fecit de Saturno in ſcripto latino, hoc modo. </s> + <s xml:id="s3081" xml:space="preserve">Si Mars in duobus annis ambulat <lb/>per omnia 12. ſigna, neceſſe eſt igitur, vtin menſibus duobus ambulet per vnum ſi-<lb/>gum, cum menſes .2. ſint duo decima pars annorum duorum. </s> + <s xml:id="s3082" xml:space="preserve">Sedibi ſtatim in ipſo <lb/>initio commitcit errorem graduum ferè .7. dicens, quod medius motus Martis inue-<lb/>niebatur ſignorum .4. & gra .17. cum eo tempore dictus medius motus non eſſet reue <lb/>ra plus quam ſign .4. grad .10. mi .36. verum hoc ad ea, quæ ſequuntur exigui eſt mo <lb/>menti. </s> + <s xml:id="s3083" xml:space="preserve">Is poſtea 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 ſignorum .9. gra .27. min .53. & tamen reuera erat <reg norm="tantum" type="context">tantũ</reg> ſigno <lb/>rum .9. gra .21. mi .29. ſed 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, ſimiliter errat, nam <lb/>centrum epicycli nunquam poni debet vbi eſt linea medij motus, niſi ſit in auge, aut <lb/>in oppoſito aug is eccentrici, quia debebat collocare ipſum centrum <reg norm="tanto" type="context">tãto</reg> poſt linec<unsure/><reg norm="am" type="context">ã</reg> <lb/>medij motus, quanta erat æquatio centri, quia medium centrum Martistunc erat mi <lb/>nus ſignis ſex, & aux eccentrici eius erat in ſexto minuto grad .16. Leonis. </s> + <s xml:id="s3086" xml:space="preserve">Tamen <lb/>hoc etiam leue eſt. </s> + <s xml:id="s3087" xml:space="preserve">Præſupponamus igitur quod centrum epicycli cſſet in grad .28 <lb/>Capricorni, vt ipſe credidit, ideſt gradibus .7. vlterius quam erat reuera. </s> + <s xml:id="s3088" xml:space="preserve">Ait poſtea <lb/>ſe comperiſſe Martem ambulaſſe ſigna .4. & grad .22. eius epicycli, ſed non explicat <lb/>an intelligat de argumento medio, an de vero, quod vocatur æquatum, nam ſi intel <lb/>ligatur de medio, hoc eſſe non poteſt, cum <reg norm="medium" type="context">mediũ</reg> eſſet ſignorum .4. gra .24. mi .35. <lb/>ſed ſi intelligatur de vero, vt iure credendum eſt (alioquin etiam erraſſet) cer-<lb/>tè falſum eſt. </s> + <s xml:id="s3089" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, verum, erat ſignorum .4. grad .29. minu .39. </s> + <s xml:id="s3090" xml:space="preserve">Itaque Mars non di-<lb/>ſtabat à linea veri motus epicycli amplius gradibus .30. & minu .21. ipſius epicycli, <lb/>& æquatio argumentiſecundo correcta erat gra .44. minu .2. à quo ſubtracta æquatio <lb/>ne centri, quæ erat gr .5. minu .4. (cum centrum epicycli deberet tanto ſpacio eſſe <lb/>poſt lineam medij motus quantum ſupra dixi) ſupererant 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. Piſcium. </s> + <s xml:id="s3091" xml:space="preserve">Sed præſuppoſito <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>ipſum, quod medius motus eſſet grad .28. </s> + <s xml:id="s3092" xml:space="preserve">Capricorni, & quod Mars eſſet non <reg norm="ſolum" type="context">ſolũ</reg> <lb/>vbi hic ait, ſed etiam in prima linea contingentiæ epicycli, ideſt in prima linea ma <lb/>ximæ æquationis argumenti, & præſuppoſito etiam quod dicta æquatio eſſet æqua-<lb/>lis illi, quam haberet ad medium Aquarij ſcilicet grad .47. quum centrum epicycli <lb/>eſt in oppoſito augis, manifeſtum eſt, quod eiuſmodi linea contingentiæ non tranſi <lb/>ret vltra grad .15. Piſcium, & 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 ſi verum eſſet, & <var>.o.c.</var> etiam eſſet partium .54. ſecundum diſtantiam pro-<lb/>ximiorem centro mundi, ſemidiameter epicycli eſſet eiuſmodi partium .52. minut <num value="49">.<lb/>49.</num> & quum Mars eſſet in <var>.g.</var> ideſt in oppoſito veræ augis epicycli, dum centrum epi <lb/>cycli eſſet in eiuſmodi diſtantia à terra, diſtantia <var>.o.g.</var> ideſt à terra ad Martem non <lb/>eſſet plus, quam vna ſola pars ex dictis, cum minut .11. cum partes .52. minu .49. <lb/>ad .54. ſint vt ſinus anguli gra .78. qui eſt partium .97814. ad ſinum totalem partium <lb/>100000. </s> + <s xml:id="s3095" xml:space="preserve">Nam iam ſupra dixi, quod triangulus <var>.o.c.i.</var> eſt rectangulus. </s> + <s xml:id="s3096" xml:space="preserve">Hinc ſeque- + <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">vniꝰ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/> + partis, & min .11. <lb/>reſpectu <var>.o.c.</var> par-<lb/>tium .54. </s> + <s xml:id="s3097" xml:space="preserve">Colloca <lb/>retur ſemidiame-<lb/>ter terrę cum ſpiſ-<lb/>ſitudine aeris, i-<lb/>gnis, cęlorum Lu-<lb/>næ, Mercurij, Ve-<lb/>neris, & Solis, <reg norm="prae terquam" type="simple">prę<lb/>terquam</reg> quod vt <lb/>inter Solem, & <lb/>terram ſunt circa <lb/>605. diametri ip <lb/>ſius terræ, inter <lb/>terram, & Mar-<lb/>tem cum eſſet in <lb/>auge ſui epicycli, <lb/>& epicyclus in au <lb/>ge eccentrici, in-<lb/>uenirentur cir--<lb/>ca .60000. dia--<lb/>metri eiuſdem ter <lb/>ræ, & tamen ea diſtantia ſiue interuallum non poteſt continere .5000. diametri ter-<lb/>ræ. </s> + <s xml:id="s3098" xml:space="preserve">Et quod plus eſt, hic tam vaſtum facit hunc ſuum 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. menſ. </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 ſint 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 falſis ſuppoſitis huius) Videamus ordine ſcientifico vbi poterat eſſe verus lo-<lb/>cus Martis, aut vero proximus, die .29. Mai anni .1514. quem hic exempli cauſa ſu-<lb/>mit. </s> + <s xml:id="s3102" xml:space="preserve"><reg norm="Idque" type="simple">Idq́</reg> tam ad defenſionem tabularum Alfonſi, quam ephemeridum ex eis col <lb/>lectarum. </s> + <s xml:id="s3103" xml:space="preserve">quæ quidem exactæ ſunt, vt quiſque peritus ſacile videre poterit, non au-<lb/>tem calculatæ à tam ſtupidis hominibus, vt à vero aberrent etiam gradibus .46. vt <lb/>hic ait ſe depræhendiſſe.</s> + </p> + <p> + <s xml:id="s3104" xml:space="preserve">Primum igitur ſupponemus eoſdem illos terminos, quos ipſe nec d@bet, nec po <lb/>teſt negare, præter ea quæ ſupra ſuppoſita ſunt, nempe quod ſemidiameter epicycli <lb/>ſit partium .39. minu .30. & eccentricitas partium .6. talium qualium eſt ſemidiame-<lb/>ter deferentis diuiſus in .60. & quod dicto tempore aux eccentrici Martis eſſet cir-<lb/>ca minutum .5. grad .16. Leonis, ſcilicet 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">centrũ</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 oſtendam, neque tabulas, neque ephemerides errare, ne quidem vno gra-<lb/>du, ac ne quidem multis minutis, non modò tam monſtruoſa differentia, vt ipſe <lb/>ait.</s> + </p> + <p> + <s xml:id="s3106" xml:space="preserve">Quare primum nobis ſcientificè inueniendum eſt, quanta eſſet diſtantia <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"/> + præciſe ideſt interuallum inter centrum mundi, & centrum epicycli Martis in huiuſ-<lb/>modi ſitu.</s> + </p> + <p> + <s xml:id="s3107" xml:space="preserve">Fingemus igitur eccenticum Martis ſignificatum per <var>.p.c.m.</var> cuius centrum ſit <var>.r.</var> <lb/>& lineam augis <var>.p.r.o.m.</var> in qua <reg norm="centrum" type="context">centrũ</reg> mundi ſit <var>.o.</var> centrum autem verum epicycli, <lb/>comprehendatur ab angulo <var>.p.o.c.</var> qui ſit graduum .151. min .30. ſecundum ſuppoſi-<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> quouſque 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 cognoſcemus angulum <var>.r.o.k.</var> tanquam reliquum ex duobus rectis, qui <lb/>erit gra .28. min .30. & ſimiliter angu-<lb/> + <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> + lum <var>.o.r.k.</var> tanquam reſiduum vnius <lb/>recti, qui erit gra .61. min .30. cuius ſi-<lb/>nus ideſt <var>.o.k.</var> erit partium .8788 1. et <var>.k.<lb/>r.</var> vt ſinus anguli <var>.r.o.k.</var> partium .47715 <lb/>talium qualium <var>.o.r.</var> eſſet 100000. ſed <lb/>vt <var>.o.r.</var> eſt .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/>cſt <reg norm="partium" type="context">partiũ</reg> 60. eiuſmodi, ſi ab eius qua-<lb/>drato ſubtractum fuerit quadratum ip <lb/>ſius <var>.r.k.</var> reliquum erit nobis <reg norm="quadratum" type="context">quadratũ</reg> <lb/>ipſius <var>.k.c.</var> cuius radix, ideſt <var>.k.</var> erit par-<lb/>tium .59. min .56. à qua <var>.c.k.</var> ſubtrahen-<lb/>do poſtea <var>.k.o.</var> partium .5. minu .16. re-<lb/>manebit <var>.o.c.</var> partium .54. min .40. pro <lb/>diſtantia 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 poſtea epicyclum <var>.f.n.g.</var> <lb/>in quo argumentum verum graduum <num value="149">.<lb/>149.</num> minu .39. ſit arcus <var>.f.n.</var> vbi Mars inueniatur in <var>.n.</var> per quem punctum tranſeat 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, ideſt ducendo ſinum <var>.n.h.</var> arcus <var>.n.g.</var> qui arcus tanquam reliquus <lb/>argumenti veri, iam præſuppoſiti, ex dimidio circulo, erit graduum 30. minu .21. & <lb/><var>n.h.</var> eius ſinus partium .50528. ſinus ſimiliter anguli <var>.n.c.h.</var> et <var>.c.h.</var> tanquam ſinus an-<lb/>guli <var>.c.n.h.</var> reſtantis ex uno recto grad .59. minu .39. erit partium .86295. <reg norm="talium" type="context">taliũ</reg> qua-<lb/>lium <var>.c.n.</var> ſinus totus eſſet partium .100000. ſed vt partium .39. & min .30. ſinus <var>.c.h.</var> <lb/>erit partium .34. min .5. et <var>.n.h.</var> partium .19. mi .57. reliquum poſtea <var>.h.o.</var> ex <var>.o.c.</var> par-<lb/>tium .20. min .35. quia iam ſupra inuenimus <var>.o.c.</var> eſſe partium eiuſmodi .54. minu .40. <lb/>vnde <var>.o.n.</var> vt radix quadrata ſummæ 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 poſtea partium, <lb/>69552. talium qualium <var>.n.o.</var> partium .100000. & ſumpta dicta <var>.n.h.</var> vt ſinus dictarum <lb/>partium, dabit nobis angulum <var>.n.o.h.</var> quæſitum gra .44. min .4. qui per tabulas Alfon <lb/>ſi inuentus eſt gra .44. min .2. par huic, vt dici poteſt. </s> + <s xml:id="s3112" xml:space="preserve">Quiangulus gra .44. minu .4. <lb/>collectus cum angulo veri centri iam ſuppoſito graduum .151. minu .20. & cum an-<lb/>gulo augis eccentrici Martis, ſimiliter ſuppoſitæ grad .135. min .5. dabit nobis ſum-<lb/>mam veræ diſtantiæ Martis à principio Arietis grad .330. min .29. quod aliud non <lb/>ſignificat, niſi quod Mars inuenietur in minu .29. primi gradus Piſcium. </s> + <s xml:id="s3113" xml:space="preserve">Et Stofle-<lb/>rus in ſuis 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/>eſt minut .5. tan <lb/> + <anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/> + <reg norm="tum" type="context">tũ</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 ſumere ip-<lb/>ſum angulum æ-<lb/>quationis à tabu <lb/>lis propter duas <lb/>rationes, <reg norm="primum" type="context">primũ</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">ĩ</reg> ſua <lb/>theorica Martis. <lb/></s> + <s xml:id="s3116" xml:space="preserve">Vbi <reg norm="inuenit" type="wordlist">ĩuenit</reg> quod <lb/>linea <var>.o.n.</var> tranſit <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 eſt, vt videa-<lb/>tur quod dictæ tabulæ rectè ſupputatæ ſunt, ſuper dictis ſuppoſitis.</s> + </p> + <p> + <s xml:id="s3119" xml:space="preserve">Sed vt videat quantus ſit medius motus Martis die .29. </s> + <s xml:id="s3120" xml:space="preserve">Mai colligit fruſtatim, <reg norm="quod" type="simple">ꝙ</reg> <lb/>eleganter colligere poterat, vna opera in columellis ipſius medij motus eiuſmodi <lb/>ſtellæ per eram eiuſdem temporis, quæ erat .2. primarum ſexagenarum .33. ſecunda <lb/>rum .32. tertiarum, et .52. quartarum.</s> + </p> + <p> + <s xml:id="s3121" xml:space="preserve">Primum deinde ſuppoſitum quod ſcribit, ſcilicet, quod diameter epicycli ſum-<lb/>ptus in longitudine media ſit ſignorum .2. & grad .19. vti ſuperfluum eſt, ita etiam fal <lb/>ſum, nam eiuſmodi diameter in dicto loco, non occupat ad centrum mundi plus <reg norm="quam" type="context">quã</reg> <lb/>gra .66. min .28. ideſt ſigna .2. gra .6. min .28. quia proportio <var>.o.c.</var> ad ſemidiametrum <lb/>epicycli in eiuſmodi loco eſt ut partium .60. minu .18. ad partes .39. min .30. quę duę <lb/>lineæ intellectę, vt latera vnius trianguli rectanguli, habebunt pro baſi aliam lineam <lb/>partium ſimilium .72. mi .5. </s> + <s xml:id="s3122" xml:space="preserve">Quæ intellecta vt ſinus totus dabit ſemidiametrum epi-<lb/>cycli partium .54798. tanquam ſinum ſubiectum 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 ſpatium duorum ſignorum cum gradibus. <lb/>19 quæ quidem ratio alia eſſe non poteſt, niſi quod cum iſte inuenerit, in <reg norm="comentarijs" type="context">comẽtarijs</reg> <lb/>Theoricarum, ſemidiametr um huiuſmodi epicycli eſſe partium .39 min .30. talium <lb/>qualium ſunt .60. illæ quæ ſunt ſemidiametri huius eccentrici, dictas igitur partes .39 <lb/>mi .30. hic putauit eſſe gradus Zodiaci, & propterea dixit <reg norm="diametrum" type="context">diametrũ</reg> huiuſmodi epicy <lb/>cli eſſe <reg norm="ſignorum" type="context">ſignorũ</reg> <reg norm="duorum" type="context">duorũ</reg>, & gra .19. qui numerus .79. duplus eſt numero .39. <reg norm="cum" type="context">cũ</reg> dimidio <lb/>hoc autem dixit accidere in longitudinibus medijs, quia ſi hic intellexiſſet de pro-<lb/>portione horum duorum diametrorum, quæ eſt ut .120. ad .79. non ſpecificaſſet 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 ipſa proportio nullo modo alteratur exiſtente epicyclo vbi volue <lb/>ris ipſius circunferentiæ eccentrici, ſed angulus in centro mundi, cui ſubiacet dictus <lb/>diameter epicycli, bene alteratur, propter inæqualem diſtantiam centri epicycli <lb/>ab ipſo centro mundi. </s> + <s xml:id="s3124" xml:space="preserve">At ſi de tali angulo inferre voluiſſet, iam probaui ipſum <reg norm="con- tinere" type="context">cõ-<lb/>tinere</reg> ſolum gra .66. minu .28. exiſtente centro epicycli in longitudinibus medijs & <lb/>non gra .79. vt ipſe dicit.</s> + </p> + <p> + <s xml:id="s3125" xml:space="preserve">Omitto poſtea, quod vbi mentionem facit coniunctionum Solis cum Marte au-<lb/>gium & earum oppoſitorum, non explicat an intelligat de veris an de medijs. </s> + <s xml:id="s3126" xml:space="preserve">Nam <lb/>ſi 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 moleſtum eſſe inciperet ſi diutius te detinerem in his conten-<lb/>tionibus aſtronomicis, vlterius non diſputabo. </s> + <s xml:id="s3128" xml:space="preserve">Satis enim hactenus explicaui <reg norm="ſenten- tiam" type="context">ſentẽ-<lb/>tiam</reg> meam, vt oſtendiſſe videor quam mihi iucundum ſit 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">quũ</reg> petitione tua occaſionem mihi de deris <lb/>efficiendi, vt tum amici tui (amant enim te omnia ſublimia ingenia) tum alij, ſi <reg norm="quam" type="context">quã</reg> <lb/>falſam opinionem ex huius Benedicti Altæuillæ ſcriptis ſumpſiſſent, <reg norm="eam" type="context">eã</reg> relinquant, <lb/>& per te hoc beneficium à me conſequantur, & huiuſmodi occaſionem, & iuuandi <lb/>hominum ſtudia & tibi gratum faciendi, honorificum, & per gratum mihi fuiſſe in-<lb/>telligant. </s> + <s xml:id="s3130" xml:space="preserve">Vale & me vt ſoles 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 diuiſionis 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 ſcire cupis, vir doctiſſime, hoc vnum eſt, vt ex literis tuis ac-<lb/>cepi, vnde ſit vt priſci noſtri probatione numeri nouenarij potius quam ſepte-<lb/>narij vſi fuerint, & qua ratione non idem proueniat ex probatione numerorum <lb/>octonarij, ſenarij, vel quinarij, aut cuiuslibet alterius: </s> + <s xml:id="s3134" xml:space="preserve">Vnde pariter oriatur quod in <lb/>partitionis probatione neceſſum ſit probationum euentus multiplicare cum proba-<lb/>tione diuiſoris, ac eam quæ eſt producti poſtea cum probatione fractionis in ſum-<lb/>mam colligere, & c. </s> + <s xml:id="s3135" xml:space="preserve">Ad hæc in primis reſpondeo, cum aliquoties accidere poſſit ta <lb/>les probationes nos fallere poſſe, <reg norm="idque" type="simple">idq́;</reg> fi in tali ſumma ſimilis numerus, ut puta ſe-<lb/>ptem, aut nouem, plus vel minus æquo iuſtouè poſitus fuerit, attamen per raro eueni <lb/>re poteſt, vt quis per nouenarium potius quam per ſeptenarium decipiatur. </s> + <s xml:id="s3136" xml:space="preserve">Exem-<lb/>pli gratia, ponamus ſummam eſſe .100. quam numerus nouenarius vndecies ſolum <lb/>ingreditur, at ſeptenarius quatuordecies, vnde quis <reg norm="ſępius" type="simple">ſępiꝰ</reg> ex ſeptenario, hacratione, <lb/>quam ex nouenario numero ſe poſſe errare facile depræhendet, efſi ex probatione <lb/>nouenarij magis quam ſeptenarij, vt practici ſcribunt, duabus de cauſis errare poſſi-<lb/>mus. </s> + <s xml:id="s3137" xml:space="preserve">Alia tamen ratio mihi ſuppetit, ob quam credibile eſt ipſos potius nouena-<lb/>rio adiutos fuiſſe, quam ſeptenario, quæ eſt ob ſui cum velocitatem tum <reg norm="facilitatem" type="context">facilitatẽ</reg>, <lb/><reg norm="neque" type="simple">neq;</reg> enim in ſeptenario eſt 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 ſit, quam numerorum ordines diuidere (ſi de ſummis primo loquamur) aut è <lb/>ſumma ſuperſluum ordinum colligere, & videre an idemmet ſuperfluum ex eadem <lb/>ſumma emanet, attamen cum modus, qui in hoc adhiberi poteſt in nouenario <reg norm="quam" type="context">quã</reg> <lb/>in ſeptenario velocior ſit, & ob id probationem nouenarij ſeligunt potius quam <lb/>ſeptenarij.</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, quæfalſa eſt, opinione conſiſtere, nonidem, & cum octona-<lb/>rio, ſenario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra <lb/>tio, quæ in ſeptenario, 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">exemplũ</reg> <lb/>hos tres or dinum numeros velle ſupputare, quorum primus ſit .679. ſecundus .846. <lb/>& tertius .935. & illorum <reg norm="ſummam" type="context">ſummã</reg> .2460. nunc maiorem numerum primi ordinis ab <lb/>octonario menſi, proijciendo, remanebit .7. deinde maiorem numerum demendo à <lb/>ſecundo or dine, reſiduum erit .6. ac ſi idem in tertio ordine fecerimus, erit nobis re-<lb/>liquum .7. </s> + <s xml:id="s3141" xml:space="preserve">Demum tria hæc reſidua in vnum collecta .20. efficient, à quibus ſi nume <lb/>rum maiorem ab octonario menſum dempſeris, ſupererunt .4. & totidem à ſumma <num value="2460">.<lb/>2460.</num> remanebunt, reiecto maiori numero ab octonario menſo. </s> + <s xml:id="s3142" xml:space="preserve">Atque idem me-<lb/>dio quouis alio numero, euenire poteſt.</s> + </p> + <p> + <s xml:id="s3143" xml:space="preserve">Cuius ratio tam perſe clara atque euidens eſt, quod ſi ſummam trium <reg norm="reliquorum" type="context">reliquorũ</reg>, <lb/>quæ eſt .20. à ſumma .2460. ſubduxeris, remanebunt .2440. pro ſumma trium nume <lb/>rorum dictorum trium ordinum ab octonario menſorum, cui numero addito .16. pro <lb/>maiori numero ſummę <reg norm="reliquorum" type="context">reliquorũ</reg>, qui ab octonario menſus ſit, ſupererunt .4. </s> + <s xml:id="s3144" xml:space="preserve">At ſi per <lb/><reg norm="ſenarium" type="context">ſenariũ</reg> <reg norm="experimentum" type="context context">experimẽtũ</reg> feceris, remanebit <var>.o.</var> & ſic de reliquis per ordinem <reg norm="procedendo" type="context">procedẽdo</reg>.</s> + </p> + <p> + <s xml:id="s3145" xml:space="preserve">Verum poſſes ſciſcitari, quare velocius, exceſſus ordinum, potius per <reg norm="noue- narium" type="context">noue-<lb/>nariũ</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 ſummæ, deinde alias duas. </s> + <s xml:id="s3146" xml:space="preserve">Exem-<lb/>plum ſit primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc ſumma ſit dua <lb/>rum figurarum, ſupputantur & ipſæ, è quibus prodeunt .4. & conſimilis erit proba-<lb/>tio numeri .67. facta per .9. quod idem eſt, ac ſi quis diuidat .67. per .9. ex quo reli-<lb/>qui erunt ſemper .4.</s> + </p> + <p> + <s xml:id="s3147" xml:space="preserve">At quo ratio huiuſce perſpicuè dignoſci poſſit, in primis ſciendum eſt, cuique <lb/>ex ſe cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium ſu <lb/>perare, & ex hoc ſequitur, ſex denarios continere in ſe ſex nouenarios, & ſex vni-<lb/>tates.</s> + </p> + <p> + <s xml:id="s3148" xml:space="preserve">At ſex vnitates, vna cum .7. faciunt .13. & quia in .13. eſt denarius, igitur in illo erit <lb/>vnitas ſupra .9. </s> + <s xml:id="s3149" xml:space="preserve">Quæ vnitas addita ternario, præbet nobis ſuperfluum, per quod .67. <lb/>ſuperat .54. iunctum cum .9. ſcilicet ſummam .63.</s> + </p> + <p> + <s xml:id="s3150" xml:space="preserve">Idem dicinon poteſt de octonario, ſeptenario, vel ſenario, & de reliquis, quo-<lb/>niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-<lb/>merum exceſſus maioris numeri, qui à numero probationis menſus eſt. </s> + <s xml:id="s3151" xml:space="preserve">Et quod di <lb/>co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta ſub-<lb/>tractionis, multiplicationis, & partitionis ſeu diuiſionis.</s> + </p> + <p> + <s xml:id="s3152" xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus ſit probationem euentus <lb/>cum diuiſionis probatione multiplicare, & productum cum fractionis probatione <lb/>ſupputare, ſeu aggregare, tibi non erit ignotum, quoties animaduerteris, quod <lb/>productum ipſius euentus cum diuiſore, adiunctum fractioni, perpetuo ſe æquat nu <lb/>mero diuiſibili. </s> + <s xml:id="s3153" xml:space="preserve">Et quoniam numeri probationum ſunt partes, quæ remanent ex <lb/>ipſis totis, detractis maioribus numeris ab eo dimenſis, quo pro communi men-<lb/>ſura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eſt vt ex ip-<lb/>ſarum remanentibus partibus, velut ex ipſis 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 ſphericorum.</head> + <head xml:id="head377" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3154" xml:space="preserve">QVod diebus præteris tibi ſignificaui, idem nunc confirmo, ſcilicet ſphærico-<lb/>rum triangulorum operationem ſæpe nos fallere, vt exempli gratia, ſi pro <lb/>poſitus 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/>ſi reliquos angulos cum tertio latere etiam cognoſcere voluerimus, ex methodo .11 <lb/>primi Copernici propoſitum 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> falſus eſſet, eo quod operatio paruorum triangulorum in <lb/>cauſa eſt, quotieſcunque eorum latera tam breuia ſint, ut non eccedant vnum gra-<lb/>dum, quare ipſorum 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> ipſius anguli <var>.B.</var> oportet poſt quam inuenerimus <lb/>angulum <var>.C.</var> mediante arcu <var>.D.E.</var> ſupponere alium polum in <var>.B.</var> deinde producere. </s> + <s xml:id="s3158" xml:space="preserve">B <lb/>A. vſque ad <var>.d.</var> et <var>.B.C.</var> vſque ad <var>.e.</var> imaginando <var>.B.d.</var> et <var>.B.e.</var> duas quartas eſſe magno-<lb/>rum circulorum, extendendo poſtea <var>.d.e.</var> vſque ad interſectionem cum <var>.A.C.</var> & <reg norm="eum" type="context">eũ</reg> <lb/>dem ordinem proſequendo, </s> + <s xml:id="s3159" xml:space="preserve">tunc <var>.e.d.</var> nobis oſtendet 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, eſto, exempli gratia, quod nobis datus ſit 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 ſi ordi-<lb/>nem .11. dicti lib. ſe quemur, obtinebimus intentum, hoc modo ſcilicet ſupponendo <lb/>in <var>.A.</var> polum, & non in <var>.B.</var> ducendo etiam <var>.A.B.</var> et <var>.A.C.</var> ſed <var>.A.B.</var> <reg norm="vſque" type="simple">vſq;</reg> ad gra .90. du-<lb/>cendo poſtea <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 ſemicirculi magnorum circulorum. </s> + <s xml:id="s3162" xml:space="preserve">quare <var>.C.<lb/>D.</var> nobis cognitus erit gra .44. min .50. & ſic etiam angulus <var>.D.C.f.</var> gra .57. min .14. ex <lb/>4. dicti lib. poſtea habebimus <var>.F.l.</var> gra .60. min .54. & angulum <var>.f.</var> gra .53. mi .24. aggre <lb/>gatum poſtea <var>.f.C.</var> cum <var>.C.B.</var> habebimus <var>.f.B.</var> gra .150. min .24. qui ſi a ſemicirculo <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">cũ</reg> ſit 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 oppoſito. </s> + <s xml:id="s3163" xml:space="preserve">Vnde ex dicta .4. co-<lb/>gnoſcemus angulum <var>.B.</var> gra .40. min .31. <lb/>qui ferè æ qualis eſt ſuperiori iam inuen-<lb/>to, nec ab ipſo differt niſi per min .9. quæ <lb/>quidem differentia parua eſt reſpectu 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 ſupra 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> ſit 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. falſus eſſet, qui <reg norm="quidem" type="context">quidẽ</reg> poſtea nobis daret <var>.D.E.</var> gra .45 <lb/>minu .16. falſum ſimiliter.</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> eſt quin paſſiones circuli innumerabiles penè ſint, quę <lb/>quidem omnes ferè caſu inueniuntur, vt mihi nunc accidit, quam tibi mitto, <lb/>hæc autem eſt, quòd quadratum lineæ <var>.a.g.</var> in figura hic ſubſcripta ſemper æquale <lb/>eſt ei producto, quod fit ex <var>.a.e.</var> in diametro circuli <var>.g.c.b.</var> ſimul ſumpto cum quadra <lb/>to inſcriptibili in dicto circulo, & ſimul cum quadrato lineæ <var>.a.b.</var> <reg norm="contingentis" type="context">contingẽtis</reg> ipſum <lb/>circulum, ſupponendo <var>.a.g.</var> per centrum ipſius circuli tranſire.</s> + </p> + <p> + <s xml:id="s3166" xml:space="preserve">Pro cuius demonſtratione à centro <var>.e.</var> duco ſemidiametrum <var>.e.c.</var> <reg norm="perpendicularem" type="context">perpendicularẽ</reg> <lb/>ipſi <var>.g.a.</var> & à puncto <var>.c.</var> ad <var>.a.</var> duco <var>.c.a.</var> quæ ſecabit circunferentiam ipſius circuli in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.d.</var> eo, quod angulus <var>.c.</var> acutus eſt. </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 eſt quadrato <var>.a.b.</var> productum autem <var>.a.c.</var> in <var>.d.c.</var> æquale eſt quadrato inſcriptibili in <lb/>circulo <var>.g.c.b.</var> ex .130. primi Vitellionis, <reg norm="in" type="wordlist">ĩ</reg> qua propoſitione ipſe Vitellio ſupplet pro <lb/>eo, quod in quinta propoſitione libri de lineis ſpirabilibus Archimedis deſideratur, <lb/>ſed quadratum <var>.a.c.</var> æquale eſt ijs duobus productis. per .2. ſecundi Eucli. ergo qua-<lb/>dratum <var>.a.c.</var> æquale erit quadrato inſcriptibili in circulo <var>.d.c.g.</var> & quadrato <var>.a.b.</var> ſed <lb/>quadratum lineæ <var>.a.c.</var> æquale eſt duobus quadratis, hoc eſt 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> & lineę <var>.e.c.</var> hoc eſt <lb/>lineæ <var>.e.g.</var> quod idem eſt, æqualia erunt duobus iam dictis, hoc eſt inſcriptibili, <lb/>& ei, quod fit ex <var>.a.b.</var> ſed quadratum lineæ <var>.a.g.</var> æquale eſt 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 eſt producto <var>.a.e.</var> in <lb/>diametrum. </s> + <s xml:id="s3169" xml:space="preserve">Quare quadratum lineæ <var>.a.g.</var> æquale eſt quadrato circunſcriptibili, & <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 demonſtrare 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 eſ-<lb/>ſe quadrato circunſcriptibili, & quadrato lineæ <var>.<lb/>a.b.</var> ducendo lineam <var>.e.b.</var> quæ æqualis eſt lineæ <var>.<lb/>e.g.</var> tali methodo, hoc eſt, conſiderando, quod <lb/>quadratum inſcriptibile ſemper duplum eſt qua <lb/>drato ſemidiametri, vel medietati circumſcri-<lb/>ptibili, quod quidem nihil aliud eſt, niſi æquale <lb/>eſſe ijs duobus quadratis, hoc eſt lineæ <var>.e.b.</var> & li-<lb/>neæ <var>.e.g.</var> ſed quadratum lineæ <var>.a.e.</var> æquale eſt iis <lb/>duobus quadratis, hoc eſt 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 eſt quadrato circunſcriptibili, ſimul 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 propoſitionum de quibus agit <lb/>Cardanus capite primo libro .16. de <lb/>ſubtilitate.</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 ſubtilitate ita ſcribit, quod ſi diame-<lb/>tros producatur extra quantumlibet, alia verò diametro in centro ſecetur ad <lb/>rectos, ex huius fine <reg norm="&c." type="unresolved">&c.</reg> quæ quidem ſecundum illum eſt vndecima proprietas cir <lb/>culi, quoniam te id non intelligere ſcribis, <reg norm="idemque" type="simple">idemq́;</reg> dicis etiam de duodecima, & ſi-<lb/>militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, ſeu <lb/>ellipſi, & hyperboli, tibi breuiter reſpondebo.</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, diuiſum <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">ducanturq́;</reg> ab ijſdem punctis tot perpendiculares diametro <var>.d.g.</var> quæ ſint <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> vſque ad concurſum 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 vſque 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 ſint ipſi <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. eiuſdem <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> parallelę, & 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 eſt igitur propoſitum.</s> + </p> + <p> + <s xml:id="s3177" xml:space="preserve">Duodecima vero <reg norm="proprietas" type="simple">ꝓprietas</reg> eſt, ut ſi fuerit circulus <var>.a.b.e.q.</var> cuius duo diametriad <lb/>rectos coniuncti ſint <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 ſi ab extremo <var>.b.</var> diametri <var>.q.b.</var> ducta fuerit <var>.b.n.u.</var> extra circulum, ſeu <var>.b.u.n.</var> in <lb/>tra circulum, vt in ſubiecta figura patet, ita vt ſecta ſit à circunferentia circuli in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.n.</var> vel à diametro in puncto <var>.u.</var> ſemper id quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit qua-<lb/>drato inſcriptibili in dicto circulo, hoc autem diuerſimodè cognoſci poteſt, tribus <lb/>enim modis ego inueni, quorum primus ita ſe habet. </s> + <s xml:id="s3179" xml:space="preserve">Nam ſi 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> ſimi <lb/>les inuicem, eo, quod angulus <var>.b.</var> communis ambobus exiſtit, & angulus <var>.b.n.e.</var> æqua <lb/>lis eſt 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, ſed ex quinta primi angulus <var>.b.<lb/>e.a.</var> ęqualis eſt 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> ęquatur duobus <lb/>rectis, ſed ex .13. eiuſdem 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. eiuſdem reliquus angulus <var>.b.u.e.</var> æqualis erit reliquo angulo <lb/><var>b.e.n.</var> latera igitur erunt proportionalia ex .4. ſexti, vnde ita ſe habebit <var>.u.b.</var> ad <var>.b.<lb/>e.</var> vt <var>.b.e.</var> ad <var>.b.n.</var> ex .16. ſexti igitur <reg norm="verum" type="context">verũ</reg> erit propoſitum.</s> + </p> + <p> + <s xml:id="s3182" xml:space="preserve">Sed ſi punctus <var>.u.</var> intra circulum fuerit, triangulus <var>.b.e.n.</var> ſimilis 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> ęqualis eſt 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. ſexti eadem proportio erit ipſius <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. eiuſdem patebit propoſitum.</s> + </p> + <p> + <s xml:id="s3186" xml:space="preserve">Secundus autem modus ita ſe habet, ducta <var>.q.n.</var> habebimus duo triangula ortho-<lb/>gonia ſimilia inuicem <var>.b.q.n.</var> et <var>.b.u.o.</var> eo quod angulus <var>.b.</var> communis ambobus exi-<lb/>ſtit, </s> + <s xml:id="s3187" xml:space="preserve">quare ex .4. ſexti ita ſe habebit <var>.u.b.</var> ad <var>.b.o.</var> vt <var>.q.b.</var> ad <var>.b.n.</var> vnde ex .15. eiuſdem <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. eiuſdem, <reg norm="quod" type="simple">ꝙ</reg> <lb/>fit ex <var>.q.b.</var> in <var>.b.o.</var> ęquatur quadrato <var>.b.e.</var> quia <var>.b.e.</var> media proportionalis eſt inter dia <lb/>metrum & ſemidiametrum eiuſdem circuli. ex .4. eiuſdem, </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 ipſius <var>.b.e</var>.</s> + </p> + <p> + <s xml:id="s3190" xml:space="preserve">Tertius modus adiungitur, & eſt quod cum quadratum <var>.u.b.</var> exiſtente <var>.u.</var> extra cir-<lb/>culum æquale ſit ei, quod ſit ex <var>.u.b.</var> in <var>.b.n.</var> ſimul ſumpto cum eo, <reg norm="quod" type="simple">ꝙ</reg> fit ex <var>.u.b.</var> in <var>.u.n.</var> <lb/>ex ſecunda ſecundi, & 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. ſcilicet et <var>.o.b.</var> ſed 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 ſe <lb/>ipſam ex .6. ſecundi, </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> ſci <lb/>licet. et <var>.o.e.</var> & ei quod fit ex <var>.a.<lb/>u.</var> in <var>.u.e.</var> ſed quod fit ex <var>b.u.</var> in <var>.u <lb/>n.</var> æquale eſt 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">qđ</reg> fit ex <var>.u.b.</var> in <var>.b.n.</var> æqua-<lb/>le ſit <reg norm="duobus" type="simple">duobꝰ</reg> quadratis <var>.o.b.</var> et <var>.o.<lb/>e</var>. </s> + <s xml:id="s3192" xml:space="preserve">quare & quadrato ipſius <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">pũctũ</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> ſed vice qua-<lb/>drati <var>.o.e.</var> dicemus <reg norm="quadratum" type="context">quadratũ</reg> <var>.o.<lb/>u.</var> cum eo quod fit ex <var>.a.u.</var> in <var>.u.e.</var> <lb/>ex .5. ſecundi, id eſt quadratum <var>.<lb/>o.u.</var> <reg norm="cum" type="context">cũ</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> ideſt quadrato <lb/><var>b.u.</var> ex Pitagorica ſimul <reg norm="cum" type="context">cũ</reg> pro-<lb/>ducto <var>.b.u.</var> in <var>.u.n.</var> ideſt producto <lb/><var>n.b.</var> in <var>.b.u.</var> quod æquale eſt qua <lb/>drat <var>o.b.u.</var> cum producto <var>.b.u.</var> in <lb/><var>u.n.</var> ex .3. ſecundi.</s> + </p> + <p> + <s xml:id="s3195" xml:space="preserve">Circa tres paſſiones commu-<lb/>nes poſtea circulo hyperboli, & <lb/>defectioni notandum eſt <reg norm="primam" type="context">primã</reg> <lb/>patere ex .36: primi Pergei, ſe- + <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. eiuſdem, </s> + <s xml:id="s3196" xml:space="preserve">propterea quod in .37. probat mediante maiori <lb/>diametro ipſius 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 niſi circulo conuenit; </s> + <s xml:id="s3198" xml:space="preserve">pace ipſius Cardani dictum ſit.</s> + </p> + <p> + <s xml:id="s3199" xml:space="preserve">Quapropter ſit circulus <var>.q.o.b.</var> cuius diameter ſit <var>.q.b.</var> contingentes vero ab extre <lb/>mitate diametri ſint <var>.d.b.</var> et <var>.q.g.</var> per punctum autem <var>.o.</var> quoduis, ipſius <reg norm="circunferentiæ" type="context">circũferentiæ</reg>, <lb/>tranſeant <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> ęquale eſ-<lb/>ſe 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> ſeu <var>.b.q.g.</var> rectus eſt ex .17. tertij Eucli. et <var>.b.o.q.</var> ſimiliter re-<lb/>ctus ex .30. ipſius lib. angulus verò <var>.b.q.d.</var> ſeu <var>.q.b.g.</var> communis eſt. </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 ſequetur <lb/>propoſitum ex .16.6. Eucli.</s> + </p> + <p> + <s xml:id="s3204" xml:space="preserve">Sed ſi circa diametrum <var>.q.b.</var> mente fingamus aliquam elipſim, quætangat ipſum <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">pũ</reg> <lb/>ctus <var>.o.</var> erit extra <reg norm="circunferentiam" type="context">circunferentiã</reg> <lb/>ipſius defectionis, </s> + <s xml:id="s3205" xml:space="preserve">quare ipſa cir <lb/>cunferentia ſecabit <var>.b.g.</var> vel <var>.q.<lb/>d.</var> in alio puncto, vnde ipſi 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 ſum, ipſum <lb/>Cardanum dicere hyperbolem <lb/>ita vocari, eo quod angulus con <lb/>tentus ab axe ipſius figuræ, & à <lb/>latere trigoni in hyperbole ma-<lb/>ior ſit quam in parabole, quod <lb/>eriam confirmat paulo inferius, <lb/>nam hoc verum non eſt, imo fal <lb/>ſiſſimum. </s> + <s xml:id="s3207" xml:space="preserve">Talis enim ſectio ita <lb/>nominata fuit, hoc eſt hyperbo <lb/>les, ſimili ratione, qua elipſis ſeu <lb/>defectio etiam vocata fuit, nam <lb/>ſicut in ipſa defectione quadra-<lb/>tum ordinatę <var>.l.m.</var> minor eſt pro <lb/>ducto lineæ <var>.e.m.</var> in <var>.e.t.</var> per figu <lb/>ram ſimilcm producto <var>.d.e.</var> in <var>.e.<lb/>t.</var> quæ eandem obtineat <reg norm="altitu- dinem" type="context">altitu-<lb/>dinẽ</reg> ipſius <var>.e.m.</var> vt ipſe Pergeus <lb/>monſtrat in .13. primi lib. ita in <lb/>hyperbole <reg norm="dictum" type="context">dictũ</reg> quadratum ex <lb/>cedit quantitatem illius figuræ, <lb/>per ſimilem dictæ vt in .12. <reg norm="ipſius" type="simple">ipſiꝰ</reg> <lb/>Pergei facilè videre eſt. </s> + <s xml:id="s3208" xml:space="preserve">ſed <reg norm="prae ter" type="simple">prę<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 ſupradicto capite, multæ aliæ ſunt, cum corollarium primæ tertij Eu-<lb/>cli. ſit paſſio propria ipſius circuli, & idem dico de propoſitione .3. 4. 7. 8. 9. 11. 12. <lb/>13. 14. 15. 17. 18. 19. 20. 30. 31. ipſius tertij lib. nec non de .8. 9. ct .10. tertijdecimi, & <lb/>de prima .3. 4. 5. 6. et .7. quartidecimi eiuſdem. </s> + <s xml:id="s3209" xml:space="preserve">Idem infero de ea quod ſcrip ſi Ma <lb/>rio Nizzolio, Franciſco Vimercato, Franciſco Contareno, Angelo Agrimenſori, & <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">Illuſtri vire, Philiberto Pingonio Sabaudo Cuſiacenſium <lb/>Baroni.</head> + <p> + <s xml:id="s3210" xml:space="preserve">CVm antea meo nomine Sebaſtianus noſter omnia ferè tibi retuliſſet, inter alia, <lb/>quæ relin quebantur tibi <reg norm="dicenda" type="context">dicẽda</reg>, hoc vnum erat, quod ſi abſque lumine ſupe-<lb/>riori, in quem finem facta fuerint corpora cœleſtia ſcire deſideras, & humanam ra-<lb/>tionem ſequi volueris, putandum tibi non erit ea ſolum effecta eſſe, vt tam vile cor <lb/>pus, vt eſt terra aquis irrigata, animalia, & plantas regant, cum ea corpora ſint diuina, <lb/>in numero incompręhenſibilia, maximis ma gnitudinibus, & motibus velocisfimis, <lb/>prædita, id etiam minus putabunt hij, qui opinionem Ariſtarchi Samij, & Nicolai <lb/>Copernici ſequuntur, quorum ratione fieri non poteſt, vt credant, eius, quod ex vni <lb/>uerſo reliquum eſt, 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 ſi centra aliorum epicyclorum <lb/>planetarum tali regimine priuarentur, id quod nullo modo cum ratione conſentit, <lb/>ſi tam vera eſt ea opinio, quemadmodum <reg norm="rationabiliorem" type="context">rationabiliorẽ</reg> eam <reg norm="exiſtimant" type="context">exiſtimãt</reg>. </s> + <s xml:id="s3212" xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> quid <lb/>quam valet opinio Ariſtotelis, qui corpora cœleſtia, ab ortu, & interitu libera eſſe <lb/>fentit. </s> + <s xml:id="s3213" xml:space="preserve">dicens ſuperioribus fęculis, à noſtris antiquis nullam vnquam animaduerſam <lb/>fuiſſe alterationem in cœlo, cum non videat ſi quis eſſet in cœlo, <reg norm="neque" type="simple">neq;</reg> etiam obſerua <lb/>re poſſet alterationes quæ in terra, & circa terram fiunt, quæ in partibus, & non in to <lb/>to ſpectantur: </s> + <s xml:id="s3214" xml:space="preserve">vnde etiam fieri poteſt, vt in cœlo ſint particulares alterationes, <lb/>quæ à nobis tamen, qui ab illis longè diſtamus, non compręhendantur, terra, <reg norm="mareque" type="simple">mareq́;</reg> <lb/>(quamuis minimum reſpectu ipſius terræ) ratione totius ita ſe ſemper <reg norm="habuerunt" type="context">habuerũt</reg> <reg norm="quem" type="context">quẽ</reg> <lb/>admodum ſeſe habere corpora cœleſtia videmus, ſed alteratio, ratione tantum ali-<lb/>quarum minimarum partium quaſi inſenſibilium, ſi cum toto comparentur fit. </s> + <s xml:id="s3215" xml:space="preserve">Quis <lb/>enim ſcit, vt iam tibi dixi, quin, quemadmodum Luna circa terram voluitur, <reg norm="ipſaque" type="simple">ipſaq́;</reg> <lb/>terra ſit veluti centrum epicycli maioris eiuſdem, vt Ariſtarchus Samius, & Nico-<lb/>laus Copernicus cenſuerunt; </s> + <s xml:id="s3216" xml:space="preserve">ſic etiam Saturnus, Iupiter, Mars, Venus, <reg norm="atque" type="simple">atq;</reg> Mercu <lb/>rius circa alia huiuſmodi corpora, huic terræ ſimilia, in orbem agantur, quaſi ſpecu-<lb/>la, lumen Solis ſuo centro ex reflexione, deferentia (fuppoſita dico vera illorum opi <lb/>nione) <reg norm="Nollem" type="context">Nollẽ</reg> tamen tibi è mente excidere, vt aliàs te monui, <reg norm="quod" type="simple">ꝙ</reg> ſi communis opinio <lb/>vera eſt, neceſſario <reg norm="fatendum" type="context">fatendũ</reg> ſit corpus ſolare, <reg norm="dum" type="context">dũ</reg> in æquatore reperitur moru diurno <lb/>quolibet horę minuto, magis <reg norm="quam" type="context">quã</reg> <reg norm="decem" type="context">decẽ</reg> & <reg norm="ſeptem" type="context">ſeptẽ</reg> mille milliaria <reg norm="peragere" type="simple">ꝑagere</reg>, ideſt paulo mi <lb/>nus quam .18000. milliaria, Saturnum verò cum ſimiliter eſt in æquatore, eodem <reg norm="tem- poris" type="context">tẽ-<lb/>poris</reg> ſpatio, quaſi tercentamille milliaria Italica conficere, & ſic 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">cũ</reg> ſimplici gyro terræ circa ſuum + <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 ſufficit ad recipiendum lumen, & influentias illo-<lb/>rum corporum. </s> + <s xml:id="s3218" xml:space="preserve">Et ita, veluti princeps corporum vniuerſi, intra vnum an-<lb/>num circa eam vertitur. </s> + <s xml:id="s3219" xml:space="preserve">Ita etiam ſuſſiceret, vt ipſa terra circa dictum diuinum <lb/>corpus ſolare, interſecando axem diurnum cum axe annuali (cum ab eo lumen, ca-<lb/>lorem, & influentiam ſuſcipere 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 ipſos, nullę ſunt, quia quęlibet pars (vt inquiunt) <lb/>retinet naturam totius, præterquam <reg norm="quod" type="simple">ꝙ</reg> aer, & aqua, quæ ipſam terram <reg norm="circundant" type="context">circundãt</reg>, pla <lb/>nè eundem naturalem impetum motus obtineant, quitanto lentior eſt, quanto lon <lb/>gius diſtat aer ab ipſa terra, ſecundum etiam talem opinionem, nulla neceſſitas fo-<lb/>ret, vt locus fixarum terminaretur aliquibus ſuperficiebus, conuexa ſcilicet, & 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 ſit lux aliqua, quæ à corpore lucido non proue-<lb/>niat, mihi facilè ad conſiderandum videtur. </s> + <s xml:id="s3222" xml:space="preserve">hic enim oportet, vt nos ad id <lb/>quod perpetuò videmus referamus, exiſtimo autem te velle dicere lumen, non lu-<lb/>cem, quia propriè lux, qualitas ea viſibilis appellatur, quæ eſt 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 ipſum corpus reperitur, à luce, quæ in dicto corpore manet emanans. </s> + <s xml:id="s3224" xml:space="preserve">vnde pa-<lb/>tet, nullam lucem abſque corpore ſubiecto eſſe poſſe, id quod cum fieri quîret, <reg norm="idem" type="context">idẽ</reg> <lb/>de quolibet alio accidente dici poſſet, id eſt quod ex ſe, & abſque aliquo ſubiecto <lb/>ſubſiſteret.</s> + </p> + <p> + <s xml:id="s3225" xml:space="preserve">Lumen deinde à luce proficiſci patet, <reg norm="quod" type="simple">ꝙ</reg> penetrat diaphanum, neque aliquo mo-<lb/>do ſuum actum oſtendit, niſi, aut per incidentiam, aut ratione opaci, ex reflexione, <lb/>cuius ſuperficiei colorem induit. </s> + <s xml:id="s3226" xml:space="preserve">Atque hæc eſt cauſa, vt inter crepuſculum matu-<lb/>tinum, aut veſpertinum, nox etiam ſi ſit ſerena, adeo obſcura nobis appareat, quam-<lb/>uis totum vniuerſum diaphanum, extra conum vmbræ, quæ ex terra prouenit ſit vn <lb/><reg norm="dique" type="simple">diq;</reg> radijs luminoſis Solis colluſtratum; </s> + <s xml:id="s3227" xml:space="preserve">qui quidem radij, non niſi à ſuamet reflexio <lb/>ne à Luna, & ab alijs ſtellis (vt corporibus opacis, quæ reſiſtunt lumini, ne vlterius <lb/>penetrare poſſit, vnde retrò redit) comprehenduntur.</s> + </p> + <p> + <s xml:id="s3228" xml:space="preserve">Ais etiam propria viſus obiecta plura eſſe, nominans pro vno, colorem, & lucem <lb/>pro alio. </s> + <s xml:id="s3229" xml:space="preserve">Ego autem reſpondeo, obiectum oculi eſſe vnicum tantum, ideſt lumen. <lb/></s> + <s xml:id="s3230" xml:space="preserve">Quod ad lucem ſpectat, iam tibi dixi, eam eſſe quandam qualitatem in corpore luci <lb/>do, & non extra ipſum poſitam, à quo quidem corpore, cum non exeat, oculi obie-<lb/>ctum eſſe nequit, ſed lumen quidem ab ipſa luce productum. </s> + <s xml:id="s3231" xml:space="preserve">Color etiam, qui eſt in <lb/>corpore colorato, obiectum oculi eſſe non poteſt, cum dictum corpus non deſerat, <lb/>ſed lumen quidem ab eodem corpore reflexum, & huiuſmodi corporis colore tin-<lb/>ctum: </s> + <s xml:id="s3232" xml:space="preserve">vnde tam lumen incidens, quam reflexum colore eſt ſemper imbutum.</s> + </p> + <p> + <s xml:id="s3233" xml:space="preserve">Illud quidem coloratum eſt qualitate lucis corporis lucidi, a ut medij, per quod <lb/>tranſit, ſed 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">priisq́;</reg> ſuperficiebus corporum coloratorum, eo <reg norm="quod" type="wordlist">qđ</reg> pauca corpora tam opaca repe-<lb/>riuntur, ut immediatè lumen à ſuperficie propriè <reg norm="reflectant" type="context">reflectãt</reg>, ſed 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 propoſitum redeamus, dico lum en tantum eſſe viſus obiectum, quod ſi <lb/>colore eſt imbutum, aut tale eſt ratione color ris lucis, quæ eum mittit, aut ratione me <lb/>dij per quem tranſit, aut ratione corporis, vnde reflectitur, etſi ſuperficies corporis <lb/>vnde lumen reflectitur eſſet omnino priuata colore, ſub aſpectum non caderet, vt <lb/>etiam cum huiuſmodi ſuperficies læuigata, & polita eſt ſecundum <reg norm="continuitatem" type="context">continuitatẽ</reg> ſua-<lb/>rum partium, videlicet, vt ſpeculi radio tamen non profundante, & ideo perfectiffi <lb/>morum quorundam ſpeculorum ſuperficies non cernuntur, ſed lumen tantum re-<lb/>flexum, colore alicuius alterius ſuperficiei, aut à luce, corporis lucidi, aut à me-<lb/>dio per quem tranſit, conſpicitur. </s> + <s xml:id="s3236" xml:space="preserve">Ego verò non aſſero colorem non eſſe quid di-<lb/>uerſum à lumine, ſed imagineris lumen eſſe veluti animam, aut ſubſtantiam & colo <lb/>rem corporis formam accidentalem, cum nullum lumen à ſenſu viſus percipi poſ-<lb/>ſit, quod aliquo modo colore non ſit imbutum: </s> + <s xml:id="s3237" xml:space="preserve">& eundem reſpectum quem ſonus <lb/>ad auditum, lumen ad oculum habet, quia vt ſonus ſecundum eam velocitatem, quæ <lb/>à motione aeris, aut aquæ, ex colliſione <reg norm="aliorum" type="context">aliorũ</reg> corporum producitur ad euitan dum <lb/>vacuum, a cutus, vel grauis ſentitur, ita lumen originem ducens à corpore lucido per <lb/>medium diaphanum aeris, aut aquæ, aut alterius huiuſmodi corporis ad oculum tran <lb/>ſit colorem lucis, aut medij per quod tranſit, aut vnde reflectitur induit.</s> + </p> + <p> + <s xml:id="s3238" xml:space="preserve">Quod verò Luna nullum ex ſe habeat lumen, ſufficiens inditium eſt nos ipſam <lb/>tantò magis obſcuram videre, quantò magis in cono vmbræ terræ immergitur, & <lb/>ſi eo tempore ipſam videmus rubeo colore affectam, hoc enim accidit, quia radij ſo <lb/>lares vndequaque refranguntur à vaporibus ipſam terram circundantibus, quæ qui-<lb/>dem refractio fit verſus axem coni vmbræ terræ, </s> + <s xml:id="s3239" xml:space="preserve">& propterea vmbra dicti coni non <lb/>eſt æqualiter obſcura, ſeu tenebroſa, circa vero <reg norm="axem" type="context">axẽ</reg> ipſius coni, magis quam circa eius <lb/><reg norm="circunferentiam" type="context context">circũferentiã</reg>, obſcura <reg norm="verum" type="wordlist">vr̃</reg>, & quia corpus lunare tale eſt, vt facillimè recipiat <reg norm="qualecum" type="context">qualecũ</reg> <lb/>que lumen, quod etiam manifeſtè videtur dum ipſa Luna reperitur ſecundum lon-<lb/>gitudinem inter Solem, & Venerem, quod pars Lunæ lumine Solis deſtituta, à lumi <lb/>ne Veneris aliquantulum illuſtratur, quod ego ſæpè vidi, & multis oſtendi. </s> + <s xml:id="s3240" xml:space="preserve">Propte-<lb/>rea dum ipſa Luna in cono vmbræ terræ reperitur adhuc videtur. </s> + <s xml:id="s3241" xml:space="preserve">Rubedo etiam il-<lb/>la nubium poſt Solis occaſum, vel ante ortum, aliunde non prouenit, niſi à qualitate <lb/><reg norm="vaporum" type="context">vaporũ</reg>, per quos ſolares radij tranſeunt, à quibus vaporibus, tali colore ipſi radij <lb/>afficiuntur, eomet modo quo radius, cuiuſuis corporis lucidit, <reg norm="tranſiens" type="context">trãſiens</reg> per vitrum, ſeu <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/>diuerſas eleuationes. Et de quibuſdam erroribus Nico-<lb/>lai Tartaleæ, circa idem.</head> + <head xml:id="head387" style="it" xml:space="preserve">Fllustri D. Ioſepho Cambiano ex Ruffia Dominis, aquiti <lb/>ſtrenuo, & tormentis bellicis Serenißimi Ducis <lb/>Sabaudia Prafecto.</head> + <p> + <s xml:id="s3242" xml:space="preserve">EXcogitaui quędam dum ocio frui licuit per abſentiam Ducis Sereniſſimi, <lb/>quæ ad te ſcribere placuit, vt ſi probaueris in lucem quandoque profer-<lb/>re non dubitem, ſi deſpexeris, ocius ſupprimam, ſunt autem huiuſmodi.</s> + </p> + <p> + <s xml:id="s3243" xml:space="preserve">Vnde fiat vt tormentum bellicum vehementi<unsure/> feriat ictu ſuperius delato <lb/>quam orizontali, vt Tartalea ſcribit, quæſito ſecundo libr. primi quæſitorum, à ne-<lb/>mine adhuc (quod ſciam) traditum eſt.</s> + </p> + <p> + <s xml:id="s3244" xml:space="preserve">Rationes verò Tartaleæ nullius ſunt momenti, quia ſi validæ eſſent, ſequeretur <lb/>vt inclinata bombarda, adeo vt angulus ſub orizonte factus æqualis eſſet ei, qui ſu <lb/>pra orizontem eſt, ictum bombardę in vtroque huiuſmodi ſitu eundem eſſe <reg norm="futurum" type="context">futurũ</reg>. <lb/></s> + <s xml:id="s3245" xml:space="preserve">& ſi aliqua differentia oriretur ratione granitatis pilæ ab ipſa bombarda emiſſæ, hoc <lb/>fieret, vt ſcilicet 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 cauſa vnde fiat, vt bom <lb/>barda eleuata vehementius feriat, quàm ea quæ eſt minus alta, eadem eſt ferè, in ge-<lb/>nere, cum ea, qua aliquod corpus materia magis denſa, ſed ſimile & ęquale alteri cor <lb/>pori materiæ minus denſæ velocius mouetur ab vna eademq́ue, aut æquali vi <lb/>compulſum. </s> + <s xml:id="s3247" xml:space="preserve">Eſt eadem etiam in ſpecie ei, qua maiorem effectum producit <lb/>puluis, qui in locis ſubterraneis ponitur quum vaſis optimè colligatis ferro in-<lb/>cluditur. </s> + <s xml:id="s3248" xml:space="preserve">Eſt etiam ſimilis ei, qua longius impellitur pila, qua ludimus, ab ali-<lb/>quo inſtrumento ligneo, quando percutitur contra, quam cum ſecundum ſuum mo-<lb/>tum proijcitur. </s> + <s xml:id="s3249" xml:space="preserve">Id quod inde fit, quia virtus mouens maiori vi, & intenſiori huiuſ-<lb/>modi corpus percutit, quia corpus quod moueri debet, quanto magis reſiſtit virtu-<lb/>ti mouenti (certum tamen terminum præſcribendo) in exiguo eo temporis ſpatio, <lb/>tanto maiorem virtutem colligit, quæ ipſum deinde tanto cum impetu mo-<lb/>uet, & tanto magis impellens concomitatur, vt maiorem effectum efficiat, quam ſi <lb/>ad mouendum ſeſe facilè reddidiſſet. </s> + <s xml:id="s3250" xml:space="preserve">Atque hoc ſupradictis ictibus eleuatis acci-<lb/>dit, quia grauitas pilæ, ea eſt quæ reſiſtens 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">ĩmpelleret</reg>. </s> + <s xml:id="s3251" xml:space="preserve">Et quia huiuſmodi 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 ſufficeret, 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, ſed vt dixi, tanta cum veloci-<lb/>tate <reg norm="adaugetur" type="simple">adauget̃</reg>, vt huiuſmodi virtus longè ſuperet <reg norm="reſiſtentiam" type="context context">reſiſtẽtiã</reg> <reg norm="ponderis" type="context">põderis</reg> pilæ, & ſic eſt cau <lb/>fa, ut effectus, quod <reg norm="experientia" type="context">experiẽtia</reg> innot eſcit <reg norm="producatur" type="simple">producat̃</reg>. </s> + <s xml:id="s3252" xml:space="preserve">Sed ea ratio, qua ſeſe <reg norm="idem" type="context">idẽ</reg> author <lb/>in tertio quæſito ad aliquod impoſſibile, circa iter ipſius pilæ Legatum Hiſpanum + <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 eſt ſemper dicendum, quod <reg norm="quam" type="context">quã</reg> <lb/>to velocior ſit quædam pila, tanto rectius moueatur, quia ei dici poſſet, vſque ad cer <lb/>tum quendam terminum velocitatis, per tantum ſpatij eam aptam eſſe, vt recta per <lb/>fectè moueatur, ſed ſi velocius iret, non tamen futurum, vt per idem ſpatium re-<lb/>ctius moueretur, ſed quod per longius ſpatium recta motum perageret, & ſic nihil <lb/>haberet quod replicaret, præter quam quod ipſe ſupponit id quod in 18. quæſito <lb/>negat, in quo ait pilam uicinam orificio, non adeo uelocem eſſe, quam cum aliquan-<lb/>tulum ab eodem eſt rem ota, ratione reſiſtentiæ ſui 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 diſtat à recto huiuſmodi <reg norm="angulus" type="simple">angulꝰ</reg>, ſiue ſit acutus ſiue obtuſus, tanto minorem uim <lb/>habet, eodem planè ferè modo quem tertio capite mei tractatus de rebus mechani <lb/>cis deſcripſi. </s> + <s xml:id="s3255" xml:space="preserve">Quia in ictibus eleuatis, iter inclinationis violentæ ipſius pilæ verſus <lb/>terminum ad quem, incipiendo à loco ipſius pilæ cum itinere inclinationis natura-<lb/>lis, angulum obtuſum, & in ictibus inclinatis acutum conſtituit. </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">cũ</reg> <lb/>putet indifferenter aliquod corpus impellere, aut percutere maiori <reg norm="cum" type="context">cũ</reg> impetu cum <lb/>eſt in itinere recto. </s> + <s xml:id="s3257" xml:space="preserve">Quia ſequeretur quod aliquod corpus graue perpendiculari-<lb/>ter ſurſum verſus proiectum, in qualibet parte ſui itineris, ſemper fortius percute-<lb/>ret, quam in qualibet parte itineris alterius cuiuſuis eleuationis obliquæ, quod <reg norm="quam" type="context">quã</reg> <lb/>ſit falſum, tibi conſiderandum relinquo.</s> + </p> + <p> + <s xml:id="s3258" xml:space="preserve">Eſt etiam falſa ea ratio, quam in quarto quæſito idem adducit, quia aer in motu <lb/>non tantum durat, quantum ipſe putat, imò huiuſmodi violenta agitatio, citò ceſſat <lb/>& citius etiam, quam ſi extra aliquam bombardam cum tanta violentia impulliſſet <lb/>ſaceum plumis plenum.</s> + </p> + <p> + <s xml:id="s3259" xml:space="preserve">Ratio etiam quam in .18. quæſito de eo, quod pila pertran ſeat illud corpus cyllin <lb/>dricum aereum adducit, eſt planè vana, quia ſtatim aer, qui prius in <reg norm="bombarda" type="context">bõbarda</reg> erat <lb/>incluſus, extra ipſam <reg norm="erumpit" type="context">erũpit</reg>, cedit, à <reg norm="pilaque" type="simple">pilaq́;</reg> diuiditur, vt ſi nunquam eam figuram in-<lb/>duiſſet, neque aer ambiens ei reſiſtit. </s> + <s xml:id="s3260" xml:space="preserve">Sed quod velocior ſit in certa quadam diſtan <lb/>tia, quam in principio erat, ſi hoc <reg norm="verum" type="context">verũ</reg> eſſet, ab alia cauſa dependeret, quæ partim ſi <lb/>milis eſſet ei, quæ efficit, vt corpora in motibus naturalibus, cum longius diſtant à ter <lb/>mino vnde naturaliter ſeſe mouerunt, ſint velociora, quia per aliquod ſpatium hu-<lb/>iuſmodi corpus moueretur quemadmodum motu naturali cietur.</s> + </p> + <p> + <s xml:id="s3261" xml:space="preserve">Ratio autem eius quare pila, aut globus bombardæ ſibiletab eodem in ſeptimo <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 ſignum, ictus altius tenderet, quod <lb/>poteſt etiam eſſe falſum, cum hocnon ſit neceſſarium, quia pila dum deſcendit, for-<lb/>taſſe tangeret ſcopum.</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. ſuarum tabularum Prute-<lb/>nicarum, à Nicolao Copernico ſumpſit pag .64. à tergo in libr. reuolutionum <lb/>cœleſtium, ſed ipſe Stadius eam non intellexit, omitto, quod mutauerit characte-<lb/>res ipſius figurę, vt illa ſua videatur, quod nihil refert, alterat etiam <reg norm="demonſtrationem" type="context">demonſtrationẽ</reg>, <lb/>ſed ipſum putare <var>.i.K.</var> perpendicularem à centro circuli ſemper dependere, eſt intol <lb/>lerabilis error; </s> + <s xml:id="s3264" xml:space="preserve">nec vnquam verificatur hoc, niſi quando punctum <var>.K.</var> interſectionis <lb/>diametrorum parallelorum, forte reperitur in axe mundi. </s> + <s xml:id="s3265" xml:space="preserve">Reliqua verò ſuæ demon <lb/>ſtrationis, ſi non intelligis, minimè miror, eo quod ipſemet Stadius ſeipſum confun <lb/>dit. </s> + <s xml:id="s3266" xml:space="preserve">Veram autem demonſtratio nem huiuſmodi 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 ſit maxima ecclipticæ declinatio, vt puta <var>.a.c.</var> ſi latitudo <reg norm="etiam" type="context">etiã</reg> <lb/>ſtellæ nobis data fuerit, vt puta <var>.c.e.</var> cognitus nobis erit totalis arcus <var>.a.e.</var> & eius ſinus <var>.<lb/>e.m.</var> & quia notus etiam nobis eſt ſinus arcus <var>.a.c.</var> hoc eſt <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 ſinus arcus <var>.e.p.</var> cognitus etiam nobis eſt ſinus <var>.<lb/>q.g.</var> declinationis <var>.a.g.</var> datæ, cui æqualis eſt <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> ſint, 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 ſcientia anguli <var>.c.</var> et <var>.e.</var> ſunt æquales, cum ex .29. dicti lib. <lb/><reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> æqualis ſit 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">quorũ</reg> <lb/>vnuſquiſque æqualis eſt angulo <var>.ω.x.t.</var> & ſic de alijs dico, co quod vnuſquiſque <reg norm="eorum" type="context">eorũ</reg> <lb/>æqualis eſt angulo <var>.m.</var> vnde cum cognitum nobis ſit latus <var>.n.c.</var> et <var>.c.i.</var> et <var>.t.e.</var> <reg norm="notum" type="context">notũ</reg> etiam <lb/>nobis erit <var>.e.K.</var> ex .19. ſeptimi, eo <reg norm="quod" type="simple">ꝙ</reg> <lb/>ex .4. ſexti ſunt inuicem proportio-<lb/> + <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> + nalia, detrahendo poſtea <var>.e.K.</var> ab <var>.<lb/>e.l.</var> cognito, vel ècontra, hoc ab il-<lb/>lo, nobis innoteſcet <var>.K.l.</var> ſinus longi <lb/>tudinis ſtellæ.</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. ſcri-<lb/>bit, hoc eſt, Solem maiorem <lb/>eſſe Luna, ſolum .1644. vici-<lb/>bus, </s> + <s xml:id="s3271" xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</reg> cum affirmet So-<lb/>lem maiorem eſſe terra (vt etiam <lb/>in Almageſto videre eſt) 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 cognoſcendam latitudinem ſtellæ, <reg norm="eiusque" type="simple">eiusq́</reg> declinationem, Monteregius in <lb/>10. propoſitione .8. li. </s> + <s xml:id="s3274" xml:space="preserve">Almageſti <reg norm="methodum" type="context">methodũ</reg> ſatis docuit, ſed ſi alia aliqua metho <lb/>do hoc idem cognoſcere voluerimus, oportebit nos prius altitudinem poli cogno-<lb/>ſcere, deinde altitudinem meridianam ipſius ſtellæ, nec non horam, <reg norm="quando" type="context">quãdo</reg> ipſa ſtel <lb/>la in meridiano ſupra terram reperitur, qua hora mediante, illicò cognoſcemus pun <lb/>ctum ecclipticæà meridiano interſecto, eo tempore, quo ſtella cœlum mediat ſu-<lb/>pra terram. </s> + <s xml:id="s3275" xml:space="preserve">Et quia ex cognita altitudine poli, illico cognoſcitur altitudo æqua-<lb/>toris, cuius altitudinis differentia ab altitudine ſtellæ eſt declinatio ipſius ſtellæ, ha-<lb/>bebimus ideo eius declinationem cognitam; </s> + <s xml:id="s3276" xml:space="preserve">qua mediante ad <reg norm="cognoſcendum" type="context">cognoſcendũ</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 aſtri <var>.u.o.</var> verò eius declinatio ab æquatore, et <var>.e.a.</var> arcus ęcclipticæ inter <lb/>æquatorem, & meridianum, hoc eſt minor quarta, et <var>.a.u.</var> aſcenſio recta ipſius arcus, <lb/>et <var>.u.e.</var> ſit declinatio puncti <var>.e.</var> ęcclipticę ab æquatore, <reg norm="reſiduum" type="context">reſiduũ</reg> vero declinationis ſtel-<lb/>lę ſit <var>.o.e.</var> quæ <reg norm="omnia" type="wordlist">oĩa</reg> nobis cognita erunt, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.t.</var> polus ęcclipticus, à quo per <var>.o.</var> vſque ad <lb/>ęcclipticam tranſeat quarta <var>.t.i.</var> in qua quęrendus 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 ſunt, cum angulo <var>.a.</var> declinatio <lb/>nis ęclipticę, & cum angulo <var>.u.</var> recto, vnde ex .4. primi Copernici, cognoſcemus 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> cognoſcemus <reg norm="angulum" type="context">angulũ</reg> <lb/>e. et deinde <var>.i.</var> vt <reg norm="rectum" type="context">rectũ</reg>, & latus <var>.o.e.</var> ergò ex <reg norm="eadem" type="context">eadẽ</reg> .4. cognoſcemus <reg norm="arcum" type="context">arcũ</reg> <var>.o.i.</var> quæſitum, <lb/>& ſimiliter arcum <var>.e.i.</var> qui coniunctus vel <reg norm="demptus" type="context">dẽptus</reg> ab <var>.a.e.</var> tribuet nobis longitudinem <lb/>ſtellę, ſed quia huiuſmodi operatio in paruis triangulis valde fallit. </s> + <s xml:id="s3280" xml:space="preserve">Ideo tibi ſua-<lb/>deo alia methodo, hoc facere, hoc eſt 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 ſunt, cum angulo <var>.p</var>. </s> + <s xml:id="s3281" xml:space="preserve">Nam <var>.o.p.</var> eſt complementum de <lb/>clinationis ſtellæ, et <var>.p.t.</var> eſt arcus coluri ſolſtitiorum inter duos polos, & angulus <var>.p.</var> <lb/>reſiduum ex recto <var>.t.p.a.</var> duorum colurum dempto angulo. a, <var>p.u.</var> cognito aſcenſionis <lb/>recte, vnde angulus <var>.u.o.s.</var> vt contrapoſitus cognitus remanet. </s> + <s xml:id="s3282" xml:space="preserve">angulus verò <var>.u.</var> rectus <lb/>eſt, & 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">cũ</reg> an-<lb/>gulo <var>.a.s.i.</var> reſiduo ex duobus rectis. </s> + <s xml:id="s3284" xml:space="preserve">Et quia <lb/>etiam angulus <var>.s.a.i.</var> cognitus eſt, cum ſit an <lb/>gulus maximę 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">cũ</reg> arcu <var>.a.i.</var> ſed <lb/><var>a.i.</var> erit longitudinis ſtellæ dempto poſtea <var>.<lb/>s.i.</var> ex <var>.s.o.</var> iam inuento habebimus arcum <var>.i.<lb/>o.</var> latitudinis ipſius ſtellæ.</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 ſcribo non vt ipſis vta-<lb/>ris, ſed potius vt tibi morem <reg norm="geram" type="context">gerã</reg>, cum bre <lb/>uiſſima methodus ſit illa, <reg norm="quam" type="context">quã</reg> Monteregius <lb/>ſcripſit <reg norm="in" type="wordlist">ĩ</reg> .10. <reg norm="propoſitione" type="simple">ꝓpoſitione</reg> .8. li. in Almageſt.</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 deſignari poßit alios duos circulos <lb/>propoſitos 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 quæſiuiſti, vt modum tibi ſcribere vel-<lb/>lem, quo circulus deſignari poſſit circunſcribens alios duos propoſitos circulos. <lb/></s> + <s xml:id="s3288" xml:space="preserve">Qua in re vt tibi ſatisfaciam quod maximè cupio ita rem accipe.</s> + </p> + <p> + <s xml:id="s3289" xml:space="preserve">Propoſiti circuli ſint, aut inter ſe contigui, aut interſecantes vel ſeparati. </s> + <s xml:id="s3290" xml:space="preserve">Eſto <reg norm="pri- mum" type="context">pri-<lb/>mũ</reg> contiguos eſſe, qui ſint <var>.d.b.</var> et <var>.f.q.</var> <reg norm="quorum" type="context">quorũ</reg> <var>.d.b.</var> maior ſit et <var>.f.q.</var> minor, <reg norm="eorum" type="context">eorũ</reg> vero <lb/>centra ſint .a et <var>.o.</var> <reg norm="punctum" type="context">punctũ</reg> autem <reg norm="contingentię" type="context">cõtingentię</reg> ſit <var>.i</var>. </s> + <s xml:id="s3291" xml:space="preserve"><reg norm="Nunc" type="context">Nũc</reg> <reg norm="protrahatur" type="simple simple">ꝓtrahat̃</reg>. <var>b.a.o.q.</var> per <reg norm="centra" type="context">cẽtra</reg> eo <lb/>rum ab vna circunferentia ad aliam, quę quidem linea tranſibit per punctum <var>.i.</var> ex <lb/>11, tertij Eucli. </s> + <s xml:id="s3292" xml:space="preserve">deinde à diametro maiori abſcindatur <var>.i.e.</var> ad æqualitatem minoris <lb/>ſemidiametri, quo facto ſumatur diſtantia inter <var>.e.</var> et <var>.b.</var> circino mediante <reg norm="factoque" type="simple">factoq́</reg> cen <lb/>tro <var>.o.</var> ſcindatur, alio circini pede, circunferentia maioris circuli in puncto <var>.u.</var> à quo ſi <lb/>mente concipiemus duas lineas <var>.u.a.d.</var> et <var>.u.o.f.</var> tranſeuntes per eorum centra <var>.a.</var> et <var>.o.</var> <lb/>vſque ad circunferentias in punctis <var>.d.</var> et <var>.f.</var> ipſę <reg norm="erunt" type="context">erũt</reg> inuicem ęquales, eo quod <var>.e.i.</var> <reg norm="sum- pta" type="context">sũ-<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> ſed <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 ſemidiameter, con-<lb/>tiguus erit ipſis propoſitis circulis ex conuerſo .11. iam dictæ. </s> + <s xml:id="s3294" xml:space="preserve">Idem dico pro circu-<lb/>lis ſe inuicem ſecantibus.</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 ſi circuli propoſiti ſeiuncti fuerint, ſumatur <var>.b.i.</var> diameter maioris, qui fiat ſe-<lb/>midiameter vnius circuli circa centrum <var>.o.</var> & hic circulus vocetur <var>.h.x.</var> coniunga-<lb/>tur deinde ſemidiameter <var>.o.i.</var> minoris circuli cum ſemidiametro <var>.a.i.</var> circuli maio-<lb/>ris, & ex huiuſmodi compoſita linea, fiat vnus ſemidiameter <var>.a.x.</var> circuli <var>.x.n.</var> concen <lb/>trici cum maiori, & à puncto <var>.x.</var> interſectionis horum circulorum (poſito quod ſe in-<lb/>uicem interſecent) ducantur per eorum centra <var>.x.a.</var> et <var>.x.o.</var> vſque ad ipſorum circun-<lb/>ferentias in punctis <var>.d.</var> et <var>.f.</var> duę <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, & ſemidiametri am-<lb/>borum circulorum, facto deni <lb/>que centro <var>.x.</var> vnius circuli, cu <lb/>ius ſemidiameter ęqualis ſit <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 diſtantia duorum <lb/>propoſitorum circulorum tanta fuerit, quod ſecundi circuli nequeant ſe inuicem <lb/>tangere, vel ſecare, tunc alia via incedendum erit, quę talis eſt & 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="ſignentur" type="context">ſignẽtur</reg> duo puncta ab ipſo ęquidi <lb/>ſtantia <var>.K.</var> et <var>.p.</var> diſtantia vero <var>.a.K.</var> facta ſit ſemidiameter eſſe vnius circuli <var>.K.x.</var> circa <lb/>centrum <var>.a.</var> diſtantia autem <var>.o.p.</var> ſemidiameter alterius circuli <var>.p.x.</var> circa cen-<lb/>trum <var>.o.</var> qui quidem circuli ſe inuicem ſecent in puncto <var>.x.</var> à quo cum ductę fue-<lb/>rinc <var>.x.a.d.</var> et <var>.x.o.f.</var> per centra dictorum circulorum, ipſe erunt <reg norm="inuicem" type="context">inuicẽ</reg> ęquales, eo <reg norm="quod" type="wordlist">qđ</reg> <lb/>cum <var>.b.K.</var> æqualis ſit <var>.q.p.</var> igitur <var>.x.d.</var> et <var>.q.p.</var> erunt inuicem ęquales, ſed <var>.f.x.</var> æqualis eſt <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 ſi <var>.x.</var> centrum fuerit vnius circuli, cuius ſemidia-<lb/>mer ſit vna dictarum, problema ſolutum erit.</s> + </p> + <p> + <s xml:id="s3299" xml:space="preserve">Talis etiam ſoiutio commo-<lb/>da erit ad inueniendum dictum <lb/> + <anchor type="figure" xlink:label="fig-0275-02a" xlink:href="fig-0275-02"/> + circulum cuiuſuis magnitudinis, <lb/>dato tamen <reg norm="quod" type="simple">ꝙ</reg> eius diameter, ma <lb/>ior ſit <var>.b.z.</var> cum in noſtra poteſta <lb/>te ſit accipere puncta <var>.K.</var> et <var>.p.</var> pro <lb/>xima vel remota ab ipſo <var>.z.</var> ad li-<lb/>bitum. </s> + <s xml:id="s3300" xml:space="preserve">Vnde abſque vlla diuiſio <lb/>neipſius <var>.q.b.</var> per medium, ſatis <lb/>erit ſignare puncta <var>.K.</var> et <var>.p.</var> dua-<lb/>bus diſtantijs mediantibus <var>.b.K.</var> <lb/>et <var>.q.p.</var> inuicem æqualibus, & <lb/>etiam propoſitis.</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 ſuperficialem ellipſi ſimilem, 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 ſuperficialem ellipſi ſimilem, 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> ſemiaxis maior <var>.a.e.</var> verò minor, ad angulum rectum inuicem coniuncti, <lb/>tunc <var>.a.e.</var> producatur vſque ad <var>.o</var>. </s> + <s xml:id="s3303" xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.a.o.</var> maior ſit quam diſtantia inter <var>.o.</var> et <var>.c.</var> quę <lb/>quidem <var>.a.o.</var> poſſet etiam dari, deſcribatur poſtea circulus <var>.a.d.b.</var> circa centrum <var>.o.</var> à <lb/>quo puncto protrahatur ſemidiameter <var>.o.b.</var> quæ cum <var>.a.o.</var> angulum rectum conſti-<lb/>tuat, quę <var>.o.b.</var> erit æquidiſtans <var>.e.c.</var> ex .28. primi, ducatur poſtea <var>.b.c.d.</var> et <var>.o.t.d.</var> vnde <lb/>angulus <var>.t.c.d.</var> ęqualis erit angulo <var>.o.b.d.</var> ex .29. eiuſdem. </s> + <s xml:id="s3304" xml:space="preserve">ex quinta autem anguli <var>.b.</var> <lb/>et <var>.d.</var> ſunt inuicem æquales, </s> + <s xml:id="s3305" xml:space="preserve">quare etiam <lb/>& anguli <var>.d.</var> et <var>.c.</var> inuicem ęquales erunt, <lb/> + <anchor type="figure" xlink:label="fig-0276-01a" xlink:href="fig-0276-01"/> + & ex .6. eiuſdem <var>.t.c.</var> ęqualis erit <var>.t.d.</var> duca <lb/>tur poſtea <var>.d.x.h.</var> perpendicularis lineæ <var>.c.<lb/>e.</var> ita diſtans ſub ipſa <var>.c.e.</var> vt arcus circula-<lb/>ris circa <var>.t.</var> delineatus ex ſemidiametro <var>.t.<lb/>d.</var> aptus ſit eam ſecare, ſumpto poſtea <var>.r.</var> <lb/>tam diſtante ab <var>.e.</var> vt <var>.t.</var> reperitur ab ipſo <lb/>e. et <var>.z.</var> ab <var>.e.</var> vt <var>.o.</var> ab eodem, ducendo po-<lb/>ſtea duos alios arcus magnitudinis <reg norm="priorum" type="context">priorũ</reg> <lb/>circa centra <var>.r.</var> et <var>.z.</var> habebimus propoſi-<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 cuiuſuis magnitudinis, vt <lb/>in ſecunda figura videre eſt, poſito tamen quod eorum diameter, minor ſit minore <lb/>axe ipſius figurę, quorum circulorum vnus ſit <var>.c.d.</var> circa <var>.t.</var> eius centrum, deinde in axe <lb/>minori ſumatur <var>.a.x.</var> æqualis <var>.c.t.</var> & protrahatur <var>.t.x.</var> quę per ęqualia diuidatur in pun-<lb/>cto <var>.n.</var> à quo poſtea 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> vſque ad interſectionem 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 ſecunda figura, ſumpto <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, ipſum, centrum erit alterius arcus oppoſi-<lb/>ti, poſſemus etiam <reg norm="abſque" type="simple">abſq;</reg> diuiſione ipſius, <var>t.x.</var> conſti <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 propoſite portionis datæ ſphæræ.</head> + <head xml:id="head397" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3308" xml:space="preserve">VTaxem propoſitæ alicuius datæ ſphæræ inuenire poſſis ita tibi operandum eſt <lb/>vt gratia exempli. </s> + <s xml:id="s3309" xml:space="preserve">Propoſita nobis eſt ſphæra <var>.c.i.e.t.</var> diametri cognitæ. </s> + <s xml:id="s3310" xml:space="preserve">pro <lb/>poſita etiam eſt nobis eius portio <var>.n.e.u.</var> axis <var>.e.a.</var> cognitæ minoris ſemidiametro, da-<lb/>ta etiam nobis eſt proportio alterius portionis minoris hemiſphærio <var>.i.e.t.</var> ad por-<lb/>tionem <var>.n.e.u.</var> quæritur nunc quantus ſit axis <var>.e.x.</var> ſecundæ portionis hoc eſt deſidera-<lb/>mus cognoſcere proportionem <var>.e.x.</var> ad <var>.e.a.</var> vel ad diametrum ipſius ſpheræ.</s> + </p> + <p> + <s xml:id="s3311" xml:space="preserve">Cuius gratia reperiatur primò proportio <reg norm="circunferentiæ" type="context">circũferentiæ</reg> maioris circuli ipſius <reg norm="ſphae ræ" type="simple">ſphę<lb/>ræ</reg> adeius diametrum, quæ ferè eſt vt .22. ad .7. ex Archimede.</s> + </p> + <p> + <s xml:id="s3312" xml:space="preserve">Quo facto, inueniatur quantitas ſuperficialis huiuſmodi maioris circuli, quæ ſem-<lb/>per æqualis eſt producto quod fit ex ſemidiametro in dimidium circunferentiæ ip-<lb/>fius circuli, ex eodem Archimede. </s> + <s xml:id="s3313" xml:space="preserve">Et ſic cognoſcemus quartam partem ſuperficiei <lb/>ſphæricæ ſphærę propoſite ex .31. primi lib. de ſphæra, & cyllindro Archimedis.</s> + </p> + <p> + <s xml:id="s3314" xml:space="preserve">Deinde ſumatur tertia pars producti, quod fit ex ſemidiametro in ſuperficiem <lb/>maioris circuli, & habebimus conum, cuius baſis erit circulus maior, altitudo verò <lb/>ſemidiameter propoſitæ ſphæræ ex .9. duodecimi Eucli.</s> + </p> + <p> + <s xml:id="s3315" xml:space="preserve">Quadruplum poſtea huiuſmodi coni, erit quantitas ſoliditatis, ſeu corporeitas to <lb/>tius ſphærę ex .32. dicti lib. Archimedis.</s> + </p> + <p> + <s xml:id="s3316" xml:space="preserve">Imaginemur poſtea <reg norm="in" type="wordlist">ĩ</reg> ſphærica portione <var>.n.e.u.</var> <reg norm="lineam" type="context">lineã</reg> <var>.e.u.</var> à <reg norm="summitate" type="context">sũmitate</reg> ad <reg norm="extremitatem" type="context">extremitatẽ</reg> <lb/>baſis, cuius <var>.e.u.</var> quantitatem cognoſcemus, hoc modo ſcilicet, 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 eſt 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 eſt producto quod <lb/>fit ex <var>.c.a.</var> in <var>.a.e.</var> ex .34. tertij <reg norm="eiuſdem" type="context">eiuſdẽ</reg>, <lb/>& quadrato <var>.a.e.</var> hoc eſt producto, <lb/>quod fit ex <var>.c.e.</var> in <var>.e.a.</var> ex .3. ſecundi <lb/>eiuſdem.</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 poſtea <var>.e.u.</var> ponamus eam <lb/>vnius circuli ſemidiametrum eſſe, cu <lb/>ius ſuperficialis quantitas etiam inue <lb/>niatur, vt ſupra dictum eſt, quæ qui <lb/><reg norm="dem" type="context">dẽ</reg> æqualis erit ſuperficiei portionis <lb/><var>n.e.u.</var> ex .40. primi li. </s> + <s xml:id="s3319" xml:space="preserve">Archimedis de <lb/>ſphæra, & 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 ſe-<lb/>midiametri datæ ſphæræ, & habebi-<lb/>mus ſoliditatem vnius coni æqualis <lb/>aggregato ſoliditatis portionis <var>.n.e.<lb/>u.</var> ſimul ſumptę<unsure/>, <reg norm="cum" type="context">cũ</reg> ſoliditate vnius co <lb/>ni, cuius axis ſit <var>.a.o.</var> <reg norm="reſiduum" type="context">reſiduũ</reg> ſemidia-<lb/>metri noſtræ ſphæræ 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"/> + ſis verò eadem quæ eſt portionis, cuius diameter eſt <var>.n.u.</var> ex .9. 12. Eucli. & ex .42. id-<lb/>eſt vltima primi Archimedis de ſphæra, & 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> eſt <lb/>axis et <var>.n.u.</var> diameter baſis, qui quidem conus nobis cognitus eſt, cum <var>.a.n.</var> ſemidia-<lb/>meter eius baſis, nobis cognita ſit ex .34. 3. Eucli. </s> + <s xml:id="s3322" xml:space="preserve">& ſic quantitas eius baſis, & ita ter-<lb/>tia pars <var>.a.o.</var> eius axis, quę multiplicata cum dicta baſi, cuius <var>.n.u.</var> eſt diameter, produ <lb/>cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-<lb/>gre gato, relinquet nobis ſoliditatem portionis <var>.n.e.u.</var> vnde cognoſcemus proportio <lb/>nem iſtius portionis ad totam ſphæram propoſitam.</s> + </p> + <p> + <s xml:id="s3323" xml:space="preserve">Sed cum nobis propoſita ſit proportio portionis <var>.n.e.u.</var> ad portionem <var>.i.e.t.</var> cogno <lb/>ſcemus etiam ſoliditatem huius ſecundę portionis <var>.i.e.t.</var> & ſimiliter <reg norm="proportionem" type="context">proportionẽ</reg> hu-<lb/>ius ad totam ſphęram, & ad <reg norm="reſiduum" type="context">reſiduũ</reg> <reg norm="etiam" type="context">etiã</reg> ipſius ſphęrę hoc eſt 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="vſque" type="simple">vſq;</reg> quo <var>.e.f.</var> æqualis ſit <var>.e.o.</var> ſemidiame <lb/>tro ſphęrę, 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 ſit proportioni portionis <var>.i.c.t.</var> ad portionem <var>.i.e.t.</var> quod quidem hoc modo efficie<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">pũ-<lb/>cto</reg> <var>.f.</var> in qua accipiemus duas lineas <var>.f.p.</var> et <var>p.q.</var> inuicem ita relatas, vt ſe habent in pro <lb/>portione duæ iam dictæ portiones, hoc eſt, vt <var>.i.c.t.</var> portio ad portionem <var>.i.e.t.</var> ducen <lb/>do poſtea <var>.q.e.</var> et <var>.p.h.</var> parallelam ad ipſam <var>.q.e.</var> diuiſam habebimus <var>.f.e.</var> in eadem pro <lb/>portione vt dictum eſt ex .2. ſexti, & .11 quinti Euclidis, vnde <var>.c.e</var>: <var>e.f.</var> et <var>.f.h.</var> nobis co <lb/>gnitę erunt.</s> + </p> + <p> + <s xml:id="s3326" xml:space="preserve">Oportebit nos nunc cognoſcere quantitatem <var>.c.x.</var> hoc modo, videlicet, quęramus <lb/>quadratum, cuius <var>.c.x.</var> eius ſit radix, cui quadratum lineę <var>.c.e.</var> cognitum, ita ſit propor-<lb/>tionatum, vt eſt linea <var>.x.f.</var> ad lineam <var>.f.h.</var> quę nobis cognita eſt, quod rectè factum erit <lb/>ex eo, quod ſcripſit Archimedes in .4. ſecundi de ſphęra, & cyllindro.</s> + </p> + <p> + <s xml:id="s3327" xml:space="preserve">Sed quia Archimedes eo in loco ſupponit id, quod necipſe, nec alius adhuc inue <lb/>nit, niſi via naturali, hoc eſt tres partes ęquales 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 ſemidiametro, diuidemus<lb/>q́ue <var>.f.e.</var> in puncto <var>.h.</var> vt ſupra factum fuit, applicabimus poſtea <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> quęremus 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">ctaq́;</reg> <var>.d.c.</var> quæ à diametro <var>.e.c.</var> deducta ab <var>.c.</var> incohando vſque ad <var>.x.</var> relinquat nobis <var>.<lb/>x.f.</var> ęqualem <var>.c.m</var>.</s> + </p> + <p> + <s xml:id="s3329" xml:space="preserve">Cuius rei ratio eſt, quia quadratum <var>.c.e.</var> ſe habet ad quadratum <var>.c.d.</var> vt <var>.c.e.</var> ad <var>.c.<lb/>b.</var> ex .4. et .18. ſexti Eucl. </s> + <s xml:id="s3330" xml:space="preserve">ſed ex .4. ita ſe habet <var>.m.c.</var> ad <var>.c.g.</var> vt <var>.e.c.</var> ad <var>.b.c.</var> & cum ſit <var>.c.<lb/>g.</var> ęq alis <var>.f.h.</var> ſi <var>.c.m.</var> ęqualis fuerit <var>.f.x.</var> habebimus propoſitum. </s> + <s xml:id="s3331" xml:space="preserve">Quod ſi quis per di-<lb/>ſcretum 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> propoſitæ ſphæræ eſſe ut decem, <lb/><reg norm="proportionemque" type="simple">proportionemq́;</reg> reſiduę portionis <var>.i.c.t.</var> ad ſecundam <var>.i.e.t.</var> hoc eſt <var>.f.h.</var> ad <var>.h.e.</var> ſeſqui-<lb/>alteram eſſe, vnde <var>.e.h.</var> bis tertia erit ìpſius <var>.f.h.</var> <reg norm="totaque" type="simple">totaq́;</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 poſtea quadratum lineæ <var>.c.x.</var> cui quadratum <var>.c.e.</var> hoc eſt .100. ita pro-<lb/>portionatum ſit vt <var>.f.x.</var> ad <var>.f.h.</var> hoc eſt ad .3. ſi autem cogitauerimus <var>.c.x.</var> eſſe nouem <lb/>partium talium qualium <var>.c.e.</var> eſt decem, eius quadratum erit .81. et <var>.x.f.</var> erit .6. par-<lb/>tium talium qualium <var>.c.f.</var> eſt .15. dicendo poſtea ſi .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 eſt .6. dabit .4. integra cum <num value="86">.<lb/> + <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> + 86.</num> centeſimis, ſed nos vellemus no-<lb/>bis prouenire tria, eo <reg norm="quod" type="simple">ꝙ</reg> ita eſt <var>.f.h.</var> <lb/>qua propter deſcendere 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> poſtea ex regula de <lb/>tribus, ſi 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">cũ</reg> 296. de .300. hoc eſt .2. <lb/>cum circa .49. quinquageſimis, quæ <lb/>quidem quantitas, cum propinquiſſi <lb/>ma ſit 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> reſiduum, hoc eſt 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 inſularijs.</head> + <head xml:id="head399" style="it" xml:space="preserve">Excellentißimo D. lo. Baptiſtæ Fæmello Ciui <reg norm="Decurionique" type="simple">Decurioniq́ꝫ</reg> Tau-<lb/>rinenſi Philoſopho, Medico, & in Accademia eius <lb/>Ciuitatis Medicinæ Practicæ Ordinario, <reg norm="Pri- marioque" type="simple">Pri-<lb/>marioq́ꝫ</reg> profeßori celeberrimo.</head> + <p> + <s xml:id="s3336" xml:space="preserve">DIj perdant tuas adeo moleſtas, & aſſiduas curas, quæ te nimis à ſuauiori-<lb/>bus ſtudijs diſtrahunt, & 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 promiſi, nunc mitto.</s> + </p> + <p> + <s xml:id="s3339" xml:space="preserve">Negari quidem non poteſt, quin fuerit laborioſum opus Porcachij, & Benedicti <lb/>Bordonij, hoc eſt inſularium, qui rectè etiam feciſſent, cum loqui eos oportebat de <lb/>terminis ſphæræ ratione ſitus locorum, ſi ſeipſos alicuius excellentis Coſmographi <lb/>conſilio ſubmiſiſſent. </s> + <s xml:id="s3340" xml:space="preserve">Conſidera quæſo, quomodo admitti poſſit, id quod ait Por-<lb/>cachius initio ſui operis, ideſt Iſlandiam ſub Polo arctico iacere, inter auſtrum, & <lb/>boream: </s> + <s xml:id="s3341" xml:space="preserve">omittamus etiam quod idem in Proęmio lib. ſecundi, vbi ait Biarmiam, <lb/>(& non Iſlandiam) eſſe ſub dicto polo arctico: </s> + <s xml:id="s3342" xml:space="preserve">in <reg norm="eodemque" type="simple">eodemq́;</reg> principio repetit ipſam <lb/>Iſlandiam inter auſtrum, & boream per centum leucas Germanicas extendi, dein-<lb/>de verſus occidentem, ea duo ſtupenda miracula conſpici. </s> + <s xml:id="s3343" xml:space="preserve">Vide quæſo, quomodo <lb/>incolę ſub aliquo ex polis, habere poſſint occidentem, orientem, magiſtrum, <reg norm="auſtrum" type="context">auſtrũ</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 poteſt, vt in-<lb/>ſula Iſlandiæ ſit ſub polo, eius tamen dies, & nox maior non ſit longior ſpatio <reg norm="trium" type="context">triũ</reg> <lb/>menſium? </s> + <s xml:id="s3345" xml:space="preserve">vt ipſe pagina .62. in proęmio ſecundi lib. affirmat, quamuis hoc à Bordo <lb/>ne deſumat. </s> + <s xml:id="s3346" xml:space="preserve">In quo <reg norm="vterque" type="simple">vterq;</reg> fallitur, <reg norm="ſentientes" type="context">ſentiẽtes</reg> huiuſmodi diem ab ingreſſu Solis, in <lb/>principium geminorum incipere, & in egreſſu à Leone terminari, ideſt à .12. Maij ad <lb/>14. Auguſti, quaſi ſi ab æquatore finis Leonis ita declinaret, vt principium gemino-<lb/>rum, & finis Aquarij, vt initium Sagittarij, nam ratio poſtulat, tantum de-<lb/>clinari ab æquatore finem quantum initium diei, vbi maximus dies .24. horas ex <lb/>cedit, & ſic dico de noctibus: </s> + <s xml:id="s3347" xml:space="preserve">vnde in huiuſmodi regione, vbi per tres menſes conti <lb/>nuos Sol radios emittit, huiuſmodi dies à medietate Tauri incipit, & in medietate <lb/>Leonis terminatur, quæ quidem loca æqualem declinationem habent, & ſic nox <lb/>trium menſium 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 ſolſtitiales eſſe circa .24. Iunij. </s> + <s xml:id="s3349" xml:space="preserve"><reg norm="quod" type="wordlist">Qđ</reg>, an <reg norm="tunc" type="context">tũc</reg> eſſet <reg norm="verum" type="context">verũ</reg>, <lb/>tu ipſe 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">inueniẽdũ</reg> <reg norm="orientem" type="context">orientẽ</reg>, & <reg norm="occidentem" type="context">occidẽtem</reg> præſcribit <lb/>in eodem proęmio pag .63. eſt tædioſus, cum ſemper expectare nos cogat æquino-<lb/>ctij tempus, cum alij modi reperiantur breuiores, qui in qualibet reuolutione primi <lb/>mobilis obſeruari poſſunt, quorum vnus erit mediante inuentione lineæ meridiane <lb/>orizontalis, eo modo, quo ſcriptum eſt ab antiquis mediante Sole, aut Luna, quæ <lb/>luminaria in quolibet alio loco, <reg norm="præterquam" type="context">præterquã</reg> ſub polo efficiunt, vt extremitas vmbræ <lb/>rectæ <reg norm="gnomonum" type="context">gnomonũ</reg> <reg norm="gyrum" type="context">gyrũ</reg> <reg norm="oxigonium" type="context">oxigoniũ</reg>, ſeu <reg norm="eclipticum" type="context">eclipticũ</reg> ducat, ideſt in ijs locis, <reg norm="quorum" type="context">quorũ</reg> zenit eſt <lb/>inter polum, & circulum arcticum, quemadmodum facit, vt alijs, exiſtentibus ipſis <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 ſub polo vi-<lb/>detur, ideſt vt multis rationibus, vt ipſe dicit, fieri non poſſit, ut fiat immediata quę <lb/>dam, & ſubita mutatio à continuo die ad continuam noctem abſque eo quod ijs, <lb/>ſaltem ſemel conceſſa ſint dies, & nox terminata duodecim horarum, eſt magis ad <lb/>mirandum impoſſibile, quod imaginari poſſimus, nam neceſſarium eſſet, ut orizon-<lb/>habitatorum ſub polo ſecaret æquatorem contra id, quod ſuperius admiſerat, id-<lb/>eſt <reg norm="orizontem" type="context context">orizõtẽ</reg> Biarmiæ, eſſe <reg norm="eundem" type="context context">eũdẽ</reg> <reg norm="cum" type="context">cũ</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 inſula Taprobana ad finem pag .186. admirabi <lb/>lia ſunt, ſcribens eiuſdem inſulę habitatoribus, Lunam ſuper terram non apparere <lb/>ab octauo uſque ad decimumſextum diem: </s> + <s xml:id="s3353" xml:space="preserve">pręter quam, quod etiam ſcribit, in <lb/>eadem inſula, tramuntanam non uideri, quod falſum eſt, quia hæc à polo arctico <lb/>circiter quatuor gradibus diſtat noſtris temporibus. </s> + <s xml:id="s3354" xml:space="preserve">unde ab ijs qui ſunt ſub æqua-<lb/>tore, cum ea ſupra orizontem eſt, conſpici poteſt, cum ijſdem ſingulis diebus oria-<lb/>tur, & occidat. </s> + <s xml:id="s3355" xml:space="preserve">Idem etiam pro re admirabili ſcribit, uideri Canopum, qui à po-<lb/>lo antarctico plus quam quadraginta gradibus diſtat.</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 Philalthęus tam eximius Mathematicus ſit, ut ipſum Anto-<lb/>nius Berga facit, ego quidem non uideo. </s> + <s xml:id="s3357" xml:space="preserve">In ſuis enim commentariis de <lb/>Cœlo, dicit primum, Pyramidem, quę 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"/> + ſex baſibus conſtare, 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 ſpectet: </s> + <s xml:id="s3359" xml:space="preserve">Omitto etiam quod idem neget aſtrologiam pag .74. <lb/>79. & quod etiam dicat pag .89. </s> + <s xml:id="s3360" xml:space="preserve">Deum eſſe ad orientem, non conſiderans aliqui-<lb/>bus populis noſtrum 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">Aſtrologiam eſſe antiquiorem Aſtronomia eſt falſiſſi-<lb/>mum, quia iudiciaria ſemper præſupponit cognitionem ſitus ſtellarum, quæ ab A-<lb/>ſtronomia petitur. </s> + <s xml:id="s3363" xml:space="preserve">Mouebit tibi riſum quod ait pag .307. his verbis.</s> + </p> + <quote> + <s xml:id="s3364" xml:space="preserve">Verum propriè media dicitur illa, quæ rectam ſphæram omninò habet, quæ eun <lb/>dem polum orizontis & mundi obtinet, quæ orizontem habet diuidentem <reg norm="ſphæram" type="context">ſphærã</reg> <lb/>æquè ſecundum angulos rectè.</s> + </quote> + <p> + <s xml:id="s3365" xml:space="preserve">Paulo inferius continuans ſermonem de ſphæra recta, ait.</s> + </p> + <quote> + <s xml:id="s3366" xml:space="preserve">Et niſi tumor terræ, & gibum eſſet, ijs perpetuus eſſet dies ſine nocte.</s> + </quote> + <p> + <s xml:id="s3367" xml:space="preserve">Linea verò .56. ait habitatores ſphęrę rectè habere .4. ſolſtitia, ſeſe ipſum huius <lb/>rei planè ignarum prodens .310. autem pag. ſic ſcribit.</s> + </p> + <quote> + <s xml:id="s3368" xml:space="preserve">Quoniam repercutiuntur radij, & peridem centrum tranſeunt, ob id ſtupam ap <lb/>poſitam centro radius accendit.</s> + </quote> + <p> + <s xml:id="s3369" xml:space="preserve">Quem quidem errorem ab Euclide deſumit, 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 ſubſtantiam corporis lucidi & corpoream, ſubijciam 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 ſi ſolum inci-<lb/>dat ſine repercuſſu, neorecta incidere iuuerit.</s> + </quote> + <p> + <s xml:id="s3375" xml:space="preserve">Quod falſum eſt cum radius incidens longè magis quam reflexus calefaciat. </s> + <s xml:id="s3376" xml:space="preserve">In fi <lb/>ne autem .405. ſic ſcribit.</s> + </p> + <quote> + <s xml:id="s3377" xml:space="preserve">Sol in ortu & in occaſu longius apparet, iccircò reuolui creditur. </s> + <s xml:id="s3378" xml:space="preserve">Hinc etiam in <lb/>abſide ſtare putatur, & in oppoſito abſidis, vnde ſolſtitia vocant, ſed nobis in Can-<lb/>cro, antipodibus verò in Capricorno tum Sol abeſſe longius apparet vtriſque.</s> + </quote> + <p> + <s xml:id="s3379" xml:space="preserve">An hoc quid peius dici poteſt? </s> + <s xml:id="s3380" xml:space="preserve">Circa vero .40. lineam pag .459. ſic ſcribit.</s> + </p> + <quote> + <s xml:id="s3381" xml:space="preserve">Si enim alij planetæ, & ſtellæ fixę reciperent à Sole lumen, dum accederent ad So <lb/>lem, vel recederent, aut contra, Sol ad eas appropinquaret, & abſcederet, eaſdem-<lb/>lucis viciſſitudinis ſubiret, quas Luna.</s> + </quote> + <quote> + <s xml:id="s3382" xml:space="preserve">Hoc autem nondum depręhenſum eſt, quin etiam Mercurius, Venus, ſuo interpo <lb/>ſitu, Solem occultarent nobis, vt Luna.</s> + </quote> + <quote> + <s xml:id="s3383" xml:space="preserve">Paulo inferius ſic ait. </s> + <s xml:id="s3384" xml:space="preserve">Rurſus æquè Saturnus, Jupiter, Mars, ſubire deliquium, <lb/>more Lunæ, aut ſaltem obiectu terræ inter Solem & ipſos, quia tum ob interpoſitam <lb/>terram non poſſent haurire lumen à Sole.</s> + </quote> + <p> + <s xml:id="s3385" xml:space="preserve">Hæc verò omnia, talia ſunt, qualia ab ijs qui incipiunt intelligere ſphæram 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 ſi aſtra lumen à Sole acciperent eiuſdem caloris eſſent. </s> + <s xml:id="s3388" xml:space="preserve">Itaque om <lb/>nia ſiccarent, & nulla eſſent frigidæ conſtitutionis contra Aſtrologos.</s> + </quote> + <p> + <s xml:id="s3389" xml:space="preserve">Quia hac ratione, Luna, quæ negari non poteſt, quin ab ipſo Sole <reg norm="lumem" type="context">lumẽ</reg> accipiat, <lb/>eiuſdem 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. quaſi 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 ſit effecta quam ad illuminandam ſuperficiem huius excrementi ip-<lb/>ſius vniuerſi ad vtilitatem hominum, imò, vt rectius dicam, <reg norm="animalium" type="context">animaliũ</reg>. </s> + <s xml:id="s3391" xml:space="preserve">vide etiam <lb/>pag .632. et .633. vbi Ariſtotelem de implendo loco non intellexit, cum citet ſphæ-<lb/>ram, loco pyramidis, & inter .46. et .47. lineas dicat <reg norm="quadratum" type="context">quadratũ</reg> eſſe quid multiplex, <lb/>cum ſit vnicum tantum in ſpecie, quia ſpecies eſt quadrilateri, & quadranguli, ſed <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. ſcribens.</s> + </p> + <p> + <s xml:id="s3394" xml:space="preserve">Pyramis, ſiue planum, ſiue ſolidum, habet acutiſſimum, & in .2. libr. de anima <lb/>pag .215. dicat de die poſſe videri ſtellas in ſpeculo poſito in vaſe aqua pleno, quod <lb/>reuera eſt valde abſurdum. </s> + <s xml:id="s3395" xml:space="preserve">Alios eiuſdem errores tibi non patefacio, quia iam ni-<lb/>hil amplius otij mihi eſt, ſed eos tu ipſe perſpicere, & cognoſcere 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 ſine ijs exploratum mihi erat. </s> + <s xml:id="s3397" xml:space="preserve">Sed conce <lb/>do tantum eſſe dicere vbi eſt maius lumen, minus non diſcerni, quantum inter <lb/>diu ſtellas non videri: </s> + <s xml:id="s3398" xml:space="preserve">immo eſt etiam magis vniuerſale, quia idem multis aliis lumi-<lb/>nibus, præter ea quæ ſunt ſtellarum, ea ratione contingit, quia ingrediente per pupil-<lb/>lam, tam lumine maiori, quam minori, reflexum ipſius maioris in oculo, in ſitu mino <lb/>ris, efficit, vt ipſum minus confundatur, & diſtingui nequeat, quemadmodum aper-<lb/>te cognoſci poteſt in aliquo cubiculo, cuius parietes dealbati ſint, in quo, vnicum <lb/>tantum ſit exiguum foramen, per quod aliqua lumina reflexa ab obiectis extrinſecis <lb/>intra ipſum cubiculum ingredi poſſint, vnde imagines <reg norm="obiectorum" type="context">obiectorũ</reg> in parietibus con-<lb/>ſpiciuntur, ſed ſi per idem foramen ingrederetur etiam primarius radius Solis, re-<lb/>flexus huiuſmodi radij efficeret, vt dictæ imagines, magis aut minus euaneſcerent, <lb/>prout dictus reflexus radij ſolaris, maiori, minoríue vi polleret.</s> + </p> + <p> + <s xml:id="s3399" xml:space="preserve">Ad hoc tamen propoſitum, nolo tibi ſilentio inuolui mirabilem quendam effe-<lb/>ctum eiuſmodi rei. </s> + <s xml:id="s3400" xml:space="preserve">Hoc eſt vt fiat foramen illud rotundum, magnitudinis tamen <lb/>vnius ſpecilli, quod foramen obturetur mediante vno illorum ſpecillorum, quæ pro <lb/>ſenibus (non breuis viſionis) conficiuntur, hoc eſt quorum ambæ ſuperficies con <lb/>uexæ ſunt, non autem concauæ. </s> + <s xml:id="s3401" xml:space="preserve">Deinde opponatur folium album papiri, adeo di <lb/>ſtans à foramine, vt extrinſeca obiecta in eo appareant. </s> + <s xml:id="s3402" xml:space="preserve">Quæ quidem obiecta ſi à <lb/>Sole illuſtrata fuerint, tam clara, & diſtincta videbuntur, vt nihil pulchrius dele-<lb/><reg norm="ctabiliusque" type="simple">ctabiliusq́;</reg> videri poterit, inuerſa tamen. </s> + <s xml:id="s3403" xml:space="preserve">Sed ſi ea directa videre voluerimus. </s> + <s xml:id="s3404" xml:space="preserve">hoc <lb/>optimè faciemus, mediante reflexione alicuius ſpeculi 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 ſequatur ac<unsure/>tatem in qua <lb/>calor viguerit.</head> + <head xml:id="head405" xml:space="preserve">NOBILISSIMO, NECNON INGENIOSISSIMO <lb/>Gabrieli Buſchæ, Mediolanenſi.</head> + <p> + <s xml:id="s3405" xml:space="preserve">QVod dixi hyemem valde frigidam ſequi <reg norm="æſtatem" type="context">æſtatẽ</reg>, in qua calor viguerit, inde na <lb/>ſcitur, quia calor terrę, aquæ, & aeris, non eſt naturalis horum corporum, vt <lb/>eſt frigus, cum calor à Sole procedat, qui ea calefacit ſuo 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 ſortiatur, & cum eandem poſtea deſerit, ad aliam partem æquatoris <reg norm="tranſmigrans" type="context">tranſmigrãs</reg> <lb/>terra ad ſuam qualitatem reddit, maiori cum impetu, eo modo, quo res in mo-<lb/>tibus localibus naturalibus, qui etiam terminos ſibi pręfixos, & conſtitutos exce-<lb/>dunt, hinc etiam hyeme fit glacies, ex calefacta prius aqua, quæ durior poſtea eſt <lb/>atque frigidior alia. </s> + <s xml:id="s3406" xml:space="preserve">Aeſtas etiàm quæ ſequitur hyemem valde frigidam, non <lb/>erit admodum calida, quia Sol inueniens contrarium naturale valde potens, non <lb/>tam facile illud pellere poteſt, vnde etiam ſi in Geminis, Cancro, & Leone, moram <lb/>trahat, non ſufficit tamen ut magnum calorem imprimere poſſit. </s> + <s xml:id="s3407" xml:space="preserve">Vnde ſequitur duas <lb/>æſtates quarum una ſequatur aliam, in eodem loco, uehementi calore præditas eſ-<lb/>ſe 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 ſcribit Tattalea quęſito quinto primi lib. necnon quæſito <lb/>21. et .24. maxima cum ratione eſſe uidetur, non tamen ea quam ipſe in <lb/>quinto profert, quia uerum non eſt, vt quanto aliquid fit calidius, <reg norm="tanto" type="context">tãto</reg> ue-<lb/>hementius attrahat, eo quod ſi etiam huiuſmodi res, in eodem calore, in <lb/>quo ſemel 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 conſequens circumcirca <reg norm="undique" type="simple">undiq;</reg> <lb/>trudit, & partes uaſis debiliores cedunt. </s> + <s xml:id="s3411" xml:space="preserve">dum uerò dictum corpus re frigeratur, re-<lb/>ſtringitur, & dum in unum cogitur, ſi reperiatur in uaſe, quod aer, aqua, aut aliud <lb/>aliquod corpus ingredi nequeat, dictum uas à quo circundatur frangit, ne aliqua <lb/>pars loci uacua remaneat, ſed ſi aliquod corpus ingredi poteſt, illud ipſum ad ſe at-<lb/>trahit, quemadmodum uidere licet in cucurbitulis. </s> + <s xml:id="s3412" xml:space="preserve">Vnde ſequitur eam propoſi-<lb/>tionem, qua dicitur, calidi eſt attrahere, ueram non eſſe, quia ſi 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 ſequitur, ut <lb/>quanto calidior facta fuerit materia aliqua, aliquo loco determinata, redeundo po-<lb/>ſtea ad ſuam priorem frigiditatem, tanto minori loco indigeat, ſimiliter etiam <lb/>è conuerſo accidit, ut quanto frigidior <reg norm="reperitur" type="simple">reꝑitur</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"/> + ſtea egeat ipſa 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">tuasq́;</reg> dubitationes conſideraui, quas quidem non inutiles <lb/>inueni, quo uerò ad primam, dico te oportere illud Theorema ſpeculari or <lb/>dine huiuſmodi methodi, uidelicet quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> habuerimus <reg norm="angulum" type="context">angulũ</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, ſi ab aliquo puncto inter ipſas poſito, puta <var>.u.</var> quod etiam uocetur <var>.i.</var> du <lb/>ctę fuerint .4. lineę ipſis dictis lateribus, hac ſcilicet <reg norm="conditione" type="context">cõditione</reg>, <reg norm="quod" type="wordlist">qđ</reg> duę ex dictis .4. ſint <lb/>parallelę ipfis <reg norm="lateribus" type="simple">lateribꝰ</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> reliquę uero duę <lb/>ſeccent ipſa 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, quę <var>.E.V.</var> ad <var>.E.I.</var> <lb/>Nam ſcimus proportionem <lb/><var>E.i.</var> ad <var>.E.i.</var> eandem eſſe quę <lb/><var>e.i.</var> ad <var>.e.A.</var> ex fimilitudine <lb/><reg norm="triangulorum" type="context">triangulorũ</reg>, ſimiliter <reg norm="propor" type="simple">ꝓpor</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> quę <lb/><var>e.a.</var> ad <var>.e.u.</var> </s> + <s xml:id="s3418" xml:space="preserve">quare aggregata <lb/>ex iſtis erunt inuicem <reg norm="aequa- lia" type="simple">ęqua-<lb/>lia</reg>, uel ſi mauis ex ęqua pro <lb/>portionalitate, quod idem <lb/>eſt, ita ſe 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">Suppoſito nunc plano orizontali <var>.V.E</var>. </s> + <s xml:id="s3420" xml:space="preserve"><reg norm="Altitudineque" type="simple">Altitudineq́;</reg> inacceſſibili <var>.A.E</var>. </s> + <s xml:id="s3421" xml:space="preserve">Duę ue-<lb/>rò ſtationes oculorum ſint <var>.V.</var> et <var>.I.</var> lineę autem uiſuales ſint <var>.V.A.</var> et <var>.I.A</var>. </s> + <s xml:id="s3422" xml:space="preserve">Et quadra-<lb/>tum geometricum ſit <var>.b.e</var>. </s> + <s xml:id="s3423" xml:space="preserve">Supponatur nunc pro prima dubitatione, quod in am-<lb/>babus ſtationibus filum perpendiculare ſeccet latus <var>.e.c.</var> non autem <var>.b.c.</var> (nam quan-<lb/>do in ambabus ſtationibus filum ſecat latus <var>.b.c.</var> nullum tibi dubium oritur, imo ma <lb/>nifeſtè 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> ſumpto <var>.E.</var> pro <var>.b.</var> et <var>.I.V.</var> pro punctis ſecatis à filo, ex <reg norm="euidenti" type="context">euidẽti</reg> ſimilitudine trian-<lb/>gulorum quadrati cum triangulis <var>.A.E.V.</var> et <var>.A.E.I.</var>) Sed cum in pręſenti caſu repe-<lb/>riatur triangulum <var>.u.e.a.</var> minus, in ſtatione remotiori, ſimile triangulo maiori <var>.V.E.<lb/>A.</var> & triangulum maius <var>.i.e.a.</var> proximioris ſtationis, ſimile triangulo minori <var>.I.E.A.</var> <lb/>(quod in alio iam dicto, caſu non accidit, ut unum triangulorum, minus ſcilicet, ſi-<lb/>mile ſit uno triangulorum, maiori ſcilicet & è conuerſo) Non omnino abſque ratio <lb/>ne dubitas quo pacto fieri poſſit ut <var>.a.e.</var> remotioris ſtationis ad <var>.a.e.</var> propinquioris ita <lb/>ſe habeat quema dmodum <var>.I.E.</var> ad <var>.E.V</var>. </s> + <s xml:id="s3424" xml:space="preserve">Quapropter ſi pręcedentem 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 inſpexeris, omnis tua dubitatio euaneſcet. in qua figura apertè vi debis cor-<lb/>reſpondentiam talium triangulorum inter ſe, nec magis, nec minus quam in infra-<lb/>ſcripta hic figura cernere licet, quamuis in hac, triangula quadrati, ſeparata ſint ab <lb/>imaginarijs <var>.A.E.V.</var> et <var>.A.E.I.</var> in ſupradicta 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 ſe 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 proſequi, vt in cognitionem <lb/>peruenias ipſius <var>.I.E.</var> ipſius <var>.A.E.</var> & ipſius <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 ſtatione latus <var>.b.c.</var> in remotiori vero latus <var>.c.e.</var> ſecatur à fi <lb/>lo (pro ſecunda dubitatione) </s> + <s xml:id="s3428" xml:space="preserve">Tunc oportet imaginatione conſiderare latus <var>.b.c.</var> in <lb/>re motiori ſtatione diſtentum eſſe vſque ad filum in puncto <var>.n.</var> vbi videbis <reg norm="triangulum" type="context">triangulũ</reg> <var>.<lb/>u.b.n.</var> ſimile triangulo <var>.A.E.V.</var> ita vt <var>.i.b.a.</var> ſimile ſuo <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> correſpondebit ipſi <var>.A.E.</var> et <var>.b.n.</var> ipſi <var>.E.V.</var> et <var>.b.a.</var> ipſi <var>.E.I.</var> <lb/>quapropter iubeo, vt quæras quantum ſit latus <var>.b.n.</var> ex regula de tribus, dicens ſi <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> ſimile eſt <var>.u.b.n.</var> reperto autem <lb/>latere <var>.b.n.</var> ex quo dempto <var>.b.a.</var> breuioris diſtantię, reſiduum reſpondebit ipſi <var>.I.V.</var> vt <lb/>ſcis, vnde proſequendo operationem tibi cognitam, obtinebis intentum, hoc eſt co <lb/>gnoſces reliqua interualla. </s> + <s xml:id="s3430" xml:space="preserve">Nihil enim miror demonſtrationem Tartaleæ circa hu <lb/>iuſmodi operationem te minime ſatisfeciſſe.</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 propoſitio illius ſcriptoris, de quo nuper <reg norm="mecum" type="context">mecũ</reg> locutus es, vt <lb/>mihi dixiſti, tua ſit, hoc enimego, nec affirmare, nec negare audeo, quamuis in mul <lb/>tis cum tua manuſcripta concordet. </s> + <s xml:id="s3432" xml:space="preserve">Nam ſępæ cogitationes hominum in idem co-<lb/>incidunt, vt pluries cenſuit Ariſto.</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 ſubiecta.</head> + <head xml:id="head411" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3433" xml:space="preserve">AMor erga te meus ſanè ſingularis, nullo modo <reg norm="permittit" type="simple">ꝑmittit</reg>, vt ea quæ Tartaleę ſcri <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 ſunt circa quæ <lb/>dam illius Authoris problemata, quorum primum ab ipſo Tartalea <lb/>ſcriptum in .3. quæſito libr .4. tale eſt, is vult locare .3500. homines, eodem modo, <lb/>quo præſupponit locatos eſſe .1000. ita vt quilibet hominum ordo ſiue vt vulgo di-<lb/>citur filtia ſit .49. quapropter multiplicat quadratum ipſius .49. quod eſt .2401. <reg norm="cum" type="context">cũ</reg> <lb/>numero .3500. propoſito, productum verò .8403500. diuidit per .1000. vt proue-<lb/>niat .8403. cuius radix quadrata eſt .91. pro numero hominum vniuſcuiuſque ordinis <lb/>propoſiti 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> ſit vna filtia ſiue ordo .49. <reg norm="hominum" type="wordlist">hoĩum</reg>, cuius <reg norm="quadratum" type="context">quadratũ</reg> ſit <var>.b.c.</var> 2401. imaginemur <reg norm="etiam" type="context">etiã</reg> re <lb/>ctangulum <var>.A.B.</var> 3500. hominum, quod ſupponemus ſimile 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"/> + ſit quadratum ipſius <var>.D.B</var>. </s> + <s xml:id="s3436" xml:space="preserve">Nunc ſupponendo <var>.A.B.</var> ſimile <var>.a.b.</var> clarum erit ex diffini-<lb/>tione ſimilium figurarum, quod eadem proportio erit <var>.A.D.</var> ad <var>.D.B.</var> quę <var>.a.d.</var> ad <var>.d.<lb/>b.</var> hoc eſt <var>.A.D.</var> ad <var>.D.C.</var> vt <var>.a.d.</var> ad <var>.d.c.</var> hoc eſt <var>.A.B.</var> ad <var>.B.c.</var> vt <var>.a.b.</var> ad <var>.b.c.</var> ex prima <lb/>ſexti, vel .18. ſeu .19. ſeptimi, </s> + <s xml:id="s3437" xml:space="preserve">tunc cum dixerimus ſi <var>.a.b.</var> ita reſpondet ad <var>.b.c.</var> ergo <var>.A.<lb/>B.</var> correſpondet 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. ſexti vel .20. ſeptimi, cuius <lb/>prouentus radix quadrata erit quod quærebatur.</s> + </p> + <p> + <s xml:id="s3439" xml:space="preserve">Sed aliter idem poſſe fieri ſpeculatus ſum, hoc eſt multiplicando numerum .49. <lb/>ordinis .1000. hominum <reg norm="cum" type="context">cũ</reg> radice quadrata numeri .3500. propoſiti, productum ve-<lb/>rò diuidere per radicem quadratam ipſius .1000. vnde prouentus .91. erit numerus <lb/>vnius ordinis .3500. numeri <reg norm="propoſiti" type="simple">ꝓpoſiti</reg>.</s> + </p> + <p> + <s xml:id="s3440" xml:space="preserve">Cuius <reg norm="operationis" type="simple">oꝑationis</reg> ſpeculatio eſt iſta. <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> ſua <lb/>radix et <var>.a.d.</var> rectangulum propoſi-<lb/>tum ipſius .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">rectangulũ</reg> <lb/>ipſius numeri .3500. propoſiti, ſimile <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">Cũ</reg> enim <var>.a.b.</var> æquale ſit <lb/><var>a.d.</var> et <var>.A.B</var>: <var>A.D.</var> <reg norm="tunc" type="context">tũc</reg> <var>.a.c.</var> erit media <lb/>proportionalis inter <var>.a.e.</var> et <var>.e.d.</var> & ſic <lb/><var>A.C.</var> erit etiam media proportiona <lb/>lis inter <var>.A.E.</var> et <var>.E.D.</var> per .16. ſexti, <lb/>ſeu .20. ſeptimi, & quia proporrio. A <lb/>E. ad <var>.E.D.</var> æqualis eſt proportioni <var>.<lb/>a.e.</var> ad <var>.e.d.</var> cum <var>.A.D.</var> ſupponatur ſi-<lb/>mile <var>.a.d.</var> ergo proportio <var>.A.E.</var> ad <var>.A<lb/>C.</var> ęqualis erit proportioni <var>.a.e.</var> ad <var>.a.<lb/>c.</var> quę medietates ſunt <reg norm="totorum" type="context">totorũ</reg> æqua-<lb/>lium, rectè igitur fiet ſi procedamus <lb/>ex regula de tribus, dicendo ſi <var>.a.c.</var> <lb/><reg norm="correſpondet" type="context">correſpõdet</reg> <var>.a.e.</var> tùc <var>.A.C.</var> <reg norm="correſpon" type="context">correſpõ</reg> <lb/>det <var>.A.E.</var> ex ſupradictis .15. ſexti. vel <lb/>20. ſeptimi.</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 ſe <lb/>patet, quod eſt inuenire <reg norm="pauimentum" type="context">pauimentũ</reg> <lb/>ſeu aream quadratam, in qua poſſint <lb/>locari quot homines volueris, ita in <lb/>ter ſe ſiti, ut vnuſquiſque occupet <num value="7">.<lb/>7.</num> pedes ipſius areę in longitudinem <lb/>et .3. per latitudinem à lateribus.</s> + </p> + <p> + <s xml:id="s3445" xml:space="preserve">Seu ex propoſito hominum nume <lb/>ro inuenire numerum ipſorum loca-<lb/>bilem in aliqua area quadrata, ita, <lb/>vt vnuſquiſque occupet .21. pedes <lb/>quadratos ipſius 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 eſt <reg norm="multiplicando" type="context">multiplicãdo</reg> radicem quadratam pro-<lb/>poſiti numeri hominum per .21. & productum item multiplicando per eandem radi <lb/>cem, & huiuſmodi 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 huiuſmodi numeri radicem <lb/>quadratam hoc eſt .60. per .21. hoc <lb/> + <anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/> + eſt per productum quod fit ex .7. <reg norm="cum" type="context">cũ</reg> <lb/>3. & reſultabit nobis .1260. quod ſi <lb/>multiplicabitur, per .60. hoc eſt per <lb/>eandem radicem, reſultabit nobis <num value="75600">.<lb/>75600.</num> cuius producti radix qua-<lb/>drata eſt ferè .275. qua diuiſa 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 eſt iſta, cogitemus nu <lb/>merum .3600. propoſitum eſſe qua <lb/>dratum <var>.a.b.</var> (ſed non areæ) cuius ra <lb/>dix .60. ſit <var>.a.c.</var> & quia hic numerus <num value="60">.<lb/>60.</num> intelligitur eſſe hominum, quo-<lb/>rum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> occupat .21. pedes <lb/>quadratos ſuperficiales ex ſuppoſi-<lb/>to, </s> + <s xml:id="s3449" xml:space="preserve">& propterea multiplicatur, 60. <lb/>cum .21. vnde nobis veniat .1260. <lb/>quadrati ſuperficiales pro vnoquo-<lb/>que ordine, & <reg norm="quia" type="simple">ꝗa</reg>.b.c. vt. latus qua-<lb/>drati <var>.a.b.</var> habet tot ordines homi-<lb/>num ſimiliter, hoc eſt .60. igi-<lb/>tur multiplicando .60. cum .1260. <lb/>habebimus totalem ſuperficiem <var>.a.<lb/>b.</var> ex .75600. quadratis ſuperficiali-<lb/>bus, quæ quadrata imaginemur lo-<lb/>cata eſſe in quodam totali quadra-<lb/>to, quod ſit <var>.e.f.</var> cuius radix ſit <var>.e.</var> g <num value="275">.<lb/>275.</num> pedum qui diuidantur per .3. <lb/>hoc eſt per numerum pedum latitu-<lb/>dinis & prouenient nobis .91. pro <lb/>numero <reg norm="hominum" type="context">hominũ</reg> <reg norm="vniuſcuiuſque" type="simple">vniuſcuiuſq;</reg> ordi-<lb/>nis, diuidendo poſtea latus <var>.f.g.</var> per <lb/>numerum ſpatij inter vnum, & <reg norm="alium" type="context">aliũ</reg> <lb/>ordinem, quod eſt .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 eſt multiplicando <reg norm="nume- rum" type="context">nume-<lb/>rũ</reg> <reg norm="propoſitum" type="context">propoſitũ</reg> <reg norm="hominum" type="context">hominũ</reg> <reg norm="cum" type="context">cũ</reg> rectangulo .21. vnde venietnobis <reg norm="productum" type="simple context">ꝓductũ</reg> .75600 quod pro <lb/><reg norm="ductum" type="context">ductũ</reg> ſi accipiemus vt <reg norm="quadratum" type="context">quadratũ</reg>, cuius radix erit .275. quæ diuidatur <reg norm="per" type="simple">ꝑ</reg> .3. habebi-<lb/>mus <reg norm="propofitum" type="simple context">ꝓpofitũ</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">qđ</reg> loco <reg norm="multiplicandi" type="context">multiplicãdi</reg> <var>.a.c.</var> (hoc eſt <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">productũ</reg> etiam multiplicare per <var>.b.c.</var> (hoc eſt .60.) breuius erit <lb/>multiplicare totum numerum .3600. per .21. cętera verò facere, vt diximus.</s> + </p> + <p> + <s xml:id="s3452" xml:space="preserve">Sed <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iſtarum operationum, aliquid imperfectionis patitur, eo quod <reg norm="cum" type="context">cũ</reg> <lb/>aliquis cuperet quadratum perfectum ſuperficiale habere, <reg norm="abſque" type="simple">abſq;</reg> aliquo defectu, vel <lb/>exceſfu, aliquid aliud adhuc facere oporteret, hoc eſt, inuentum cum fuerit quadra <lb/>tum <var>.e.f.</var> cum ſuis radicibus <var>.e.g.</var> et <var>.g.f.</var> pedum .275. vnaquaque, vt in dicto exemplo <lb/>factum eſt, oportebit <reg norm="numerum" type="context">numerũ</reg> quærere minorem ipſo .275. ſed proximiorem men-<lb/>furabilem ab .3. & ab .7. quod facilè fiet ſi diuiſerimus .275. per .21. detrahendo fra-<lb/>cta diuiſionis ab ipſo .275. quæ quidem fracta in hoc exemplo ſunt .2. vnde remane-<lb/>bit .273. pro numero laterum quadrati ſuperficialis, in quo poſſent locari .3549. ho-<lb/>mines, eo ordine quo ſupra dictum eſt, quorum ſcilicet vnuſquiſque obtineat .21. <lb/>pedes ſuperſiciales.</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 Muſico celeberrimo.</head> + <p> + <s xml:id="s3453" xml:space="preserve"><emph style="sc">OPinio</emph> Hectoris Euſonij Cypriane mi dilectiſſime, vera non eſt, quod ali <lb/>quisrectè poſſit intelligere rationes conſonantiarum muſicæ, ablque co <lb/>gnitione illarum mediante ipſo ſenſu, imo nemo <reg norm="pont" type="context">põt</reg> calere <reg norm="theoriam" type="context">theoriã</reg> mu <lb/>ſices, niſi aliquo <reg norm="mon" type="context">mõ</reg> verſatus ſit in praxi. </s> + <s xml:id="s3454" xml:space="preserve"><reg norm="Quando" type="wordlist">Qũo</reg> enim cognoſci <reg norm="poterunt" type="context">poterũt</reg> <lb/>quid nam ſint diapaſon, diapente, diateſſeron, ditonus, ſemiditonus, hexacordum <lb/>maius, aut minus, & conſonantiæ ex ijs cum diapaſon compoſitæ, abſque earum <lb/>praxi? </s> + <s xml:id="s3455" xml:space="preserve">vnde ſequetur <reg norm="neque" type="simple">neq;</reg> etiam cognoſci poſſe interualla diſſonantia. </s> + <s xml:id="s3456" xml:space="preserve">Et purus <lb/>practicus non intelliget quid ſit octaua, quinta, quarta, tertia maior, tertia minor, <lb/>ſexta maior, ſexta minor, decima maior, decima minor, vndecima, duode-<lb/>cima, decimatertia maior, aut minor, aut decimaquinta, & aliæ, ita vt ad <lb/>comparandam perfectionem muſicæ neceſſarium ſit, & thęoriam & praxim ad-<lb/>diſcere. </s> + <s xml:id="s3457" xml:space="preserve">Cum pręterea Ludouicus Folianus apertè monſtrarit (etiam ſi id à diato-<lb/>nico ſintono Ptolomei deſumpſerit) reperiri duos tonos, maiorem, & minorem, id-<lb/>eſt ſeſquioctauum, & ſeſquinonum, & tria ſemitonia, maius, minus, & mini-<lb/>mum, ideſt ſeſquiquintumdecimum, qui eſt maius, ſeſquiuigeſimum quartum id-<lb/>eſt minimum, & mediocre, vt .27. ad .25. quæ proportio ſuperbipartiens vigeſi-<lb/>maſquintas appellatur, & cum cognouerit ſemiditonum conſonantem eſſe ſeſqui-<lb/>quintum, ditonum ſeſquiquartum, & hexachordum minus, vt .8. ad .5. quæ propor-<lb/>tio dicitur ſupertripartiens quintas, & hexachordum maius, vt .5. ad .3. hęc autem vo <lb/>catur ſuperbipartienstertias; </s> + <s xml:id="s3458" xml:space="preserve">omnium ſimplicium conſonantiarum cognitioni, ex-<lb/>tremam impoſuit manum. </s> + <s xml:id="s3459" xml:space="preserve">Et quia tibi etiam oſtendere promiſi 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 ſeruari, ideo ad te mitto ſeptem hic ſubſcripta exempla, in <lb/>quorum primo, & ſecundo, inter dieſim, et <var>.b.</var> in ſuperiori, agnoſces interuallum mi <lb/>nimi ſemitonij, & ſi ibi ſit dieſis, 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 dieſim et <var>.b.</var> ſit ſemitonium minimum, facilè agnoſces <lb/>ſi ſubtraxeris <reg norm="decimam" type="context">decimã</reg> <reg norm="minorem" type="context">minorẽ</reg> à maiori, <reg norm="quam" type="context">quã</reg> facit ſuperius <reg norm="cum" type="context">cũ</reg> inferiori, ideſt <reg norm="cum" type="context">cũ</reg> baſſu.</s> + </p> + <p> + <s xml:id="s3461" xml:space="preserve">Qua quidem modulatione tu etiam vſus 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 ſermo-<lb/>ne compoſitis, quæ Moreta vocantur aliquando vſus ſum.</s> + </p> + <p> + <s xml:id="s3464" xml:space="preserve">Sed in tertio exemplo inuenies ſemitonium maius, neceſſariò genitum in ſupe-<lb/>riori, ſi ſextam maiorem cum baſſu eſſicere volueris, quia tenor, à ditono cum <lb/>ſuperiori ad diapentem, & ad vniſonum cum baſſu procedit, vbi quieſcit, progre-<lb/>diendo poſtea baſſus ad ſemiditonum cum tenore, </s> + <s xml:id="s3465" xml:space="preserve">tunc ſi à proportione huius ſep-<lb/>timæ, quæ eſt vt .9. ad .5. hoc eſt ſuperquadripartiensquintas demptum fuerit hexa-<lb/>chordum maius, ſeu ſexta maior, quæ eſt vt .5. ad .3. remanebit proportio .27. ad .25. <lb/>quæ maior eſt quam .32. ad .30.</s> + </p> + <p> + <s xml:id="s3466" xml:space="preserve">In quarto <reg norm="exemplo" type="context">exẽplo</reg> habebis ſemitonium minus in ſuperiori, quod quidem remanet <lb/>ex ſubtractione ditoni <reg norm="conſonantis" type="context context">cõſonãtis</reg> ab diateſſaron <reg norm="compręhenſa" type="context">cõpręhenſa</reg> à ſuperiori cum tenore.</s> + </p> + <p> + <s xml:id="s3467" xml:space="preserve">In quinto exemplo videbis tonum minorem, & tonum maiorem ſucceſſiuè vnum <lb/>poſt alium in tenore, detrahendo primo <reg norm="ſemiditonum" type="context">ſemiditonũ</reg> à diateſſaron, quod ſuperius fa-<lb/>cit cum tenore, vel detrahendo diapente ab hexachordo maiori, quod facit tenor <lb/>cum baſſu, vnde remanet tonus minor ſeſquinonus, detrahendo poſtea diateſſaron <lb/>à diapente, quod ſuperius facit cum tenore, remanebit tonus maior ſeſquioctauus.</s> + </p> + <p> + <s xml:id="s3468" xml:space="preserve">In ſexto exemplo deinde videbis tenorem aſcendere per duos tonos minores ſuc <lb/>ceſſiuè vnum poſt alium in tenore, ſi <reg norm="demp" type="context">dẽp</reg> ſeris <reg norm="ſemiditonum" type="context">ſemiditonũ</reg> à diateſſaron <reg norm="cum" type="context">cũ</reg> ſuperiori.</s> + </p> + <p> + <s xml:id="s3469" xml:space="preserve">In .7. <reg norm="exemplo" type="context">exẽplo</reg> demum videbis <reg norm="ſuperiorem" type="context">ſuperiorẽ</reg> aſcendere per duos tonos maiores ſucceſ-<lb/>ſiuè <reg norm="vnum" type="context">vnũ</reg> poſt <reg norm="alium" type="context">aliũ</reg>, ſi dempſeris diateſſaron à diapente, quod facittenor <reg norm="cum" type="context">cũ</reg> ſuperiori.</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 ſubiecto.</head> + <head xml:id="head415" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3470" xml:space="preserve">QVod aliàs tibi dixi, verum eſt, quod neceſſarium nullo modo ſit, vt modulan-<lb/>do, deſinat cantilena in eodem tono (quod Græci phthongum appel-<lb/>lant) à quo incępit. </s> + <s xml:id="s3471" xml:space="preserve">immo neceſſariò ſemper ferè, altius, aut depræſ-<lb/>ſius terminatur, per differentiam alicuius interualli æqualis, vel multiplicis ipſi com <lb/>mati ſeſquioctuageſimæ, quod quidem comma, quamuis cantabile non ſit, inſenſi-<lb/>biliter tamen generatur, & toties ab aliqua parte ipſius cantilenæ poſſet <reg norm="dictum" type="context">dictũ</reg> com-<lb/>ma gcnerari, verſus acutum, vel graue, quod in fine ipſius cantilenę, vocis phtongus <lb/>reperiatur diſtans à primo per interuallum alicuius toni ſeſquinoni, ſeu ſeſquioctaui <lb/>plus, minúsue, vt in ſubſcripto exemplo clarè videre potes in prima figura, vbi ſu-<lb/>perius à <var>.g.</var> primę cellulæ ad <var>.g.</var> ſecundæ, intereſt vnum <reg norm="comma" type="context">cõma</reg>, eo quod progrediens <lb/>ſuperius in prima cellula ipſius cantilenæ à quarta ad quintam cum tenore, aſcendit <lb/>per tonum ſeſquioctauum, à prima cellula deinde ad ſecundam, tenor aſcendit ſimi-<lb/>liter per tonum ſeſquioctauum cum tranſeat à quinta ad quartam, quod facit cum <lb/>ſuperiori, in ſecunda cellula poſtea, cum ſuperius deſcendat à maiori ſexta ad quin <lb/>tam, quod facit cum baſſu, ſeu à quarta ad tertiam minorem, quod facit cum teno-<lb/>re, </s> + <s xml:id="s3472" xml:space="preserve">tunc deſcendit per tonum ſeſquinonum, 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, ſed reperitur per <reg norm="vnum" type="context">vnũ</reg> <reg norm="coomma" type="context">coõma</reg> altius, <reg norm="quod" type="wordlist">qđ</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/><reg norm="comma" type="context">cõma</reg> eſt differentia inter <reg norm="tonum" type="context">tonũ</reg> <reg norm="ſeſquioctauum" type="context">ſeſquioctauũ</reg> & <reg norm="ſeſquinonum" type="context">ſeſquinonũ</reg>, vt alias tibi <reg norm="demonſtraui" 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 tranſeat à tertia minori ad quartam, quod facit cum ſuperiori, deſcendit per <lb/>tonum ſeſquinonum, vnde in tertia cellula altius remanet quam in prima per <reg norm="vnum" type="context">vnũ</reg> <lb/>comma, in qua tertia cellula, cum iterum tranſeat ſuperius à quarta ad quintam, <reg norm="quod" type="simple">ꝙ</reg> <lb/>facit cum tenore, eleuatur per tonum ſeſquioctauum, proſequendo deinde tali ordi <lb/>ne, vidc<unsure/>bis in quarta cellula cantilenam auctam per duo commata, in ſexta, <reg norm="autem" type="wordlist">aũt</reg> cel-<lb/>lula per tria commata, in octaua verò per .4. commata, vnde hac merhodo, ſi can-<lb/>tilena prolixior debito eſſet, vel ſi 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 ſuperant tonum ſeſquinonum, & ſi eſſent .10. commata<unsure/> ſuperarent tonum <lb/>ſeſquioctauum, eo quod aggregatum ex .9. commatibus continetur ſub iſtis duobus <lb/>terminis hoc eſt .150094635296999121. et .134217728000000000. quæ qui-<lb/>dem proportio maior eſt proportione ſeſquinona, ſumma verò .10. commatum con <lb/>tinetur ſub .12157665459056928801. et .10737418240000000000. quæ pro <lb/>portio maior eſt tono ſeſquioctauo, quod autem dico de aſcenſu cantilenæ, idem aſ-<lb/>ſero de eiuſdem deſcenſu, & hoc non tantum per interuallum illius commatis, quod <lb/>eſt differentia toni maioris à minori, ſed etiam per illud quod eſt differentia ſemito <lb/>nij maioris à minori, vt in ſecundo exemplo hic ſubſcripto videre eſt in deſcenſu <lb/>cantilenæ per comma & comma, vt differentia inter ſemitonia maiora<unsure/> & minora, <lb/>vbi in prima cellula diſcedens baſſus à quinta cum ſuperiori, & ab vniſono cum te-<lb/>nore deſcendens ad tertiam minorem cum ipſo tenore, facit cum ſuperiori <reg norm="ſeptimam" type="context">ſeptimã</reg> <lb/>maiorem, quæ eſt vt .9. ad .5. ſuperquadripartiensquintas ſcilicet, à qua diſcedens <lb/>poſtea ſuperius, vt faciat cum baſſu ſextam maiorem, deſcendit per ſemitonium ma <lb/>ius, à qua ſexta maiori deſcendens baſſus, & aſcendens per quartam, efficit cum di-<lb/>cto ſuperiori <reg norm="tertiam" type="context">tertiã</reg> maiorem, à qua diſcedens ſuperius, vt efficiat quartam cum ipſo <lb/>baſſu (qui quidem baſſus tranſit in tenorem) aſcendit per ſemitonium 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 deſcendit, & ſic toties facere poſſet, vt poſtremo 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 ſupradictum eſt, clrca inſtrumenta artificialia non accidit, qua <lb/>propter organa, & clauicimbula concordantur certo quodam ordine, ita vt omnes <lb/>conſonantiæ, excepta diapaſon, ſeu octaua, ſint imperfectæ, hoc eſt, aut diminutę, <lb/>aut ſuperantes à inſto, vt exempli gratia, omnes quintæ ſunt diminutæ, quartæ verò <lb/>ſunt exceſſiuę, quod quidem fit, vt tertiæ, & ſextæ, non multum auribus diſſonent, <lb/>eo quod ſi quintæ omnes, & quartæ, perfectæ eſſent, </s> + <s xml:id="s3476" xml:space="preserve">tunc omnes ſextę, & tertiæ in-<lb/>tollerabiles eſſent, & à perfectis differrent per vnum comma, quod manifeſtum no-<lb/>bis erit hoc modo, accipiamus tres diapentes, ſeu quintas, conſequenter ſucceſſiuas <lb/>vnam poſt aliam, hoc eſt tres proportiones ſeſquialteras, quarum aggregatum erit <lb/>vt .27. ad .8. quæ proportio, dicitur tripla ſupertripartiensoctauas, & 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 ſecundo ela <lb/>mi, </s> + <s xml:id="s3477" xml:space="preserve">tunc talis tertiadecima valde odioſa eſſet ſenſui auditus, à qua, ſi dempta ſuerit <lb/>diapaſon, ſeu octaua, remaneret quoddam hexachordum maius, ſeu ſexta maior, au-<lb/>ribus valde inimica, ſub proportione .13. ad 8. ſed hæc proportio differret à propor <lb/>tione ſuperbipartientetertias perfecti hexachordi maioris, hoc eſt ſextæ maioris <lb/>conſonantis, per proportionem ſeſquioctuageſimam, hoc eſt per vnum comma, <lb/>quod quidem eſt etiam differentia aggregati trium ſeſquialterarum, à tertiadeci-<lb/>ma maiori conſonanti, hoc eſt exceſſus proportionis triplæ ſupertripartientis octa-<lb/>uas, ſupra triplam ſeſquitertiam, quæ eſt ſumma ipſius duplæ cum ſuperbi partien-<lb/>tetertias.</s> + </p> + <p> + <s xml:id="s3478" xml:space="preserve">A tali ſumma igitur trium ſeſquialterarum efficitur tertiadecima maior diſſonans <lb/>excedens conſonantem per vnum comma (cuius proportio eſt .81. ad .80.) quæ con-<lb/>fonans continetur in proportione .10. ad .3. vt ſupra dixi.</s> + </p> + <p> + <s xml:id="s3479" xml:space="preserve">Hæcigitur eſt vera ratio, propter quam debemus comma diſtribuere in organis <lb/>& clauicymbalis, cum ab aggregato trium quintarum producatur talis exceſſus ſu-<lb/>pra perfectam, ſeu conſonantem tertiamdecimam maiorem, quod quidem aggre-<lb/>gatum, cum demptum fuerit à quintadecima, relinquet nobis tertiam minorem <lb/>diſſonantem, & mancam, per eundem exceſſum à conſonanti. </s> + <s xml:id="s3480" xml:space="preserve">quæ quidem tertia <lb/>minor diſſonans ſubtracta à diapente ſeu quinta perfecta, relinquet nobis tertiam <lb/>maiorem diſſonantem, qu@ conſonantem excedit per eundem exceſſum comma-<lb/>tis, & hæc demum tertia maior diſſonans, dempta ex diapaſon, ſeu octaua, relin-<lb/>quet nobis hexachordum minus, hoc eſt ſextam minorem diſſonantem, & muti-<lb/>lam à conſonanti per eundem exceſſum commatis. </s> + <s xml:id="s3481" xml:space="preserve">De huiuſmodi verò commatis <lb/>diſtributione doctiſſimè ſcripſit Excellentiſſimus Zarlinus in ſecunda parte Inſtitu-<lb/>tionum Harmonicarum.</s> + </p> + <p> + <s xml:id="s3482" xml:space="preserve">Sed quia ſenſus auditus non poteſt exactè cognoſcere debitam quantitatem ex-<lb/>ceſſus, vel defectus, intendendo vel remittendo chordas inſtrumentorum, ideo hanc <lb/>viam ſequutus ſum.</s> + </p> + <p> + <s xml:id="s3483" xml:space="preserve">Sit exempli gratia, hic ſubſcriptus ordo lignorum tangentium ſeu pinarum inci-<lb/>piens ab <var>.G.</var> deſinens ad <var>.g.</var> ita quod inter ipſos terminos ſit ea conſonantia quæ vo-<lb/>catur vigeſimaſecunda, quæro primum <var>.b.</var> inter <var>.D.E.</var> quod eſt nigrum ipſius Ela-<lb/>mi grauiſſimum, quod groſſo modo facio conſonans cum <var>.G.</var> grauiſſimo per ſex-<lb/>tam <reg norm="minorem" type="context">minorẽ</reg>, deinde <reg norm="cum" type="context">cũ</reg> ipſo pt<unsure/>imo <var>.b.</var> ipſius elami concordo ſuum octauum & quin-<lb/>tumdecimum, quo perfectius poſſum, deinde accipio <var>.b.</var> molle ſecundum ipſius. b <lb/>fabmi quod concordo cum <var>.b.</var> primo ipſius Elami per quintam imperfectam, dein-<lb/>de cum hoc <var>.b.</var> ſecundo ipſius bſabmi concordo ſecundum <var>.f.</var> per quintam ſimiliter <lb/>imperfectam, cum quo <var>.f.</var> poſtea concordo tertium <var>.c.</var> per ſimilem quintam, quem <lb/>tertium <var>.c.</var> poſtea confero <reg norm="cum" type="context">cũ</reg> ſecundo <var>.b.</var> ipſius elami, ita quod inter ſe conſonent per <lb/>ſextam maiorem tolerabilem, & ſi ſic 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"/> + eſt <var>.b.</var> ſecundum ipſius bfabmi, f. ſecundum, et <var>.c.</var> tertium, ſed ſi dictum tertium <var>.c.</var> <lb/>valde diſſonans eſſet cum <var>.b.</var> ſecundo ipſius elami, </s> + <s xml:id="s3484" xml:space="preserve">tunc ipſum <var>.c.</var> intendo, aut re-<lb/>mitto, quouſque aliquo modo ſit conſonans per ſextam maiorem aliquantulum ex <lb/>ceſſiuam cum <var>.b.</var> ſecundo ipſius elami, cum quo poſtea <var>.c.</var> conſonare aliquantulum fa <lb/>cio <var>.f.</var> ſecundum per quintam defectiuam, & cum hoc<unsure/> demum <var>.b.</var> ſecundum ipſius <lb/>bfabmi, quo facto concordo ſecundum <var>.c.</var> cum tertio per octauam, cum quo ſecun-<lb/>do <var>.c.</var> poſtea concordo tertium <var>.g.</var> per talem quintam, quod ipſum tertium <var>.g.</var> cum ſe-<lb/>cundo <var>.b.</var> ipſius bfabmi conſonet tolerabiliter per ſextam maiorem <reg norm="aliquantulum" type="context">aliquãtulum</reg> ex-<lb/>ceſſiuam, </s> + <s xml:id="s3485" xml:space="preserve">deinde cum iſto tertio <var>.g.</var> concordo tertium <var>.d.</var> per talem quintam, ita quod <lb/>ipſum <var>.3.d.</var> concordet tolerabiliter cum <var>.2.f.</var> per ſextam maiorem exceſſiuam, poſtea <lb/>cum hoc <var>.3.d.</var> concordo <var>.2.d.</var> per octauam perfecte, cum quo <var>.2.d.</var> poſtea concordo <var>.<lb/>3.a.</var> per quintam, vt in alijs <reg norm="factum" type="context">factũ</reg> eſt, ita vt <reg norm="cum" type="context">cũ</reg> <var>.2.c.</var> conſonet talis ſexta maior, vt ſupra <lb/>dictum eſt, cum quo <var>.3.a.</var> poſtea concordo <var>.3.e.</var> per quintam, vt dictum eſt, ita quod <lb/>cum <var>.3.g.</var> faciat ſextam maiorem vt ſupra, poſtea 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 eſt, ita quod <reg norm="cum" type="context">cũ</reg> <lb/><var>2.d.</var> faciat ſextam maiorem ſimilem alijs ſuperius dictis, cum quo <var>.b.</var> quadrato tertio <lb/>concordo tertium nigrum ipſius <var>.f.</var> per quintam, ita quod cum <var>.3 a.</var> faciat ſextam ma-<lb/>iorem, vt ſupra, 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 ſextam dictam, demum <reg norm="cum" type="context">cũ</reg> <lb/>hoc concordo <var>.4.g.</var> nigrum per quintam, ita quod faciat cum <var>.3.b.</var> quadrato ſextam <lb/>dictam, & ſic ad vltimam quintam peruenio, ſupra quod <var>.g.</var> nigrum nulla quinta am-<lb/>plius reperitur, poſtea cum iſtis 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 eſt, quod perſectiores quæque conſonan <lb/>tiæ, ita in harmonica diuiſione ſibi inuicem conueniant, vt dia paſon cum diapente, <lb/>cum diapaſondiapente, cum ditono, cum hexachordo maiori cum bisdiapaſon, <reg norm="cum" type="context">cũ</reg> <lb/>decimaleptima maiori. </s> + <s xml:id="s3487" xml:space="preserve">Nam in ipſa diapaſon, harmonicè <reg norm="locantur" type="context">locãtur</reg> diapente in par <lb/>te grauiori, & diateſlaron in acutiori. </s> + <s xml:id="s3488" xml:space="preserve">In diapente verò harmonicè locantur ditonus <lb/>in parte grauiori, & ſemiditonus 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 diapaſondia-<lb/>pente, harmonic<unsure/>è locantur diapaſon in parte grauiori, & diapente in acutiori. <lb/></s> + <s xml:id="s3492" xml:space="preserve">In bisdiapaſon, ha@monicè locantur decima maior in parte grauiori & hexachor-<lb/>dum minus in acutiori. </s> + <s xml:id="s3493" xml:space="preserve">In decimaſeptima maiori, harmonicè locantur diapaſondia-<lb/>pente in parte grauiori, & hexachordum maius in parte acutiori. </s> + <s xml:id="s3494" xml:space="preserve">Ita quod ronus <lb/>ſeſquioctauus in ditono, proportionalis eſt ipſi ditono in diapente. </s> + <s xml:id="s3495" xml:space="preserve">Tonus verò ſeſ-<lb/>quinonus in ipſo ditono, proportionalis eſt triemitonio, vel ſeſquitonio ſeu ſemidi-<lb/>tono (quod idem eſt) in diapente. </s> + <s xml:id="s3496" xml:space="preserve">Ditonus autem in diapente, proportionalis eſt <lb/>ipſi diapente in diapaſon. </s> + <s xml:id="s3497" xml:space="preserve">Seſquitonus verò in diapente, proportionalis eſt diateſ-<lb/>ſaron in diapaſon. </s> + <s xml:id="s3498" xml:space="preserve">Et ſic de ſingulis. </s> + <s xml:id="s3499" xml:space="preserve">Ita quod tonus ſeſquioctauus in ditono, dito-<lb/>mus in diapente, diateſſaron in hexachordo maiori, diapente in diapaſon, diapaſon <lb/>in diapaſondiapente, decimamaior in bisdiapaſon, diapaſondiapente 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"/> + ſeptima maiori, omnia ſibi inuicem ſunt proportionalia, idem etiam dico de reli-<lb/>quis partibus, cum relatæ fuerint ad ſua tota.</s> + </p> + <p> + <s xml:id="s3500" xml:space="preserve">Nec alienum mihi videtur à propoſito inſtituto, ſpeculari modum generationis <lb/>ipſarum ſimplicium <reg norm="conſonantiarum" type="context">conſonantiarũ</reg>; </s> + <s xml:id="s3501" xml:space="preserve">qui quidem modus fit ex quadam æquatione per <lb/>@uſſionum, ſeu æquali concurſu vndarum aeris, vel conterminatione earum.</s> + </p> + <p> + <s xml:id="s3502" xml:space="preserve">Nam, nulli dubium eſt, quin vniſonus ſit prima principalis <reg norm="audituque" type="simple">audituq́</reg> amiciſſima, <lb/>nec non magis propria conſonantia; </s> + <s xml:id="s3503" xml:space="preserve">& ſi intelligatur, vt punctus in linea, vel vnitas <lb/>in numero, quam immediate ſequitur diapaſon, ei ſimillima, poſt hanc verò diapen <lb/>te, <reg norm="cæteræque" type="simple">cæteræq́;</reg>. </s> + <s xml:id="s3504" xml:space="preserve">Videamus igitur ordinem concurſus percuſſionum terminorum, ſeu <lb/>vndarum aeris, vnde ſonus generatur.</s> + </p> + <p> + <s xml:id="s3505" xml:space="preserve">Concipiatur igitur mente monochordus, hoc eſt chorda diſtenta, quæ cum diuiſa <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 ſonum pro-<lb/>feret, & ambæ formabunt vniſonum, 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 ſimul eant, & æqua <lb/>liter concurrant, abſque ulla interſectione, vel fractione illarum inuicem.</s> + </p> + <p> + <s xml:id="s3508" xml:space="preserve">Sed cum ponticulus ita diuiſerit chordam, vt relicta ſit eius tertia pars ab vno la-<lb/>tere, ab alio vero, duę tertię, </s> + <s xml:id="s3509" xml:space="preserve">tunc maior pars, dupla erit minori, & <reg norm="ſonabunt" type="context">ſonabũt</reg> ipſam dia <lb/>paſon conſonantiam, percuſſiones vero terminorum ipſius, tali proportione ſe inui-<lb/>cem habebunt, ut in qualibet ſecunda percuſſione minoris portionis ipſius chordæ, <lb/>maior percutiet, ſeu concurret cum minori, eodem temporis inſtanti, cum ne-<lb/>mo ſit qui neſciat, quod quo longior eſt chorda, etiam tardius moueatur, </s> + <s xml:id="s3510" xml:space="preserve">quare <lb/>cum longior dupla ſit breuiori, & eiuſdem intenſionis 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 diuiſerit chordam, ut ab uno latere relinquantur duæ <lb/>quintæ partes, ab alio verò tres quintæ, ex quibus partibus generatur conſonantia <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 eſt tempus maioris interualli ad tempus minoris <lb/>erit <reg norm="ſeſquialterum" type="context">ſeſquialterũ</reg> quare non <reg norm="conuenient" type="context">cõuenient</reg> ſimul, niſi 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 ipſius chordæ <lb/>in numerum interuallorum motus ipſius portionis, æquale erit producto numeri <lb/>portionis maioris in numerum interuallorum ipſius maioris portionis; </s> + <s xml:id="s3516" xml:space="preserve">quæ quidem <lb/>producta ita ſe habebunt, vt in diapaſon, ſit binarius numerus; </s> + <s xml:id="s3517" xml:space="preserve">in diapente verò <lb/>ſenarius; </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 ſemiditono tricenarius, demum in hexachordo minori quadra <lb/>genarius: </s> + <s xml:id="s3520" xml:space="preserve">qui quidem numeri non abſque mirabili analogia conueniunt inuicem.</s> + </p> + <p> + <s xml:id="s3521" xml:space="preserve">Voluptas autem, quam auditui afferunt conſonantiæ fit, quia leniuntur ſenſus, <lb/>quemadmodum <reg norm="contra" type="context">cõtra</reg>, dolor qui à diſſonantijs oritur, ab aſperitate naſcitur, id quod <lb/>facilè vide<unsure/>re poteris cum conchordantur organorum fiſtulæ.</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">Franciſco Ferrario Anciſa Iuriſconſulto <reg norm="ſenatorique" type="simple">ſenatoriq́ꝫ</reg> apud <lb/>ſubalpinos 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 comparaſti loquuti ſu <lb/>mus. </s> + <s xml:id="s3523" xml:space="preserve">Cum vero nonnunquam de mirabili iuſtitiæ commutatíuæ inſtitu <lb/>to non ingratus incidiſſet ſermo, dixi modum, quo formam ſuam à proportionali-<lb/>tate arithmetica diſiuncta, & non a coniuncta deſumat, à nemine literis proditum <lb/>eſſe, libet autem nunc per otium latius explicare. </s> + <s xml:id="s3524" xml:space="preserve">dixi enim à diſiuncta, & non con-<lb/>iuncta proportionalitate, quia in coniuncta, ſeu continua nullo pacto fieri poteſt <lb/>talis commutatio, cum ſemper quatuor terminos ad minus tranſeat, vt nunc vide-<lb/>bimus.</s> + </p> + <p> + <s xml:id="s3525" xml:space="preserve">Exempli gratia, Petrus ex ſuis bonis tribuat Ioanni aliquid valoris quinquagin <lb/>ta aureorum.</s> + </p> + <p> + <s xml:id="s3526" xml:space="preserve">Vnde priuſquam Ioannes aliquid ex ſuis bonis retribuat Petro, bona ipſius Pe-<lb/>tri diminuta erunt per quinquaginta aureos, bona verò ipſius Ioannis, aucta toti-<lb/>dem aureis.</s> + </p> + <p> + <s xml:id="s3527" xml:space="preserve">Ecce nunc quo pacto conftituti ſunt .4. termini in proportionalitate aritmetica, <lb/>per quos ſit talis permutatio, ſed nondum æquata, niſi 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 ſignificentur bona Ioannis <var>.B.</var> vero Petri, prius quam Petrus aliquid ex bo <lb/>nis ſuis tribuat Ioanni. </s> + <s xml:id="s3530" xml:space="preserve">Tunc Petrus ſecat partem vnam ex <var>.B.</var> <reg norm="eamque" type="simple">eamq́;</reg> dat ipſi Ioan-<lb/>ni, vnde ipſi Petro remanet <var>.D</var>. </s> + <s xml:id="s3531" xml:space="preserve">Ioanni autem <var>.C.</var> quatuor igitur termini conſtituti <lb/>ſunt <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">aũt</reg> ſecundus, ſed <lb/>B. et <var>.A.</var> ſunt in ſua naturali mediocritate abſque defectu vel exceſſu ſui ipſius. </s> + <s xml:id="s3532" xml:space="preserve">Non <lb/>ita tamen ſe 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 iſti .4. termini conſtituti ſunt in ipſa aritmetica proportionalitate, nam <lb/>eadem quantitate qua <var>.D.</var> diminuta eſt à <var>.B.</var> eadem <var>.C.</var> aucta eſt ſupra <var>.A</var>.</s> + </p> + <p> + <s xml:id="s3534" xml:space="preserve">Sed quia <var>.B.</var> et <var>.A.</var> tantummodo iuſti ſunt termini <var>.C.</var> uerò et <var>.D.</var> iniuſti, vt ad ſuam <lb/>priorem æqualitatem reuertantur, oportebit ex <var>.C.</var> ſecare aliquam partem æqualis <lb/>valoris ei, qua <var>.C.</var> ſuperat <var>.A.</var> vel qua <var>.D.</var> minor eſt <var>.B.</var> & ipſam partem addere ipſi <var>.D.</var> <lb/>vt bona Petri reuertantur ad priorem ſuam quantitatem ipſius <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 eſt, vt talis proportionalitas ſit coniuncta (vt inquit Eu <lb/>f<gap/>atius ſeu Michael Epheſius, <lb/>ſuper 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 ipſa <lb/>diſiuncta ſit, ut diximus, vbi <lb/>non eſt neceſſe quod <var>.A.</var> æqualis ſit <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 ſpacio extra <lb/>Cœlum, Cœ<reg norm="lique" type="simple">liq́;</reg> figura.</head> + <head xml:id="head419" style="it" xml:space="preserve">Illust. Ioanni Paulo Capra Nouarienſi Sabaudia Ducis boſpicij <lb/>Magistro, viro ingeny praſtantia, & morum cando-<lb/>re, non minus quam familia nobili-<lb/>tate conſpicuo.</head> + <p> + <s xml:id="s3536" xml:space="preserve">SI vera eſſet animorum illa tranſmigratio quam ſibi Italicæ ſapientiæ Pa-<lb/>ter Pythagoras effinxerat, tuam, <reg norm="meamque" type="simple">meamq́;</reg> exiſtimarem 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, ſi ſuper aliquod punctum, quaſi <reg norm="mathematicum" type="context">mathematicũ</reg>, quieſceret, poſſet eſſe perpetuus, <lb/>cum aliquando eſſet mota, ſupponendo etiam eandem eſſe perfectè rotundam, & <lb/>lęuigatam. </s> + <s xml:id="s3538" xml:space="preserve">Reſpondeo huiuſmodi 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/>reſiſtentiam facit ſtringitur, eſt etiam reſiſtentia partium illius corporis moti, quæ <lb/>cum motæ ſunt, natura, impetum habent efficiendi iter directum, vnde cum ſimul <lb/>iunctæ ſint, & earum vna continuata cum alia. </s> + <s xml:id="s3539" xml:space="preserve">dum circulariter mouentur patiuntur <lb/>violentiam, & in huiuſmodi motu per vim vnitæ manent, quia quanto magis mo-<lb/>uentur, tanto magis in ijs creſcit naturalis inclinatio recta eundi, vnde tanto magis <lb/>contra ſuammet naturam voluuntur, ita vt ſecundum naturam quieſcant, quia cum <lb/>eis proprium ſit, quando ſunt motæ, eundi recta, quanto violentius voluuntur, tan-<lb/>to magis vna reſiſtit alteri, & quaſi retrò reuocat eam, quam antea reperitur habere.</s> + </p> + <p> + <s xml:id="s3540" xml:space="preserve">Ab eiuſmodi inclinatione rectitudinis motus partium alicuius corporis rotundi <lb/>fit, vt per aliquod temporis ſpacium, trochus cum magna violentia ſeipſum circun-<lb/>agens, omninò rectus quieſcat ſuper illam cuſpidem ferri quam habet, non incli-<lb/>nans ſe verſus mundi centrum, magis ad vnam <reg norm="partem" type="context">partẽ</reg>, quam ad aliam, cum quælibet <lb/>ſuarum partium in huiuſmodi motu non inclinet omnino verſus <reg norm="mundi" type="context">mũdi</reg> centrum, ſed <lb/>multo magis per tranſuerſum ad angulos rectos cum linea directionis, aut verticali, <lb/>aut orizontis axe, ita vt neceſſariò huiuſmodi corpus rectum ſtare debeat. </s> + <s xml:id="s3541" xml:space="preserve">Et quod <lb/>dico ipſas partes non omninò inclinare verſus mundi centrum, id ea ratione dico, <lb/>quia non abſolutè ſunt unquam priuatæ huiuſmodi inclinatione, quę efficit vt ipſum <lb/>corpus eo puncto nitatur. </s> + <s xml:id="s3542" xml:space="preserve">Verum tamen eſt, quod quanto magis eſt velox, tan-<lb/>to minus premit ipſum punctum, imò ipſum 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">qđ</reg> <lb/>apertè patet <reg norm="ſumendo" type="context">ſumẽdo</reg> <reg norm="exemplum" type="context context">exẽplũ</reg> pilę alicuius arcus, aut <reg norm="alicuius" type="simple">alicuiꝰ</reg> alterius <reg norm="inſtrumenti" type="context">inſtrumẽti</reg>, ſeu ma <lb/>chinæ miſſilis, quæ pila quanto eſt velocior, in motu violento, tanto maiorem pro-<lb/>penſionem habet rectius eundi, vnde verſus mundi centrum tanto minus inclinat, <lb/>& hanc ob cauſam leuior redditur. </s> + <s xml:id="s3544" xml:space="preserve">Sed ſi clarius, hanc veritatem videre cupis, <lb/>cogita illud corpus, Trochum ſcilicet, dum velociſſime circunducitur ſecari, ſeu <lb/>diuidi in multas partes, vnde uidebis illas omnes, non illico uerſus 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"/> + deſcendere, ſed recta orizontaliter, vt ita dicam, moueri. </s> + <s xml:id="s3545" xml:space="preserve">Id quod à nemine ad-<lb/>huc (quod ſciam) in trocho eſt obſeruatum. </s> + <s xml:id="s3546" xml:space="preserve">Ab huiuſmodi motu trochi, aut hu-<lb/>ius generis corporis, clarè perſpicitur, quàm errent peripatetici circa motum uio-<lb/>lentum alicuius corporis, qui exiſtimant aerem qui ſubintrat ad occupandum locum <lb/>à corpore relictum, ipſum corpus impellere, cum ab hoc, magis effectus contrarius <lb/>naſcatur.</s> + </p> + <p> + <s xml:id="s3547" xml:space="preserve">Quod deinde ampullæ iungantur in aqua, non fit ratione ſimpathiæ, de qua lo-<lb/>quitur Fracaſtorus, nam per accidens iunguntur, quia cum alia ad aliam accedit, quę <lb/>libet earum tentat aſcendere ab ea parte, à qua inuicem hærent, quemadmodum <lb/>efficiunt iuxta labrum vaſis, ea enim ſuperficies a quæ vicina circunferentiæ vaſis ali <lb/>quantulum aſcendit in vaſe, qui non eſt omnino plenus.</s> + </p> + <p> + <s xml:id="s3548" xml:space="preserve">Ad id deinde quod de claritate noctium ſcribis, miror cur non videas, quod <reg norm="quam" type="context">quã</reg> <lb/>to magis obſcura nox apparet, non dico ratione nubium, ſed diſtantiæ Solis ſub <lb/>orizonte ab eodem orizonte, tanto magis claram, & luminoſam ſeſe nobis oſten-<lb/>dit Luna in quintadecima, quia cum Sol eſt in Sagittario, & Capricorno, Luna eſt <lb/>in Geminis, & in Cancro, vnde in media nocte, eius radius per valde exiguam <reg norm="quam" type="context">quã</reg> <lb/>titatem vaporum tranſit, quia tunc ipſa eſt valde propinqua axi orizontis, & præ-<lb/>terea in huiuſmodi tempore anni & noctis, aer eſt magis purgatus, quàm in qualibet <lb/>alia temporis parte, quia hieme Sol non poteſt excitare multos vapores, & ij, qui <lb/>at tolluntur, nocte à frigore ſtatim congellati ratione grauitatis <reg norm="decidunt" type="context">decidũt</reg>, </s> + <s xml:id="s3549" xml:space="preserve">unde rema-<lb/>net aer multo clarior, qua ratione apparent ſtellæ minutæ, & Cœlum ijſdem ma-<lb/>gis ornatum, quàm in quolibet alio anni tempore.</s> + </p> + <p> + <s xml:id="s3550" xml:space="preserve">Dicere deinde, quemadmodum hic mundus eſt ætatis ſeptem, aut octomillium <lb/>annorum, ita nunc potuiſſet eſſe (ſi 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 ſe habet, ac ſi diceremus, quemadmodum hic mundus eſt tan-<lb/>tæ magnitudinis, ita etiam quinquagies maior eſſe potuiſſet, ergo eſt ſpatium, aut <lb/>interuallum corporeum, quod eum capere potuiſſet.</s> + </p> + <p> + <s xml:id="s3553" xml:space="preserve">Illud, nihil, Ariſtotelis extra Cęlum, nullo modo nobis inſeruit pro eiuſdem Cœ <lb/>li ſphęrica rotunditate, cum cuiuſque alterius ex infinitis figuris Cęlum ipſum eſſe <lb/>poſſit, ſecundum ſuam ſuperficiem conuexam. </s> + <s xml:id="s3554" xml:space="preserve">Nam Cœlum ea ratione ſphęricum <lb/>non eſt, quod magis ſit capax, quia ei innumerahiles alias figuras adeo magnas po <lb/>terat concedere cauſa diuina: </s> + <s xml:id="s3555" xml:space="preserve">ſed ſphæricum eſt effectum, ne partem aliquam habe <lb/>ret ſui termini ſuperfluam, quia nullum corpus à breuiori termino quam à ſphærico <lb/>terminari poteſt.</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 appenſa eſt ſitula, longè facilius axi inuoluitur, ſi ipſi axi affixa ſit 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, ſi ipſa rotæ extrema circunferentia, ex materia minori, <lb/>& denſiori, ac proinde grauiori conſtabit. </s> + <s xml:id="s3559" xml:space="preserve">Cuius rei ratio multiplex eſt. </s> + <s xml:id="s3560" xml:space="preserve">Nem-<lb/>pe quia omne corpus graue, aut ſui natura, aut vi motum, in ſe 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 ſeparatum à virtute mouente per aliquod temporis <lb/>ſpatium ex ſeipſo moueatur. </s> + <s xml:id="s3561" xml:space="preserve">nam ſi ſecundum naturam motu cieatur, ſuam veloci-<lb/>tatem ſemper augebit, cum in eo, impetus & impreſſio ſemper 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">eaq́;</reg> eam remouendo rota ſtatim non quieſcet, ſed per aliquod temporis ſpatium <lb/>circunuertetur.</s> + </p> + <p> + <s xml:id="s3563" xml:space="preserve">Secunda cauſa eſt, quia quoduis gr aue corpus, aut per naturam, aut per vim mo-<lb/>tum, rectitudinem itineris naturaliter appetat, quod clarè cognoſcere 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 inſito 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 ipſius rotæ, tanto facilius cir-<lb/>cunuoluatur, & ex ſeipſo tanto longiori tempore moueatur, quanto longius diſtat à <lb/>centro, cum eius iter tanto minus ſit curuum. </s> + <s xml:id="s3565" xml:space="preserve">Hanc igitur ob cauſam, rota, quanto <lb/>maior erit, <reg norm="eiuſque" type="simple">eiuſq́;</reg> pondus tanto magis vicinum circunferentiæ, tanto magis durabit <lb/>impetus motus aſſumptus.</s> + </p> + <p> + <s xml:id="s3566" xml:space="preserve">Tertia cauſa eſt, quod funis dum circunuoluitur, vicinius axi mathematico reuo-<lb/>lutionis, quam corpus graue circunferentiæ rotæ, ratione vectis, cum rota eſt in mo <lb/>tu, eius impetus non obtinet reſiſtentiam æqualem à contrario pondere aquæ in ſitu <lb/>la poſitæ.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head xml:id="head422" style="it" xml:space="preserve">De machina, qua aquam impellit & ſubleuat.</head> + <head xml:id="head423" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3567" xml:space="preserve">VNde ſit vt in fonte mandauerim, <lb/> + <anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/> + vas ſeu mortarium in quod in-<lb/>greditur inſtrumentum, quod aquam <lb/>impellit, diametrum ſuæ concauitatis, <lb/>habere non oportere maiorem dia-<lb/>metro fiſtulæ, per quam debet aſcende <lb/>re aqua, ratio eſt, quia ſi maius eſſet, <lb/>neceſſarium eſſet aliquod inſtrumen-<lb/>tum quo aqua impelleretur multo gra <lb/>uius toto corpore aqueo, quod aptum <lb/>eſſet implere aliquam fiſtulam adeo <lb/>altam, vt eſt fons, quæ tamen eſſet <lb/>adeo lata vt eſt 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 fiſtula, ſeu <lb/>hirundo, per quam aſcendit aqua <var>.f.</var> <lb/>mortarium verò ſit <var>.a.u.</var> quod tam <reg norm="altum" type="context">altũ</reg> <lb/>ſit vt <var>.f.</var> ſed <var>.f.</var> anguſtior ipſo <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 vaſa, ma-<lb/>nifeſtum erit, quod aqua ipſius <var>.f.</var> ſuffi-<lb/>ciens erit ad <reg norm="reſiſtendum" type="context">reſiſtẽdum</reg> toti aquæ <reg norm="ipſius" type="simple">ipſiꝰ</reg> <lb/><var>a.u.</var> & aqua <var>.a.u.</var> reſiſtet aquæ <var>.f.</var> quam-<lb/>uis aqua <var>.a.u.</var> maioris quantitatis ſit, & <lb/>ponderis ipſa <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 ſuo pondere, </s> + <s xml:id="s3570" xml:space="preserve">propterea quod pondus diuiditur proportionaliter ſupra ba-<lb/>ſim vaſis.</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, ſeu <reg norm="truncus" type="context">trũcus</reg> coni concaui aqua <lb/>plenus, cuius orificij diameter ſit <var>.b.d.</var> & multiplex diametro <var>.m.n.</var> infimæ baſis. </s> + <s xml:id="s3572" xml:space="preserve">co-<lb/>gitemus etiam <var>.b.d.</var> diuiſum in tot partes, quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis ſit <var>.m.n.</var> imagi-<lb/><reg norm="nemurque" type="simple">nemurq́;</reg> tot lineas perpendiculares deſcendere verſus mundi centrum ad puncta <var>r.<lb/>c.m.</var> et <var>.t.x.m.</var> vt in ſubſcripta hic figura videre eſt, per quas cogitemus tot ſuperfi-<lb/>cies curuas <reg norm="conicasque" type="simple">conicasq́;</reg>, inter quas, mente concipienda eſt aqua, quę pondere ſuo quie <lb/>ſcet ſupra maiorem ſuperficiem illa, quæ æque diſtans eſſet mundi centro, ſeu quam <lb/>ſupra baſim <var>.m.n.</var> vt exempli gratia conſideretur aqua inter <var>.g.m.</var> et <var>.s.r.</var> cuius pondus <lb/>diſtribuitur fecundum latitudinem <var>.m.r.</var> quæ maior eſt <var>.g.s.</var> cogitemus igitur <var>.m.c.</var> æ-<lb/>qualem eſſe <var>.g.s.</var> manifeſtum erit, quod <var>.m.c.</var> non ſuſtinebit 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/>ſus mundi centrum, quapropter fundus ſeu baſis <var>.m.n.</var> non ſuſtinet aliud pondus <reg norm="quam" type="context">quã</reg> <lb/>aquæ <var>.f.m.</var> ſed ſi quis hoc in dubium reuocaret dicens, quod aqua circunſcribens ſi-<lb/>tum corporis aquei <var>.f.m.</var> impellit lateraliter dictum corpus aqueum, reſpondendum <lb/>eſt, quod ex æquo huius corporis <var>.f.m.</var> aqua impellit etiam aquam circunſtantem, <lb/>eo, quod ſunt 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 vaſa <var>.a.u.</var> et <var>.f.</var> dico quod ſicut aqua <var>.f.</var> ſufficit ad <reg norm="reſiſtendum" type="context">reſiſtendũ</reg> aquæ <lb/><var>a.u.</var> ita quodlibet aliud pondus ęquale <var>.f.</var> cuiuſuis materiæ, in fiſtula <var>.f.</var> poſitum, ſuffi-<lb/>ciens erit, dummodo illud corpus ita ſit adæquatum concauitati fiſtulæ <var>.f.</var> quod non <lb/>permittat tranſitum aliquem aquæ vel <lb/> + <anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/> + aeris inter conuexum ipſius corporis, <lb/>& deuexum fiſtulæ <var>.f.</var> & hoc ex ſe ſatis <lb/>patet, ſed in vaſe <var>.a.u.</var> cum ex hypothe <lb/>ſi latius ſit ipſo <var>.f.</var> nullum aliud corpus <lb/>ſufficiens erit ad reſiſtendum aquæ ip-<lb/>ſius <var>.f.</var> quin tam graue ſit, quam tota <lb/>aqua <var>.a.u.</var> exiſtente <var>.a.u.</var> tam alto quam <lb/>f. </s> + <s xml:id="s3574" xml:space="preserve">Vnde ſi aqua ipſius <var>.f.</var> nil plus eſſet <lb/>quam vna tantummodo libra, & vas <var>.a.<lb/>u.</var> exiſteret latius ipſo <var>.f.</var> in decupla pro <lb/>portione, </s> + <s xml:id="s3575" xml:space="preserve">tunc in ipſo <var>.a.u.</var> oporteret <lb/>corpus adæquatum ipſi concauitati po <lb/>nere, cuius pondus eſſet decem libra-<lb/>rum, vt ſufficeret ad ſuſtinendum <reg norm="aquam" type="context">aquã</reg> <lb/>ipſius <var>.f.</var> & ad im<unsure/><reg norm="pellendum" type="context">pellendũ</reg> ipſam <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"/> + denſius eſſe aqua, vt maius <reg norm="interuallum" type="context">interuallũ</reg> <lb/>non occupet, quam <var>.o.e.</var> corpus igitur <lb/><var>o.e.</var> ſufficiens 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 poſtulas eſt problema ab alijs iam ſcriptum, ſed illud tibialio <lb/>medio ſoluam.</s> + </p> + <p> + <s xml:id="s3578" xml:space="preserve">Proponitur vas <reg norm="plenum" type="context">plenũ</reg> liquore aliquo, puta aqua, <reg norm="quod" type="simple">ꝙ</reg> tres habeat fiſtulas <lb/>ad baſim, quarum vnaquæque poſſit euacuare ipſum vas, inæquales ta-<lb/>men, ita quod prima tam lata ſit, vt ſpatio vnius horæ poſſit ipſum euacuare to-<lb/>tum; </s> + <s xml:id="s3579" xml:space="preserve">ſecunda vero ſpatio duarum horarum, tertia autem ſpatio trium hora-<unsure/>-<lb/>rum. </s> + <s xml:id="s3580" xml:space="preserve">Tunc quæritur quanto tempore omnes tres fiſtulæ ſimul apertæ euacua-<lb/>bunt ipſum vas. </s> + <s xml:id="s3581" xml:space="preserve">Ad hoc volo vt quæratur primo quanta pars aquæ vnaquęquę fi-<lb/>ſtula euacuabit in aliquo dato tempore, quod facilè eſt, vt puta, prima fiſtu-<lb/>la, ſpatio dimidiæ horæ euacuabit dimidium vas, eo quod ſpatio integræ horæ po-<lb/>teſt totum euacuare, ſecunda fiſtula, eodem temporis ſpatio, euacuabit quartam <lb/>partem ipſius vaſis, tertia verò fiſtula, eodemmet ſpatio temporis dimidiæ horæ, <lb/>euacuabit ſextam partem ipſius vaſis, quæ omnia fracta ſimul collecta faciunt vnde-<lb/>cim duodecimas partes totius vaſis, vnde manifeſtum erit, quod omnes fiſtulæ pari-<lb/>ter apertæ, ſpatio dimidię horæ euacuabunt vndecim duodecimas partes totius a-<lb/>quæ, ſed nos cupimus ſcire, quanto tempore, totum vas euacuabitur, apertis omni <lb/>bus fiſtulis, quapropter dicemus ita; </s> + <s xml:id="s3582" xml:space="preserve">Si vndecim duodecimæ partes conſumunt mi-<lb/>nuta .30. temporis, quantum conſument omnes partes aquæ? </s> + <s xml:id="s3583" xml:space="preserve">quæ ſunt .12. quare ex <lb/>regula de tribus prouenient nobis minuta .32. cum .8. vndecimis vnius minuti, hoc <lb/>eſt cum .43. ſecundis horæ ferè, vel ſiaccipiemus tres quartas vnius horæ, </s> + <s xml:id="s3584" xml:space="preserve">tunc pri-<lb/>ma fiſtula emittet tres quartas partes totius aquæ, ſecunda, tres octauas <reg norm="eiuſdem" type="context">eiuſdẽ</reg> aquę, <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 abſu-<lb/>mit .45. minuta temporis, ergo illud ſolum integrum abſumet idem vt ſupra hoc eſt <lb/>min .32. cum .8. vndecimis vnius minuti vel .43. ſecundis. </s> + <s xml:id="s3586" xml:space="preserve">Cuius rei ſpeculatio <reg norm="tam" type="context">tã</reg> con <lb/>iuncta eſt operationi, quòd vna cognita, reliqua ſtatim cognoſcitur.</s> + </p> + <p> + <s xml:id="s3587" xml:space="preserve">Idem eueniet de implendo vaſe tribus ſimilibus fiſtulis mediantibus.</s> + </p> + <p> + <s xml:id="s3588" xml:space="preserve">Secundum quæſitum ab alijs traditum, tuum etiam, aliter quoque poteſt ſolui, <lb/>propterea non prętermittam tibi ſatisfacere.</s> + </p> + <p> + <s xml:id="s3589" xml:space="preserve">Problema itaque tale eſt, vt ſit vas aliquod in <reg norm="quod" type="wordlist">qđ</reg> infunditur aqua per tres fiſtu-<lb/>las, ſed dum infunditur aqua, eadem egreditur per duas alias fiſtulas in fundo <lb/>vaſis poſitas, ſed tres ſuperiores ſint inuicem proportionatæ, vt ſupradictum <lb/>eſt, primaq́ue inferiorum talis ſit, vt ſpatio .4. horarum poſſit totum vas euacua-<lb/>re, ſecunda autem poſſit ſpatio .6. horarum idem facere, vnde ex ſupradictis, vas im <lb/>plebitur à tribus fiſtulis ſuperioribus, clauſis exiſtentibus inferioribus, ſpatio tempo <lb/>ris <reg norm="minutorum" type="context">minutorũ</reg> .32. <reg norm="cum" type="context">cũ</reg> .8. vndecimis hoc eſt min .32. cum .43. ſecundis, deinde per duas <lb/>fiſtulas inferiores poſſet euacuari ſpatio <reg norm="temporis" type="context">tẽporis</reg> horarum .2. et. mi .24. exſupradictis.</s> + </p> + <p> + <s xml:id="s3590" xml:space="preserve">Supponamus igitur omnes fiſtulas operari ſpatio temporis minutorum .32. cum <lb/>ſecundis .43. </s> + <s xml:id="s3591" xml:space="preserve">tunc manifeſtum eſt quod vas non implebitur, eo ſpatio min .32. cum <lb/>ſecundis .43. ſed tanta aqua deficiet, quanta ab inferioribus fiſtulis eo ſpatio tempo <lb/>ris min .32. ſecun .43. poteſt euacuari, </s> + <s xml:id="s3592" xml:space="preserve">quare proportio partis vaſis vacuæ, ad totum <lb/>vas, erit vt min .33. ferè ad horas .2. min .24. quod per ſe patet, </s> + <s xml:id="s3593" xml:space="preserve">tunc ſi 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 vaſe 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 ſi .111. indigent mi-<lb/>nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-<lb/>tum vas omnibus fiſtulis 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 aſcendendum ignis, & ad <reg norm="deſcendendum" type="context">deſcendendũ</reg> quicquid graue natum eſt, ita ad <lb/>ſpeculandum humanus intellectus. </s> + <s xml:id="s3596" xml:space="preserve">nec quieſcit, dum poteſt, eſt enim ver-<lb/>ſatile, <reg norm="agitandoque" type="simple">agitandoq́;</reg> ſeſe cauſis rerum immiſcere, & abditum aliquid rimari, <lb/>conatur, & eſt in nobis, quaſi Diogenes quidam in Dolio.</s> + </p> + <p> + <s xml:id="s3597" xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias ſcilicet nouas circuli paſſiones, <lb/>quæ ita ſe <reg norm="habent" type="context">habẽt</reg>. </s> + <s xml:id="s3598" xml:space="preserve">Sit circulus <var>.a.b.c.</var> in quo ſit <var>.a.d.</var> latus quadrati inſcriptibilis in ipſo <lb/>circulo, ct <var>.b.c.</var> ſit 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. ſit ſimiliter <var>.a.f.</var> contingens ipſum circulum in puncto <var>.a.</var> quæ <lb/>protracta ſit vſque ad punctum <var>.f.</var> interſectionis cum diametro protracto, quod ita <lb/>eueniet cum anguli <var>.a.e.f.</var> et <var>.f.a.e.</var> minores ſint duobus rectis, eo quod angulus <var>.f.a.e.</var> <lb/>acutus ſit, cum <var>.a.d.</var> tranſeat 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> ipſius, æqualis erit produ-<lb/>cto ipſius <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">ſed ex .31. eiuſdem angulus <var>.f.a.<lb/>c.</var> æqualis eſt 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. ſexti <lb/>eandem proportionem <var>.f.c.</var> ad <var>.c.e.</var> quæ <var>.f.a.</var> ad <var>.a.e.</var> ſed <var>.a.f.</var> eſt æqualis ſemidiametro <lb/>circuli propoſiti, </s> + <s xml:id="s3604" xml:space="preserve">propterea quod ſi producta fuerit à puncto <var>.a.</var> ad centrum <var>.o.</var> ſemi <lb/>diameter <var>.a.o.</var> hæc cum <var>.o.e.</var> faciet dimidium angulirecti, cum ex ſuppoſito <var>.a.d.</var> la-<lb/>tus ſit quadrati inſcriptibilis in ipſo 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 ſimiliter medietas recti ex .32. primi, </s> + <s xml:id="s3606" xml:space="preserve">quare ex .6. eiuſdem <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> ſit. vt <var>.f.a.</var> ad <var>.a.e.</var> erit ſimiliter vt <var>.b.c.</var> ad <var>.a.d.</var> <lb/>hoc eſt ut dupli ad duplum, vnde ex .15. ſexti <lb/>manifeſtum erit propoſitum, ex quo alia paſ-<lb/> + <anchor type="figure" xlink:label="fig-0302-01a" xlink:href="fig-0302-01"/> + ſio oritur, hoc eſt, quod productum <var>.f.c.</var> in <var>.a.<lb/>d.</var> æ quale ſit qua drato ipſius <var>.a.c.</var> ratio eſt, quia <lb/>quadratum <var>.a.c.</var> æ quale eſt producto <var>.b.c.</var> in <var>.c.<lb/>e.</var> eo quod <var>.a.c.</var> media proportionalis eſt inter <var>.<lb/>b.c.</var> et <var>.c.e.</var> ex ſimilitudine 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 ſunt <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, ſequitur etiam, quod <var>.a.c.</var> ſit 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> eſt latus octago-<lb/>ni igitur tale latus <reg norm="medium" type="context">mediũ</reg> proportionale erit <lb/>inter latus quadrati. et <var>.f.c.</var> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> circuli, quę <lb/>quidem <var>.f.c.</var> eſt una portio diametri quadrati circunſcriptibilis ipſum circulum inter <lb/>circulum & angulum ipſius 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 ſolarium, non fiat in cen <lb/>tro ſpeculi ſpharici, & aliquid contra Cardanum, & <lb/>de moturadiorum ſolarium.</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 ſolares, quià diuerſis punctis ipſius ſolaris corpo-<lb/>ris veniunt, tranſeuntes per centrum ſpeculi ſphærici concaui, quamuis à ſuper-<lb/>ficie ſpeculi ad centrum ipſum reflectantur, vt alíâs tibi dixi, nihilominus nullo mo <lb/>do poſſunt aliquod obiectum incendere duabus ex cauſis, quarum vna eſt, quia cum <lb/>Sol valde remotus ſit à nobis, valde etiam acutus generatur angulus coni radiorum <lb/>in centro ſpeculi, vnde à parua ſuperficie ipſius ſpeculi reflectuntur, </s> + <s xml:id="s3609" xml:space="preserve">quare pauciſſi-<lb/>mi radij ſunt qui reflectantur in ipſo centro, & propterea non ſufficiunt ad combu <lb/>ſtionem alicuius obiecti. </s> + <s xml:id="s3610" xml:space="preserve">Alia verò cauſa eſt, quod quamuis multi, & ſufficientes <lb/>radij fuiſſentad <reg norm="comburendum" type="context context">cõburendũ</reg> velociter quoduis obiectum. </s> + <s xml:id="s3611" xml:space="preserve">impoſſibile tamen omnino <lb/>eſset, vt aliquod obiectum comburerent, propterea quod cum radij incidentes de-<lb/>beant per centrum tranſire, obiectum combuſtibile, vt opacum, obſtaret ipſis radijs, <lb/>ne vlterius tranſirent, vnde nulla fieret reflexio, ſed etiam ſi dicti radij in centro re <lb/>flexi, ſufficerent ad combuſtionem, incidentes hoc magis eſſicerent. </s> + <s xml:id="s3612" xml:space="preserve">& ita abſque <lb/>vllo ſpeculo, omnia & in quolibet loco comburerentur, quod manifeſtè falſum eſt. <lb/></s> + <s xml:id="s3613" xml:space="preserve">Deſine igitur mihi citate Lucillum Philalteum, qui in philoſophia mathematica <lb/>fuit omnium imperitiſſimus. </s> + <s xml:id="s3614" xml:space="preserve">Verum ſpeculum vſtorium illud eſt quod ab Alhazem <lb/></s> + <s xml:id="s3615" xml:space="preserve">Deinde à Vitellìone deſcribitur.</s> + </p> + <p> + <s xml:id="s3616" xml:space="preserve">Quod deinde verum ſit, <reg norm="vmbram" type="context">vmbrã</reg> vniuſcuiuſque corporis opaci à Sole productam <lb/>ſemper eſſe centum <reg norm="nouemque" type="simple">nouemq́;</reg> vicibus maiorem diametro eiuſdem corporis, nego.</s> + </p> + <p> + <s xml:id="s3617" xml:space="preserve">Imaginemur <var>.s.</var> 1. diametrum eſſe illius circuli, quo vltimi radij ſolares veniunt tan <lb/>gentes corpus cuius diameter ſit <var>.c.e.</var> et <var>.a.i.</var> ſit diameter alterius circuli eiuſdem cor-<lb/>poris ſolaris à quo vltimi radij veniunt tangentes corpus, cuius diameter ſit <var>.f.g.</var> in <lb/>eadem diſtantia, & eodem ſitu prioris corporis. </s> + <s xml:id="s3618" xml:space="preserve">Tunc conus vmbræ ipſius <var>.f.g.</var> ſit <var>.f.<lb/>g.q.</var> & ipſius <var>.c.e.</var> ſit <var>.c.n.e.</var> centrum autem ſolare ſit <var>.o.</var> conorum verò axes ſint <var>.t.n.q</var>. <lb/></s> + <s xml:id="s3619" xml:space="preserve">tunc ex ſuppoſito <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 ſolari, 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 ſi vmbra <var>.t.q.</var> tanto maior eſt <var>.f.g.</var> quanto .109. eſt <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> quę <var>.n.t.</var> ad <var>.t.<lb/>c.</var> ſed cum angulus <var>.t.</var> communis ſit ambobus triangulis <var>.q.t.f.</var> et <var>.n.t.c.</var> ſequitur ex .6. <lb/>ſexti dictos triangulos æ quiangulos eſſe. </s> + <s xml:id="s3623" xml:space="preserve">Vnde ſi anguli <var>.t.n.c.</var> et <var>.t.q.f.</var> æ quales inui <lb/>cem ſunt, ergo <var>.q.f.</var> æquidiſtans erit <var>.n.c.</var> quod eſt impoſſibile, quia nunc demonſtra-<lb/>uimus ipſas concurrere in puncto <var>.u</var>. </s> + <s xml:id="s3624" xml:space="preserve">Quare non eſt 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 ſubtilitate.</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 eſt, quæ ſit cauſa, quod nos videmus <lb/>radium ſolarem tardiſſimè moueri, cum alias tibi dixerim ipſum qualibet hora cir-<lb/>ca terram quindecim gradus perficere, reſpondeo, quod radius ille quem videmus, <lb/>exempli gratia, in aliquo cubiculo, nunquam eſt idem numero, ſed quia ipſi radij <lb/>nullo modo differunt inter ſe, niſi in numero, proptera putamus eundem ſemper eſſe, <lb/>cum ſemper alius, atque alius ſit, quorum vnuſquiſque (de illis loquor, qui ad hunc <lb/>terræ globum perueniunt) circa terram reuoluitur ſpatio .24. horarum, & cum quili <lb/>bet circulus diuidatur in .360. gradus, quorum vigeſimaquarta pars eſt .15. verum <lb/>eſt 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 ſcribit in .3. lib. pag .282. non nego <lb/>quin pulchrum ſit, ſed alijs pulchrioribus modis poſſumus illud idem de-<lb/>monſtrare; </s> + <s xml:id="s3627" xml:space="preserve">cogita igitur ſuperficiem rectangulam, cuius medietas ſit <reg norm="triam" type="context">triã</reg> <lb/>gulus rectangulus <var>.a.b.g.</var> vnde ex ſuppoſito nobis cognita erit ſuperficies <lb/>ipſius trianguli, tanquam dimidium totius parallelogrammi rectanguli cogniti. <lb/></s> + <s xml:id="s3628" xml:space="preserve">Quare ex .25. ſecundi triangulorum <reg norm="Monteregij" type="context">Mõteregij</reg>, cognita nobis <reg norm="erunt" type="context">erũt</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 ſit ſuperficies trianguli ſimul <reg norm="cum" type="context">cũ</reg> baſi <var>.a.g.</var> cognita etiam <lb/>nobis fit perpendicularis <var>.b.d.</var> à puncto <var>.b.</var> ad baſim, & conſequenter cognitum no-<lb/>bis erit productum ipſius <var>.a.d.</var> in <var>.d.g.</var> & quia nobis cognita eſt <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> ſimiliter nobis cognita erit ex quinta ſecundi <lb/>Eucl. </s> + <s xml:id="s3630" xml:space="preserve">vnde ex penultima primi habebimus propoſitum.</s> + </p> + <p> + <s xml:id="s3631" xml:space="preserve">Poſſumus item circulum mente concipere cuius <var>.a.g.</var> ſit diameter, & ab eius cen-<lb/>tro <var>.e.</var> protracta cum fuerit <var>.e.b.</var> quæ nobis cognita erit, vt medietas ipſius <var>.a.g.</var> de cu <lb/>ius potentia, dempta <reg norm="cum" type="context">cũ</reg> fuerit potentia <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>b.o.</var> remanebit nobis potentia ipſius <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">huiuſmodi 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 ſubſcriptus <var>.a.b.c.u.</var> ſuperficiei cognitę ſimul cum diametro <var>.a.<lb/>c.</var> extendatur imaginatione <var>.b.c.</var> vſque ad, f. ita quod <var>.c.f.</var> æqualis ſit <var>.c.u.</var> intelligan-<lb/><reg norm="turque" type="simple">turq́;</reg> quadrata <var>.g.f</var>: <var>g.u.</var> ct <var>.u.f.</var> vnde <reg norm="summa" type="context">sũma</reg> <reg norm="quadratorum" type="context">quadratorũ</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="ſummam" type="context">ſummã</reg> <var>.g.u</var>:u.b: et <var>.u.f.</var> cognoſce-<lb/>mus, cui <reg norm="summæ" type="context">sũmæ</reg> addito ſuplemento <var>.d.e.</var> æ quali <var>.u.<lb/>b.</var> dabit nobis <reg norm="cognitum" type="context">cognitũ</reg> quadrarum <var>.g.f.</var> totale, qua <lb/> + <anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/> + re cognoſcetur eius radix <var>.b.f.</var> cognita igitur <var>.b.f.</var> <lb/>cum pro ducto <var>.b.u.</var> illico ex .5. ſecundi cognoſce-<lb/>tur <var>.b.c.</var> et <var>.c.f.</var> forte cognita <var>.b.f.</var> diuiſa <reg norm="per" type="simple">ꝑ</reg> æqualia <lb/>in puncto <var>.t.</var> & per inæqualiz in <reg norm="puncto" type="context">pũcto</reg> <var>.c</var>. </s> + <s xml:id="s3636" xml:space="preserve">Nam qua <lb/><reg norm="dratum" type="context">dratũ</reg> ipſius <var>.t.f.</var> cognitum, ęquatur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.b.u.</var> <lb/><reg norm="cum" type="context">cũ</reg> quadrato ipſius <var>.t.c.</var> <reg norm="dempto" type="context">dẽpto</reg> igitur rectangulo, <var>b.<lb/>u.</var> ex quadrato ipſius <var>.t.f.</var> relinquetur quadratum <lb/><reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.t.c.</var> cognitum & eius radix <var>.t.c.</var> qua addita ipſi <lb/>medietati <var>.b.t.</var> & <reg norm="dempta" type="context">dẽpta</reg> ex medietate <var>.f.t.</var> relinque-<lb/>tur propoſitum.</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 eiuſdem 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, ſimul cum proportione late <lb/>rum <var>.b.c.</var> et <var>.b.a.</var> nobis data ſit. </s> + <s xml:id="s3639" xml:space="preserve">cum autem ſcire voluerimus eius ſuperficiem <var>.b.u.</var> cla-<lb/>rum eſt, quod cum nobis data ſit proportio <var>.b.c.</var> ad <var>.b.a.</var> illico cognoſcemus <reg norm="etiam" type="context">etiã</reg> pro-<lb/>portionem quadrati ipſius <var>.b.c.</var> ad quadratum ip-<lb/> + <anchor type="figure" xlink:label="fig-0305-02a" xlink:href="fig-0305-02"/> + ſius <var>.b.a.</var> cum dupla ſit ei quæ <var>.b.c.</var> ad <var>.b.a.</var> ita etiam <lb/>& aggregati dictorum quadratorum ad quadra-<lb/>tum ipſius <var>.b.a.</var> hoc eſt nota erit nobis proportio <lb/>quadrati ipſius <var>.a.c.</var> diagonalis ad quadratum ip-<lb/>ſius <var>.a.b.</var> idem dico de quadrato <var>.b.c.</var> ideſt quod <lb/>proportio quadrati ipſius <var>.a.c.</var> ad quadratum <var>.b.c.</var> <lb/>cognita nobis erit, ſed <var>.a.c.</var> data nobis fuit, qua-<lb/>re cognoſcemus etiam omnia dicta quadrata eo-<lb/><reg norm="rumque" type="simple">rumq́;</reg> radices <var>.a.b.</var> et <var>.b.c.</var> </s> + <s xml:id="s3640" xml:space="preserve">quare & ſuperficiem 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 poteſt <lb/>ſolui, propterea, quod cum nobis cognita ſit ba-<lb/>ſis trianguli cum ſumma reliquorum laterum, & <lb/><reg norm="cum" type="context">cũ</reg> angulo oppoſito baſi ipſius reliqua cognita no <lb/>bis emergunt ex .15. problemate ſecundi lib. de Triangulis ipſius 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 ſi 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">triangulũ</reg> <var>.a.b.c.</var> <reg norm="orthogonium" type="context">orthogoniũ</reg>, & fiat <reg norm="rectangulum" type="context">rectangulũ</reg> <var>.b.u.</var> vt in mea figura <lb/>ſecundi exempli hic vides. </s> + <s xml:id="s3644" xml:space="preserve"><reg norm="producaturque" type="simple">producaturq́;</reg> <var>.b.c.</var> quouſque <var>.c.f.</var> æqualis ſit <var>.c.u.</var> vnde <var>.b.f.</var> <lb/>cognita nobis erit ex hypotheſi, </s> + <s xml:id="s3645" xml:space="preserve">quare cognoſcemus etiam quadratum <var>.g.f.</var> à quo <lb/><reg norm="demptum" type="context">demptũ</reg> cum fuerit <reg norm="aggregatum" type="context">aggregatũ</reg> quadratorum <var>.g.u.</var> et <var>.u.f.</var> nobis <reg norm="cognitum" type="context">cognitũ</reg> (nam quadra <lb/>ta <var>.g.u.</var> et <var>.u.f.</var> æqualia ſunt quadrato ipſius <var>.a.c.</var> diagonalis datę) remanebit aggrega-<lb/>tum <reg norm="ſupplementorum" type="context context">ſupplemẽtorũ</reg> cognitum, </s> + <s xml:id="s3646" xml:space="preserve">quare eius medietas cognoſcetur ideſt <var>.b.u.</var> vndæ ex <num value="5">.<lb/>5.</num> ſecundi Eucli. vt ſuperius diximus cognoſcetur etiam <var>.b.c.</var> et <var>.c.f.</var> diſtinctæ.</s> + </p> + <p> + <s xml:id="s3647" xml:space="preserve">Idem aſſero de <reg norm="exemplo" type="context">exẽplo</reg> Gemmæ Friſij à Stifelio citato in Appendice regulæ falſi.</s> + </p> + <p> + <s xml:id="s3648" xml:space="preserve">Sit gratia exempli rectangulum hicſubſcriptum <var>.a.b.</var> datæ ſuperficiei data etiam <lb/>nobis ſit proportio <var>.a.e.</var> ad <var>.e.b.</var> laterum producentium, <reg norm="cogitemusque" type="simple">cogitemusq́;</reg> <var>.a.e.</var> producta <lb/>vſque ad <var>.o.</var> ita vt <var>.e.o.</var> æqualis ſit ipſi <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 ſexti ſeu .18. vel .19. ſeptimi vel .15. quinti <lb/>eadem proportio erit ipſius <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/>ſcemus quadratum <var>.b.o.</var> & eius <reg norm="radicem" type="context">radicẽ</reg> <var>.e.o.</var> & ex ea <lb/>demregula cognoſcemus <var>.a.e.</var> cum cognita nobis ſit <var>.e.o.</var> ſimul 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 ſit figura infinitorum angulorum hoc eſt <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 ſecundi libri dicat circulum eſſe figuram <lb/>poligoniam, non eſt ita mirandum, nam & alij multi doctiſſimi viri hanc <lb/>veritatem cognouerunt, de Leone Baptiſta Alberto nihil dicam, cum ipſe fateatur <lb/>hoc accepiſſe à philoſophis, vt etiam refert Ariſt. de ſphæratertio de cœlo. </s> + <s xml:id="s3651" xml:space="preserve">conſi-<lb/>dera quæſo in circulo, quod cum angulus contingentiæ ſit angulus, quamuis <reg norm="omnium" type="context">omniũ</reg> <lb/>acutorum rectilineorum anguſtiſſimus, vnde ex communi ratione ſequitur reliquum <lb/>ex duobus rectis rectilineis eſſe angulum, & ſi omnium obtuſorum rectilineorum ſit <lb/>ampliſſimum, tanto magis igitur erit angulus, id quod remanet ex duobus rectis re <lb/>ctilineis, detractis <reg norm="cum" type="context">cũ</reg> fuerint duobus angulis contingentiæ, qui quidem angulus erit <lb/>in quouis puncto circunferentiæ ipſius circuli, idem intelligendum eſt de ſphæra, <lb/>cuius angulus eſt reſiduum ex quatuor rectis ſolidis, detractis cum fuerint quatuor <lb/>angulis contingentiæ <reg norm="ſolidisque" type="simple">ſolidisq́;</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 eſt, eo quod quandoque bonus dormitat Homerus. </s> + <s xml:id="s3653" xml:space="preserve">Puto enim il-<lb/>lud problema ab ipſo Monteregio non fuiſſe viſitatum. </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">methodũ</reg> a me aliter <reg norm="etiam" type="context">etiã</reg> <reg norm="excogitatam" type="context">excogitatã</reg> in eadem ipſius 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="Propoſitum" type="context">Propoſitũ</reg> ſit nobis triangulum <var>.a.b.g.</var> cuius baſis data ſit cum area, ſeu perpendi-<lb/>culari <var>.a.d.</var> cum angulo etiam <var>.a.</var> ad cognoſcendum autem <var>.a.b.</var> et <var>.b.g.</var> cogitemus circu <lb/>lum <var>.a.b.q.g.</var> circunſcribere ipſum triangulum cuius diameter <var>.p.q.</var> ad rectos ſe-<lb/>cet baſim <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 ipſo cognoſces, & <lb/><reg norm="angulus" type="simple">angulꝰ</reg> <var>.p.b.m.</var> ſimiliter cognoſcetur, <lb/></s> + <s xml:id="s3656" xml:space="preserve">quare ex .29. primi eiuſdem Montere <lb/>gij cognoſcemus <var>.p.m.</var> et <var>.p.b.</var> (nam <var>.b.<lb/>m.</var> datum fuit, vt dimidium totius ba-<lb/>ſis <var>.b.g.</var>) ducta poſtea <var>.b.q.</var> ex <reg norm="eadem" type="context">eadẽ</reg> .29. <lb/>cognoſcemus <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">cũ</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 cognoſcemus <var>.a.t.</var> quæ ſinus <lb/>eſt arcus <var>.a.p.</var> vnde cognitus erit arcus <lb/><var>a.p.</var> ſed arcus <var>.p.g.</var> cognitus eſt median <lb/>te angulo <var>.p.b.g.</var> cognito, qui quidem <lb/>arcus <var>.p.g.</var> ſi coniunctus fuerit cum arcu <var>.p.a.</var> cognoſcemus compoſitum <var>.a.g.</var> & eius <lb/>chorda ſimiliter (hoc eſt <reg norm="ſecundum" type="context">ſecundũ</reg> latus) qua cognita, illico cognoſcemus chordam <lb/><var>a.b.</var> hoc eſt tertium latus trianguli propoſiti.</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/>henſibilem eſſe, vide igitur, quod ſcribit in quinto lemmate in decimam <lb/>propoſitionem libr .2. de inſidentibus aquæ Archimedis, volens demonſtra-<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 eſt aliquo modo pro-<lb/>lixum, mediante linea <var>.c.p.</var> cum ſuis partibus, citans etiam antecedens lemma extra <lb/>propoſitum, eo quod nec in antecedente lemmate, nec in alio, ipſe vnquam proba <lb/>uerit proportionem <var>.c.d.</var> ad <var>.d.q.</var> eſſe, vt <var>.l.b.</var> @d.b.m. ſed ne putes me falli, tibi demon <lb/>ſtrabo non eſſe neceſſarium ducere lineam <var>.c.m.p.</var> vel <var>.q.p.</var> eo quod <reg norm="cum" type="context">cũ</reg> per quintam <lb/>lib. de quadratura parabolę Archimedis, ita ſit <var>.c.d.</var> ad <var>.d.e.</var> vt <var>.l.b.</var> ad <var>.b.m.</var> exiſtente <lb/><var>a.c.</var> dupla ipſi <var>.d.c.</var> et <var>.e.c.</var> dupla ipſi <var>.g.c.</var> et <var>.l.d.</var> dupla ipſi <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> ideſt <var>.g.c.</var> ad <var>.g.d.</var> vt <var>.b.d.</var> ideſt <var>.l.b.</var> <lb/>ad <var>.b.m.</var> ſed <var>.c.g.</var> ad <var>.g.d.</var> eft vt <var>.c.e.</var> ad <var>.e.a.</var> ratio eſt, quia componendo ita eſt <var>.c.d.</var> ad <var>.d.<lb/>g.</var> vt <var>.c.a.</var> ad <var>.a.e.</var> & hoc eſt, quia permutando, ita eſt <var>.a.c.</var> ad <var>.d.c.</var> vt <var>.a.e.</var> ad <var>.d.g.</var> & hoc <lb/>verum eſt ex .19. quinti eo quod totius <var>.a.c.</var> ad totum <var>.d.c.</var> eft vt abſciſſi <var>.e.c.</var> ad abſciſ <lb/>ſum <var>.g.c.</var> vt ſupradixi.</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 vniuerſaliori modo potes probare, quod ita ſit <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 ſimiliter <var>.y.</var> in <lb/>tali ſitu 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> diuiſa eodem modo etiam ſit à puncto <var>.s.</var> ex ſupradicta 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">totiꝰ</reg> <var>.a.c.</var> ad totum <var>.y.c.</var> vt abſcisſi <var>.s.c.</var> ad abſciſsum <var>.s.c.</var> </s> + <s xml:id="s3661" xml:space="preserve">quare reſidui <lb/><var>a.e.</var> ad reſiduum <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 ſupradictam <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 ſequitur, <lb/>primum, ſecundum, tertium, & quartum lemma ſuperflua 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/>poteſt demonſtrari, hoc mode. </s> + <s xml:id="s3665" xml:space="preserve">Nam ſuperius demonſtrauimus 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 æquidiſtantium ad <var>.h.e.</var> quibus rationibus mediantibus codem modo ſcies, <reg norm="quod" type="simple">ꝙ</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="æquidiſtantibus" type="context">æquidiſtãtibus</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, ſed cum ſit <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> & euerſim <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 eſt propoſitum.</s> + </p> + <p> + <s xml:id="s3667" xml:space="preserve">Non video etiam, quare ipſe ducat lineam <var>.s.r.</var> cum in ipſo contextu nihil ſaciac <lb/>de dicta <var>.s.r</var>.</s> + </p> + <p> + <s xml:id="s3668" xml:space="preserve">Comentum poſtea contextus <var>.P.</var> pulchrius eſſet, ſi diceret, quod cum ita ſit totius, <lb/><var>l.a.</var> ad totum <var>.a.d.</var> ſic ſe habebit abſciſſum <var>.a.i.</var> ad abſciſſum <var>.a.z.</var> eo quod ita eſt, vt ſcis, <lb/>hoc eſt in proportione dupla, ergo reſidui <var>.i.l.</var> ad reſiduum <var>.d.z.</var> erit vt totius <var>.a.l.</var> ad <lb/>totum <var>.a.d.</var> hoc eſt in proportione dupla.</s> + </p> + </div> + <div type="letter" level="4" n="5"> + <head xml:id="head438" style="it" xml:space="preserve">De Viſu.</head> + <head xml:id="head439" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3669" xml:space="preserve">RAtio vnde ſiat, vt videamus diſtinctè omnes eolores, cum in qualibet aeris par <lb/>te, quo lumina reſlexa poſſunt peruenire mixta ſint, & non diſtincta, oritur à <lb/>paruitate ipſius pupillæ oculorum, & à magna expanſione virtutis viſiuæ in ſuperſi-<lb/>cie concaua orbis continentis humores diaphanos oculorum per ramuſculos nerui <lb/>optici remotè ab ipſa pupilla. </s> + <s xml:id="s3670" xml:space="preserve">& quamuis radii luminoſi frangantur ab vnoquoque <lb/>humore diuerſimodè, hoc nihilominus maximè iuuat ad diſtinctionem radiorum, <lb/>ſed & ſi directè procederent, idem ferè eueniret, non tamen ſuis locis, cogita exem-<lb/>pli gratia lineam <var>.a.u.e.</var> vt communis ſectio cuiuſdam plani ſecantis ſphæram oculi, <lb/>per centrum ipſius, & pupillæ, et <var>.o.</var> punctum ſit proximum centro ipſius pupillæ, <lb/>ſed interius aliquantulum, extra <reg norm="autem" type="context">autẽ</reg> <reg norm="oculum" type="context">oculũ</reg>, ſint varij colores, vt <var>.c.n.t.</var> in dicto plano.</s> + </p> + <p> + <s xml:id="s3671" xml:space="preserve">Iam nulli dubium eſt quod lumina quæ producuntur ab <var>.c.n.t.</var> ad <var>.o.</var> in ipſo <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 diſtincta, procedendo igitur vlteriusipſi radij citra <var>.o</var>. </s> + <s xml:id="s3672" xml:space="preserve">tunc <reg norm="diſgregantur" type="context">diſgregãtur</reg>, <lb/>& ſeparantur abinuicem, & <reg norm="cum" type="context">cũ</reg> perueniunt ad lineam <var>.a.u.e.</var> ſentiuntur diſtincti alij <lb/>ab alijs. </s> + <s xml:id="s3673" xml:space="preserve">Cuius quidem rei, exemplum manifeſtum accipere poſſumus à quouis <lb/>cubiculo ex omni parte clauſo, quod tranſitum <reg norm="nullum" type="context">nullũ</reg> permittat radijs luminoſis, ni <lb/>ſi per aliquod paruum foramen, in quo foramine, & extra ipſum cubiculum, omnes <lb/>radij mixti erunt, ſed in obiecto pariete ipſius cu-<lb/> + <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> + biculi videb untur diſtincti, vnde ſequitur, 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> & ſuus contrapoſitus ſimili-<lb/>ter, & per conſequens linea <var>.e.u.a.</var> breuior erit, & <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.o.</var> propinquius etiam erit ipſi lineæ <var>.a.u.e.</var> <lb/>quæ omnia efficiunt, vt nobis obiectum <var>.c.t.</var> <reg norm="paruum" type="context">paruũ</reg>, <lb/>& minus diſtinctum, ſeu magis confuſum 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/>hæmiſphærij.</head> + <head xml:id="head441" style="it" xml:space="preserve">Anſelmo Fucaro.</head> + <p> + <s xml:id="s3674" xml:space="preserve">GRatæ mihi tuæ literæ <reg norm="fuerunt" type="context">fuerũt</reg>, quibus <reg norm="oſtendis" type="context">oſtẽdis</reg> non <reg norm="paruum" type="context">paruũ</reg> <reg norm="deſiderium" type="context">deſideriũ</reg> <reg norm="ſciendi" type="context">ſciẽdi</reg> vnde <lb/>fiat, quod cum dies illuceſcit, & eſt ſerena pars Cœli, circa axem orizontis <lb/>demiſſior appareat, quam aliæ partes, <reg norm="cum" type="context">cũ</reg> ab alijs (quod <reg norm="ſciam" type="context">ſciã</reg>) ſatis expreſſum <reg norm="non" type="context">nõ</reg> fue <lb/>rit, ſed quia de eo à me aliquid ſcire deſideras dicam quod mihi <reg norm="verum" type="wordlist">vr̃</reg>. </s> + <s xml:id="s3675" xml:space="preserve">Scias non <reg norm="ſolum" type="context">ſolũ</reg> <lb/><reg norm="multitudinem" type="context">multitudinẽ</reg> <reg norm="obiectorum" type="context">obiectorũ</reg> <reg norm="oppoſitorum" type="context">oppoſitorũ</reg> efficere, vt aliqua res alia longius diſtare videat, <lb/>vt alij <reg norm="putarunt" type="context">putarũt</reg>, ſed etiam diuerſitates colorum, quamobrem cum decipiamur, cre-<lb/>dentes Cœlum eſſe præditum colore cęruleo, cum is color, aeri, non Cœlo <lb/>conueniat, & videntes huiuſmodi colorem circa axem <reg norm="orizontis" type="context">orizõtis</reg> magis denſum, <reg norm="quam" type="context">quã</reg> <lb/>verſus ipſum orizontem, ratione exiguæ reſlexionis, à pauca quantitate vaporum <lb/>inter noſtrum ſitum, & reſlexionis locum, iudicamus Cœlum proximiorem eſſe cir-<lb/>ca dictum axem, quam ſint aliæ partes; </s> + <s xml:id="s3676" xml:space="preserve">præterquam, quod is color, qui videtur <lb/>terminare, aut impedire radium viſualem (aduertas tamen me hac in re platonicum <lb/>non eſſe) eo ſemper propinquior eſſe videtur, qui ei locum dat, & hanc ob cauſam <lb/>videntes nos <reg norm="denſitatem" type="context context">dẽſitatẽ</reg> cęrulei 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">contrariũ</reg> <lb/>euenire poteſt, ideſt vt eis magis profundum videa <lb/>tur <reg norm="Cœlum" type="context">Cœlũ</reg>, circa axem orizontis, quam vicinum gy <lb/>ro <reg norm="eiuſdem" type="context">eiuſdẽ</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 ſeſe magis <reg norm="obſcuram" type="context">obſcurã</reg> oculo <reg norm="demonſtrat" type="context">demõſtrat</reg>, & <lb/><reg norm="eam" type="context">eã</reg> <reg norm="propinquiorem" type="context">propinquiorẽ</reg> quę ſeſe <reg norm="clariorem" type="context">clariorẽ</reg> oſtendit, vt ei <reg norm="ent" type="context">ẽt</reg> <lb/>cont<unsure/>ingere poteſt, qui <reg norm="ſubſcriptam" type="context">ſubſcriptã</reg> <reg norm="ſiguram" type="context">ſigurã</reg> <reg norm="cubicam" type="context">cubicã</reg> non <lb/><reg norm="quidem" type="context">quidẽ</reg> <reg norm="ductam" type="context">ductã</reg> ſecundú <reg norm="ordinem" type="context">ordinẽ</reg> opticè, ſed ita, vt om-<lb/>nia latera oppoſita <reg norm="inuicem" type="context">inuicẽ</reg> ſint parallela, proſpiciet, <lb/>ideſt <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 ſequitur, vt aliquando quadratum <var>.a.o.</var> <lb/>videbitur citra, et <var>.i.n.</var> vltra <reg norm="dictum" type="context">dictũ</reg> cubum aliquando <lb/>verò èconuerſo.</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 Piſano.</head> + <p> + <s xml:id="s3678" xml:space="preserve">SI omnia vno colore conſtarent, & corporum vmbræ à luminibus non di-<lb/>ſtinguerentur, neque diuerſitas ſitus, lumina, quæ veniunt ad oculum non al-<lb/>teraret; </s> + <s xml:id="s3679" xml:space="preserve">perinde eſſet, ac ſi eſſemus cœci. </s> + <s xml:id="s3680" xml:space="preserve">Miror quod cum in Ariſtotele ſis <lb/>verſatus, in tuis tamen ſcriptis philoſophum à Mathematico ſepares, quaſi mathe-<lb/>maticus non ſit adeò philoſophus, vt eſt naturalis, & metaphyſicus, cum multo ma <lb/>gis quam ij philoſophus ſit appellandus, ſi ad veritatem ſuarum concluſionum reſpi <lb/>ciamus. </s> + <s xml:id="s3681" xml:space="preserve">Verum <reg norm="quidem" type="context">quidẽ</reg> eſt, te in huiuſmodi errore <reg norm="ſolum" type="context">ſolũ</reg> non verſari; </s> + <s xml:id="s3682" xml:space="preserve">ſed grauius eſt, <lb/>quod cum vos videatis etiam res morales ſub philoſophię <reg norm="appellationem" type="context">appellationẽ</reg> cadere, non <lb/>animaduertatis diuinas ſcientias mathematicas etiam philoſophiæ nomine ornan-<lb/>das eſſe. </s> + <s xml:id="s3683" xml:space="preserve">Quod ſi eiuſdem nomen penitius conſiderare velimus, inueniemus aper-<lb/>tè, mathematico magisillud ipſum quàm cuilibet alio conuenire, cum nullus ex alijs <lb/>tam certo ſciat id quod affirmat quam mathematicus, neque aliquis ſit, qui in co-<lb/>gnitionis, & ſcientiæ cupiditatem magis ducatur, vt apertè patet, cum nec etiam ipſi <lb/>ſenſui det locum, neque aliquid præſupponat, quod non ſit ita verum & intellectui <lb/>notum, vt nulla quæuis porentia, illud eſſe falſum oſtendere queat. </s> + <s xml:id="s3684" xml:space="preserve">Sed quia <lb/>Græci, qui ad placitum nomina rebus impoſuerunt, voluerunt etiam, non ſolum <lb/>mathematica, ſed etiam naturalia, metaphyſica, & moralia, ſub communi philoſo-<lb/>phiæ nomine contineri. </s> + <s xml:id="s3685" xml:space="preserve"><reg norm="Vtaunt" type="context">Vtaũt</reg> tibi ſatisfaciam authoritate Ariſtotelis, quem tanto-<lb/>pere colis, primum conſidera, nunquam eum de philoſopho <reg norm="mentionem" type="context">mẽtionem</reg> facere quin <lb/>prius aperiat de quo philoſopho loquatur, atque hoc ſemper præſtat, exceptis qui-<lb/>buſdam locis, vt cap .2. lib. 4. <reg norm="Metaphyſicorum" type="context">Metaphyſicorũ</reg>, vbi de philoſopho in genere <reg norm="loquens" type="context">loquẽs</reg>, <lb/>ait, proprium eſſe philoſophi. vt res omnes ſpeculetur atque hoc in principio quin <lb/>ti textus aſſerit, cum in quarto iam oſtenderit mathematicum eſſe philoſophum: <lb/></s> + <s xml:id="s3686" xml:space="preserve">omitto quod in .2. textu ſecundi phyſicorum idem affirmet, æquum eſſe appellare <lb/>philoſophiam ſcientiam veritatis, & finem ſpeculatiuæ exiſtere veritatem. </s> + <s xml:id="s3687" xml:space="preserve">An non <lb/>idem in primo cap .6. metaphiſicæ philoſophiam ſpeculatiuam, mathematicis phy <lb/>ſicis & ſupernaturalibus rebus contineri? </s> + <s xml:id="s3688" xml:space="preserve">An non idem paulo inferius ſcribit phyſi-<lb/>cam primam futuram, ſi aliæ ſubſtantiæ quam naturales non reperirentur? </s> + <s xml:id="s3689" xml:space="preserve">conſidera <lb/>deinde quid dicat in fine tertij cap. lib. 11. quo loco nil clarius eſſe poteſt, lege etiam <lb/>quæ .6. cap. eiuſdem libri ab eodem adducuntur, & quæ in .8. cap .12. libri textu .44. <lb/>apertè ponuntur. </s> + <s xml:id="s3690" xml:space="preserve">Quod ſi hæc tibi non ſufficiunt, vereor ne tuus morbus deſpe-<lb/>ratus euadat.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head444" style="it" xml:space="preserve">De imaginatione ſpecierum.</head> + <head xml:id="head445" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3691" xml:space="preserve">QVod dixi domino Tadeo eſt, quod aliquas particularium ſpecies, perfectè & <lb/>integrè imaginari poſſumus, alias non item, id tibi melius exemplo innote-<lb/>ſc<unsure/>et. </s> + <s xml:id="s3692" xml:space="preserve">Proponatur tibi triangulus æquilaterus datæ magnitudinis, <reg norm="datique" type="simple">datiq́;</reg> coloris, hu <lb/>iuſmodi enim particularis, potes imaginatione tibi fingere integram ſpeciem, tota <lb/><reg norm="lemque" type="simple">lemq́;</reg> ei adæquatam, ſed ſi aliquam ſpeciem aliquando vniuerſaliorem 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 ęquilateri, tali magnitudine, <lb/>ſed non præfinito colore conſtantis, hoc minime præſtare poſſes. </s> + <s xml:id="s3693" xml:space="preserve">quia nullam rem <lb/>viſibilem priuatam colore imaginari poſſumus. </s> + <s xml:id="s3694" xml:space="preserve">nec etiam potes imaginari <reg norm="ſpeciem" type="context">ſpeciẽ</reg> ali <lb/>cuius trianguli æquilateri, indeterminatæ magnitudinis, & indefiniti coloris, quæ <lb/>cuilibet particulari cuiuſuis magnitudinis, & coloris poſtea applicari queat. </s> + <s xml:id="s3695" xml:space="preserve">Species <lb/>deinde alicuius trianguli ęquicruri, aut vnius trianguli laterum inęqualium, aut <reg norm="triam" type="context">triã</reg> <lb/>guli in genere, aut tandem figuræ, conſiderato tu ipſe, an poſſit ſub imaginationem <lb/>cadere. </s> + <s xml:id="s3696" xml:space="preserve">Poſſumus quidem huiuſmodi ſpeciem (ratione mediante) intelligere, vn <lb/>de quamlibet ſpeciem rei particularis viſibilis, compoſitæ, ex figura, magnitudine, <lb/>& colore, perfectè imaginari poſſimus, & huiuſmodi conceptus erit ſpecialiſſima <lb/>ſpecies, quia in infinito ſuorum 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 ſe <lb/>partem aliquam habeat, quod aliquid aliud non obtineat, quin dicta differentia ſit <lb/>ſpecifica, eius tamen ſolum partis quæ differt ab alia duorum indiuiduorum, vnius, <lb/><reg norm="eiuſdemque" type="simple">eiuſdemq́;</reg> ſpeciei. </s> + <s xml:id="s3698" xml:space="preserve">quia ſi eſt in magnitudine nulla planè magnitudo reperitur, quę <lb/>ſua ſpecie non ſit dotata, quod ſi non eſſet, inter res omnes nulla æqualitas <lb/>eluceret: </s> + <s xml:id="s3699" xml:space="preserve">& ſi in figura, & colore, idem affirmo, aliter nulla res fimilis eſſet <lb/>alteri, neque aliqua ſimilitu lo reperiretur. </s> + <s xml:id="s3700" xml:space="preserve">Idem de quolibet alio obiecto <lb/>ſenſibili dico. </s> + <s xml:id="s3701" xml:space="preserve">Ratio autem eorum omnium quæ dixi eſt, quia imaginatiua nihil <lb/>aliud intellectui oſtendere poteſt, quam id quod recipit à ſenſu, & cum ſenſus, <lb/>alio modo moueri non poſſit quam ſupradicto, hanc ob cau ſam verum eſt, <reg norm="quicquid" type="simple">ꝗcquid</reg> <lb/>ſcripſi. </s> + <s xml:id="s3702" xml:space="preserve">Vnde triangulum ęquilaterum datę magnitudinis, erit genus triangulorum <lb/><reg norm="ęquilaterum" type="context">ęquilaterũ</reg> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> datæ magnitudinis, ſed <reg norm="diuerſorum" type="context">diuerſorũ</reg> <reg norm="colorum" type="context">colorũ</reg>, erit <reg norm="etiam" type="context">etiã</reg> ſpecies trianguli <lb/>æquilateri indeterminatæ magnitudinis, & hic deinde erit ſpecies trianguli, & hic <lb/>poſtea ſpecies figuræ. </s> + <s xml:id="s3703" xml:space="preserve">Idem de alijs omnibus rebus per gradus dico, quę ſicut à ſen <lb/>ſu, ita etiam ab imaginatione longè recedunt, adeo vt has ſpecies ſpecialiſſimas tan-<lb/>tum, ideſt eas ſolùm, quas hic ſuperius deſcripſi, integrè capere poſſit: </s> + <s xml:id="s3704" xml:space="preserve">at verò gene <lb/>ra, quanto vniuerſaliora ſunt, ab eadem imaginatione, tanto longius diſtant.</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 ſunt, quàm partes ipſius Lunę magis perſpicuæ, à qui <lb/>bus, lumen non refleſſum, ſed penetrans, nobis occultatur; </s> + <s xml:id="s3706" xml:space="preserve"><reg norm="quemadmodum" type="context">quemadmodũ</reg> <lb/>via lactea, nihil aliud eſt, quam pars octaui orbis magis opaca, à qua lumen Solis re-<lb/>fleſſum, ſeſe nobis oſtendit. </s> + <s xml:id="s3707" xml:space="preserve">Quod autem Maurolicus ſcribit folio .64. cap. de aſtro <lb/>rum fulſionibus, circa <reg norm="Lunam" type="context">Lunã</reg>, eſt falſum; </s> + <s xml:id="s3708" xml:space="preserve">primo, quia non conſiderat <reg norm="differentiam" type="context">differentiã</reg> inten-<lb/>ſionis luminum inter Venerem, & Lunam, cum <reg norm="lumem" type="context">lumẽ</reg> illius ſit magis intenſum, quam <lb/>Lunæ, quia quilibet qui ſano ſit <reg norm="onculo" type="context">õculo</reg>, facile poteſt compræhendere, ſi Lu-<lb/>na eſſet, vbi eſt Venus, aut Venus vbireperitur Luna (quibus in locis eiuſdem ma <lb/>gnitudinis nobis apparerent) ipſa Luna à Venere longè ſuperaretur, & excedere-<lb/>tur ſplendore, & lumine, ita vt ſi etiam verum eſſet, quod per tres gradus inter-<lb/>ualli ſeſe nobis proderet ſexageſima pars luminis (quod in quadraturis nec in vllo <lb/>alio ſitu verum e<unsure/>uadit, reſpectu ad Solem, ideſt vt tres gradus differentiæ ſitus, con <lb/>ſtituant ſexag eſimam partem differentiæ luminis reſpectu noſtri) 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 conſpiceretur, quia non ſufficit extenſio luminis, cum eiuſdem inten <lb/>ſio ſit etiam neceſſaria. </s> + <s xml:id="s3709" xml:space="preserve">Sed id quoque tibi dico, quod etiam ſi dicta ſexageſima <lb/>pars totius luminis lunaris, eadem intenſione ſplendoris, & luminis Veneris, in tali <lb/>diſtantia trium graduum à Sole prædita eſſet, non eam <reg norm="tamen" type="wordlist">tamẽ</reg> videremus, ratione ob <lb/>liquitatis curuę, & ſphæricę ſuperficiei Lunæ, reſpectu noſtri, in huiuſmodi ſitu: </s> + <s xml:id="s3710" xml:space="preserve">id <reg norm="qui" type="simple">ꝗ</reg> <lb/>tibi ita demonſtratum volo.</s> + </p> + <p> + <s xml:id="s3711" xml:space="preserve">Pars ſuperficialis lunaris globi, quæ nos reſpicit ſit <var>.a.p.u.</var> quam accipere poſſu-<lb/>mus pro medietate ipſius ſuperficiei totalis, eo quod reſpectu noſtri viſus, inſenſibi <lb/>liter, ab ipſa medietate differat, pars autem à Sole viſa ſit <var>.u.q.a.</var> cogitemus etiam cir <lb/>culum <var>.a.p.u.q.</var> vnum eſſe ex maioribus ipſius globi, cuius ſuperficies <reg norm="tranſeat" 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 ipſius Lunæ à Sole receptum, ita quod pars illumi <lb/>nata, erit medietas <var>.u.q.a.</var> exceſſus verò, cum noſtro viſui incompræhenſibilis ſit, pro <lb/>nihilo reputetur, cuius cauſa eſt, maxima illa diſtantia, quæ inter Solem, & Lunam <lb/>reperitur, quamuis Sol maior ſit 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 ipſius <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 ſexageſima pars ipſius <var>.a.p.u.</var> <reg norm="quod" type="wordlist">qđ</reg> <lb/>à vero ſenſibiliter non diſcedit. <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 eſt quod linea <var>.d.u.</var> ſe-<lb/>cabit <reg norm="arcum" type="context">arcũ</reg> <var>.t.u.</var> ſed ita propinqua <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.u.</var> quod erit ei ferè contingens, <lb/>vnde abſque ſenſibili 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, ſub an-<lb/>gulo <var>.t.d.u.</var> quem quidem angulum <lb/>oportet nos videre, cuius magnitu-<lb/>dinis exiſtat, reſpectu 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> vſque 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> ſed quia, cum dia-<lb/>meter <var>.a.u.</var> tam breuis ſit reſpectu di <lb/>ſtantiæ à terra, tempore interlunij, <lb/>vnde minor <reg norm="centeſima" type="context">cẽteſima</reg> parte ipſius di-<lb/>ſtantiæ exiſtit, <reg norm="ſequitur" type="simple">ſequit̃</reg> nos poſſe <reg norm="abſque" type="simple">abſq;</reg> <lb/>ſenſibili errore cogitare, à puncto <var>.d.</var> <lb/>ad quoduis punctum ipſius diametri <lb/>omnes lineas ad angulos rectos cum <lb/>ipſo diametro, & inſenſibilis 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 ſinu arcus <var>.t.u.</var> qui eſt graduum .3. hoc <lb/>eſt ſexageſima pars ſemicirculi graduum .180. quapropter <var>.t.i.</var> erit partium .5233. ta-<lb/>lium qualium <var>.o.u.</var> eſt .100000. cuius <var>.t.i.</var> quadratum demptum cum fuerit à quadra-<lb/>to ſemidiametri <var>.o.t.</var> relinquet nobis quadratum ipſius <var>.o.i.</var> quæ quidem <var>.o.i.</var> vt radix <lb/>quadrata, erit partium .99862. talium qualium ſemidiameter eſt .100000. vnde <var>.i.u.</var> <lb/>reſiduum diametri, remanebit partium .138. </s> + <s xml:id="s3717" xml:space="preserve">Vel ſic, cum cognitus ſit nobis arcus <var>.<lb/>t.u.</var> illicò cognoſcemus ſinum arcus <var>.p.t.</var> <reg norm="complementum" type="context">complementũ</reg> vnius quartæ, qui ſinus æ qua-<lb/>lis erit ferè arcui <var>.o.i.</var> partium .99862. vnde <var>.i.u.</var> erit, vt dictum eſt, partium .138. quę <lb/>quidem <var>.i.u.</var> æqualis eſt ferè ſinui arcus <var>.u.e.</var> & ita etiam <var>.u.e.</var> </s> + <s xml:id="s3718" xml:space="preserve">quare ſi diuiſa fuerit to-<lb/>ta <var>.a.u.</var> <reg norm="partium" type="context">partiũ</reg> .200000. per .138. proueniet nobis .1449. & ſic 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 poteſt, 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">Poſt eas literas quas proximè ad te dedi, Franciſcus Monardus mihi retulit tuas <lb/>nonnullas dubitationes circa noſtrum Theorema Arithmeticum .116. quarum <lb/>prima eſt, quod ſi numerus <var>.a.</var> <reg norm="cogitatus" type="simple">cogitatꝰ</reg>, eſſet æqualis .4. </s> + <s xml:id="s3721" xml:space="preserve">tunc ipſe non eſſet multiplex <lb/>ipſi .4. de quo tamen nullam mentionem feci. </s> + <s xml:id="s3722" xml:space="preserve">Idem etiam inquis, ſi <var>.a.</var> fuiſſet .5. 6. 7. <lb/>nec non .1. 2. et .3. </s> + <s xml:id="s3723" xml:space="preserve">Cui reſpondi, quod <reg norm="quanuis" type="context">quãuis</reg> nullam fecerim mentionem de æqua <lb/>litate ipſius <var>.a.</var> cum .4. nihiltamen refert, </s> + <s xml:id="s3724" xml:space="preserve">propterea quod quando ita fuiſſet, nihi-<lb/>lominus eaſdem conditiones ſubiret, <reg norm="quemadodum" type="context">quemadodũ</reg> ſi fuiſſet duplus, triplus, aut qua <lb/>druplus. eo quod à genere multiplici, æqualitas, formam diuerſam non induat. </s> + <s xml:id="s3725" xml:space="preserve">Qua <lb/>re idem eueniet ſi <var>.a.</var> fuerit .4. 5. 6. 7. vt ſi eſſet .8. 9. 10. et .11. & ſic de cæteris, excepto <lb/>quod in proprijs multiplicibus, vel in ſuperantibus ipſis multiplicibus <var>.a.</var> menſurare <lb/>tur ab ipſo .4. plus quam ſemel. </s> + <s xml:id="s3726" xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eſt, quia, <lb/>vt in ſecunda ſumma, hoc eſt in tertio termino maximo, reliquus tertius terminus, <lb/>ideſt .9. non compræhendetur, ita nobis indicabit primum numerum ſumptum mi <lb/>norem eſſe quaternario. </s> + <s xml:id="s3727" xml:space="preserve">Quæ omnia, exipſa noſtra thęoria ibidem expreſſa ma-<lb/>nifeſtantur. </s> + <s xml:id="s3728" xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neſcio, quia ipſius opus <lb/>ad manus meas nunquam peruenit, ſatis 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 ſcripta ego nunquam vidi, vt Leonardi Piſani, Proſdocimi, Petri Borghi, <lb/>Fratris Lucæ, Ioannis Sfortunati, <reg norm="cæterorumque" type="simple">cæterorumq́;</reg> ſimilium.</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">Illuſtri Uiro Franciſco Mendo Zzæ</head> + <p> + <s xml:id="s3730" xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noſter ſufficie-<lb/>bat. </s> + <s xml:id="s3731" xml:space="preserve">Sed quia, quæ nunc à me petis, talia ſunt, vt ſine tripartita <reg norm="aequa- liter" type="simple">ęqua-<lb/>liter</reg> aliqua data proportione non poſſit aliquis exactè intentum perfice-<lb/>re, nihilominus, ſuppoſita di <lb/> + <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> + cta diuiſione, reliqua facilia <reg norm="erunt" type="context">erũt</reg>. </s> + <s xml:id="s3732" xml:space="preserve"><reg norm="Primum" type="context">Primũ</reg> <lb/>enim eſt. </s> + <s xml:id="s3733" xml:space="preserve">Propoſitam Ellipſim 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> Ellipſis propoſita <var>.a.b.d.c.</var> cu-<lb/>ius axes ſint <var>.a.b.</var> et <var>.d.c.</var> dati, ſeu <reg norm="reperti" type="simple">reꝑti</reg> ex <lb/>47. <reg norm="ſecundi" type="context">ſecũdi</reg> Pergei, <reg norm="ſintque" type="simple">ſintq́;</reg> duo circuli <var>.a.e.<lb/>b.f.</var> et <var>.g.d.h.c.</var> circa eaſdem diametros, <lb/><reg norm="tunc" type="context">tũc</reg> proportio <var>.a.b.</var> ad <var>.d.c.</var> <reg norm="dimidium" type="context">dimidiũ</reg> erit <lb/>proportionis circulorum ex .2. 12. Eu-<lb/>clid. </s> + <s xml:id="s3735" xml:space="preserve">ſed proportio <var>.a.b.</var> ad <var>.d.c.</var> æqualis <lb/>eſt proportioni maioris circuli ad Elli <lb/>pſim .ex .5. Archimedis in lib. de cono<lb/>idalibus, quapropter proportio Elli-<lb/>pſis ad minorem circulum altera me-<lb/>dietas erit totius proportionis circulo-<lb/>rum, hoc eſt maioris ad minorem, qua <lb/>re Ellipſis media proportionalis erit <lb/>inter eos circulos. </s> + <s xml:id="s3736" xml:space="preserve">Nunc verò cum <lb/>ex Archimede repertę fuerint duæ fi-<lb/>guræ rectilineæ æquales duobus circu <lb/>lis iam dictis, & inter has, reperta fue <lb/>rit alia media proportionalis propoſi-<lb/>tum obtinebimus.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head453" style="it" xml:space="preserve">Spheroidem propoſitam cubare.</head> + <head xml:id="head454" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3737" xml:space="preserve">PRopoſita ſphæroides erit, aut prolata, aut oblonga, ſit prius prolata, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.a.b.</var> <lb/>diameter circuli, qui eam per æqualia ſecat, circa quam <var>.a.b.</var> vt circa axem in-<lb/>telligatur ſphæroides oblonga, cuius ſpiſſitudo ſit <var>.d.c.</var> axis prolatæ, cogitemus <reg norm="nunc" type="context">nũc</reg> <lb/>duas ſphæras <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 eſt duas ſphæras, & duas ſphæroides, quas probabo continuas proportio-<lb/>nales inuicem eſſe.</s> + </p> + <p> + <s xml:id="s3739" xml:space="preserve">Conſideremus igitur duos conos rectos, quorum <var>.a.b.</var> diameter ſit eorum baſium, <lb/>altitudo autem maioris, æqualis ſit ſemidiametro majori, hoc eſt 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 ſit ſemidiametro minori, hoc eſt medietati <var>.d.c.</var> vnde <lb/>habebimus proportionem coni maioris ad conum minorem, <reg norm="eandem" type="context">eãdem</reg> quæ eſt 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. eiuſdem manifeſtum eſt, ſed conus minor, eſt quarta pars ſphæroidis prolatæ <lb/>ex .29. </s> + <s xml:id="s3741" xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eſt etiam quarta pars <lb/>ſphæræ, ex .32. primi lib. de ſphæra, & cyllindro, </s> + <s xml:id="s3742" xml:space="preserve">quare ex communi ſcientia, <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio erit ſphæræ maioris ad ſphæroidem prolatam, quæ <var>.a.b.</var> ad <var>.d.c.</var> ſed pro-<lb/>portio <var>.a.b.</var> ad <var>.d.c.</var> eſt tertia pars proportionis maioris ſphæræ ad <reg norm="minorem" type="context">minorẽ</reg>. </s> + <s xml:id="s3743" xml:space="preserve">Conſidere <lb/>mus <reg norm="nunc" type="context">nũc</reg> alios duos conos rectos, vnius & <reg norm="eiuſdem" type="context">eiuſdẽ</reg> baſis, <reg norm="cuius" type="simple">cuiꝰ</reg> diameter ſit <var>.d.c.</var> ſed altitu <lb/>do maioris, æqualis ſit ſemidiametroſphęrę maioris, altitudo verò minoris, ſit æqua <lb/>lis ſemidiametro minoris ſphæræ, 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 eſt <var>.o.b.</var> ad <var>.o.d.</var> hoc eſt vt <var>.a.b.</var> ad <var>.d.c.</var> & ex dictis <reg norm="pro poſitionibus" type="simple">ꝓ<lb/>poſitionibus</reg> ita ſe habebit ſphæroides oblonga ad ſphęram minorem vt <var>.a.b.</var> ad <var>.d.<lb/>c.</var> hoc eſt tertia pars proportionis ſphæræ maioris ad minorem. </s> + <s xml:id="s3744" xml:space="preserve">Quare proportio <lb/>ſphæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris <reg norm="ſphae ræ" type="simple">ſphę<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 eſt inter <var>.a.b.</var> & ſuas duas tertias partes vna media pro por-<lb/>tionalis, quæ ſit <var>.K.</var> & ex Archimede, inuentum ſit quadratum ęquale circulo, cuius <lb/>ſit <var>.K.</var> diameter. </s> + <s xml:id="s3747" xml:space="preserve">Vnde proportio circuli (cuius <var>.a.b.</var> eſt diameter) ad circulum cu-<lb/>ius <var>.K.</var> eſt diameter, ſeſquialtera 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 ſphærę maiori, ex corellario .32. primi de ſphęra & <lb/>cyllindro, cuius corporis, latus cubus ſit <var>.m</var>.</s> + </p> + <p> + <s xml:id="s3749" xml:space="preserve">Idem facere oportebit mediante <var>.d.c.</var> minoris ſphærę, cuius corporis cubica ra-<lb/>dix ſit <var>.n</var>.</s> + </p> + <p> + <s xml:id="s3750" xml:space="preserve">Nunc verò inter <var>.m.</var> et <var>.n.</var> inueniantur duę medię proportionales <var>.s.t.</var> & ex <var>.s.</var> pro-<lb/>ducatur cubus, qui ęqualis erit ſphęroidi prolatæ propoſiti, cubus vero <var>.t.</var> æqualis <lb/>erit ſphęroidi oblongę, cuius axis eſſet <var>.a.b</var>.</s> + </p> + <p> + <s xml:id="s3751" xml:space="preserve">Si autem ſphęroides oblonga nobis propoſita fuiſſet, eodem methodo ſoluere-<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 deſignare.</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 ſupra datam lineam, vt Do-<lb/>minum Gaſparem docui, facillimus eſt.</s> + </p> + <p> + <s xml:id="s3753" xml:space="preserve">Sit enim linea <var>.b.a.</var> 46. propoſitionis primi Euclidis, <reg norm="poſitoque" type="simple">poſitoq́;</reg> pede immobli circi-<lb/>ni in puncto <var>.a.</var> ſecundum quantitatem lineæ <var>.a.b.</var> propoſitę fiat circulus, ſimiliter cir-<lb/>ca punctum <var>.b.</var> alius circulus eiuſdem magnitudinis, </s> + <s xml:id="s3754" xml:space="preserve">erecta deinde ſola <var>.a.c.</var> perpendi <lb/>culari ipſi <var>.a.b.</var> ex puncto <var>.a.</var> ipſa ſecabitur à circunferentia circuli. cuius centrum eſt <var>.<lb/>a.</var> in puncto <var>.c.</var> vnde <var>.a.c.</var> æqualis erit <var>.a.b.</var> poſito demum pede immobili ipſius circi <lb/>ni in puncto <var>.c.</var> ſecundum longitudinem ipſius <var>.c.a.</var> fiat alius circulus, qui æqualis erit <lb/>reliquis duobus circulis cum eorum ſemidiametri æquales ſint, & hic vltimo factus <lb/>ſecabit circulum, cuius <reg norm="centrum" type="context">centrũ</reg> eſt <var>.b.</var> in <reg norm="puncto" type="context">pũcto</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 ſunt inuicem ęqualia ex condi-<lb/>ti onibus circuli, angulus autem <var>.a.</var> rectus effectus fuit, </s> + <s xml:id="s3756" xml:space="preserve">tunc ſi 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. eiuſdem concludemus etiam reliquos angulos rectos eſſe.</s> + </p> + <p> + <s xml:id="s3758" xml:space="preserve">Circa verò id quod mihi ſcripſiſti de igne perpetuo putans nugas eſſe, quod Ro-<lb/>mæ inuentæ fuerint lucernę ardentes in ſepulchris 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, ſed multi homines fide digniſſimi. </s> + <s xml:id="s3760" xml:space="preserve">Prętera cum aisid nulla ratione poſſe <lb/>fieri. </s> + <s xml:id="s3761" xml:space="preserve">Reſpondeo quod maxima ratione poſſibile eſſe puto, quam quidem ra-<lb/>tionem ita eſſe oportet, quod primum lucerna ſit perfectè circuncluſa, vt <lb/>materia in ea conſtituta nullo modo exire poſſit, </s> + <s xml:id="s3762" xml:space="preserve">deinde quod materia inflamabilis <lb/>talis ſit, vt excrementum fuliginoſum ex flamma tranſmiſſum, tangendo ſuperfi-<lb/>ciem deuexam ipſius lucernæ, aptum ſit in <reg norm="priſtinum" type="context">priſtinũ</reg> <reg norm="humorem" type="context">humorẽ</reg> conge<unsure/>lari, ſiue transfor-<lb/>mari, vnde materia prima per tres formas perpetuò tranſibit, hoc eſt per humorem, <lb/>ſiue oleum tale, vt diximus, per ignem, ſeu flammam, & per vaporem, ſeu exhala-<lb/>tionem fuliginoſam aptam condenſari, 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/>propoſitam proportionem.</head> + <head xml:id="head458" style="it" xml:space="preserve">Michaeli Angelo Muciaſco.</head> + <p> + <s xml:id="s3763" xml:space="preserve"><emph style="sc">QVod</emph> mihi proponis, tale eſt, vt ſcilicet tibi modum ſcribam diuidendi <lb/>triangulum propoſitum ſecundum datam proportionem à linea tranſeun <lb/>te per punctum notatum extra triangulum.</s> + </p> + <p> + <s xml:id="s3764" xml:space="preserve">Tr<unsure/><reg norm="iangulum" type="context">iangulũ</reg> <reg norm="igitur" type="simple">igit̃</reg> à te mihi propoſitum ſit <var>.n.o.u.</var> conſidero <reg norm="primum" type="context">primũ</reg> quod ſi <lb/>quis ipſum diuiſerit 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 tranſiens <lb/>alia linea à puncto <var>.p.</var> propoſito, ita quod efficiat duo triangula <var>.m.r.e.</var> et <var>.r.s.x.</var> inui-<lb/>cem æqualia, problema ſolutum 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 reſiduum <var>.m.n.u.x.</var> etiam ęquale <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 propoſitum expe-<lb/>ditum fuiſſer, hoc eſt, <reg norm="quod" type="simple">ꝙ</reg> ſi à puncto p. <lb/>protracta eſſet linea <var>.p.m.</var> quę trian-<lb/>gulum <var>.n.o.u.</var> in duas partes inuicem <lb/>ita proportionatas diuiſiſſet, vt ſe ha <lb/>bet <var>.A.</var> et <var>.B.</var> ita ſe haberet <reg norm="productum" type="context">productũ</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 eſt diffi-<lb/>cile ſpeculari, ex methodo .24. ſexti, + <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 ſit proportio <var>.A.</var> ad <var>.B.</var> vt data, cognita etiam <lb/>nobis erit coniuncta, hoceſt <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> & ſimiliter productorum. </s> + <s xml:id="s3768" xml:space="preserve">Quæſiui poſtea modum inueniendi duas <lb/>dictas lineas <var>.m.o.</var> et <var>.o.x.</var> & cognoui quod ſi producta fuerit <var>.p.i.</var> æquidiſtans li-<lb/>neæ <var>.o.x.</var> <reg norm="producendoque" type="simple">producendoq́</reg> <var>.o.n.</var> quouſque cum <var>.p.i.</var> ſe interſecarent in puncto <var>.i.</var> inuenien <lb/>do poſtea 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> poteſt fieri, quod cognitum dico, eo quod nobis <lb/>cognita eſt proportio data, & rectangulum etiam <var>.n.o.</var> in <var>.o.u.</var> deinde ſecando ab <var>.o.<lb/>n.</var> partem æqualem lineæ iam inuentæ, quæ ſit <var>.o.t</var>. </s> + <s xml:id="s3769" xml:space="preserve">Inueniendo poſtea, ex .28. ſexti <lb/>lineam <var>.o.m.</var> cuius productum in <var>.m.t.</var> æquale ſit producto <var>.t.o.</var> in <var>.o.i.</var> vnde ex .15. eiuſ <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 ſe ha-<lb/>beret <var>.m.i.</var> ad <var>.m.o.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> ſed ex .4. ſexti, 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. ſexti productum <var>.<lb/>o.x.</var> in <var>.m.o.</var> æquale eſſet producto. p, i. in <var>.o.t.</var> & ſic 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 ſi punctum <var>.m.</var> caderet in punctum <var>.n.</var> idem eſſet, ſi vorò punctum <var>.m.</var> tranſiret <lb/>n. oporteret nos facere hoc in latere <var>.n.u.</var> ipſum quærendo in linea <var>.n.u.</var> ducendo pri <lb/>mum lineam <var>.p.i.</var> <reg norm="æquidiſtantem" type="context">æquidiſtantẽ</reg> <var>.u.x.</var> & producendo <var>.u.n.</var> ad partem <var>.u.</var> proſequendo, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſuperius iam dictum eſt.</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, ſecundum aliquam datam <lb/>proportionem à linea tranſeunte per punctum propoſitum.</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 ſit, quæ <var>.A.</var> ad <var>.B.</var> vt ſupra. </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 ſit <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 ſecabit 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 ſi intra dictos terminos tranſibit, vt in prima figura videre potes, <lb/>problema ſolutum erit, eo quod <lb/> + <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> + ſi à puncto <var>.i.</var> protracta fuerit <var>.p.<lb/>d.</var> pa rallela ad <var>.u.x.</var> habebimus <lb/>ex prima ſexti 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 eſt vt <var>.A.</var> ad <var>.B.</var> ſed <reg norm="triangulus" type="context">triãgulus</reg> <lb/><var>i.e.d.</var> æqualis eſt 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"/> + ſcientia. </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 eſt vt <var>.A.</var> ad <var>.B.</var> ex .11. eiuſdem.</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 ſi punctum <var>.q.</var> fuerit extra ut in .2. figura videre eſt. </s> + <s xml:id="s3778" xml:space="preserve">tunc manifeſtum erit, <reg norm="quod" type="simple">ꝙ</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">dictũ</reg> <var>.q.t.u.</var> quapropter <lb/>cum diuiſus fuerit triangulus <var>.e.<lb/>x.t.</var> mediante linea <var>.o.n.K.</var> ita <reg norm="quod" type="simple">ꝙ</reg> <lb/><reg norm="quadrilaterum" type="context">quadrilaterũ</reg> <var>.e.n.K.t.</var> ſit æquale <lb/>triangulo <var>.q.t.u.</var> ex doctrina præ <lb/>cedenti, habebimus propoſitum.</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 ſi quadrilaterum dictum eſſet fruſtum alicuius <reg norm="trianguli" type="context">triãguli</reg> ut in figura <var>.A.</var> hic ſub <lb/>ſcripta videre eſt, ſuppoſita, <var>b.d.</var> parallela ad <var>.u.p.</var> ita faciendum eſſet, ducendo <lb/>ſcilicet parallelam <var>.u.x.</var> ad <var>.b.p.</var> quæ producatur vſque ad concurſum cum <var>.b.d.</var> <lb/>in puncto <var>.x.</var> <reg norm="ſitque" type="simple">ſitq́;</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> ſit 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> ſe ha-<lb/>beat vt <var>.t.a.</var> ad <var>.a.e.</var> quo facto ex doctrina <reg norm="prae cedenti" type="simple">prę<lb/>cedenti</reg> diuidatur totum parallelogram--<lb/>mum <var>.p.x.</var> mediante linea <var>.o.q.</var> ſecundum <lb/>quod ſe habet <var>.t.r.</var> ad <var>.r.e</var>. </s> + <s xml:id="s3782" xml:space="preserve">Atque ita ſolu-<lb/>tum erit problema, vt exte ipſo 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 ſi nullum latus parallelum reliquo erit, ita faciendum erit. </s> + <s xml:id="s3784" xml:space="preserve">ſi ſit tale quadrila <lb/>terum <var>.b.d.u.p.</var> oportet vt ipſum conuertamus in triangulum, producendo duo <lb/>quęuis eius latera oppoſita uſque ad interſectionem ut pote <var>.u.p.</var> et <var>.d.b.</var> in puncto <var>.x.</var> <lb/>quo facto, ſupponemus <var>.o.</var> eſſe punctum datum, proportio verò data ſit <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> ipſa <var>.t.i.</var> ſe 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"/> + ſe habet ad triangulum <var>b.p.x.</var> ducatur poſtea <var>.o.q.</var> quæ diuidat totale triangulum <var>.d.<lb/>u.x.</var> in duas partes inuicem ita proportionatas, ut ſe habent <var>t.r.</var> et <var>.r.e.</var> quæ quidem <lb/>partes ſint <var>.c.d.u.q.</var> et <var>.c.q.x.</var> ut in primo problemate tibi monſtraui, & habebis pro-<lb/>poſitum, dato quod punctum <var>.c.</var> ſit inter <lb/>b. et <var>.d</var>.</s> + </p> + <p> + <s xml:id="s3785" xml:space="preserve">Sed ſi forte linea <var>.o.q.</var> ſecabit <var>.b.x.</var> hoc <lb/> + <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> + eſt ſi punctum <var>.c.</var> eſſet inter <var>.b.</var> et <var>.x.</var> mani-<lb/>feſtum eſt, quod <var>.c.q.</var> ſecaret <var>.b.p.</var> in pun-<lb/>cto <var>.y.</var> vnde in tali caſu, alio modo ope-<lb/>randum eſſet, hoc eſt ducendo <var>.b.u.</var> quæ <lb/>diuideret quadrilaterum in duo triangu-<lb/>la, & ut ſe haberet triangulum <var>.b.d.u.</var> ad <lb/>triangulum <var>.b.p.u.</var> vellem vt ita ſecaretur <lb/><var>t.i.</var> in puncto <var>.n.</var> vt ita ſe haberet <var>.t.n.</var> ad <var>.n.<lb/>i.</var> ut dictum eſt de iſtis duobus triangulis, <lb/></s> + <s xml:id="s3786" xml:space="preserve">deinde prout ſe habet <var>.n.r.</var> ad <var>.r.i.</var> ita ſeca-<lb/>res triangulum <var>.b.p.u.</var> mediante linea <var>.o.<lb/>K.</var> ex doctrina primi problematis, & ita haberes propoſitum.</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, ſeu hexagonum, vel alias quaſuis multilateras figuras propoſitas its <lb/>diuidere, vt dictum eſt 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 ſecare volumus <reg norm="mediante" type="context">mediãte</reg> linea <var>.o.<lb/>q.</var> in duas partes inuicem ſe habentes, vt ſe habent <var>.t.r.</var> et <var>.r.i.</var> oportet igitur ut ipſum <lb/>pentagonum reducas ad quadrilaterum <var>.x.a.d.u.</var> quod diuidatur ſecundum præce-<lb/>dentem doctrinam, vt ſe habet <var>.t.r.</var> ad <var>.r.e.</var> <lb/>vnde ſi punctum <var>.q.</var> incidit inter <var>.p.</var> et <var>.u</var>. </s> + <s xml:id="s3789" xml:space="preserve">tunc <lb/>habebis propoſitum, ſi 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> ſecabit <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">diuidaturq́;</reg> <var>.t.i.</var> in puncto <var>.n.</var> ita <lb/>vt <var>.t.n.</var> ad <var>.n.i.</var> ſe 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">hũc</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 propoſitum. </s> + <s xml:id="s3791" xml:space="preserve">Idem dico de hexa <lb/>gono, reducendo ipſum ad pentagonum, & <lb/>item de eptagono, ipſum reducendo ad exa <lb/>gonum, & idem infero de infinito ipſarum <lb/>ſuperficialium 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 ſuperficie pla-<lb/>na, non tamen inuicem æqu idiſtantes, proponis etiam <var>.n.t.</var> in eadem ſuperfi-<lb/>cie, quæ vnamquamque priorum ſecat, proponis etiam lineam <var>.h.</var> tali conditione, <lb/>quod nulli dictarum ſit parallela, </s> + <s xml:id="s3793" xml:space="preserve">deinde ſcire cupis qua arte aliquis poſſet ducere <var>.<lb/>c.u.</var> parallelam ad <var>.h.</var> ita quod ſecando <var>.n.t.</var> conſtituat 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/>vſque ad concurſum 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> poſtea <lb/>ex .25. ſexti Eucli. conſtitues <reg norm="triangulum" type="context">triãgulum</reg> <var>.i.u.e.</var> ſimile triangulo <var>.i.c.n.</var> æquale tamen <lb/>triangulo <var>.i.t.n.</var> & ſolutum erit problema. <lb/></s> + <s xml:id="s3796" xml:space="preserve">Velſic, 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 poſtea <var>.e.u.</var> paralle-<lb/>lam lineę <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">& ſo-<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. ſexti 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">triangulũ</reg> <var>.i.t.n.</var> ex pri-<lb/>ma ſexti, et .11. quinti. </s> + <s xml:id="s3800" xml:space="preserve">Vnde <lb/>ex .9. eiuſdem <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">Thaodoſio à Raifestaim.</head> + <p> + <s xml:id="s3802" xml:space="preserve"><emph style="sc">DVritandvm</emph> profecto non eſt, quin quotidie hominibus ſtudioſis ali-<lb/>quid noui deſit, quemadmodum, quod tibi nunc occurrit, mihi non-<lb/>nunquam accidit, hoc eſt inuenire orizontem, cui aliqua propoſita ſtel <lb/>la oriatur cum gradu ipſius longitudinis. </s> + <s xml:id="s3803" xml:space="preserve">pro <reg norm="cuius" type="simple">cuiꝰ</reg> rei operatione te prius <lb/>ſcire oportebit vtrum ſtella in ſignis aſcendentibus, vel deſcendentibus reperiatur, <lb/>hoc eſt in ſignis, quę à Capricorno ad Cancrum procedunt, vel in illis, quę à Can-<lb/>cro ad Capricornum numerantur, </s> + <s xml:id="s3804" xml:space="preserve">propterea quod ſi in ſignis aſcendentibus inue-<lb/>nitur, ſciendum eſt, quod ſupra talem orizontem polus mundi auſtralis attollitur, <lb/>ſed ſi in ſignis deſcendentibus reperitur, </s> + <s xml:id="s3805" xml:space="preserve">tunc polus borealis eleuatur ſupra 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 vigeſimi gradus Cancri, quapropter polus borealis eleuatur ſupra ori-<lb/>zontem, cui ipſa oritur cum eodem gradu, & minuto eclipticę illius ſigni. </s> + <s xml:id="s3806" xml:space="preserve">ſed <lb/>quia volumus etiam ſcire veram quantitatem arcus eleuationis huiuſmodi poli, pro <lb/>pterea accipiemus in tabula generali Monteregij numerum qui vocatur radix aſcen <lb/>ſionum, èregione numeri longitudinis ipſius ſtellæ, qui quidem numerus in præſen <lb/>ti exemplo erit gra .107. cum minutis .53. qui eſt <reg norm="cuiuſdam" type="context">cuiuſdã</reg> arcus æquatoris, qui inci-<lb/>pit in principio Arietis, & in circulo latitudinis deſinit, hoc eſt ab orizonte quæſi-<lb/>to, ita quod talis numerus erit aſcenſio obliqua huiuſmodi puncti eclipticæ illi ori-<lb/>zonti, qua aſcenſione mediante, ſimul cum gradu, & minuto longitudinis in tabulis <lb/>aſcenſionum obliquarum, inueniemus gradum, & minutum altitudinis pollaris, <reg norm="quod" type="wordlist">qđ</reg> <lb/>quærebatur, eodem ordine ac methodo, quo vtimur ad inueniendum in tabulis po-<lb/>ſitionum, polum circuli poſitionis alicuius aſtri, mediante declinatione & diſtantia <lb/>à meridiano ciuſdem aſtri, vt ſcis. </s> + <s xml:id="s3807" xml:space="preserve">Vnde in præſenti exemplo eleuatio poli borea <lb/>lis ſupra talem orizontem erit gra .7. cum minutis .45.</s> + </p> + <p> + <s xml:id="s3808" xml:space="preserve">Sed ſi ſtella fuerit in medietate aſcendente, tunc certi erimus polum auſtralem ſu <lb/>per dictum orizontem attolli, nam idem eſt quærere altitudinem vnius <reg norm="polorum" type="context">polorũ</reg> mun <lb/>di à tali orizonte, quod diſtantiam dicti poli à circulo ſecundum quem longitudo <lb/>terminatur, qui etiam latitudinis dicitur, eo quod tunc temporis talis circulus vnus <lb/>& idem eſt cum orizonte. </s> + <s xml:id="s3809" xml:space="preserve">Sumatur ergo exempli gratia ſtella, quæ in ore piſcis au <lb/>ſtralis eſt, quę, pro nunc, ſit 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 ſtella oritur cum eiuſmodi puncto eclipticæ, depreſſum eſſe à parte <lb/>auſtrali ſub <reg norm="illoque" type="simple">illoq́</reg> polo, ſed quia propoſitum eſt ſcire etiam quantitatem huiuſmo-<lb/>di depræſſionis, reperiemus in tabula generali gradum, & minutum æquatoris, cor-<lb/>reſpondentem 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 eſt aſcen. obli. ad dictum orizontem, vbi polus auſtralis attollitur, & deſcenſio <lb/>obliqua, vbi polus borealis eleuatur. </s> + <s xml:id="s3812" xml:space="preserve">Quapropter ſi à .317. gradibus cum minutis <lb/>46. demptus fuerit dimidius circulus gra .180. remanebunt gra .137. cum minutis .46 <lb/>& punctus oppoſitus gradibus .20. cum .14. minutis Aquarij eſt in eodem numero <lb/>Leonis, & mediantibus iſtis gradibus .137. min .46. aſcenſionis, cum grad .20. min <num value="14">.<lb/>14.</num> Leonis inueniemus eleuationem poli borealis ab orizonte in tabulis aſcenſio-<lb/>num obliquarum Monteregij, hoc eſt gra .17. min .53. & eadem altitudo erit poli <lb/>auſtralis ſupra orizontem à quo Fomahant cum dicto puncto eclipticæ oritur, in qua <lb/>longitudine dicta ſtella reperitur.</s> + </p> + <p> + <s xml:id="s3813" xml:space="preserve">Sed ſi propoſitus nobis fuerit punctus eclipticæ, cum quo aliqua ſtella oritura ſit, <lb/>& oporteat inuenire vbi, hoc eſt orizontem huiuſmodi ortus, eleuatione poli arti <lb/>ci, ſeu antarctici ſupra 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 ſtella <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 ſtella oritura ſit <var>.e.</var> orizon de <lb/><reg norm="mum" type="context">mũ</reg> <var>.o.e.</var> vbi ſtella oriri poſſit <reg norm="cum" type="context">cũ</reg> <lb/>puncto <var>.e</var>. </s> + <s xml:id="s3815" xml:space="preserve">Nam cum ſtella 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 ſupponatur arcus <var>.a.<lb/>o.</var> eſſe circuli latitudinis, et <var>.a.o.</var> <lb/>Iatitudo ipſius ſtellæ, & angulus <lb/>a. rectus erit, & quia punctum <var>.e.</var> <lb/>datur, ergo arcus <var>.a.o.</var> & arcus <var>.a.<lb/>e.</var> ſimul <reg norm="cum" type="context">cũ</reg> angulo <var>.a.</var> recto cogni-<lb/>ti ſunt, vnde ex .11. primi lib. co<lb/>pernici, angulus <var>.a.e.o.</var> cognoſce-<lb/>tur, & angulus <var>.q.e.o.</var> ſimiliter, vt <lb/>reſiduum ex duobus rectis quo. e <lb/>mediante cum angulo <var>.q.</var> declina <lb/>tionis ab æquatore, <reg norm="medianteque" type="simple">medianteq́;</reg> <lb/>latere <var>.q.e.</var> cognito, cognitus quo <lb/>que nobis erit angnlus <var>.e.t.q.</var> ex <lb/>12. eiuſdem. </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 poſtea gradum eclipticę, cum quo ſtella data oriatur ad orizontem pro <lb/>poſitum, nullius eſt difficul@atis.</s> + </p> + <p> + <s xml:id="s3819" xml:space="preserve">Ponamus exempli gratia, aliquem ſcire velle gradum eclipticæ, cum quo canicu-<lb/>la oritur ad orizontem, cui polus boreus eleuatur per gradus .44. quæ canicula ſup-<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/>ſeptentrionalis, </s> + <s xml:id="s3821" xml:space="preserve">deinde inuenire eius aſcenſionem rectam ex doctrina .4. problema <lb/>tis eiuſdem 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 aſcenſionalium ſub polo .44. accipiemus differen-<lb/>tiam aſcenſionum, qua differt recta ab obliqua, quæ in præſenti exemplo erit gra <num value="5">.<lb/>5.</num> min .55. quæ dempta ab aſcenſione recta ſtellæ, vt præſens exemplum exigit, relin <lb/>quet nobis aſcenſionem obliquam ſtellæ propoſitæ ad polum. gra .44. quæ erit gra <num value="102">.<lb/>102.</num> minu .47. qua mediante, in tabulis aſcenſionum obliquarum poli .44. habebi-<lb/>mus gradum & minutum eclipticę cum quo ſtella oritur. </s> + <s xml:id="s3823" xml:space="preserve">quod in caſu noſtro erit <lb/>gra .1. min .8. Leonis, ſed ſi tecum non fuerint tabulæ dictæ, potes eleganter omnia <lb/>hæc perficere via triangulorum ſphæricorum.</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> propoſitus <reg norm="autem" type="wordlist">aũt</reg> orizon ſit <var>.o.c.d.</var> <lb/>& ſtella data ſit <var>.o.</var> in orientali parte orizontis, circulus verò <var>.a.o.</var> ille ſit, qui <reg norm="tranſiens" type="context">tranſiẽs</reg> <lb/>per polos eclipti<unsure/>cæ & per centrum ſtellæ terminat longitudinem ipſius ſtellæ, & in <lb/>ipſo ſit eius latitudo. </s> + <s xml:id="s3826" xml:space="preserve">Nunc propoſitum ſit inuenire arcum <var>.d.q.</var> eo quod illicò ſcie <lb/>mus punctum <var>.d.</var> qua propter oportet nos prius cognoſcere arcum <var>.d.a.</var> qui demptus, <lb/>vel additus arcui <var>.a.q.</var> prius cognito ex ſuppoſito (nam data nobis eſt longitudo, & <lb/>latitudo ſtellæ) dabit nobis <var>.d.q</var>.</s> + </p> + <p> + <s xml:id="s3827" xml:space="preserve">Cum igitur voluerimus arcum <var>.d.a.</var> cognoſcere, ita faciemus. </s> + <s xml:id="s3828" xml:space="preserve">nam <var>.q.a.</var> cognitus <lb/>nobis eſt ex ſuppoſito vt dictum eſt. </s> + <s xml:id="s3829" xml:space="preserve">angulus quoque <var>.a.q.b.</var> qui declin tionis eclipti <lb/>cæ ab æquatore eſt, angulus deinde <var>.a.</var> (trianguli <var>.a.b.q.</var>) rectus eſt, 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> reſiduus ex duobus re-<lb/>ctis in duobus primis hic ſubſcriptis <lb/>figuris nobis itidem cognitus erit, <lb/>etiam & arcus <var>.b.o.</var> reſiduus ſiue com <lb/>poſitus ex ar cu <var>.a.o.</var> cognito ex ſup-<lb/>poſito <reg norm="cum" type="context">cũ</reg> ſit 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/>gnoſcimus latus <var>.o.b.</var> & angulum <var>.o.<lb/>b.e.</var> nec non angulum <var>.b.e.o.</var> qui eſt <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">Conſideremus <lb/>deinde triangulum <var>.a.o.d.</var> cuius angu <lb/>lus <var>.a.</var> rectus eſt, & angulus <var>.a.o.d.</var> <reg norm="cum" type="context">cũ</reg> <lb/>latere <var>.a.o.</var> etiam cognitus, vnde ex <lb/>ſupradicta .4. nobis cognitus erit ar-<lb/>cus <var>.a.d.</var> & conſequenter cognoſce-<lb/>mus at cum <var>.d.q.</var> eius reſiduum, ſeu <lb/>compoſitum, 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 ſi hac via inuenire deſideras, <lb/>cui orizonti propoſita ſtella oriatur <lb/>cum eodem eclipticę puncto <var>.a.</var> lon-<lb/>gitudinis, hoc aliud nihil eſſet, quam <lb/>cognoſcere 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 ſci-<lb/>ret vnum illorum angulorum quem <lb/>æquator efficit cum orizonte, reli-<lb/>qua illicò ei innoteſcent, ſed dictus <lb/>angulus <var>.b.</var> iam diximus quomodo <lb/>cognoſcatur.</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 ſcire velle <reg norm="punctum" type="context">pũctum</reg> <lb/> + <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> + eclipticę, cum quo Procyon oritur <lb/>polo .44. o. dato, quod ſtella in gra <num value="19">.<lb/>19.</num> cum min .24. </s> + <s xml:id="s3835" xml:space="preserve">Cancri, reperiatur <lb/>diſtans ab ecliptica per gra .16. min <num value="10">.<lb/>10.</num> meridiem verſus. </s> + <s xml:id="s3836" xml:space="preserve">vnde <reg norm="arcus" type="simple">arcꝰ</reg> <var>.a.q.</var> <lb/>erit gra .70. min .36. <reg norm="eiusque" type="simple">eiusq́;</reg> ſinus par-<lb/>tium 94321. talium qualium totalis <lb/>eſt .100000. arcus verò <var>.a.o.</var> gra .16. <lb/>minut .10. ſinus erit 27845. angu-<lb/>lus autem <var>.a.q.e.</var> declinationis zodia <lb/>ci ab ęquatore grad .23. min .30. cu-<lb/>ius ſinus eſt .39875. </s> + <s xml:id="s3837" xml:space="preserve">Quare ex ſupra-<lb/>dictis rationibus angulus <var>.a.b.q.</var> erit <lb/>gra .82. mi .24. cuius <reg norm="ſinus" type="simple">ſinꝰ</reg> erit .99122. <lb/>arcus vero <var>.a.b.</var> gra .22. minu .17. cu-<lb/>ius ſinus erit .37945. </s> + <s xml:id="s3838" xml:space="preserve">angulus deinde <lb/><var>o.e.b.</var> trianguli <var>.o.e.b.</var> eſt gra .46. mi <var>.<lb/>o.</var> altitudinis æquatoris ab orizonte, <lb/>cuius ſinus eſt .71934. angulus ſimili <lb/>ter <var>.o.b.e.</var> medio coniuncti, quibus <lb/>rectus perſicitur, arcus etiam <var>.o.b.</var> no <lb/>tus eſt grad .6. min .7. cuius ſinus eſt <num value="10655">.<lb/>10655.</num> cum ſit 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 ſupradicta an-<lb/>gulus <var>.e.o.b.</var> hoc eſt <var>.a.o.d.</var> erit. grad <num value="36">.<lb/>36.</num> min .39. cuius ſinus 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="ſinus" type="simple">ſinꝰ</reg> <lb/>erit .81998. arcus verò <var>.d.o.</var> gra .19. <lb/>min .51. cuius ſinus erit .33957 ar-<lb/>cus autem gra .11. min .42. cuius <reg norm="ſinus" type="simple">ſinꝰ</reg> <lb/>erit .20270. vnde <reg norm="arcus" type="simple">arcꝰ</reg> <var>.d.q.</var> reſiduus <lb/>ex <var>.a.q.</var> erit gra .58. min .54. complementum <reg norm="autem" type="wordlist">aũt</reg> quartæ erit gra .31. mi .6. hoc eſt gra <num value="1">.<lb/>1.</num> ſigni Leonis. cum min .6.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head471" style="it" xml:space="preserve">De ſphæroide duplæ ſpbær æ propoſit æ.</head> + <head xml:id="head472" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3841" xml:space="preserve">MOdus autem inueniendi ſphæroidem ex dato axe, quod duplum ſit ſphæ-<lb/>ra propoſita, talis eſt.</s> + </p> + <p> + <s xml:id="s3842" xml:space="preserve">Sit exempli gratia <var>.a.b.c.</var> ſphæra propoſita. cuius ſemidia meter ſit <var>.o.c.</var> ſemiaxis <lb/>vero ſphæroidis ſit <var>.d.x.</var> cuius dimidium ſit <var>.u.x.</var> </s> + <s xml:id="s3843" xml:space="preserve">tunc ex doctrina .9. ſexti 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 ſicut ſe 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 ſphæræ ita ſe habcat ad <var>.e.f.</var> ex .10. ſexti, quæ <lb/><var>e.f.</var> erit reliqua axis quæſita. </s> + <s xml:id="s3845" xml:space="preserve">Vnde conſtituta cum fuerit ellipſis <var>.d.f.t.e.</var> ex dictis axi-<lb/>bus, </s> + <s xml:id="s3846" xml:space="preserve">deinde circumuertendo ellipſim circa maiorem axem, conſtituemus ſphæroi-<lb/>dem oblongam, ſi autem circumuertemus ipſam circa minorem axim conſtituemus <lb/>ſphæroidem prolatam.</s> + </p> + <p> + <s xml:id="s3847" xml:space="preserve">Quod autem talis operatio rationalis ſit, nulli dubium erit, que<unsure/>tieſcunque co-<lb/>gnoſcet 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 ſit cono <var>.e.u.f.</var> ex lemmate collecto ab <lb/>11. duodecimi, conus <var>.e.d.f.</var> duplus exiſtit 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 ſphæra, & cyllindro ſphæra <var>.a.c.b.q.</var> quadrupla ſit cono <var>.a.<lb/>c.b.</var> ipſa conſequenter dupla erit cono <var>.e.d.f.</var> ſed ex .29. primi de conoidalibus, dimi <lb/>dium ſphæroidis <var>.e.d.f.t.</var> hoc eſt <var>.e.d.f.</var> dupla eſt cono <var>.e.d.f</var>. </s> + <s xml:id="s3850" xml:space="preserve">Quare talis medietas <lb/>æqualis eſt ſphæræ propoſitæ, totaq́ue ſphæroides dupla erit ſphærę datæ. </s> + <s xml:id="s3851" xml:space="preserve">Quod <lb/>autem dico de proportione dupla, idem infero de qualibet alia, ſumendo <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">Sphęram autem inuenire quæ dimidia ſit ſphæroidis propoſitæ nullius erit nego-<lb/>tij, quotieſcunque inuentus fuerit modus diuidendi vnam datam proportionem in <lb/>tres æquales partes.</s> + </p> + <p> + <s xml:id="s3853" xml:space="preserve">Sit propoſita ſphæroides <var>.e.f.d.t.</var> cuius axes ex conſequentia dantur <var>.e.f.</var> et <var>.d.t.</var> quę <lb/>quidem ſphæroides ſit primo oblonga, et <var>.u.x.</var> ſit dimidium axis maioris. </s> + <s xml:id="s3854" xml:space="preserve">imagine-<lb/>tur etiam conus <var>.e.u.f.</var> vt ſupra. </s> + <s xml:id="s3855" xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc <lb/>eſt, vt ſphæra <var>.a.b.c.q.</var> ſit dimidium ipſius ſphæroidis, vnde conus <var>.a.c.b.</var> æqualis erit <lb/>cono <var>.e.u.x.</var> vt ſupra demonſtratum eſt, & ſit <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 viſum ſuperius fuit, quod eadem proportio erat ipſius <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> ſed <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 eſt <var>.o.b.</var>) medię proportionales ſunt, eo quod cum <var>.g.h.</var> media <lb/>proportionalis ſit inter <var>.u.x.</var> et <var>.o.c.</var> & proportio <var>.o.b.</var> ad <var>.e.x.</var> æqualis ſit ei, quæ <var>.u.x.</var> <lb/>ad <var>.g.h.</var> hoc eſt 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 quotieſcunque 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> ipſa <var>.o.c.</var> vel <var>.o.b.</var> erit ſemi <lb/>diameter ſphæræ quæſitę. </s> + <s xml:id="s3860" xml:space="preserve">eodem modo faciendum erit ſi ſphęroides 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 ſe habet, duo ſcilicet triangula inuenire, æqualia dua-<lb/>bus ſuperficiebus rectilineis propoſitis, quę quidem triangula ſint eiuſdem <lb/>alritudinis, & quod <reg norm="vnunquodque" type="context">vnũquodque</reg> habeat angulum æqualem angulo pro<unsure/>poſito, & <reg norm="quod" type="simple">ꝙ</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 propoſitæ ſuperflcies <var>.c.y.</var> duo verò anguli dati ſint <var>.r.s.</var> <lb/>cum voluerimus inuenire duo triangula (quæ ſint <var>.a.i.u.</var> et <var>.n.t.x.</var> ) tali conditio-<lb/>ne prædita, quod angulus, a. æqualis ſit angulo <var>.s.</var> & angulus <var>.t.</var> angulo <var>.r.</var> & quod <lb/>angulus <var>.x.</var> ſimul 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">gulũ</reg> <var>.a.i.u.</var> æquale ſit ſuperficiei <var>.<lb/>c.</var> reliquum verò ſuperficiei <var>.y.</var> <lb/>Ex duabus ſuperficiebus <var>.c.</var> et <var>.y.</var> <lb/>conſtituemus duo quadrata, per vl <lb/>timam ſecundi Eucli. accipiemus, <lb/></s> + <s xml:id="s3863" xml:space="preserve">deinde duo latera tetragonica ip-<lb/>ſorum quadratorum, & inuenie-<lb/>mus tertiam lineam in continua <lb/>proportionalitate cum illis lateri-<lb/>bus ex .10. ſexti, ſeruabimus po-<lb/>ſtea extremas illarum, quæ ſint <var>.z.</var> <lb/>et <var>.l.</var> quarum proportio, <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quæ inter duas propoſitas ſuperfi-<lb/>cies reperitur ex .18. ſexti, accipie <lb/>mus, deinde lineam aliquam cu-<lb/>inſuis longitudinis, quæ ſit <var>.q.g.</var> ſu-<lb/>pra quam conſtituemus in puncto <lb/>q. angulum <var>.m.q.g.</var> ęqualem angu-<lb/>lo <var>.s.</var> & angulum <var>.m.q.K.</var> æqualem <lb/>angulo <var>.r.</var> ex .23. primi, poſtea ve-<lb/>rò à quouis puncto ipſius lineæ <var>.q.<lb/>m.</var> puta <var>.o.</var> ducetur <var>.o.f.</var> vſque ad <var>.q.<lb/>g.</var> quorſum volueris, producendo <lb/>ipſam <reg norm="vſque" type="simple">vſq;</reg>. ad <var>.d.</var> ita quod propor-<lb/>tio <var>f.o.</var> ad <var>.o.d.</var> ſit vt <var>.z.</var> ad <var>.l.</var> ex .10. <lb/>ſexti, ducendo poſtea 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> ſecatur ab <var>.q.K.</var> in puncto <var>.b.</var> <lb/>protrahemus <var>.b.o.p.</var> vnde ex ſimi-<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 eſt vt <var>.z.</var> ad <var>.l.</var> hoc eſt vt <var>.c.</var> ad <var>.y</var>. </s> + <s xml:id="s3864" xml:space="preserve">quare <reg norm="triangulum" type="context">triangulũ</reg> <var>.p.q.o.</var> ita erit proportio <lb/><reg norm="natum" type="context">natũ</reg> triangulo <var>.o.q.b.</var> vt <var>.c.</var> ad <var>.y.</var> conſtituo deinde ex .25. fexti duo triangula ſimi-<lb/>lia duobus <var>.p.q.o.</var> et <var>.o.q.b.</var> <reg norm="æqualiaque" type="simple">æqualiaq́</reg> <var>.c.</var> et <var>.y.</var> quę ſint <var>.a.i.u.</var> et <var>.n.t.x.</var> ſecetur poſtea <var>.q.<lb/>g.</var> in puncto <var>.æ.</var> ita, quod <var>.q.æ.</var> æqualis ſit <var>.i.a.</var> duco poſtea <var>.æ.e.</var> æquidiſtantem. ad <var>.p.b.</var> <lb/>& ſic 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> eaſdem 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/>uerſione in Ptolomeum. Deincendio carbo-<lb/>num à vento.</head> + <head xml:id="head476" style="it" xml:space="preserve">Clariβimo Dominico Moreſino.</head> + <p> + <s xml:id="s3865" xml:space="preserve"><emph style="sc">SCio</emph> propoſitam tibi quæſtionem te diu agitauiſſe, nectamen ſolutio-<lb/>nem aſſequi potuiſſe, aduerte igitur ipſam falſam, ideſt impoſſibilem <lb/>eſſe, quemadmodum etiam decimumoctauum quæſitum propoſitum à <lb/>Cardano Tartaleæ, ab ipſo Tarralea ſolutum minimè fuit. </s> + <s xml:id="s3866" xml:space="preserve">Quiquidem <lb/>Tartalea vult circulum deſcribi circa triangulum per quintam libri quarti Euclidis, <lb/>vt in fine ferè quintæ partis ſuarum menſurarum affirmat, neque videt in quinta <lb/>quarti Euclidem vti vndecima primi, & in vndecima primi, quarta aut octaua eiuſ-<lb/>dem, quas, ipſe Euclides oſtenſiuè non demonſtrauit. </s> + <s xml:id="s3867" xml:space="preserve">Quapropter oportebat Tar-<lb/>taleam demonſtraſſe omnes propoſitiones ad hoc neceſſarias oſtenſiuè <reg norm="vſque" type="simple">vſq;</reg> ad pri-<lb/>mas indemonſtrabiles, quia ad demonſtrandam ſcientificè <reg norm="aliquam" type="context">aliquã</reg> propoſitionem, <lb/>aut à propoſitione in propoſitionem vſque ad prima principia vniuerſalia ( vt ali-<lb/>quando ego feci) eſt retrogradandum, aut ab ipſis principijs incipiendum ſucceſſi-<lb/>uè eouſque progrediendo donec ad propoſitionem quam demonſtrare volumus <lb/>perueniamus.</s> + </p> + <p> + <s xml:id="s3868" xml:space="preserve">Quod ad Ptolomeum in geographia attinet, dico eum mihi non ſatisfacere, cum <lb/>ſumit portionem arcus circuli maioris inter vnam ciuitatem, & aliam, ea ratione <lb/>quam deſcribit. </s> + <s xml:id="s3869" xml:space="preserve">Quod ſi vſus fuiſſet modo Menelai, ab ipſomet deinde in <reg norm="ſuum" type="context">ſuũ</reg> Al-<lb/>mageſtum vſurpato, aut Monteregij triangulorum ſphęricorum, 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 ſuum illud inſtrumentum geometricum attinet, eſt <reg norm="imperfectum" type="context">imperfectũ</reg>, <lb/>vt oſtendi domino Pandulfo.</s> + </p> + <p> + <s xml:id="s3871" xml:space="preserve">Motum autem aeris, aut mauis ventum, accendere ignem, non ſolum ratione an <lb/>tiperiſtaſis, quam affers euenit, ſed etiam quia à carbonibus accenſis 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 propoſitionis .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. propoſitionem lib. 2. Monteregij. cum necſcias <lb/>reperire diametrum circuli circunſcriptibilis circa propoſitum 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 ſit b aſis tantummodo ſimul cum angulo, qui ipſi baſi opponitur.</s> + </p> + <p> + <s xml:id="s3873" xml:space="preserve">Imagineris igitur triangulum datum eſſe obtuſiangulum <var>.a.b.g.</var> cuius baſi <var>.b.<lb/>g.</var> ſit nobis data ſimul cum angulo <var>.a.</var> ei oppoſito, obtuſoq́ue; </s> + <s xml:id="s3874" xml:space="preserve">Conſidera etiam cir-<lb/>culum <var>.a.b.g.q.</var> ipſum trian gulum circunſcribentem, cuius diameter <var>.q.e.p.</var> tranſeat <lb/>per <var>.m.</var> punctum medium ipſius <var>.b.g.</var> <reg norm="tunc" type="context">tũc</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">angulumq́</reg> <var>.q.e.g.</var> duplum eſſe an <lb/>gulo <var>.q.p.g.</var> ex .19. eiuſdem, vnde æqualem angulo <var>.a.</var> qui etiam duplus eſt 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 ſinum <var>.m.g.</var> arcus <var>.g.p.</var> & quia <var>.m.g.</var> vt <lb/>dimidium ipſius <var>.b.g.</var> tibi data eſt, cognoſces <lb/>etiam <var>.p.g.</var> vt <var>.m.g.</var> & ſic 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> eſt æquale ei quod fit ex <var>.b.m.</var> <lb/>in <var>.m.g.</var> ideo cum diuiſum 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, ſi <reg norm="cum" type="context">cũ</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 ſignificati, talis eſt. <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/>ſitus <var>.a.c.</var> minor autem <var>.b.d.</var> cum ergo <lb/>volueris inuenire punctum circunfe-<lb/>rentiæ correſpondentem 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 ſit <var>.h.</var> <reg norm="latusque" type="simple">latusq́</reg> <reg norm="tetragonicum" type="context">tetragonicũ</reg> pro-<lb/>ducti <var>.a.e.</var> in <var>.e.c.</var> quod ſit <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 ſecundi axis ita ſe habeat, quæ po <lb/>ſtea iuncta axi maiori, ad angulosrectos in puncto <var>.e.</var> dabit ſitum puncti <var>.f.</var> quæſiti ex <lb/>dicta .21. primi lib. Pergei, ſed talis modus prolixus eſt.</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 propoſitus maior axis <var>.q.p.</var> minor verò <var>.e.c.</var> ad angulos rectos ſe inuicem <lb/>ſecantes in puncto <var>.o.</var> deſcribatur circulus <var>.q.n.p.a.</var> cuius diameter ſit 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> deſignetur poſtea ſeparatim circulus <var>.u.b.n.</var> cuius diameter æqualis ſit ſe <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 ſit <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/>ſecabitur à minimo circulo in puncto <var>.i.</var> cum itaque longitudo <var>.u.i.</var> menſurata fue-<lb/>rit in <var>.u.b.</var> maximi circuli à puncto <var>.u.</var> habebimus propoſitum.</s> + </p> + <p> + <s xml:id="s3882" xml:space="preserve">Cuius reiratio eſt, quia <var>.u.b.</var> mediocris circuli diuiditur à gyro minimi in puncto <lb/>i. eadem proportione, qua diuiſa eſt <var>.u.n.</var> in puncto <var>.c.</var> quod manifeſtum eſt exſimi-<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> ſed ita <lb/>eſſe oportet parallelas maximi circuli, quotieſcunque circunferentia ipſius ellipſis <lb/>tranſitura ſit per <var>.c.</var> vt in .51. cap. meæ gnomonicæ oſtenſum 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 deſignandi 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, quotieſcunque nobis propoſita <lb/>fuerint duo puncta <var>.a.</var> et <var>.b.</var> ſimul 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="conſtituant" type="context">conſtituãt</reg> <lb/>angulum æqualem dato, & ipſæ directè <reg norm="con" type="context">cõ</reg> <lb/>iunctæ conſtituant lineam æqualem da-<lb/>tæ. </s> + <s xml:id="s3884" xml:space="preserve">Nam ducatur linea indefinita per <lb/>puncta propoſita, cuius lineæ, pars illa, quę <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 ſit <var>.o.c.</var> reliqua verò ſit <var>.o.e.</var> </s> + <s xml:id="s3885" xml:space="preserve">deinde aperiatur circinus quantum <var>.<lb/>o.c.</var> <reg norm="poſitoque" type="simple">poſitoq́;</reg> vno pede in <var>.b.</var> <reg norm="deſignentur" type="context">deſignẽtur</reg> cum altero duo arcus <var>.n.K.</var> poſito iterum vno <lb/>pede in <var>.a.</var> deſignentur alij duo arcus inter-<lb/>ſecantes 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> tranſibit, quam <var>.n.K.</var> mente <reg norm="con- cipio" type="context">cõ-<lb/>cipio</reg>, vt axis minor <reg norm="vnius" type="simple">vniꝰ</reg> ellipſis, cuius <var>.e.c.</var> <lb/>ſit axis maior, quibus axibus mediantibus <lb/>deſignetur ellipſis <var>.n.c.K.e.</var> conſidero dein <lb/>de <var>.a.b.</var> vt <reg norm="chordanvnius" type="context">chordãvnius</reg> circuli, ſeu portio-<lb/>nis circularis, quæ capax ſit vnius anguli <lb/>æqualis angulo <var>.d.</var> propoſito, ex .32. tertij <lb/>Euclid. cuius circunferentia, circunferen-<lb/>tiam ipſius ellipſis ſecabit in <reg norm="duobus" type="simple">duobꝰ</reg> punctis <lb/>quorum <reg norm="vnum" type="context">vnũ</reg> ſit <var>.i.</var> à quo protractæ cum fue <lb/>rint duæ lineæ <var>.a.i.</var> et <var>.i.b.</var> habebis propoſitum, 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 monſtrauit <lb/>neſcio quis, ni fallor, talis eſt. </s> + <s xml:id="s3888" xml:space="preserve">Deſignato meridiano <var>.l.b.m.q.</var> <reg norm="ductisque" type="simple">ductisq́;</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> ſit <lb/>verticalis <var>.b.q.</var> vero <reg norm="orizontalis" type="context">orizõtalis</reg>, <reg norm="ductoque" type="simple">ductoq́;</reg> diametro <var>.f.h.</var> tropici Cancri ſe-<lb/>cundum altitudinem poli datam, <reg norm="deſcriptoque" type="simple">deſcriptoq́;</reg> dimidio circulo <var>.f.z.h.</var> ipſius paralleli, <lb/><reg norm="inuentoque" type="simple">inuentoq́;</reg> puncto <var>.z.</var> horæ propoſitę, & 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 poſtea <var>.f.ω.</var> et <var>.<lb/>r.t.</var> vſque 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 poſtea gnomone <var>.g.s.</var> in orizontis axe, <reg norm="ductaque" type="simple">ductaq́;</reg> vmbrarum linea <var>.s.K.</var> parallela <lb/>orizontali, <reg norm="ductaque" type="simple">ductaq́;</reg> <var>.y.g.K.</var> ad terminandam <var>.s.K.</var> delineas deinde ſeparatim circulum <lb/><var>q.x.b.n.</var> magnitudinis prioris, qui quidem circulus ſignificet orizontem ipſum, 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">cẽ<lb/>tro</reg> <var>.g.</var> diſtantia, vt ſunt in diametro orizontali prioris circuli, ducis poſtea 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/>tranſire facis circunferentiam circuli per <reg norm="quintam" type="context">quintã</reg> quarti Euclidis, poſtea in dicto dia-<lb/>metro accipis punctum <var>.t.</var> ita diſtans à centro, & ex eadem parte, vt in priori circulo, <lb/>à quo puncto ducis <var>.t.u.</var> parallelam <var>.x.n.</var> vſque 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> ipſius circularis circunferentiæ <var>.g.u.</var> inde-<lb/>terminatam, quam poſtea terminas in puncto <var>.K.</var> ita quod <var>.g.K.</var> æqualis ſit <var>.s.K</var>. </s> + <s xml:id="s3891" xml:space="preserve">Dicis <lb/>poſtea punctum <var>.K.</var> in eodem ſitu reperiri, reſpectu duorum diametrorum <var>.q.b.</var> me-<lb/>ridiani. et <var>.l.m.</var> verticalis, vt decet, & oportet punctum horæ propoſitę exiſtere.</s> + </p> + <p> + <s xml:id="s3892" xml:space="preserve">Quod quidem dico eſſe falſum, propterea quod perpendiculares quas cogita-<lb/>mus cadere à punctis circunferentiæ cuiuſuis paralleli ſupra 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 ſecantem æquatorem, omnes caduntin gyro elliptico, oxygonio, ſeu de-<lb/>fectionali, & non circulari. </s> + <s xml:id="s3893" xml:space="preserve">Vnde per ſupradicta tria puncta <var>.n.ω.x.</var> oporteret tranſi <lb/>re talem circunferentiam, & non <reg norm="circularem" type="context">circularẽ</reg>, quæ circunferentia eſſet vnius ellipſis, <lb/>cuius minor axis in diametro <var>.b.q.</var> eſſet .ab <var>.ω.</var> vſque ad <var>.i.</var> terminum ſini <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="tranſiſ- ſet" type="context">trãſiſ-<lb/>ſet</reg> per punctum <var>.c.</var> medium inter <var>.ω.</var> et <var>.i.</var> quę 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">tiguaq́;</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 ſinus altitudinis horę, et <var>.t.u.</var> æqualis eſſet <var>.r.z.</var> communi ſectioni paralle <lb/>li cum almicantarat ex .34. primi Euclid. et <var>.u.g.</var> æqualis eſſet <var>.o.y.</var> communi ſectioni <lb/>almicantarat cum meridiano, vel cum azimut illius horæ ex .4. primi, cum <var>.g.t.</var> æqua <lb/>lis ſit ipſi <var>.o.r.</var> et <var>.t.u.</var> ipſi <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 ſe haberent, diſtan-<lb/>tia verò inter <var>.K.</var> et <var>.g.</var> <reg norm="iam" type="context">iã</reg> rectè ſum-<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">plurimũ</reg> <lb/>(in gyro circulari ſumptum) extra <lb/>puncta interſectionum ipſius 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> falſos, & non æquales il-<lb/>lis, qui fiunt ab azimut horæ cum <lb/>verticali, & cum meridiano, quæ <lb/>omnia ex cap .52. meæ gnomonicę <lb/>facilè videre potes.</s> + </p> + <p> + <s xml:id="s3897" xml:space="preserve">Nectacere volo quod punctum <lb/>u. verum, hoc eſt ellipticum, inue-<lb/>niri poſſet ea via quam ſcripſi in <lb/>eodem .52. cap. qua mediante do-<lb/>cui demum inuenire punctum <var>.π.</var> <lb/>orizontis, quamuis in præſenti ca<lb/>ſu <var>.ω.λ.</var> perpendicularis eſſet ſupra <lb/>minorem axem ipſius ellipſis, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> ſupra maiorem axem, quod ta-<lb/>men minimè mutat ordinem, imò <lb/>rationes eędem ſunt, tam in vna, <lb/>quam in alia operatione, ſed vt il-<lb/>licò <reg norm="idipsum" type="context">idipsũ</reg> habeas, fac vt <var>.t.u.</var> æqua <lb/>lis. ſit <var>.r.z</var>. </s> + <s xml:id="s3898" xml:space="preserve">& tunc punctum <var>.K.</var> erit <lb/><reg norm="quæſitum" type="context">quæſitũ</reg>, quod ego in .52. cap. meę <lb/>gnomonicę, ijs verbis ſignificaui.</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">medioq́;</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 ſcribere promiſi delineandi lineas horarias communes in <lb/>pariete perpendiculariter ad orizontem rectum, declinantem à meridiano, <lb/>ſumendus eſt ex .46. cap. meæ gnomonicæ, hocſcilicet ordine.</s> + </p> + <p> + <s xml:id="s3901" xml:space="preserve">Sit exempli gratia, orizon hic ſubſcriptus. or. oc <var>.M.S.</var> diuiſus à meridiana <var>.M.<lb/>S.</var> et verticali ſeu æquinoctiali. or. oc. </s> + <s xml:id="s3902" xml:space="preserve">Sitq́ue <var>.e.t.</var> communis ſectio muri cum ori-<lb/>zonte, et <var>.g.n.</var> ſit gnomon perpendicularis ipſi muro, vnde ex dictis in mea gnomo-<lb/>nica, cognoſcemus in ipſa murali orizontali totam <var>.e.t.</var> inter meridianam orizonta-<lb/>lem, & æquinoctialem orizontalem, cognoſcemus etiam partem <var>.g.t.</var> ipſius æ-<lb/>quinoctialis orizontalis, quam quidem accipiamus in rectitudine ipſius muralis ori <lb/>zontalis, quæ quidem ſit <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> deſignetur vna medietas circuli verſus <var>.e.</var> cuiuſuis magnitudinis, quæ di-<lb/>uiſa in .12. partes ęquales, ſignificabit <reg norm="medietatem" type="context">medietatẽ</reg> æquatoris, <reg norm="protrahanturque" type="simple">protrahanturq́;</reg> lineæ oc <lb/>cultæ à centro <var>.G.</var> per ſectiones circunferentię dimidij circuli, quæ fignificabunt <reg norm="con" type="context">cõ</reg> <lb/>munes ſectiones 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> ſignifi-<lb/>cabit communem ſectionem æquatoris cum muro propoſito, & erit ęquędiſtans me <lb/>ridianæ murali ex .6. vndecimi Eucli. eo quod ex .19. eiuſdem <reg norm="vnaquæque" type="simple">vnaquæq;</reg> illarum, per <lb/>pendicularis eſt tali orizonti. </s> + <s xml:id="s3903" xml:space="preserve">Videantur nunc puncta communia iſti <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> ſignificat punctum axis mundi, & meridianæ in mu-<lb/>ro propoſito, eo quod in tali ſitu ſphæræ rectæ, dictum punctum reperitur in orizon <lb/>re, cum <var>.M.s.</var> non ſolum ſit meridiana orizontalis, ſed etiam axis mundi, </s> + <s xml:id="s3904" xml:space="preserve">deinde nul <lb/>li dubium eſt quin meridiana muralis ſit 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">linearũ</reg> horariarum erit <lb/>ſub orizontali <var>.e.t.G.</var> alterum <lb/>vero <reg norm="dimidium" type="context">dimidiũ</reg> ſupra ipſam, opor <lb/>tet quod quę ſupra ſunt produ-<lb/>cantur à parte .oe. ſub orizonta-<lb/>li, ab alia parte meridianæ, & <lb/>talis erit effigies horologij mura <lb/>lis in hoc ſphęrę ſitu, hoc eſt ver <lb/>ſus quartam orientalem auſtra-<lb/><reg norm="lemque" type="simple">lemq́;</reg>, vnde orizontalis <var>.e.t.</var> <lb/>erit ſemper horæ .6. matutinæ, <lb/>ſecunda verò ab ipſa erit horæ. <lb/>7. tertia <reg norm="autem" type="context">autẽ</reg> horæ .8. & ſic 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">Quotieſcunque verò angu-<lb/>lus <var>.n.g.e.</var> minor erit maxima <lb/>Solis declinatione, & Sol fuerit <lb/>in parte auſtrali ab æquatore <reg norm="cum" type="context">cũ</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 ipſo Sole à mane vſque ad veſperam.</s> + </p> + <p> + <s xml:id="s3908" xml:space="preserve">Huius quidem rei ſpeculatio, vnicuique manifeſta erit, qui rationes .46. cap. no-<lb/>ſtræ gnomonicæ prius intellexerit, vbi manifeſtè apparet proportionem ſemidiame <lb/>tri horologij (ſi ita eam appellare licet) ad ſemidiametrum æquatoris horarij ſem-<lb/>per eſſe, vt <var>.e.t.</var> ad <var>.t.g.</var> hoc eſt 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 ſemper iuſto breuiores ſint.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head xml:id="head487" style="it" xml:space="preserve">Declar atio quorundam verborum noſtræ<unsure/> Gnomonicæ. <lb/><reg norm="Defenſioque" type="simple">Defenſioq́ꝫ</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 ſignificas te totum <num value="52">.<lb/>52.</num> caput meæ gnomonicæ intelle-<lb/>xiſſe, præter illa verba, quæ etiam <lb/>ſuperioribus diebus ad te ſcripſi, <lb/>hoc eſt.</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">medioq́;</reg> azimut Solis, pariter ho-<lb/>rologia fabricari poterunt.</s> + </quote> + <p> + <s xml:id="s3913" xml:space="preserve">Quapropter nealiquid tibi de-<lb/>ſit, ſcire debes, menihil aliud, eo <lb/>in loco inferre voluiſſe, quàm <reg norm="quod" type="wordlist">qđ</reg> <lb/>punctum horæ propoſitæ in plano <lb/>horologij orizontali reperiri po-<lb/>teſt, ope longitudinis vmbræ gno-<lb/>monis, & eius declinationis à ver-<lb/>ticali linea, ſeu à meridiana orizon <lb/>tali, iam in ipſo horologij plano <lb/>ductis.</s> + </p> + <p> + <s xml:id="s3914" xml:space="preserve">Exempli gratia, ſit analemma <var>.<lb/>l.q.m.b.</var> in quo <var>.l.m.</var> ſit verticalis <var>.q.<lb/>b.</var> verò orizontalis <var>.f.n.h.</var> autem ſit <lb/>ſemicirculus, cuiuſuis paralleli æqui <lb/>noctiali, cuius diameter fit <var>.f.h.</var> et <var>.<lb/>n.</var> ſit Solis locus in ipſo parallelo: </s> + <s xml:id="s3915" xml:space="preserve">n.<lb/>r. autem ſit rectus ſinus arcus <var>.f.n</var>: et <var>.<lb/>r.o.z.</var> ſectio communis ipſius almi-<lb/>cantarat cum meridiano, et <var>.s.a.</var> <reg norm="con" type="context">cõ</reg> <lb/>munis ſectio 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 ſinus alti-<lb/>tudinis ipſius 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 cognoſcitur longitudo vmbræ gnomonis hoc eſt <var>.s.a</var>.</s> + </p> + <p> + <s xml:id="s3916" xml:space="preserve">Cum autem dico, <reg norm="medioque" type="simple">medioq́;</reg> azimut Solis, nihil aliud ſigniſicare volo, niſi angu-<lb/>lum, quem terminat linea azimutalis horologij, hoc eſt vmbra gnomonis cum li-<lb/>nea meridiana, ſeu cum verticali in ipſo plano horologij. </s> + <s xml:id="s3917" xml:space="preserve">qui quidem anguli, æqua <lb/>les ſunt ijs, qui in triangulo conſtituto 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 eſt, angulus verò terminatus ab <var>.n.r.</var> et <var>.o.z.</var> il <lb/>le eſt quem conſtituit azimut cum verticali, vel ipſi æqualis, vt coalternus, reliquus <lb/>verò in <reg norm="puncto" type="context">pũcto</reg> <var>.o.</var> ille eſt <reg norm="quem" type="context">quẽ</reg> azimut facit <reg norm="cum" type="context">cũ</reg> meridiano, vel ipſi ęqualis vt coalternus.</s> + </p> + <p> + <s xml:id="s3918" xml:space="preserve">Vnde quotieſcunque volueris in aliquo plano, orizonti parallelo, lineas hora-<lb/>rias ducere, iudico optimum fore ſi ſeparatim deſignatæ fuerint hæ tres figuræ, hoc <lb/>eſt analemma meridianum, vel azimutale, vt ita dicam, </s> + <s xml:id="s3919" xml:space="preserve">deinde parallelus <reg norm="inſeruiens" type="context">inſeruiẽs</reg> <lb/>pro tropicis, vt ego feci cap .51. meæ gnomonicæ, quæ duæ figuræ, <reg norm="ſufficientes" type="context">ſufficiẽtes</reg> <reg norm="erunt" type="context">erũt</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 ſeparatam <lb/>deſignaui, quę erit circulus <var>.H.I.<lb/>K.</var> eiuſdem magnitudinis cum ana-<lb/>lemmate, in quo, ductis duobus <lb/>dia metris inuicom ad rectos, quo-<lb/>rum vnus <var>.H.I.</var> ſignificet orizon-<lb/>talem lineam, reliqua verò <reg norm="ver- ticalem" type="context">ver-<lb/>ticalẽ</reg>, ducatur poſtea <var>.s.a.</var> tam di-<lb/>ſtans ab <var>.H.I.</var> quanta eſt 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/>ſius zenit, accipiantur arcus æqua-<lb/>les ijs ipſorum azimut, quos termi-<lb/>nat zenit, & ipſi almicantarat, vt <lb/>exempli gratia, accipiemus arcum <lb/><var>k.L.</var> æqualem arcui <var>.L.z.</var> ipſius ana-<lb/>lemmatis, ducta poſtea linea occul <lb/>ta <var>.o.L.</var> ſignabimus azimutalem <var>.s.a.</var> <lb/>in puncto <var>.a.</var> vbi hæ duæ lineæ ſein <lb/>uicem ſecant, & ſic habebimus iu-<lb/>ſtam <reg norm="quantitatem" type="context">quãtitatem</reg> ipſius 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> ſumatur <var>.o.r.</var> à centro <var>.<lb/>o.</var> ęqualis ei quę in analemate repe <lb/>ritur, quæ vna portio eſt commu-<lb/>nis ſectionis 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> vſque ad circunferentiam, in qua accipiatur <var>.r.n.</var> æqua <lb/>lis ei quæ eſt in parallelo, ducatur poſtea <var>.o.n.M.</var> & habe bimus triangulum <var>.o.r.n.</var> ſi-<lb/>milem <reg norm="æqualemque" type="simple">æqualemq́;</reg> triangulo iam ſupradicto. </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 ęqualis, quem azimut con-<lb/>ſtituit cum verticali, ita quod ſi talis circulus <var>.H.k.I.</var> eſſet planum horologij orizon-<lb/>talis, ſuppoſito <var>.o.</var> pro pede gnomonis, ſecando poſtea <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 ſit ad partem ſibi conuenientem, reſpectu <var>.o.<lb/>k.</var> ipſa pro verticali ſuppoſita, quod tibirelinquo, cum hoc facillimum ſit, 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 ſignificas Chriſtophorum 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 eſt in pagin .161. <lb/>ita inquit.</s> + </p> + <quote> + <s xml:id="s3927" xml:space="preserve">Non enim deſunt, qui vel omninò negent, inter quos eſt Ioannes Baptiſta Be-<lb/>nedictus in ſua gnomonica cap .70. et .71. vbialia, & multo longiore ratione cona-<lb/>tur arcus ſignorum deſcribere, vel certe dubitent, hoc modo rectè poſſe deſcribi ar-<lb/>cus ſignorum, cum rationem non videant, qua hæc noſtra deſcriptio quam quidem <lb/>omnes ſcriptores ſine vlla demonſtratione tradunt nitatur.</s> + </quote> + <p> + <s xml:id="s3928" xml:space="preserve">Abſque dubio raptim tranſcurrit illa capita .70. 71. </s> + <s xml:id="s3929" xml:space="preserve">Reuerendus Clauius alio quin <lb/>non ſcripſiſſet, quòd ego alia & multo longiore ratione conatus ſim arcus ſignorum <lb/>deſcribere & c. præſertim cum eadem prorſus ratio, quæ ibi à me tradita eſt, illa ſit, <lb/>quam ip ſe ſuis ſcriptis inſeruit.</s> + </p> + <p> + <s xml:id="s3930" xml:space="preserve">Meus igitur modus in dictis capitibus traditus, minime diſcrepat ab eo, ſed ab il-<lb/>lorum modo, quorum opinio eſt 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. poſitæ, æqualia eſſe interuallis <var>.e.h</var>:h.u:u.n:n.m. et <var>.m.d.</var> præcedentis figuræ, <lb/>qui etiam ſupponunt <var>.t.e.</var> meæ figuræ .75. eſſe directè coniuncta cum linea <var>.e.h.u.n.<lb/>m.d.</var> & propterea verſus finem .73. pag. dixi.</s> + </p> + <quote> + <s xml:id="s3932" xml:space="preserve">Aduertat autem quam diligentiſſime quiſque ne ſe decipi patiatur à ſubſcripta fi <lb/>gura ſemicirculi <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 vetuſtioribus (quorum ſcripta ad meas manus <lb/>peruenerunt) qui ſumentes interualla <var>e.h</var>:h.u. & c. figuræ. pag .75. æqualia illis figu-<lb/>ræ pag .74. <reg norm="putauerunt" type="context">putauerũt</reg> lineam <var>.t.e.</var> directè coniunctam eſſe cum <var>.e.h.</var> & c. quod quidem <lb/>maximi erroris cauſa erat, </s> + <s xml:id="s3934" xml:space="preserve">& propterea cap .71. verum modum oſtendi, ſeruando il <lb/>lam eandem ſuppoſitionem, hoc eſt quod interſtitia <var>.e.h</var>: <var>h.u</var>: & c. figurę pag .75. æqua <lb/>lia ſint interſtitijs <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">Suppoſito deinde <var>.f.e.b.</var> lineam meridianam eſſe in plano orizontali, cęterę lineę <lb/>horarię erunt prędictę.</s> + </quote> + <p> + <s xml:id="s3936" xml:space="preserve">Stantibus igitur his ſuppoſitis, vt habeantur omnia ſcientificè, 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) poſuit in figura horologij, vt ipſe docuit <lb/>propoſitione ſecunda, lib. ſecundi, cuius baſis eſt triangulum <var>.H.I.F.</var> ſuæ figurę (exem <lb/>pli gratia pro quinta hora poſt 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, eſt in ſua <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 ſunt, quamuis in plano æquatoris <lb/>tres ſint, & c.</s> + </quote> + <p> + <s xml:id="s3938" xml:space="preserve">Angulus verò <var>.e.</var> quem dico rectum eſſe, in ſua figura eſt 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, eſt ſua <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 ſua eſt triangulum <var>.D.F.H.</var> vnde mea <var>.f.t.</var> reſpon-<lb/>det ſuę <var>.H.D.</var> & mea <var>.f.d.</var> ſuæ <var>.H.F.</var> & mea <var>.t.d.</var> ſuę <var>.D.F</var>. </s> + <s xml:id="s3940" xml:space="preserve">Quartum autem triangulum <lb/><var>f.t.e.</var> in mea figura, reſpondet ſuo <var>.H.D.I.</var> & meum punctum <var>.t.</var> ſuo. </s> + <s xml:id="s3941" xml:space="preserve">D, Nunc triangu <lb/>lum rectangulum, quod dico ſeparatim conſtituere, eſt illud tertium dictum corre-<lb/>ſpondens ſuo <var>.D.F.H.</var> vt ipſe facit in <reg norm="ſequenti" type="context">ſequẽti</reg> figura, quod ipſe vocat <var>.D.C.H.</var> & <reg norm="meus" type="simple">meꝰ</reg><unsure/> <lb/>radius <var>.t.x.</var> in ſua figura, ille eſt qui terminatur ab <var>.D.</var> & ab initio Tauri, & Virginis.</s> + </p> + <p> + <s xml:id="s3942" xml:space="preserve">Et quamuis ego non ſcripſerim talem ſiguram, vt ipſe fecit, nihilominus ipſam <lb/>verbis deſcripſi eomet modo, & propterea dixi.</s> + </p> + <quote> + <s xml:id="s3943" xml:space="preserve">Quam <reg norm="diuiſionem" type="context">diuiſionẽ</reg>, ſi in triangulo ſeorſum deſcripto inuenire voluerimus, res erit <lb/>inuentu facillima, cum rectum angulum <var>.f.t.d.</var> (reſpondentem ſuo <var>.H.D.C.</var>) prędicti <lb/>trianguli tertij ea ratione diuiſerimus, & 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 ſua, cum eademmet ipſa ſit.</s> + </p> + <p> + <s xml:id="s3945" xml:space="preserve">Citaui etiam Munſterum cap .30. eo quod in ea impreſſione, quam tunc prę mani <lb/>bus habui, vidi in ea figura, quam ipſe vocat fundamentum horologiorum, literam <lb/>c. poſitam eſſe loco <var>.f.</var> et <var>.f.</var> loco <var>.c.</var> quod cauſæ fuit, vt omnia mendoſa viderentur, re <lb/>centiores autem impreſſiones correctæ ſunt.</s> + </p> + <p> + <s xml:id="s3946" xml:space="preserve">Rurſus alio in loco mihi accidit vt repręhenderim Alexandrum Piccolomineum <lb/>in libris de ſphęra, qui quidem dicebat eas figuras ſuperficiales, quæ paucioribus an <lb/>gulis circunſcriberentur, capaciores eſſe alijs, dummodo earum periphæriæ eſſent <lb/>æquales.</s> + </p> + <p> + <s xml:id="s3947" xml:space="preserve">Nunc autem correctę ſunt eo in loco impreſſiones, & qui non viderit primas, pu-<lb/>tabit me immeritò ipſum repræhendere.</s> + </p> + <p> + <s xml:id="s3948" xml:space="preserve">Idem etiam dico de eo capite ipſius Piccolominei, in ijſdem libris, vbi tractat de <lb/>modo, quo vſi ſunt antiqui ad diuidendum zodiacum in .12. ſigna, quod erat circa <lb/>finem quarti libri.</s> + </p> + <p> + <s xml:id="s3949" xml:space="preserve">Nunc verò, in recentioribus impreſſionibus, illud caput poſitum non eſt. </s> + <s xml:id="s3950" xml:space="preserve">Impreſ <lb/>ſiones autem illæ, vbi talia dixit, duæ fuerunt, quarum prima erat anni .1540. ſecun <lb/>da verò .1552 Venetijs apud Andream Puteum.</s> + </p> + <p> + <s xml:id="s3951" xml:space="preserve">Alius verò locus ipſius Reuerendi Clauij, contra meas repręhenſiones, eſt circa <lb/>finem pag .298. & circa .299. vbi ita ſcribit.</s> + </p> + <quote> + <s xml:id="s3952" xml:space="preserve">Ex his liquido conſtat, non rectè à Ioan. Baptiſta Benedicto in ſua gnomonica ca <lb/>pit .49. repręhendi hancrationem deſcribendi horologij declinantis, qua omnes fe-<lb/>rè alij ſcriptores vtuntur, quoniam, vt ex demonſtratione à nobis allata conſtat, re-<lb/>ctè per eam lineæ ho rarię 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/>eſt, ſed nulla ratione noſtro contrarius, quia nos conſtituimus <var>.D.E.F.</var> angulum de-<lb/>clinationis plani à verticali circulo propriè dicto, ipſe autem loco huius anguli aſſu-<lb/>mit angulum declinationis eiuſdem plani à Meridiano circulo, vnde mirum non eſt <lb/>modum ipſius à noſtro diſcrepare. </s> + <s xml:id="s3954" xml:space="preserve">Quod ſi <reg norm="conſtitueremus" type="context">cõſtitueremus</reg> <var>.D.E.F.</var> angulum decli-<lb/>natio nis plani à Meridiano, ut ipſe (quemadmodum forſitan ab alijs putauit fieri) <lb/>& in reliqua deſcriptione progrederemur, vt tradidimus, proculdubio <reg norm="horologium" type="context">horologiũ</reg> <lb/>declinans perperam deſcriberetur, vt rectè docet.</s> + </quote> + <p> + <s xml:id="s3955" xml:space="preserve">Optimè ſcripſiſſet Reuerendus Clauius, ſi verum fuiſſet, quod antiqui ſumerent <lb/>declinationem ſuperius <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. meę gnomonicę ſumere 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 ſidera primum in Munſtero cap .16. ſuæ horologiographiæ, vbi clarè docet <lb/>accipere angulum compræhenſum inter meridianum, & planum propoſitum, vbi <lb/>etiam ponit quandam figuram ædificij cum pariete ſuper quo deſignatum eſt quod <lb/>dam horologium, & vbi ſe manifeſtè declarar, ita dicens.</s> + </p> + <quote> + <s xml:id="s3958" xml:space="preserve">Nam ipſarum partium complementum. propoſitum indicabit angulum, quan-<lb/>tus videlicet fuerit arcus eiuſdem circuli <var>.d.e.f.g.</var> à puncto <var>.g.</var> vſque ad productam li-<lb/>neam meridianam interceptus, qui vnà cum ipſo <var>.f.g.</var> quadrantem integrare videtur, <lb/>vt in ſequenti figura: </s> + <s xml:id="s3959" xml:space="preserve">quoniam arcus <var>.f.g.</var> eſt ſexaginta partium, qualium <var>.e.</var> f<unsure/>. <reg norm="quadrans" type="context">quadrãs</reg> <lb/>nonaginta, vnde concluditur reliquam partem hoc eſt, datum inclinationis <reg norm="angulum" type="context">angulũ</reg>, <lb/>fore partium triginta ſimilium.</s> + </quote> + <p> + <s xml:id="s3960" xml:space="preserve">Orontius verò cap .13. <reg norm="ijſdem" type="context">ijſdẽ</reg> vtitur verbis, cum figura ſimili ad reliqua autem ipſius <lb/>R. Clauij, videnda nondum mihi otium fuit. </s> + <s xml:id="s3961" xml:space="preserve">quod ſi dabitur, tibi libenter<unsure/> dicam <lb/>quid ſentiam.</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/>ſuper 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 ſuper cyllindro immobili, ad orizontemq́ue <lb/>perpendiculariter erecto difficile tibi non erit, (quod à me poſtulaſti) ſi <lb/>modum .53. cap. meæ gnomonicæ obſeruaueris, accipiendo tamen pro <lb/>linea orizontali in tabula non aliquam rectam lineam, ſed circularem, <lb/>ſimilemque circunferentiæ ipſius cyllindri, dico autem ſimilem, eo quod ſi gn o-<lb/>mon <var>.o.x.</var> ſupra tabulam ſignatus, & perpendicularis ipſi orizontali circulari <var>.b.i.x.</var> <lb/>eſſet dimidia, vel tertia vel quarta pars gnomonis cyllindro infixi, oporteret, vt <lb/>ſemidiameter 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 ſemidiametri cyllindri, vt <lb/>omnes arcus huiuſmodi circuli in <lb/>ter ipſos azimut intercepti ſimiles <lb/>ſint arcubus cyllindri, quod à <lb/>te ipſo facilè videre ſcientificè po <lb/>teris. </s> + <s xml:id="s3963" xml:space="preserve">reliqua nihil mutanda erunt <lb/>ab eo, quod ſcripſi circa figuram <num value="53">.<lb/>53.</num> cap. vt dixi. </s> + <s xml:id="s3964" xml:space="preserve">Vnde inuenta <lb/>cum fuerit diſtantia 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 ſemper ab orizontali per <lb/>pendiculariter deſcendit, illicò <lb/>punctum <var>.t.</var> horæ propoſitæ 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 horarię lineæ inuenta fuerint, quæ à Solis ſi-<lb/>tu in diuerſis parallelis efficiuntur, ſi voluerimus ipſam lineam <reg norm="horariam" type="context">horariã</reg> ducere, ſcien <lb/>dum primò eſt ipſam lineam horariam eſſe communem ſectionem circuli horarij, <lb/>illius horæ cum ſuperficie cyllindrica, </s> + <s xml:id="s3966" xml:space="preserve">& propterea ellipticam, vt oſtendit Serenus <lb/>in .19. primi lib. quod etiam ellicere poſſumus ab eo, quod Archimedes in .10. pro-<lb/>poſitione libr. de conoidalibus, ſcribit. </s> + <s xml:id="s3967" xml:space="preserve">Quapropter oporter nos inſtrumen-<lb/>tum prius componere, modo circini, ſed trium crurum, quæ omnia in eadem <lb/>plana ſuperficie ſint, ea tamen arte factum, vt quodlibet illorum poſſimus pro-<lb/>longare, necnon contrahere, ut cum duo extrema firmata fuerint, media poſ-<lb/>ſit circunduci circa centrum, ſeu punctum commune illarum interſectionum <reg norm="ſimulque" type="simple">ſimulq́;</reg> <lb/>poſſit produci, necnon abbreuiari vel augeri, & diminui, vt mediante ſua extremi-<lb/>tate inſeriori poſſimus delineare gyrum ellipticum horarium, dum <reg norm="centrum" type="context">cẽtrum</reg> ipſorum <lb/>crurum adhæreat extremitati gnomonis, reliquæ vero extremitates ipſorum <reg norm="crurum" type="context">crurũ</reg> <lb/>ſint ſupra puncta inuenta ipſius horæ. </s> + <s xml:id="s3968" xml:space="preserve">oportet etiam vt hoc inſtrumentum à tergo <lb/>ipſorum crurum habeat in ſuperiori parte ſuperficiem quandam <reg norm="ſemicircularem" type="context">ſemicircularẽ</reg>, quę <lb/>ſit vice vnius partis illius ſuperficiei, in qua ſupponuntur omnia crura inſtrumenti, <lb/>& hoc quantum fieri poteſt, quod quidem fieri debet, ne crus medium, hoc eſt mo <lb/>bile, exeat à tali ſuperficie, ſeu declinet ab ea, quæ ſemper ſupponitur in ſitu circuli <lb/>horarij talis horæ. </s> + <s xml:id="s3969" xml:space="preserve">oportet etiam, vt iuxta circunferentiam dimidij circuli ſint duo <lb/>gyri eiuſdem materiæ inter ſe parum diſtantes, ita ut crura poſſint moueri, intra hos <lb/>gyros, & dimidium circulum, & quod inter hos gyros locatæ ſint duæ cochleæ, ſeu <lb/>duo helices, vt quando voluerimus, poſſimus fir-<lb/>mare ipſa crura extrema, dum eorum extremitates <lb/> + <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> + fuerint ſupra puncta inuenta illius horæ, </s> + <s xml:id="s3970" xml:space="preserve">deinde in <lb/>dorſo iſtius inſtrumenti, circa centrum coniunctio <lb/>nis, rectè factum erit ſi 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 inſtrumentum excogitaui ad fugiendum <lb/>tædium inueniendi dictam ellipticam ex punctis.</s> + </p> + <p> + <s xml:id="s3972" xml:space="preserve">Nunc autem ſciendum eſt, quod vnus tantum-<lb/>modo gnomon ſufficiens non erit pro tota die æſti-<lb/>ua, neque duo, niſi valde breues fuerint reſpectu <lb/>ſemidiametri cyllindri, & in ſitu medio quartarum <lb/>meridionalium noſtro 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="diuiſusque" type="simple">diuiſusq́;</reg> à duobus diametris <var>.d.e.</var> et <var>.c.<lb/>f.</var> quarum <var>.c.f.</var> ſit pro meridiana: </s> + <s xml:id="s3974" xml:space="preserve">d.e. autem pro verticali, <reg norm="ſitque" type="simple">ſitq́;</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> ſeptentrionalis, <reg norm="computeturque" type="simple">computeturq́;</reg> maxima. <lb/></s> + <s xml:id="s3976" xml:space="preserve">Solis amplitudo ab <var>.f.</var> verſus <var>.e.</var> quæ terminetur ab <var>.q.</var> ita <reg norm="quodarcus" type="simple">quodarcꝰ</reg> <var>.f.q.</var> minor ſit <reg norm="quam" type="context">quã</reg> <lb/>graduum .45. aliter impoſſibile eſſet duobus <reg norm="tantummodo" type="context">tantũmodo</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> vſque ad contingentem <var>.q.p.</var> vnde <var>.u.i.</var> longitu <lb/>do erit vniuſcuiuſque 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">Huiuſmodi rei ratio per ſe nota erit quotieſcunque cogitauerimus verum arcum <lb/><var>e.b.</var> amplitudinis æſtiuæ, <reg norm="protractaque" type="simple">protractaq́;</reg> <var>.o.b.</var> quę parallela erit <var>.q.p.</var> vnde cum Sol tem-<lb/>pore æſtiuo orietur, tunc radios ſuos emittet via iſtarum æquidiſtantium linearum.</s> + </p> + <p> + <s xml:id="s3979" xml:space="preserve">Sed ſi 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="vnuſquiſque" type="simple">vnuſquiſq;</reg> poteſt eſſe maior <lb/>tertia parte ſemidiametri cyllin-<lb/>dri, ſed ſi voluerimus ſcire <reg norm="quan- tum" type="context">quan-<lb/>tũ</reg> ad plus poſſit eſſe longus vnuſ-<lb/>quiſque 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">quadratũ</reg> <var>.o.a.h.u.</var> ex <lb/>ſemidiametro dicti circuli, a dia-<lb/>metro poſtea <var>.o.h.</var> huiuſmodi qua <lb/>drati ſubtrahatur ſemidiameter <var>.<lb/>o.e.</var> circuli, reſiduum verò <var>.e.h.</var> ip-<lb/>ſius diametri <var>.o.h.</var> quadrati, erit <reg norm="lon" type="context">lõ</reg> <lb/>gitudo gnomonis, vbi ſimul appa <lb/>ret huiuſmodi rei ratio, eo quod <lb/>cum gnomon <var>.e.h.</var> orientalis deſi-<lb/>net operari, illico meridionalis <var>.<lb/>f.g.</var> ſubintrabit, poſt hunc verò occidentalis <var>.d.K.</var> monſtrabit reliquum diei.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head491" style="it" xml:space="preserve">Earundem line arum deſcriptio ſaper conum rectum.</head> + <head xml:id="head492" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s3981" xml:space="preserve">CVm ſuper datum conum re-<lb/> + <anchor type="figure" xlink:label="fig-0339-02a" xlink:href="fig-0339-02"/> + ctum idem facere volueris <lb/>eſto conus <var>.A.</var> & R. qui diuiſus ima <lb/>ginatione ſit à quodam plano per <lb/>axem, & communis ſectio ſit trian <lb/>gulus <var>.A.</var> & R. in quo plano cogite <lb/>mus gnomonem infixum ad rectos <lb/>vbi volueris, qui ſit <var>.p.t.o.</var> cogitem <lb/>etiam <var>.l.t.m.</var> aliud eſſe planum (in <lb/>quo ſit gnomon) quod conum ſe <lb/>cet, quæ quidem ſectio, circularis <lb/>erit, ex .4. primi Pergei. </s> + <s xml:id="s3982" xml:space="preserve">imagine-<lb/>mur etiam ſuperficiem <var>.p.s.</var> eſſe azi <lb/>mut in quo gnomonreperirur, ſu-<lb/><reg norm="perficiemque" type="simple">perficiemq́;</reg> <var>.e.s.</var> azimut propoſitæ <lb/>horę, <reg norm="angulumque" type="simple">angulumq́;</reg> <var>.e.o.a.</var> contrapoſi <lb/><reg norm="tum" type="context">tũ</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 ver<lb/>ſus baſim tranſit per medium latitudinis ipſius gnomonis, concipiamus etiam mente <lb/><var>e.a.</var> communem ſectionem eſſe trianguli ſupra dicti cum azimut horæ, necnon pun-<lb/>ctum <var>.K.</var> eſſe commune radio Solis <var>.o.a.</var> & ſuperficiei conicæ, quod quidem eſt illud <lb/>quod quæritur, hoc ſcilicet modo. </s> + <s xml:id="s3984" xml:space="preserve">Primum cognoſcimus angulum <var>.p.A.t.</var> vt medie <lb/>tas anguli totius coni, & angulum <var>.p.</var> rectum, vnde <var>.t.</var> tam intrinſecus, quam extrinſe-<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> cognoſcatur, ſi <lb/>gnomon <var>t.o.</var> fixus fuerit in ſuperficie conica, ita qd cum latere <var>.A.t.</var> eſſiciat <reg norm="angulum" type="context">angulũ</reg> <lb/><var>A.t.o.</var> & lateraliter faciat angulosrectos cum ſuperficie conica, ad quod efficiendum <lb/>nulla eſt difficultas, cognoſcendo deinde <var>.A.t.</var> ſimul cum angulis <var>.A.</var> et <var>.t.</var> intrinſecis <lb/>trianguli ortogonij <var>.A.p.t.</var> cognoſcemus <var>.p.t.</var> et <var>.A.p.</var> vnde etiam tota <var>.o.p.</var> ſed cogno <lb/>ſcendo <var>.o.p.</var> cum angulo <var>.p.o.e.</var> (angulus enim <var>.p.o.e.</var> cognoſcitur ex hypotheſi cum <lb/>ſit inter azimut Solis & azimut gnomonis) cum angulo <var>.o.p.e.</var> recto cognoſcemus <var>.p.<lb/>e.</var> et <var>.o.e.</var> </s> + <s xml:id="s3986" xml:space="preserve">deinde cum nobis nota ſit <var>.o.e.</var> cum angulo altitudinis Solis <var>.e.o.a.</var> & angu-<lb/>lo <var>.o.e.a.</var> recto cognoſc emus longitudinem azimutalis <var>.e.a.</var> necnon quantitatem <var>.a.o.</var> <lb/>Imaginata poſtea <var>.a.q.</var> æquidiſtante <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="cognitiſque" type="simple">cognitiſq́;</reg> cum angulo recto <var>.p.</var> cogno <lb/>ſcemus <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"/> + gnoſcemus <var>.A.i.</var> et <var>.q.i.</var> quę <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> ſemper <lb/>ſunt in eadem ſuperficie ſecante co <lb/>num, quæ etiam ſecat ſuperficiem <lb/>trianguli <var>.A.q.x.</var> ad rectos ex .18. vn <lb/>decimi, cum linea <var>.u.n.</var> perpendicu <lb/>laris ſit ſuperficiei trianguli <var>.A.q.i.</var> <lb/>ex .8. dicti, quia parallela eſt <var>.l.p.</var> quę <lb/>perpendicularis eſt ſuperficiei <reg norm="trian- guli" type="context">triã-<lb/>guli</reg> <var>.o.p.q.</var> ex .4. eiuſdem, ſequitur, <lb/>quod talis ſectio ( quæ intelligatur <lb/>per <var>.u.K.i.n.</var>) ſemper erit elliptica, <lb/>vel parabole, ſeu hyperbole, <reg norm="prout" type="simple">ꝓut</reg> <lb/>linea <var>.o.i.q.</var> ſecabit latus coni, oppo <lb/>ſitum lateri <var>.A.i.</var> diſtento in ipſa ſuperficie conica, ſeu ad ſuperiorem partem produ <lb/>ctum, velipſi 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> ſecare dictum oppoſitum latus lateri <var>.A.i.</var> <lb/>verſus baſim, vnde ſectio <var>.u.K.i.n.</var> erit elliptica. </s> + <s xml:id="s3990" xml:space="preserve">quod facile cognitu eſt <reg norm="mediante" type="context">mediãte</reg> com <lb/>paratione angulorum <var>.A.q.i.</var> et <var>.q.A.i.</var> interſe, eo quod ſi eſſent ęquales, dicta ſect o <lb/>barabola eſſet ex .27. primi Eucli. et .11. primi Pergei, ſed ſi angulus <var>.A.q.i.</var> maior eſ-<lb/>ſet angulo <var>.q.A.i.</var> ſectio eſſet ellipſis, ex ultimo poſtulato primi Euclid. </s> + <s xml:id="s3991" xml:space="preserve">& ex .13. pri-<lb/>mi Pergei, ſed ſi dictus angulus <var>.A.q.i.</var> minor eſſet angulo <var>.A.</var> tunc ſectio eſſet hyper-<lb/>bole ex dicto poſtulato & ex .12. primi Pergei. </s> + <s xml:id="s3992" xml:space="preserve">Sit ergo primum vt <reg norm="dictum" type="context">dictũ</reg> eſt, hoc <lb/>eſt, quod ſectio eſſet oxygonia, ideſt elliptica, ſeu defectio (quod idem eſt,) ſepa-<lb/>ratim oportebit nos ellipſim deſignare <reg norm="ſimilem" type="context">ſimilẽ</reg> <reg norm="ęqualemque" type="context simple">ęqualẽq́;</reg> ei, quæ eſt <var>.u.K.i.n.</var> <reg norm="quod" type="wordlist">qđ</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/>difficile non erit, quotieſcunque ſuos axes inuenerimus, maiorem ſcilicet, & 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 ſit <var>.i.A.b.</var> quem diui-<lb/>demus per æqualia mediante <var>.A.q.</var> conſtituendo <var>.A.i.</var> huius anguli æqualem <var>.A.i.</var> ſu-<lb/>perficiei conicæ et <var>.A.q.</var> diuidentem, æqualem parti <var>.A.q.</var> axis coni, ducendo poſtea <lb/>ab <var>.i.</var> per <var>.q.</var> lineam vnam quouſque concurrat <var>.A.b.</var> in puncto <var>.b.</var> habebimus <var>.i.b.</var> pro <lb/>maiori axi ipſi ellipſis, quod per ſe clarum eſt, cuius medietas ſit <var>.i.c.</var> ſed <var>.i.q.</var> ipſius <var>.i.<lb/>b.</var> æqualis eſt ipſi <var>.q.i.</var> ipſius coni, ex quarta primi Eucli. et <var>.q.b.</var> ipſius <var>.i.b.</var> æqualis alte <lb/>ri parti inuiſibili. </s> + <s xml:id="s3993" xml:space="preserve">Reliquum eſt, 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">æqualisq́;</reg> ei quæ eſt coni, & diui <lb/>ſa ſimiliter in <var>.a.</var> quæ <var>.u.n.</var> ipſius coni nobis cognita eſt ex lateribus <var>.A.u.</var> et <var>.A.n.</var> & ex <lb/>angulo coni, et <var>.a.q.</var> æqualis eſt <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 ipſius <var>.f.c.</var> quæ pro-<lb/>ducti ipſius <var>.i.q.</var> in <var>.q.b.</var> ad productum ipſius <var>.i.c.</var> in <var>.c.b.</var> & cum cognita nobis ſint <lb/>hæc tria producta hoc eſt <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 ſeipſa, cognoſcemus <reg norm="etiam" type="context">etiã</reg> <lb/>quartum ipſius <var>.f.c.</var> & fic <var>.f.c.</var> <reg norm="eiuſque" type="simple">eiuſq́;</reg> duplum <var>.f.r.</var> cogniti nobis itaque cum ſint hi duo <lb/>axes <var>.i.b.</var> et <var>.f.r.</var> formabimus ellipſim. </s> + <s xml:id="s3995" xml:space="preserve">Deinde producemus axim <var>.b.i.</var> à part <var>e.i.</var> quo-<lb/>uſque <var>.i.o.</var> æqualis ſit ei quæ extra conum eſt, 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/>ſecabit in puncto <var>.K.</var> vnde habebimus quantita-<lb/>tem ipſius <var>.o.K.</var> et <var>.K.i.</var> rectam. </s> + <s xml:id="s3996" xml:space="preserve">inde mediante cir-<lb/>cino ſi acceperimus rectam diſtantiam ab <var>.i.</var> ad <var>.K.</var> <lb/>in ellipſi, </s> + <s xml:id="s3997" xml:space="preserve">deinde firmando pedem circini in pun-<lb/>cto <var>.i.</var> in ſuperficie conica, & cum alio ſignando <lb/>lineam vnam curuam ad partem <var>.K.</var> in ſuperficie <lb/>conica, ſumendo poſtea interuallum <var>.o.K.</var> extra el <lb/>lipſim, </s> + <s xml:id="s3998" xml:space="preserve">deinde firmando vnum pedem circini in <lb/>extre mitate gnomonis, cum alio poſtea ſignan-<lb/>do aliam lineam curuam in ſuperficie ipſius coni, <lb/>quæ primam ſe cet in puncto <var>.K.</var> hoc erit punctum <lb/>quæſitum horę propoſitæ in ſuperficie conica <lb/>propoſita.</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 ſi talis ſectio fuerit parabole, vel hyperbo <lb/>le, tunc mediante ſuo diametro <var>.i.q.</var> cum baſi <var>.u.<lb/>q.n.</var> cognita, deſignabimus ipſam ſectionem <var>.u.i.</var> n <lb/>ope mei <reg norm="inſtrumenti" type="context">inſtrumẽti</reg> in calce meę gnomonicæ de <lb/>ſcripti, </s> + <s xml:id="s4000" xml:space="preserve">deinde diuiſa <var>.u.q.</var> in <var>.a.</var> <reg norm="pro" type="simple">ꝓ</reg><reg norm="ductaque" type="simple">ductaq́;</reg> <var>q.i.</var> <reg norm="vſque" type="simple">vſq;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/> + ad <var>.o.</var> <reg norm="ductaque" type="simple">ductaq́;</reg> <var>.o.a.</var> habebimus punctum <var>.K</var>. </s> + <s xml:id="s4001" xml:space="preserve">Reli-<lb/>qua facienda ſunt, vt dictum eſt de ellipſi.</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 eiuſ-<lb/>dem horæ propoſitę, ducemus ab vno ad a-<lb/>liud, lineam horariam mediante circino trium <lb/>crurum, quem tibi ſcripſi nudius tertius pro cyl <lb/>lindro, quæ <reg norm="quidem" type="context">quidẽ</reg> linea crit portio gyri ellipſis, <lb/>ſeu hyperbolę, vel parabolę, vt à te ipſo 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 Sabaudiœ 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">qualiq́;</reg> voluntate, ſed ne <lb/>tua pulcherrima ſtudia aliquo modo imperfecta <reg norm="relinquantur" type="simple">relinquant̃</reg>, vel ego tibi <lb/>deeſſe videar, dum Problemata geographica Magni Ptolomei conſi-<lb/>deras, aduerte, quod ſi putares in figura .6. cap. libr .7. geographię eiuſ-<lb/>dem (vt multi credunt) lineam <var>.V.*.</var> ſecare circunferentiam <var>.A.D.</var> in puncto <var>.G.</var> ita <lb/>vt punctus <var>.G.</var> ſit tropici æſtiui, ideſt 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 conſidera quæ nunc tibi ſcribo.</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="ſupponaturque" type="simple">ſupponaturq́;</reg> ſemidiameter <var>.E.D.</var> eſſe <lb/>partium 120. quarum <var>.E.Z.</var> in alio ſemidiametro <var>.C.E.</var> ei orthogonaliter coniuncto, <lb/>talium ſit .17. in ſemidiametro 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. ſimilium.</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/>ſectione lineę ductæ per puncta <var>.Z.D.</var> in puncto <var>.*.</var> ducatur demum <var>.V.*.</var> quæ ſeca-<lb/>bit circunf rentiam <var>.A.D.</var> in puncto <var>.G</var>. </s> + <s xml:id="s4007" xml:space="preserve">Quæratur nunc quantitas ipſius <var>.G.D.</var> <lb/>Ad quod efficiendum quærenda primum eſt quantitas ipſius <var>.S.*.</var> quam illico co <lb/>gnoſcemus ex regula de tribus, cum dixerimus, ſi@ 17. dat nobis .120. quid dabit .41. <lb/>(nam duo triangula <var>.Z.E.D.</var> et <var>.Z.S.*.</var> ſunt inuicem ſimilia, cum <var>.S.*.</var> parallela ſit <lb/>ipſi <var>.E.D.</var>) vnde <var>.S.*.</var> proueniet nobis ex ſimilibus partibus .289. cum fracto, quod <lb/>r<unsure/>eijciamus ob minorem laborem.</s> + </p> + <p> + <s xml:id="s4008" xml:space="preserve">Producantur poſtea <var>.V.*.</var> et <var>.E.D.</var> vſque ad eorum concurſum in puncto <var>.ω.</var> quæ-<lb/><reg norm="remusque" type="simple">remusq́;</reg> quanta ſit <var>.E.ω.</var> ex eadem regula, cum dixerimus, ſi .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> ſunt inuicem ſimilia 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 ipſius <var>.E.V.</var> quod eſt .4096. cum quadrato ipſius <var>.<lb/>E.ω.</var> quod eſt .213444. & habebimus quadratum ipfius <var>.V.ω.</var> talium <reg norm="partium" type="context">partiũ</reg> .217540.</s> + </p> + <p> + <s xml:id="s4010" xml:space="preserve">Dicemus poſtea ſi .217540. dat nobis .4096. quid dabit quadratum ipſius <var>.V.ω.</var> vt <lb/>ſinus totus quod eſt .10000000000. vnde veniet pro quadrato ipſius <var>.V.E.</var> talium <lb/>partium, ſuperficialium ſcilicet .18827211. cuius radix erit .13721. & erit ſinus 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 ſinus erit partium .99054.</s> + </p> + <p> + <s xml:id="s4011" xml:space="preserve">Nunc autem quia angulus <var>.E.V.ω.</var> eſt acutus, imaginemur <var>.E.Ŕ.</var> ductam eſſe ad re<lb/>ctos ipſi <var>.V.ω.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> etiam ducta ipſa <var>.E.G</var>. </s> + <s xml:id="s4012" xml:space="preserve">Vnde habebimus angulum <var>.Ŕ.E.V.</var> gra-<lb/>duum .7. min .53. eius vero ſinus <var>.Ŕ.V.</var> partium .13721. (propter ſimilitudinem trian<lb/>gulorum <var>.E.Ŕ.V.</var> et <var>.ω.E.V.</var>) talium ſcilicet, qualium <var>.E.V.</var> fuerit .100000. </s> + <s xml:id="s4013" xml:space="preserve">Sed qua-<lb/>lium <var>.E.V.</var> eſt .64. talium erit .8. cum tribus quartis, cuius <var>.Ŕ.V.</var> quadratum erit par<lb/>tium .76. cum dimidio ſimilium ſed ſuperſicialium, quo quidem quadrato dempto <lb/>ex quadrato ipſius .64. quod eſt .4096. remanebit quadratum ipſius <var>.E.Ŕ.</var> partium <num value="2871">.<lb/>2871.</num> quo etiam quadrato <var>.E.Ŕ.</var> dempto ex quadrato <var>.E.G.</var> partium .14400. remane<lb/>bit quadratum ipſius <var>.Ŕ.G.</var> partium .11529. cuius radix <var>.Ŕ.G.</var> erit partium .107. <reg norm="talium" type="context">taliũ</reg> <lb/>qualium <var>.E.G.</var> eſt .120. ſed qualium <var>.E.G.</var> erit .100000. talium <var>.Ŕ.G.</var> erit partium <num value="89166">.<lb/>89166.</num> quæ vt ſinus anguli <var>.Ŕ.E.G.</var> habebit pro ipſo angulo, gra .63. min .5. qui colle<lb/>cti cum gra .7. min .53. anguli <var>.V.E.Ŕ.</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 Philoſopbo Franciſco Vimercato.</head> + <p> + <s xml:id="s4014" xml:space="preserve"><emph style="sc">QVoniam</emph> non videbatur quieſcere animus tuus, cum paucis ab hinc die-<lb/>bus tibi ſiſcitanti reſpondiſſem, nec tamen rationem omnium, quæ dixe-<lb/>ram exactè explicare per tem poris anguſtiam potuiſſem, cogitaui ad te <lb/>per hanc occaſionem ſcribens, & iam dicta repetere, & omnium tibi ra-<lb/>tionem ſubiungere, & vt mihi plenius ſatisfaciam, & tibi commodè perlegenti faci <lb/>lius ſit veritatem intueri. </s> + <s xml:id="s4015" xml:space="preserve">Scripſiſti enim in tuis diſputationibus, vir doctiſſime, quod <lb/>omnis res viſa per <reg norm="ſpeculum" type="context">ſpeculũ</reg> <reg norm="quodcunque" type="context">quodcũque</reg>, ſub breuiſſimis lineis <reg norm="compræhendatur" type="context">cõpræhendatur</reg> à vifu.</s> + </p> + <p> + <s xml:id="s4016" xml:space="preserve">Propoſitio hæc non eſt vniuerſaliter vera (quamuis etiam ab alijs omnibus pro ta <lb/>li poſita ſit) cum in ſpeculis concauis non ſemper verificetur, vt nunc tibi demon-<lb/>ſtrabo.</s> + </p> + <p> + <s xml:id="s4017" xml:space="preserve">Eſto 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 ſit communis ſectionis ſup crficiei re <lb/>flexionis, & ſphæricę alicuius ſpeculi ſphærici <lb/>concaui, & punctum contingentiæ ſit <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 viſa primò ſit in ipſa circunferen-<lb/>tia huiuſmodi circuli in puncto <var>.n.</var> oculus vero in <lb/>puncto <var>.q.</var> ipſius circunferentię. </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> ſimul ſumptas longiores eſſe omnibus alijs lineis exeuntibus ab ip <lb/>ſis punctis <var>.q.n.</var> quæ in aliquo puncto dictæ circunferentiæ ſimul 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 ſimul ſumptas, eſſe minores dua <lb/>bus ſimul ſumptis <var>.q.b.</var> et <var>.n.b</var>. </s> + <s xml:id="s4020" xml:space="preserve">Nam ex .20. tertij Eucli. cognoſcimus angulos <var>.q.b.n.</var> <lb/>et <var>.q.o.n.</var> inuicem æquales eſſe, & ſimiliter 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 eiuſdem habemus angulos contra ſe poſitos, <lb/>circa <var>.a.</var> eſſe etiam inuicem ęquales. </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"/> + ſexti, habebimus proportionem <var>.a.b.</var> ad .a<lb/>o. eandem eſſe, quæ <var>.a.n.</var> ad <var>.a.q.</var> & ſic .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> ſed cum <var>.a.n.</var> maior ſit <var>.q.a.</var> ex .18. primi, <lb/>eo quod angulus <var>.b.q.n.</var> (qui æqualis eſt angulo <var>.<lb/>b.n.q.</var> ex .5. eiuſdem) maior eſt angulo <var>.a.n.q.</var> <lb/>qui pars eſt ipſius <var>.b.n.q.</var> ergo latera ſimul ſum-<lb/>pta <var>.a.b.n.</var> maiora erunt lateribus <var>.a.o.q.</var> ſed 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 ſequi-<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 ſi oculus eſſet in <var>.u.</var> quemadmodum in ſubſcripta hic <reg norm="ſecunda" type="context">ſecũda</reg> figura videre eſt, <lb/>res autem viſibilis in <var>.n.</var> ambo extra dictum circulum, eſto etiam primum <var>.b.u.</var> æqua-<lb/>lis <var>.b.n.</var> probabo ſimiliter <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 eſt angu-<lb/>lo <var>.b.</var> eo quod ſi circulum <var>.u.b.n.</var> cogitemus circunſcribere triangulum <var>.u.b.n.</var> ducen-<lb/>do vſque ad ſuam 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> ſed <reg norm="cum" type="context">cũ</reg> angulus <var>.u.o.n.</var> exterior trian <lb/>guli <var>.u.o.s.</var> exiſtat, ipſe maior erit angulo <var>.s.</var> ex .16. primi. </s> + <s xml:id="s4027" xml:space="preserve">duco poſtea <var>.o.q.</var> parallelam <lb/>ad <var>.u.s.</var> quæ ſecabit <var>.a.u.</var> in puncto <var>.q.</var> & habebimus angulum <var>.a.o.q.</var> ęqualem angulo <var>.<lb/>n.s.u.</var> ex .29. eiuſdem, hoc eſt angulo <var>.n.b.u.</var> fed ex ſu-<lb/> + <anchor type="figure" xlink:label="fig-0344-02a" xlink:href="fig-0344-02"/> + pradictis rationibus, lineæ <var>.q.b.n.</var> ſimul ſumptæ maio-<lb/>rem efficient longitudinem, quam <var>.q.o.n</var>. </s> + <s xml:id="s4028" xml:space="preserve">Nunc cum <lb/>ipſi <var>.q.b.</var> addita fuerit <var>.u.q.</var> & vice <var>.q.o.</var> ſumpta fuerit ali-<lb/>qua linea minor ipſa <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 eſt ambabus <var>.o.<lb/>q.</var> et <var>.q.u.</var> ex .20. primi, ita enim habebimus <reg norm="propoſitum" type="context">propoſitũ</reg>. <lb/></s> + <s xml:id="s4030" xml:space="preserve">ſed breuiori modo hoc ipſum videbis ex pręcedenti, <lb/>& ex .21. primi Euclid. </s> + <s xml:id="s4031" xml:space="preserve">Nam ex præcedenti <var>.u.b.n.</var> lon-<lb/>gior eſt ipſa <var>.u.s.n.</var> ex .21. autem primi <var>.u.s.n.</var> longior eſt <lb/>ipſa <var>.u.o.n.</var> ergo verum eſt propoſitum.</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, ſit vt in hac ſubſcripta tertia figura vide <lb/>re eſt <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/>manifeſtabitur verum eſſe propoſitum, 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 viſibilis <reg norm="oculusque" type="simple">oculusq́;</reg> ambo fuerint intra circulum, <reg norm="tunc" type="context">tũc</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 ipſa <var>.u.o.n.</var> <reg norm="nunc" type="context">nũc</reg>. <lb/></s> + <s xml:id="s4036" xml:space="preserve">Quod etiam affirmo de <var>.u.b.p.</var> ſimiliter etiam eueniet ſi vnus terminorum <var>.u.</var> vel <var>.n.</var> <lb/>fuerit intra circunferentiam, reliquus verò extra ipſam.</s> + </p> + <p> + <s xml:id="s4037" xml:space="preserve">Conſideremus nunc hic inſraſcriptam .4. figuram vbi <var>.d.b.p.</var> ſit circunferentia oxy <lb/>gonia ſeu elliptica (quod idem eſt) cuius maior axis ſit <var>.d.p.</var> in quo, duo termini <var>.u.n.</var> <lb/>ſint centra eius generationis: </s> + <s xml:id="s4038" xml:space="preserve">b.x. verò ſit minor axis. </s> + <s xml:id="s4039" xml:space="preserve">Imaginemur etiam circulum <var>.<lb/>b.o.x.</var> cuius ſemidiameter ſit <var>.c.b.</var> non maior medietate minoris axis, ne circunferen-<lb/>tia huiuſmodi circuli ſecet circunferentiam oxygoniam. </s> + <s xml:id="s4040" xml:space="preserve">Cogitemus etiam circu-<lb/>lum <var>.b.e.</var> cuius ſemidiameter, minor non ſit minori axe <var>.b.x.</var> ipſius oxygoniæ, ne ſe <lb/>inuicem ſecent huiuſmodi circunferentiæ, ſint etiam ambo eorum centra in linea <var>.b.<lb/>x.</var> minoris axis, & punctum <var>.b.</var> ſit commune vnicuique earum periphæriarum, vnde <lb/>minor circulus, totus intra, maior autem, totus extra ipſam <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 tranſire alium circulum, cuius cen-<lb/>trum in axe <var>.b.x.</var> ſit <var>.t.</var> <reg norm="omnesque" type="simple">omnesq́;</reg> iſti circuli imaginentur trium diuerſorum ſphærico-<lb/>rum ſpeculorum, vnde pro genera <lb/>tione <reg norm="ipſius" type="simple">ipſiꝰ</reg> oxygonię, ſeu 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> ęqualem eſſe longitudini <var>.u.b.<lb/>n.</var> & ei, quæ eſt <var>.u.o.n.</var> (vt minor ip <lb/>ſa <var>.u.r.n.</var> ex .21. primi Euclidis) mi-<lb/>nor ipſa <var>.u.b.n.</var> & longitudinem <var>.u.<lb/>e.n.</var> (vt maior ipſa <var>.u.r.n.</var> ex eadem <num value="21">.<lb/>21.</num> primi Eucli.) maior ipſa <var>.u.b.n</var>. <lb/></s> + <s xml:id="s4042" xml:space="preserve">Sed ſi quis vellet hoc demonſtrare <lb/>ope circuli, <reg norm="vnius" type="simple">vniꝰ</reg> <reg norm="tantummodo" type="context context">tãtũmodo</reg> ſpeculi, <lb/><reg norm="multiplicando" type="context">multiplicãdo</reg> ipſas oxygonias <reg norm="quem- admodum" type="wordlist">quẽ-<lb/>admodum</reg> de ipſis circulis fecimus, obtineret ſimiliter propoſitum.</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 eſt dubitatio tua, <lb/> + <anchor type="figure" xlink:label="fig-0345-02a" xlink:href="fig-0345-02"/> + vtrum (<reg norm="cum" type="context">cũ</reg> circulus minor hoc <lb/>eſt <var>.b.o.</var> habeat ſuum centrum in mi <lb/>nori axe inter centrum oxygoniæ, <lb/>et .b: exiſtente <var>.b.</var> extremo axis mi-<lb/>noris, <reg norm="communeque" type="simple">communeq́;</reg> ambobus circun-<lb/>ferentijs circuli ſcilicet & oxigonię) <lb/>dictus circulus minor, plura puncta <lb/>communia habeat cum ipſis 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="reſpondeo" type="context">reſpõdeo</reg> quod <lb/>quotieſcunque 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 ſcilicet minoris axis, exiſtente <var>.b.</var> <lb/>ſua 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, ipſas circunferentias inuicem contiguas eſſe oportebit in puncto <var>.b.</var> <lb/>tantummodo.</s> + </p> + <p> + <s xml:id="s4045" xml:space="preserve">Eſto primum quod centrum <var>.c.</var> commune exiſtat, vt dictum eſt. </s> + <s xml:id="s4046" xml:space="preserve">ſit etiam centrum <lb/>vnius circuli, cuius diameter ſit <reg norm="idem" type="context">idẽ</reg> <reg norm="cum" type="context">cũ</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/>ipſi <var>.g.c.</var> vſque 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="ſecabitque" type="simple">ſecabitq́;</reg> gyrum circuli <var>.b.o.</var> minoris in puncto <var>.t.</var> quod di <lb/>co eſſe intra oxygoniam, <reg norm="ſeparatumque" type="simple">ſeparatumq́;</reg> ab <var>.f</var>. </s> + <s xml:id="s4049" xml:space="preserve">Quapropter duco <var>.c.e.</var> quæ ſecabit cir-<lb/>cunferentiam circuli minoris in <reg norm="puncto" type="context">pũcto</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 conſidero, quod ex ra-<lb/>tionibus ab Archimede adductis in <lb/> + <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> + quinta propoſitione libri de conoi-<lb/>dalibus, & ſphæroidibus, eadem <lb/>proportio erit <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.g.c.</var> ad <var>.b.c.</var> quę <lb/>ipſius <var>.e.a.</var> ad <var>.f.a.</var> vnde permutando <lb/>ita erit ipſius <var>.g.c.</var> ad <var>.e.a.</var> vel <var>.b.c.</var> ad <lb/><var>f.a.</var> hoc eſt ipſius <var>.e.c.</var> ad <var>.e.a.</var> vt <var>.o.c.</var> <lb/>ad <var>.f.a.</var> ſed ex ſimilitudine triangu-<lb/>lorum, & ex .11. quinti, ita <reg norm="etiam" type="context">etiã</reg> erit <lb/>ipſius <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 ſequitur <var>.o.i.</var> æqualem eſſe <var>.f.a.</var> <lb/>ſed ex .14. tertij Eucli <var>.t.a.</var> minor eſt <var>.<lb/>o.i</var>. </s> + <s xml:id="s4052" xml:space="preserve">Quare minor etiam erit ipſa <var>.f.<lb/>a</var>. </s> + <s xml:id="s4053" xml:space="preserve">Vnde punctum <var>.t.</var> intra oxygo-<lb/>niam erit, & conſequenter ſepara-<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 ſi centrum circuli minoris <lb/>fuerit inter <var>.c.</var> et <var>.b.</var> hoc eſt eccentri-<lb/>cum ipſius oxygoniæ, ipſe 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 diſtans erit punctum <var>.t.</var> à puncto <var>.f.</var> quod <lb/>erit propoſitum.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head xml:id="head499" style="it" xml:space="preserve">Alterius dubitationis ſolutio.</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 diſtantes <lb/>vltra ſuperficiem ipſius ſpeculi videantur, vt obiecta citra ipſam ſuperficiem <lb/>reperiuntur.</s> + </p> + <p> + <s xml:id="s4057" xml:space="preserve">Pro cuius rei ſcientia, tres cognitiones nos primum habere oportet, quarum pri-<lb/>ma eſt. </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">Secũda</reg>. </s> + <s xml:id="s4060" xml:space="preserve">vn-<lb/>de efficiatur, quod angulus reflexionis, ſemper æqualis ſit angulo incidentiæ.</s> + </p> + <p> + <s xml:id="s4061" xml:space="preserve">Terria demum. </s> + <s xml:id="s4062" xml:space="preserve">Vnde naſcatur quod radius incidentiæ ſimul cum radio reflexio-<lb/>nis ſit in quodam plano ſecante ſuperficiem ſpeculi ſemper ad rectos, quod qui-<lb/>dem planum vocatur ſuperficies reflexionis. </s> + <s xml:id="s4063" xml:space="preserve">Huiuſmodi tres paſſiones, ab omnibus <lb/>ſpecularijs conſideratæ ſunt, ſed rationes ab illis traditæ, mihi non ſatisfaciunt.</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 vniuerſaliter euenire à breuitate aggregati radiorum incidentiæ reflexio-<lb/><reg norm="nisque" type="simple">nisq́;</reg>. </s> + <s xml:id="s4065" xml:space="preserve">Sed hoc naſcitur potius ab eo, quod cum radius incidentiæ non poſſit ſuper <lb/>ficiem corporis opaci penetrare, reflectit, vt citra ipſam <reg norm="cum" type="context">cũ</reg> angulo æquali ei, quem <lb/>faceret cum eadem ſuperficie vltra ipſam ſi tranſiuiſſet.</s> + </p> + <p> + <s xml:id="s4066" xml:space="preserve">Exempli gratia ſit <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> ſuperficies ipſius <lb/>ſpeculi <var>.d.</var> verò ſit punctum ipſius ſuperficiei, à quo ad oculum reflectitur imago ip-<lb/> + <anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/> + ſius <var>.a</var>. </s> + <s xml:id="s4067" xml:space="preserve">Nunc ſi radius <var>.a.d.</var> incidentiæ, recta <lb/>incederet ſub <var>.c.e.</var> efficeret angulum <var>.e.d.h.</var> <lb/>æqualem angulo <var>.c.d.a.</var> eius contrapoſito, <lb/>ſed quia impeditur ipſæ radius ab opacitate <lb/>ipſius ſpeculi <var>.c.e.</var> ne vlterius incedat, propte <lb/>rea reflectitur ab ipſa ſuperficie ſpeculi, con-<lb/>ſtituens cum ipſa angulum <var>.e.d.b.</var> æqualem <lb/>angulo <var>.e.d.h.</var> ſed quia angulus <var>.c.d.a.</var> eſt <reg norm="etiam" type="context">etiã</reg> <lb/>ęqualis ipſi angulo <var>.e.d.h.</var> propterea angulus <var>.e.d.b.</var> ęqualis exiſtit angulo <var>.c.d.</var> a; </s> + <s xml:id="s4068" xml:space="preserve">per <lb/>accidens igitur ſequitur <var>.a.d.</var> et <var>.d.b.</var> ſimul ſumptas, breuiorem facere longiludinem <lb/>omni alia, quæ ab ipſa ſuperficie <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 eſt efficere angulum <var>.e.d.b.</var> æqualem angulo <var>.e.d.h.</var> vnde ex accidenti po <lb/>ſtea ſequitur, ipſum æqualem eſſe angulo <var>.c.d.a.</var> & deinde <reg norm="quod" type="wordlist">qđ</reg> lineæ <var>.a.d.</var> et <var>.d.b.</var> con-<lb/>ſtituant longitudinem breuiorem. </s> + <s xml:id="s4070" xml:space="preserve">Quare illud quod omnes putabant eſſe primum <lb/>& perſe, vltimum eſt, & 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 ſuperficies, quæ vocatur reflexionis, in qua ſunt duæ lineę, hoc eſt <lb/>incidentię, <reg norm="reflexionisque" type="simple">reflexionisq́;</reg>, ſemper ſit perpendicularis ſuperficiei ipſius ſpeculi: </s> + <s xml:id="s4072" xml:space="preserve">Hæc <lb/>eſt ratio, quia cum quilibet radius incidentiæ, perpendicularis ipſi ſuperficiei ſpe-<lb/>culi, in ſeipſo reflectit, ex ijſdem dictis rationibus, hoc eſt, quia cum tali angulo vult <lb/>reflecti, cum quali tranſiret, ita etiam purandum eſt, quodradius incidens obliquus, <lb/>cum in ſeipſum non poſſit redire, quia non eſt perpendicularis ſuperficiei ſpeculi, <lb/>reflectitur tamen per planum erectum ipſi ſuperficiei ſpeculi, vt in eo, cui magis re-<lb/>ſiſtit ſuperficies corporis opaci, quàm alicui alij plano ipſius infiniti inclinatorum <lb/>planorum, ab vtraque parte ipſius plani perpendicularis, quod vnum etiam tan-<lb/>tummodo eſt, & in quo, radius maiorem vim obtinet reflectendi, ſeu in eo, in quo <lb/>radius ipſe cum maiori reſiſtentia repercutitur à ſuperficie corporis opaci.</s> + </p> + <p> + <s xml:id="s4073" xml:space="preserve">Poſtremo <reg norm="ſciendum" type="context context">ſciẽdũ</reg> vnde oriatur, <reg norm="quod" type="simple">ꝙ</reg> rei viſibilis imago, à ſpeculo plano reflexa, ſem <lb/>per in catheto incidentiæ videatur.</s> + </p> + <p> + <s xml:id="s4074" xml:space="preserve">Pro cuius rei ratione cognoſcendum primò eſt, quo modo fit perfecta <reg norm="ſimplexque" type="simple">ſimplexq́;</reg> <lb/>viſio, & non reflexa, deinde proſequemur ad reliqua huius tertiæ propoſitionis.</s> + </p> + <p> + <s xml:id="s4075" xml:space="preserve">Animaduertendum igitur eſt, quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> obiectum aliquod viſibile aſpi <lb/>cimus, nos nunquam perfectè illud comprehendere poſſumus, niſi in puncto con-<lb/>curſus, ſeu interſectionis axium viſualium, ſeu radialium ( vt ita loquar ) <reg norm="quam" type="context">quã</reg> <reg norm="inter- ſectionem" type="context">inter-<lb/>ſectionẽ</reg>, nos efficimus ope reuolutionis oculorum <reg norm="adinuicem" type="context">adinuicẽ</reg>, hoc eſt voluendo vnum <lb/>verſus alium, ita vt in ſitu ipſius obiecti, ſeinuicem ſecent axes iam dicti, </s> + <s xml:id="s4076" xml:space="preserve">tunc enim <lb/>vtroque oculo mediante, exacte rem perſpicimus, cęteris .8. circunſtantijs non ob-<lb/>ſtantibus.</s> + </p> + <p> + <s xml:id="s4077" xml:space="preserve">Vnde ſtantibus oculis in tali ſitu, altero reſpectu alterius, ſi eorum alter tectus; <lb/></s> + <s xml:id="s4078" xml:space="preserve">ſeu velatus fuerit, tune alio tantummodo oculo mediante, videbimus obiectum, <lb/>in ea diſtantia, 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, ſecundum diſtantiam obiecti, oculum accommodat ad recipien-<lb/>dum quam exactiſſimè ſpeciem ipſius obiecti, & hoc voluendo ambos oculos, vnum <lb/>verſus alium, ita quod interſectio axium ſit in ſitu ſeu loco dicti obiecti, nam tunc vi <lb/>dent ambo vel aliquis eorum ſolus, in tali diſtantia exactè obiectum videbit.</s> + </p> + <p> + <s xml:id="s4080" xml:space="preserve">Vnde ſequitur obiectum viſibile, compræhenſibile non eſſe ab vno tantummodo <lb/>oculo in quolibet ſitu axis ipſius oculi, ſed in eo, vbi alius axis interſecatur à dicto. <lb/></s> + <s xml:id="s4081" xml:space="preserve">Quæ quidem interſectio poteſt fieri propinqua, vel remota à viſu, ad certos tamen <lb/>terminos vſque.</s> + </p> + <p> + <s xml:id="s4082" xml:space="preserve">De huiuſmodi axium viſualium interſectione ſcribit Alhazem in .2. et .15. propo <lb/>ſitione tertij lib. Vitellio verò in .32. et .45. eiuſdem.</s> + </p> + <p> + <s xml:id="s4083" xml:space="preserve">Quod igitur dico, verum eſt, ideſt, quod ſi vno tantummodo oculo aſpiciemus <lb/>obiectum aliquod, ipſum nunquam perfectè proſpicietur, niſi cum oculus ita ſitus <lb/>fuerit, vt eius axis cum axe alterius in loco obiecti ſe inuicem ſecent, quamuis alter <lb/>oculus nihil videat, <reg norm="cum" type="context">cũ</reg> <reg norm="autem" type="wordlist">aũt</reg> duobus oculis in tali ſitu <reg norm="conſtitutis" type="context">cõſtitutis</reg> <reg norm="obiectum" type="context">obiectũ</reg> videmus, vnum <lb/>tantummodo nobis cernere videbimur, & ſi extra talem punctum interſectionis ip-<lb/>ſum obiectum poſitum fuerit, tunc duo talia, obiecta nobis apparebunt, ſed huiuſ <lb/>modi rei cauſam alias tibi manifeſtabo.</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"/> + ſpeculi ſuperficiem eſſe <var>.g.h.</var> in figura <var>.B.</var> <lb/>obiectum autem viſibile <var>.b.</var> oculos vero <var>.a.</var> <lb/>et <var>.u.</var> punctum autem <var>.n.</var> in ſuperficie ſpecu <lb/>li, à quo imago ipſius <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/>ſit <reg norm="communis" type="context">cõmunis</reg> ſectio ſuperficiei reflexionis <lb/>radiorum <var>.b.n.a.</var> et <var>.c.f.</var> ſit communis ſectio <lb/>ſuperficiei reflexionis radiorum <var>.b.t.u.</var> qua <lb/>rum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> ſuperficies reflexionis, ere-<lb/>cta eſt ad ſuperficiem ſpeculi <var>.g.h.</var> vt ſupra <lb/>diximus. </s> + <s xml:id="s4085" xml:space="preserve">Nunc ex .19. vndecimi Eucl. ſequitur communem ſectionem harum dua-<lb/>rum ſuperficierum. (b.c.d. ſcilicet) ad rectos etiam eſſe ſupra ſuperficiem ſpeculi <var>.g.<lb/>h.</var> cum qua <var>.b.c.</var> quælibet linearum <var>.a.n.</var> vel <var>.u.t.</var> reflexarum ( productę cum fuerint ) <lb/>ſeinuicem interſecabunt eo quod duo anguli <var>.d.c.n.</var> et <var>.d.n.c.</var> ſimul collecti minores <lb/>ſunt 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, ipſis con-<lb/>trapoſiti, æquales ſint angulis <var>.b.n.c.</var> et <var>.b.t.c.</var> incidentiæ, quorum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> ex .32. <lb/>primi, minor eſt 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 huiuſmodi catheti <var>.b.c.d.</var> in quo interſecabi-<lb/>tur à linea <var>.a.n.</var> in eodem ſecabitur à linea <var>.u.t.</var> & quod punctum dicti concurſus, tan-<lb/>tum depreſſum erit ſub ſuperficie ſpeculi <var>.g.h.</var> quantum <var>.b.</var> ſupra ipſam reperietur. <lb/></s> + <s xml:id="s4087" xml:space="preserve">Nam anguli <var>.b.n.c.</var> et <var>.d.n.c.</var> ſunt inuicem æquales, <reg norm="angulique" type="simple">anguliq́;</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 ipſius 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 ſit: </s> + <s xml:id="s4089" xml:space="preserve">d.c. igitur etiam erit, & ipſum <reg norm="vnum" type="context">vnũ</reg> <lb/>& idem, quod erit propoſitum.</s> + </p> + <p> + <s xml:id="s4090" xml:space="preserve">Nunc autem cum hi duo radij ſeinuicem ſecent in puncto <var>.d.</var> ergo in ipſo puncto <var>.<lb/>d.</var> videbimur nobis videre <reg norm="imaginem" type="context">imaginẽ</reg> obiecti .b: <reg norm="cum" type="context">cũ</reg> ope <reg norm="duorum" type="context">duorũ</reg> <reg norm="iſtorum" type="context">iſtorũ</reg> <reg norm="radiorum" type="context">radiorũ</reg> <var>.n.a.</var> et <var>.t.<lb/>u.</var> ita inuicem <reg norm="ſitorum" type="context">ſitorũ</reg>, videamur nobis <reg norm="imaginem" type="context">imaginẽ</reg> proſpicere. </s> + <s xml:id="s4091" xml:space="preserve">Vnde ſi in tali caſu, 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> ipſo <lb/>loco <var>.d.</var> & non in alio ex ſuperius dictis rationibus.</s> + </p> + <p> + <s xml:id="s4092" xml:space="preserve">Et ſi ſtantibus ijs terminis volueremus pupillam oculi <var>.u.</var> verſus aliam <var>.a.</var> ad aſpi-<lb/>ciendum punctum <var>.n.</var> in ſuperficie <var>.g.h.</var> ipſius ſpeculi, hoc eſt ſi fecerimus quod axes <lb/>viſuales ſeinuicem ſecarent in ipſo puncto <var>.n</var>. </s> + <s xml:id="s4093" xml:space="preserve">tunc videremur nobis videre duas <lb/>imagines ipſius obiecti <var>.b.</var> intra ſpeculum, eo quod obiectum, propter hoc non <lb/>ceſſaret reflectere ad oculos ab ipſis punctis <var>.n.</var> et <var>.t.</var> quapropter recipiendo ra-<lb/>dium <var>.t.u.</var> in ſitu axis oculi <var>.u.</var> & radium <var>.n.a.</var> in ſitu axis oculi <var>.a.</var> hi axes ex neceſſitate <lb/>(vt probauimus ) ſeinuicem ſecant in puncto <var>.d.</var> vnde vnam tantummodo imaginem <lb/>ipſius obiecti nobis apparebit.</s> + </p> + <p> + <s xml:id="s4094" xml:space="preserve">Ex his igitur omnibus potes facilè videre omnem imaginem, cuiuſuis obiecti, re-<lb/>flexam à ſpeculo, reperiri in ipſo catheto incidentiæ, cum ipſe ſemper ſit communis <lb/>ſectio duarum ſuperficierum reflexionis, in quo catheto concurrunt ipſæ axes vi-<lb/>ſuales.</s> + </p> + <p> + <s xml:id="s4095" xml:space="preserve">Exijſdem etiam dictis rationibus facile compræhendere poteris, vnde fiat, vt vi-<lb/>deamus imaginem reflexam à ſpeculis ſphęricis concauis citra <reg norm="ipſorum" type="context">ipſorũ</reg> ſuperficiem, & <lb/>non vltra. </s> + <s xml:id="s4096" xml:space="preserve">Quod <reg norm="nunquam" type="context">nunquã</reg> euenit, niſi quando <reg norm="punctum" type="context">punctũ</reg> <var>.d.</var> interſectionis <reg norm="ipſorum" type="context">ipſorũ</reg> <reg norm="radiorum" type="context">radiorũ</reg> <lb/>viſualium (quod alio in loco non fit, niſi in catheto incidentiæ hoc eſt in communi <lb/>ſectione duarum ſuperficierum reflexionis. </s> + <s xml:id="s4097" xml:space="preserve">Dato quod obiectum non ſit in vna ea-<lb/>demq́ue ſuperficie, in qua reperti fuerint axes viſuales, hoc eſt dato, <reg norm="quod" type="simple">ꝙ</reg> ambo axes <lb/>viſuales non ſint in vna <reg norm="eademque" type="simple">eademq́;</reg> ſuperficie reflexionis) reperitur citra & non vltra ſu <lb/>perficiem ipſius ſpeculi.</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 ſemper ſu-<lb/>perficies reflexionum perpendiculares eſſe, velad rectos ſecare ſuperficiem ipſius <lb/>ſpeculi, ipſarum communes ſectiones cum ſuperficie ſpeculi ſphęrici, ſemper erunt <lb/>circunferentiæ magnorum circulorum illius ſphæræ, cuius portio eſt ſpeculum <lb/>propoſitum, vt etiam Vitellio affirmat in prima ſexti libri. </s> + <s xml:id="s4099" xml:space="preserve">Vnde vnuſquiſque ca-<lb/>thetus incidentiæ tranſibit per centrum ſpeculi, cum ipſe ſit communis ſectio dua-<lb/>rum ſuperficierum reflexionis, </s> + <s xml:id="s4100" xml:space="preserve">quare in ipſo catheto erit punctum interſectionis ip <lb/>ſorum axium viſualium ex neceſſitate, vt videbimus, ſi 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, ſint duæ ſuperficies reflexionis ſpeculi ſphærici concaui <var>.b.n.c.a.</var> <lb/>et <var>.b.t.c.u.</var> <reg norm="obiectumque" type="simple">obiectumq́;</reg> ſit <var>.b.</var> oculi autem ſint <var>.a.u.</var> punctum verò ſuperficiei ſpeculi, à <lb/>quo obiectum emittit reflexionem ſuę <lb/>imaginis ad oculum <var>.a.</var> ſit <var>.n.</var> <reg norm="punctum" type="context">pũctum</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> ſit <lb/>t. communis autem ſectio harum dua-<lb/>rum ſuperficierum ſit <var>.b.c.</var> ſed <var>.x.</var> <reg norm="centrum" type="context">centrũ</reg> <lb/>ſit ſpeculi, radius verò incidentię ſuper <lb/>ficiei <var>.b.n.c.</var> erit <var>.b.n.</var> cuius reflexus ſit <var>.n.<lb/>a.</var> radij autem alterius ſuperficiei erunt <lb/><var>b.t.</var> et <var>.t.u</var>. </s> + <s xml:id="s4102" xml:space="preserve">Imaginemur nunc duos ſemi <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 ſup-<lb/>poſito.</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 ſuppoſitis, ſi 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 ſupradictis nos ipſam videre in <reg norm="communi" type="context">cõmuni</reg> concurſu ipſorum <lb/>axium viſualium, qui axes cum reperiantur vnà cum ipſis radijs reflexis <var>.n.a.</var> et <var>.t.u.</var> <lb/>ex neceſſitate ſeinuicem <reg norm="ſecabunt" type="context">ſecabũt</reg> in catheto <var>.b.c.</var> cum extendantur in ipſis ſuperficie-<lb/>bus reflexionum, quæ ſuperficies nihil aliud commune inuicem habent, quam cathe <lb/>tum dictum <var>.b.c.</var> ſit igitur in puncto <var>.d</var>.</s> + </p> + <p> + <s xml:id="s4104" xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eſt, quod quotieſcunque vnam tantummo <lb/>do imaginem obiecti <var>.b.</var> videmus, dato quod duæ ſuperficies reflexionis ſint, & non <lb/>vna tantum, tunc angulos <var>.n.</var> et <var>.t.</var> ſemper 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. ſexti Euclid. quod eadem proportio erit ipſius <var>.b.n.</var> ad <var>.n.<lb/>d.</var> quę ipſius <var>.b.x.</var> ad <var>.x.d.</var> & ipſius <var>.b.t.</var> ad <var>.t.d.</var> ſimiliter, </s> + <s xml:id="s4107" xml:space="preserve">quare ipſiusb <var>.n.</var> ad <var>.n.<lb/>d.</var> erit vt ipſius <var>.b.t.</var> ad <var>.t.d</var>. </s> + <s xml:id="s4108" xml:space="preserve">Vnde ſequitur <var>.b.n.</var> æqualem eſſe ipſi <var>.b.t.</var> et <var>.n.d.</var> <lb/>ipſi <var>.t.d.</var> vt à medio circulo <var>.E.</var> potes videre, quamuis etiam <var>.b.</var> non eſſet extremum <lb/>diametri, ſed vbicunque volueris in ipſo diametro, vel <reg norm="etiam" type="context">etiã</reg> protracta, eo quod pun-<lb/>ctum <var>.n.</var> & punctum <var>.t.</var> in eodem ſemicirculo, vel in æqualibus ſemicirculis, non <reg norm="poſsent" type="context">poſsẽt</reg> <lb/>aliter in ipſa circunferentia locari, <reg norm="eandem" type="context">eãdem</reg> ſeruando 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 ſitu exiſtente puncto <var>.t.</var> ipſa <var>.b.t.</var> eſſet aut maior <lb/>aut minor ipſa <var>.b.n.</var> et <var>.t.d.</var> aut minor, aut maior ipſa <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 ipſius <var>.b.t.</var> ad <var>.t.d.</var> quam ipſius <var>.b.n.</var> ad <var>.n.d.</var> & non <lb/>eadem.</s> + </p> + <p> + <s xml:id="s4111" xml:space="preserve">Nunc è conuerſo ſi <var>.b.n.</var> et <var>.b.t.</var> ſunt ſibi inuicem æquales, & ſic <var>.n.d.</var> cum <var>.t.d.</var> ſequi-<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 ijſdem ſpeculationibus potes etiam videre vnde accidat quod partes ſuperio <lb/>res alicuius obiecti reflexæ à tali ſpeculo concauo videntur nobis inferiores eſſe, & <lb/>inferiores appareant ſuperiores, & dextræ ſiniſtræ, & ſiniſtræ dextræ. </s> + <s xml:id="s4113" xml:space="preserve">quod autem <lb/>hucuſque demonſtraui de ſpeculis planis, & ſphæricis concauis, ratiocinare tu ijſdem <lb/>medijs circa ſphærica conuexa, vbi clarè videbis puncta huiuſmodi ſpeculi conuexi, <lb/>à quibus reflectitur imago obiecti ad ambos oculos, ſemper oportere æquidiſtantia <lb/>eſſe à <reg norm="puncto" type="context">pũcto</reg> communi ipſius ſuperficiei ſpeculi, & catheto incidentiæ, dum unam tan <lb/>tummodo imaginem ipſius obiecti videmus, & à diuerſis ſuperficiebus reflexionum.</s> + </p> + <p> + <s xml:id="s4114" xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi ſuccurrit, hoc eſt quod poſſet ali-<lb/>quis duos ſitus inuenire, vnum pro oculo, alterum verò pro obiecto, reſpectu alicu-<lb/>ius ſpeculi concaui, ſphęroidis prolatæ, vt reflexio ipſius obiecti videretur, vt linea <lb/>diuidens per æqualia ipſum ſpeculum. </s> + <s xml:id="s4115" xml:space="preserve">Reſpectu verò alicuius ſpeculi concaui ſphæ-<lb/>roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota ſuperficie ipſius ſpecu-<lb/>li, vnde tota ſuperficies ipſius ſpeculi videretur colorata illo colore cuius eſſet <lb/>obiectum, quæ quidem paſſiones <reg norm="pendent" type="context">pendẽt</reg> à .48. tertij lib. ipſius Pergei, vt ex te ipſo fa <lb/>cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s> + </p> + <p> + <s xml:id="s4116" xml:space="preserve">Opinio autem mea, quam ſcire cupis de imagine obiecti reflexa, quam putas eſ-<lb/>ſe in ſuperficie ſpeculi, hæc eſt, quod nec in ſuperficie, nec ultra, nec citra eam eſt ip <lb/>ſa imago, quod autem vltra non ſit, hoc puto nulli dubium eſſe. </s> + <s xml:id="s4117" xml:space="preserve">eadem etiam ra-<lb/>tione non erit citra ſuperficiem ſpeculi concaui, quamuis ipſam nos compræhenda-<lb/>mus in concurſu radiorum viſualium, tam ab vno ſpeculo quam ab alio reflexione <lb/>facta. </s> + <s xml:id="s4118" xml:space="preserve">Sed quòd ipſa neque ſit in ipſa ſpeculi ſuperficie, manifeſtum erit ex hoc, <reg norm="quod" type="simple">ꝙ</reg> <lb/>duo ſpectantes in eodem ſpeculo, duas diuerſas imagines vident, tres, <reg norm="autem" type="wordlist">aũt</reg> tres, qua-<lb/>tuor, quatuor, & ſic deinceps, vnde tot eſſent imagines ſupra ſuperficiem ſpeculi, <lb/>quot obiecta, <reg norm="quod" type="simple">ꝙ</reg> tamen ita non eſt, nec plus eſt in vno loco ipſa 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"/> + niſi in obiecto ipſo, lumen enim abipſo obiecto reflexum, ſeipſum diffundit vndi-<lb/>que, & radijipſius lu<unsure/>minis reflexi, vt plurimum ſeinuicem ſecant. </s> + <s xml:id="s4119" xml:space="preserve">Vnde in ipſo ae-<lb/>re funt omnes miſti. </s> + <s xml:id="s4120" xml:space="preserve">Quapropter natura ſagacifſima pupillam oculi animalibus tam <lb/>paruam conſtruxit ad ſuperficiem tam amplæ ſphæræ ipſius oculi, vt diſtinctæ vide-<lb/>rentur omnia obiecta.</s> + </p> + <p> + <s xml:id="s4121" xml:space="preserve">Nolo etiam tibi tacere, quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> oculorum pupillæ poſitæ fuerint inter <lb/>cathetum incidentiæ, & ſuperficiem <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"/> + ſpeculi ſphærici 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, ſed duas nec non confu-<lb/>sè, </s> + <s xml:id="s4123" xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</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">cũ</reg> <lb/>ambobus axibus viſualibus, eo quod <lb/>axes vifuales nunquam poſſunt inui-<lb/>cem interſecari poſt viſum, ſed ſem-<lb/>per ante ipſum, 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 ſi obiectum inci <lb/>derit in eadem ſuperficie, in qua duo <lb/>axes viſuales, vel radij reflexi <reg norm="reperiuntur" type="context">reperiũtur</reg>, hoc eſt in vna <reg norm="eademque" type="simple">eademq́;</reg> ſuperficie 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 ipſo catheto ſed extra, in qua interſectione fit viſio vnius tan-<lb/>tummodo imaginis, quod antiqui non animaduerterunt. </s> + <s xml:id="s4127" xml:space="preserve">Hoc autem dico deſpe-<lb/>culo ſphærico concauo.</s> + </p> + </div> + <div type="letter" level="4" n="4"> + <head xml:id="head501" style="it" xml:space="preserve">Speculatio cuiuſdam propoſitionis aritbmetica.</head> + <head xml:id="head502" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4128" xml:space="preserve">SPeculatio vltimæ propoſitionis quam numerorum via inueni, hæc eſt. </s> + <s xml:id="s4129" xml:space="preserve">Imagi-<lb/>nemur triangulum <var>.r.e.o.</var> abſciſum à circulo, in cuius circunferentia ſit punctum <lb/>r. ſuperioris anguli ipſius trianguli, vel etiam non ſit abſciſum dummodo protrahan <lb/>tur lineæ <reg norm="vſque" type="simple">vſq;</reg> ad circunferentiam, à quo ad oppoſitum 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">ꝙ</reg>. <var>K.o.</var> æqualis ſit <var>.f.e.</var> vnde hæc .4. lineæ ſecabuntur 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">ꝙ</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 tranſire per centrum <var>.s.</var> ipſius circuli, ſi-<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 eſt <var>.e.a</var>: <var>i.d.</var> et <var>.o.q.</var> & vbi circulus ſecat <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> ſimi-<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">productumq́</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> ſunt in-<lb/>u<unsure/>icem ſimiles, ſed productum <var>.g.r.a.</var> ſimul cum producto <var>.g.r.q.</var> duplum eſt producto <var>.<lb/>g.r.d.</var> ex prima ſexti, eo quod <var>.a.r.q.</var> dupla eſt <var>.d.r.</var> & ideo productum <var>.e.r.u.</var> ſimul <reg norm="cum" type="context">cũ</reg> <lb/>producto <var>.o.r.n.</var> duplum erit producto <var>.i.r.x.</var> quod quidem æquale eſt producto <var>.g.r.<lb/>d.</var> ex ſimilibus rationibus iam ſupradictis. </s> + <s xml:id="s4131" xml:space="preserve">Nunc ex ſimilibus 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 ſecun-<lb/>dis. </s> + <s xml:id="s4133" xml:space="preserve">Quod eſt propoſitum.</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 huiuſmodi demonſtrat ione facilè videre poteris non eſſe generaliter verum, <lb/>id quod Nicolaus Tartalea inquit .43. quæſito vltimæ partis ſuorum tractatuum, hoc <lb/>eſt centrum circuli <var>.r.n.g.</var> ſemper eſſe in perpendiculari, quæ à puncto <var>.r.</var> ad lineam <var>.e.<lb/>o.</var> tranſit, protracta ipſa <var>.e.o.</var> quantum volueris, imò in quacunque alia linea ipſum eſ <lb/>ſe poteſt, nec non in aliqua parallela ipſi <var>.e.o.</var> quemadmodum ex te ipſo, medianti-<lb/>bus, hic ſupradictis rationibus videre poteris, vnde ex neceſſitate ſequitur illud pro <lb/>blema ſemper ferè falſum 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 ſpeculatio 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 demonſtrandum <lb/>breuitatem radiorum incidentium, & reflexorum in ſpeculo plano, nunc ad <lb/>te ſcribo, quamuis prolixior ali quantulum ſit eo, quod ab antiquis traditus eſt.</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> ſectione ſuperficiei reflexionis <reg norm="cum" type="context">cũ</reg> ſpe-<lb/>culo <var>.r.a.</var> verò et <var>.a.b.</var> pro radijs dictis, qui ſemper <reg norm="faciunt" type="context">faciũt</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 iiſdem punctis <var>.b.r.</var> ad aliud punctum, <lb/>quod volueris ipſius lineæ <var>.p.h.</var> quas probabo <reg norm="longiores" type="conjecture">longiores'</reg> (ſimul ſumptas) eſſe priori-<lb/>bus. </s> + <s xml:id="s4138" xml:space="preserve">Imaginemur igitur duas perpendiculares, ſeu cathetos <var>.b.i.</var> et <var>.q.r.a.</var> punctis <var>.b.<lb/>r.</var> ad <var>.p.h.</var> <reg norm="abſciſſaque" type="simple">abſciſſaq́</reg> ſit linea <var>.o.b.</var> in puncto <var>.x.</var> ita quod <var>.b.x.</var> æqualis ſit ipſi <var>.b.a.</var> quod <lb/>nulli dubium erit poſſe effici, cum <var>.o.b.</var> <reg norm="longiot" type="context">lõgiot</reg> ſit <var>.b.a.</var> co quod opponatur angulo ob-<lb/>tuſo ipſius trianguli <var>.b.a.o.</var> quę <var>.o.b.</var> ſimiliter protrahatur vſque ad <var>.d.</var> ita quod <var>.b.d.</var> <lb/>æqualis ſit <var>.x.b.</var> </s> + <s xml:id="s4139" xml:space="preserve">deinde protrahatur <var>.o.i.</var> quouſque <var>.i.h.</var> æqualis ſit <var>.a.i</var>. </s> + <s xml:id="s4140" xml:space="preserve">In alia parte po-<lb/>ſtea idem faciendum eſt ſecando <var>.a.r.</var> in puncto <var>.u.</var> ita quod <var>.u.r.</var> æqualis ſit <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">productũ</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> exiſtis <lb/>rationibus. </s> + <s xml:id="s4141" xml:space="preserve">Nam cum quadratum ipſius <var>.o.b.</var> æquale ſit duobus quadratis <var>.o.i.</var> et <var>.i.<lb/>b.</var> ex penultima primi Eucli. ipſa quadrata <var>.o.i.</var> et <var>.i.b.</var> æqualia erunt producto <var>.o.d.</var> in <lb/><var>o.x.</var> ſimul ſumpto cum quadrato <var>.b.x.</var> ex .6. ſecundi, hoc eſt ipſi producto ſimul ſum-<lb/>pto cum quadrato <var>.b.a.</var> hoc eſt ipſi producto ſimul ſumpto cum duobus quadratis <var>.a.<lb/>i.</var> et <var>.i.b.</var> ſed quia productum <var>.o.h.</var> in <var>.o.a.</var> ſimul ſumpto cum quadrato <var>.a.i.</var> ęquatur qua <lb/>drato <var>.o.i.</var> ideo productum <var>.o.h.</var> in <var>.o.a.</var> ſimul ſumptum 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> ſimul ſumpto <reg norm="cum" type="context">cũ</reg> duobus quadratis dictis <lb/>hoc eſt ipſius <var>.a.i.</var> et <var>.i.b.</var> quę 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 ipſi <var>.o.b.</var> & habebimus proportionem <lb/>ipſius <var>.a.b.</var> ad <var>.a.i.</var> maiorem eſſe ea quæ eſt ipſius <var>.a.e.</var> ad eandem <var>.a.i.</var> cum <var>.a.b.</var> maior <lb/>ſit ipſa <var>.a.e.</var> vt oppoſita angulo obtuſo, quapropter proportio <var>.x.b.</var> ad <var>.a.i.</var> maior erit <lb/>ea quæ eſt <var>.o.b.</var> ad <var>.o.i</var>. </s> + <s xml:id="s4144" xml:space="preserve">Iam enim ſcis proportionem <var>.o.b.</var> ad <var>.o.i.</var> eſſe, vt <var>.a.e.</var> ad <var>.a.i.</var> ex <lb/>ſimilitudine 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">tũc</reg> ex .27. quinti <reg norm="permutando" type="simple context">ꝑmutãdo</reg> <reg norm="proportio" type="simple">ꝓportio</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="eiuſdem" type="context">eiuſdẽ</reg> <reg norm="componendo" type="context context">cõponẽdo</reg> maior <reg norm="proportio" type="simple">ꝓportio</reg> erit <var>.o.d.</var> ad <var>.o.b.</var> ea quę eſt <var>.o.h.</var> ad. o <lb/>i. & <reg norm="permutando" type="context">permutãdo</reg> maior ipſius <var>.o.d.</var> ad <var>.o.h.</var> ea quæ <var>.o.b.</var> ad <var>.o.i.</var> & ex .33. maior ipſius <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 eſt <var>.a.r.</var> ad <var>.a.q.</var> ex ſimilitudine <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 eſt <var>.o.b.</var> ad <var>.o.i.</var> & exijſdem <lb/>ſupradictis rationibus maior erit proportio ipſius <var>.s.a.</var> ad <var>.p.a.</var> ea quæ eſt <var>.a.r.</var> ad <var>.a.q.</var> <lb/>ſed 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 eſt <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 ipſius <var>.a.s.</var> ad <lb/><var>a.p.</var> ca quæ <var>.o.d.</var> ad <var>.o.h.</var> ſed cum ex .15 <lb/>ſexti, eadem ſit 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> quę <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 ipſius <var>.o.d.</var> ad <var>.o.h.</var> quæ <var>.o.a.</var> <lb/>ad <var>.o.x.</var> & ipſius <var>.a.o.</var> ad <var>.a.u.</var> quemad-<lb/>modum ipſius <var>.a.s.</var> ad <var>.a.p</var>. </s> + <s xml:id="s4149" xml:space="preserve">Quare maior proportio erit ipſius <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 ſequitur <var>.o.x.</var> maiorem eſſe <var>.a.u.</var> ex .8. quinti, ergo <var>.b.x.o.r.</var> longior erit <lb/>ipſa <var>.b.a.u.r</var>. </s> + <s xml:id="s4151" xml:space="preserve">Quod eſt propoſitum.</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 ellipſis tranſire per duo puncta <var>.r.</var> et <var>.b.</var> ſupponendo ipſa puncta, ea eſle, quæ ita <lb/>axem diuidunt, vt ſingula produ-<lb/> + <anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/> + cta fectionum ſint, vt inquit Per-<lb/>geus. </s> + <s xml:id="s4154" xml:space="preserve">imaginemur, etiam <var>.p.h.</var> con <lb/>tiguam eſſe ipſi ellipſi in <reg norm="puncto" type="context">pũcto</reg> <var>.a.</var> <lb/>vnde ſi protractæ fuerint duæ <var>.r.a.</var> <lb/>et <var>.b.a.</var> habebimus ex .48. tertijip-<lb/>ſius 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/>poſtea ad quoduis punctum ipſius <lb/><var>p.h.</var> duas <var>.b.o.</var> et <var>.r.o.</var> certi erimus, <lb/>quod ſecabuntur à gyro oxygo-<lb/>nio, quarum vna ſecta ſit in pun-<lb/>cto <var>.i.</var> ducta poſtea <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 ipſa <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 ſpeculum vstorium.</head> + <head xml:id="head506" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4156" xml:space="preserve">VErum ſpeculum vſtorium, illud non eſt, quod ab Euclide traditum fuit, & <reg norm="quod" type="simple">ꝙ</reg> <lb/>tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie <lb/>ſphærica concaua ſeinuicem in centro ſpeculi interſecare. </s> + <s xml:id="s4157" xml:space="preserve">Nam cum omnes lineę <lb/>recte à centro, & cir cunferentia alicuius ſphæræ terminatæ, ſint eidem circunferen-<lb/>tiæ perpendiculares, ſequeretur ex neceſſitate radios incidentiæ etiam perpendicu <lb/>lares eidem ſuperficiei eſſe, cum anguli incidentiæ ſemper æquales ſint angulis re-<lb/>flexionis, vnde etiam ex neceſſitate ſequeretur punctum corporis lucidi, à quo radij <lb/>luminoſi excunt, in centro ſpeculi reperiri. </s> + <s xml:id="s4158" xml:space="preserve">quod quidem falſiſſimum eſt.</s> + </p> + <p> + <s xml:id="s4159" xml:space="preserve">Alia etiam via poſſum hanc oſtendere impoſſibilitatem, & tibi probabo, quod <lb/>in nullo aliquo puncto poſſunt inuicem conuenire ipſi 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> ſectio ſuperficiei reflexionis cum ſpeculo, cuius centrum ſit <var>.o.</var> <lb/>punctum verò lucidum ſit <var>.g.</var> <reg norm="protrahaturque" type="simple">protrahaturq́</reg> <var>.g.o.a</var>. </s> + <s xml:id="s4161" xml:space="preserve">Nunc autem primum dico, quod <lb/>radij reflexi à punctis diuerſarum <reg norm="diſtantiarum" type="context">diſtantiarũ</reg> ab <var>.a.</var> non <reg norm="coincident" type="context">coincidẽt</reg> inuicem in aliquo <lb/>puncto lineę <var>.g.o.a</var>: ſint ergo duo puncta <var>.u.</var> et <var>.r.</var> diuerſarum <reg norm="diſtantiarum" type="context">diſtantiarũ</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> ſit <var>.r.e.</var> protrahatur <lb/><var>u.e.</var> quam dico effe non poſſe radium reflexum ab <var>.u.</var> quotieſcunque eius incidens <lb/>deſcendat 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">reflexũ</reg> eſſe ipſius <var>.g.u.</var> igitur anguli <var>.g.u.o.</var> et <var>.o.u.e.</var> erunt inuicem æquales, & ſic <lb/>etiam erunt duo <var>.g.r.o.</var> et <var>.o.r.e.</var> vnde ex tertia ſexti & .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 demonſtra-<lb/>bo, eo quod cum <var>.g.u.</var> maior ſit <var>.g.r.</var> ex .8. tertij, erit ex .8. quinti proportio ipſius <var>.g.u.</var> <lb/>ad <var>.r.e.</var> maior proportione ipſius <var>.g.r.</var> ad <var>.r.e.</var> ſed ex .7. tertij <var>.u.e.</var> minor eſt <var>.r.e.</var> erit igi-<lb/>tur ex dicta .8. quinti maior proportio <reg norm="ipſius" type="simple">ipſiꝰ</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 ipſius <var>.g.u.</var> ad <var>.u.e.</var> <lb/> + <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> + quam ipſius <var>.g.r.</var> ad <var>.r.e.</var> ergo non æqualis, <lb/>quapropter impoſſibile eſt <var>.u.e.</var> eſſe radium <lb/>reflexum incidentis radij <var>.g.u</var>. </s> + <s xml:id="s4163" xml:space="preserve">Vnde ſequi <lb/>tur concurſum radiorum reflexorum à ſpe-<lb/>culo ſphærico concauo non eſſe in vno, & <lb/>e<unsure/>odem puncto ipſius catheti incidentiæ, <lb/>quando à ſitu non æquidiſtanti ab ipſo ca-<lb/>theto <reg norm="reflectuntur" type="context">reflectũtur</reg>, ex hac ſpeculatione <reg norm="etiam" type="context">etiã</reg> <lb/>videre licet, verum eſſe id quod in .3. Epiſto <lb/>la tibi ſcripſi nempe, quod quotieſcunque <lb/>axes viſuales, vel radij reflexi, in vna ea-<lb/><reg norm="demque" type="simple">demq́</reg> ſuperficie reflexionis fuerint, </s> + <s xml:id="s4164" xml:space="preserve">tunc <lb/>imago obiecti nullo modo videbitur in ca-<lb/>theto incidentiæ, in ſpeculo ſphærico 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 ſolutio.</head> + <head xml:id="head508" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4165" xml:space="preserve">NOn abſque ratione dubitas, vtrum etiam in ſphæricis ſpeculis conuexis idem <lb/>accidat, hoc eſt, an radij reflexi à punctis inęqualis diſtantiæ à catheto inciden <lb/>tiæ conueniant inuicem in eodem catheto.</s> + </p> + <p> + <s xml:id="s4166" xml:space="preserve">Ad quod reſpondeo, non concurrere in dicto catheto, ſed extra ipſum, & ſimi-<lb/>liter extra ipſum vide bitur imago.</s> + </p> + <p> + <s xml:id="s4167" xml:space="preserve">Pro cuius rei ratione, imaginemur ſuperficiem reflexionis alicuius ſpeculi ſphæ-<lb/>rici conuexi <var>.b.d.h.g.</var> cuius communis ſectio cum ſuperficie ſphærica ſit 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> ſit ſe <lb/>midiameter circuli <var>.d.g.h.</var> et <var>.o.c.</var> ſit plus medietate ipſius <var>.o.g.</var> <reg norm="accipiaturque" type="simple">accipiaturq́;</reg> linea <var>.e.c.</var> <lb/>minor ipſa <var>.o.c.</var> ſed maior ipſa <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 ipſum aliquantulum <lb/>plus quam <var>.c.g.</var> ſed minus quam <var>.c.<lb/>o.</var> ſignando circunferentiam <var>.d.e.h.</var> <lb/>in puncto <var>.e.</var> quod ex .7. tertij poſſi-<lb/>bile eſt, protrahatur poſtea <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 poſtea cum fuerit <var>.c.<lb/>e.K.</var> indeterminatè, <reg norm="habebimus" type="simple">habebimꝰ</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">qđ</reg> ſi <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 ſecabit cathetum <var>.b.o.</var> in puncto <var>.c.</var> intra ſpeculum, nec dubitandum eſt quin <lb/>linea <var>.e.b.</var> ſectura ſit <var>.b.o.</var> eo quod cum angulus <var>.o.e.c.</var> ſit maior angulo <var>.e.o.c.</var> ex .19. <lb/>primi, & ſimiliter angulus <var>.b.e.f.</var> ſequitur 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 eſt <lb/><reg norm="neceſſarium" type="context">neceſſariũ</reg> <reg norm="semper" type="context">sẽper</reg> exiſtat, 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/>ſa neceſſariò debeat ſemper maior eſſe ipſa <var>.c.g.</var> clarum eſt 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> ſit vt reflectatur. </s> + <s xml:id="s4172" xml:space="preserve">Accipiamus <lb/>nunc <var>.p.c.</var> minorem medietate ipſius <var>.b.c.</var> & à puncto <var>.p.</var> imaginemur tangentem <var>.p.q.</var> <lb/>in puncto <var>.q.</var> prorractaq́ue ſit <var>.b.q.</var> vt radius incidentiæ, </s> + <s xml:id="s4173" xml:space="preserve">tunc dico, radium reflexum <lb/>ipſius <var>.b.q.</var> <reg norm="non" type="context">nõ</reg> concurrere in eodem puncto <var>.c.</var> ipſius catheti, ſi vero dixeris <reg norm="quod" type="simple">ꝙ</reg> ſic. </s> + <s xml:id="s4174" xml:space="preserve">Eſto <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. ſexti Vitellionis proportio <var>.b.i.</var> ad <var>.i.c.</var> erit, vt <var>.b.o.</var> ad <var>.o.c.</var> & ſimiliter erit <lb/>ipſius <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/>ſius <var>.b.p.</var> ad <var>.p.c.</var> erit vt ipſius <var>.b.i.</var> ad <var>.i.c.</var> ſed quia <var>.p.b.</var> vt pars ipſius <var>.b.i.</var> minor eſt ip-<lb/>ſa, ergo ex .14. dicti <var>.p.c.</var> minor erit ipſa <var>.c.i.</var> hoc eſt totum minus ſua parte, quod eſt <lb/>impoſſibile, </s> + <s xml:id="s4177" xml:space="preserve">quare non in ipſo catheto videbitur imago ipſius 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 ipſis ſpeculis ſphæricis conuexis, hoc eſt <lb/>quod poſſibile ſit aliquoties, radium reflexum concurrere cum catheto incidentiæ <lb/>extra ſpeculum inter puncta <var>.g.</var> et <var>.p.</var> vt exempli gratia .ſi 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> ipſius concurſus cum catheto incidentiæ eſſet <lb/>inter <var>.g.</var> et <var>.p.</var> eo quod <reg norm="cum" type="context">cũ</reg> linea <var>.p.q.</var> debeat @iui lere <reg norm="angulum" type="context">angulũ</reg> <var>.b.</var> q, c. <reg norm="per" type="simple">ꝑ</reg> ęqualia, oportebit <lb/>c. poſitum eſſe inter <var>.g.</var> et <var>.p.</var> quia angulus <var>.g.q.p.</var> maior eſt 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 poſtea <var>.q.<lb/>p.</var> <reg norm="vſque" type="simple">vſq;</reg> ad <reg norm="alteram" type="context">alterã</reg> <reg norm="partem" type="context">partẽ</reg> circunferen-<lb/>tiæ ipſius circuli. </s> + <s xml:id="s4180" xml:space="preserve"><reg norm="argumentando" type="context context">argumẽtãdo</reg> dein-<lb/>de mediante vltima ſexti, illud <reg norm="idem" type="context">idẽ</reg> <lb/>po@es etiam ſcire ex .22. quinti Alha <lb/>zeni. & ex .26. ſexti Vitellionis. </s> + <s xml:id="s4181" xml:space="preserve">vn-<lb/>de ſi ad ambas pupillas venerint ra <lb/>dij reflexi ipſius obiecti <var>.b.a.</var> duobus <lb/>punctis huiuſmodi ſpeculi, ita di-<lb/>ſtantibus à puncto <var>.g.</var> vt <var>.q</var>. </s> + <s xml:id="s4182" xml:space="preserve">tunc com <lb/>mune punctum concurſus axium vi <lb/>ſualium erit in catheto inter <var>.g.p.</var> <lb/>vbi apparebit imago ex ſuperius di <lb/>ctis rationibus, ita vt <reg norm="non" type="context">nõ</reg> ſolum con <lb/>cauis, ſed 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 poteſt euenire, vt tibi alias dixi, eo quod ſi <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> ſectione <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg> <var>.l.t.x.</var> reflexionis & <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg> ſpeculi, <reg norm="punctumque" type="context context simple">pũctũq́;</reg> <lb/>lucidum <var>.l.</var> <reg norm="protractoque" type="simple">protractoq́;</reg> catheto <var>.l.r.t.</var> <reg norm="lineisque" type="simple">lineisq́;</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 ſint, & ſic 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 ſemper in puncto <var>.t.</var> <reg norm="conuenient" type="context">conueniẽt</reg> omnes radij reflexi ipſius + <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 ſphærico concauo, ſeu conuexo, non <lb/>omnes radij reflexi conueniunt in vno, <reg norm="eodemque" type="simple">eodemq́;</reg> puncto catheti incidentiæ, quemad <lb/>modum in planis accidit, in quibus ſemper vnum, & idem punctum eſt ipſis commu <lb/>ne in ipſo incidentiæ catheto.</s> + </p> + <p> + <s xml:id="s4184" xml:space="preserve">Non prætermittam etiam hunc alium breuiorem modum ſpeculandi <reg norm="æqualita- tem" type="context">æqualita-<lb/>tẽ</reg> depreſſionis imaginis ſub ſpeculo plano, ei quæ ſupra reperitur ipſius obiecti, in ca <lb/>theto incidentiæ, quemadmodum nu nc <lb/>vltimò diximus, hoc eſt 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 eiuſdem obiecti reflexa à pun-<lb/>cto <var>.m.</var> reperiatur in linea <var>.z.m.t.</var> & <lb/>iſtæ duæ lineę ſeinuicem ſecent in <lb/>puncto <var>.t.</var> ipſius catheti, exiſtente <var>.<lb/>r.t.</var> æquali <var>.r.l.</var> vt nunc vidimus, er-<lb/>go ſemper imago reflexa à ſpecu-<lb/>lo plano, nobis apparebit <reg norm="in" type="wordlist">ĩ</reg> ipſo ca <lb/>theto, tam vltra ſpeculum, quam ci <lb/>tra ipſum, <reg norm="repertum" type="simple context">reꝑtũ</reg> fuerit <reg norm="ipsum" type="context">ipsũ</reg> <reg norm="obiectum" type="context">obiectũ</reg> <lb/>quod nec Alhazem, nec Vitellio, <lb/>nec alius aliquis (quod ſciam) ad huc ſcientificè demonſtrauit. </s> + <s xml:id="s4185" xml:space="preserve">exempla enim vel ex <lb/>perientia non faciunt ſcire. </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 ſint duobus rectis cum <lb/>æquales ſint 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 ecclipſibus Lunaribus.</head> + <head xml:id="head510" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4187" xml:space="preserve">ROtunditas vmbræ in ecclipſi-<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, & ſi terra eſ-<lb/>ſet <reg norm="etiam" type="context">etiã</reg> cuiuſuis alterius figurę, <reg norm="quam" type="context">quã</reg> <lb/>ſphæricę, dummodo aqua impleret <lb/><reg norm="locum" type="context">locũ</reg> ſphęriceitatis à terra <reg norm="derelictum" type="context">derelictũ</reg>, <lb/>nihilominus vmbra eſſet rotunda, <lb/>quę 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 ſentiat in libro de magnitudi-<lb/>ne terrę, & 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">Sciẽdũ</reg> enim eſt, quod omne cor <lb/>pus in ſe habens <reg norm="aliquantulum" type="context">aliquantulũ</reg> opaci-<lb/>tatis, ſemper debilitat <reg norm="radium" type="context">radiũ</reg> lumino <lb/>ſum, & <reg norm="tanto" type="context">tãto</reg> magis, <reg norm="quanto" type="context">quãto</reg> magis in ip <lb/>ſo corpore radius penetrat, <reg norm="etiam" type="context">etiã</reg> & ſi <lb/>ad rectos incideret ipſe radius ſupra <lb/><reg norm="ſuperficiem" type="context">ſuperficiẽ</reg> <reg norm="ipſius" type="simple">ipſiꝰ</reg> corporis. </s> + <s xml:id="s4189" xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gra <lb/>tia, eſto <var>.q.p.</var> corpus a <reg norm="queum" type="context">queũ</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"/> + deſcendat radius <var>.b.d.K.s.f.</var> ad <reg norm="libitum" type="context">libitũ</reg> hoc eſt rectè vel obliquè, cuius pars <var>.b.d.</var> in ipſo <lb/>aere exiſtat. </s> + <s xml:id="s4190" xml:space="preserve">Nunc manifeſtum erit partem <var>.b.d.</var> ipſius radij clariorem ſeu minus im <lb/><reg norm="peditam" type="context">peditã</reg> eſſe quam <var>.d.K.</var> quod ex eo etiam cognoſcere poſſumus quia <var>.b.d.</var> reflectitur à <lb/>puncto <var>.d.</var> ſuperficiei corporis a quei, quapropter minus luminoſa remanebit pars <var>.d.<lb/>K.</var> cum non tota claritas <var>.b.d.</var> deſcendat in corpore aqueo, ſed vna eius pars reflecta-<lb/>tur, reliqua verò tantummodò deſcendat, </s> + <s xml:id="s4191" xml:space="preserve">deinde pars <var>.K.s.</var> ex neceſſitate debilior <lb/>erit ipſa <var>.d.K.</var> eo quod ſuccedit poſt ipſam <var>.d.K.</var> propter hoc etiam, quia cum corpus <lb/>aqueum habeat aliquantulum opacitatis, radius <var>.d.K.</var> ab omni puncto ipſius ſpiſſitu-<lb/>dinis a quæ continuo reflectitur, quę quidem reflexio eſt illud lumen cęruleum, quod <lb/>in profunditate ipſius aquę nobis apparet. </s> + <s xml:id="s4192" xml:space="preserve">Cum igitur reflexio ipſa ſemper detra-<lb/>hat ab ipſo radio luminoſo, reſiduum verò ſit id quod penetrat, ideo <var>.K.s.</var> erit vna <lb/>pars tantummodò luminis ipſius <var>.d.K</var>: in <var>.s.f.</var> verò aliqua pars luminis ipſius <var>.K.s.</var> & ſic <lb/>continuò debilitatur radius, ita quod ad nihilum vſque deuenit, & vltra tale cor-<lb/>pus remanebit vmbra, quaſi ſi ipſum corpus eſſet perfectè opacum, cuius rei cauſa, <lb/>eſt illa continua reflexio, vt diximus, quæ continuò adimit aliquid ex ipſo radio, <lb/>nec permittit eum totum tranſire.</s> + </p> + <p> + <s xml:id="s4193" xml:space="preserve">Quapropter mirandum non eſt 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/>ſuperficie corporis lunaris, ab alia etiam ratione prouenire poſſet. </s> + <s xml:id="s4195" xml:space="preserve">Imaginemur enim <lb/>aggregatum terrę, <reg norm="marisque" type="simple">marisq́;</reg> eſſe tantummodò aqueum, quod quidem eſſet perfectè <lb/>ſphæricum ratione centri grauitatis, <reg norm="ſupponamusque" type="simple">ſupponamusq́;</reg> <reg norm="ipsum" type="context">ipsũ</reg> eſſe valde diaphanum, ita <lb/>quod radij ſolares ipſum penetraſſent. </s> + <s xml:id="s4196" xml:space="preserve">Tunc dico quod in ſuperficie corporis luna-<lb/>ris produceret vmbram. </s> + <s xml:id="s4197" xml:space="preserve">Pro cuius intelligentia cogitemus ſubſcriptam hic figuram <lb/><var>b.h.q.a.e.</var> eſſe ſphęram aliquam cryſtallinam, & ad partem <var>.b.h.q.</var> ſit radius lumino-<lb/>ſus ſolaris qui ipſam illuminet, cuius radij extremitates ſint <var>.d.b.l.</var> et <var>.p.q.r.</var> ſupponen-<lb/>do <var>.d.l.</var> et <var>.p.r.</var> terminos eſſe vnius plani ſecantis ipſum radium per axem, </s> + <s xml:id="s4198" xml:space="preserve">tunc vide-<lb/>bis ipſum radium <var>.b.p.q.d.</var> <reg norm="tranſeun- tem" type="context">tranſeũ-<lb/> + <anchor type="figure" xlink:label="fig-0358-01a" xlink:href="fig-0358-01"/> + tem</reg> ipſam ſphæram, congregari ſeu <lb/><reg norm="condenſari" type="context">condẽſari</reg>, ob vniformem refractio-<lb/>nem, vſque ad punctum <var>.o.</var> deinde; <lb/>propter rectitudinem ipſius diffu-<lb/>ſionis, vltra punctum <var>.o.</var> ipſum dila-<lb/>tari, diſgregari, ſeu rarefieri, <reg norm="quouſque" type="simple">quouſq;</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 vmbroſa remanerent. di-<lb/><reg norm="ſtantiaque" type="simple">ſtantiaq́;</reg> ab <var>.o.</var> ad ſuperficiem ſphęri <lb/>cam corporis <var>.b.e.d.q.</var> non ſolum <reg norm="non" type="context">nõ</reg> <lb/>maior eſt diametro ipſius ſphæræ; <lb/></s> + <s xml:id="s4199" xml:space="preserve">imo minor, vt à te ipſo experiri po-<lb/>tes. </s> + <s xml:id="s4200" xml:space="preserve">Poſito igitur aliquo obiecto <lb/>opaco in loco <var>.K.o.g.</var> eius ſuperficies <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">Poſito dein <lb/>de ipſo obiecto in loco <var>.n.y.x.m.</var> <reg norm="eius" type="simple">eiꝰ</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 deſtitutæ <reg norm="interuallumque" type="simple">interuallumq́;</reg> tantummodò inter <var>.y.x.</var> illuminatum erit, ſed ſi in <lb/>loco <var>.c.u.</var> poſitum fuerit, </s> + <s xml:id="s4202" xml:space="preserve">tunc totum <var>.c.u.</var> illuminatum erit, ſed debili modo propter <lb/>detractionem factam à reflexione in ſuperficie corporis ſphærici, vt ſupra 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">Poſito 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, ſed <var>.z.H.</var> pauciſſimum habebit lumen, pro-<lb/>pter diſgregationem radiorum. </s> + <s xml:id="s4204" xml:space="preserve">Poſito poſtea ipſo obiecto in loco <var>.t.l.r.s.</var> tanto <lb/>minus lumen habebit pars <var>.l.r.</var> propter dictam <reg norm="diſgregationem" type="context">diſgregationẽ</reg>, ſeu <reg norm="diſſipationem" type="context">diſſipationẽ</reg> radio <lb/>rum, & ſic ſucceſſiuè quanto remotius poſitum fuerit ipſum 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 ſcilicet, qui per ſphæram chryſtallinam tranſibunt, ſed videbi-<lb/>tur vmbra ipſius ſphęrę in obiecto propoſito, cum nullum actum illuminationis in <lb/>eo loco obiecti habeant radij tranſeuntes per dictam ſphęram. </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 deſtituta.</s> + </p> + <p> + <s xml:id="s4207" xml:space="preserve">Quòd vero tolerabilior ſit oculis radius reflexus Solis à ſuperſicie aquæ, quàm <lb/>à ſuperficie alicuius ſpeculi, oritur ab eo, quod ſupra diximus, hoc eſt, quod ma-<lb/>gna parsipſius luminis penetrat in aquam, & non totum reflectit, quod quidem non <lb/>accidit ſpeculis 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/>cuiuſuis trianguli ſupra 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 cuiuſuis triangu-<lb/>li orthogonij, longiora ſint tertio latere, per diametrum circuli in eo in-<lb/>ſcripti, ab alijs iam demonſtratum fuit. </s> + <s xml:id="s4209" xml:space="preserve">Sed quòd quælibet duo latera <lb/>cuiuſuis trianguli longiora ſint tertio per latus tetragonicum, quadrupli <lb/>producti cuiuſuis lineæ deſcendentis ab angulo contento à dictis duobus lateribus <lb/>ad oppoſitam partem circuli inſcripti, in partem extrinſecam ipſius lineæ, nullus <lb/>(quod ſciam) vnquam ſcripſit, 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 deſcribatur circulus <var>.<lb/>u.s.n.</var> & puncta contingentiæ ſint eadem <var>.u.s.n.</var> à puncto vero <var>.a.</var> deſcendat linea <var>.a.<lb/>i.e.</var> quæ terminetur à circunferentia in puncto <var>.e.</var> ipſius 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">tetragonicũ</reg> qua-<lb/>drupli producti ipſius <var>.a.e.</var> in <var>.a.i</var>. </s> + <s xml:id="s4212" xml:space="preserve">Nam certi ſamus ex vltima parte penultimæ ter-<lb/>tij Eucli <var>.n.c.</var> et <var>.s.c.</var> æquales inuicem eſſe, & ſimiliter <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"/> + ipſo <var>.b.c.</var> per <var>.a.u.</var> et <var>.a.n.</var> quæ duæ <lb/>partes ſunt inuicem æquales di-<lb/>cta ratione, & quadratum lineæ <lb/>æqualis aggregato earum, eſſet qua <lb/>druplum quadrato cuiuſuis earum <lb/>ex .4. ſecundi, ſed ex penultima ter <lb/>tij, productum <var>.a.e.</var> in <var>.a.i.</var> æquale eſt <lb/>quadrato ipſius <var>.a.u.</var> vel ipſius <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 eſt igitur quod <var>.a.b.</var> cum <var>.a.c.</var> longiores ſint ipſa <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 propoſitum.</s> + </p> + <p> + <s xml:id="s4214" xml:space="preserve">Illud etiam non eſt ſpernendum, quod quotieſcunque data fuerint omnia latera <lb/>alicuius trianguli, illicò poſſumus cognoſcere puncta <var>.u.n.s.</var> contingentiæ circuli in <lb/>ſcripti, ope vltimæ partis penultimæ tertij, eo quod ex illa iam ſcimus, 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 vnaquęque nota erit, cum illarum quælibet, medietas ſit reſidui cogniti, detra <lb/>hendo poſtea vnam <reg norm="illarum" type="context">illarũ</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> ęqualis <var>.b.s.</var> vel <var>.c.<lb/>s.</var> vnde ſimiliter nobis innoteſcet <lb/>punctum <var>.s.</var> cum duobus punctis <var>.u.</var> <lb/>ct <var>.n.</var> à quibus duobus punctis, ſi <lb/>duę perpendiculares ad talia latera <lb/>ductæ fuerint, vbi hæe perpendicu <lb/>lares ſeinuicem ſecabunt, ibi <reg norm="cen- trum" type="context">cen-<lb/>trũ</reg> circuli inſcriptibilis erit in trian <lb/>gulo propoſito.</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 eſt niſi <lb/>quod cum Iris videtur, non eodem loco ab omnibus videtur, quia reflexio eſt, & <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 Inſtrumento oxygonio, ſeu elliptico.</head> + <head xml:id="head514" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4216" xml:space="preserve">QVod aliquando à me audiuiſti falſum non eſt, ſcilicet poſſibile eſſe (vt <lb/>ſpeculatus ſum) particulare inſtrumentum fabricari ad deſignandum oxy-<lb/>goniam, ſeu ellipticam ſectionem, quæ à Pergeo defectio appellatur, quod quidem <lb/>inſtrumentum valde diuerſum eſt ab alijs, quę aliàs inueni, pro ipſis conicis ſectio <lb/>nibus delineandis. </s> + <s xml:id="s4217" xml:space="preserve">Occaſionem <reg norm="autem" type="wordlist">aũt</reg> huiuſimodi inſtrumenti inueniendi mihi præ <lb/>buit <reg norm="ſecunda" type="context">ſecũda</reg> dubij ſolutio <reg norm="quam" type="context">quã</reg> feci ann .1568. grauiſſ. philoſopho Franciſco Vimer <lb/>cato, <reg norm="nam" type="context">nã</reg> <reg norm="cum" type="context">cũ</reg> viderim in ea figura <var>.f.a.</var> ſemper <reg norm="æqualem" type="context">æqualẽ</reg> eſſe <var>.o.i.</var> ſuæ parallelæ ſcilicet, <lb/>vnde cum recta linea fuerit protracta per <var>.o.</var> et <var>.f.</var> ipſa foret ſemper <reg norm="ęquidiſtans" type="context">ęquidiſtãs</reg> <var>.d.p.</var> ex <lb/>33. primi Eucli. </s> + <s xml:id="s4218" xml:space="preserve">Venit mihi in mentem modus conſtruendi hoc ſubſcriptum inſtru-<lb/>mentum, tali ordine, videlicet, <reg norm="coniungendo" type="context">coniungẽdo</reg> ſeptem hic ſubnotatas 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> ſimul, hoc modo, ſcilicet <reg norm="ſabricando" type="context">ſabricãdo</reg> quadrila-<lb/>terum æquilaterum <var>.c.e.s.l.</var> hac conditione, quod immobili exiſtente puncto <var>.c.</var> in li <lb/>nea <var>.z.r.</var> reliqua omnia mobilia exiſtant, hoc eſt quod <reg norm="punctum" type="context">punctũ</reg> <var>.s.</var> moueatur per di-<lb/>ctam lineam <var>.z.r.</var> & immobili exiſtente puncto <var>.e.</var> vt extremum lineæ <var>.e.h.</var> hoc eſt <lb/>coniuncto extremo <var>.e.</var> lineæ <var>.e.h.</var> cum angulo <var>.c.e.s.</var> reliqua puncta lineæ ipſius <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> ſit compo-<lb/>ſita ex duplo vnius lateris ipſius quadrilateris. </s> + <s xml:id="s4219" xml:space="preserve">Oportet deinde quod punctum <var>.f.</var> <lb/>ſemper vnum, & idem ſit ipſius 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 ſectonis + <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"/> + deſignabit, puncta verò <var>.o.</var> et <var>.K.</var> vt puncta laterum <var>.c.e.</var> et <var>.s.e.</var> æquædiſtantia à <lb/>punctis <var>.c.</var> et <var>.s.</var> <reg norm="eadem" type="context">eadẽ</reg> ſemper ſint, ita tamen vt puncta lineæ <var>.u.n.</var> ſemper diuerſa exi <lb/><reg norm="ſtant" type="context">ſtãt</reg>, & quodlibet ipſius quadrilateri latus, æquale ſit medietati maioris axis ipſius <lb/>oxygoniæ ſectionis delineandæ, et <var>.c.o.</var> ſeu <var>.s.K.</var> (quod idem eſt) ſit æqualis medie <lb/>tati axis minoris dictæ ſectionis, et <var>.z.r.</var> æqualis duplo <var>.e.h.</var> vnde, quando puncta <var>.e.</var> et <var>.<lb/>l.</var> coniuncta ſimul erunt, ſimiliter coniunctæ ſimul 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 eſt in aliquo extremorum maioris axis, & cum punct is <var>.s.</var> coniunct is fuerit cum <lb/>centro <var>.c.</var> punctus <var>.f.</var> parallelę <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 eſt, vt ſint duo <reg norm="puncta" type="context">pũcta</reg> fer <lb/>rea, ad firmandum ipſam <var>.z.r.</var> ſuper ſubiectam lineam ſignificantem maiorem axem <lb/>propoſitę ſectionis.</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 conſtituta ſit alia parallela ad <var>.z.<lb/>r.</var> & in æquali diſtantia ab ipſa quemadmodum <var>.u.n.</var> diſtat 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="vſque" type="simple">vſq;</reg> <lb/>ad <var>.c.</var> deſignabimus quartam partem ſectionis, conuerſo poſtea inſtrumento, hoc eſt <lb/>poſito puncto <var>.r.</var> vbi prius erat <var>.z.</var> et <var>.z.</var> vbi erat <var>.r.</var> aliam delineabimus quartam, & <lb/>ſic ad oppoſitam partem ipſius <var>.z.r.</var> faciendum erit. </s> + <s xml:id="s4224" xml:space="preserve">Hoc inſtrumentum poſſumus <lb/>etiam ita conſtruere, 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">axiũ</reg> propoſita, oxygoniam <lb/>deſignare. </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 poſtulas, non eſt admodum difficile, cupis enim triangulum <lb/>orthogonium, exempli gratia <var>.o.i.e.</var> in figura <var>.A.</var> ita conſtituere, vt di-<lb/>uiſum ſit à perpendiculari <var>.a.i.</var> & quod proportio <var>.o.e.</var> ad <var>.o.i.</var> ſit vt <var>.o.i.</var> ad <lb/><var>i.e.</var> & quod quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> ſit vt <var>.e.i.</var> ad <var>.e.a.</var> & quadra <lb/>tum <var>.o.i.</var> ad quadratum <var>.e.i.</var> ſit .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/>tieſcunque <var>.o.e.</var> fuerit diameter alicuius circuli, <reg norm="diuiſaque" type="simple">diuiſaq́;</reg> in puncto <var>.a.</var> ſecundum pro <lb/>portionem habentem medium <reg norm="duoque" type="simple">duoq́;</reg> extrema, protracta deinde perpendiculari <var>.a.<lb/>i.</var> ad <var>o.e.</var> uſque ad circunferentiam, <reg norm="coniunctæque" type="simple">coniunctæq́;</reg> <var>.o.i.</var> et <var>.i.e</var>: tale triangulum, omnia <lb/>ſupradicta in ſe continebit.</s> + </p> + <p> + <s xml:id="s4228" xml:space="preserve">Nam ex .30. tertij angulus <var>.i.</var> rectus erit, & ex .8. ſexti <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> ſed quia ex diuiſione facta in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.a.</var> etiam <var>.o.a.</var> erit media proportio-<lb/>nalis inter totum & reſiduum, 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. eiuſdem <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> eſt, quę ipſius <var>.o.e.</var> ad <lb/><var>o.a</var>. </s> + <s xml:id="s4230" xml:space="preserve">tunc videbis ex .18. ſexti, 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> ſint inuicem ſimiles <lb/>ex ſupradicta .8. ſexti, </s> + <s xml:id="s4231" xml:space="preserve">tunc videbis ex <lb/>18. et .17. eiuſdem dictos <reg norm="triangulos" type="context">triãgulos</reg> ean <lb/>dem habere inter ſe proportionem, quę <lb/>eſt inrer quadrata ipſius <var>.o.i.</var> et <var>.i.e.</var> vnde <lb/>ex prima ſexti ita ſe 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 ſit 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> ipſius <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">tũc</reg> <lb/>habebimus ad partem <var>.o.u.i.</var> triangulum <var>.o.u.i.</var> diuiſum in duo triangula ſimilia ipſi <lb/>totali triangulo. </s> + <s xml:id="s4234" xml:space="preserve">Idem etiam dico ad partem <var>.o.n.i.</var> vnde ex tali ſimilitudine habe-<lb/>bimus <var>.o.u.</var> mediam proportionalem inter <var>.o.i.</var> et <var>.o.a.</var> et ſic <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. ſexti, quadratum <var>.o.u.</var> æquale erit producto ipſius <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> ſed ex prima eiuſdem, ea <lb/>dem proportio eſt ipſius <var>.o.a.</var> ad <var>.o.e.</var> quæ producti ipſius <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 hęc <lb/>eſt alia circuli paſſio.</s> + </p> + <p> + <s xml:id="s4238" xml:space="preserve">Reliqua verò difficultas quam te habere ſcribis, eſt, quare cum duæ lineę <lb/><var>a.u.</var> et <var>.b.s.i.</var> ſint inuicem ęquales, diuiſæ verò non æquali modo, ſed tali, quod <var>.a.</var> <lb/>maior ſit quam <var>.u.</var> et <var>.b.s.</var> maior quam <var>.i.</var> quomodo poteſt fieri, quod ſi <var>.u.</var> maior fue-<lb/>rit <var>.i.</var> proportio <var>.a.</var> ad <var>.i.</var> maior ſit quam ipſius <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 ſi 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="ipſaque" type="simple">ipſaq́;</reg> abſciſa fuerit <var>.t.</var> æqualis <var>.i.</var> & ab <var>.<lb/>b.s.</var> abſciſa <var>.s.</var> æqualis <var>.n.</var> habebimus <var>.a.</var> et <lb/>b. inuicem æ quales, vnde habebis ma-<lb/>iorem propor tionem ipſius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/>s. ad <var>.n.</var> quod cum clarum per ſe ſit, tibi <lb/>relinquo. </s> + <s xml:id="s4240" xml:space="preserve">ſed ex .27. quinti, proportio <lb/>b. ad. s, maior erit quam <var>.t.</var> ad <var>.n.</var> & ex <lb/>28. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <reg norm="proportio" type="simple">ꝓportio</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 ipſius <var>.b.s.</var> ad <var>.n.t.</var> quam <var>.s.</var> ad <var>.n.</var> <lb/>ergo ex .33. maior erit ipſius <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 eſt maior ipſius <var>.a.</var> ad <lb/>i. quam <var>.b.s.</var> ad <var>.u.</var> quod eſt propo-<lb/>ſitum.</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/>diſtinctione orbium cęleſtium, ortum <lb/>habet à communi opinione motuum <lb/>fixarum. </s> + <s xml:id="s4242" xml:space="preserve">Nam cum putauerint philo-<lb/>ſophi ipſas moueri, ſemper eandem <reg norm="ſeruando" type="context">ſeruãdo</reg> inuicem diſtantiam, non ſine ratione <lb/>crediderunt eas fixas eſſe eodem in orbe, idem etiam poſtea de planetis opinaue-<lb/>runt. </s> + <s xml:id="s4243" xml:space="preserve">Hoc eſt, vnumquemque, aliquo in orbe, fixo exiſtere.</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/>propoſitam ſecundum 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, eſt quidem poſſibile, non tamen adhuc inuentum, quo <lb/>niam nemo ad <reg norm="hunc" type="context">hũc</reg> vſque diem diuiſit vnam datam proportionem in tres <lb/>æquales partes, ſed ſi 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="cupiſque" type="simple">cupiſq́;</reg> <lb/>ſcire modum diuidendi ipſam parabolem vna mediante linea parallela ipſi baſi, ita <lb/>vt eandem habeat proportionem tota parabola ad partem abſciſſam, quæ eſt inter <var>.<lb/>p.</var> et <var>.q</var>. </s> + <s xml:id="s4246" xml:space="preserve">Ad quod faciendum, ſupponendum primò datam proportionem inter <var>.<lb/>p.</var> et <var>.q.</var> diuiſam 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 circunferentię <lb/>ipſius figuræ ducatur parallela baſi <var>.x.e.</var> poſtea verò per puncta media harum dua-<lb/>rum <reg norm="æquidiſtantium" type="context">æquidiſtantiũ</reg> protrahatur <var>.g.b.</var> quæ diameter erit ſectionis, ex 28. ſecundi Per-<lb/>gei, </s> + <s xml:id="s4248" xml:space="preserve">diuidatur deinde hæc diameter in puncto <var>.a.</var> ita quod eadem proportio ſit ipſius <lb/><var>b.g.</var> ad <var>.b.a.</var> quæ ipſius <var>.p.</var> ad <var>.u.</var> quod tibi facile erit, ſecando à linea <var>.p.</var> partem <var>.i.</var> æqua <lb/>lem ipſi <var>.u.</var> tali modo poſtea diuidendo <var>.b.g.</var> ex .12. ſexti, ducatur a puncto <var>.a.</var> ipſa <var>.d.<lb/>h.</var> parallclam ipſi <var>.x.e.</var> & habebitur propoſitum.</s> + </p> + <p> + <s xml:id="s4249" xml:space="preserve">Pro cuius reiratione, ſcies primum quod <var>.h.d.</var> diuiſa erit à diametro <var>.b.g.</var> per æqua <lb/>lia ex .7. primi Pergei, vel ſi cogitabimus aliquam lineam tangentem ipſam parabo <lb/>lam in puncto <var>.b</var>. </s> + <s xml:id="s4250" xml:space="preserve">tunc ex quinta ſecundi ipſius Pergei habebimus ipſam eſſe paralle-<lb/>lam <var>.e.x.</var> & ex .30. primi Eucli. erit ſimiliter æquidiſtans <var>.d.h.</var> vnde ex .46. primi eiuſ-<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 ſuper <lb/>ficiei totalis parabolæ <var>.x.b.e.</var> ad trigonum <var>.x.b.e.</var> quæ portionis <var>.h.b.d.</var> ad ſuum <reg norm="tri- gonum" type="context">tri-<lb/>gonũ</reg>, eo quod <reg norm="tam" type="context">tã</reg> vna quàm alia erit ſeſquitertia, <reg norm="eius" type="simple">eiꝰ</reg> <reg norm="etiam" type="context">etiã</reg> medietates ſic ſe <reg norm="habebunt" type="context">habebũt</reg>.</s> + </p> + <p> + <s xml:id="s4252" xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolę ad medietatem partia <lb/>lem ipſius, æqualis erit proportioni trianguli <lb/><var>g.b.e.</var> ad triangulum <var>.a.b.d.</var> ſed ex .20. primi <lb/>Pergei, eadem eſt proportio quadrati ipſius <var>.<lb/> + <anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/> + g.e.</var> ad quadratum ipſius <var>.a.d.</var> quæ <var>.b.g.</var> ad <var>.b.a.</var> <lb/>hoc eſt, vt <var>.g.e.</var> ad <var>.a.o.</var> ex ſimilitudine triangu-<lb/>lorum, & quia <var>.b.g.</var> ad <var>.b.a.</var> eſt ſicut <var>.p.</var> ad <var>.u.</var> ita <lb/>igitur erit quadrati ipſius <var>.g.e.</var> ad quadratum <lb/>ipſeus <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. ſexti Euclid. </s> + <s xml:id="s4254" xml:space="preserve">ſed cum ex .24. eiuſdem <lb/>proportio trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> compoſita ſit ex proportione <var>.g.e.</var> ad <var>.a.<lb/>d.</var> er. ex <var>.g.b.</var> ad <var>.b.a.</var> hoc eſt <var>.g.e.</var> ad <var>.a.o.</var> & <lb/>quia <reg norm="proportio" type="simple">ꝓportio</reg> <var>.g.e.</var> ad <var>.a.o.</var> æqualis eſt 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 eſt ei quæ <var>.p.</var> ad <var>.n.</var> hoc eſt 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> compoſita erit ex ca quę <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 ſemper erit ſeſquialtera proportioni <lb/>ipſius <var>.b.g.</var> ad <var>.a.b.</var> eo quod cum ſit proportio totalis portionis ad partialem vt trian-<lb/>guli <var>.b.g.e.</var> ad <var>.b.a.d.</var> & hæc ſeſquialtera proportioni ipſius <var>.g.e.</var> ad <var>.a.o.</var> hoc eſt vt ip-<lb/>ſius <var>.b.g.</var> ad <var>.b.a.</var> ideo proportio ipſarum portionum erit ſimiliter ſeſquialtera pro-<lb/>portioni diametrorum.</s> + </p> + <p> + <s xml:id="s4257" xml:space="preserve">Deinde ſi protractæ fuerint <var>.b.d.</var> et <var>.g.e.</var> quouſque 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 ſupra diximus.). <var>a.d.</var> media proportionalis ſit inter <var>.<lb/>g.e.</var> et <var>.a.o.</var> & proportio <var>.g.z.</var> ad <var>.g.e.</var> vt ipſius <var>.a.d.</var> ad <var>.a.o.</var> eo quodipſius <var>.g.z.</var> ad <var>.a.d.</var> <lb/>& ipſius <var>.g.e.</var> ad <var>.a.o.</var> eſt vt ipſius <var>.b.g.</var> ad <var>.b.a.</var> ex ſimilitudine triangulorum, ideo di-<lb/>ctæ <reg norm="proportiones" type="simple">ꝓportiones</reg> erunt <reg norm="inuicem" type="context">inuicẽ</reg> æquales. </s> + <s xml:id="s4258" xml:space="preserve">Vnde permutatim ita erit ipſius <var>.g.z.</var> ad <var>.g.e.</var> <lb/>vt ipſius <var>.a.d.</var> ad <var>.a.o.</var> & ut ipſius <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 eſt, quę <lb/>cubi ipſius <var>.g.e.</var> ad cubum ipſius <var>.a.d.</var> & ex <reg norm="con" type="context">cõ</reg> <lb/>ſequenti, vt cuborum earundem baſium, eo <lb/>quod cum ſit, ex .36. vndecimi Euclid. pro-<lb/>portio cubi ipſius <var>.g.e.</var> ad cubum ipſius <var>.a.d.</var> <lb/>tripla ei quæ ipſius <var>.g.e.</var> ad <var>.a.d.</var> ideo æqualis <lb/>erit ei quę trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> cum proportio horum duorum triangu <lb/>lorum compoſita ſit (vt ſupra 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, ſed trianguli ita ſe in <lb/>uicem habenr, vt parabolę, </s> + <s xml:id="s4260" xml:space="preserve">quare ipſæ para-<lb/>bolæ ſeinuicem habebunt, vt cubi ipſarum <lb/>baſium.</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 propoſitæ.</head> + <head xml:id="head521" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4261" xml:space="preserve">CVbum fabricare æqualem propoſitæ pyramidi quadrilateræ, nullius erit diffi-<lb/>cultatis, ſuppoſita tamen pro reperta diuiſione cuiuſuis datæ proportionis in <lb/>tres partes æquales. </s> + <s xml:id="s4262" xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus ſerratile d-ui <lb/>ſibile eſſe in tres pyramides quadrilateras æquales, ſcimus etiam quod cuilibet py-<lb/>ramidi quadrilateræ poteſt reperiri ſuum ſerratile. </s> + <s xml:id="s4263" xml:space="preserve">Sit igitur propoſita pyramis qua <lb/>drilatera <var>.m.g.f.h.</var> cuius ſerratile ita inueniemus, ducendo primum <var>.h.i.</var> parallelam <lb/>ipſi <var>.g.f.</var> et <var>.f.i.</var> ipſi <var>.g.h.</var> in ſuperficie trianguli <var>.f.g.h.</var> et <var>.m.K.</var> ipſi <var>.g.h.</var> in ſuperficie <lb/>trianguli <var>.m.g.h.</var> & æqualem dictæ <var>.g.h.</var> ducetur poſtea <var>.K.h.</var> et <var>.K.i.</var> & habebimus cor <lb/>pus <var>.f.K.g.</var> ſerratile, & triplum pyramidi propoſitæ. </s> + <s xml:id="s4264" xml:space="preserve">Nunc duplicemus ipſum, du-<lb/>cendo <var>.K.x.</var> in ſuperficie trianguli <var>.i.k.h.</var> parallelam, <reg norm="æqualemque" type="simple">æqualemq́;</reg> ipſi <var>.i.h.</var> et <var>.m.y.</var> <lb/>in ſuperficie trianguli <var>.f.m.g.</var> parallelam, ę<reg norm="qualemque" type="simple">qualemq́;</reg> ipſi <var>.f.g.</var> ducatur poſtea <var>.g.y.</var> et <var>.h.<lb/>x.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis erit ipſi <var>.f.m.</var> vnde habebimus corpus <var>.f.x.</var> parallelepe-<lb/>pidum, & ſexcuplum ipſi pyramidi propoſitæ.</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 ſextæ parti ſuperficiei <var>.f.i.g.h.</var> quod per <lb/>ſe 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 baſim <var>.f.g.h.</var> quę ſit <var>.n.e.</var> qua mediante, cum quadrato <var>.u.n.</var> fabri <lb/>cetur ſolidum 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 ſint duæ mediæ proportionales <var>.r.s.</var> inter <var>.n.e.</var> et <var>.n.p.</var> quarum <var>.s.</var> ſit <lb/>proximior ipſi <var>.u.p.</var> ex qua <var>.s.</var> ſi conſtitutus fuerit cubus, habebimus propoſitum.</s> + </p> + <p> + <s xml:id="s4268" xml:space="preserve">Pro cuius rei ratione, cogitemus corpus <var>.u.e.</var> productum eſſe vſque ad <var>.a.o.</var> per lon-<lb/>gitudem <var>.s.</var> latus dicti cubi, qui quidem cubus ſit <var>.d.b.</var> vnde proportio corporis <var>.u.e.</var> <lb/>ad corpus <var>.e.o.</var> erit, vt ſuperficiei <var>.p.e.</var> ad ſuperficiem <var>.t.e.</var> ex .33. undecimi, ipſæ verò <lb/>ſuperficies ſibi inuicem erunt vt <var>.n.e.</var> ad <var>.e.a.</var> ex prima ſexti, </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 ipſius <var>.s.</var> ad <var>.n.p.</var> ſed cum ex .33 vndeci-<lb/>mi, proportio cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> ſit vt <reg norm="quadratum" type="context">quadratũ</reg> <var>.q.b.</var> ad quadratum <var>.o.a.</var> & cum <lb/>proportio <var>.q.b.</var> ad <var>.o.a.</var> dupla ſit ei quæ <var>.q.o.</var> ad <var>.o.t.</var> ex .18. ſexti, 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 eſt ei quæ <var>.s.</var> ad <var>.n.p.</var> ſed 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 eſt pyramidi propoſitæ.</s> + </p> + <p> + <s xml:id="s4271" xml:space="preserve">Sed ſi oportebit cubum maiorem vel minorem ipſa 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 ſe habeat ad quadratum <var>.u.n.</var> quam volueris, quo mediante ſimul cum <lb/>altitudine pyramidis conſequemur propoſitum.</s> + </p> + <p> + <s xml:id="s4273" xml:space="preserve">Aduertendum tamen quod fabri-<lb/>care ipſum corpus ſerratile <var>.k.f.h.</var> & ſo <lb/> + <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> + lidum <var>.f.x.</var> neceſſarium non eſt, niſi pro <lb/>demonſtratione. </s> + <s xml:id="s4274" xml:space="preserve"><reg norm="idemque" type="simple">idemq́;</reg> dico de alijs <lb/>ſolidis, nam pro ſimplici operatione <lb/>huiuſmodi problematis, abſque ali-<lb/>qua re neceſſaria ad ſpeculandum, 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">eiꝰ</reg> <lb/>alitudinem à <reg norm="puncto" type="context">pũcto</reg> <var>.m.</var> vſque ad ſuper <lb/>ficiem baſis <var>.f.g.h.</var> quæ ſit <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 ipſius baſis <var>.f.<lb/>g.h.</var> quod latus ſit <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">quarũ</reg> <var>.<lb/>s.</var> proximior ſit <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 propoſito <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 propoſito vna tantummodò <lb/>ſtatione, non ſolum poſſibile eſt ſed etiam facile.</s> + </p> + <p> + <s xml:id="s4277" xml:space="preserve">Sit exempli gratia murus rectus <var>.a.d.</var> ſitus 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, ſeu ſcala altimetra <lb/>vel aliquod ſimile inſtrumentum, quo mediante à ſitu <var>.o.</var> videbimus punctum <var>.q.</var> <lb/>quod volueris ipſius muri, <reg norm="dexteram" type="context">dexterã</reg> <lb/>verſus, inferius tamen. ipſo <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="ſiniſtram" type="context">ſiniſtrã</reg> <lb/>cum eodem angulo <var>.n.o.q.</var> oporte-<lb/>bit nos inuenire punctum aliquod <var>.<lb/>p.</var> in dicta ſuperficie 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">duoq́;</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> ſint 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> extrinſe <lb/>cus trianguli <var>.p.q.n.</var> <reg norm="reſiduusque" type="simple">reſiduusq́;</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> ſunt inuicem æquales, & ex .32. eiuſdem, æquales ſunt extrinſeco <var>.q.n.<lb/>g.</var> & ex 27. <var>n.u.</var> erit parallela ipſi <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, ſumendo duo illa puncta in ſuprem a linea <lb/>orizontali ipſius muri ad ſuperiorem partem aſpiciendo, quemadmodum ad infe-<lb/>riorem, quod vnum & idem erit, dummodò non aſpiciamus orizontaliter, eo quod <lb/>nos oportet ſuperficiem conicam producere, linea viſuali mediante. </s> + <s xml:id="s4282" xml:space="preserve">cognoſcere au<lb/>tem angulum <var>.q.n.p.</var> facile erit, conſtituendo primò inſtrumentum in ſitu trianguli <var>.<lb/>o.n.q.</var> <reg norm="aſpiciendoque" type="simple">aſpiciendoq́;</reg> punctum <var>.c.</var> in ſuperficie <var>.n.q.o.</var> & ſic in alia parte, exiſtente in-<lb/>ſtrumento in ſitu trianguli <var>.o.p.n.</var> aſpicere 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 ſi ſitus puncti <var>.n.</var> talis eſſet, vt ab eo non poſſet aliquis murum videre ad re-<lb/>ctos angulos, aſpiceremus punctum <var>.q.</var> ſub orizontali ab oculis noſtris, in orizontali <lb/>tamen puncti <var>.n.</var> ita quod angulus <var>.o.n.q.</var> rectus exiſtat, quo facto obſeruando angu-<lb/>lum <var>.n.o.q.</var> eo mediante, medianteq́ue <var>.n.o.</var> cum angulo <var>.o.n.q.</var> cognoſcemus <lb/>quantitatem diſtantiæ <var>.n.q.</var> idem etiam faciendum eſt cum alio puncto <var>.p.</var> quod <lb/>volueris, & mediantibus duobus punctis inuicem proximis <var>.c.e.</var> cognoſcatur 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/>cognoſcemus reliqua trianguli <var>.<lb/>q.p.n</var>. </s> + <s xml:id="s4284" xml:space="preserve">Conſtituendo poſtea angu-<lb/>lum <var>.q.n.u.</var> æqualem angulo <var>.n.q.p.</var> <lb/>propoſitum 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/>ſterent 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">pũcta</reg> <var>.c.e.</var> proxima <var>.n.</var> in ijſdem <lb/>ſuperficiebus 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> ſectiones dictarum ſuperficierum cum ſuperficie pauimenti ſupra quam fit <lb/>ſtatio.</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 baſi ſecundum 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/>ſi ſecundum vnam datam proportionem, nullius tibi erit difficultatis, con <lb/>ceſſa <reg norm="tamen" type="wordlist">tamẽ</reg> pro inuenta diuiſione cuiuſuis propoſitę 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> ſecandus vt dictum eſt, accipiatur latus <lb/>ipſius, quod ſit <var>.a.c.</var> <reg norm="ipſumque" type="simple">ipſumq́;</reg> diuidatur in puncto <var>.d.</var> ſecundum illam proportionem <lb/>quam deſideras, hoc eſt ipſius <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 ſit <var>.a.i.</var> </s> + <s xml:id="s4288" xml:space="preserve">tunc ſi conus <var>.a.b.c.</var> ſectus fue-<lb/>rit à plano per punctum <var>.i.</var> parallelo baſi, habebimus quod quærebamus.</s> + </p> + <p> + <s xml:id="s4289" xml:space="preserve">Cuius rei ratio, primò eſt, quia quotieſcunque conus aliquis ſectus fuerit ab ali-<lb/>quo plano parallelo baſi ipſius, pars ſuperior ſimilis ſemper erit totali cono, quod <lb/>ita probo, cogitemus conum ſectum 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 ſectio triangularis erit, quæ <lb/>ſit <var>.a.b.c.</var> et <var>.b.c.</var> diameter erit baſis.</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 ſectionem huiuſmodi trianguli cum <lb/>plano parallelo ipſi baſi, </s> + <s xml:id="s4291" xml:space="preserve">tunc tale <reg norm="planum" type="context">planũ</reg>, <lb/>circulare erit ex .4. primi ipſius Pergei <var>.K.<lb/>i.</var> verò, eius diameter erit, et <var>.a.m.</var> <reg norm="ſuus" type="simple">ſuꝰ</reg> axis.</s> + </p> + <p> + <s xml:id="s4292" xml:space="preserve">Cum verò <var>.a.l.</var> ſit perpendicularis ipſi <lb/>baſi conitotalis, eo quod rectus ſupponi-<lb/>tur, ideo eadem <var>.a.m.l.</var> erit perpendicula <lb/>ris eriam ipſi ſecundo plano circulari, ex <lb/>conuerſa .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"/> + ſecunda definitione eiuſdem 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 ipſi <var>.b.c.</var> ex .28. primi, quod etiam poteſt con <lb/>cludi mediante .16. vndecimi, cum <var>.k.i.</var> et <var>.b.c.</var> ſint communes ſectiones 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 poſtea ad <var>.a.</var> communes <lb/>ſunt 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. ſexti, eadem <lb/>proportio erit ipſius <var>.m.i.</var> ad <var>.l.c.</var> & ipſius <var>.m.k.</var> ad <var>.l.b.</var> vt ipſius <var>.a.m.</var> ad <var>.a.l</var>. </s> + <s xml:id="s4296" xml:space="preserve">Quare ex <lb/>vndecima quinti, ita erit ipſius <var>.m.k.</var> ad <var>.l.b.</var> vt ipſius <var>.m.i.</var> ad <var>.l.c.</var> & ex .13. eiuſdem, ita <lb/>erit ipſius <var>.k.i.</var> ad <var>.b.c.</var> vt <var>.m.i.</var> ad <var>.l.c.</var> ſed ipſius <var>.m.i.</var> ad <var>.l.c.</var> eſt vt ipſius <var>.a.m.</var> ad <var>.a.l.</var> quod <lb/>iam dictum eſt, vnde ex .11. dicta, ita erit ipſius <var>.k.i.</var> ad <var>.b.c.</var> vt ipſius <var>.a.m.</var> ad <var>.a.l.</var> & ex <lb/>16. dicti ita erit ipſius <var>.a.m.</var> ad <var>.k.i.</var> vt ipſius <var>.a.l.</var> ad <var>.b.c</var>. </s> + <s xml:id="s4297" xml:space="preserve">Quare ex definitione ab Eu-<lb/>cli. poſita in .11, lib. pars coni ſuperior ſimilis erit cono totali.</s> + </p> + <p> + <s xml:id="s4298" xml:space="preserve">Deinde ſciendum eſt illud quod Euclid. ſcribit in .10. duodecimi lib. hoc eſt, <reg norm="quod" type="simple">ꝙ</reg> <lb/>proportio duarum pyramidum inuicem <lb/>ſimilium, triplicata eſt ei diametrorum <lb/> + <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> + ſuarum baſium, hoc eſt, 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> ſed ita eſt ipſius <var>.a.c.</var> ad <var>.a.i.</var> vt ipſius <var>.b.<lb/>c.</var> ad <var>.k.i.</var> ex .4. ſexti cum trianguli <var>.a.b.c.</var> <lb/>et <var>.a.k.i.</var> ſint æquianguli, quod ex ijs, quę <lb/>ſuperius diximus facile compręhenditur. <lb/></s> + <s xml:id="s4299" xml:space="preserve">Quare <reg norm="proportio" type="simple">ꝓportio</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 abſciſſam <var>.a.k.i.</var> ſed eadem proportio <lb/>ipſius <var>.a.c.</var> ad <var>.a.i.</var> erat etiam tertia pars pro <lb/>portionis ipſius <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 abſciſſam, æqualis erit pro-<lb/>portioni ipſius <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, ſed ex rationibus, quas tibi ſcribo conſidera, quod quo <lb/><reg norm="tieſcunque" type="simple">tieſcunq;</reg> craſſities vel <reg norm="denſitas" type="context">dẽſitas</reg> <reg norm="vaporum" type="context">vaporũ</reg>, ſeu altitudo, maior eſſet ea, quę nunc re-<lb/>peritur, </s> + <s xml:id="s4302" xml:space="preserve">tunc minor differentia eſſet inter maiorem <reg norm="minoremque" type="simple">minoremq́;</reg> calorem Solis, quam <lb/>nunc ſentiamus. </s> + <s xml:id="s4303" xml:space="preserve">Pro cuius rei euidentia, imaginemur in hac ſubſcripta figura, li-<lb/>neam <var>.o.a.</var> pro ſemidiametro terræ, et <var>.a.c.</var> pro craſſitie vaporum, vt nunc ſe <lb/>habet, et <var>.a.d.</var> pro maiori craſſitie, imaginemurq́ue lineam <var>.a.b.</var> quaſi perpen-<lb/>dicularem ad <var>.o.a.</var> quæ abſciſſa ſit 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 ipſius <var>.a.b.</var> ad <var>.a.d.</var> quam ipſius <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> ſecabit <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 ipſi <var>.d.b.</var> ex. ſecunda par-<lb/>te ſecundæ ſexti, vnde ex prima parte <lb/>ciuſdem, ita eritipſius <var>.b.i.</var> ad <var>.i.a.</var> vt <var>.d.<lb/>c.</var> ad <var>.c.a.</var> & coniunctim ita erit ipſius <var>.b.<lb/>a.</var> ad <var>.a.i.</var> vt ipſius <var>.d.a.</var> ad <var>.a.c.</var> & permu <lb/>tatim ipſius <var>.a.b.</var> ad <var>.a.d.</var> erit, vt <var>.a.i.</var> <lb/>ad <var>.a.c.</var> ſed cum <var>.a.u.</var> maior ſit ipſa <var>.a.i.</var> <lb/>vt omne totum maius eſt ſua parte. <lb/></s> + <s xml:id="s4305" xml:space="preserve">maior proportio erit ipſius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> quam ipſius <var>.a.i.</var> ad <var>.a.c.</var> hoc eſt quam <lb/>ipſius <var>.a.b.</var> ad <var>.a.d</var>. </s> + <s xml:id="s4306" xml:space="preserve">Verum igitur eſt <lb/>propoſitum.</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 reſpectu altitudinis ipſius.</head> + <head xml:id="head529" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4307" xml:space="preserve">QVodà me poſtulas deinde, ita ſe habet. </s> + <s xml:id="s4308" xml:space="preserve">Inquis enim, quod cum differentia <lb/>inter maiorem, <reg norm="minoremque" type="simple">minoremq́;</reg> calorem, oriatur etiam ex differentia maioris <lb/>quantitatis vaporum ad minorem, per quam quantitatem vaporum rranſit lumen <lb/>Solis (vt alias etiam tibi dixi) velles nunc ſcire quantitatem ipſius differentię, quæ <lb/>inter duas Solis datas altitudines ſupra 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, ſupponatur etiam quod angulus <var>.z.o.d.</var> vel <var>.z.a.b.</var> qui ſunt inuicem fe-<lb/>rè æquales, ſit angulus diſtantiæ Solis à zenit, <var>z.a.</var> verò ſit ſpiſſitudo vaporum, et <var>.a.<lb/>b.</var> radius tranſiens 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 eſt 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, velipſo minore, vnde cogno <lb/>ſcemus differentiam duarum <var>.a.b.</var> quæ qui-<lb/>dem inæquales inuicem erunt, eo quod ſup <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 eſt quam <lb/>Alhazem docetinuenire, cognoſcimus <reg norm="etiam" type="context">etiã</reg> <lb/><var>o.a.</var> vt ſemidiametrum 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">erũt</reg> <lb/>ſimul cum angulo <var>.o.a.b.</var> reſiduo duorum re <lb/>ctorum, eo quod reliquus <var>.z.a.b.</var> datus eſt. <lb/></s> + <s xml:id="s4312" xml:space="preserve">Quare <var>.a.b.</var> cognita erit reſpectu <var>.o.a.</var> et <var>.o.<lb/>b.</var> et <var>.a.z.</var> quæ eſt eorum differentia. </s> + <s xml:id="s4313" xml:space="preserve">Nunc <lb/>ſi idem faciemus cum alia <var>.a.b.</var> ſub diuerſo <lb/>angulo, habebimus propoſitum.</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, baſibus oppoſiti, ijſdem baſibus propor <lb/>tionales eſſent, cuius opinionis cauſa fuit quod nunquam viderit vel me <lb/>minerit eius quod Ptolomeus ſcripſit lib. primo Almageſti, vbi de diſpro <lb/>portionalitate chordarum <reg norm="arcuumque" type="simple">arcuumq́;</reg> tractat, vel quod ſcribit Vitellio lib. primo pro <lb/>poſitione .35. ſeu lib. quarto, propoſitione .21. quod idem eſt. </s> + <s xml:id="s4315" xml:space="preserve">Sed nec ego tibi pro <lb/>ponam id quod ſcribit Nicolaus Tartalea diuiſioni .28. quinti capitis quartæ partis <lb/>ſuorum tractatuum, eo quod non exactè ſcientificè ſcripſerit, nec vniuerſaliter, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> talis propoſitio poſſit ſcientificè ſcribi, 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 poſtea <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 vniuerſaliori modo <lb/>poſſumus hoc ſpeculari. </s> + <s xml:id="s4317" xml:space="preserve">Nam manifeſtè ſcimus, 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> cuiuſuis amplitudinis, quorum latera <var>.e.a</var>: <var>e.b</var>: <lb/>et <var>.e.c.</var> ſint 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">ꝙ</reg> <lb/>faciat angulum <var>.g.b.c.</var> æqualem angulo <var>.e.b.a.</var> protracta poſtea <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 eſt angulo <var>e.b.a.</var> ex quinta eiuſdem, ideo ex .32. dicti, et .4. ſexti, horum <lb/>duorum triangulorum latera, erunt inuicem proportionalia. </s> + <s xml:id="s4320" xml:space="preserve">Conſtituto deinde <var>.g.</var> <lb/>centro, & ſecundum ſemidiametrum <var>.g.b.</var> vel <var>.g.c.</var> quod idem eſt, deſcripto circu-<lb/>lo <var>.b.i.c.</var> necnon circulo <var>.b.c.a.</var> circa centrum <var>.e.</var> ope ſemidiametri <var>.e.b.</var> et <var>.e.a.</var> vn <lb/>de iſte circulus eritillo maior, cum <var>.e.b.</var> maior ſit <var>.g.b.</var> ex .14. quinti. cum ex .14. tertij <lb/><var>a.b.</var> longior ſit <var>.c.b.</var> ſed ex vltima definitione tertij, arcus <var>.b.i.c.</var> et <var>.b.c.a.</var> erunt in-<lb/>uicem ſimiles, hoc eſt 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> ſed proportio diame-<lb/>tri ad circunferentiam eſt vt diametri ad cir <lb/>cunferentiam, vt ſupra diximus; </s> + <s xml:id="s4321" xml:space="preserve">Quare ex <lb/>proportionum æqualitate, vt ſemidiametri <lb/>ad circunferentiam erit, vt ſemidiametri <lb/>ad circunferentiam, & per eandem propor <lb/>tionum ęqualitatem, proportio <var>.e.b.</var> ad <reg norm="arcum" type="context">arcũ</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> ſed <lb/>arcus <var>.b.i.c.</var> maior eſt 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 ſcicntia. </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 ſexti et .12. quinti, proportio anguli <var>.a.e.b.</var> ad an-<lb/>gulum <var>.c.e.b.</var> maior erit quam chordæ, ſiue baſis <var>.a.b.</var> ad chordam ſiue baſim <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 Franciſco Venerio.</head> + <p> + <s xml:id="s4324" xml:space="preserve">EGo enim non tantum miror ea quæ mihi ſcripſiſti de opinione Ortenſij <lb/>quantum quod Antonius Berga putat nubes à Sole ſupenſas teneri, id pla <lb/>nè falſum eſt, vera cauſa huiuſmodi effectus, alia nulla eſt, niſi earundem <lb/>raritas hoc eſt, cum rariores ſint ipſo aere ſubiecto, </s> + <s xml:id="s4325" xml:space="preserve">propterea ſupra <reg norm="ipsum" type="context">ipsũ</reg> <lb/>natant & ſtant ſub eo qui rarior ipſis eſt, eo quod corpora rariora poſita in medio <lb/>non tam raro, aſcendunt, & denſiora in medio minus denſo deſcendunt. </s> + <s xml:id="s4326" xml:space="preserve">Nam ſi <lb/>Sol ipſas nubes ſuſpenſas in aere teneret, hoc interdiu tantummodo fieret, ſed no <lb/>ctu, cur non deſcendunt vſque ad terram, & in eodem loco ſemper manent? </s> + <s xml:id="s4327" xml:space="preserve">Scien-<lb/>dum igitur eſt nubes aſcendere in altum quouſque inueniant aerem eiuſdem ra-<lb/>ritatis cuius ipſæ ſunt. </s> + <s xml:id="s4328" xml:space="preserve">Raritas enim & denſitas non ſunt res viſibiles niſi per acci-<lb/>dens, quemadmodum etiam leuitas, & grauitas, opacitas verò & diaphaneitas ma <lb/>gis <reg norm="compræhenduntur" type="context">compræhendũtur</reg>, opacitas enim ex reflexione radiorum luminoſorum, diapha <lb/>neitas verò compræhenditur ex penetratione ipſorum radiorum, opacitas autem nu <lb/>bis non eſt denſitas, cum valde diuerſa ſit denſitas ab opacitate, ſicut raritas ab dia-<lb/>phaneitate, vt aliàs dixi. </s> + <s xml:id="s4329" xml:space="preserve">Et quando dicit, quod Sol calefaciendo aerem ipſam nu <lb/>bem ambientem, rarefaciat eum magis quam ipſam nubem reſpondeo, hoc verum <lb/>non eſſe, proptere<unsure/>a quodradius Solis non multum calefacit ea corpora, quæ ip ſi per <lb/>mittunt liberum tranſitum. </s> + <s xml:id="s4330" xml:space="preserve">vnde corpora quanto magis diaphana ſunt tanto minus <lb/>ab ipſo radio luminoſo calefiunt, ſed ea quæ magis opaca ſunt, magis etiam calefiunt <lb/>& per conſequens magis rarefiunt, cum calidi ſit per ſe rarefacere, & non attrahere, <lb/>vt ipſe & 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/>cuiuſdam libramenti, & de quadam ſimboleita-<lb/>te circuli cum ellipſi.</head> + <head xml:id="head535" style="it" xml:space="preserve">Angelo Ferrario Serenißimi Ducis Sabaudia <lb/>Agrimenſori expertißimo.</head> + <p> + <s xml:id="s4331" xml:space="preserve"><emph style="sc">TIbi</emph> in mentem veniet, quod cum ſuperioribus diebus in villa lucenti, in <lb/>qua degebat Sereniſſimus Dux noſter, dum viridarium ad æquilibrium <lb/>reducebas, eſſemus, à te quæſiui an ſcires vnde fieret, vt ſtante libramen-<lb/>to ad angulos rectos ſupra ſuum pedem, funis quæ extrema eiuſdem li-<lb/>bramenti cum pede in formam trianguli æquicruris coniungit, magis diſtentus exi-<lb/>ſteret, quam cum dictum libramentum cum pede obliquum remanet, ita vt huiuſ- + <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 ſcalenum conſtitueret.</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 iſocellum, & aliquan <lb/>do ſcalenum, eſſe <var>.d.e.c.</var> eſto etiam quod in figura <var>.A.</var> dictus triangulus <var>.d.e.c.</var> ſit iſo-<lb/>cellus, & in figura <var>.B.</var> ſcalenus. </s> + <s xml:id="s4333" xml:space="preserve">Tunc quæſiui à te an ſcires rationem, quare <lb/>funis <var>.d.e.c.</var> in figura <var>.A.</var> eſſet diſtenſus, & in figura <var>.B.</var> laxus quemadmodum vide-<lb/>bamus. </s> + <s xml:id="s4334" xml:space="preserve">cum mihireſponderis, neſcio quid, quod nunc memoria <reg norm="non" type="context">nõ</reg> teneo, ſed quia <lb/>pollicitus ſum metibi eam afferre, propterea nunc ad te mitto. </s> + <s xml:id="s4335" xml:space="preserve">Scias ergo huiuſ-<lb/>modirationem nihil aliud eſſe niſi quod in figura <var>.A.</var> duæ lineæ <var>.c.e.</var> et <var>.d.e.</var> ſimul è <lb/>directo iunctæ longiores ſint illis, quę reperiuntur in figura <var>.B.</var> ſed quia funis tam in <lb/>figura <var>.B.</var> quam in figura <var>.A.</var> vnus, & idem eſt, ideo in figura <var>.B.</var> laxatus eſt, & non in <lb/>tenſus, ut in figura <var>.A</var>. </s> + <s xml:id="s4336" xml:space="preserve">Sed vt huiuſmodi veritatis certam notitiam habeas, infraſcri <lb/>ptum circulum mente concipe <var>.f.e.i.</var> cuius ſemidiameter, æqualis ſit <var>.b.e.</var> & diame-<lb/>ter ſit <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> figurę <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">alicuiꝰ</reg> ellipſis <lb/>cuius quidem figuræ ponamus <var>.d.</var> et <var>.c.</var> <lb/>centra ipſius circunſcriptionis eſſe, cu <lb/>ius <reg norm="circunferentia" type="context">circunferẽtia</reg>, nullidubium eſt, quin <lb/>extra propoſitum circulum tranſitura, <lb/>& in vno tantummodo puncto ipſum <lb/>circulum tactura ſit, qui exiſtat <var>.e.</var> <lb/> figuræ <var>.A.</var> ſeparatum tamen à puncto <lb/>e. figuræ <var>.B</var>. </s> + <s xml:id="s4338" xml:space="preserve">Tunc ſi protracta fue-<lb/>rit linea <var>.d.e.</var> figuræ <var>.B.</var> vſque 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 ſit linea <lb/><var>g.c</var>. </s> + <s xml:id="s4339" xml:space="preserve">tunc <reg norm="manifeſtum" type="context">manifeſtũ</reg> erit duas lineas <lb/><var>d.e.</var> et <var>.e.c.</var> figuræ <var>.A.</var> ſimul iunctas, <lb/>æquales eſſe duabus <var>.d.g.</var> et <var>.g.c.</var> ſi-<lb/>mul poſitis, vt etiam ex .52. tertij <lb/>Pergei facilè videre eſt, ſed ex .21. <lb/>primi Euclid. iam certò ſcimus <var>.d.g.c.</var> longiores eſſe <var>.d.e.c.</var> ſiguræ <var>.B.</var> ergo <var>.d.e.c.</var> figu-<lb/>ræ <var>.A.</var> longiores ſunt <var>.d.e.c.</var> figuræ <var>.B.</var> quod eſt propoſitum.</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 ſuccurrit, tibi libenter ſignifico, hoc eſt, quod <lb/>ſicut in ellipſi duæ lineæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> ſimul iunctæ, ſunt ſemper 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 ſunt in <lb/>potentia duabus <var>.d.e.e.c.</var> figurę <var>.B</var>.</s> + </p> + <p> + <s xml:id="s4341" xml:space="preserve">Manifeſtum enim primum eſt ex penultima primi in figura <var>.A.</var> quadratum <var>.e.c.</var> <lb/>æquale eſſe duobus quadratis ſcilicet <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 ſunt quadratis <var>.e.b.</var> figuræ <var>.A.</var> et <var>.e.<lb/>b.</var> figurę. B et <var>.b.c.</var> et <var>.b.d.</var> hoc eſt 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> figurę <var>.B.</var> æqualia ſunt 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 eſt <reg norm="vtrum" type="context">vtrũ</reg> <reg norm="duplum" type="context">duplũ</reg> quadrati <var>.a.e.</var> <reg norm="cum" type="context">cũ</reg> duplo qua <lb/>drati <var>.b.a.</var> <reg norm="cum" type="context">cũ</reg> duplo quadrati <var>.b.c.</var> ſit æquale duplo quadrati <var>.a.e.</var> <reg norm="cum" type="context">cũ</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 quàm 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> ſimul cum duplo qua-<lb/>drati <var>.b.c.</var> ęquale ſit quadrato <var>.a.c.</var> cum quadrato <var>.a.d.</var> ſed hoc manifeſtum eſt .ex .10. <lb/>ſecundi Euclidis, dato quod <reg norm="punctum" type="context">punctũ</reg> <var>.a.</var> ſit inter <var>.f.</var> et <var>.d.</var> ſed ſi fuerit inter <var>.d.</var> et <var>.b.</var> hoc <lb/>manifeſtum erit ex .9. ſecundi dicti, nihilominus accipe hunc alium modum.</s> + </p> + <p> + <s xml:id="s4347" xml:space="preserve">Sit hic ſubſcriptum quadratum <var>.D.</var> ex <var>.a.c.</var> in ſeipſa producta, cuius diameter ſit <lb/><var>a.n.</var> <reg norm="protrahanturque" type="simple">protrahanturq́</reg> parallelę <var>.d.h</var>: <var>b.K</var>: <var>l.m.o.</var> et <var>.r.q.s.</var> <reg norm="eique" type="simple">eiq́;</reg> addatur <var>.c.p.</var> ad <var>.a.c.</var> æqua-<lb/>lis tamen <var>.d.a.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> protracta <var>.p.u.</var> vſque 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 eſt, <reg norm="vtrum" type="context">vtrũ</reg> <lb/>hęc duo quadrata æqualia ſint 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> ſint <var>.K.o.</var> et <var>.h.l.</var> videndum nunc eſt utrum reſiduum ęquale <lb/>ſit duobus quadratis lineę <var>.a.b.</var> quorum vnum ſit <var>.m.b.</var> alterum verò <var>.l.p.</var> quod ſupe-<lb/>rat <var>.l.c.</var> et <var>.s.p.</var> figuræ <var>.D.</var> per ſupplementum <var>.o.t.</var> cui æquale eſt parallelogrammum <var>.h.<lb/>m.</var> figuræ <var>.D.</var> ſed ſi punctus <var>.a.</var> poſitus fuerit inter <var>.d.</var> et <var>.b.</var> conſtituto quadrato <var>.d.u.</var> <reg norm="cum" type="context">cũ</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> ęquari duplo quadratorum <var>.l.n.</var> et <var>.r.l.</var> nam in quadrato <var>.r.n.</var> ipſa duo quadra-<lb/>ta <var>.l.n.</var> et <var>.r.l.</var> capiuntur, reliquum eſt igitur vt videamus an duo ſupplementa <var>.l.t.</var> et <var>.l.<lb/>s.</var> cum quadrato <var>.d.r.</var> ſint æqualia dictis <reg norm="quadratis" type="typo">q́uadratis</reg> <var>.l.n.</var> et <var>.r.l.</var> ſed quadratum <var>.d.l.</var> æ qua-<lb/>tur quadrato <var>.l.n.</var> videndum igitur eſt, <lb/>an duo ſupplementa <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> ſint æqualia duobus quadra <lb/>tis <var>.d.l.</var> et <var>.r.l.</var> ſed quadratum <var>.d.l.</var> æqua-<lb/>tur quadrato <var>.d.r.</var> & ſupplemento <var>.l.t.</var> <lb/>mediante <var>.q.l.</var> & ſupplemento <var>.r.b.</var> ſup-<lb/>plementum verò <var>.l.s.</var> ſuperat <reg norm="ſupplemem" type="context">ſupplemẽ</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 ſupplementa <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">cũ</reg> quadrato <var>.l.r.</var> verum igitur eſt duas <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales eſſe in <lb/>potentia duabus <var>d.e.e.c.</var> figurę <var>.D.</var> quæ quidem affectio circuli, à nemine fuit adhuc <lb/>(quod ſciam) 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 ſtateram appenſi, & quadam alia demonſtratione, <lb/>& quibuſdam 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 audiuiſti non credis, conſidera quæſo libram ſeu ſtateram <lb/><var>o.a.</var> cuius centrum non longitudinis ſed ponderum ſit <var>.i.</var> quę ſtatera, vt ori <lb/>zontaliter conſiſtat, oportebit pondus extremitatis <var>.o.</var> ita ſe habere <lb/>ad pondus extremitatis <var>.a.</var> ut <var>.a.i.</var> ſe habet ad <var>.o.i.</var> quod te ſcire puto, ima <lb/>ginemur nunc d uas lineas <var>.a.e.</var> et <var>.o.n.</var> paralle las <reg norm="infinitasque" type="simple">infinitasq́;</reg> & à puncto <var>.n.</var> immobili, <lb/>& fixo extra ſtateram, tranſeat 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/>ſectionis ipſius <var>.n.i.e.</var> cum <var>.a.e.</var> progredi vniformiter <reg norm="continuòque" type="simple">continuòq́;</reg> ab <var>.a.</var> per lineam <var>.a.e.</var> <lb/>vnde punctum <var>.i.</var> interſectionis ipſius <var>.n.i.e.</var> cum <var>.a.i.o.</var> ſemper 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 ſtateram <var>.o.i.a.</var> oporteat ſemper orizontalem eſſe virtute ponderis, <lb/>o. oportebit pundus <var>.o.</var> in infinitum etiam augeri, <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> pondus <var>.a.</var> nunquam <lb/>diminui voluerimus vel econtra hoc in infinitum diminui, ſi illud nunquam augeri <lb/>voluerimus.</s> + </p> + <p> + <s xml:id="s4353" xml:space="preserve">Sedre vera non putabam te indigere aliqua demonſtratione, quod linea <var>.b.h.</var> di-<lb/>uiſa ſit per æqualia à<unsure/> linea <var>.c.a.</var> cum hæc perpendicularis ſit ab <var>.a.</var> ad baſim <var>.g.d.</var> in <reg norm="triam" type="context">triã</reg> <lb/>gulo orthogonio <var>.g.a.d.</var> & cum ſit <var>.b.h.</var> perpendicularis ad <var>.a.o.</var> ex ſuppoſito quæ <var>.a.<lb/>o.</var> in ſe habet punctum medium baſis <var>.g.d.</var> nec <reg norm="non" type="context">nõ</reg> illud anguli recti <var>.a.</var> quod per ſe cla <lb/>riſſimum eſt, cum iam ſcis <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 ipſi <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. eiuſdem, angulus <var>.h.</var> æqua <lb/>bitur angulo <var>.d.</var> eo quod an gulus <var>.e.</var> rectus eſt, quemadmodum et <var>.a.</var> ſed angulus <var>.d.</var> <lb/>æqualis eſt 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 ipſi <var>.u.<lb/>a.</var> ex .6. primi, cum poſtea angulus <var>.<lb/> + <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> + o.a.d.</var> æqualis ſitangulo <var>.d.</var> ex quin<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 eſt, & ex eadem æqualis <lb/>erit angulo <var>.d.a.c.</var> vnde <var>.u.b.</var> erit <lb/>æqualis ipſi <var>.u.a.</var> ex .6. dicti, & ideo <lb/>æqualis eric ipſi <var>.u.h</var>. </s> + <s xml:id="s4356" xml:space="preserve">Reliqua ve-<lb/>rò illius propoſitionis credo ex te <lb/>omnia poſſe <reg norm="intelligere" type="context">ĩtelligere</reg>, excepto, <reg norm="quod" type="simple">ꝙ</reg> <lb/>vt tibi ſignificaui ſi à <reg norm="puncto" type="context">pũcto</reg> <var>.i.</var> com-<lb/>muni ipſi <var>.a.c.u.</var> & circunferentiæ, <lb/>ducta fuerit <var>.i.x.</var> ad <reg norm="punctum" type="context">pũctum</reg> <var>.x.</var> com <lb/>mune vni parallelæ à <reg norm="puncto" type="context">pũcto</reg> <var>.g.</var> ipſi <lb/><var>h.b.</var> & circunferentiæ, quod di-<lb/>cta <var>.i.x.</var> ad rectos erit ipſi <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"/> + ſit angulo <var>.a.h.b.</var> propter æquidiſtantiam 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> ſed angulus <var>.i.a.d.</var> com-<lb/>munis eſt 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 eſt <var>.i.x.</var> per <lb/>pendicularis erit ipſi <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 ſcripſi circa finem illius epiſtolæ, Tartaleam erraſſe in quinta propo-<lb/>ſitione primi lib. ſuæ nouæ ſcientiæ, non ſine ratione illud ſcripſi. </s> + <s xml:id="s4359" xml:space="preserve">Nam, inquit ipſe, <lb/>nullum corpus æquè graue poteſt in aliquo temporis ſpatio moueri motu naturali, <lb/><reg norm="violentoque" type="simple">violentoq́;</reg> ſimul miſtis. </s> + <s xml:id="s4360" xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve <lb/>locitatis vnius motus, ſimul eſſe cum decremento velocitatis alterius, <reg norm="eodemque" type="simple">eodemq́;</reg> tem <lb/>pore, vt manifeſtè patet in itinere corporis, ab ipſo pro exemplo aſſumpto, hoc eſt <lb/>quod velocitas motus in ſpatio <var>.c.d.</var> creſcit vt naturalis, & decreſcit vt violenta. </s> + <s xml:id="s4361" xml:space="preserve"><reg norm="nam" type="context">nã</reg> <lb/>creſcit orizontem verſus & decreſcit in remotione à linea <var>.a.b.</var> ſed ſi à puncto <var>.c.</var> ad <lb/>punctum <var>.d.</var> motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-<lb/>ſcenderet, eo quod uirtus mouens, in <var>.a.</var> poſita, nullo pacto poteſt talem effectum ef-<lb/>ficere, vnde ab ipſa natura prouenit deſcenſio 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 ſcilicet, & non ex violentia aliqua. </s> + <s xml:id="s4362" xml:space="preserve">Sed ſi <lb/>dixiſſet ipſe, illum motum eſſe purum naturalem, hoc eſſet falſum, eo quod purus <lb/>naturalis motus alicuius corporis non impediti, extra locum ſuum, ſit per lineam re <lb/>ctam, & non per curuam, vt videre eſt inter <var>.c.</var> et <var>.d</var>.</s> + </p> + <p> + <s xml:id="s4363" xml:space="preserve">In vltima propoſitione deinde eiuſdem lib. quæ .6. eſt decipitur ſimiliter, & hæc <lb/>deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau <lb/>ſam ipſius deſcenſus per ſpatium <var>.c.d</var>. </s> + <s xml:id="s4364" xml:space="preserve">Sed quia tibi ſignificaui expeditiorem viam <lb/>repeririad <reg norm="cognoſcendam" type="context">cognoſcendã</reg> proportionem inter <var>.a.h.</var> et <var>.a.e.</var> in vltima propoſitione ſe-<lb/>cundi lib. ipſius Tartaleæ, ipſam nunc tibi ſcribo. </s> + <s xml:id="s4365" xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> iam ſcis angulum <var>.h.l.i.</var> diui-<lb/>ſum eſſe per æqualia ab <var>.P.l.</var> & quod <var>.a.h.</var> et <var>.h.p.</var> ęquales inuicem ſunt 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 ſimiliter, ſed ex .3. ſexti ita eſt ipſius <var>.a.l.</var> <lb/>ad <var>.l.i.</var> vt ipſius <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> ſed <var>.a.l.</var> cogni <lb/>ta eſt ex eius quadrato, et <var>.l.i.</var> etiam, cum æqualis ſit ipſi <var>.a.i.</var> vnde ex regula de tribus <lb/>notam habebimus <var>.p.i.</var> reſpectu <var>.a.i.</var> & ita reſpectu <var>.a.e.</var> ſi hypotheſes ipſius Tartaleæ <lb/>veræ ſunt.</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 ſuperparticularem in <lb/>diſcretis.</head> + <head xml:id="head539" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4367" xml:space="preserve">QVod à me poſtulas, hoc eſt ſcientiam impoſſibilitatis diuidendi per æqualia <lb/>proportionem ſuperparticularem in numeris ſatis à Campano in .8. octaui <lb/>potes habere, Iacobus Faber Stapulenſis etiam idem tractat<unsure/> in libello ſuę muſicæ <lb/>demonſtratæ. </s> + <s xml:id="s4368" xml:space="preserve">Sed ſi etiam alia via idem deſideras, quamuis longiori, nih<unsure/>ilomi-<lb/>nus vniuerſaliori, conſidera duos numeros <var>.g.</var> et <var>.h.</var> inuicem relatos ſecundum pro-<lb/>portionem ſuperparcicularem, quam volueris. </s> + <s xml:id="s4369" xml:space="preserve">Tunc dico impoſſibile eſle, vt per <lb/>æqualia diuidatur, quod ſi dixeris poſſibile eſſe, ſit 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 diuiſibilis eſſet ſi <var>.g.h.</var> minimi fuiſſent, quod non conceditur, ſint igitur mini <lb/>mi in dicta proportione <var>.a.</var> et <var>.b.</var> quorum differentia erit vnitas, vt ſcis, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.c.</var> quadra <lb/>tum ipſius <var>.g.</var> et <var>.d.</var> quadratum ipſius <var>.K</var>. </s> + <s xml:id="s4371" xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-<lb/>tio ipſius c. ad <var>.d.</var> eadem erit quæ <var>.g.</var> ad <var>.h.</var> hoc eſt vt ipſius <var>.a.</var> ad <var>.b.</var> vnde ſi vnus termi. <lb/>norum <var>.a.</var> vel <var>.b.</var> eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex <lb/>16. eiuſdem, inter <var>.a.</var> et <var>.b.</var> reperiretur aliquis medius numerus proportionalis, quod <lb/>fieri non poteſt ex hypotheſi, cum inter <var>.a.</var> et <var>.b.</var> nullus ſit numerus, quia differunt in <lb/>ter ſe 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 ſit, ponatur quod <var>.f.</var> quadratus ſit ipſius <var>.b.</var> et <var>.e.</var> ſit productum ipſius <var>.a.</var> in <var>.b.</var> vn <lb/>de ex .18. ſeptimi, proportio ipſius <var>.e.</var> ad <var>.f.</var> erit vt. ipſius <var>.a.</var> ad <var>.b.</var> hoc eſt vt ipſius <var>.c.</var> ad <lb/>d. quapropter <var>.e.</var> erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet <reg norm="medium" type="context">mediũ</reg> <lb/>proportionale inter <var>.a.</var> et <var>.b.</var> ex .20. ſeptimi, quod eſt impoſſibile, vt iam dixi, cum <var>.a.</var> <lb/>et <var>.b.</var> ſint inui cem conſequentes, vnus poſt alium immediatè.</s> + </p> + <p> + <s xml:id="s4373" xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerſalem, <lb/>hoc eſt quod hac methodo poſſumus in cognitionem <lb/>vcnire, quod non ſolum in duas æquales partes diui-<lb/> + <anchor type="figure" xlink:label="fig-0378-01a" xlink:href="fig-0378-01"/> + di non poſſit, ſed 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/>ipſo cognoſces ope <reg norm="cuborum" type="context">cuborũ</reg> vice <reg norm="quadratorum" type="context">quadratorũ</reg>, opevero <lb/><reg norm="cenſuum" type="context">cenſuũ</reg> <reg norm="cenſuum" type="context context">cẽſuũ</reg>, ve<unsure/>l qui cognouerit eam <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe indiuiſibilem per æqualia, illicò etiam cognoſcet <lb/>indiuiſibilem eſſe per quatuor partes, ope verò pri-<lb/>morum relatorum, cognoſcet non eſſe diuiſibilem per <lb/><reg norm="quinque" type="simple">quinq;</reg> partes, & ſic de cęteris, ſed mediantibus ijs <lb/>quas ſcripſi de iſtis dignitatibus in libro <reg norm="Thęorematum" type="context">Thęorematũ</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 Illuſtriſſimus Daniel Barbarus ſcri <lb/>bit in quinta parte ſuæ perſpectiuæ, ſi ſupra aliquo im <lb/>mobili, atque magno pariete facere volueris, te opor <lb/>tebit hoc ex reflexione radij ſolaris 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 circunſcribentis 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 eſt triangulum, cuius baſis cum angulo ſibi op <lb/>poſito dantur. </s> + <s xml:id="s4377" xml:space="preserve"><reg norm="Vellesque" type="simple">Vellesq́;</reg> diametrum circuli apti eum triangulum circnn-<lb/>ſcribere inuenire in diſcreto.</s> + </p> + <p> + <s xml:id="s4378" xml:space="preserve">Sit igitur triangulum <var>.a.b.g.</var> cuius baſis <var>.b.g.</var> ſimul cum angulo <var>.a.</var> ei op-<lb/>poſito data ſit in numeris. </s> + <s xml:id="s4379" xml:space="preserve">Imaginetur ergo circulas circunſeribens ipſum triangu-<lb/>lum <var>.b.p.g.q.</var> cuius diameter ſit <var>.q.p.</var> perpendicularis eius baſi <var>.b.g.</var> vnde <var>.b.g.</var> diuiſa <lb/>erit per æqualia ab ipſo 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 ſit angulo <lb/><var>q.p.g.</var> medietati anguli ipſius portionis ex .19. tertij, ita quod angulus <var>.q.e.g.</var> nobis <lb/>cognitus erit, & ſimiliter arcus <var>.g.q.</var> & conſequenter ar-<lb/>cus <var>.p.g.</var> reſiduum medij circuli, & ſic <var>.m.g.</var> eius ſinus re <lb/> + <anchor type="figure" xlink:label="fig-0379-01a" xlink:href="fig-0379-01"/> + ctus, & etiam chorda <var>.p.g.</var> vt dupla ſinus dimidij arcus <var>.<lb/>p.g.</var> & ſic <var>.p.m.</var> eius ſinus verſus, vel vt tertium latus trian <lb/>guli orthogonij <var>.p.g.m.</var> vnde nobis cognita erit propor <lb/>tio ipſius <var>.b.g.</var> (quæ dupla eſt ipſi <var>.m.g.</var>) ad <var>.m.p.</var> & quia <lb/>productum <var>.p.m.</var> in <var>.m.q.</var> æquale eſt 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 ipſius <var>.b.g.</var> ad <var>.q.p.</var> qua <lb/>mediante cognoſcemus diametrum ſecundum partes il <lb/>las quibus propoſita ſuerit <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 ſed 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">QVotieſcunque etiam inuenire voluerimus triangulum aliquem, puta <var>.n.q.o.</var> <lb/>æqualem triangulo <var>.t.</var> (exempli gratia) propoſito, qui habeat angulum <var>.n.</var> æ-<lb/>qualem angalo <var>.a.</var> dato, latera vero continentia ipſum angulum <var>.n.</var> ſint 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, ſupra quam conſtituemus triangulum <var>.m.n.p.</var> æqualem triangulo <var>.t.</var> hac metho-<lb/>do, hoc eſt prolungando latus <var>.r.z.</var> trianguli <var>.t.</var> quod ſit <var>.r.e.</var> ita vt duplum ſit ipſi <var>.r.z.</var> <lb/>ducendo poſtea <var>.c.e.</var> habebimus ex .38. primi triangulum <var>.t.</var> eſſe dimidium totius <lb/>trianguli <var>.r.c.e.</var> deſignabimus deinde ex .44. dicti ſuperficiem <var>.p.n.m.b.</var> parallelo <lb/>grammam <reg norm="æqualemque" type="simple">æqualemq́;</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/>poſtea <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 poſtea <var>.n.p.</var> ita vt <var>.n.K.</var> <lb/>ſe 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 ſi oportuerit, <lb/></s> + <s xml:id="s4382" xml:space="preserve">deinde eas ad æqualitatem ſe-<lb/>can<unsure/>do ipſis <var>.x.</var> et <var>.y.</var> efficiendo <lb/>exempli gratia quod <var>.n.i.</var> ſit <lb/>æqualis ipſi <var>.x.</var> et <var>.n.u.</var> ipſi <var>.y.</var> du <lb/>cendo poſtea <var>.u.i.</var> deinde à puncto <var>.m.</var> ducendo <var>.m.K.</var> æquidiſtanter <var>.u.i.</var> ex .31. <lb/>primi. </s> + <s xml:id="s4383" xml:space="preserve">& ſic habebimus ex .4. ſexti 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 poſtea ex .9. eiuſ-<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 ſit <var>.n.o.</var> duces poſtea <lb/><var>o.q.</var> parallelam ipſi <var>.m.K.</var> & ha <lb/>bebis propoſitum, eo quod <reg norm="cum" type="context">cũ</reg> <lb/>ſit 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 ſexti, 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. eiuſdem <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. ſexti ſit 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 ſit quadrato alterius.</s> + </p> + <p> + <s xml:id="s4385" xml:space="preserve">Vt exempli gratia ſi fuerint duæ lineæ <var>.e.d.</var> et <var>.e.f.</var> opor-<lb/><reg norm="teretque" type="simple">teretq́;</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">iũ<lb/> + <anchor type="figure" xlink:label="fig-0380-02a" xlink:href="fig-0380-02"/> + gere</reg>, ita longam, vt productum totius compoſiti <var>.e.c.</var> vel <var>.<lb/>f.b.</var> in <var>.f.c.</var> vel <var>.e.b.</var> eſſet æquale quadrato ipſius <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="quotieſcun- que" type="context">quotieſcũ-<lb/>que</reg> <var>.e.d.</var> coniuncta erit cum <var>.e.f.</var> ad rectos, <reg norm="diuiſaque" type="simple">diuiſaq́;</reg> per me <lb/>dium à puncto <var>.a.</var> à quo ducta <var>.a.d.</var> deinde ſecundum ſemi-<lb/>diametrum <var>.a.d.</var> deſignato circulo <var>.b.d.c.</var> & protracta <var>.e.f.</var> <lb/>à qua volueris parte vſque ad circunferentiam in <reg norm="puncto" type="context">pũcto</reg> <var>.c.</var> <lb/>ſeu in puncto <var>.b.</var> habebimus intentum, eò quod ſi produ-<lb/>cta fuerit <var>.e.f.</var> etiam ab alia parte, vſque ad circunferentiam, habebimus <var>.b.e.</var> æqua-<lb/>lem ipſi <var>.f.c.</var> ex communi conceptu, & productum <var>.e.c.</var> in <var>.e.b.</var> æqualem quadra-<lb/>to ipſius <var>.e.d.</var> ex .34. tertij, cum ex .3. eiuſdem <var>.e.d.</var> medietas ſit chordæ arcus dupli <lb/><var>b.d</var>.</s> + </p> + <p> + <s xml:id="s4387" xml:space="preserve">De lapſu verò lapidis verſus mundi centrum, dum ipſum attingere, ac præterire <lb/>poſſet, de quo me interrogas. </s> + <s xml:id="s4388" xml:space="preserve">Dico Nicolaum Tartaleam, nec non Franciſcum <lb/>Maurolicum rectè ſenſiſſe, malè verò Alexandrum Piccolhomineum, & exemplum <lb/>Maurolici optimum eſſe, quod tamen ſi capere non potes, crede ſaltem authoritati <lb/>bus talium virorum, qui tantum in ijs ſcientijs ſuperant ipſum Alexandrum Piccol-<lb/>homineum, quantum à Sole cætera ſuperantur aſtra.</s> + </p> + <p> + <s xml:id="s4389" xml:space="preserve">Lapis igitur ille tranſiret centrum, <reg norm="reddiretque" type="simple">reddiretq́;</reg>, cum diminutione tamen motus im <lb/>preſſi, eo fermè modo vt ſcribunt iudicioſiſſimi illi viri, donec poſt multas reddi-<lb/>tiones ſurſum, <reg norm="deorſumque" type="simple">deorſumq́;</reg> quieſceret 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 ſilum illum (exempli adducti ab illis doctiſſimis viris) cui pon <lb/>dus appenſum eſt, æqualem eſſe axi orizontis, hoc eſt eius extremitatem immobi-<lb/>lem eſſe in primo mobili, & in ipſo zenit tui orizontis, </s> + <s xml:id="s4391" xml:space="preserve">tunc arcus motionis ipſius la <lb/>pidis per tantum interuallum, quantum eſt diameter terræ, inſenſibiliter differret à <lb/>linea recta, & cum lapis diſtans à centro mundi per ſemidiametrum terræ, iret re-<lb/><reg norm="diretque" type="simple">diretq́;</reg>, vt ſcis, ergo idem faceret ſi ſilum longius eſſet per dictum terræ ſemidia-<lb/>metrum, ita vt poſſetipſum centrum attingere, nam differentia illa ſemidiametri <lb/>terræ, ferè nulla eſt reſpectu ſemidiametri ipſius 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/>deſcriptus æquiangulus ſit.</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 ſuper datam lineam deſi-<lb/>gnatum, æquiangulum non eſſe. </s> + <s xml:id="s4393" xml:space="preserve">Fingamus hic ſubiectam ſiguram ſimi-<lb/>lem ei quæ à Durero ponitur, in qua primò, ducta ſit 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 ſuppoſito, arcus <var>.a.b.</var> eſt <lb/>ſexta pars totius circunferentiæ, angulus vero <var>.b.o.d.</var> rectus eſt, eo quod <var>.b.o.q.</var> rectus <lb/>etiam ſit, </s> + <s xml:id="s4394" xml:space="preserve">quare angulus <var>.d.o.a.</var> reſiduus ex recto erit graduum .60. talium, ut rectus <lb/>eſt .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 ſinus 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> eſt partium .51762. talium qualium <var>.a.d.</var> vel <var>.a.c.</var> ſemidiameter eſt .100000.</s> + </p> + <p> + <s xml:id="s4396" xml:space="preserve">Nam ſinus dimidij arcus <var>.a.o.</var> (exi <lb/>ſtente <var>.a.o.</var> graduum .30.) eſt <reg norm="partium" type="context">partiũ</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 ſi <lb/>nus eſt <var>.a.e.</var> erit graduum .26. min .38 <lb/>qui quidem angulus, ſumptus 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 ſumma ex duobus <lb/>rectis gra .180. reliquum erit gra .93. <lb/>min .22. ideſt 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 ſic, cum inuentus fuerit angu-<lb/>lus <var>.a.d.o.</var> gra .26. min .38. ſi 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> reſiduo ex re cto dempto angulo <var>.a.o.e.</var> grad .60. qui <var>.e.a.o.</var> eſt 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> cognoſcemus vt ſinus anguli <var>.d.o.n.</var> gra .45. nam <lb/>angulus ei contrapoſitus <var>.q.o.p.</var> eſt 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. ſed <var>.d.o.</var> eſt partium .115270. qualium <var>.a.d.</var> eſt <num value="100000">.<lb/>100000.</num> nam <var>.e.d.</var> vt ſinus anguli <var>.e.a.d.</var> gra .63. min .22. eſt partium .89389. <var>o.e.</var> ve-<lb/>ro eſt partium .50000. talium qualium <var>.a.o.</var> eſt .100000. vt ſinus anguli <var>.e.a.o.</var> gra .30. <lb/>ſed vt <var>.a.o.</var> eſt <reg norm="partium" type="context">partiũ</reg> .51762. </s> + <s xml:id="s4400" xml:space="preserve">hoc eſt vt <var>.a.d.</var> eſt .100000. ipſa <var>.o.e.</var> erit <reg norm="partium" type="context">partiũ</reg> .25881 <lb/>quæ iuncta cum fuerit cum <var>.e.d.</var> efficiet <var>.d.o.</var> partium .115270. vt dictum eſt, quapro-<lb/>pter cum <var>.d.n.</var> ſit partium .70710. talium qualium <var>.d.o.</var> fuerit .100000. ipſa <var>.d.n.</var> erit <lb/>partium .81507. talium qualium <var>.d.o.</var> erit .115270. ideſt qualium <var>.d.a.</var> vel <var>.d.u.</var> erit <lb/>100000. quæ quidem <var>.d.n.</var> eſt ſinus anguli <var>.d.u.n.</var> graduum ſcilicet .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 ſupra, cui applica angu-<lb/>lum <var>.o.d.n.</var> gra .45. min <var>.o.</var> ſimul cum <lb/>angulo <var>.u.d.n.</var> reſiduo ex recto gra <lb/>duum .35. minu .24. & conficies an-<lb/>gulum <var>.a.d.u.</var> grad .107. minu .2. & <lb/>habebis propoſitum, 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 ſuppu-<lb/>tata ſint, ex ſumma omnium angulo <lb/>rum patere poteſt. </s> + <s xml:id="s4403" xml:space="preserve">nam collectis om <lb/>nibus quinque angulis <var>.a.c.d.f.u.</var> ſi-<lb/>mul, hoc eſt grad .108. minu .22. cum <lb/>gra .107. min .2. cum grad .12. effi-<lb/>cient grad .540. min <var>.o.</var> ſumma æqua <lb/>lis ſex 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/>ſecundi 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 ſecundi Euclid. aliter à Comandino & Mau-<lb/>rolico demonſtratæ fuerint, nihilominus mihi etiam viſum eſt non nihil <lb/>meo moræ in eas tibi ſcribere, vt ſenſibiliter quoque cognoſcas il-<lb/>las veras eſſe.</s> + </p> + <p> + <s xml:id="s4405" xml:space="preserve">Eſto linc<unsure/>a.a.b. pro nona propoſitione, diuiſa per æqualia in <var>.c.</var> per inæqualia verò <lb/>in <var>.d.</var> quadratum autem <var>.a.d.</var> ſit <var>.d.e.</var> quadratum verò <var>.d.b.</var> ſit <var>.d.i.</var> quadratum <var>.a.c.</var> <lb/>ſit <var>.c.f.</var> & quadratum <var>.c.d.</var> ſit <var>.c.K.</var> clarum enim erit <var>.K.h.</var> æqualem exiſtere ipſi <var>.a.c.</var> <lb/>ſeccetur igitur <var>.e.h.</var> in <var>.g.</var> ita vt <var>.h.g.</var> ęqualis exiſtat ipſi <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">duplũ</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 ſint 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">aũt</reg> propoſitione, <reg norm="quadratum" type="context">quadratũ</reg> totalis lineæ <var>.a.d.</var> ſit <var>.d.e.</var> & lineæ <var>.b.d.</var> ſit <var>.b.i.</var> et <lb/><var>c.d.</var> ſit <var>.d.n.</var> et <var>.a.c.</var> ſit <var>.c.f.</var> et <var>.f.e.</var> ſit <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 Aſtrologo Peritißimo.</head> + <p> + <s xml:id="s4407" xml:space="preserve"><emph style="sc">POstqvam</emph> tua doctiſſima ſcripta perlegi, conſideraui, quod ſi à mul-<lb/>titudine exhalationum in regione elementari acciderit anno .1572. & <lb/>1573. vt totos ſex menſes ab omnibus per vniuerſum terrarum orbem <lb/>viſa fuerit ſtella illa, quæ eſt in angulo ſeptentrionali 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 ipſo lucifero videretur rutilantior <reg norm="atque" type="simple">atq;</reg> cæterarum (abſque vlla <lb/>aſpectus diuerſitate) magis ſcintillans. </s> + <s xml:id="s4408" xml:space="preserve">Quî fieri poterat, vt ſtellæ quæ ab illa pa-<lb/>rum diſtant, alioqui multo maiores, non etiam illa clariores apparuerint? </s> + <s xml:id="s4409" xml:space="preserve">ſed ſi ali-<lb/>quis diceret eam exhalationem non ita fortaſſe dilatari, vt inter nos, & aliam ali-<lb/>quam ſtellam interponeretur. </s> + <s xml:id="s4410" xml:space="preserve">Tunc ego reſponderem neceſſariò ſequi debere ta-<lb/>lem exhalationem, tantam latitudinem occupare, quod aliquibus populis aliam <lb/><reg norm="aliquam" type="context">aliquã</reg> <reg norm="ſtellam" type="context">ſtellã</reg> circunuicinãhac ipſa de qua mentionem ſecimus redderet lucidiorem. <lb/></s> + <s xml:id="s4411" xml:space="preserve">Sed cum hoc perſpectum fuerit nulli, ſequebatur lucem illam ab ipſis exhalatio-<lb/>nibus elementaribus haud poſſe oriri: </s> + <s xml:id="s4412" xml:space="preserve">quod nobis ſcintillatio illa maxima perma-<lb/>gno fuit inditio, ſi phas eſt credere, <reg norm="nam" type="context">nã</reg> quo magis aliquod coeleſte corpus ſcintillat, <lb/>eo longius à nobis diſtare.</s> + </p> + <p> + <s xml:id="s4413" xml:space="preserve">Verum quoniam efflagitaſti à me vt aliquid circa huiuſce rei ſpeculationem tibi <lb/>ſcribam, idcirco tibi morem gerere volens paucis ſubiungam.</s> + </p> + <p> + <s xml:id="s4414" xml:space="preserve">Conſidera primo hanc ſubſcriptam primam figuram, in qua <var>.c.a.e.</var> ſignatur pro <lb/>Globo terreſtri cuius <var>.i.</var> centrum ſit et <var>.u.o.n.</var> pro conuexo ignis, ſed <var>.K.x.s.</var> pro orbe <lb/>octauo <var>.x.</var> autem pro ſtella iam ſuperius dicta, quæ ſemper fuit, eſt, & erit, quamuis <lb/>cæteris tribus nunc obſcurior ſit. </s> + <s xml:id="s4415" xml:space="preserve">Accipiantur deinde duo loca in ſuperſicie terræ, <lb/>quę ſint <var>.c.</var> et <var>.e.</var> diametraliter inuicem oppoſita, ita quod circa eorum orizontes poſ <lb/>ſibile ſit ſtellam <var>.x.</var> videre, radijs ipſius ſtellæ mediantibus <var>.x.n.e.</var> et <var>.x.u.c.</var> <reg norm="quorum" type="context">quorũ</reg> par-<lb/>tes <var>.n.c.</var> et <var>.u.e.</var> ita breues ſint, reſpectu eorum <reg norm="totorum" type="context">totorũ</reg>, vt vix ſexcenteſima pars ſit vna <lb/><reg norm="quæque" type="simple">quæq;</reg> illarum, nec non <var>.c.e.</var> ita breuis reſpectu ſemidiametri octauæ ſphæræ, quod <lb/>vix ſit vna ex partibus decemmillibus, vt ſcis, ſequitur quod recta terminata ab <var>.u.</var> <lb/>et <var>.n.</var> minor ſenſibiliter non ſit ipſo terræ diametro <var>.c.e.</var> cum duo hæc interualla ex <lb/>triangulorum ſimilitudine ſe habeant vt <var>.x.i.</var> ad <var>.x.o.</var> hoc eſt 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 ſit. </s> + <s xml:id="s4416" xml:space="preserve">ductę poſtea <reg norm="cum" type="context">cũ</reg> fuerint duæ diagonales <var>.e.u.</var> et <var>.n.c.</var> <reg norm="termi- nabunt" type="context">termi-<lb/>nabũt</reg> angulos <var>.n.e.u.</var> et <var>.e.n.c.</var> <reg norm="inuicem" type="context">inuicẽ</reg> ferè ęquales, <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 ſublimatam eſſe ad ſupremas par-<lb/>tes elementaris regionis circum circa lineam <var>.o.i</var>: </s> + <s xml:id="s4418" xml:space="preserve">tunc clarum eſſet quod ſi ratione hu <lb/>iuſmodi exhalationis ſtella <var>.x.</var> ita lucida viſa fuerit tam aſpicientibus ab <var>.e.</var> quam ab <lb/>c. exhalatio minoris latitudinis quam <var>.u.n.</var> eſſe non poterat, hoc eſt, quam terræ dia-<lb/>meter, cum idem in longitudine ferè ſit, ſed punctum <var>.u.</var> ſatis videri poteſt 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 diuerſa diafana tranſeuntium. </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> vſque 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> ſed <var>.s.x.</var> ſubiacet 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 ſenſibiliter ac ſi eſſent in centro <var>.i</var>. </s> + <s xml:id="s4420" xml:space="preserve">Et cum ſuperius di-<lb/>xerimus angulos <var>.s.e.x.</var> et <var>.k.c.x.</var> ſenſibiliter mi nime differre ab angulis <var>.c.n.e.</var> et <var>.e.<lb/>u.c.</var> ſi cognouerimus quantitatem iſtorum, cognita etiam nobis erit quantitas <lb/>ill orum.</s> + </p> + <p> + <s xml:id="s4421" xml:space="preserve">Cum igitur ſemidiameter elementaris regionis maior ſit ſemidiametro terræ, vt <lb/>33. ad vnum, & cogitata <var>.c.n.</var> vt dicta ſemidiameter, quia ſenſibiliter ab ea minime <lb/>diffe<unsure/>rt, nunc ſi ſupponatur dicta <var>.n.c.</var> vt <reg norm="baſistrianguli" type="context">baſistriãguli</reg> orthogonij eſſe <reg norm="partium" type="context">partiũ</reg> .100000 <lb/>& dixerimus ſi <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">partiũ</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 ſtellam om-<lb/>nes aliæ non diſtant huiuſmodi 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 ſequuti ſunt <lb/>opinionem exhalationum, ad ſer <lb/>uandam nullitatem diucrſitatis <lb/>aſpectus, affirmant poſſe tam lon <lb/>ge à terra <reg norm="aſcendere" type="context">aſcẽdere</reg> exhalationes, <lb/>imo nec attingere ſupremas tertię <lb/>regionis a eris partes, ita ut non <reg norm="in- grediantur" type="context">in-<lb/>grediãtur</reg> ſuum <reg norm="igneum" type="context">igneũ</reg> orbem, qui <lb/>quidem orbis ſecundum illorum <lb/>opinionem incipit non valde lon-<lb/>gè à ſuperſicie terræ, vt <reg norm="iam" type="context">iã</reg> in mea <lb/>conſideratione contra Antonium <lb/>Bergam probaui, ſed demus, <reg norm="quod" type="simple">ꝙ</reg> <lb/>dictæ exhalationes aſcenderint <lb/>per decem ſemidiametros terræ, <lb/>diſcurrendo poſtea ſic, cum <var>.c.n.</var> ut <lb/>decem, nobis dat <var>.c.e.</var> vt duo, <reg norm="quid" type="simple">ꝗd</reg> <lb/>dabit nobisipſa <var>.c.n.</var> vt .100000. <lb/>& proueniet nobis <var>.c.e.</var> vt .200000. <lb/>cuius ſinus 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> & ſic eorum arcus <var>.s.x.</var> et <var>.k.x.</var> <lb/>ſed quis vnquam dubitabit <reg norm="quod" type="simple">ꝙ</reg> in <lb/>tanto interuallo à dicta ſtella non <lb/>fint aliæ multæ ipſa 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 aſpectus <lb/>diuerſitatis 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, exiſtente 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 ipſo oculo non videretur ob terræ globoſitatem. </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 ſecunda figura videre eſt, in qua <lb/>ducantur <var>.c.b</var>: <var>i.b</var>: et <var>.i.u.</var> quæ <var>.i.u.</var> ſecabit 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> ſimiliter in pun-<lb/>cto <var>.y.</var> quod nulli dubium eſt, cum <var>.c.u.</var> æqualis ſit <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> ſed ex .4. primi <var>.c.y.</var> æqua <lb/>lis erit ipſi <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 ſuppoſita <var>.c.i.</var> decima parte ipſius <var>.c.u.</var> nemi <lb/>ni dubium erit quod cum <var>.u.i.</var> ſubtendatur angulo <lb/>recto <var>.u.c.i.</var> (iam ſupra diximus angulum <var>.c.</var> ſenſibi-<lb/>liter minime differre à recto) ipſa vt ſinus totus erit <lb/>partium .100000. cuius quadratum cum diuiſum <lb/>fuerit in partes æquales centum & <reg norm="vnam" type="context">vnã</reg>, illarum vna <lb/>æqualis erit quadrto <var>.c.i.</var> reliquę vero quadrato ip-<lb/>ſius <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 ipſius <var>.u.i.</var> eſt 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 ipſius <var>.c.u.</var> erit <lb/>partium .9900990100. cuius radix <var>.u.c.</var> erit <reg norm="partium" type="context">partiũ</reg> <lb/>99500. vnde angulus <var>.c.i.u.</var> erit graduum .84. & mi <lb/>nu .17. & angulus <var>.c.u.i.</var> qui reſpondet ſinui <var>.c.i.</var> erit <lb/>gra .5. & min .43. cuius duplum, hoc eſt angulus <var>.c.<lb/>u.b.</var> erit grad .11. min .26. æqualis ferè angulo iam <lb/>ſupradicto. </s> + <s xml:id="s4433" xml:space="preserve">ſed <var>.c.y.</var> ſinus anguli <var>.c.i.y.</var> erit ſimiliter <lb/>partium .99500. talium vt <var>.c.i.</var> ſunt .100000. ſed vt <lb/><var>c.i.</var> eſt partium .9950. </s> + <s xml:id="s4434" xml:space="preserve">tunc <var>.c.y.</var> erit partium .9900 <lb/>hoc eſt quaſi decima pars ipſius <var>.c.u.</var> </s> + <s xml:id="s4435" xml:space="preserve">quare ſi 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> abſque ſenſibili 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/>iſoperimetrarum.</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 ſupra Ptolomei Almageſtum ſufficienter traditum <lb/>ſit de magnitudinibus figurarum iſopetimetrarum, nihilominus vt tibi <lb/>morem geram, ea nunc ſcribo, quæ mihi in mentem venerunt contra <lb/><reg norm="Alexandrum" type="context">Alexãdrum</reg> Piccolhomineum, <reg norm="antequam" type="context">antequã</reg> aliquid ipſius 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 ſphæra vbi tractat de <reg norm="caeliro- tunditate" type="simple">cęliro-<lb/>tunditate</reg>, ita inquit.</s> + </p> + <quote xml:lang="it"> + <s xml:id="s4437" xml:space="preserve">Oltre di queſto, douendo il decimo cielo contenere & in ſe chiudere tutte le co-<lb/>ſe, è conueneuol coſa il penſare, che foſſe fatto di quella più capace figura che eſ-<lb/>ſer poſſa, la qual è la figura rotunda, però che ſi può trar da molti luoghi d'Euclide <lb/>che ſi come ſe noi ciimmagineremo più figure ſuperficiali talmente che tutte le li-<lb/>nee de l'vna congionte inſieme, ſieno vguali à tutte le linee pur inſiememente com <lb/>poſte di qual ſi voglia de l'altre figure, ne ſeguirà, che quella figura ſarà più capa-<lb/>ce la qual haurà manco angoli, & quella capaciſſima che ſarà 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 iſoperimetris, ſit enim triangulus æ-<lb/>quilaterus <var>.o.b.g.</var> quadratum verò <var>.b.l.</var> quorum pe-<lb/>riferiæ inuicem æquales ſint. </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 ſuperficiei eſſe ipſo triangulo. </s> + <s xml:id="s4440" xml:space="preserve">Accipio pri-<lb/>mum lineam <var>.f.h.</var> eiuſdem longitudinis quæ vnius <lb/>periferiæ dictarum figurarum, quam punctis <var>.r.K.</var> <lb/>mediantibus diuido in tres ęquas 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> ideſt 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> ſed <var>.f.l.</var> eſt quarta pars ip-<lb/>ſius <var>.f.h.</var> ergo <var>.k.i.</var> erit quarta pars ipſius <var>.k.h</var>. </s> + <s xml:id="s4441" xml:space="preserve"><reg norm="Conium" type="context">Coniũ</reg> <lb/>gantur enim ambo iſtæ figuræ vt hic inferius vides, <lb/>vnde <var>.a.g.</var> erit quarta pars ipſius <var>.b.g.</var> diuiſa poſtea <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> quę per .8. primi, nec <reg norm="non" type="context">nõ</reg> ex definitione, <lb/>perpendicularis erit ipſi <var>.b.g.</var> ergo etiam <reg norm="quadratum" type="context">quadratũ</reg> <lb/>b q. ſupra <var>.b.g.</var> <reg norm="producoque" type="simple">producoq́;</reg> <var>.o.c.</var> vſque ad <var>.m.</var> nam nul <lb/>li dubium eſt quin <var>.o.c.</var> breuior ſit <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">ducoq́;</reg> <var>t.e.p.</var> æquidiſtantem <var>.b.g.</var> <lb/>vnde habebimus duo quadrata <var>.e.g.</var> et <var>.e.b.</var> ſed <lb/>quadratum <var>.b.l.</var> æquatur quadrato ipſius <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. ſecundi, hoc eſt æquatur quadrato <var>.c.a.</var> & re-<lb/>ctangulo <var>.t.g</var>. </s> + <s xml:id="s4443" xml:space="preserve">Deinde vt ſe habet <var>.p.g.</var> ad <var>.o.e.</var> ita ſe habet <var>.u.p.</var> ad <var>.u.e.</var> ex ſimilitudine <lb/>triangulorum. </s> + <s xml:id="s4444" xml:space="preserve">Sed <var>.p.g.</var> maior eſt ipſa <var>.o.e.</var> cum <var>.p.g.</var> æqualis ſit <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. ſexti. </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 ſequitur rectangulum <var>.t.g.</var> maiorem eſſe triangu-<lb/>lo <var>.b.o.g.</var> ſed quadratum <var>.b.l.</var> eſt etiam maior ipſo rectangulo <var>.t.g.</var> ex quadrato ipſius <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 ſit periferia ipſius trianguli æquilateri. </s> + <s xml:id="s4448" xml:space="preserve">Cogita triangulum æquilaterum hic <lb/>ſubſcriptum <var>.d.l.q.</var> cuius baſis <var>.l.q.</var> diuiſa ſit per æqualia à perpendiculari <var>.d.o.</var> <reg norm="deſcri- ptumque" type="context simple">deſcri<lb/>ptũq́;</reg> ſit rectangulum <var>.o.g.</var> quod æquale erit triangulo <var>.d.l.q.</var> ſed periferia trianguli <lb/>maior eſt periferia rectanguli, nam <var>.l.q.</var> æqualis eſt <var>.o.q.</var> cum <var>.d.g.</var> ſed <var>.q.d.</var> maior eſt <var>.o.<lb/>d.</var> ex .18. primi, vnde <var>.l.d.</var> maior etiam <var>.q.g.</var> cum ex .34. dicti latera oppoſita ipſius re <lb/>ctanguli ſint inuicem æqualia, accipiamus poſtea <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 pun<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 eſt quadratum ipſius <var>.c.i.</var> æquale erit ei quod fit .ex <var>.e.c.</var> <lb/>in <var>.c.h.</var> hoc eſt rectangulo <var>.g.o.</var> hoc eſt triangulo <var>.d.l.q.</var> ſed <var>.e.h.</var> eſt dimidium perife-<lb/>rię ipſius rectanguli <var>.g.o.</var> quæ minor eſt di midio periferiæ trianguli <var>.d.l.q.</var> vt vidimus <lb/>et <var>.i.k.</var> eſt dimidium periferię quadrati ipſius <var>.i.c.</var> & minor etiam ipſa <var>.e.h.</var> ex .14. tertij <lb/></s> + <s xml:id="s4449" xml:space="preserve">quare verum eſt propoſitum.</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 ſunt inuicem æquales, poſſumus etiam breuiter videre id <lb/>quod ſupradiximus, hoc eſt, quod quadratum, maius ſit triangulo æquilatero. </s> + <s xml:id="s4451" xml:space="preserve">Nam <lb/>cum <var>.b.g.</var> ſeſquitertia ſit 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 ipſo <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.c.q.</var> ſed <var>.c.q.</var> maius eſt <reg norm="triam" type="context">triã</reg> <lb/>gulo <var>.b.o.g.</var> cum <var>.q.g.</var> quæ æqualis eſt <var>.o.g.</var> maior ſit <var>.o.c.</var> ex .18. vel penultima primi, <lb/>nam ſi <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 noſtri vſi ſunt quæ generalis eſt vt in Theone ſupra Al-<lb/>mageſtum videre eſt, medijs perpendicularibus à centris ad latera figurarum, ſed <lb/>quia <reg norm="differentia" type="context">differẽtia</reg> longitudinum ipſarum perpendicularium alio medio inueniri poteſt, <lb/>eo quo ipſi vſi ſunt, prætermittere nolo quin tibi ſcribam.</s> + </p> + <p> + <s xml:id="s4455" xml:space="preserve">Ego enim ita diſcurro.</s> + </p> + <p> + <s xml:id="s4456" xml:space="preserve">Sint duæ figuræ iſoperimetrę æquilaterę & æquiangulæ, puta primò trian-<lb/>gulum & quadratum quorum centra ſint <var>.e.</var> et <var>.o.</var> à quibus centris ad latera ſint 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 ſcis, ducantur <lb/>poſtea <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/>midiũ</reg> recti, et <var>.e.t.n.</var> tertia pars vnius recti, vt ex te ipſo 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. ſeſquialter erit angulo <var>.t.</var> quod vt clarius videas cogita lineam <var>.b.d.</var> cuius medietas <lb/>ſit <var>.c.d.</var> tertia verò pars illius ſit <var>.g.d.</var> </s> + <s xml:id="s4458" xml:space="preserve">tunc dico <var>.c.d.</var> ſeſquialteram eſſe ipſi <var>.g.d.</var> ſit enim <lb/><var>f.d.</var> duplum ipſius <var>.g.d.</var> </s> + <s xml:id="s4459" xml:space="preserve">quare <var>.f.d.</var> erunt duæ tertiæ totius lineę <var>.b.d.</var> & quia eadem pro <lb/>portio eſt 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> ſeſquialtera eſt, verum igitur erit quod an-<lb/>gulus <var>.a.</var> ſeſquialter ſit ipſi <var>.t.</var> deinde <var>.t.n.</var> eſt ſeſquitertia ipſi <var>.a.u.</var> vt ſuperius vidimus .<lb/>in eorum duplis. </s> + <s xml:id="s4461" xml:space="preserve">ſcimus etiam <var>.n.e.</var> eſſe dimidium ipſius <var>.t.e.</var> co quod cum <var>.e.t.n.</var> ſit <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 inſcriptibilis circulo cuius diameter ſit <var>.e.t.</var> </s> + <s xml:id="s4462" xml:space="preserve">quare <var>.e.t.</var> dupla erit <lb/>ipſi <var>.e.n.</var> in longitudine, ſed quadrupla in potentia: </s> + <s xml:id="s4463" xml:space="preserve">t.n. vero tripla in potentia ipſi <var>.n.<lb/>e.</var> ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, ſed <lb/><var>c.n.</var> erat ſexquitertia ipſi <var>.a.u.</var> in longitudine, hoc eſt ipſi <var>.o.u.</var> nam <var>.o.u.</var> æqualis eſt ipſi <lb/><var>a.u</var>. </s> + <s xml:id="s4464" xml:space="preserve">quare <var>.n.t.</var> erit minus quam dupla in potentia ipſi <var>.o.u.</var> hoc eſt, vt .16. ad .9. ergo <lb/>maior proportio erit ipſius <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 ipſius <var>.t.n.</var> ad <var>.n.e.</var> quam ad <var>.o.u.</var> vnde <var>.o.u.</var> longior erit <lb/>ipſa <var>.n.e.</var> quod eſt propoſitum.</s> + </p> + <p> + <s xml:id="s4466" xml:space="preserve">Sed ſi <var>.o.a.u.</var> eſſet pentagonus æquilaterus & æquiangulus, ſimiliter probabo per-<lb/>pendicularem <var>.o.u.</var> longiorem eſſe <var>.n.e.</var> ipſius trianguli æquilateri, dummodo ſint iſo-<lb/>perimetrę. </s> + <s xml:id="s4467" xml:space="preserve">Sit enim <var>.a.u.</var> dimidium lateris pentagoni ex ſuppoſito, cuius centrum ſit <lb/>o. </s> + <s xml:id="s4468" xml:space="preserve">tunc proportio <var>.t.n.</var> ad <var>.a.u.</var> erit ſuperbipartienstertias, vt ex ordine iam hic ſupradi <lb/>cto à te facillimè elicere potes, hoc eſt, 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">duorũ</reg> rectorum, hoc <lb/>eſt duæ quintæ vnius recti, vnde angulus <var>.a.</var> reſiduum 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> & conſequenter latus <var>.o.u.</var> ma-<lb/>ius latere <var>.a.u.</var> ſed <var>.t.n.</var> minor eſt quam tripla in potentia ad <var>.a.u.</var> eo quod erit vt .25. <lb/>ad .9. cum in longitudine ſit vt .5. ad .3. ſed dicta <var>.t.n.</var> tripla eſt in potentia ad <var>.e.n.</var> qua-<lb/>re <var>.a.u.</var> maior erit ipſa <var>.e.n.</var> ſed <var>.o.u.</var> maior eſt ipſa <var>.a.u.</var> vt diximus, igitur multo magis <var>.<lb/>o.u.</var> maior eſt ipſa <var>.a.u.</var> vt <reg norm="diximus" type="simple">diximꝰ</reg> & <reg norm="conſequenter" type="context context">cõſequẽter</reg> multo magis <var>.o.u.</var> maior erit ipſa <var>.n.e</var>.</s> + </p> + <p> + <s xml:id="s4470" xml:space="preserve">Quotieſcunque enim cognoſcimus proportionem anguli <var>.o.</var> ad angulum <var>.a.</var> quod <lb/>quidem facillimum eſt, nec non proportionem <var>.t.n.</var> ad <var>.a.u.</var> quod, etiam illico cogno-<lb/>ſcitur, </s> + <s xml:id="s4471" xml:space="preserve">tunc exſcientia cordarum & arcuum omnia etiam facillimè innueniuntur. <lb/></s> + <s xml:id="s4472" xml:space="preserve">Verum circa <reg norm="triangulum" type="context">triangulũ</reg> æquilaterum, & pentagonum, alium <reg norm="modum" type="context">modũ</reg> inueni, ſed 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 incommenſur abilitate, in longitudine perpendicu-<lb/>laris trianguli æquilateri cum eiuſdem latere.</head> + <head xml:id="head557" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4473" xml:space="preserve">ID quod à me poſtulas eſt omnino impoſſibile, velles enim duos numeros inueni <lb/>re inter ſe ita ſe habentes, vt ſe habent perpendicularis in triangulo æquilatero <lb/>cum vno eius laterum, quod vero hoc fieri non poſſit, conſidera in figura præcedenti <lb/>triangulum æquilaterum <var>.d.l.q.</var> cuius perpendicularis ſit <var>.d.o.</var> quæ diuidit <var>.l.q.</var> per <lb/>æqualia in <var>.o.</var> vnde ex .4. ſecundi Euclidis, quadratum <var>.l.q.</var> (ideſt <var>.d.q.</var>) quadruplum <lb/>erit quadrato <var>.o.q.</var> & ex penultima primi ęquale quadratis <var>.d.o.</var> et <var>.o.q.</var> </s> + <s xml:id="s4474" xml:space="preserve">quare erit ſeſ-<lb/>quitertium quadrato ipſius <var>.d.o.</var> & ita quadratum <var>.d.o.</var> erit triplum quadrato ipſius <var>.<lb/>o.q.</var> hæe autem proportiones non ſunt vt numeri quadrati ad numerum quadratum <lb/>quod ſi ita fuiſſent, ſequeretur ternarium numerum eſſe quadratum ex .22. octaui. <lb/></s> + <s xml:id="s4475" xml:space="preserve">Cum igitur non ſint vt numeri quadrati ad numerum quadratum, ſequitur ex ſepti-<lb/>ma decimi <var>.d.o.</var> eſſe incommenſurabilem ipſi <var>.l.q.</var> ſeu <var>.d.q.</var> in longitudine.</s> + </p> + <p> + <s xml:id="s4476" xml:space="preserve">Vel dicamus ita, proportio quadrati ipſius <var>.l.q.</var> ad quadratum ipſius <var>.o.d.</var> eſt in ge <lb/>nere ſuperparticulari, cum ſit ſeſquitertia, vnde quadratum ipſius <var>.d.o.</var> numeris da-<lb/>ri non poteſt, eo quod ſi dabilis fuiſſet, ſequeretur, quod inter quadratum ipſius. l<unsure/> <var>.<lb/>q.</var> & ipſius <var>.d.o.</var> eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex <lb/>octaua eiuſdem vnitas diuiſibilis eſſet, quod fieri non poteſt.</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 conſideraui circa triangulum æquilaterum & pentagonum, vt <lb/>tibi ſignificaui ita ſe habet.</s> + </p> + <p> + <s xml:id="s4478" xml:space="preserve">Probandum primò eſt pentagonum altiorem eſſe triangulo ſibi iſoperimetro. <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 diſtenta ſit <var>.K.z.</var> baſis autem <var>.m.<lb/>g.</var> bifariam diuiſa ſit in puncto <var>.a.</var> <reg norm="ductaque" type="simple">ductaq́;</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> (baſes 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>) ſint inuicem æquales ex .4. eiuſdem.</s> + </p> + <p> + <s xml:id="s4481" xml:space="preserve">Accipiatur deinde vel intelligatur <var>.g.p.</var> æqualis duabus te<unsure/>rtijs ipſius <var>.a.g.</var> ducatur<lb/>q́ue <var>.b.p.</var> quam probabo maiorem eſſe duplo ipſius <var>.a.p.</var> vnde maior erit latere ipſius <lb/>trigoni æquilateris, cuius dimidium eſt <var>.a.p.</var> ſcimus enim ipſum latus ſe 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> ſit 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 ipſius <var>.a.p.</var> erit .103006. latus igitur dicti trigoni, quod <lb/>ab <var>.p.</var> erigitur, ſecabit perpendicularem <var>.a.b.</var> ſub <var>.b.</var> hoc eſt 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 iſoperimetro pentagono propoſito, <reg norm="ducaturque" type="simple">ducaturq́;</reg> <var>.u.p.</var> ducatur etiam <var>.u.n.</var> pa-<lb/>rallela ipſi <var>.b.g.</var> quæ concludet triangulum <var>.g.u.n.</var> ſimilem triangulo <var>.m.b.g.</var> eo quod <lb/>cum angulus <var>.m.b.g.</var> æqualis ſit 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="æquidiſtantes" type="context">æquidiſtãtes</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. eiuſdem, </s> + <s xml:id="s4485" xml:space="preserve">vnde ex .4. ſexti proportio <var>.g.n.</var> ad <var>.g.m.</var> erit .vt <var>.g.u.</var> <lb/>ad <var>.m.b.</var> ſed cum <var>.g.u.</var> maior ſit dimidio ipſius <var>.b.g.</var> ex .20. primi, hoc eſt maior dimi-<lb/>dio ipſius <var>.b.m.</var> ergo <var>.g.n.</var> etiam maior erit ipſa <var>.g.a.</var> quapropter maior erit ipſa <var>.g.p.</var> <lb/>cum <var>.g.p.</var> minor ſit ipſa <var>.g.a.</var> ex hypotheſi, 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 ſexti <lb/>vel quia totum maius eſt ſua parte. </s> + <s xml:id="s4486" xml:space="preserve">Triangulus igitur <var>.b.u.g.</var> maior eſt 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 ſibi iſo-<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 circunſcriptorum 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 inſcripti, in quo ſit triangu-<lb/>lus æquilaterus, longior ſit periferia ipſius trianguli æquilateri, abſque 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 demonſtrare quod valde de-<lb/>ſideras.</s> + </p> + <p> + <s xml:id="s4490" xml:space="preserve">Quapropter ſit circulus <var>.b.a.e.q.</var> in quo ſit <reg norm="triangulum" type="context">triangulũ</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 ſcire potes. </s> + <s xml:id="s4492" xml:space="preserve">Sit etiam <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.b.</var> commune tam anguli quadrati quam trianguli. </s> + <s xml:id="s4493" xml:space="preserve">vnde ſequitur quod dictus <lb/>diameter ſecabit 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 ſit arcui <var>.b.n.</var> ex .27. tertij, remanet vt arcus <var>.q.e.</var> equalis ſit arcui <var>.q.n.</var> vnde <lb/>angulus <var>.q.b.e.</var> æqualis erit angulo <var>.q.b.n.</var> ex .26. eiuſdem. </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 ipſi <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> ſeinuicem <reg norm="ſecant" type="context">ſecãt</reg> in puncto <var>.o.</var> vt ex ſe clarum patet, ducatur po <lb/>ſtea <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 ipſa <var>.q.e.</var> et <var>.q.e.</var> longior erit ipſa <var>.e.t</var>. </s> + <s xml:id="s4498" xml:space="preserve">quare <var>.q.o.</var> longior erit ipſa <var>.t.e</var>.</s> + </p> + <p> + <s xml:id="s4499" xml:space="preserve">Vt probemus poſtea <var>.b.a.o.</var> longiorem eſſe ipſa <var>.b.e.</var> producatur <var>.b.a.</var> ita quod <var>.a.<lb/>p.</var> æqualis ſit ipſi <var>.a.o.</var> <reg norm="ducaturque" type="simple">ducaturq́;</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">rectꝰ</reg> ſit, erit angulus <var>.o.a.p.</var> ſimiliter <reg norm="rectus" type="simple">rectꝰ</reg> ex .13. primi, vnde ex .5. et .32. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <lb/>angulus <var>.a.p.o.</var> erit dimidium recti, & ſimiliter, exijſdem, angulus <var>.b.q.a.</var> eſt 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> ſed angulus <var>.a.e.b.</var> æqualis eſt 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 eſt 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 ſexti, <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> ſed ex .18. primi <var>.b.o.</var> maior eſt <lb/>ipſa <var>.b.a</var>. </s> + <s xml:id="s4503" xml:space="preserve">quare ex .14. quinti <var>.b.p.</var> maior erit <lb/>ipſa <var>.b.e.</var> ſed <var>.b.p.</var> æquatur ipſis <var>.b.a.</var> cum <var>.a.</var> o <lb/>ex hypoteſi, ergo <var>.b.a.</var> cum <var>.a.o.</var> maior erit <lb/>ipſa <var>.b.e.</var> ſed <var>.q.o.</var> maior erat ipſa <var>.t.e.</var> vt ſupe <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 eſt ipſa <var>.b.e.</var> cum <var>.e.t.</var> hoc eſt dimidium <lb/>periferię ipſius quadrati, <reg norm="maius" type="simple">maiꝰ</reg> erit dimidio <lb/>periferię <reg norm="ipſius" type="simple">ipſiꝰ</reg> <reg norm="trianguli" type="context">triãguli</reg> propoſiti, </s> + <s xml:id="s4505" xml:space="preserve">quare ex 14. <lb/>dicta tota periferia dicti trianguli, ſimiliter <lb/>probarem de omnibus alijs figuris regulari <lb/>bus eodem circulo inſcriptis.</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 aliàs dixi verum eſt, intellectum ſcilicet non omninò quieſcere cir <lb/>ca illas duas Archimedis propoſitiones, quæ in translatione Tartaleæ ſunt <lb/>ſub numeris .4. et .5. & in impreſſione Baſileæ ſub 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, ſeu ſtateræ: </s> + <s xml:id="s4507" xml:space="preserve">A ſpice igitur in .4. ſupradicta, quod cum appen-<lb/>ſæ fuerint omnes illæ partes ponderum, partibus longitudinis ipſius <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 ipſi <var>.l.K.</var> hoc eſt reſpicientes 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 ſit paralle la ipſi <var>.l.k.</var> quæ diuiſa ſit in puncto <var>.i.</var> ſupra <var>.g</var>. </s> + <s xml:id="s4509" xml:space="preserve">Hinc nulli dubium erit, <lb/>cum <var>.g.</var> fuerit centrum totius ponderis appenſi ipſi <var>.l.K.</var> quod <var>.i.</var> ſimiliter erit centrum <lb/>cum directe locatum ſit ſupra <var>.g.</var> hoc eſt in eadem directionis linea, quod quidem <lb/>non indiget aliqua demonſtratione, cum per ſe ſatis pateat. </s> + <s xml:id="s4510" xml:space="preserve">Vnde ex communi <lb/>conceptu <var>.o.</var> erit centrum ponderis appenſi ipſi <var>.l.h.</var> et <var>.u.</var> erit centrum ponderis ap-<lb/>penſi. ipſi <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 eſt ipſius <var>.l.h.</var> & ipſius <var>.h.k.</var> con <lb/>tinuatorum per totam <var>.l.k</var>. </s> + <s xml:id="s4512" xml:space="preserve">Nunc ergo ſi conſideremus <var>.l.k.</var> diuiſam eſſe, hoc eſt di-<lb/>ſiunctam in puncto <var>.h.</var> inueniemus nihilominus <var>.i.</var> centrum eſſe dictorum ponderum, <lb/>& quod tantum eſt, ipſam eſſe <reg norm="continuam" type="context">continuã</reg>, quantum diuiſam 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> diuiſa vt dictum eſt, habebimus propoſitum.</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 propoſitionem tibi relinquo.</s> + </p> + <p> + <s xml:id="s4514" xml:space="preserve">Illa verò propoſitio, quam tibi dixi Archimedem tacuiſſe in huiuſmodi materia <lb/>eſt, quod ſi duo pondera æquilibrant ab extremis alicuius ſtateræ, in certis præfixis <lb/>diſtantijs à centro. </s> + <s xml:id="s4515" xml:space="preserve">Tunc dico ſi eorum vno manente alterum moueatur remotius <lb/>ab ipſo centro quod illud deſcendet, & ſi vicinius ipſi centro appenſum fuerit aſcen-<lb/>det. </s> + <s xml:id="s4516" xml:space="preserve">Hæc enim propoſitio quotidie omnibus in locis videtur, ipſam verſo4; </s> + <s xml:id="s4517" xml:space="preserve">puto Ar <lb/>chimedem prætermiſiſſe ob facilitatem, cum ab antedicta ferè dependeat.</s> + </p> + <p> + <s xml:id="s4518" xml:space="preserve">Sit exempli gratia ſtatera <var>.a.u.</var> cuius centr um ſit <var>.i.</var> & pondera <var>.u.a.</var> appenſa, ſein-<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> ſe inuicem habent. </s> + <s xml:id="s4519" xml:space="preserve">Nunc dico quod ſi pondus ipſius <var>.u.</var> <lb/>poſitum fuerit vicinius centro vt puta in <var>.o.</var> inmoto exiſtente pondere, a. quod bra-<lb/>chium <var>.i.o.u.</var> aſcendet, & è conuerſo, ſi remotius poſitum fuerit, deſcendet.</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">dictũ</reg> eſt in <var>.o.</var> vicinius <reg norm="centro" type="context">cẽtro</reg>, quapropter brachium <var>.i.o.</var> <reg norm="breuius" type="simple">breuiꝰ</reg> erit <lb/>brachio <var>.i.u.</var> vnde minor proportio erit ipſius <var>.i.o.</var> ad <var>.i.a.</var> quàm.i.u. ad eundem <var>.a.i.</var> & <lb/>conſequenter quam ponderis ipſius <var>.a.</var> (quod ſit <var>.n.e.</var>) ad pondus ipſius <var>.u</var>. </s> + <s xml:id="s4521" xml:space="preserve">Quare ſi cx <lb/>pondere <var>.n.e.</var> dempta fuerit <var>.e.</var> pars eius, ita quod reliqua pars <var>.n.</var> ſe habeat ad pondus <lb/>o. vt ſe habet. i<unsure/> <var>.o.</var> ad <var>.i.a.</var> tunc ſtatera non mouebitur; </s> + <s xml:id="s4522" xml:space="preserve">addita verò parte <var>.e.</var> ex com-<lb/>muni conceptu, a. deſcendet vnde <var>.o.</var> aſcenderet conuerſum verò ex ſimilibus 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 ſatisfacerem, cum Eu-<lb/>totius citet ſextum librum Pergei, quem nunquam vidimus, <reg norm="ſupponatque" type="simple">ſupponatq́;</reg> ea, quæ nec <lb/>ipſe nec alius vnquam quod ſcimus probauit.</s> + </p> + <p> + <s xml:id="s4524" xml:space="preserve">Deſideras enim demonſtrationem illius quod Archimedes dicit inter primam, <lb/>& ſecundam propoſitionem ſecundi libri, vbi tractat de centris grauium, propte-<lb/>rea quod illud ſupponit pro manifeſto.</s> + </p> + <p> + <s xml:id="s4525" xml:space="preserve">Sit enim figura hic ſubſcripta, ferè ſimilis parabolæ poſitæ in .2. propoſitione di <lb/>cti libri, vt in impreſſione Baſileenſi habetur, <reg norm="ſintque" type="simple">ſintq́;</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">protractisq́;</reg> <var>.f.x.</var> et <var>.u.i.</var> ad <var>.b.d.</var> quæ inuicem etiam erunt parallelę <lb/>ex .30. primi Eucli. </s> + <s xml:id="s4526" xml:space="preserve">vnde ipſæ etiam, diametri erunt ipſarum portionum: </s> + <s xml:id="s4527" xml:space="preserve">vt ex eo col <lb/>ligere eſt, quod in .49. primi lib. Pergei probatur. </s> + <s xml:id="s4528" xml:space="preserve">Imaginando poſtea ad puncta <var>.b.<lb/>f.</var> er<unsure/> <var>.i.</var> tres contingentes, manifeſtum erit punctum <var>.b.</var> illud eſſe quod terminat alti-<lb/>tudinem huiuſmodi portionis, et <var>.f.</var> et <var>.i.</var> terminantia altitudines partialium, ex .5. ſe<lb/>cundi ipſius Pergei, eo quod dictæ contingentes paralellæ erunt ipſis baſibus, vnde <lb/>trianguli inſcripti, eaſdem habebunt altitudines, quas portiones ipſæ, quod erit ex <lb/>mente Archimedis. </s> + <s xml:id="s4529" xml:space="preserve">Et ſic deinceps poteris multiplicare angulos ſiguræ rectilineæ <lb/>in parabola, quæ deſignata erit vt deſiderat Archimedes, qui quidem dicit, quod <lb/>protractæ cum fuerint aliæ deinceps poſt <var>.f.i.</var> ipſæ inuicem ęquidiſtantes <reg norm="erunt" type="context">erũt</reg>, diuiſę-<lb/>q́ue peræqualia ab <var>.d.b.</var> quod <reg norm="quanuis" type="context">quãuis</reg> <reg norm="verum" type="context">verũ</reg> ſit, <reg norm="tantum" type="wordlist/context">tñ</reg> ab Eutotio non ſatis <reg norm="demonſtratum" type="context context">demõſtratũ</reg> <lb/>eſt, cum ſupponat <var>.a.f.b.</var> æqualem eſſe ipſi <var>.b.i.c.</var> probare volens eius diametros æqua <lb/>les eſſe abſque aliqua citata ratione, quæ quidem ratio eſſet conuerſum .4. propoſi-<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 ipſius Pergei, <lb/>& propterea tibi non ſatisfacerem.</s> + </p> + <p> + <s xml:id="s4531" xml:space="preserve">Eſto igitur, ut inuenta ſit linea <var>.K.</var> cuius productum in <var>.u.i.</var> æquale ſit qua drato ip <lb/>ſius <var>.u.c.</var> inuenta etiam ſit linea <var>.h.</var> cuius productum cum <var>.f.x.</var> æquale ſit quadrato ip-<lb/>ſius <var>.a.x.</var> vnde ex conuerſo .49. primi ipſius Pergei, proportio ipſius <var>.K.</var> ad <var>.b.c.</var> erit ut <lb/>ipſius <var>.b.c.</var> ad <var>.b.d.</var> & ipſius <var>.h.</var> ad <var>.a.b.</var> vt ipſius <var>.a.b.</var> ad <var>.b.d</var>. </s> + <s xml:id="s4532" xml:space="preserve">Erit igitur ex .16. ſexti Eucl. <lb/>quadratum <var>.b.c.</var> æquale producto ipſius <var>.K.</var> in <var>.b.d.</var> & quadratum <var>.a.b.</var> æquale produ-<lb/>cto ipſius <var>.h.</var> in <var>.b.d.</var> & ex prima ſexti, ita erit ipſius <var>.K.</var> ad <var>.h.</var> vt producti quod fit ex <var>.K.</var> <lb/>in <var>.b.d.</var> ad productum ipſius <var>.h.</var> in <var>.b.d.</var> hoc eſt vt quadrati ipſius <var>.b.c.</var> ad quadratum ip <lb/>ſius <var>.b.a.</var> ex .16. et .11. quinti, hoc eſt vt quadrati ipſius <var>.u.c.</var> ad quadratum ipſius <var>.a.x.</var> <lb/>hoc eſt ut productum ipſius <var>.k.</var> in <var>.u.i.</var> ad productnm ipſius <var>.h.</var> in <var>.x.f</var>. </s> + <s xml:id="s4533" xml:space="preserve">Nunc ſi ipſius <var>.k.</var> <lb/>ad <var>.h.</var> c<unsure/>ſt vt producti ipſius <var>.K.</var> in <var>.u.i.</var> ad productum ipſius <var>.h.</var> in <var>.f.x.</var> ergo ex .24. ſexti, <lb/>& communi conceptu, proportio ipſius <var>.k.</var> ad <var>.h.</var> compoſita erit ex ea quæ ipſius <var>.u.i.</var> <lb/>ad <var>.f.x.</var> & ex ea quæ ipſius <var>.k.</var> ad <var>.h</var>. </s> + <s xml:id="s4534" xml:space="preserve">Cum ergo dempta fuerit proportio ipſius <var>.k.</var> ad <var>.h.</var> <lb/>(vt ſimplex) à proportione ipſius <var>.k.</var> ad <var>.h.</var> (vt compoſita) reliquum nihil erit. </s> + <s xml:id="s4535" xml:space="preserve">Qua-<lb/>re <var>.f.x.</var> æqualis erit ipſi <var>.u.i</var>.</s> + </p> + <p> + <s xml:id="s4536" xml:space="preserve">Sed quod <var>.f.m.</var> æqualis ſit ipſi <var>.m.i</var>. </s> + <s xml:id="s4537" xml:space="preserve">Videto in Eutotio, quia hoc ſatis ſui natura <lb/>facile eſt.</s> + </p> + <p> + <s xml:id="s4538" xml:space="preserve">Sed accipe alium modum breuiorem ad probandum <var>.f.x.</var> eſſe æqualem ipſi <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> prolungatisq́ue diametris <var>f.<lb/>x.</var> et <var>.u.i.</var> vſque ad contingentem ipſam, habebis <var>.f.e.</var> æqualem ipſi <var>.f.x.</var> et <var>.g.i.</var> ipſi <var>.u.i.</var> <lb/>Ex .35. primi Pergei, producta poſtea <var>.x.u.</var> habeb is ex .2. ſexti Eucli <var>.x.u.</var> parallelam <lb/>ipſi <var>.a.c.</var> ſed <var>.e.g.</var> parallela eſt ipſimet <var>.a.c.</var> ex quinta ſecundi ipſius Pergei, </s> + <s xml:id="s4540" xml:space="preserve">quare ex .30 <lb/>primi Euclid <var>.e.g.</var> parallela erit ipſi <var>.u.x.</var> & ex .34. eiuſdem æqualis erit <var>.e.x.</var> ipſi <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 deſideres probabo <var>.f.m.</var> æqualem eſſe ipſi <var>.m.i</var>. </s> + <s xml:id="s4542" xml:space="preserve">Iam igitur ſcis 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 ſit <var>.f.x.</var> æqualis ipſi <var>.u.i.</var> vt tibi probaui, & inuicem parallelæ ideo <var>.f.i.</var> parallela <lb/>erit ipſi <var>.x.u.</var> ex .33. primi Euclidis. </s> + <s xml:id="s4543" xml:space="preserve">Vnde ex .30. eiuſdem, parallela erit etiam ipſi <var>.a.<lb/>c.</var> ſed cum <var>.x.u.</var> diuiſa ſit ab <var>.d.b.</var> per æqualia, eo quod diuidit <var>.a.c.</var> eodem modo, quę <lb/>ipſi parallela eſt ex .2. ſexti. </s> + <s xml:id="s4544" xml:space="preserve">Reliqua tibi conſideranda relinquo. </s> + <s xml:id="s4545" xml:space="preserve">cum verò ambæ <var>.f.<lb/>x.</var> et <var>.u.i.</var> parallelæ ſint ipſi <var>.b.d.</var> ſequitur quod cum ex .34. primi <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>.f.m.</var> et <var>.m.<lb/>i.</var> æqualis ſit medietati ipſius <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 ſatisfacere Eutocium in .3. propoſitione ſecundi <lb/>lib. de centris Grauium Archimedis, cum citet .6. librum de elementis conicis, ad-<lb/>de quod ſi aliud in ipſo .6. libro ab eo citato non eſſet magis ad propoſitum, quàm <lb/>ca quæ ab ipſo citata ſunt, nihilominus adhuc irreſolutus maneres.</s> + </p> + <p> + <s xml:id="s4547" xml:space="preserve">Conſidera igitur eandem ipſam figuram præcedentem; </s> + <s xml:id="s4548" xml:space="preserve">pro alia verò parabola ſi <lb/>mili dictæ, accipe ſecundam figuram ipſius tertiæ dictæ propoſitionis. </s> + <s xml:id="s4549" xml:space="preserve">Deinde ima <lb/>ginabis duo latera <var>.o.x.</var> et <var>.o.p.</var> diuiſa eſſe per æqualia in punct is <var>.g.</var> et <var>.K.</var> <reg norm="protractisque" type="simple">protractisq́;</reg> <lb/>diametris <var>.g.y.</var> et <var>.K.u.</var> quæ, vt in præcedenti probaui, ſunt inuicem æquales, ſcire <lb/>debes quod ſimiles parabolæ inuicem aliæ non poſſunt eſſe, niſi eæ quæ diametros <lb/>proportionales ſuis baſibus habeant, <reg norm="ſimiliterque" type="simple">ſimiliterq́;</reg> poſitæ, hoc eſt, ut proportio ipſius <lb/><var>b.d.</var> ad <var>.a.c.</var> ſit eadem quæ ipſius <var>.o.r.</var> ad <var>.x.p.</var> & quod anguli ad <var>.r.</var> ſint æquales angulis <lb/>circa <var>.d</var>. </s> + <s xml:id="s4550" xml:space="preserve">Notentur ergo primum puncta communia ip ſius <var>.o.g.</var> cum <var>.y.t.</var> & ipſius <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 ſcimus <var>.f.m.</var> æqualem eſſe <var>.m.i.</var> tota <reg norm="mque" type="simple">mq́;</reg> <var>.f.<lb/>i.</var> parallelam eſſe ipſi <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">trianguliq́;</reg> <var>.x.f.n.</var> et <var>.g.y.</var> ω<unsure/>. eſſe ſimiles <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/>ſunt inuicem æquales, ex .29. verò eiuſdem anguli <var>.f.x.n.</var> et <var>.n.b.m.</var> ſimiliter 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 ſecunda figura, vnde ex .4. ſexti Eucli. proportio <var>.n.f.</var> ad <var>.m.n.</var> erit ea <lb/>dem quę <var>.f.x.</var> ad <var>.b.m.</var> & ipſius <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"/> + eiuſdem erit vt <var>.a.d.</var> ad <var>.d.b</var>. </s> + <s xml:id="s4555" xml:space="preserve">Idem etiam dico in ſecunda parabola, ſed ipſius <var>.x.o.</var> ad <lb/><var>o.r.</var> eſt vt <var>.a.b.</var> ad <var>.b.d.</var> ex .6. ſexti 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 ſecunda parabola, ita ut <var>.h.</var> eſt 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. ſexti <var>.a.x.</var> media proportionalis eſt inter <var>.h.</var> et <var>.f.<lb/>x.</var> cum ſupponatur 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 ipſius <var>.f.n.</var> ad <var>.a.b.</var> ut <var>.y.ω.</var> ad <var>.x.o.</var> & ſic <var>.f.n.</var> ad <var>.d.a.</var> <lb/>vt <var>.y.ω.</var> ad <var>.x.r.</var> ſed <var>.f.m.</var> ad <var>f.n.</var> eſt 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 ijſdem 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 conſideranda 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 propoſitionibus illius lib. nullo pacto poteris dubitare: </s> + <s xml:id="s4565" xml:space="preserve">Verum ne <lb/>in .4. aliquid tibi noui exurgat, te ſcire volo <ref id="ref-0025">corollarium .20. in libr. de quadratu<lb/>ra parabolę</ref> docere poſſibile eſſe inſcriptionem 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 poſtea animaduertendum eſt, quod prima pars, probat tantummodo de <lb/>centro trianguli, et .2. pars probat de centro pentagoni, à te ipſo deinde potes pro-<lb/>bare de centro nonanguli: </s> + <s xml:id="s4567" xml:space="preserve">& ſic de cæteris: </s> + <s xml:id="s4568" xml:space="preserve">eo quod cum probatum fuerit de centro <lb/>figuræ in medio locatæ ſi conſtitutæ poſtea fuerint ſimiles figuræ in portionibus la-<lb/>teralibus habebitur propoſitum in infinitum.</s> + </p> + <p> + <s xml:id="s4569" xml:space="preserve">Idem intelligendum eſt in .3. propoſitione quamuis exemplum vlterius non ex-<lb/>tendatur quam ad pentagonos.</s> + </p> + <p> + <s xml:id="s4570" xml:space="preserve">Sexta verò <reg norm="propoſitio" type="simple">ꝓpoſitio</reg> tibi ſacilis erit, quæ nihilominus <reg norm="pont" type="context">põt</reg> <reg norm="demonſtrari" type="context">demõſtrari</reg> hoc <reg norm="mon" type="context">mõ</reg> ſcili<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> ipſius Archimedis <reg norm="ſupponendo" type="context">ſupponẽdo</reg> <var>.a.</var> pro figura rectilinea <lb/>inſcripta in parabola, et <var>.b.</var> pro reſiduo ipſius parabolę et <var>.c.</var> pro triangulo <var>.a.b.c.</var> in me <lb/>dio ipſius parabolę et <var>.d.</var> pro triangulo <var>.r</var>. </s> + <s xml:id="s4572" xml:space="preserve">Nunc cum <var>.a.</var> maior ſit <var>.c.</var> prout totum ma-<lb/>ius eſt ſua 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 ſit <var>.d.</var> ex ſuppoſito, 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 cuiuſuis figuræ plenæ neceſſariò ſit intra <lb/>ipſam figuram, idcirco centrum reſidui ipſius parabolę intra ipſam reperietur. </s> + <s xml:id="s4573" xml:space="preserve">quod <lb/>ita <reg norm="clarum" type="context">clarũ</reg> <reg norm="per" type="simple">ꝑ</reg> ſe eſt, <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> quoduis aliud axioma, & quia <reg norm="dictum" type="context">dictũ</reg> <reg norm="centrum" type="context">centrũ</reg> ex .8. primi <lb/>de centris, neceſſariò eſt 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 eſt <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">cũ</reg> <var>.h.b.</var> maior ſit ipſa <var>.h.g.</var> prout omne totum ma-<lb/>ius eſt ſua 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 ipſa <var>.f.</var> ex .10. <reg norm="quinti" type="context">quĩti</reg>.</s> + </p> + <p> + <s xml:id="s4577" xml:space="preserve">Septima verò et .8. propoſitio 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 ſcribat ſuper duas vltimas lib. ſecundi de centris <reg norm="grauium" type="context">grauiũ</reg>, nihil <lb/>miror ipſum tibi non ſatisfacere. </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">primũ</reg> ſupponit 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>: ſupponit etiam quod proportio quæ eſt ipſius <var>.<lb/>e.b.</var> ad <var>.e.a.</var> <reg norm="eadem" type="context">eadẽ</reg> ſit quæ ipſius <var>.f.g.</var> ad tres quintas ipſius <var>.a.d.</var> & quod proportio com <lb/>poſiti dupli ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.c.</var> cum ſexcuplo ipſius <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"/> + ipſius <var>.b.e.</var> ad compoſitum quintupli ipſius <var>.a.b.</var> cum decuplo ipſius <var>.c.b.</var> cum decuplo <lb/>ipſius <var>.b.d.</var> cum quintuplo ipſius <var>.b.e.</var> eadem ſit quæ ipſius <var>.g.h.</var> ad <var>.a.d.</var> & vult proba-<lb/>re <var>.f.h.</var> eſſe duas quintas ipſius <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 ipſius <var>.a.c.</var> ad <var>.c.d.</var> & ipſius <var>.c.d.</var> ad <var>.d.e.</var> eſſe vt ipſius <lb/><var>a.b.</var> a<unsure/>d.b.c. & cętera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheſi ſit <lb/>ipſius, <var>a.b.</var> totalis ad <var>.c.b.</var> totalom vt ipſius <var>.c.b.</var> partialis (ſumptæ vt pars abſciſa ab <var>.a.<lb/>b.</var> pro nunc) ad <var>.d.b.</var> partialem (abſciſam ab <var>.c.b.</var>) erit ex .19. dicta ipſius <var>.a.c.</var> (reſidui <lb/>ex <var>.a.b.</var>) ad <var>.c.d.</var> (reſiduum ex <var>.c.b.</var>) vt ipſius <var>.a.b.</var> ad <var>.c.b.</var> & ita probabitur de pro-<lb/>portione ipſiuas <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 ſit ipſius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.c.b.</var> vt ipſius <var>.a.d.</var> ad <var>.d.e.</var> ergo ex <lb/>22. eiuſdem, ita erit ipſius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.d.b.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> & exijſdem rationibus <lb/>eadem proportio erit ipſius <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 ipſius <var>.a.d.</var> <lb/>ad <var>.d.e.</var> quę dupli primi antecedentis cum ſimplo ſecundi antecedentis ad duplum <lb/>primi conſequentis cum ſimplo ſecundi conſequentis, hoc eſt dupli ipſius <var>.a.b.c.</var> <reg norm="cum" type="context">cũ</reg> <lb/>ſimplo <var>.c.b.d.</var> ad duplum ipſius <var>.d.b.</var> cum ſimplo <var>.e.b.</var> hoc eft dupli <var>.a.b.</var> cum triplo ip-<lb/>ſius <var>.b.c.</var> cum ſimplo <var>.d.b.</var> ad duplum ipſius <var>.d.b.</var> cum ſimplo <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 ſimplo <var>.b.d.</var> ſignatum ſit charactere <var>.D.</var> ſuum verò conſequens, <lb/>hoc eſt duplum <var>.d.b.</var> <reg norm="cum" type="context">cũ</reg> ſimplo <var>.e.b.</var> ſignificetur à charactere <var>.B.</var> hinc proportio ipſius <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, ſi quis ſumeret aliquod maius antecedens æquale ſci-<lb/>licet duplo ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.e.</var> cum quadruplo ipſius <var>.b.d.</var> cum du-<lb/>plo ipſius <var>.b.e.</var> <reg norm="compararetque" type="simple">compararetq́;</reg> illud cum <reg norm="conſequente" 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 eſt maiorem <lb/>quàm ipſius <var>.a.d.</var> ad <var>.d.e.</var> ex .12. quinti.</s> + </p> + <p> + <s xml:id="s4586" xml:space="preserve">Nunc ſi ſumpta 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 ſecunda parte decimę quinti quod <var>.d.o.</var> minor erit ipſa <var>.d.e.</var> <lb/>Corrige igitur impreſſionem Baſileę locando characterem <var>.o.</var> inter <var>.d.</var> et <var>.e.</var> eo quod <lb/>ibi poſitum non fuit.</s> + </p> + <p> + <s xml:id="s4587" xml:space="preserve">Volo nunc quod dictum maius antecedens æquale ſcilicet duplo ipſius <var>.a.b.</var> cum <lb/>quadruplo ipſius <var>.b.c.</var> cum quadruplo ipſius <var>.b.d.</var> cum duplo ipſius <var>.b.c.</var> ſignificetur à <lb/>charactere <var>.A</var>. </s> + <s xml:id="s4588" xml:space="preserve">Hinc habebimus proportionem ipſius <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 poſtea habebimus <var>.A.B.</var> ad <var>.B.</var> vt <var>.a.o.</var> ad <var>.d.o.</var> & proportionalitate <lb/>euerſa 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"/> + ſe continet id quod Archimedes ſcribit, hoc eſt duplum ipſius <var>.a.b.</var> <reg norm="quadruplum" type="context">quadruplũ</reg> ipſius <lb/><var>b.c.</var> ſexcuplum ipſius <var>.b.d.</var> & triplum ipſius <var>.b.e</var>. </s> + <s xml:id="s4591" xml:space="preserve">Conſequens verò <var>.A.</var> continet du <lb/>plum ipſius <var>.a.b.</var> quadruplum ipſius <var>.b.c.</var> quadruplum ipſius <var>.b.d.</var> & duplum ipſius <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 ſuppoſito deinde ipſius Archimedis & ex conuerſa proportionalitate in .19. <lb/>dicta, verum eſt id quod dicit Archimedes, videlicet quod eadem proportio eſt <lb/>ipſius <var>.a.d.</var> ad <var>.g.h.</var> quod quintupli ipſius <var>.a.b.</var> cum quintuplo ipſius <var>.b.e.</var> cum decuplo <lb/>ipſius <var>.b.c.</var> cum decuplo ipſius <var>.b.d.</var> (quod quidem antecedens ſignificetur per <var>.V.</var>) <lb/>ad duplum ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.c.</var> cum ſexcuplo ipſius <var>.b.d.</var> cum triplo <lb/>ipſius <var>.b.e.</var> hoc eſt 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 ipſius <var>.a.d.</var> ad <var>.g.h.</var> ſed ſuperius vbi ſignatum eſt <var>.T.</var> iam <lb/>probatum fuit ita eſſe <var>.A.B.</var> ad <var>.A.</var> vt ipſius <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 ſcribit, hoc eſt quod ita erit ipſius <var>.V.</var> ad <var>.A.</var> vt ipſius <var>.a.o.</var> ad <var>.g.h</var>.</s> + </p> + <p> + <s xml:id="s4595" xml:space="preserve">Clarum per ſe etiam eſt, id quod Archimed. dicit hoc eſt quod <var>.V.</var> ad <var>.A.</var> eſt 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 compoſito <var>.V.</var> ad quodlibet in-<lb/>gredientium in compoſito <var>.A.</var> ſit 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="eiuſdem" type="context">eiuſdẽ</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 ſcriptum eſt, rurſus quoniam <var>.o.a.</var> quia oportet dicere <lb/>Rurſus quoniam <var>.o.d</var>.</s> + </p> + <p> + <s xml:id="s4599" xml:space="preserve">Archimedes igitur verum dicit, quod ipſius <var>.o.d.</var> ad <var>.d.a.</var> eſt vt ipſius <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"/> + conuerſa 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 ſpeculationis probatum iam fuit ita eſſe ipſius <var>.d.a.</var> ad <var>.d.e.</var> <lb/>vt ipſius <var>.D.</var> ad <var>.B.</var> vbi notatum eſt <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> ſe habeat ut <var>.D.</var> ad <var>.A.</var> erit ex conuerſa proportionalitate iam <lb/>dicta <var>.d.e.</var> ad <var>.d.o.</var> vt <var>.A.</var> ad <var>.D.</var> per euerſam vero erit <var>.d.e.</var> ad <var>.a.o.</var> vt <var>.A.</var> ad ſuum reſi-<lb/> + <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> + duum. </s> + <s xml:id="s4603" xml:space="preserve">quod reſiduum componitur ex ſimplo <var>.b.c.</var> cum triplo <var>.b.</var> cum duplo <var>.b.o.</var> quod <lb/>à te ipſo videre poteris detrahendo numeros ipſarum quantitatum quæ in <var>.D.</var> <lb/>reperiuntur, ex numeris earundem, quæ in <var>.A.</var> quod quidem reſiduum ſigniſicetur <lb/>à charactere <var>.E</var>. </s> + <s xml:id="s4604" xml:space="preserve">Vnde ex conuerſa proportionalitate verum dicit Archime. hoc eſt <lb/>quod ita ſe 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 ſit <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 ſuppoſito, ideo ex <lb/>17. quinti verum dicit Archim. </s> + <s xml:id="s4606" xml:space="preserve">hoc eſt quod ita erit ipſius <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. eiuſdem eadem proportio erit tripli ipſius <var>.c.d.</var> ad triplum <lb/>ipſius <var>.d.b.</var> quæ dupli ipſius <var>.d.e.</var> ad. duplum ipſius <var>.e.b.</var> vt inquit Archi.</s> + </p> + <p> + <s xml:id="s4607" xml:space="preserve">Ex qua .13. compoſitum ex <var>.a.c.</var> cum triplo ipſius <var>.c.d.</var> cum duplo ipſius <var>.d.e.</var> ean-<lb/>dem proportionem habebit ad <reg norm="compoſitum" type="context">compoſitũ</reg> ipſius <var>.c.b.</var> cum triplo <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.d.b.</var> cum duplo <lb/>ipſius <var>.e.b.</var> quam ipſius <var>.d.e.</var> ad <var>.e.b</var>. </s> + <s xml:id="s4608" xml:space="preserve">Sed horum compoſitorum primum ſignificetur <lb/>per <var>.H.</var> ſecundum verò ſignificatum fuit per <var>.E.</var> vnde <var>.H.</var> ad <var>.E.</var> ſe habebit vt <var>.d.e.</var> ad <lb/><var>e.b.</var> ſed <var>.E.</var> ad <var>.A.</var> iam dictum eſt eſſe vt <var>.o.e.</var> ad <var>.d.e.</var> vbi ſignatum eſt <var>.λ</var>. </s> + <s xml:id="s4609" xml:space="preserve">quare ex .23. <lb/>quinti eadem proportio erit ipſius <var>.o.e.</var> ad <var>.e.b.</var> quæ <var>.H.</var> ad <var>.A.</var> vt ipſe inquit.</s> + </p> + <note position="left" xml:space="preserve">X</note> + <!-- note-0400-02 --> + <p> + <s xml:id="s4610" xml:space="preserve">Ex .18. poſtea eiuſdem 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 eſt quod ſi collectæ fuerint omnes partes compoſiti <var>.H.A.</var> hoc <lb/>eſt duplum <var>.a.b.</var> cum duplo <var>.b.e.</var> cum quadruplo <var>.b.c.</var> cum quadruplo <var>.b.d.</var> cum ſimplo <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> & ſexcuplum <lb/><var>b.c.</var> vt ipſe dixit. </s> + <s xml:id="s4612" xml:space="preserve">Quod autem hoc verum ſit, cum diſtinctæ fuerint omnes partes, <lb/>vt in ſubſcriptis his lineis videre eſt, videbis quod ſi ex <var>.H.</var> detracta fuerit ſimplex <var>.a.<lb/>c.</var> quæ quidem poſtea iuncta vni ex partibus quadrupli <var>.b.c.</var> ipſius <var>.A.</var> reſultabit nobis <lb/>vna inte gra <var>.a.b</var>. </s> + <s xml:id="s4613" xml:space="preserve">Vnde habebimus triplum ipſius <var>.a.b.</var> & in <var>.A.</var> remanebit triplum ip <lb/>ſius <var>.c.b</var>. </s> + <s xml:id="s4614" xml:space="preserve">Deinde ſi ex <var>.H.</var> auferatur triplum ipſius <var>.c.d.</var> & ipſum addatur tribus parti-<lb/>bus quadrupli <var>.b.d.</var> ipſius <var>.A.</var> habebimus tres vices <var>.b.c.</var> quæ ſi iungantur tribus, quæ <lb/>remanebant in <var>.A.</var> vt dixi, habebimus ſexcuplum ipſius <var>.b.c.</var> & in <var>.A.</var> remanebit ſim <lb/>plum <var>.b.d.</var> cum duplo ipſius <var>.b.e</var>. </s> + <s xml:id="s4615" xml:space="preserve">Vnde ſi ex <var>.H.</var> demptum fuerit duplum ipſius <var>.d.<lb/>e.</var> quod quidem iungatur cum duplo ipſius <var>.b.e.</var> habebimus duplum ipſius <var>.b.d.</var> quod <lb/>coniunctum cum ſimplo <var>.b.d.</var> quod in <var>.A.</var> relictum fuerat, habebimus triplum ipſius <lb/><var>d.b</var>. </s> + <s xml:id="s4616" xml:space="preserve">Verum igitur eſt quod inquit Archimedes, hoc eſt, quod <var>.H.A.</var> eſt triplum ip-<lb/>ſius <var>.a.b.</var> ſexcuplum ipſius <var>.b.c.</var> & triplum ipſius <var>.b.d</var>.</s> + </p> + <p> + <s xml:id="s4617" xml:space="preserve">Verum etiam dicit ex eo (vt ſupra probatum eſt) quod <var>.a.c</var>: <var>c.d</var>: et <var>.d.e.</var> ſe <reg norm="habebant" type="context">habebãt</reg> <lb/>in continua proportionalitate, </s> + <s xml:id="s4618" xml:space="preserve">quare ex conuerſa proportionalitate erunt ſibi 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> ſint continuæ proportionales in ea proportione <lb/>in qua ſunt <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> & ſic etiam <var>.c.b.</var> ad <var>.e.b</var>. </s> + <s xml:id="s4620" xml:space="preserve">Vnde ex .24. eiuſdem <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/>ſumpto, & cum <var>.b.d.</var> ad <var>.e.b</var>. </s> + <s xml:id="s4621" xml:space="preserve">Quare ex conuerſa proportionalitate, vt ſe habet <var>.e.d.</var> <lb/>ad <var>.d.a.</var> ita ſe habebit <var>.e.b.</var> <reg norm="cum" type="context">cũ</reg> <var>.d.b.</var> ad <var>d.b.</var> <reg norm="cum" type="context">cũ</reg> <var>.b.c.</var> duplicato & <reg norm="cum" type="context">cũ</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 eſt <var>.b.e.</var> cum <var>.d.b.</var> conſequens 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"/> + eſt <var>.d.b.</var> cum duplo <var>.b.c.</var> cum ſimplo <var>.b.a.</var> vocetur <var>.N</var>.</s> + </p> + <p> + <s xml:id="s4623" xml:space="preserve">Animaduertendum tamen eſt quod impreſſio mendoſa eſt 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 eſt 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 ſe habet <var>.a.e.</var> ad <var>.d.a.</var> ita ſe 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 ſcriptum eſt <lb/><q open="„" close="">ad vtrunque ſimul <var>.b.d</var>: <var>d.a.</var> cum dupla <var>.b.c.</var></q> <lb/>dicendum eſt ita, <lb/>ad vtranque ſimul <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 ſicut ſe haber <var>.e.a.</var> ad <var>.d.a.</var> ita ſe habebit duplum <var>.M.N.</var> <lb/>ad duplum <var>.N</var>. </s> + <s xml:id="s4628" xml:space="preserve">Quod quidem verum eſt ex .13. quinti, huiuſmodi verò antecedons <lb/>& conſequens, Archi. manifeſtat ex ſuis partibus, ſumendo 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/>quę ſimul 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 quàm du-<lb/>plum ipſius <var>.M.N</var>.</s> + </p> + <p> + <s xml:id="s4629" xml:space="preserve">Verum etiam ſcribit, vbi dicit, quod proportio <var>.e.a.</var> ad tres quintas ipſius <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 ſuppoſito ita ſe habeat <var>.f.<lb/>g.</var> ad tres quintas ipſius <var>.a.d.</var> quemadmodum <var>.b.e.</var> ad <var>.e.a.</var> erit ex .16. quinti verum <reg norm="quod" type="simple">ꝙ</reg> <lb/>dicit Archimed. </s> + <s xml:id="s4631" xml:space="preserve">hoc eſt, ita ſe habere <var>.b.e.</var> ad <var>.f.g.</var> vt <var>.e.a.</var> ad tres quintas ipſius <var>.a.d</var>.</s> + </p> + <p> + <s xml:id="s4632" xml:space="preserve">Et per .11. eiuſdem verum etiam erit quod ſicut ſe habet <var>.e.b.</var> ad <var>.f.g.</var> ita ſe habe-<lb/>bit <var>.A.</var> ad tres quintas dupli <var>.N.</var> quod quidem duplum <var>.N.</var> ſignificetur per <var>.Q</var>.</s> + </p> + <p> + <s xml:id="s4633" xml:space="preserve">Sed ſuperius iam demonſtratum fuit (vbi <var>.X.</var>) quod <var>.o.b.</var> ad <var>.b.e.</var> ita ſe habebat vt <lb/><var>H.A.</var> ad <var>.A.</var> & <reg norm="nunc" type="context">nũc</reg> demum probatum fuit ita eſſe <var>.A.</var> ad tres quintas ipſius <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 ipſius <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 ſuis partibus videre eſt) ita ſe habet vt tres ad duo ex .13. <lb/>quinti, vt inquit Archimedes.</s> + </p> + <p> + <s xml:id="s4636" xml:space="preserve">Ipſe etiam dicit proportionem <var>.H.A.</var> ad tres quintas ipſius <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> diuiſa 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> ipſius <var>.Q.</var> <lb/><reg norm="æqualem" type="context">æqualẽ</reg> eſſe duabus tertijs vniuſcuiuſque quintæ partis <var>.H.A.</var> vnde tres quintæ ipſius <lb/>Q. erunt, ex communi conceptu, ſex tertiæ vnius quintæ ipſius <var>.H.A.</var> hoc eſt duæ <lb/>quintæ. ipſius <var>.H.A</var>. </s> + <s xml:id="s4638" xml:space="preserve">Quare <var>.o.b.</var> ita ſe 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 ſe habere vt <var>.H.A.</var> ad tres quintas ipſius <lb/>Q. (vbi <var>.Y.</var>) ſed iam probatum fuit (vbi. ω) quod <var>.o.a.</var> ad <var>.h.g.</var> erat etiam vt <lb/>quinque ad duo, hoc eſt quod <var>.f.h.</var> erit duæ quintę ipſius <var>.a.b</var>. </s> + <s xml:id="s4639" xml:space="preserve">Quod eſt propoſitum.</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 propoſitione ſecundi lib. de ponderibus Archi. hoc modo intelli<lb/>gendus eſt, vt ſi diceret, <lb/>Sit paraboles <var>.a.</var> cuius baſis ſit <var>.a.c.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.d.e.</var> recta parallela dictæ baſi <var>.a.c.</var> <reg norm="diameterque" type="simple">diameterq́;</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 ipſi <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 ſit et <var>.a.c.</var> baſis, 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> diuiſa erit per æqua <lb/>lia à diametro <var>.b.f</var>. </s> + <s xml:id="s4644" xml:space="preserve">Quare verum dicit ex quinta ſecundi ipſius Pergei hoc eſt 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 poſtea quod diuiſa cum fuerit pars diametri quę inter <var>.d.e.</var> et <var>.a.c.</var> poſita eſt <lb/>(hoc eſt <var>.g.f.</var>) per quinque partes æquales, <reg norm="quarum" type="context">quarũ</reg> partium media ſit <var>.h.k.</var> diuiſa etiam <lb/>imaginatione ſit in puncto <var>.i.</var> ita quod proportio ipſius <var>.h.i.</var> ad <var>.i.K.</var> eadem ſit quæ in-<lb/>ter duo ſolida quorum vnum (illud ſcilicet à quo relatio incipit, hoc eſt antecedens) <lb/>pro ſua baſi teneat quadratum ipſius <var>.a.f.</var> cuius etiam ſolidi altitudo compoſita ſit ex <lb/> + <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> + duplo ipſius <var>.d.g.</var> cum ſimplo <var>.a.f</var>. </s> + <s xml:id="s4646" xml:space="preserve">Aliud verò ſolidum habeat pro ſua baſi quadra-<lb/>tum ipſius <var>.d.g.</var> eius verò altitudo compoſita ſit ex duplo ipſius <var>.a.f.</var> cum ſimplo <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, oſtendet punctum <var>.i.</var> centrum eſſe <lb/>portionis abſciſſę à tota ſectione, quod <reg norm="fruſtum" type="context">fruſtũ</reg> <reg norm="nominatur" type="simple">nominat̃</reg> <reg norm="ſignatum" type="context">ſignatũ</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="ſitque" type="simple">ſitq́;</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 poſt <var>.o.n.</var> <lb/>hoc eſt quod ea proportio quæ eſt ipſius <var>.o.n.</var> ad <var>.n.t.</var> eadem ſit ipſius <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 ſibi 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> verſus <var>.g.</var> alia linea abſciſſa ſit, cui li-<lb/> + <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> + neæ, ita proportionata ſit <var>.f.h.</var> vt <var>.t.m.</var> eſt ad <var>.t.n.</var> quæ quidem linea ſignata ſit <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 poſtea quod diameter <var>.b.f.</var> erit fortaſſe a xis vel aliqua reliquarum diame-<lb/>trorum, quod quidem in .46. primi Pergei videre eſt, cum omnes diametri ſint in-<lb/>uicem paralleli ipſi axi.</s> + </p> + <p> + <s xml:id="s4652" xml:space="preserve">Cum poſtea dicit, quod <var>.a.f.</var> et <var>.d.g.</var> ſunt intentæ ductæq́ue, ibi vult id em infer-<lb/>re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipſius Pergei videre li-<lb/>cet, vnde ex .20. eiuſdem proportio <var>.b.f.</var> ad <var>.b.g.</var> erit vt quadrati <var>.a.f.</var> ad quadratum <lb/>ipſius <var>.d.g.</var> vt ipſe 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">dratũ</reg> <var>.x.n.</var> ex .18. ſexti 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 ipſius <var>.m.n.</var> ad quadratum ipſius <var>.n.x.</var> eandem habebit proportionem, <lb/>quam quadratum ipſius <var>.a.f.</var> ad quadratum ipſius <var>.d.g</var>. </s> + <s xml:id="s4655" xml:space="preserve">Vnde ex .18. & ex communi <lb/><reg norm="ſcientia" type="context">ſciẽtia</reg>, eadem proportio erit ipſius <var>.m.n.</var> ad <var>.n.x.</var> quę ipſius <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 ipſius <var>.m.n.</var> ad cubum ipſius <var>.n.x.</var> erit vt cubi ipſius <var>.a.<lb/>f.</var> ad cubum ipſius <var>.d.g.</var> vt etiam dicit ex communi ſcientia, nec non ex .36. vndecimi.</s> + </p> + <p> + <s xml:id="s4657" xml:space="preserve">Inquit poſtea quod proportio totius ſectionis <var>.a.b.c.</var> ad portionem <var>.d.b.e.</var> eadem <lb/>eſt quæ cubi ipſius <var>.a.f.</var> ad cubum ipſius <var>.d.g.</var> quod verum eſt, vt aliàs tibi monſtraui in <lb/>diuiſione parabolæ ſecundum aliquam propoſitam proportionem.</s> + </p> + <p> + <s xml:id="s4658" xml:space="preserve">Quando autem dicit quod proportio cubi ipſius <var>.m.n.</var> ad cubum ipſius <var>.n.x.</var> eadem <lb/> + <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/> + eſt quæ ipſius <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 ſe <lb/>habebit totalis ſectio <var>.a.b.c.</var> ad portionem <var>.d.b.c.</var> vt <var>.m.n.</var> ad <var>.n.t.</var> & ex .17. eiuſdem ita <lb/>erit ipſius <var>.m.t.</var> ad <var>.t.n.</var> vt fruſti <var>.a.d.e.c.</var> ad ſectionem <var>.d.b.e.</var> quemadmodum ipſe di-<lb/>cit. </s> + <s xml:id="s4660" xml:space="preserve">Sed quia ſuperius, vbi <var>.A.</var> ipſa <var>.f.h.</var> (quæ eſt tres quintæ ipſius <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 ipſius fruſti <var>.a.e.</var> ad ſectionem <var>.d.b.<lb/>e.</var> vt tres quintę ipſius <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 ſupradicti, quod pro ſua baſi habeat <lb/>quadratum ipſius <var>.a.f.</var> altitudinem verò compoſitam ex duplo ipſius <var>.d.g.</var> cum ſimplo <lb/><var>a.f.</var> ad cubum ipſius <var>.a.f.</var> eadem erit quæ dupli ipſius <var>.d.g.</var> cum ſimplo <var>.a.f.</var> ad <var>.a.f</var>. </s> + <s xml:id="s4662" xml:space="preserve">Quod <lb/>quidem verum eſt ex .33. vndecimi & ex prima ſexti.</s> + </p> + <p> + <s xml:id="s4663" xml:space="preserve">Sed ſuperius (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 conuerſa pro portion alitate ita erit ipſius <var>.x.n.</var> ad <var>.n.<lb/>m.</var> vt ipſius <var>.d.g.</var> ad <var>.a.f.</var> ſed dupli <var>.x.n.</var> ad ſimplum <var>.x.n.</var> eſt 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. eiuſdem ita erit <lb/>dupli <var>.x.n.</var> cum ſimplo <var>.m.n.</var> ad <var>.m.n.</var> vt dupli <var>.d.g.</var> cum ſimplo <var>.a.f.</var> ad <var>.a.f</var>. </s> + <s xml:id="s4665" xml:space="preserve">Quare ſolidi <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 inſius <var>.a.f.</var> ex .11. quinti erit vt dupli <var>.x.n.</var> <reg norm="cum" type="context">cũ</reg> ſimplo <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. demonſtratum fuit ita eſſe ipſius <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 poſtea quod eadem proportio erit inter cubum <var>.d.g.</var> & corpus illud quod <lb/>pro baſi habeat quadratum inſius <var>.d.g.</var> altitudinem verò vt dictum eſt, quæ eſt inter <lb/><var>d.g.</var> & compoſitum ex duplo <var>.a.f.</var> cum ſimplo <var>.d.g.</var> quod compoſitum eſt altitudo di <lb/>cta, & <reg norm="verum" type="context">verũ</reg> dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius <var>.a.f</var>. <lb/></s> + <s xml:id="s4669" xml:space="preserve">Quare etiam quemadmodum <var>.t.n.</var> ſe habet ad duplum ipſius <var>.o.n.</var> cum ſimplo <var>.t.n.</var> <lb/>ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de <var>.x.n.</var> cum <var>.m.n</var>.</s> + </p> + <p> + <s xml:id="s4670" xml:space="preserve">Diſponantur <reg norm="nunc" type="context">nũc</reg> omnia tali ordine, ita vt <var>.u.</var> primum ſit corpus quod pro ſua ba <lb/>ſi habeat quadratum ipſius <var>.a.f.</var> & c.</s> + </p> + <p> + <s xml:id="s4671" xml:space="preserve">Et <var>.y.</var> ſit cubus ipſius <var>.a.f.</var> et <var>.s.</var> ſit cubus ipſius <var>.d.g.</var> et <var>.z.</var> ſit corpus quod baſim ha-<lb/>bet quadratum ipſius <var>.d.g.</var> altitudinem verò vt ſupradictum eſt, et <var>.p.</var> ſit compoſitum <lb/>dupli <var>.n.x.</var> cum ſimplo <var>.m.n.</var> et <var>.l.</var> ſit compoſitum dupli ipſius <var>.n.o.</var> cum ſimplo <var>.t.n.</var> <lb/>Sed <var>.u.</var> locata ſit 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 ipſius <var>.u.</var> ad <var>.y.</var> vt <var>.y.</var> ad <var>.m.n.</var> & ipſius <var>.y.</var> ad <var>.s.</var> vt <var>.m.n.</var> ad <var>.<lb/>n.t.</var> quod ſuperius iam demonſtratum fuit, vbi, δ. et <var>.s.</var> ad <var>.z.</var> ita ſe 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 ſe 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 ſe 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 ſe <lb/>habet <var>.h.i.</var> ad <var>.i.K.</var> ita ſe habebit <var>.p.</var> ad <var>.l.</var> vt ipſe 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> ſe habebit ad <var>.h.K.</var> vt quintuplum <lb/>ipſius <var>.p.l.</var> ad <var>.p.l.</var> & ex .22. eiuſdem ita ſe habebit <var>.f.g.</var> ad <var>.i.k.</var> vt quintuplum ipſius <var>.p.<lb/>l.</var> ad <var>.l.</var> quintuplum autem ipſius <var>.p.l.</var> compoſitum eſt ex quintuplo ipſius <var>.n.m.</var> cum <lb/>decuplo ipſius <var>.n.x.</var> cum quintuplo ipſius <var>.n.t.</var> cum decuplo ipſius <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 ſcientia quod <var>.g.f.</var> ad <var>.f.k.</var> eſt ut quintuplum ipſius <lb/><var>p.l.</var> ad duplum ipſius <var>.p.l.</var> eo quod ſuperius ſuppoſitum 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/>poſitum eſt ex duplo ipſius <var>.m.n.</var> cum duplo ipſius <var>.n.t.</var> cum quadruplo ipſius <var>.n.x.</var> & <lb/>cum quadruplo ipſius <var>.x.o</var>.</s> + </p> + <p> + <s xml:id="s4676" xml:space="preserve">Ex conuerſa proportionalitate deinde ita ſe habet, <var>i.K.</var> ad <var>.i.k.</var> ad <var>.f.g.</var> vt <var>.l.</var> ad quin-<lb/>tuplum ipſius <var>.p.l.</var> et <var>.k.f.</var> ad <var>.f.g.</var> vt duplum ipſius <var>.p.l.</var> ad quintuplum ipſius <var>.p.l</var>. </s> + <s xml:id="s4677" xml:space="preserve">Vnde <lb/>ex .24. quinti <var>.i.f.</var> ſe habebit ad <var>.f.g.</var> vt <reg norm="duplum" type="context">duplũ</reg> ipſius <var>.p.l.</var> cum ſimplo <var>.l.</var> ad quintuplum <lb/>ipſius <var>.p.l</var>. </s> + <s xml:id="s4678" xml:space="preserve">Deinde ex conuerſa proportionalitate quintuplum ipſius <var>.p.l.</var> ſe habebit <lb/> + <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/> + ad duplum ipſius <var>.p.l.</var> cum ſimplo <var>.l.</var> vt <var>.f.g.</var> ad <var>.f.i</var>. </s> + <s xml:id="s4679" xml:space="preserve">Sed compoſitum dupli ipſius <var>.p.l.</var> <lb/>cum ſimplo <var>.l.</var> æquale eſt duplo ipſius <var>.m.n.</var> cum quadruplo ipſius <var>.x.n.</var> cum ſexcuplo <lb/>ipſius <var>.o.n.</var> cum triplo ipſius <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 ſumpta fuit <var>.i.r.</var> ad quam ita ſe haberet <var>.f.h.</var> hoc eſt tres quintæ ip-<lb/>ſius <var>.f.g.</var> vt <var>.m.t.</var> ad <var>.t.n</var>. </s> + <s xml:id="s4681" xml:space="preserve">Quare ex conuerſa proportionalitate ita ſe habebit <var>.i.r.</var> ad tres <lb/>quintas ipſius <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> ſumpta fuit æqualis ipſi <var>.b.g.</var> et <var>.m.n.</var> ipſi <lb/><var>b.f.</var> ideo <var>.m.o.</var> ex communi ſcientia æ qualis erit ipſi <var>.g.f</var>. </s> + <s xml:id="s4683" xml:space="preserve">Vnde proportio <var>.r.i.</var> ad tres <lb/>quintas ipſius <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 ſe habere <var>.i.f.</var> ad <var>.f.g.</var> vt duplum <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.p.l.</var> cum ſim-<lb/>plo <var>.l.</var> ſe habet ad quintuplum ipſius <var>.p.l.</var> hoc eſt <var>.i.f.</var> ad <var>.m.o.</var> vt duplum ipſius <var>.p.l.</var> cum <lb/>ſimplo <var>.l.</var> ad quintuplum ipſius <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/>propoſitione ſupponit. </s> + <s xml:id="s4686" xml:space="preserve">Vnde ex rationibus ibi allegatis ſequitur <var>.f.r.</var> eſſe duas quin-<lb/>tas ipſius <var>.m.n.</var> hoc eſt ipſius <var>.f.b</var>. </s> + <s xml:id="s4687" xml:space="preserve">Quapropter punctum <var>.r.</var> centrum erit ponderis to-<lb/>tius ſectionis parabolæ ex .8. ſecundi lib. de ponderibus eiuſdem Archimedis.</s> + </p> + <p> + <s xml:id="s4688" xml:space="preserve">Inquit nunc Archimedes, quod exiſtente <var>.q.</var> centro ponderis ipſius parabolæ <var>.d.<lb/>b.e.</var> partialis, centrum fruſti 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 ipſius <var>.r.f.</var> quæ reperitur inter centrum <var>.r.</var> & centrum <lb/>huius fruſti æqualis eſt proportioni totius parabolæ ad partialem. </s> + <s xml:id="s4689" xml:space="preserve">Quod quidem ve <lb/>rum eſt ex .8. primi libri eiuſdem.</s> + </p> + <p> + <s xml:id="s4690" xml:space="preserve">Inquit etiam punctum <var>.i.</var> illud eſſe, eo quod cum probatum ſit <var>.f.r.</var> duas quintas eſ-<lb/>ſe ipſius <var>.f.b.</var> ideo <var>.b.r.</var> tres quintas erit ipſius <var>.b.f.</var> vt ipſe 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> ſimiliter tres quintæ eſt ipſius <var>.d.b.</var> ex .8. prædicta. </s> + <s xml:id="s4692" xml:space="preserve">Quare <var>.q.r.</var> tres quintæ <lb/>erit ipſius <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 ſe habeat vt abſciſ-<lb/>ſum <var>.b.g.</var> ad abſciſſum <var>.q.b.</var> ex .7. et .8. dicti primi libri eiuſdem ideo reſiduum <var>.f.g.</var> ex <lb/><var>f.b.</var> ad reſiduum <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 ſub. β. probauimus ita ſe habere fruſtum <var>.a.d.e.c.</var> ad parabolam <var>.d.b.e.</var> vt <lb/><var>m.t.</var> ad <var>.t.n.</var> ſed vt <var>.m.t.</var> ad <var>.t.n.</var> ita aſſ umpta fuit (vbi <var>.A.</var>). <var>i.r.</var> ad quam ſic ſe haberet <var>.f.<lb/>h.</var> hoc eſt tres quintæ ipſius <var>.f.g.</var> hoc eſt <var>.q.r</var>. </s> + <s xml:id="s4695" xml:space="preserve">quare ex .11. quinti prop ortio fruſti <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">Exiſtente igitur <var>.r.</var> centro totius pa <lb/>rabolæ et <var>.q.</var> centro partialis, ergo <var>.i.</var> centrum erit fruſti propoſiti.</s> + </p> + <p> + <s xml:id="s4697" xml:space="preserve">Sed ſi nullo ſolido intercedente, voluerimus centrum <var>.i.</var> fruſti <var>.a.e.</var> citius inuenire, <lb/>inueniemus primò centrum <var>.r.</var> totius figuræ ex .8. ſecundi eiuſdem conſtituendo <var>.b.r.</var> <lb/>tres quintas totius axis <var>.b.f.</var> & centrum <var>.q.</var> parabolæ <var>.d.b.e.</var> partialis ſimiliter.</s> + </p> + <p> + <s xml:id="s4698" xml:space="preserve">Nunc igitur manifeſtum eſt nobis, eandem proportionem fore ipſius <var>.q.r.</var> <lb/>ad <var>.r.i.</var> quæ fruſti <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 ſe habebit <var>.q.i.</var> ad <var>.i.r.</var> vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ſed vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ita ſe <lb/>habet <var>.m.n.</var> ad <var>.n.t.</var> eo quod vnaquæque harum duarum proportionum ſeſquialtera <lb/>eſt proportioni <var>.f.b.</var> ad <var>.b.g.</var> eo. quod <var>.f.b.</var> ad <var>.b.g.</var> ita ſe 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 ſe habebit vt <var>.g.i.</var> ad <var>.r.i.</var> vnde diſiunctim <var>.m.t.</var> ad <var>.t.n.</var> ita ſe 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 ſe habeat ad <var>.r.q.</var> vt <var>.t.n.</var> ad <var>.t.m.</var> vt <lb/>habeatur centrum fruſti.</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">Illuſtri Domino Horatio Muto.</head> + <p> + <s xml:id="s4702" xml:space="preserve"><emph style="sc">INter</emph> ea quæ olim contra Antonium Bergam, ſermone Italico ſcripſi, <lb/>hoc vnum erat, quod ip ſe Berga non viderat quendam notatu dignum <lb/>errorem ipſius Pi ccolhominei, vbi ipſe 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 ipſum putare eandem proportionem inter diametros, <lb/>quæ inter ſphæras ipſas eſſe, nec amplius recordari eius quod ſcripſerat pag .24.</s> + </p> + <p> + <s xml:id="s4704" xml:space="preserve">Piccolhom. igitur ibi <reg norm="ſupponens" type="context">ſupponẽs</reg> centrum <var>.D.</var> eſſe magnitudinis aquæ, & intra ſphæ <lb/>ram terreſtrem, putat omnino cauſam eſſe vt terra ſuperet aquam magnitudine, qua-<lb/>ſi quod ſi punctum <var>.D.</var> vt centrum ſphæræ aquæ, vnum <reg norm="idemque" type="simple">idemq́;</reg> eſſet cum puncto <var>.E.</var> <lb/>extremo diametri ipſius terræ, ſphæra <var>.A.G.H.</var> ſphæræ <var>.A.B.E.</var> dupla eſſe deberet, <lb/>quod quidem nullo pacto fieri poteſt, quamuis etiam proportio <var>.A.H.</var> ad <reg norm="diametrum" type="context">diametrũ</reg> <lb/><var>A.E.</var> ſuperbipartiensſeptimas exiſteret, quæ minor eſſet quam ſeſquitertia, ita quod <lb/>quando etiam <var>.D.E.</var> maior medietate ipſius <var>.D.H.</var> fuiſſet, nihilominus tamen <lb/>terra minor eſſet aqua, eo quod proportio dupla minor eſt, quam tripla ad <reg norm="propor- tionem" type="context">propor-<lb/>tionẽ</reg> ſuperbipartientenſeptimas, & maior <reg norm="quam" type="context">quã</reg> tripla ad proportionem ſeſquiquar-<lb/>tam. </s> + <s xml:id="s4705" xml:space="preserve">Vnde ſi Piccolhom. ſuppòſuiſſet proportionem ipſius <var>.D.H.</var> ad <var>.C.E.</var> eſſe <lb/>ſeſquiquartam, rectè profectò dixiſſet, ſed dicere quod ubicunque exiſtat <reg norm="punctum" type="context">punctũ</reg> <var>.<lb/>D.</var> intra ſphæram terreſtrem, ſequitur ipſam eſſe maiorem aquea, verum non eſt.</s> + </p> + <p> + <s xml:id="s4706" xml:space="preserve">Scripſi 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 exiſtimare non poteſt, quod vmbra quæ facit ori-<lb/>ri e cclipſes Lunæ, producta ſit à terra, & ab aqua ſimul, vt ab vno corpore aggre-<lb/>gato exijs duobus elementis, & ad vnam communem ſphæreceitatem reductis, pro <lb/>pterea quod cum vmbra produci debeat à corporibus opacis, quorum opacitas effi-<lb/>cit illa corpora vmbroſa, aqua autem, ſit corpus diaphanum, & tranſparens, 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 eſt, quod ra-<lb/>dius luminoſus non poteſt multum in profundum mergi, vt probaui in .8. epiſtola ad <lb/>Vimercatum, altera verò eſt, quod cum ſphærica ſit aqua maris, ſupponatur etiam <lb/>quod ſub ea nulla terræ portio eſſet, & quod radij ſolares ipſam, non ſecus ac pilam <lb/>ex criſtallo fabręfactam penetrarent, cum autem ipſi radij, tam ab una, quam ab alia <lb/>parte ſup erficiei huiuſmodi globi <reg norm="frangantur" type="context">frãgantur</reg>, ob diſſimilem diaphaneitatem inter ae <lb/>rem & aquam, ipſi ſeinuicem interſecarent, vt poſt pilam criſtallinam videre eſt, de-<lb/>inde procedentes, diſgregarentur, <reg norm="diſciparenturque" type="simple">diſciparenturq́;</reg> quouſque nullam vim illuminatio <lb/>nis haberent, quod quilibet experiri poterit mediante aliquo vaſe uitreo ſphærico, <lb/>aqua pleno, cuiuſuis magnitudinis, ſoli expoſito.</s> + </p> + <p> + <s xml:id="s4709" xml:space="preserve">Rationes etiam quas eodem loco Piccolho. adducit ad probandum quod ſi quis <lb/>in fundo maris exiſteret, nullum uideret lumen, nihil ualent. </s> + <s xml:id="s4710" xml:space="preserve">Quarum prima eſt, <lb/>ubi ita dicit.</s> + </p> + <quote> + <s xml:id="s4711" xml:space="preserve">Ille qui ſe in aquam mergit, cum maiorem lucem, quæ ſupra aquam eſt, relin-<lb/>quat, iudicat pro magno temporis ſpatio locum illum obſcurum, quemadmodum <lb/>accidit quando per multum temporis ſpatium fixis oculis in corpore Solis intuiti ſu <lb/>mus, ab eodem poſtea eoſdem amouentes, omnia obſcura nobis videntur.</s> + </quote> + <p> + <s xml:id="s4712" xml:space="preserve">Ipſe autem non conſiderat quod talis obſcuritas quæ ſequitur viſionem maioris <lb/>luminis, parum durat, immo cito euaneſcit, ſed in aqua nunquam reuertimur ad vi-<lb/>dendum, ne que veſtigium aliquod luminis ibi videtur, in fundo maris dico, quem-<lb/>admodum nobis nuntiauerunt hi qui margaritas expiſcantur in imis partibus ingen <lb/>tium æquorum indicorum.</s> + </p> + <p> + <s xml:id="s4713" xml:space="preserve">Secunda uerò ratio ipſius Piccolhom. eſt ubi ita dicit.</s> + </p> + <quote> + <s xml:id="s4714" xml:space="preserve">Altera cauſa quod nobis obſcurus appareat locus ſub aqua, eſſe poteſt obſtacu-<lb/>lum quod aquæ habent ab opacitate terræ ſub eorum fundo, etenim ſicut <reg norm="chriſtallum" type="context">chriſtallũ</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="perſpicuum" type="context">perſpicuũ</reg> ſiue <reg norm="tranſparens" type="context">tranſparẽs</reg> ſit, nihilominus propter obſtaculum plumbi ſub <lb/>ipſo poſiti, efficit vt radij viſuales repercuſſi reuertantur. ita etiam quamuis aqua ſit <lb/>corpus tranſparens, nihilominus propter obſtaculum terræ opacæ, quæ ſubſidet in <lb/>fundo maris efficere poteſt obſcuras partes illas ſub aqua, illis hominibus qui in <lb/>ipſa aqua mèrguntur.</s> + </quote> + <p> + <s xml:id="s4715" xml:space="preserve">In hac ſecunda ratione decipitur Piccolhom. </s> + <s xml:id="s4716" xml:space="preserve">Primum quia ſi vſque ad imam par <lb/>tem maris, Solis radius ferri poſſet, ille qui ibi eſſet, attollens oculos ſurſum Solem <lb/>cerneret, </s> + <s xml:id="s4717" xml:space="preserve">deinde aſpiciendo ipſum fundum Maris, videret illum, ratione reflexio-<lb/>nis luminis ab ipſo fundo, & ex eadem ratione ſpeculi ab ipſo adducta, quæ contra <lb/>ipſum eſt.</s> + </p> + <p> + <s xml:id="s4718" xml:space="preserve">Decipitur etiam cum dicatradios viſuales à ſpeculo ſeu plumbo repercuti, eo <reg norm="quod" type="simple">ꝙ</reg> <lb/>non radij viſuales ſunt hi qui reflectuntur, ſed ſunt radij luminoſi primarij, ſeu ſecun <lb/>darij qui non ab oculis exeunt ſed à corpore lucido.</s> + </p> + <p> + <s xml:id="s4719" xml:space="preserve">Scripſi etiam quod ſi verum eſſet proportionalitatem continuam <reg norm="quantitatum" type="context">quãtitatum</reg> ele-<lb/>mentorum ex proportione decupla conſtare, ignem pro maximo, terram verò pro <lb/>minimo terminorum ſumentes, totum aggregatum ex terra, aqua, aere, & igne, ita <lb/>eſſet maius terra, quemadmodum mille centum & vndecim ad vnum, vnde ſemidia <lb/>meter regionis elementaris eſſet quaſi aut paulo maior decuplo ſolum ſemidiame-<lb/>tro terræ, vnde inter conuexum ignis, & concauum minimi, ſeu inferioris orbis lu-<lb/>naris, relinqueretur quidam orbis vacuus ſpiſſitudinis vnius interualli plus quam vi-<lb/>ginti terræ ſemidiametrorum, quod ſpatium vacuum orbiculariter, maius exi-<lb/>ſteret ipſa totali regione elementari plus quam trigeſies millies, immo ſi ſemidia <lb/>meter dicti primi orbislunaris maior eſſet terreſtri vt trigintanouem ad unum, <reg norm="dictus" type="simple">dictꝰ</reg> <lb/>orbis vacuus maior eſſet elementari regione plus quam .58208. ad vnum, proportio <lb/>nalitatem igitur continuam quæ ex decupla proportionalitate reſultat in elementis <lb/>eſſe putare eſt 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 ſolidis (quaſi Bergę ſint quædam corpora quæ ſolida non extent) poſſit <lb/>dari medium æquale in geometrica proportione.</s> + </p> + <p> + <s xml:id="s4721" xml:space="preserve">Sed vbi Plato ad ſermonem de numero elementorum ſe confert, poſtquam ra-<lb/>tione creationis ignis, & terrę ſe propoſuiſſe 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 ſuperficialium primò, deinde in quatuor, ratione corporearum <lb/>vtitur, ita dicens.</s> + </p> + <quote> + <s xml:id="s4722" xml:space="preserve">Vinculorum verò ideſt aptiſſimum atque pulcherrimum quod exſe, & ex ijs quę <lb/>aſtringunt, 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 inſerre vult de proportionalitate geometrica trium termino-<lb/>rum, in qua ijdem ita ſe habent, vt medius, primi, <reg norm="vltimique" type="simple">vltimiq́;</reg> vice fungatur, ita vt vtriuſ-<lb/>que ipſorum extremorum particeps fiat, cum productum quod à medio termino in <lb/>ſeipſo progignitur idem ſit ei quod ab extremis fuit, vnde medius, potentia idem eſt <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 ſe habet <lb/>ad poſtremum vt primum ad medium, <reg norm="viciſſimque" type="simple">viciſſimq́;</reg> vt poſtremum cum medio, ita me-<lb/>dium cum primo congruit, </s> + <s xml:id="s4726" xml:space="preserve">tunc quod medium eſt, & primum fit & <reg norm="poſtremum" type="context">poſtremũ</reg>, po-<lb/>ſtremum 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">animaduertendũ</reg> eſt 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 cenſent, <lb/>quia de numeris, magnitudinibus, <reg norm="viribusque" type="simple">viribusq́;</reg>, aut ut dici ſolet, virtutibus mentionem <lb/>fecerit. </s> + <s xml:id="s4728" xml:space="preserve">Plato enim nihil aliud inferre voluit, quam eandem paſſionem (ut ipſe reci-<lb/>tat) inter medium <reg norm="extremaque" type="simple">extremaq́;</reg> vnius proportionalitatis continuæ geometricæ, tam <lb/>in quantitate, quam in qualitate <reg norm="reſultaturam" type="context">reſultaturã</reg>, cum tres termini eiuſdem eſſent ſpe-<lb/>ciei, & quia quantitas in duas principes <reg norm="primariasque" type="simple">primariasq́;</reg> partes, ideſt in continuam, & <lb/>diſcretam diuiditur, hanc ob cauſam Plato hoc præcipuè ſignificat numerorum ma-<lb/><reg norm="gnitudinisque" type="simple">gnitudinisq́;</reg> vocabulis vtens, quibus vniuerſum quantitatis genus complectitur.</s> + </p> + <p> + <s xml:id="s4729" xml:space="preserve">Cum verò ait vires, uniuerſum qualitatis genus inferre uult. </s> + <s xml:id="s4730" xml:space="preserve">Quia proportio & <lb/>proportionalitas tam continua quam diſcreta, non ſolum interterminos quanti, ſed <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">cũ</reg> geometrica magis ſim <lb/>bola eſt quam cum Arithmetica Plato minime intelligat, ex eiuſdem uerbis cum ita <lb/>ſcribit manifeſtè patet.</s> + </p> + <quote> + <s xml:id="s4732" xml:space="preserve">Quando enim medium ita ſe habet ad poſtremum ut primum ad medium, <reg norm="uiciſ- ſimque" type="simple">uiciſ-<lb/>ſimq́;</reg> ut poſtremum 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/>poſtremum, & non ad medium, ita ſe habet geometricè ut differentia inter primum <lb/>& medium ad differentiam inter medium & ultimum.</s> + </p> + <p> + <s xml:id="s4734" xml:space="preserve">Quod ſi clarum eſt ipſum 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 eſt & primum fit & poſtremum, poſtremum 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 oſtendere uult, quam ſimilitudinem quæ inter huiuſmodi medium & <lb/>extrema intercedit, cum ipſum medium ad poſtremum, quem primus ad ſeipſum, <lb/>eundem reſpectum habeat, in quo eſt ſimilis primo, & contra ad primum <reg norm="terminum" type="context">terminũ</reg>, <lb/>eundem reſpectum, quem poſtremum ad ſeipſum habet, </s> + <s xml:id="s4738" xml:space="preserve">unde hac ratione ultimum <lb/>repręſentat, uolens Plato inferre de conuenientia quę inter media elementa, & ex-<lb/>trema intercedit, ut aquæ inter aerem, & terram, cum aqua, ratione ſuæ frigiditatis, <lb/>terrę, ratione uero ſuæ humiditatis aeri ſimilis euadat. </s> + <s xml:id="s4739" xml:space="preserve">Aer uero qui inter ignem, <lb/><reg norm="aquamque" type="simple">aquamq́;</reg> ponitur quod ad caliditatem attinet cum igne, quod uero ad humidita-<lb/>tem ſpectat cum aqua communicet.</s> + </p> + <p> + <s xml:id="s4740" xml:space="preserve">Sed quia Plato multis in rebus doctrinam Pythagoricam ſequutus eſt, Pythago-<lb/>rici aut em omnia numeris metiebantur, & de omnire ſecundum numerorum ratio <lb/>nem diſſerebant, <reg norm="uidensque" type="simple">uidensq́;</reg> Plato quod inter duos numeros ſuperficiales, <reg norm="inuicemque" type="simple">inuicemq́;</reg> <lb/>ſimiles exiſtentes, unum tantum numerum medium in proportionalitate continua <lb/>geometrica cadere poteſt, ideo ſubiungit.</s> + </p> + <quote> + <s xml:id="s4741" xml:space="preserve">Quod ſi uniuerſi corpus latitudinem habere debuiſſet, nullam uerò profundita-<lb/>tem, unum ſanè, tum ad ſeipſum, tum ad extrema uincienda interiectum medium <lb/>ſuffeciſſet.</s> + </quote> + <p> + <s xml:id="s4742" xml:space="preserve">Sequitur poſtea ſic.</s> + </p> + <quote> + <s xml:id="s4743" xml:space="preserve">Sed cum ſoliditatem mundus requireret, ſolida uerò non uno, ſed duobus ſem-<lb/>per modis copulentur, inter ignem, & terram, Deus, Aerem, Aquamq́ue 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 ſolidos, & inuicem <lb/>ſimiles, <reg norm="vnus" type="simple">vnꝰ</reg> <reg norm="tantum" type="context context">tãtũ</reg> medius proportionalis intercedere <reg norm="non" type="context">nõ</reg> poteſt, ſed duo neceſſariò re <lb/>quiruntur (vt exijs quæ Euclid .8. lib. 16. 17. 18. et .19. propoſitione proponit viden<lb/>tur) ita <reg norm="dictante" type="context">dictãte</reg> ratione inter igneum, <reg norm="terreumque" type="simple">terreumq́;</reg> corpus duo corpora interiecta <reg norm="eſsent" type="context">eſsẽt</reg>, <lb/>non ratione proportionalitatis continuæ in quantitate <reg norm="eorundem" type="context">eorũdem</reg> corporum, ſed pro <lb/>pter ſimilitudinem connexionis, cum productum ex duobus medijs proportionali-<lb/>bus æquale ſit producto ab extremis, & idem reſpectus, quem primum ipſorum qua <lb/>tuor ad ſecundum habet, ſecundi ad tertium extet, vnde ſecundum primo ſimile <lb/>euadit, & contra, reſpectus qui eſt quarti ad tertium, ſit etiam tertij ad ſecundum, vn-<lb/>de ipſum tertium, ratione vltimi ſubit, & eius imaginem induit, & hanc ob cauſam <lb/>ſic ſcribit Plato.</s> + </p> + <quote> + <s xml:id="s4745" xml:space="preserve">Propterea ex huiuſmodi rebus numero quaternario concluſis, mundi corpus con <lb/>flatum eſt, ea connexum comparatione qua dixi. </s> + <s xml:id="s4746" xml:space="preserve">Ex quo ſeipſum 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 cuiuſuis generis eſſe noluiſſe, clarè perſpicitur, ſed huiuſmo <lb/>di ſimilitudine, in eo quod media elementa cum extremis conueniunt eſt vſus, quæ <lb/>quidem conuenientia, nullibi maior, quam in proportionalitate continua geome-<lb/>trica reperitur. </s> + <s xml:id="s4748" xml:space="preserve">Sed etiam ſi Plato de huiuſmodi corporea elementorum magnitu-<lb/>dine ſeipſum intelligi voluiſſet, ſi ſemidiameter regionis elementaris ex ęquo vt .39 <lb/>ad vnum, reſpectu ſemidiametri terræ fuiſſet, aqua, ipſam terram, magis quam tri-<lb/>geſies, & octies, non ſolum decies, & aer quoque eandem magis quam .1500. & <lb/>ignis magis quam .55000. partibus magnitudine ſuperaret.</s> + </p> + <p> + <s xml:id="s4749" xml:space="preserve">Subſtantia vero rerum quas ſcripſeram circa finem illius conſiderationis talis fuit.</s> + </p> + <p> + <s xml:id="s4750" xml:space="preserve">Nunc autem tempus eſſe videtur, vt ego etiam, ne tantum deſtruxiſſe, ſed etiam <lb/>conſtruxiſſe videar aliquid pro veritate diſſeram.</s> + </p> + <p> + <s xml:id="s4751" xml:space="preserve">Non eſt igitur dubium, ſolidæ doctrinæ viris, quin præſtantiſſimus Piccolo. ſe-<lb/>cutus ſit tutam viam ad explorandum, quod terra maior ſit quam aqua, metiendo <lb/>vtriuſque horum corporum ſuperficiem detectam. </s> + <s xml:id="s4752" xml:space="preserve">Omittamus autem compenſa-<lb/>tionem illam curuitatis, & concauitatis vallium, & montium, <reg norm="&c." type="unresolved">&c.</reg> quam ipſe Piccolo. <lb/>propè finem ſexti cap. vellet dare fluminibus, ſtagnis, fontibus, & eiuſmodi aquis. <lb/></s> + <s xml:id="s4753" xml:space="preserve">eo enim in loco labitur Piccolo. vbi non conſiderat, quod eiuſmodi obliquis ſuper-<lb/>ficiebus non reſpondent anguli ſolidi centri ſphæræ, qui reſpiciunt eorum baſim ad <lb/>rectos angulos. </s> + <s xml:id="s4754" xml:space="preserve">Sed poſtquam Piccolo. comperit ſuperficiem terrę detectam, eſſe <lb/>maiorem apparente ſuperficie ſphærica aquæ, proculdubio poterat concludere ter-<lb/>ram eſſe maiorem aqua, ſicuti fecit, <reg norm="etiam" type="context">etiã</reg> ſi aqua profunda eſſet pyramidaliter <reg norm="vſque" type="simple">vſq;</reg> <lb/>ad mundi centrum, ideſt .3500. milliaria, ſupponendo tantum eſſe huius globi ſemi <lb/>diametrum.</s> + </p> + <p> + <s xml:id="s4755" xml:space="preserve">Verum quia poſſet aliquis dubitare circa diligentiam Piccolo. in hiſcæ duabus ſu <lb/>perficiebus dimetiendis, viſum eſt mihi non alienum ſequi aliam viam pro hac veri <lb/>tate probanda, ſupponendo verum eſſe, quod non vnus ſolus metitus fuerit, ſed mul <lb/>ti, ideſt ſupponendo <reg norm="verum" type="context">verũ</reg> eſſe quod maris profunditas menſurari poſſit, & præterea, <lb/>quod non modo ipſius maris maxima profunditas non perueniat ad quingentos paſ <lb/>ſus, ſicuti refert Piccolo. in fine ſui tractatus, & mihi aſſeruerunt Hiſpani multi, & <lb/>Luſitani præſtantiſſimi nautæ, tum Venetijs, tum Parmæ, in Aula Sereniſſimæ quon <lb/>dam Principis, inter quos, Venetijs fuit Illuſtris Rodericus Guzmanus, Dominus <lb/>Franciſcus Lopes, Dominus Garzias de Seuilia, <reg norm="multique" type="simple">multiq́;</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="recenſerem" type="context">recenſerẽ</reg> moleſtum eſſet. </s> + <s xml:id="s4757" xml:space="preserve">Sed etiam ſupponendo quod maxima <lb/>pelagi profunditas ſit, non modo .500. paſſuum, ſed etiam .500. millium paſſuum, vt <lb/>dixi, & quod mare ſit huius profunditatis, non vno in loco tantum, aut multis, ſed <lb/>quod ſupra totam etiam faciem terræ, mare tantę profunditatis ipſam terram vn-<lb/>dique operiret, ideſt, quod vbicunque nunc terra detecta eſt, eſſet aqua, ſpiſſitudi-<lb/>nis .500. millium paſſuum. </s> + <s xml:id="s4758" xml:space="preserve">Atque vt planius intelligar ſupponendo quod ſicuti to-<lb/>tus huius globi ſemidiameter eſt <reg norm="milliarium" type="context">milliariũ</reg> .3500. </s> + <s xml:id="s4759" xml:space="preserve">Terreſtris partis ſemidiameter <lb/>eſſet <reg norm="tantum" type="wordlist">tm̃</reg> .3000. & reliquum ſemidiametri, id eſt quingenta milliaria eſſet craſſitudo <lb/>ſiue profunditas orbis aquei, in quo nihil neceſſe eſſet laborare in dimetiendis fon-<lb/>tibus, fluminibus, lacubus, ſtagnis, paludibus, & huiuſmodi particulis nullius momen <lb/>ti apud peritos, nec curare ſubterraneas aquas cauernarum, aut aliorum terræ cauo-<lb/>rum, ſeu terræ porroſitatum, quæ omnia ſunt circa ipſius terræ ſuperficiem. </s> + <s xml:id="s4760" xml:space="preserve">Quia ve <lb/>riſimile non eſt naturam eiuſmodi caua ſiue ſpong oſitates produxiſſe demiſſius li-<lb/>bramenti maris. </s> + <s xml:id="s4761" xml:space="preserve">Supponendo igitur ea quæ nunc dicta ſunt, terra tamen eſſet ferè <lb/>duplo maior aqua, hoc eſt, vt .12. ad .7. </s> + <s xml:id="s4762" xml:space="preserve">Quod quidem, cuiuis mathematicæ philoſo-<lb/>phiæ mediocriter perito, ſupputatu facillnnum eſt. </s> + <s xml:id="s4763" xml:space="preserve">Cum proportio diametrorum, <lb/>ſeu ſemidiametrorum, tertia pars exiſtat proportionis eorundem ſphærarum. </s> + <s xml:id="s4764" xml:space="preserve">Sed <lb/>vt parum periti minore labore ſupputare poſſint.</s> + </p> + <p> + <s xml:id="s4765" xml:space="preserve">Primum ſciendum eſt, quod ſupponendo diametrum globi, ex terra, & aqua com <lb/>poſiti, eſſe .3500. milliarium, & ſemidiametrum puræ terreſtris partis eſſe .3000. tan <lb/>tum, eiuſmodi proportio erit ut .7. ad .6. quia communis maior numerator horum <lb/>duum ſemidiametrorum erit .500. qui in maiorem ingredietur ſepties, in minorem <lb/>a utem ſexies. </s> + <s xml:id="s4766" xml:space="preserve">Et eiuſmodi proportio ſuperparticularis, vocatur ſeſquiſexta, cuius <lb/>triplum erit vt .57. cum ſexta parte ad .36. & idem erit inter dictum globum compo-<lb/>ſitum, & partem terreſtrem ſimplicem. </s> + <s xml:id="s4767" xml:space="preserve">Quare ſubtrahendo puram, ſeu ſimplicem <lb/>partem terreſtrem, ex compoſito, reliqua pars erit, vt .21. cum ſexta, pro quantitate <lb/>aquei orbis, ad quam, terreſtris 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> ſolum de parte illa maximæ e ius profunditatis, quæ nuſquam <lb/>ad quingentos paſſus peruenit, ſed de ficto illo orbe aqueo, profunditatis .500. paſ-<lb/>ſuum, qui totum terreſtrem orbem circundaret, & tegeret, ſupponendo quod per <lb/>quingentos paſſus profunditatis, quidquid eſt terra, eſſet aqua, ideſt ſuppoſito quod <lb/>ex totius orbis compoſiti ſemidiametro exiſtente .3500. milliarium, purę terræ ſemi <lb/>diameter eſſet milliarium .3499. cum dimidio. </s> + <s xml:id="s4769" xml:space="preserve">Supponendo igitur, vt ſupradixi. <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 eiuſmodi fictum orbem aqueum, quæ quidem partes, <lb/>ſunt ampliores ſuperficię aquæ, vt obſeruauit Piccolo. atque alij <reg norm="præſtantes" type="context">præſtãtes</reg> viri, ideo <lb/>ſequetur, vt terra ſit 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 compoſitum circundant: </s> + <s xml:id="s4773" xml:space="preserve">reſpondeo quod <lb/>non modò ei concedo computari eiuſmodi aquam, ſed ſupponendo etiam quodto <lb/>tus locus à vaporibus occupatus, qui attolluntur .52. milliaria ſupra ſuperficiem <lb/>huius globi, vt iam ſupradictum eſt, totus eſſet aqueus, & amplius, ſupponendo quod <lb/>orbis hic aqueus eſſet ſpiſſitudinis, ſiue altitudinis quingentorum milliarium ſupra <lb/>totum ipſum globum compoſitum. </s> + <s xml:id="s4774" xml:space="preserve">Tamen terra eſſet maior ipſa aqua ferè duplo; <lb/></s> + <s xml:id="s4775" xml:space="preserve">qua dere, quiſque eiuſmodi ſupputationum 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 ſolum maiorem eſſe aqua, ſed aqua & præte-<lb/>rea aere, ſi aer non tam altè pertingit, quam multi alij præter Piccolo. ſentiunt, qui <lb/>dicuntinde euenire quod aerea humiditas non tam altè aſcendere poteſt, quoniam <lb/>humiditas ipſa grauitatem ſecum affert, præterquam quod nubium ſitus oſtendit ſu <lb/>pra eas materiam eſſerariorem quam ſint ipſę nubes, infra vero denſiorem. </s> + <s xml:id="s4777" xml:space="preserve">Corpo-<lb/>ra enim eouſque aſcendunt donec inueniunt conſtitutionem mediam formæ æqua-<lb/>lis (vt ita dicam) ſuis. </s> + <s xml:id="s4778" xml:space="preserve">Quare materia illa quæ impropriè ignis vocatur (non enim <lb/>eſt ignis) incipit carere humiditate (qua mediante aer definitur) circa quinquage-<lb/>ſimum ſecundum milliarium ſupra ſuperficiem terræ, vt iam ſupradixi à Vitellione <lb/>demonſtratum fuiſſe. </s> + <s xml:id="s4779" xml:space="preserve">Ariſto. autem affert <reg norm="rationem" type="context">rationẽ</reg> quare nubes altius <reg norm="non" type="context">nõ</reg> <reg norm="tranſcendant" type="context context">tranſcẽdãt</reg>. <lb/></s> + <s xml:id="s4780" xml:space="preserve">Vnde apparet tertiam aeris regionem impropriè aerem appellari, ſi humiditate ca-<lb/>ret, vt ait Ariſt. qua mediante aer definitur, immo potius retinet ignis naturam, vt <lb/>etiam aſſerunt interpretes Ariſtotelis in primum Meteororum. </s> + <s xml:id="s4781" xml:space="preserve">Qui Ariſto. in locis <lb/>ſupra citatis itidem oſtendit ſe etiam huius modi eſſe opinionis.</s> + </p> + <p> + <s xml:id="s4782" xml:space="preserve">Quod autem attinet ad probandum quod ſuperficies terrę detecta ſit altior quam <lb/>ſuperficies detecta aquæ, id tam clarum eſt ſua ſponte philoſophis, qui ſciunt quid ſit <lb/>altum, quidue demiſſum, quod ſuperfluum eſſet quidquid ſuper hoc dicerem præ <lb/>terquam, quòd conſtat ex demonſtratione ab Ariſto. ſacta textu 31. li .2. de cœlo, in <lb/>quo agit de corporibus in aqua poſitis, vnde eiuſmodi veritas planiſſimè aperitur. <lb/></s> + <s xml:id="s4783" xml:space="preserve">Omittimus etiam quod præſtantes Moderni omnes, eam pro manifeſtiſſima <reg norm="ponunt" type="context">ponũt</reg>, <lb/>ſicutiapud omnes ſani iudicij homines reuera exiſtimatur.</s> + </p> + <p> + <s xml:id="s4784" xml:space="preserve">Hæc enim ſunt quæ in fine illius conſiderationis ſcripſeram.</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 Auguſtino Michele, Patritio Veneto, ad cor <lb/>roborandam opinionem antiquorum, vbi tot authoritates, <reg norm="totque" type="simple">totq́;</reg> rationes adducit, <lb/>vt nil amplius dici poſſit. </s> + <s xml:id="s4786" xml:space="preserve">Atego ſenſum, <reg norm="rationemque" type="simple">rationemq́;</reg>, & non authoritatem <reg norm="aliquam" type="context">aliquã</reg> <lb/>ſequutus ſum: </s> + <s xml:id="s4787" xml:space="preserve">cum verò dico ſenſum, de ſenſu illorum intelligo, qui profunditatem <lb/>maris metiti ſunt, vt non mihi ſolum, ſed, & Piccolo. & alijs permultis retulerunt, <lb/>de ratione vero à me adducta, aliorum ſit iudicium.</s> + </p> + <quote> + <s xml:id="s4788" xml:space="preserve">Sediſte mirabilis & Excellentiſſimus vir, verba mea non accepit in eo ſenſu, vt <lb/>ego ſcripſi, ita vt omnino alienas conſequentias ſibi confingat, <reg norm="quemadmodum" type="context">quemadmodũ</reg> pag. <lb/>3. ſui tractatus inquit, me non concedere naturam produxiſſe in magna quantitate, <lb/>atque immenſa, id totum, quod bonum, & neceſſarium eſt. </s> + <s xml:id="s4789" xml:space="preserve">Hanc enim conſequen <lb/>tiam ipſe colligit ex eo, quod ego pag .19. meæ conſiderationis contra Antonium <lb/>Bergam ſcripſeram, quod videntur multa corpora alijs nobiliora, nihilominus mi-<lb/>nora, eo quod quantitas non ſequitur nobilitatem, neque ab ea pender, ita vt res <lb/>illa quæ nobilior eſt, neceſſarium ſit vt etiam maior exiſtat. </s> + <s xml:id="s4790" xml:space="preserve">Sed Excellentiſſimus <lb/>iſtæ vir ſcribit ita me dixiſſe.</s> + </quote> + <p> + <s xml:id="s4791" xml:space="preserve">Multa immo infinita corpora ſunt nobilia, & neceſſaria, nihilominus ſunt paruę <lb/>molis.</s> + </p> + <p> + <s xml:id="s4792" xml:space="preserve">Vide igitur quantum hoc diſtat ab illo.</s> + </p> + <p> + <s xml:id="s4793" xml:space="preserve">Præterea cap .12. aliam conſequentiam facit, quam ego non tam amplam facio. <lb/></s> + <s xml:id="s4794" xml:space="preserve">„ Ipſe enim me inferre vult in alijs terrę partibus cauernas non reperiri, eo quod Mon <lb/>„ tes ſint cauernoſi. </s> + <s xml:id="s4795" xml:space="preserve">Aſpice quæſo. pag .29. meæ conſiderationis, & clarè videbis me <lb/>nullo modo negare illas concauitates ſeu porroſitates terræ extra montana loca, <lb/>circa ſuperficiem terræ, vſque ad æquilibrium, orbiculariter, infimæ profunditatis <lb/>maris.</s> + </p> + <p> + <s xml:id="s4796" xml:space="preserve">Sed putare inferius has porroſitates reperiri, cum nulla ratio nobis perſuaſibilis + <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 ſit, idoneum nullo pacto eſſet. </s> + <s xml:id="s4797" xml:space="preserve">Rationes autem ab ipſo Ex <lb/>cellentiſſimo Auguſtino adductas circa huiuſmodi rem, alij dijudicent, de authori-<lb/>tatibus verò, nihil dicam, quia ab illis petendæ ſunt, qui profitentur tales facultates, <lb/>quorum vnius tantummodo authoritas præualere deberet, contra omnes alias <reg norm="eorum" type="context">eorũ</reg> <lb/>qui nunquam attigerunt ſummis labris orificia harum <reg norm="ſcientiarum" type="context">ſcientiarũ</reg>. </s> + <s xml:id="s4798" xml:space="preserve">Vt ſi exempli gra-<lb/>tia non ſolum authoritas illorum virorum, quos ipſe recenſuit, ſufficiens eſſet vt pu <lb/>ta Pioccolo. </s> + <s xml:id="s4799" xml:space="preserve">Naibodæ, Bordini, Clauij, <reg norm="reliquorumque" type="simple">reliquorumq́;</reg> fautorum verę opinionis, <lb/>ſed Franciſci Maurolici tantummodo, qui in primo Dialogo ſuæ coſmographiæ ita <lb/>inquit.</s> + </p> + <quote> + <s xml:id="s4800" xml:space="preserve">Exiſtimo autem totum terræ corpus rigidum eſſe ſaxum, nam ſi arena eſſet, aue <lb/>gleba fragilis, ita humorem imbiberet, vt cum eo quaſi confunderetur; </s> + <s xml:id="s4801" xml:space="preserve">huc ac-<lb/>cedit, quod ſi mineræ, ac rupes, quæ ſunt grauiſſimæ partes in ipſa plerunque ſuper-<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 ſint propinquiora.</s> + </quote> + <p> + <s xml:id="s4803" xml:space="preserve">Hæcigitur ſola authoritas, inſtar reliquarum omnium ſufficere poſſet. </s> + <s xml:id="s4804" xml:space="preserve">Verum <lb/>de authoritatibus minime curandum eſt, vbi ſenſus, <reg norm="ratioque" type="simple">ratioq́;</reg> vera illis opponuntur.</s> + </p> + <p> + <s xml:id="s4805" xml:space="preserve">Quod autem numerus animalium aquatilium maior exiſtat numero terreſtrium, <lb/>ſatis reſpondimus pag .41. noſtræ conſiderationis.</s> + </p> + <p> + <s xml:id="s4806" xml:space="preserve">Sed in cap .14. Excellentiſſimus Auguſtinus ita inquit (vt etiam ſuperius dixerat) <lb/></s> + <quote> + <s xml:id="s4807" xml:space="preserve">quod certiorem cognitionem homo non habet illa, quæ à ſenſu prouenit. </s> + <s xml:id="s4808" xml:space="preserve">Et quod <lb/>nemo eſt qui aſpiciat terram, & aquam, quod hanc maiorem illa non iudicet, & <reg norm="non" type="context">nõ</reg> <lb/>exiſtimet.</s> + </quote> + </p> + <p> + <s xml:id="s4809" xml:space="preserve">Quod autem certiorem cognitionem homo non habeat illa, quæ à ſenſu proue-<lb/>nit, concedendum non cenſeo. </s> + <s xml:id="s4810" xml:space="preserve">Nam omnis cognitio mathematica (cum primum <lb/>gradum certitudinis obtineat) ab ipſo ſenſu fieret, quod omnino alienum <lb/>eſt à veritate. </s> + <s xml:id="s4811" xml:space="preserve">Senſus enim nunquam vidit incommenſurabilitates magnitudinum, <lb/>vel incoincidentias linearum non tangentium cum curuitate hyperbolica, aut angu <lb/>lum contingentiæ aliquem, nec (vt vno verbo dicam) aliquam concluſionem ma-<lb/>thematicam, quam volueris. </s> + <s xml:id="s4812" xml:space="preserve">Neque per ſenſum eſt ſcire, inquit Ariſtoteles. </s> + <s xml:id="s4813" xml:space="preserve">Co-<lb/>gnitio igitur ſenſitiua, certior non eſt illa, quæ per habitum ſcientiſicum acquiritur.</s> + </p> + <p> + <s xml:id="s4814" xml:space="preserve">Ad reliqua verò, ſupponamus nos tunc fuiſſe in Arca Noe, <reg norm="cum" type="context">cũ</reg> aquæ cooperiebant <lb/>omnia cacumina montium, vbi nullum terræ veſtigium videbatur, </s> + <s xml:id="s4815" xml:space="preserve">quare proculdu <lb/>bio aquam iudicaremus, atque exiſtimaremus maiorem terra, <reg norm="dum" type="context">dũ</reg> nulla aliare vtere-<lb/>mur niſi ſenſu abſque alio diſcurſu intellectuali, ut reliqua illa animalia irrationalia, <lb/>quæ nobiſcum erant in dicta arca. </s> + <s xml:id="s4816" xml:space="preserve"><reg norm="Non" type="context">Nõ</reg> ſufficit igitur ſuperficiem aquæ tantummodo <lb/>aſpicere, quia neque tunc temporis, aqua erat maior terra, etiam ſi non ſolum tot <lb/>cubitis attolleretur ſupra cacumina montium, ſed quingenta milliaria, vt ſupradi-<lb/>ximus.</s> + </p> + <p> + <s xml:id="s4817" xml:space="preserve">Ratio autem illa, ex infinitis, ab ipſo, eo in loco adducta, talis eſt.</s> + </p> + <quote> + <s xml:id="s4818" xml:space="preserve">Aqua eſt eccentrica ad terram, & pro <reg norm="centro" type="context">cẽtro</reg> habet centrum grauitatis terræ, aqua <lb/>igitur maioris eſt amplitudinis ipſa terra.</s> + </quote> + <quote> + <s xml:id="s4819" xml:space="preserve">Hanc etiam conſequentiam alijs relinquo Philoſophis dijudicandam.</s> + </quote> + <p> + <s xml:id="s4820" xml:space="preserve">Subſequitur poſtea dicens.</s> + </p> + <quote> + <s xml:id="s4821" xml:space="preserve">Præterea proprius locus terræ, eſt ſuperſicies 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 ſuperficies aeris, & non aquæ exiſtet.</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 ſi locus terræ eſt ſub aqua, ergo locus aquæ proprius eſt ſub aere, & <lb/>non ſub terra, vnde non erit rationabile putare maiorem copiam aquarum exiſtere <lb/>in cauernis ſubterraneis, quam ſupra ſuperficiem terræ. </s> + <s xml:id="s4825" xml:space="preserve">Adde quod locus illarum <lb/>aquarum non eſſet ſuperficies aeris, ſed terræ, vnde non minus locus aquę 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 eſt. </s> + <s xml:id="s4829" xml:space="preserve">Ergo aqua maior eſt terra.</s> + </p> + <quote> + <s xml:id="s4830" xml:space="preserve">Hæc enim conſequentia veriſſima eſſet. </s> + <s xml:id="s4831" xml:space="preserve">Sed nullus vnquam Philoſophus (vt Phi-<lb/>loſophus dico) concedet totam materiam elementarem, in quatuor æquales partes <lb/>eſſe diuiſam.</s> + </quote> + <p> + <s xml:id="s4832" xml:space="preserve">Cap. verò .21. inquit me dixiſſe non ſuffecturam paucam ſpiſſitudinem. </s> + <s xml:id="s4833" xml:space="preserve">Eo enim <lb/>in loco pag .26. mei tractatus contradicens ipſi Bergæ, dixi, quod ſecundum ipſum <lb/>Bergam non ſufficeret pauca ſpiſſitudo.</s> + </p> + <p> + <s xml:id="s4834" xml:space="preserve">Similiter etiam dixi, quod ſecundum ipſum, quanto remotius diffunditur lumen <lb/>fortaſſe tantò magis illuminat. </s> + <s xml:id="s4835" xml:space="preserve">Putans ipſe Berga quod in propinquo debilius exi-<lb/>ſteret dictum lumen. </s> + <s xml:id="s4836" xml:space="preserve">Et propter ea dixi, quod apud ipſum fortaſſe nihil valet illa <lb/>propoſicio, 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 Auguſtinus inquit, vnum tantummodo ele <lb/>mentum non ſufficere ad generationem miſtorum. </s> + <s xml:id="s4839" xml:space="preserve">Hoc enim concedo, ſed hoc ni-<lb/>hil ad me ſpectat, eo quod meum reſponſum ad Bergam, erat circa tranſitum 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 ſcribit me dixiſſe, iudicare, oportebat <lb/>ſcribere, 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 expertusſum, difficillimum omnes Thy <lb/>pographi errores corrigere, neque (vt fertur) Argi oculi ſufficerent.</s> + </p> + <p> + <s xml:id="s4842" xml:space="preserve">Hactenus enim in mei defenſionem hæc ſubiungere volui.</s> + </p> + <p> + <s xml:id="s4843" xml:space="preserve">Ad defenſionem autem Piccolo. <reg norm="aliorumque" type="simple">aliorumq́;</reg> virorum meæ opinionis, nec non de <lb/>proportione duplicata profunditatis maris ad ſuam amplitudinem, ex conſequentia <lb/>pyramidali: </s> + <s xml:id="s4844" xml:space="preserve"><reg norm="alijsque" type="simple">alijsq́;</reg> ſimilibus rationibus, prodeant alij. </s> + <s xml:id="s4845" xml:space="preserve">Huiuſmodi tamen Doctiſſi-<lb/>mi viriingenium, memoriam, nec non doctrinam valde admiror, atque obſeruo.</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 Agrimenſores.</head> + <head xml:id="head567" style="it" xml:space="preserve">Anſelmo Roſemburg Agrimenſori Caſareo.</head> + <p> + <s xml:id="s4846" xml:space="preserve"><emph style="sc">MEthodvs</emph> quàm mihi ſcribis in Prouincia tua maximè in vſu eſſe, nimis <lb/>longa atque prolixa eſt, Pedemontani verò Agrimenſores in produ-<lb/>ctione fractorum, valde breui methodo vti ſolent, quam libenter tibi <lb/>ſcribo, 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 menſuram <lb/>vocare Trabucum, cuius ſextam 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">punctũ</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">Quotieſcunque igitur multiplicant trabucum, per trabucum nulli dubium eſt <lb/>quin producant trabucum ſuperficialem ſcilicet.</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 ſuperficialia rectangula oblonga, quorum lon <lb/>gitudo eſt ipſius trabuchi, latitudo vero lineæ dictarum ſpecierum.</s> + </p> + <p> + <s xml:id="s4851" xml:space="preserve">Dum vero multiplicant pedem per pedem, nulli dubium eſt quin producant pe-<lb/>dem quadratum, ſed apud ipſos non vocatur quadratum, quamuis reuera ita ſit, ſed <lb/>illud vocant duas vncias, quæ quidem ſunt rectangula oblonga iam hic ſupradicta, <lb/>quarum vniuſcuiuſque longitudo ſit vnius trabuchi, latitudo vero vnius duodecimæ <lb/>partis ipſius pedis linearis.</s> + </p> + <p> + <s xml:id="s4852" xml:space="preserve">Productum autem pedis per vnciam, vocant duo puncta, quæ etiam ſunt duo re-<lb/>ctangula oblonga, vt ſupra.</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> ſunt duo re-<lb/>ctangula oblonga, vt dictum eſt, quæ omnia ſcientificè videbimus.</s> + </p> + <p> + <s xml:id="s4854" xml:space="preserve">Pro cuius rei cognitione, ſit, 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> poſtea 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 ſexta pars ipſius <var>.a.e.</var> et <var>.i.o.</var> duodecima ipſius <var>.e.i.</var> et <var>.o.u.</var> duodecima <lb/>ipſius <var>.i.o.</var> et <var>.u.t.</var> duodecima ipſius <var>.o.u</var>. </s> + <s xml:id="s4856" xml:space="preserve">Sit etiam <var>.a.b.</var> æqualis <var>.a.e.</var> lineæ & ſic <var>.e.d</var>: <var>i.<lb/>f</var>: <var>o.g</var>: <var>o.n.</var> & c. <reg norm="terminenturque" type="simple">terminenturq́;</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 ſupra, 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">trabuchũ</reg>, nulli dubium eſt quin ſit quadratum <var>.<lb/>a.d.</var> vt ſuperius diximus.</s> + </p> + <p> + <s xml:id="s4858" xml:space="preserve">Productum autem trabuchi cum pede erit <var>.d.i.</var> ſexta pars ipſius <var>.a.d.</var> cum <var>.e.i.</var> ſit ſex <lb/>ta ipſius <var>.a.e.</var> ex prima ſexti vel .18. aut .19. ſeptimi, ſiue 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 eſt <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> ſexta pars eſt ipſius <var>.f.i.</var> ex ſuppoſito, et <var>.i.o.</var> duo-<lb/>decima ipſius <var>.e.i.</var> proportio igitur <var>.e.i.</var> ad <var>.o.i.</var> dupla eſt proportioni ipſius <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 eſt ipſius <var>.f.o.</var> eo quod ſi <var>.i.o.</var> vel <var>.f.g.</var> (quod idem eſt) duplo <lb/>maius eſſet ipſo 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 ipſi <var>.K.e.</var> ex .15. ſexti <lb/>vel .20. ſeptimi quod quidem <var>.f.o.</var> duplo maius eſſet ipſo 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 ſi mauis, ita dicas <lb/><var>e.K.</var> ſexta pars eſt ipſius <var>.d.i.</var> ex iam dictis propoſitionibus <var>.f.o.</var> autem eſt duodecima <lb/>ipſius <var>.d.i.</var> ex ijſdem, cum exſuppoſito <var>.i.o.</var> duodecima ſit ipſius <var>.e.i.</var> </s> + <s xml:id="s4863" xml:space="preserve">quare <var>.e.K.</var> <reg norm="duplum" type="context">duplũ</reg> <lb/>erit ipſius <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. ſit <var>.K.o.</var> quod probabimus ex ijſdem rationibus <lb/>duplum eſſe ipſius <var>.g.u.</var> puncti rectanguli oblongi. </s> + <s xml:id="s4865" xml:space="preserve">Nam <var>.l.o.</var> ſexta pars ſimiliter eſt <lb/>ipſius <var>.g.o.</var> et <var>.o.u.</var> duodecima ipſius <var>.o.i.</var> </s> + <s xml:id="s4866" xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.o.n.</var> dupla eſt propor <lb/>tioni <var>.g.o.</var> ad <var>.o.l.</var> ſequitur ergo ex prædictis rationibus <var>.k.o.</var> duplum eſſe ipſius <var>.g.u.</var> <lb/>vel ſic, vtlin præcedenti, cum <var>.K.o.</var> ſit ſexta pars ipſius <var>.f.o.</var> ex dictis propoſitionibus <var>.<lb/>g.u.</var> verò duodecima eiuſdem <var>.f.o.</var> ex ijſdem, nam <var>.o.u.</var> duodecima eſt ipſius <var>.o.i.</var> ergo <lb/><var>K.o.</var> duplo maius eſt ipſo <var>.g.u</var>.</s> + </p> + <p> + <s xml:id="s4867" xml:space="preserve">Ex ijſdemmet rationibus productum <var>.l.u.</var> pedis cum puncto duplum eſt ipſius <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 eſt quadra-<lb/>tum, duplum eſſe ipſius <var>.c.t.</var> attomi rectanguli oblongi. </s> + <s xml:id="s4869" xml:space="preserve">Nam <var>.i.n.</var> eſt pars vna ex .72. <lb/>ipſius <var>.c.u.</var> et <var>.u.t.</var> pars vna ex .144. ipſius <var>.o.i.</var> ex ſuppoſito, </s> + <s xml:id="s4870" xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.u.t.</var> <lb/>dupla eſt proportioni ipſius <var>.c.u.</var> ad <var>.n.i.</var> ex dictis igitur rationibus <var>.o.n.</var> duplo maius <lb/>eſt ipſo <var>.c.t</var>. </s> + <s xml:id="s4871" xml:space="preserve">Vel ſi placet dicas <var>.n.o.</var> eſt vna pars ex .72. ipſius <var>.f.o.</var> exſupradictis, eo <lb/>quod <var>.n.i.</var> ita ſe habet ad <var>.f.i.</var> vt vnitas ad .72. ſed ex ijſdem rationibus <var>.c.t.</var> pars vna ex <lb/>144. eſt ipſius <var>.f.o.</var> eo quod ita ſe 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 ipſo <var>.<lb/>c.t</var>.</s> + </p> + <p> + <s xml:id="s4873" xml:space="preserve">Propoſitum ſit nobis nunc, exercitij gratia, quærere ſuperficiem alicuius rectan <lb/>guli, cuius vnum latus ſit <reg norm="trabuchorum" type="context">trabuchorũ</reg> .3. pedum .2. & vnciarum .3. aliud vero latus ſit <lb/>trabuchorum .2. pedum .3. vnciarum vero .2.</s> + </p> + <p> + <s xml:id="s4874" xml:space="preserve">Huiuſmodi autem methodo mediante, multiplicando primum latus <reg norm="dictum" type="context">dictũ</reg> .3. 2. 3. <lb/>per numerum trabucorum ſecundi lateris .2. ſcilicet producentur nobis primò trabu <lb/>cha ſuperficialia .6. pedes .4. & vnciæ .6. omnia rectagula, vt dictum eſt. </s> + <s xml:id="s4875" xml:space="preserve">Multiplican-<lb/>do deinde idem primum latus .3. 2. 3. per pedes .3. ſecundi lateris. </s> + <s xml:id="s4876" xml:space="preserve">Ex trabuchis .3. <lb/>primi lateris cum .3. pedibus ſecundi, producentur .9. pedes rectanguli, hoc eſt <lb/>vnus trabuchus cum tribus pedibus rectangulis. </s> + <s xml:id="s4877" xml:space="preserve">Ex pedibus autem huius .2. cum <lb/>ijſdem alterius lateris .3. producentur .12. vnciæ rectangulæ ideſt vnus pes rectangu-<lb/>lus. </s> + <s xml:id="s4878" xml:space="preserve">Exijſdem pedibus .3. ſecundi 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 eſt vna vncia cum .6. punctis rectangulis. </s> + <s xml:id="s4879" xml:space="preserve">Deinde ex <lb/>multiplicatione vnciarum .2. ſecundi lateris, cum .3. trabuchis primi lateris, produ-<lb/>centur .6. vnciæ. </s> + <s xml:id="s4880" xml:space="preserve">Ex multiplicatione poſtea dictarum .2. vnciarum ſecundi 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 ijſdem .2. vncijs ſecundi lateris cum .3. primi, producentur .12. atto-<lb/>mi, ideſt 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 cauſa, vt dixi, quomodo conueniat calculus iſte cum <lb/>calculo ordinario communi?</s> + </p> + <p> + <s xml:id="s4885" xml:space="preserve">Nam quotieſcunque dicta latera, fracta fuerint in vncias, primum latus erit <lb/>vnciarum .243. ſecundum autem .182. productum vero vnius in alterum erit vn-<lb/>ciarum quadratarum .44 226. quod quidem productum cum diuiſum fuerit per <num value="5184">.<lb/>5184.</num> vncias quadratas vnius trabuchi quadrati, prouentus erit .8. trabucho-<lb/>rum, reliquus verò numerus, ſiue fractus, erit vnciarum quadratarum .2754. qui <lb/>cum diuiſus fuerit per numerum .144. vnciarum vnius pedis quadrati, prouenient <lb/>pedes .19. quadrati cum vncijs .18. ſuperabundantibus, dicti autem pedes .19. ſignifi-<lb/>cant tres pedes rectangulos oblongos cum vno pede quadrato, hoc eſt cum duabus <lb/>vncijs rectangulis oblongis, vt ſupra.</s> + </p> + <p> + <s xml:id="s4886" xml:space="preserve">Videndum nunc eſt, vtrum illæ .18. vnciæ æquipolleant tribus punctis rectangu-<lb/>lis oblongis: </s> + <s xml:id="s4887" xml:space="preserve">ſed hoc manifeſtè videre eſt, ex hoc, quia quęlibet vncia rectangula <lb/>oblonga componitur ex .72. quadratis, punctum autem rectangulum oblongum, <reg norm="cum" type="context">cũ</reg> <lb/>ſit duodecima pars ipſius vnciæ rectangulæ oblongæ, ipſum componetur ex .6. vn-<lb/>cijs quadratis .18. igitur vncijs quadratis, triplum erit ipſius puncti rectanguli dicti. <lb/></s> + <s xml:id="s4888" xml:space="preserve">Vnde clarè patet, quod, quotieſcunque voluerimus ſcire proportionem ipſarum vn <lb/>ciarum quadratarum ſuperabundantium, ad punctum rectangulum oblongum, ſi <lb/>dixerimus ex regula de tribus, ſi .72. (vncia rectangula oblonga) dat .18. quid <reg norm="dabunt" type="context">dabũt</reg> <lb/>12? </s> + <s xml:id="s4889" xml:space="preserve">puncta rectangula oblonga, quarum vnaquæque eſt duodecima pars ipſius vn-<lb/>ciæ rectangulæ oblongæ, in præſenti autem caſu prouenient .3. pro quarto termino <lb/>quæſito, & habebimus propſitum.</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 ſolutio quam neſcio quis te docuit, valde diuerſa eſt 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 ſocij, Ludouicus, Hieronymus, Franciſcus, & Lau <lb/>rentius quorum primus, Ludouicus ſcilicet, poſuit aureos .6000. Hierony <lb/>mus verò aureos .5000. Franciſcus autem .2000. & Laurentius .1000. quorum ſum-<lb/>ma faciebat aureos .14000. </s> + <s xml:id="s4893" xml:space="preserve">interim tamen de tali ſumma Ludouicus recepit aureos <lb/>2000. Hieronymus verò .1000. Franciſcus autem .900. & Laurentius .800. quapro-<lb/>pter in ſumma reſidua Ludouicus non habebat niſi aureos .4000. Hieronymus <reg norm="etiam" type="context">etiã</reg> <lb/>4000. Franciſcus .1100. & Laurentius .200. quorum ſumma erat .9300. </s> + <s xml:id="s4894" xml:space="preserve">Nunc au-<lb/>tem iſti ſocij cupiunt augere hanc ſummam per aureos .20000. tali tamen conditio-<lb/>ne quod quilibet tantum tribuat vt in totali ſumma, tantam partem unus habeat, <lb/>quantam alter.</s> + </p> + <p> + <s xml:id="s4895" xml:space="preserve">Hoc autem problema tam ſacile eſt, & cum ſuo theoremate ita coniunctum, quod <lb/>miror amicum noſtrum illud illico non vidiſſe.</s> + </p> + <p> + <s xml:id="s4896" xml:space="preserve">Accipe igitur illos aureos .20000. & eos collige cum ſumma .9300. vnde habebis <lb/>aureos .29300. pro <reg norm="summa" type="context">sũma</reg> totali, cuius quarta pars erit .7325. <reg norm="quam" type="context">quã</reg> <reg norm="vnuſquisque" type="simple">vnuſquisq;</reg> poſtea <lb/>habebit in dicta ſumma. </s> + <s xml:id="s4897" xml:space="preserve">Sed ut reperias quantitatem aureorum quam quilibet <lb/>prius debet contribuere, vt poſtea habeat aureos .7325. in dicta ſocietate. </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 ipſius <lb/>Ludouici. </s> + <s xml:id="s4899" xml:space="preserve">Demptis ſimiliter aureis .4000. ex dictis .7325. <reg norm="remanebunt" type="context">remanebũt</reg> .3325. pro con <lb/>tributione ipſius Hieronymi. </s> + <s xml:id="s4900" xml:space="preserve">Deinde ſi ex .7325. extracti fuerint aurei .1100. relin-<lb/>quent .6225. pro contributione Franciſci. </s> + <s xml:id="s4901" xml:space="preserve">Demptis demum .200. ex .7325. reſidui <lb/>erunt .7125. pro contributione Laurentij, & ſic quilibet habebit æqualem portio-<lb/>nem in totaliſumma.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head570" style="it" xml:space="preserve">Speculatio cuiuſdam Methodire ductionis numiſmatum <lb/>unius ſpeciei 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 ſit, quod ſumma <reg norm="mendietatis" type="context">mẽdietatis</reg> cuiuſuis <lb/>numeri illorum numiſmatum, quæ hic vocantur Blanci, cum ſexta parte eiuſ <lb/>dem medietatis, ſemper ſit numerus florenorum huius prouinciæ. </s> + <s xml:id="s4903" xml:space="preserve">Vt exempli gra <lb/>tia, quotieſcunque reducere voluerimus .48. Blancos in Florenos, ſi medietati ip-<lb/>ſius .48. hoc eſt .24. adiecta fuerit ſexta pars ipſius medietatis, quæ eſt .4. </s> + <s xml:id="s4904" xml:space="preserve">tunc habebi <lb/>mus .28. & ita dicemus quod .48. Blanci conſtituunt Florenos .28. quod quidem <lb/>verum eſt.</s> + </p> + <p> + <s xml:id="s4905" xml:space="preserve">Huiuſmodi autem rei ſpeculatio ita ſe habet. </s> + <s xml:id="s4906" xml:space="preserve">Nam vnuſquiſque Blancus diuidi-<lb/>tur in .7. æquales partes, quarum .12. conſtituunt vnum Florenum, horum verò nu-<lb/>miſmatum communis menſura, vocatur Groſſus, vt ſcis, ex quo ſequitur, quod ſi + <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, noſtram figuram .79. Theorematis Arithmetici <var>.x.u.o.<lb/>e.n.</var> ſupponendo ambo producta <var>.u.x.</var> et <var>.n.e.</var> inuicem ęqualia exiſtere, & vnumquod-<lb/>que eſſe groſſorum .336. ſit 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/>ſexti vel .20. ſeptimi Euclidis eandem fore proportionem <var>.o.u.</var> ad <var>.o.e.</var> quæ <var>.o.n.</var> ad <var>.o.<lb/>x.</var> ſed <var>.o.n.</var> eſt ſumma medietatis ipſius <var>.o.x.</var> cum ſexta parte dictæ medietatis, ita igi-<lb/>tur erit <var>.o.u.</var> ipſius <var>.o.e.</var> hoc eſt ſumma medietatis <var>.o.e.</var> <reg norm="cum" type="context">cũ</reg> ſexta parte medietatis eiuſ-<lb/>dem, quæ ſumma in præſenti exemplo erit .28.</s> + </p> + <p> + <s xml:id="s4909" xml:space="preserve">Hac enim ſpeculatione mediante, poteris methodum inuenire conuertendi Flo-<lb/>renos in Blancos. </s> + <s xml:id="s4910" xml:space="preserve">Vt ſi nobis propoſiti fuerint Floreni .28. </s> + <s xml:id="s4911" xml:space="preserve">Voluerimusq́ue inuenire <lb/>quot Blancos faciant, ſuppoſita menſura communi, iam ſupradicta. </s> + <s xml:id="s4912" xml:space="preserve">Nam duplica-<lb/>bimus numerum Florenorum, à quo duplo detrahemus ſeptimam partem, <reg norm="reliquum" type="context">reliquũ</reg> <lb/>verò erit numerus quæſitus.</s> + </p> + <p> + <s xml:id="s4913" xml:space="preserve">Huiuſmodi autem rei ratio eſt, quia, cum in ſupradicta figura, proportio <var>.o.e.</var> ad <lb/><var>o.u.</var> ęqualis exiſtat ei, quæ <var>.o.x.</var> ad <var>.o.n.</var> atque etiam <var>.o.x.</var> ſit minor duplo ipſius <var>.o.n.</var> <lb/>per ſeptimam partem ipſius dupli <var>.o.n.</var> minor erit <var>.o.e.</var> duplo ipſius <var>.o.u.</var> per <reg norm="ſeptimam" type="context">ſeptimã</reg> <lb/>partem eiuſdem dupli ipſius <var>.o.u</var>.</s> + </p> + <p> + <s xml:id="s4914" xml:space="preserve">Idem affirmo de quauis conuerſione aliorum numiſmatum, quorum ſemper <var>.o.x.</var> <lb/>maior ſit <var>.o.n.</var> verò minor. </s> + <s xml:id="s4915" xml:space="preserve">Vt ſi <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> menſuraretur ab <var>.o.n</var>.</s> + </p> + <p> + <s xml:id="s4916" xml:space="preserve">Si cuperemus ſcire quot <var>.o.x.</var> ſint in <var>.o.n</var>. </s> + <s xml:id="s4917" xml:space="preserve">Primo dicemus in <var>.o.n.</var> reperiri ſummam <lb/>medietatis ſex ſeptimorum ipſius <var>.o.x.</var> collectæ cum vna ſeptima parte ipſius <var>.o.x.</var> ſeu <lb/>(vt ita dicam) cum tertia ipſius medietatis. </s> + <s xml:id="s4918" xml:space="preserve">Vnde dempta ſeptima parte ipſius .42. <lb/>quæ eſt .6. <reg norm="collectaque" type="simple">collectaq́</reg> cum medietate reſidui, quæ eſt .18. habebimus .24. res, quarum <lb/>vnaquæque æqualis erit ipſi <var>.o.x</var>.</s> + </p> + <p> + <s xml:id="s4919" xml:space="preserve">Sed ſi 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> ipſius <var>.o.u.</var> & habebit propoſitum. </s> + <s xml:id="s4920" xml:space="preserve">Nam ita ſe 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 ſcire à me deſideras, eſt, quod cum vendideris libram vnam <lb/>mercis pro .4. ſolidis, & lucratus fueris .2. cum quarta parte vnius pro ſingu-<lb/>lis decem libris, ſcire velles quantum lucri facturus eſſes in libris <reg norm="decem" type="context">decẽ</reg> dan-<lb/>do ſingulam libram pro .6. ſolidis.</s> + </p> + <p> + <s xml:id="s4922" xml:space="preserve">Nulli dubium eſt quin decima pars de .2. cum quarta vnius ſit lucrum libræ vnius. <lb/></s> + <s xml:id="s4923" xml:space="preserve">Quæ decima pars ſunt <reg norm="nouem" type="context">nouẽ</reg> quadrageſimæ partes, & hæc ſubducta à ſolidis .4. reli-<lb/>qui erunt ſolidi .3. cum .31. quadrageſimis partibus pro ſorte vnius libræ. </s> + <s xml:id="s4924" xml:space="preserve">Quę ſors <lb/>ſubtracta à ſolidis .6. remanebunt ſol .2. cum .9. quadrageſimis lucri pro libra, quod <lb/>multiplicatum per .10. proueniunt ſol .22. cum quarta parte vnius, & tantum aſcen-<lb/>deret lucrum, quod fieri poſſet in libris decem ſi quamlibet, ſol .3. cum .31. quadra <lb/>geſimis nobis conſtaret.</s> + </p> + <p> + <s xml:id="s4925" xml:space="preserve">Vel ſic multiplicemus ſortem 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. ſortem cum lucro vnius librę quod eſt .4. pro-<lb/>ductum erit .40. differens à primo ſol .2. cum quarta parte, multiplicemus pariter <lb/>per .10. precium .6. ſolidorum proueniens erit .60. à quo deducendo productum ſor-<lb/>tis librarum .10. quod erat ſol .37. cum tribus quartis ſupererunt ſol .22. cum quar-<lb/>ta parte, vt ſupra.</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 diſtinctionem orbium, quæiam inualuit, nonteneas, ſed putes <lb/>totum eſſe quoddam continuum excipiens corpora ſtellarum, nouum <reg norm="non" type="context">nõ</reg> <lb/>eſt, nam nonnulli ſolidæ doctrinæ Philoſophi idem cenſuerunt. </s> + <s xml:id="s4927" xml:space="preserve">Sed <lb/>quod attinet ad dignitates planetarum in ſignis zodiaci, ſcias huiuſmo-<lb/>di ordinem me compręhendere eſſe deſumptum ab ordine antiquo orbium <reg norm="ipſorum" type="context">ipſorũ</reg> <lb/>planetarum, quiquidem ordo erat, vt ſtatim poſt Lunam ſuccederet Sol, poſt So-<lb/>lem Mercurius, tum Venus deinde Mars, poſtea Iupiter, & tandem Saturnus per <lb/><reg norm="eoſdemque" type="simple">eoſdemq́;</reg> orbes, retro redibant, atque hoc cognoſcitur conſtituendo 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">eundemq́;</reg> tribuentes Saturno, Piſces, Ioui, <lb/>Arietem, Marti, Taurum, Veneri, & Gemellos, Mercurio, ſeptem Planetas cum <lb/>duodecim ſignis zodiaci concordes reddebant.</s> + </p> + <p> + <s xml:id="s4928" xml:space="preserve">Quod deinde Ariſtoteles in libris de ſenſu & ijs quæ ſenſibus percipiuntur, dicit <lb/>pupillam oculi eſſe nigram, non ita ſe habet, nam idem eſt, ac ſi quis diceret <reg norm="nigrum" type="context">nigrũ</reg> <lb/>eſſe illud medium, quod permitteret tranſitum lumini per ſuam diaphaneitatem, nul <lb/>lum lumen à ſeipſo reflectens, & etiam ac ſi quis diceret nigrum eſſe aerem alicuius <lb/>cubiculi vndequaque clauſi tenebroſi.</s> + </p> + <p> + <s xml:id="s4929" xml:space="preserve">Quod etiam idem Ariſtoteles volens adducere cauſam, cur oculus magis mate-<lb/>riam aquæ, quam aeris participet, dicensidea ratione fieri, quod aqua magis quam <lb/>aer ſeruari poſſit, eodem libro ſcribit, eſt reuera admirandum. </s> + <s xml:id="s4930" xml:space="preserve">Ibi enim clarè de-<lb/>monſtrat ſe planè ignorare, & conſtructionem oculi, & cauſam diuerſitatis eorum <lb/>humorum tam in ſubſtantia, quam in figura, quæ non aliunde dependet quam quod <lb/>diuerſam refractionem radiorum luminoſorum producat, qui per pupillam ingre-<lb/>diuntur, vt ad proprios <reg norm="ſibique" type="simple">ſibiq́;</reg> deſtinatos locos dirigantur radij, vt à virtute viſiua per <lb/>fectius ſen tiantur.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head576" style="it" xml:space="preserve">De ratione Frigiditatis locorum umbroſorum.</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 ſentitur calor in locis expoſitis Soli, tan-<lb/>to minus ſentiatur in vmbra, vbi Solis radius non reflectitur, eſt quia cum ra <lb/>refactus eſt aer à vehementi calore radij ſolaris, ſeipſum colligit, & condenſatur in <lb/>locis, à quibus à calore, ratione rarefactionis, non expellitur, & quia naturaliter ca-<lb/>lor ſequitur rarum, rarum calorem, & frigidum <reg norm="densum" type="context">densũ</reg>, & <reg norm="densum" type="context">densũ</reg> frigidum, vt vnicui <lb/>que ſanę mentis patet, hanc ob cauſam ſequitur rem ita ſe habere vt diximus. </s> + <s xml:id="s4932" xml:space="preserve">Poſſu <lb/>mus etiam abſque dubio credere huiuſmodi ratione fieri, vt frigus matutini tempo <lb/>ris, in crepuſculo maius eſſe eo, quod noctu viguit. </s> + <s xml:id="s4933" xml:space="preserve">Nam materia conſiſtens in co-<lb/>no vmbræ terræ, ſemper denſior eſt ea, quæ extra reperitur, imo noua materia con <lb/>tinuo condenſatur, propter motum vmbrę, quæ ſemper corpori ſolari opponitur. </s> + <s xml:id="s4934" xml:space="preserve">hęc + <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 condenſatio dico ſemper fit in crepuſculo matutino, hoc eſt in parte co <lb/>ni à Sole pulſa, in parte vero contrari a ipſius coni hoc eſt in parte crepuſculi ve-<lb/>ſpertini, contrarium accidit, quia potius aliquantulum in hac parte materia coni ra <lb/>rificatur, quia extrinſeca condenſatur, in parte vero matutina extrinſeca rarificatur; <lb/></s> + <s xml:id="s4935" xml:space="preserve">& propterea intrinſeca conde nſatur.</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/>caſu 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="Ariſtippum" type="context">Ariſtippũ</reg> tempeſtate maris ad incognita littora delatum, cum in are-<lb/>na vidiſſet <reg norm="quaſdam" type="context">quaſdã</reg> figuras geometricas delineatas <reg norm="exultantem" type="context">exultantẽ</reg> lętitia dixiſſe: </s> + <s xml:id="s4937" xml:space="preserve">Hæc <lb/>ſunt hominum veſtigia. </s> + <s xml:id="s4938" xml:space="preserve">Nam conſonum rationi non erat, vt huiuſmodi figuræ ca-<lb/>ſu eſſent impreſſæ: </s> + <s xml:id="s4939" xml:space="preserve">neque etiam credendum eſt ingentem hanc ma chinam tanto or <lb/>dine conſtantem fortuitò eſſe productam, cum nulla quantumuis minima eiuſdem <lb/>particula, dummodo nitatur ordine, aliquo modo caſu effecta fuerit; </s> + <s xml:id="s4940" xml:space="preserve">cum caſus ni-<lb/>hil producat, quod regulam & ordinem ſeruet. </s> + <s xml:id="s4941" xml:space="preserve">Non eſt igitur producta caſu admi <lb/>randa correſpondentia, quæ eſt obiectorum cum potentijs, luminis cum oculo, ſo-<lb/>ni cum auditu, ſaporis cum guſtatu, odoris cum odoratu, qualitatum tangibilium <reg norm="cum" type="context">cũ</reg> <lb/>tactu. </s> + <s xml:id="s4942" xml:space="preserve">Si diligenter deinde cuiuſlibet rei naturalis operationem conſiderabimus, <lb/>eas tanta arte conſtructas videbimus, vt cogamur fateri aliquam prudentiſſimam, <lb/>& ſagaciſſimam mentem eas formaſſe, ſi ergo quælibet <reg norm="mundi" type="context">mũdi</reg> pars tanta cum ratione <lb/>& ordine eſt conſtructa: </s> + <s xml:id="s4943" xml:space="preserve">quomodo fieri poterit, vt de toto ipſo mundo id in dubium <lb/>vocemus, <reg norm="certiſſimeque" type="simple">certiſſimeq́;</reg> non credamus diuiniſſimam aliquam <reg norm="mentem" type="context">mẽtem</reg> eſſe à qua exqui-<lb/>ſitiſſima huius vniuerſi harmonia, quæ ex tot <reg norm="tantisque" type="simple">tantisq́;</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 ſcribis dico, quod excrementa quæ ex corpore ſano prodeunt <lb/>in ſua <reg norm="ipſorum" type="context">ipſorũ</reg> qualitate ſenſibili ita ſe <reg norm="habent" type="context">habẽt</reg> ad <reg norm="facultatem" type="context">facultatẽ</reg> illius partis eiuſdem <lb/>corporis, ut <reg norm="eam" type="context">eã</reg> non lędant, <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> efficeret <reg norm="ſputum" type="context">ſputũ</reg>, ſi eſſet <reg norm="amarum" type="context">amarũ</reg>, aut quod ex <lb/>cernitur naſo <reg norm="fętidum" type="context">fętidũ</reg> eſſet. </s> + <s xml:id="s4945" xml:space="preserve">Imagineris igitur <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> poſſit eſſe <reg norm="verum" type="context">verũ</reg> id quod <reg norm="idem" type="context">idẽ</reg> <lb/><reg norm="amicus" type="simple">amicꝰ</reg> noſter ait. </s> + <s xml:id="s4946" xml:space="preserve">Pręterea ſi aliquid tibi in <reg norm="oculum" type="context">oculũ</reg> inciderit, an neſcis quomodo ſtatim <lb/>affatim affluat humor, vt id foras <reg norm="propellat" type="simple">ꝓpellat</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 abſque myſterio naturam in tot miſerijs ſenectutem poſuiſſe, cum <lb/>ſæpiſſimè ſenex mori deſideret, ut huius vitæ calamitatibus liberetur, vnde fit, vt <lb/>cum eius aduentum ſentiat, minus affligatur. </s> + <s xml:id="s4949" xml:space="preserve">Dicito etiam eidem, naturam non <lb/>fuiſſe tam ſolicitam de quibuſdam partibus quemadmodum eſt de toto, vnde ma-<lb/>gis rotunda, & polita poterat eſſe ſuperficies terræ, quam nunc eſt, quia natura ma <lb/>gis reſpicit 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 riſerim, id quod ſcribis te inter-<lb/>rogaſſe eum Philoſophum naturalem, vnde fit, vt ventus ſit frigidus, <reg norm="eumque" type="simple">eumq́;</reg> tibi re <lb/>ſpondiſſe, quod à remotiſſimis partibus veniat, <reg norm="genereturque" type="simple">genereturq́;</reg> à vaporibus terræ frigi-<lb/>dis. </s> + <s xml:id="s4951" xml:space="preserve">( cum ipſa ſit 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/>proſilit, 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, ſcito <reg norm="naturam" type="context">naturã</reg> coniunxiſſe <reg norm="frigiditatem" type="context">frigiditatẽ</reg> <reg norm="cum" type="context">cũ</reg> denſitate, & <reg norm="caliditatem" type="context">caliditatẽ</reg> <reg norm="cum" type="context">cũ</reg> raritate, <lb/>vt ſup. diximus, ita vt cum aliquod corpus <reg norm="denſatur" type="simple">denſat̃</reg>, <reg norm="frigidum" type="context">frigidũ</reg> <reg norm="reddatur" type="simple">reddat̃</reg>, & dum rarefit ma-<lb/>iorem caliditatem acquirat, & ſic econtra fit, vt quanto magis aliquod corpus refri <lb/>geratur, tanto denſius 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 ſuæ ſubtilitatis, velociter <lb/>condenſari, & rarefieri poſſit, eius partes denſiores ſemper erunt frigidæ, & hanc <lb/>obrem quilibet ventus, qui per calida loca non tranſeat, natura ſua frigidus, calidus <lb/>autem per accidens erit. </s> + <s xml:id="s4955" xml:space="preserve">Hinc fit vt vaſa 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, ſed hoc non ſequeretur, ſi in materia, qua huiuſ <lb/>modi vas conſtat, 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 cęleſtium, de albedine, <lb/>de ſphæra.</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 ſi ſit lumen reflexum Solis ab ipſa Luna, ab ea tamen <lb/>non ita reflectitur, vt à ſuperficie polita ſpeculi, <reg norm="cum" type="context">cũ</reg> eius luminis <reg norm="tantam" type="context">tantã</reg> <reg norm="quan titatem" type="context">quã<lb/>titatem</reg> ſuper ipſum corpus lunare videamus, & eo modo <reg norm="terminatam" type="context">terminatã</reg> quo <lb/>conſpicimus. per ſe lumen, cauſa oculi eſt 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> ſolaris reflexio exiſtat) <lb/>omnino deſtituta eſt, dico etiam, neque in ipſis ecclipſibus ſolaribus vel lunaribus, <lb/>in ſolaribus enim cum Soltot millia vices maior ſit Luna, Luna verò minor terra, ſe <lb/>quitur, vt terra non omnino priuata remancat lumine Lunæ, in ecclipſibus ve-<lb/>rò lunaribus Luna ſemper videtur, gratia luminis ſolaris, quamuis refracti. </s> + <s xml:id="s4958" xml:space="preserve">Mo-<lb/>tus corporum cœleſtium fit ratione ſitus, & varietatis virtutis ſtellæ in diuerſis locis, <lb/>hæc autem varietas abſque diuerſo ſitu eiuſdem ſtellæ, nec diuerſus hic ſitus abſque <lb/>motu fieri poſſet, ita vt motus ſtellarum ſit ratione diuerſitatis ſituum ipſarum, er-<lb/>go motus, & diuerſitas ſituum, fit, ob diuerſam 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> ſimilitudinem 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 ſenſum. </s> + <s xml:id="s4962" xml:space="preserve">Tertiò quia abſque reſiſtentia magis reci-<lb/>pit qualitatem aliorum colorum, quam alij colores. </s> + <s xml:id="s4963" xml:space="preserve">Quartò quia maximus <lb/>eſt omnium colorum. </s> + <s xml:id="s4964" xml:space="preserve">Quintò quia ſimplicior eſt reliquis. </s> + <s xml:id="s4965" xml:space="preserve">Sextò quia diſgregat vi-<lb/>ſum. </s> + <s xml:id="s4966" xml:space="preserve">Septimò quia qualitas quæ in niue alba eſſe videtur, nihil aliud eſt quam mul-<lb/>titudo quædam luminum reflexorum, & non albedo, ſimilis ei, quæ eſt lactis, aut <lb/>panni, quæ quidem ſeptima cauſa effecit, vt ipſam albedinem, magis quam alium <lb/>quemuis colorem cum ipſo lumine compararem, cum nihil ſit, quod eſſe ſuum <reg norm="tranſ" type="context">trãſ</reg> <lb/>mutans, aut apparenter, aut eſſentialiter, illud ipſum prius non tranſmutet in for-<lb/>mam ſibi propin quiorem, vt manifeſtè patet. </s> + <s xml:id="s4967" xml:space="preserve">Eſt etiam huius rei octaua ratio, <lb/>magni ponderis, quia ſcilicet nullus ſit color, qui magis reſiſtat lumini, aut in quem <lb/>lumen minorem impreſſionem faciat, quam albedo. </s> + <s xml:id="s4968" xml:space="preserve">Vnde ſequitur, obiecta alba, <lb/>minus eſſe combuſtibilia quam alia, cum quælibetres in ſuum 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"/> + ſibi ſimile valentius agat, vtrectè vidit Ariſtoteles cum dixit, omne contrarium @ <lb/>ſuo contrario patinatum eſt.</s> + </p> + <p> + <s xml:id="s4969" xml:space="preserve">Inter corpora, multum ſimplicitatisretinet ſphæra.</s> + </p> + <p> + <s xml:id="s4970" xml:space="preserve">Circa quod, præter rationes adductas ab Ariſtotele in libris de Cœlo, poſſumus <lb/>etiam ratiocinarià facilitate motus <reg norm="vndique" type="simple">vndiq́;</reg> ab eo quod violentiæ non reſiſtar, ab eo, <lb/>quod apta <reg norm="nataque" type="simple">nataq́;</reg> ſit quieſcere ſupra quoduis punctum ſuę ſuperficiei, ab eo quod ab <lb/>aliqua ſuperficie alterius corporis ſeſe tangi non permittat, quæ curuitate concaua <lb/>non adæquetur, niſi medio vnius puncti. </s> + <s xml:id="s4971" xml:space="preserve">Verum eſt, quod licet hæc vltima ratio <reg norm="non" type="context">nõ</reg> <lb/>ſit propria ſphæræ, eſt tamen cauſa ſimplicitatis in eo, in quo reperitur, ſed proprię <lb/>paſſiones ſphæræ ſunt ſupradictæ, præter quam quod alia eiuſdem ſphæræ eſt pro-<lb/>prijſſima, quæ eſt diſtantia eiustermini ab vno tantummodo puncto ideſt ab <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <lb/>centro, & etiam poſſe diuidere corpus aliquod medium, cum æquali reſiſtentia circa <lb/>punctum, quod prius in motu reperitur.</s> + </p> + <p> + <s xml:id="s4972" xml:space="preserve">Aequalitas autem rerum, eſt etiam valde ſimilis ſimplicitati, & vnitati.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head584" style="it" xml:space="preserve">Comparatio uiſus, & auditus.</head> + <head xml:id="head585" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s4973" xml:space="preserve">QVodad viſum & auditum attinet, magis neceſſarium eſſe viſum, & nobilio-<lb/>rem quam auditum exiſtimo, primò quia ſi quis viſu orbatus eſſet, contra <lb/>frigus, & calorem, contra famen, & ſitim nil prouidere poſſet, neque aliud quic-<lb/>quam hoc vocabulum prouidere ſignificat, neque abſque 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 ſi quis deſtitutus eſſet facultate audiendi, ſupradictas tamen operationes <reg norm="prae- ſtare" type="simple">prę-<lb/>ſtare</reg> poſſet, neque modo careret, quo animi ſui ſenſa abſque beneficio ſoni, ſed <lb/>ope figurarum & characterum alteri aperiret: </s> + <s xml:id="s4975" xml:space="preserve">neque etiam munere ſpeculandi ſcien <lb/>tias (excepta muſica) deſtitueretur. </s> + <s xml:id="s4976" xml:space="preserve">Ad ſcientiam comparandam, longè magis ne <lb/>ceſſarius eſt viſus, quam auditus præterquam, quod viſus maiorem numerum obie-<lb/>ctorum, & differentiarum rerum percipit, & inter reliquos ſenſus velociſſimè imò <lb/>in inſtanti operatur, magis remotè quam alij, & exactius ſentit, <reg norm="minusque" type="simple">minusq́;</reg> quam reli-<lb/>qui afficitur, præterquam quod ſemperagit, dummodò non dormiat animal. </s> + <s xml:id="s4977" xml:space="preserve">Præ-<lb/>terea ſeſe magis patefacit, & prodit anima per oculos, quam per aliud, cuiuslibet <lb/>ſenſus, inſtrumentum. </s> + <s xml:id="s4978" xml:space="preserve">Oculo magis quam alia corporis parte, hominis natura co-<lb/>gnoſcitur: </s> + <s xml:id="s4979" xml:space="preserve">& ſi aliquid ſpeculari volumus, quod ſine imaginatiua fieri non poteſt, <lb/>ſtatim imaginamur nos videre huiuſmodirem, ac ſi oculo fuiſſet compræhenſa, & <lb/>ab imagine quæ eſt vnum ex obiectis oculi, imaginatiua nuncupatur. </s> + <s xml:id="s4980" xml:space="preserve">Beneficio <lb/>oculorum omnes ferè ſcientiæ ſunt adinuentæ. </s> + <s xml:id="s4981" xml:space="preserve">Auditus nil aliud quam ſonum ca-<lb/>pit, auditus nunquam detulit intellectui figuram, corpus ſuperficiem, aut lineam, <lb/>materiam, formam, locum, dimenſionem, plenum inane, nec innumera alia acci-<lb/>dentia, quæ ab oculo compræhenduntur. </s> + <s xml:id="s4982" xml:space="preserve">Quæ verò viſui, & auditui ſunt commu-<lb/>nia, ſunt etiam tactui communia, vt numerus, motus, maius, & minus, ſunt tamen ali <lb/>qua oculo & tactui communia, quæ auditus non poteſt capere, vt durum, molle, acu <lb/>tum, obtuſum, aſperum, lene, planum, curuum, concauum, conuexum, magnum, <lb/>paruum, & ſupradicta, ideſt figura corpus & cętera, vt ctiam rectum, obliquum, & <lb/>ſimilia.</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">Ariſtoteles circa finem primi capitis libri de ſenſu ait mediante viſu, magis <reg norm="quam" type="context">quã</reg> <lb/>quolibet alio ſenſu, nos percipere ſenſibilia communia. </s> + <s xml:id="s4984" xml:space="preserve">Vbi eundem per ſe, & <lb/>non per accidens magis neceſſarium eſſe quam auditum, tam in ijs quæ ad victum, <lb/>quam in ijs quæ ad ſcientiam pertinent eſſe aſſerit, quia auditus intellectui confert <lb/>per accidens. </s> + <s xml:id="s4985" xml:space="preserve">Vide etiam quod idem ſcribit primo metaphyſicorum. </s> + <s xml:id="s4986" xml:space="preserve">Et ſi ad ali-<lb/>quid perfectè cognoſcendum, oculo ſeſe nobis offerrent ea omnia obiecta, quorum <lb/>ſpecies in imaginatiua formamus, ipſa imaginatiua non egeremus. </s> + <s xml:id="s4987" xml:space="preserve">Sed quia hoc <lb/>fieri non poteſt, hunc <reg norm="theſaurum" type="context">theſaurũ</reg> imaginatiuè, ſeu memoriæ ad conſeruandam imagi <lb/>nem omnium obiectorum ſenſibilium nobis dedit natura, vt ope diſcurſus <reg norm="intellectus" type="simple">intellectꝰ</reg> <lb/>circa dictas imagines, rerum veritatem venari poſſimus. </s> + <s xml:id="s4988" xml:space="preserve">Sed vt ad propoſitum re-<lb/>deamus, beneficio oculi animal liberum eſt, cum ſine ipſo locum mutare nequeat, <lb/>vt ſit tutum. tenebræ, <reg norm="priuatioque" type="simple">priuatioq́;</reg> viſus ſunt ferè vnum, & idem. </s> + <s xml:id="s4989" xml:space="preserve">Neque vllus eſt@ſen-<lb/>ſus, qui ſit magis ſimilis intellectui quam viſus: </s> + <s xml:id="s4990" xml:space="preserve">neque alij ſenſus habent obiecta vi-<lb/>ciſſim communia, quæ non ſint etiam oculo communia, ſed inter oculum, & quem <lb/>libet alium ex ſenſibus, inuenientur quidem obiecta communia, quæ cum alijs non <lb/>communicabunt, vt inter oculum & tactum, figura, acutum, obtuſum, & ſimilia, <lb/>quæ alijs ſenſibus non percipiuntur. </s> + <s xml:id="s4991" xml:space="preserve">Mediante viſu, & auditu etiam, <reg norm="compræhendum" type="context">compræhendũ</reg> <lb/>tur variæ diſtantiæ, <reg norm="ſitusque" type="simple">ſitusq́;</reg> obiectorum, nec non proportiones, & alia quę ab alijs ſen-<lb/>ſibus non compræhenduntur. </s> + <s xml:id="s4992" xml:space="preserve">Multa obiecta deinde ſunt ſubiecta guſtatui, quę alijs <lb/>accidentibus prędita ſunt, vnde cum fuerint ſemel deguſtata, talia, qualia ſunt ab o-<lb/>culo percipiuntur, quod nullus ex alijs ſenſibus præſtabit. </s> + <s xml:id="s4993" xml:space="preserve">Idem de obiectis odora-<lb/>tus dico. </s> + <s xml:id="s4994" xml:space="preserve">Senſuum nullus eſt qui maiorem ſimilitudinem gerat cum vigilia & cum <lb/>vita, quam viſus, neque aliquid eſt, quod magis repræſentet imaginem ſomni, & <lb/>mortis, quàm cęcitas.</s> + </p> + <p> + <s xml:id="s4995" xml:space="preserve">Qui ſibi oculos eruit vt melius ſpecularetur maxima ſtultitia prius obcęcatus fuit <lb/>quia ſoni magis impediunt ſpeculationem quàm lumina, imò qui commodè vult <lb/>contemplari, quantum plus poteſt nititur longius eſſe ab omni ſtrepitu, magis quàm <lb/>à locis luminoſis, & animal magis lætatur lumine quam ſono: </s> + <s xml:id="s4996" xml:space="preserve">& ad ſpeculationem <lb/>nos magis inuitat harmonia luminum variorum colorum & figurarum, quàm har-<lb/>monia ſonorum, præterquam quod inſtrumentum viſus totius corporis partium eſt <lb/>pulcherrima, & in eminentiori loco locata, ſi de inſtrumentis ſenſuum loquamur, & <lb/>veluti fineſtræ animæ. </s> + <s xml:id="s4997" xml:space="preserve">Et ſi Ariſtoteles dicat oculos & aures in vno <reg norm="eodemque" type="simple">eodemq́;</reg> orbe <lb/>exiſtere, volens inferre quod in eodem æquilibrio ſint æqualiter alta non ita ſe ha-<lb/>bet, quia (ſi de homine loquamur) oculus eſt altior aure. </s> + <s xml:id="s4998" xml:space="preserve">Beneficio huius ſenſus, eo <lb/>rum quæ abſunt, & longo iam tempore ſunt mortui, animi ſenſa, & conceptus intel-<lb/>ligimus, neque alia ratione rerum omnium memoria ſeruari poteſt. </s> + <s xml:id="s4999" xml:space="preserve">Si cabala un-<lb/>quam vera fuit, nulla alia ratione eſt deleta, quam quia alicuius ſigni viſibilis medio <lb/>conſeruata non fuerit, & quæcunque non ſcribuntur, ideſt 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 ſemper pretio fuit pictura <reg norm="quam" type="context">quã</reg> muſi-<lb/>ca: </s> + <s xml:id="s5001" xml:space="preserve">obiectis viſibilibus magis quam ijs quæ ſub 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 ſignificare Deum, <lb/>oculi medio id præſtabant.</s> + </p> + <p> + <s xml:id="s5003" xml:space="preserve">Oculus, reſpectu aliorum inſtrumentorum ſenſuum, eſt quaſi 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 naſcitur, ſtatim cæcus euadat, neque aliqua eſt 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">Ariſtoteles ad finem cap .15. lib. pri <lb/>mi poſteriorum ait, clarum eſſe quod ſi aliquis ſenſus deficiat, futurum vt aliqua <lb/>quoque ſcientia deſit. </s> + <s xml:id="s5007" xml:space="preserve">Conſidera, quot ſcientijs careret homo, ſi viſu orbaretur.</s> + </p> + <p> + <s xml:id="s5008" xml:space="preserve">Et in tertio de anima ait, eum qui non ſentit, nihil intelligere poſſe; </s> + <s xml:id="s5009" xml:space="preserve">id quod in-<lb/>de confirmat, quia nihil ſit in intellectu, quod prius non fuerit in ſenſu. </s> + <s xml:id="s5010" xml:space="preserve">Plato in ti <lb/>meo ait, oculos nobis attuliſſe rerum optimarum notitiam, & ſi oculus non fuiſſet ni <lb/>hil eorum, quæ ad cœlum ſpectant inueniri potuiſſe, & <reg norm="cognitionem" type="context">cognitionẽ</reg> diei ac noctis ab <lb/>oculis ortum duxiſſe, vt reuolutiones menſium, & annorum metiri, & tempus co-<lb/>gnoſcere, & inueſtigare ordinem naturæ vniuerſalis poſſemus; </s> + <s xml:id="s5011" xml:space="preserve">quibus <reg norm="philoſophiam" type="context">philoſophiã</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 ſpecialem differentiam inter auditum & viſum, quæ eſt, vt obiectum vi <lb/>ſus ſit permanens, & obiectum auditus tranſitorium ſiue ſucceſſiuum aut, vt alio mo <lb/>do idem dicamus, obiectum viſus particpes ſit æternitatis, illud autem quod eſt au-<lb/>ditus non item, nam auditus tempori ſubiectus eſt, viſus autem minimè. </s> + <s xml:id="s5013" xml:space="preserve">Vel ſi di-<lb/>camus operationem auditus abſque tempore fieri non poſſe cum ſit motio, operatio <lb/>verò viſus, nullo indiget tempore, cum ip ſa ſit momentanea, & propterea inſtan-<lb/>tanea. </s> + <s xml:id="s5014" xml:space="preserve">Nam momentum non eſt motus, nec inſtans 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 noſtrum videamus, & non æſtate, ratio eſt ab <lb/>eiuſdem halitus congelatione, quæ ab extrinſeco frigore fit. </s> + <s xml:id="s5016" xml:space="preserve">Prius enim <lb/>ſcire debes aerem <reg norm="attractum" type="context">attractũ</reg> in pulmone, foras deinde erumpere cum alio <lb/>vapore aliquantulum craſſiore humido, & excrementitio expulſo à natu-<lb/>ra, quæ continuò noſtrum corpus euaporare facit, vnde ſequitur dum aer foras à pul <lb/>mone pellitur, maiorem ſemper materiæ portionem, ea quæ intus attracta eſt exire: <lb/></s> + <s xml:id="s5017" xml:space="preserve">vnde ſtatim vt dicta materia foras expulſa, frigidum aerem offendit, cum conſtet ex <lb/>partibus craſſis, & obnoxiis congelationi, condenſatur in formam vaporis, ad dif-<lb/>ferentiam aeris ambientis qui in ſe eas partes craſſas non habet, à quibus <reg norm="quidem" type="context">quidẽ</reg> par-<lb/>tibus condenſatis, & redditis opacis reflectitur lumen, atque hanc ob cauſam æſtate <lb/>hoc non fit, quia calor vim condenſandi non habet.</s> + </p> + <p> + <s xml:id="s5018" xml:space="preserve">Ventus nihil aliud eſt quam quidam aeris motus, cum condenſatur, ob defectum <lb/>caloris, neque (pace Ariſtotelis dicam) eſt exhalatio ſicca. </s> + <s xml:id="s5019" xml:space="preserve">Exemplum à Vitruuio <lb/>allatum nil planè valet, quantum ſpectatad 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 eſt, quia eius beneficio, non ſolum arcemus à nobis aerem ambientem <lb/>calidum, ſed alium etiam aerem circa nos condenſamus: </s> + <s xml:id="s5021" xml:space="preserve">& quia ordo naturæ eſt hu <lb/>iuſmodi quod quemadmodum calor ſequitur raritatem <reg norm="corporum" type="context">corporũ</reg>, ſic etiam frigus <lb/>eorundem denſitatem ſequatur. </s> + <s xml:id="s5022" xml:space="preserve">Quod ſi vis vt exemplo illuſtrem, diligenter ob-<lb/>ſeruato 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 ſolaris fert, qui per vim, <lb/>dictum aerem rarefactum conſeruabat, ſtatim dictum aerem condenſari cognoſces: <lb/></s> + <s xml:id="s5023" xml:space="preserve">& quia ea condenſatio homogenea non eſt, ob diuerſas rationes, hanc ob cauſam <lb/>percipimus eam aeris impulſionem, & inæqualiter, dum verò eadem vmbra diſce-<lb/>dit, ventus, ferè, ſtatim ceſſat, & ſæpe ante quam dicta vmbra diſcedat; </s> + <s xml:id="s5024" xml:space="preserve">cuius rei cau <lb/>ſa eſt longa mora quam trahi vmbra, ita vt prius abſoluatur reditus aeris ad <reg norm="formam" type="context">formã</reg>, <lb/>quæ ei conuenit in huiuſmodi vmbra, quam faciet nubes dum Sol deregitur.</s> + </p> + <p> + <s xml:id="s5025" xml:space="preserve">Vera non ſunt ea, quæ tibi Arnoldus dixit, vt mihi tuis literis ſignificaſti. </s> + <s xml:id="s5026" xml:space="preserve">Nam ego <lb/>ita dixi, videlicet, quod quoti eſcunque aliquis aſpexerit aliquod punctum in ſuper-<lb/>ficie ſpeculi, </s> + <s xml:id="s5027" xml:space="preserve">tunc imaginem ipſius poſt dictam ſuperficie<unsure/>m videbit duplicatam, ſi <lb/>verò aſpexerit imaginem intra ſpeculum, </s> + <s xml:id="s5028" xml:space="preserve">tunc illud punctum videbit duplicatum, <lb/>huiuſmodi autem rei ratio pendet ab hijs quę ad Franciſcum Vimercatum ſcri-<lb/>pſi, quæ ſi memoria tenes, nullum tibi dubium remanebit. </s> + <s xml:id="s5029" xml:space="preserve">Nam ea tibi omnia <lb/>oſtendi.</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 ſuper primum cap. noni lib. eiuſdem Authoris, ita dicens.</s> + </p> + <quote> + <s xml:id="s5031" xml:space="preserve">Benè eſſe poteſt vt diagonalis (quadrati ſcilicet) numerorum via reperiatur, ſed <lb/>fortaſſe intercedent fracta.</s> + </quote> + <p> + <s xml:id="s5032" xml:space="preserve">Miror te non memoria tenere quid ſint numeri rationales quidúe ſurdi, <reg norm="neque" type="simple">neq;</reg> con <lb/>ſideras, non ſolum non eſſe diuiſibilem in tres æquas partes omnem proportionem <lb/>rationabilem, ſed neque in duas, vt ſunt ſuperparticulares proportiones, necnon <lb/>aliæ innumeræ, ſed cum talia ſcribis te nimis parum verſatum in iſtis rebus oſtendis.</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 eſt calculis qui fiunt de curſu Mercurij, eo quod eius ſitus nullo mo <lb/>do obſeruabilis eſt, nam ipſe nunquam nec vbiuis locorum orbis terrarum viſibilis <lb/>eſt altior .18. gradibus ſupra orizontem, ſed neque confidendum eſſet ſi <reg norm="etiam" type="context">etiã</reg> ipſum <lb/>videremus altum .20. gradibus, </s> + <s xml:id="s5034" xml:space="preserve">propterea quod magna refractio <reg norm="radiorum" type="context">radiorũ</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 eſt, propter diformem ſeu inæqualem craſſiciem vaporum quæ continuò muta <lb/>tur. </s> + <s xml:id="s5035" xml:space="preserve">Imo multoties eum. videre putabimus ſupra orizontem, exiſtente ipſo ſub <lb/>orizonte.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head xml:id="head588" style="it" xml:space="preserve">Quod Ouidius tr anſcurrit 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 ceſpitauit, cum pro itinere vnius diei efficiat, vt Phaeton à <lb/>patre edoctus ſit etiam de itinere annuali.</s> + </p> + <p> + <s xml:id="s5037" xml:space="preserve">Nam, quod Phaeton petat pro curſu vnius diei, clarè patet <reg norm="ex" type="conjecture">ẽx</reg> diuerſis locis, & pri-<lb/>mò vbi ita ſcribit 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 eſt, & 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 eſt altiſſima cęlo. <lb/></s> + <s xml:id="s5043" xml:space="preserve">Vnde mare, & terras ipſi 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 eſt & 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">Dumq́;</reg> ea magnanimus <reg norm="Phaenton" type="context">Phaẽton</reg> miratur, <reg norm="opusque" type="simple">opusq́;</reg> <lb/>Perſpicit, ecce vigil nitido patefecit ab ortu. <lb/></s> + <s xml:id="s5048" xml:space="preserve">Purpureas aurora fores, & plena roſarum. <lb/></s> + <s xml:id="s5049" xml:space="preserve">Atria, diffugiunt ſtellæ, quarum agmina cogit. <lb/></s> + <s xml:id="s5050" xml:space="preserve">Lucifer, & coeli ſtatione 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 ſi (modo credimus) vnum <lb/>Iſſe diem ſine Sole ferunt, incendia lumen Præbebant.</s> + </quote> + <p> + <s xml:id="s5053" xml:space="preserve">Quod autem à Patre inſtruatur etiam de curſu annuali, <lb/>videbitur vbi ita dicit.</s> + </p> + <quote> + <s xml:id="s5054" xml:space="preserve">Nitor in aduerſum, 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">Forſitan & lucos illic, <reg norm="vrbesque" type="simple">vrbesq́;</reg> Deorum. <lb/></s> + <s xml:id="s5058" xml:space="preserve">Concipias animo <reg norm="delubraque" type="simple">delubraq́;</reg> ditia donis <lb/>Eſſe per inſidias iter eſt, <reg norm="formasque" type="simple">formasq́;</reg> ferarum. <lb/></s> + <s xml:id="s5059" xml:space="preserve"><reg norm="Vtque" type="simple">Vtq́;</reg> viam teneas, <reg norm="nulloque" type="simple">nulloq́;</reg> errore traharis. <lb/></s> + <s xml:id="s5060" xml:space="preserve">Per tamen aduerſi, gradieris cornua Tauri. <lb/><reg norm="Aemoniosque" type="simple">Aemoniosq́;</reg> arcus, <reg norm="violentique" type="simple">violentiq́;</reg> ora Leonis. <lb/><reg norm="Sæuaque" type="simple">Sæuaq́;</reg> circuitu curuantem brachia longo. <lb/>Scorpion atque aliter curuantem brachia cancrum. <lb/></s> + <s xml:id="s5061" xml:space="preserve">Nec tibi quadrupedes animoſos 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 eſt 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 eſt lato curuamine limes. <lb/></s> + <s xml:id="s5066" xml:space="preserve"><reg norm="Zonarumque" type="simple">Zonarumq́;</reg> trium contentus fine, <reg norm="polumque" type="simple">polumq́;</reg> <lb/>Effugit auſtralem <reg norm="iunctamque" type="simple">iunctamq́;</reg> aquilonibus arcton. <lb/></s> + <s xml:id="s5067" xml:space="preserve">Hac ſit iter, manifeſta rotæ veſtigia 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 ſiniſterior 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 ſupputatione quinque corporum regularium. <lb/>De aliquibus etiam eorum ſympathijs.</head> + <head xml:id="head591" xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:id="s5071" xml:space="preserve">ID quod à me deſideras, ab alijs etiam factum eſt, ſed ne me putes laborem euita <lb/>re, non præter mittam aliquid tibi ſcribere, 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 iſtisrebus me aliquo modo exercebam.</s> + </p> + <p> + <s xml:id="s5072" xml:space="preserve">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ <reg norm="cuiuſque" type="simple">cuiuſq;</reg> <reg norm="quinque" type="simple">quinq;</reg> cor-<lb/>porum regularium ab vna <reg norm="eademque" type="simple">eademq́;</reg> ſphæra terminatorum ſeu <reg norm="circunſcriptibilium" type="context">circunſcriptibiliũ</reg> cu-<lb/>rabis primum, cognoſcere quantitatem lateris <reg norm="cuiusque" type="simple">cuiusq́;</reg> eorum, talium partium, qua-<lb/>lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici. <lb/></s> + <s xml:id="s5073" xml:space="preserve">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13. <lb/>lib. Eucli. & inuenies <var>.c.d.</var> tertiam partem ſemidiametri <var>.d.b.</var> eſſe partium .33333. æ-<lb/>qualem ſinui arcus <var>.f.e.</var> graduum .19. mi .28. qui quidem arcus <reg norm="demptus" type="context">dẽptus</reg> <reg norm="cum" type="context">cũ</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 cognoſces, ſumendo ſcilicet ſinum medietatis <var>.b.e.</var> hoc eſt ſinum 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 poſtea quadrato lateris exaedri, & quadrato totius diametri <var>.a.b.</var> reſi-<lb/>dui radix quadrata, erit <var>.a.e.</var> latus Tetraedri. </s> + <s xml:id="s5075" xml:space="preserve">Vel ſi duplicaueris ſinum 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, ſiue, <lb/>vt reſiduus totius dimidij circuli, dempto <var>.b.e.</var> iam ſupra 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/>ſius <var>.d.b.</var> & habebis <var>.f.b.</var> latus quæſitum. </s> + <s xml:id="s5077" xml:space="preserve">Vel ſi malis accipe duplum ſinus 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. ſecundi <lb/>Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s> + </p> + <p> + <s xml:id="s5079" xml:space="preserve">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli <var>g.d.<lb/>a.</var> hoc eſt ipſius arcus <var>.b.n.</var> qui tali angulo ſubiacet, quod cum pluribus modis inue-<lb/>niri poſſit, nihilominus, hunc ſeruabis, inuenies primò quantitatem <var>.d.g.</var> quæ eſt ra <lb/>dix quadrata ſummæ duorum quadratorum hoc eſt <var>.d.a.</var> et <var>.a.g.</var> quæ <var>.a.g.</var> æqualis eſt <lb/>diametro <var>.a.b.</var> vt ſcis, dices poſtea, ſi <var>.d.g.</var> correſpondet ipſi <var>.g.a.</var> cui correſpondet <var>.d.<lb/>h.</var> ſemidiametro ſphæræ? </s> + <s xml:id="s5080" xml:space="preserve">tibi veniet <var>.h.k.</var> ſinus arcus <var>.a.h.</var> hoc eſt <var>.b.n.</var> graduum .63 -<lb/>min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du <lb/>plum eſt partium .105142. pro latere Icoſaedri.</s> + </p> + <p> + <s xml:id="s5081" xml:space="preserve">Incipiendo nunc à Tetraedro, ſcire debes, quod pars <var>.a.c.</var> totius diametri <var>.a.b.</var> æ-<lb/>qualis eſt axi ipſius Tetraedri, quæ quidem <var>.a.c.</var> vt ſubſeſquialtera ipſius <var>.a.b.</var> erit par <lb/>tium .13333.</s> + </p> + <p> + <s xml:id="s5082" xml:space="preserve">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-<lb/>thodo, inueniendo primum radicem quadratam trium quartarum quadrati <lb/>ipſius <var>.a.e.</var> lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per<lb/>pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-<lb/>to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit <reg norm="partium" type="context">partiũ</reg> .141416. <lb/>& hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit <lb/>ſuperficiem quæſitam, hoc eſt baſim Tetraedri <reg norm="partium" type="context">partiũ</reg> <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg> .11546192152. <lb/><reg norm="Hanc" type="context">Hãc</reg> demum baſim 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 animaduerſionem, quæ eſt, <lb/>quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-<lb/>qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi <var>.b.f.</var> <lb/>quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo <lb/><var>b.f.</var> habebimus intentum. </s> + <s xml:id="s5084" xml:space="preserve">Et quod hoc verum ſit poſſumus ita demonſtrare.</s> + </p> + <p> + <s xml:id="s5085" xml:space="preserve">Primum, notum nobis eſt, ipſam 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, ipſum triangulum circunſcribentis, terminatur, & à baſi, vt in tertio <lb/>propoſito decimæſeptimæ quartidecimi Eucli. probatur, ex quo ſequitur proportio<lb/>nem huiuſmodi perpendicularis ad axem Tetraedri, hoc eſt ad <var>.a.c.</var> ſeſquioctauam <lb/>eſſe in potentia, ex penultima primi Eucli. </s> + <s xml:id="s5086" xml:space="preserve">Sed cum <var>.d.c.</var> tertia pars ſit ipſius <var>.d.a.</var> vt <lb/>etiam ex .2. propoſito, ſeu corollario decimæſeptimæ .14. lib. diſcurrere licet, cum ex <lb/>dicto corollario <var>.d.c.</var> ſit ſexta pars ipſius <var>.a.b</var>. </s> + <s xml:id="s5087" xml:space="preserve">Quare <var>.d.c.</var> quarta pars erit ipſius <var>.a.c.</var> vn <lb/>de <var>.a.c.</var> ſeſquitertia erit ipſi <var>.a.d.</var> in longitudine, <reg norm="ideoque" type="simple">ideoq́;</reg> quadratum ipſius <var>.a.d.</var> ad qua-<lb/>dratum ipſius <var>.a.c.</var> erit vt .9. ad .16: </s> + <s xml:id="s5088" xml:space="preserve">& ita duplum quadrati ipſius <var>.a.d.</var> hoc eſt quadra-<lb/>tum ipſius <var>.b.f.</var> ad quadratum ipſius <var>.a.c.</var> erit, vt .18. ad .16. hoc eſt ſeſquioctauum, 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 poſtea ipſius <var>.b.e.</var> erit partium .1539838576570176.</s> + </p> + <p> + <s xml:id="s5090" xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in ſemidiametrum, quod <lb/>productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-<lb/>ductum, multiplicando per .100000. ſemidiametrum ſphæræ, tibi dabit columnam <lb/>quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum <lb/>erit ipſum Octaedron partium .1333333333333.</s> + </p> + <p> + <s xml:id="s5091" xml:space="preserve">Pro Icoſaedro autem, oportet prius quantitatem perpendicularis inuenire, quæ <lb/>perpendicularis, per æqualia diuidit baſim ipſius Icoſaedri, quæ vt radix quadrata <lb/>trium quartarum quadrati lateris ipſius baſis, erit partium .91055. talium, qualium <lb/>dictum latus erit partium .105142. cuius medietas eſt .52571. quæ medietas ſi mul-<lb/>tiplicata fuerit cum dicta perpendiculari, dabit totam baſim ſuperficialem, hoc eſt <lb/>ſuperficiem vnius trianguli æquilateris partium ſuperficialium .4786852405. quo <lb/>facto, accipe quadratum duarum tertiarum ipſius, hic ſupra dictæ perpendicularis, <lb/><reg norm="ipſumque" type="simple">ipſumq́;</reg> deme ex quadrato ſemidiametri ſphæræ, hoc eſt, ex quadrato <reg norm="ipſius" type="simple">ipſiꝰ</reg> .100000 <lb/>radix poſtea quadrata reſidui, erit partium .79468. & hæc erit perpendicularis à cen <lb/>tro ſphærę ad vnam baſim ipſius Icoſaedri, quam volueris, quam perpendicularem <lb/>ſi multiplicaueris cum quantitate ſuperficiali, hic ſuperius reperta, vnius baſis, con-<lb/>ſequeris columnam trilateram partium .380401586920540. cuius tertia pars, erit <lb/>partium .126800528973513. pro vna ex .20. </s> + <s xml:id="s5092" xml:space="preserve">Pyramidibus ipſum corpus compo-<lb/>nentibus. </s> + <s xml:id="s5093" xml:space="preserve">Breuius tamen hoc efficiens, ſi multiplicaueris baſim dictam, cum tertia <lb/>parte ipſius perpendicularis, hanc poſtea pyramidem multiplicando per .20. habebis <lb/>totam corpulentiam ipſius Icoſaedri partium .2536010579470260.</s> + </p> + <p> + <s xml:id="s5094" xml:space="preserve">Pro Duodecaedro demum, accipe ſinum gra .36. qui <reg norm="gradus" type="simple">gradꝰ</reg> ſunt pro dimidio quin <lb/>tæ partis totius gyri circularis, <reg norm="qui" type="simple">ꝗ</reg> <reg norm="quidem" type="context">quidẽ</reg> ſinus, erit partium .58778. cuius quadratum <lb/>ſi <reg norm="dem" type="context">dẽ</reg> pſeris ex quadrato <reg norm="ipſius" type="simple">ipſiꝰ</reg> .100000. ſemidiametri circuli <reg norm="circunſcribentis" 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 reſidui, erit perpendicularis du-<lb/>cta à centro dicti circuli ad medium vnius lateris ipſius pentagoni, quæ perp endicu <lb/>laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778. <lb/>Nunc verò dicendo ſi .58778. dat .80902. quid nobis dabit .35684? </s> + <s xml:id="s5096" xml:space="preserve">medietas lateris <lb/>ipſius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipſius penta-<lb/>goni, ad latus ipſius Duodecaedri, quæ multiplicata cum me dietate ſupradicta ip-<lb/>ſius lat eris, hoc eſt cum .35684. producet vnum ex quinque triangulis componenti-<lb/>bus vn um pentagonum, ſeu vnam baſim ipſius Duodecaedri, quod quidem triangu <lb/>lum, erit partium .1752655344. ſu perficialium, quas ſi per quinque multiplicaueris <lb/>habeb is vnam baſim pentagonam dicti corporis partium .8763276720. </s> + <s xml:id="s5097" xml:space="preserve">Dicendum <lb/>poſtea eſt, ſi ad .80901. conuenit ſemidiameter circularis partium .100000. quid <reg norm="con" type="context">cõ</reg> <lb/>ueniet partibus .49116. dabit .60711. pro tali ſemidiametro 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, ſi dempſeris ex quadrato ipſius .100000. ſemidiametro ſphęræ, </s> + <s xml:id="s5098" xml:space="preserve">tuncradix qua-<lb/>drata reſidui, erit perpendicularis à centro ſphæræ ad centrum pentagoni partium, <lb/>79461. cuius tertia pars, ſi multiplicata fuerit cum pentagono ſupra reperto dicti cor <lb/>poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron, <lb/>quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipſius Duo <lb/>decaedri partium .2785354925791680.</s> + </p> + <p> + <s xml:id="s5099" xml:space="preserve">Nunc verò ſi experiri voluerimus vtrum iſti calculi duorum corporum maiorum <lb/>ſint rectè ſupputati, <reg norm="dicemus" type="simple">dicemꝰ</reg> ſi ad <reg norm="corpus" type="simple">corpꝰ</reg> .12. <reg norm="baſium" type="context">baſiũ</reg>, <reg norm="quod" type="wordlist">qđ</reg> eſt <reg norm="partium" type="context">partiũ</reg> .2785354925791680 <lb/>conuenit numerus partium .2536010579470260. ipſius Icoſaedri, quid conueniet <lb/>lateri cubi partium .115476. & inueniemus conuenire latus ipſius Icoſaedri partium <lb/>105138. eo quod probatum ſit in <ref id="ref-0026">.10. propoſitione .14. li. Eucl.</ref> eandem <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe corpulentiæ ipſius Duodecaedri ad corpulentiam ipſius Icoſaedri, quæ lateris <lb/>cubi ad latus Icoſaedri.</s> + </p> + <p> + <s xml:id="s5100" xml:space="preserve">Hæc autem corpora, ita ſibi inuicem, & cum eorum ſphæra harmonicè <reg norm="conueniunt" type="context">conueniũt</reg> <lb/>quemadmodum antiqui philoſophi inuenerunt, vt <reg norm="mirandum" type="context">mirandũ</reg> non ſit, ipſos credidiſ-<lb/>ſe omnia quæ natura conſtant, aliquo pacto exiſtis corporibus fieri. </s> + <s xml:id="s5101" xml:space="preserve">Conſidera quæ-<lb/>ſo quomodo conueniant inuicem Tetraedron, Octaedron, & Icoſaedron, cum uniuſ-<lb/>cuiuſque baſes ſint triangulares æquilateræ intelli gendo ſemper hæc corpora ab ea-<lb/>dem ſphæra circunſcriptibilia.</s> + </p> + <p> + <s xml:id="s5102" xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale <lb/>ſit ei perpendiculari, quæ diuidit baſim Tetraedri per æqualia, vtſupra demonſtra-<lb/>uimus.</s> + </p> + <p> + <s xml:id="s5103" xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baſis <lb/>Tetraedri ad baſim Octaedri ſeruet proportionem ſeſquitertiam, conſonantiæ dia-<lb/>teſſaron. </s> + <s xml:id="s5104" xml:space="preserve">Et proportio omnium ſuperficierum ſiue baſium Octaedri ſimul ſumpta-<lb/>rum, ad omnes baſes ipſius Tetraedri ſimul ſumptas ſit ſeſquialtera, conſonantiæ dia <lb/>pentis. </s> + <s xml:id="s5105" xml:space="preserve">Neque omittendum eſt, quod proportio Octaedriad triplum Tetraedri ſit, <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 eſt ſeſquioctaua, <lb/>vt ſupra vidimus interuallum ſcilicet harmonicum toni maioris.</s> + </p> + <p> + <s xml:id="s5107" xml:space="preserve">Harmonia verò Tetraedri, & Exaedri <reg norm="cum" type="context">cũ</reg> eorum ſphæra, talis eſt, vt proportio dia <lb/>metriſphæræ, potentia, tripla ſit lateri Exaedri, & ſeſquialtera lateri Tetraedri, ex <lb/>quo ſequitur latus Tetraedri potentia duplum exiſtere lateri Exaedri. </s> + <s xml:id="s5108" xml:space="preserve">Interuallum <lb/>enim triplum in harmonicis, componitur ex diapaſon, & diapente, & ſonat ſpeciem <lb/>diapentis. </s> + <s xml:id="s5109" xml:space="preserve">Duplum verò eſt diapaſon, ſeſquialterum autem eſt di apente, quę con-<lb/>ſonantiæ perfectiſſimæ ſunt.</s> + </p> + <p> + <s xml:id="s5110" xml:space="preserve">Proportio verò diametri ſphæræ, potentia dupla eſt lat eri Octaedri, conſonantię <lb/>diapaſon. </s> + <s xml:id="s5111" xml:space="preserve">Ex quo ſequitur proportionem lateris Tetraedri ad latus Octaedri, po-<lb/>tentia, ſeſquitertiam eſſe, hoc eſt conſonantiæ diateſſaron, & proportionem lateris <lb/>Octaedri ad latus Exaedri, potentia, ſeſquialteram eſſe, ita quod quatuor iſtæ poten <lb/>tiæ, ideſt diametri ſphæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-<lb/>ſtituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenſam .6. 4. 3. 2. (dixi <lb/>ferè, quia ditonus ſupra terminum .3. vel ſemiditonus ſub 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 ſphæræ triplus eſt longitudine ad <reg norm="perpendicularem" type="context">perpendicularẽ</reg> ductam <lb/>à centro ſphæræ ad baſim Octaedri, quæ proportio, vt ſupra dictum eſt, dicitur dia-<lb/>paſondiapente, 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 ſphæræ ſeſquialter eſt longitudine axi Tetraedri, conſonantiæ <lb/>diapentis. </s> + <s xml:id="s5114" xml:space="preserve">Axis autem Tetraedri ſeſquitertius eſt longitudinis ſemidiametro ſphæ-<lb/>ræ conſonantiæ diateſſaron. </s> + <s xml:id="s5115" xml:space="preserve">Ita quod iſti tres termini, qui ſunt, diameter ſphæræ, <lb/>axis Tetraedri, & ſemidiameter ſphæræ conſtituunt etiam valde perfectam harmo-<lb/>niam huiuſmodi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-<lb/>lentiam Tetraedri tripla eſt, conſonantiæ iam ſupradictæ diapaſondiapente. </s> + <s xml:id="s5116" xml:space="preserve">Si ve-<lb/>rò de vniſono aliquid videre deſideras, conſidera æqualitatem dupli quadrati dia-<lb/>metri ipſius ſphæræ, cum omnibus baſibus Exaedri, vel potentia diametri ſphæræ <lb/>cum duabus potentijs ſimul ſumptis, quarum vna eſt lateris Tetraedri, reliqua verò <lb/>lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baſibus Exae <lb/>dri. </s> + <s xml:id="s5117" xml:space="preserve">Nec mihi videtur ſilentio inuoluendum eſſe, antequam vlterius progrediar no<lb/>tabilem ſympatiam inter triangulum æquilaterum, & Tetraedron (<reg norm="quanuis" type="context">quãuis</reg> <reg norm="triangulum" type="context">triangulũ</reg> <lb/>corpus non ſit) non ſolum 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, ijſdem lateribns exiſtentibus, cum ex quadrato rom-<lb/>bus, vel ex pentagono ęquiangulo, pentagonum non æquiangulum & c. efficiatur) <lb/></s> + <s xml:id="s5119" xml:space="preserve">ſed quod quemadmodum latus trianguli æquilateri ſeſquitertium potentia eſt per-<lb/>pendiculari ipſum per æqualia diuidenti, ita latus Tetraedri, ſeſquialterum eſt po-<lb/>tentia axi ipſius Tetraedri, vnde cum dempta fuerit illa proportio ſeſquitertia, ex <lb/>hac ſeſquialtera relinquetur nobis proportio ſeſquioctaua, inter perpendicularem <lb/>trianguli, & axem Tetraedri (quod etiam ſupra demonſtrauimus.) </s> + <s xml:id="s5120" xml:space="preserve">Tranſeamus nunc <lb/>hęc, nec omittamus tamen ſympatias quaſdam inter Exaedron, Octaedron, & Tetra <lb/>edron, hoc eſt quod eadem proportio ſit inter corpulentias Exaedri, & Octaedri, <lb/>quæinter eorum ſuperficies, nec non, vt latus Exaedri ad ſemidiametrum ſphæræ. <lb/></s> + <s xml:id="s5121" xml:space="preserve">Proportio verò baſis Exaedri ad baſim Tetraedri, vtlatus Tetraedri ad perpendicu <lb/>larem diuidentem per æqualia eius baſim.</s> + </p> + <p> + <s xml:id="s5122" xml:space="preserve">Hactenus ſatis dictum ſit de Tetraedro, Exaedro, & Octaedro cum ſphæra. </s> + <s xml:id="s5123" xml:space="preserve"><reg norm="Dicem" type="context">Dicẽ</reg> <lb/>dum nunc cenſeo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè <lb/>omnia hæc ab antiquis philoſophis inuenta ſint, quorum primum eſt, quod tam ba-<lb/>ſis Duodecaedri, quam Icoſaedri, ab vno <reg norm="eodemque" type="simple">eodemq́;</reg> circulo circunſcriptibiles ſunt, ve <lb/>rùm, talis paſſio accidit etiam baſibus Exaedri & Octaedri. </s> + <s xml:id="s5124" xml:space="preserve">Præterea quemadmo-<lb/>dum in Duodecaedro, quilibet angulus ſolidus terminatur tribus angulis pentago-<lb/>norum æquiangulorum ita in Icoſaedro, quilibet angulus ſolidus viceuerſa 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 ſolidos angulos trian-<lb/>gulares, habet Duodecaedron, quot baſes triangulares continet Icoſaedron.</s> + </p> + <p> + <s xml:id="s5127" xml:space="preserve">Et Icoſaedron, tot ſolidos angulos <reg norm="pentagonos" type="context">pẽtagonos</reg>, quot baſes <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 ſuperficiales. </s> + <s xml:id="s5129" xml:space="preserve"><reg norm="Eademque" type="context simple">Eadẽq́;</reg> <lb/>proportio eſt omnium baſium ſimul <reg norm="ſumptarum" type="context">ſumptarũ</reg> Duodecaedri ad omnes baſes ſimul <lb/>ſumptas ipſius Icoſaedri, quæ corpulentiæ ipſius Duodecaedri ad corpulentiam <lb/>Icoſaedri (quamuis hęc paſſio accidat Exaedro cum Octaedro, vt ſpra diximus) quę <lb/>quidem proportio, eadem etiam eſt, quę lateris Exaedri ad latus Icoſaedri, vt ſu-<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 Ingenioſißimo Horologiorum Serenißimi <lb/>Sabaudiæ Ducis Artifici.</head> + <p> + <s xml:id="s5130" xml:space="preserve"><emph style="sc">NOnnvnqvam</emph> conſideraui mirabilem pulchritudinem, ſimul cum vtili-<lb/>tate coniunctam, illorum horologiorum, quæin Germania conſtruuntur <lb/><reg norm="cum" type="context">cũ</reg> mobili Rete, ſeu Aranea Aſtrolabij <reg norm="ſuper" type="simple">ſuꝑ</reg> <reg norm="Tabulam" type="context">Tabulã</reg> regionis, in <reg norm="quibus" type="simple">ꝗbus</reg> <reg norm="conti" type="context">cõti</reg> <lb/>nuo <reg norm="videntur" type="simple">vident̃</reg> oriri, <reg norm="occidereque" type="simple">occidereq́;</reg> cæleſtia ſigna, cælum mediare ſupra orizon <lb/><reg norm="tem" type="context">tẽ</reg><unsure/>, necnon ſub 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 diſtinctum habent in .24. horas, qua <lb/>propter diametrum limbi, minorem duobus palmis, ſeu ſemipede eſſe non oportet <lb/>neinterſtitia horarum iuſtò breuiora ſeu anguſtiora efficiantur, etiam ne interualla <lb/>dentium rotæ indicis nimis anguſta ſint. </s> + <s xml:id="s5132" xml:space="preserve">Sed quia talis magnitudo vt plurimum in-<lb/>commoda exiſtit. </s> + <s xml:id="s5133" xml:space="preserve">Ideo non inutile fore cogitaui, ſi modus aliquis inuentus fuerit, <lb/>vt ea omnia efficiantur in limbo diuiſo tantummodo in .12. horas æquales, <reg norm="ipſumque" type="simple">ipſumq́;</reg> <lb/>inueni, qui quidem erit, efficiendo vt Tabula (in qua deſignantur cęleſtes domus, <lb/>cum almicantarat, atque azimut) Reti ſubiectæ, mobilis ſit, tardior tamen ipſo Re-<lb/>te cum indice, pro duplo temporis, hoc eſt, quod eo tempore, quo Aranea cum in <lb/>dice circunuoluetur ſpacio .12. horarum vno gyro perfecto, ipſa Tabula efficiat tan <lb/>tummodo ſexinterſtitia horarum. </s> + <s xml:id="s5134" xml:space="preserve">Ideſt dum Tabula dicta eſſicit vnam integram re <lb/>uolutionem, Aranea, ſeu Zodiacus cum indice, duas efficiat reuolutiones. </s> + <s xml:id="s5135" xml:space="preserve">Ita quod <lb/>Aranea cum indice perficiet vnam reuolutionem ſpaci o temporis .12. horarum, Ta-<lb/>bula verò perficiet eam ſpacio temporis .24. horarum. </s> + <s xml:id="s5136" xml:space="preserve">Vnde ſequetur quod Ara-<lb/>nea ſeu Zodiacus cum indice, ſpacio .24. horarum perfectè circunuoluetur ſupra Ta-<lb/>bulam, & ita huiuſmodi horologia, in hoc nihil differrent ab illis ſupradictis. </s> + <s xml:id="s5137" xml:space="preserve">Vt au <lb/>tem facias dictam tabulam tardiorem duplo temporis Araneæ cum indice, quamuis <lb/>diuerſis modis hoc fieri poſſit, pręſtantiorem tamen iudico, ſi 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="vnaquęque" type="simple">vnaquęq;</reg> liberè poſſit volui, ſimiliter <lb/>ſi cum ea horologii particula (quę <reg norm="circumagit" type="context">circũagit</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">aũt</reg> ipſum vocabo, <emph style="sc">colinvm</emph>, qui ſubro-<lb/>ta fuſi reperitur) coniunxeris alium colinum quem, ſecundum vocabo, <reg norm="concentricum" type="context">concentricũ</reg> <lb/>verò cum primo, cum <reg norm="eoque" type="simple">eoq́;</reg> conſolidato, numerum verò dentium, tam Rotę adiunctę <lb/>quam ſecundi colini, varijs modis poteris inuenire, quorum primus erit, vt numerus <lb/>dentium ſecundæ Rotę duplus exiſtat numero dentium primę, efficiendo ſecundum <lb/>colinum eiuſdem numeri dentium quo primum, ſed quia interualla dentium huiuſ-<lb/>modi Rotę, nimis anguſta fortaſſe reſultabunt, </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> ſecun <lb/>dam <reg norm="Rotam" type="context">Rotã</reg> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> numeri <reg norm="dentium" type="context">dentiũ</reg> cuius eſt prima. <reg norm="ſecundum" type="context context">ſecũdũ</reg> vero colinum, medietatis <lb/>numeri dentium cuius erit primus. </s> + <s xml:id="s5139" xml:space="preserve">Attamen ſi primus colinus eſſet .4. dentium, ſecun <lb/>dum oporteret eſſe duorum dentium, vnde motus ſecundę Rotę non eſſet ita conti-<lb/>nuus. </s> + <s xml:id="s5140" xml:space="preserve">Quapropter alium etiam mòdum excogitaui, hoc eſt, cupiendo vt ſecundus <lb/>colinus, extribus dentibus exiſtat, ſi primus ex .4. repertus fuerit, oportebit prius ex <lb/>regula de tribus, numerum quendam inuenire quo inuento ipſum duplicare, & hunc <lb/>duplicatum numerum conueniet ſecundam Rotam habere, vt ipſa poſſit ab illo co-<lb/>lino <reg norm="trium" type="context">triũ</reg> <reg norm="dentium" type="context">dentiũ</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, ſi prima Rota conſtaret ex .36. dentibus, dicendum eſſet, ſi <lb/>4. conuenit cum .36. cum quibus conuenient .3. & inueniemus .27. cum quo numero <lb/>dicta ſecunda 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 dictę ſecundæ Rotæ, vt duplo temporis circunuoluatur quo prima. </s> + <s xml:id="s5143" xml:space="preserve">Sed <lb/>ſi primus colinus conſtaret ex .6. dentibus, exiſtente ſua Rota ex .36. <reg norm="vellemusque" type="simple">vellemusq́;</reg> <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſecundus exiſteret ex .4. </s> + <s xml:id="s5144" xml:space="preserve">tunc ſuam Rotam oporteret habere dentes .48. ex dicta re-<lb/>gula. </s> + <s xml:id="s5145" xml:space="preserve">Si autem primus colinus conſtaret ex numero impari, nihil referret, dummo-<lb/>do huiuſmodi numerus impar, ſeu par, exiſteret pars propria numeri dentium, vel <lb/>ipſius dupli primæ Rotę, hoc eſt, eſſet pars aliquota numeri dentium ipſius primæ <lb/>Rotæ vel ipſius dupli. </s> + <s xml:id="s5146" xml:space="preserve">In ijs verò horologiis in quibus duplum numeri dentium di-<lb/>ctę primę Rotę 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 conſideranti, patebit, vt nullis verbis indigeat cum <lb/>ſemper numerus dentium ſecundę Rotę multiplex eſſe debeat numero dentium ſe-<lb/>cundi colini. </s> + <s xml:id="s5148" xml:space="preserve">Idem autem non dico de prima Rota cum ſuo colino, hoc eſt, vt nu-<lb/>merus primę multiplex ſit numero ſui colini, nam hoc neceſſarium non eſt. </s> + <s xml:id="s5149" xml:space="preserve">Pona-<lb/>mus exempli gratia primum colinum conſtare ſex dentibus, ſuam vero Rotam den-<lb/>tibus .21. cuius quidem numeri, 6. non eſt pars aliquota, ſed dupli ipſius .21. ipſe .6. <lb/>eſt pars aliquota. </s> + <s xml:id="s5150" xml:space="preserve">Nunc verò ſi voluerimus numerum dentium ſecundæ Rotę inue-<lb/>nire, cuius colinus ex quinque dentibus exiſtat (ſuppoſito primo ex .6. conſtare) </s> + <s xml:id="s5151" xml:space="preserve">tunc <lb/>ex regula de tribus, diuiſo producto, quod fit ex .21. in .5. per .6. exibit .17. cum di-<lb/>midio, cuius duplum eſſet .35. qui multiplex eſt ipſi quinque. </s> + <s xml:id="s5152" xml:space="preserve">Reperto igitur nume <lb/>ro ſecundę Rotę, cum numero ipſius colini, oportet nunc ſcire modum compoſitio-<lb/>nis, ſeu coniunctionis harum rerum, hoc eſt duorum colinorum <reg norm="concentricorum" type="context">concentricorũ</reg> (ſed <lb/>de ijs ſatis iam ſuperius dictum fuit) duarum Rotarum concentricarum cum Tabula, <lb/>cum Zodiaco, & cum indice, ſeu Oſtenſore, cuius quidem Oſtenſoris medietas tan <lb/>tummodo nobis ſufficiet. </s> + <s xml:id="s5153" xml:space="preserve">Sciendum igitur nunc eſt quod cum primus colinus re-<lb/>uoluat totam primam Rotam, ſpacio temporis .12. horarum, oportet vt eius axis, ſeu <lb/>arbor voluat oſtenſorem, <reg norm="Zodiacumque" type="simple">Zodiacumq́;</reg>, eodem temporis ſpacio, & quia Rota hęc <lb/>inalterabis eſt, propter eius coniunctionem cum ſuo colino, & nos oporteat indicem <lb/><reg norm="Zodiacumque" type="simple">Zodiacumq́;</reg>, quotidie ferè, dirigere, <reg norm="ſuisque" type="simple">ſuisq́;</reg> locis collocare, ideo nos oportet, indi <lb/>cem, Zodiacum, & primam Rotam, ita cum axe, ſeu arbore coniungere, vt poſſimus <lb/>dicta omnia efficere. </s> + <s xml:id="s5154" xml:space="preserve">Pars igitur Arboris, ſeu axis dicti, quæ ingredi debet in prima <lb/>Rota, ſit rotunda, & contigua ipſi Rotæ, non autem continua, vel cum Rota conſoli-<lb/>data. </s> + <s xml:id="s5155" xml:space="preserve">Pars verò quę per foramen Zodiaci, ſeu Araneę tranſibit, ſit quadrata vſque <lb/>ad <reg norm="ſummitatem" type="context">ſummitatẽ</reg> ipſius axis (tali ſpiſſitudine, vt in claui ipſius horologij ingredi poſ-<lb/>ſit) & ita foramen ipſius Araneę, quadratum ſit, Oſtenſor autem circa axem, com <lb/>poſitus ſit tali ordine, vt circa paruum circulum volui poſſit, qui paruus circulus ha-<lb/>beat quadratum foramen, per quod tranſeat axis, qui axis aliquantulum emineat <lb/>ſupra <reg norm="oſtenſorem" type="context">oſtenſorẽ</reg>. </s> + <s xml:id="s5156" xml:space="preserve">Sub Aranea vero vel Zodiaco, locata erit Tabula, vt <reg norm="nunc" type="context">nũc</reg> dicemus, <lb/>ſed ſciendum eſt prius, quod inter Tabulam, & ſuam ſecundam Rotam, aliam lami-<lb/>nam immobilem interpoſitam eſſe oportet, quę circulare foramen habeat, per <reg norm="quod" type="wordlist">qđ</reg> <lb/>quędam breuis fiſtula tranſeat circundans axem & coniungens <reg norm="Tabulam" type="context">Tabulã</reg> cum ſua Ro-<lb/>ta, cuius quidem fiſtulæ ſuperficies concaua, rotunda ſit, ſuperficies verò extrinſe-<lb/>ca, nontota, niſi ea pars, quę ſecundam Rotam ingreditur, vt in rotundo foramine <lb/>ipſius Rotę, dicta fiſtula volui poſſit, pars vero extrinſeca quę Tabulam ingredi de-<lb/>bet, ſit 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, meridianę, & 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 ſcire modum ipſo vtendi. </s> + <s xml:id="s5159" xml:space="preserve">Qua-<lb/>propter quotieſcunque volueris, aſpice Solis locum in Zodiaco, Ephemeridibus me <lb/>diantibus, idem dico de vnoquoque reliquorum planetarum. </s> + <s xml:id="s5160" xml:space="preserve">Inuento poſtea So-<lb/>lis loco in noſtro Zodiaco horologij, manu mediante, volue oſtenſorem, ita, vt li-<lb/>nea fiduciæ tranſeat per gradum Solis, deinde, claui ipſius horologij mediante, vol-<lb/>ue indicem, ita cum Zodiaco coniunctum, vt linea fiducię, punctum, ſeu partem ho-<lb/>rę oſtendat in limbo horologii, quę quidem hora notanda eſt ſi fuerit ex ijs quę in-<lb/>cipiunt à meridie vſque ad mediam noctem, vel à media nocte vſque ad meri-<lb/>diem, </s> + <s xml:id="s5161" xml:space="preserve">tunc acu ſupradicta mediante, poſita in aliquo illorum quatuor foraminum, <lb/>circunuoluenda eſt Tabula, ita, vt extremitas lineę meridianę ſupra orizontem, ex <lb/>ęquo incidat inter duodecimam horam, & lineam fiducię, computum incipiendo à <lb/>duodecima hora, ſi vero dicta indicis hora fuerit ex ijs quę <reg norm="incipiunt" type="context">incipiũt</reg> à media nocte & <lb/>deſinunt poſtea in meridie, oportebit, acu mediante, circunuoluere Tabulam, quo-<lb/>uſque punctum extremum meridianæ ſub terra, medio loco exiſtat inter <reg norm="duodecimam" type="context">duodecimã</reg> <lb/>horam, & horam oſtenſam à linea fiducię. </s> + <s xml:id="s5162" xml:space="preserve">Quo facto continuo videbis erectam <reg norm="cae- li" type="simple">cę-<lb/>li</reg> figuram. </s> + <s xml:id="s5163" xml:space="preserve">& quia vidiſti loca planetarum in Ephemeridibus, videbis etiam <lb/>eorum loca accidentalia in domibus ſcilicetaccidentalibus, ſi aliquas fixarum in <lb/>Aranea deſiderabis, accipere poteris Ocu. ♉, cor. ♌, ſpi. ♍, Liram, Aquilam, & <lb/>Arcturum, dum locus fuerit capax. </s> + <s xml:id="s5164" xml:space="preserve">Nec te moueat, quod oportebit lineam fiducię <lb/>ſupra gra. Solis quotidie collocare, quod nihil refert. </s> + <s xml:id="s5165" xml:space="preserve">Nam oportet etiam quoti-<lb/>die cordam fuſo 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 Aſtrologia Iudiciaria.</head> + <head xml:id="head595" style="it" xml:space="preserve">Fu<unsure/>llustriſſ <var>.D.</var> Volfardo Aiſeſtain.</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 oſtendis ſciendi rationem, quare ego non vna methodo ad omnes <lb/>propoſitiones demonſtrandas vſus ſim, hoc eſt, </s> + <s xml:id="s5167" xml:space="preserve">quare non omnia ea Eucl. <lb/>Theoremata citem in vnaquaque propoſitione, quę ad <reg norm="eam" type="context">eã</reg> demonſtrandam <reg norm="faciunt" type="context">faciũt</reg>, <lb/>quemadmodum in mea Gnomonica vidiſti me aliquando omiſiſſe. </s> + <s xml:id="s5168" xml:space="preserve">Reſpondeo <reg norm="quod" type="simple">ꝙ</reg> <lb/>mathematicę demonſtrationes, hominibus Euclidis Elementa poſſidentibus, non in <lb/>digent aliqua citatione numerorum Theorematum ipſius Euclidis, & ſi aliquando <lb/>vſus ſum aliqua citatione eorundem, hoc feci propter conſuetudinem noſtri tempo <lb/>ris, vel etiam ad faciliorem intelligentiam illorum, quibus ſcribebam. </s> + <s xml:id="s5169" xml:space="preserve">Sed omnia <lb/>quamuis minima citare, vt <reg norm="faciunt" type="context">faciũt</reg> nonnulli, mihi, nimis laborioſum, <reg norm="ſuperfluumque" type="simple">ſuperfluumq́;</reg> <lb/>videtur, preſertim ijs (vt dixi) qui memoria tenent prima Elementa. </s> + <s xml:id="s5170" xml:space="preserve">Hęc igitur <lb/>eſt vna ratio. </s> + <s xml:id="s5171" xml:space="preserve">Alia, quia multoties, ita coniuncta eſt ſpeculatio cum ipſa concluſio <lb/>ne, vt mihi ſępius viſum ſit ſuperfluum, aliquid de ipſa theoria ſcribere. </s> + <s xml:id="s5172" xml:space="preserve">In iis <lb/>enim, quę dum puer eramſcripſi, videbis ſcrupuloſam illam methodum, ſed po-<lb/>ſtea, non niſi in arduis propoſitionibus me nihil eſſentiale prętermittere.</s> + </p> + <p> + <s xml:id="s5173" xml:space="preserve">Circa vero id de quo me interrogas, ſcilicet, vtrum putem omnia vera eſſe, ea <lb/>quę ſcripta reperiuntur in libris Aſtrologæ iudiciarię. </s> + <s xml:id="s5174" xml:space="preserve">Reſpondeo 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 falſa eſſe. </s> + <s xml:id="s5175" xml:space="preserve">Nam illa multitudo partium, vt pars vitę, pars Hylech, pars <lb/>futurorum, & reliquę omnium domorum cœleſtium, ſalua parte fortunę, ſunt merę <lb/>nugę. </s> + <s xml:id="s5176" xml:space="preserve">Idem dico de faciebus, ſiue decanis, de terminis, & de gradibus ipſis, vt pu-<lb/>ta azemenis, puteis, vacuis, fumoſis, & 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">Obſeruationes etiam complexionum ſeu inſluentiarum ipſorum Planetarum rectè <lb/>factæ ſunt, quę etiam à coloribus ipſorum Planetarum ferè iudicari poſſunt. </s> + <s xml:id="s5180" xml:space="preserve">Con-<lb/>iunctiones <reg norm="aſpectusque" type="simple">aſpectusq́;</reg> ipſorum inuicem, ſimiliter mirabilia faciunt, & ex maiori par <lb/>te, ea, quę de iſtis ſcribuntur vera ſunt. </s> + <s xml:id="s5181" xml:space="preserve">Reuolutiones annuę ſimiliter, cum Domino <lb/>anni. </s> + <s xml:id="s5182" xml:space="preserve">Dominum verò orbis <reg norm="Diuiſoremque" type="simple">Diuiſoremq́;</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 ſi <lb/>milia omnia nego. </s> + <s xml:id="s5184" xml:space="preserve">Antiſcia, vera ſunt, ideſt influunt, malos tamen effectus, alia <lb/>plus alia verò minus, prout aliqua eorum ſunt tetragona, alia verò trigona, alia ma-<lb/>gna, alia parua, magna ſunt, vt Arietis cum Virgine, & Librę cum Piſcibus, parua ve <lb/>rò, <reg norm="debiliaque" type="simple">debiliaq́;</reg> Geminorum cum Cancro, & Sagittarij <reg norm="cum" type="context">cũ</reg> Capricorno. </s> + <s xml:id="s5185" xml:space="preserve">Sed difuſius <lb/>hęc <reg norm="omnia" type="wordlist">oĩa</reg> videbis in meo illo particulari tractatu, de quo tibi aliàs dixi, in quo multa <lb/>videbis, quę omnia ab experientia, ex multis à me obſeruatis, comprobata ſunt, <lb/>quem quidem tractatum cum quibuſdam alijs meis ſpeculationibus in lucem prode <lb/>re cupio, ſi fieri poterit, antequam ad directionem mei Horoſcopi cum corpore <lb/>Martis Anęretę perueniam, quę quidem directio circa annum milleſimum quin-<lb/>genteſimum nonageſimum ſecundum 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 reſpiciunt, benignus lector corrigat, quiſciat multos errores in <lb/>editione irrepſiſſe, 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 ſuperficialis, erit ac</xhtml:td> + <xhtml:td xml:space="preserve">in vnitate, ſupreficialis 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 ſunt</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, ſint</xhtml:td> + <xhtml:td xml:space="preserve">tertiæ ſint</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">eſt</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">diſtinguendo</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">ſubſequens</xhtml:td> + <xhtml:td xml:space="preserve">ſubſequentem</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">ſpeculari</xhtml:td> + <xhtml:td xml:space="preserve">conſiderari</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> ſunt 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">addendũ</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">ſingulos itinere</xhtml:td> + <xhtml:td xml:space="preserve">ſingulos 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">ſumma</xhtml:td> + <xhtml:td xml:space="preserve">ſummam</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">deſideraremus</xhtml:td> + <xhtml:td xml:space="preserve">deſiderauerimus</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">diſpoſitis facit</xhtml:td> + <xhtml:td xml:space="preserve">diſpoſitis, 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">diſpoſitis, facit</xhtml:td> + <xhtml:td xml:space="preserve">diſpoſitis, 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">ſufficeret</xhtml:td> + <xhtml:td xml:space="preserve">ſuffecißet</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 ſecundam</xhtml:td> + <xhtml:td xml:space="preserve">quorum ſecundum</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">ſecundæ</xhtml:td> + <xhtml:td xml:space="preserve">ſecundi</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">diſtantius</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">ſit</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, ſi quis</xhtml:td> + <xhtml:td xml:space="preserve">patebit, quòd ſi 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">paſtam</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">ꝓponit</reg> <reg norm="non" type="context">nõ</reg> <reg norm="concludit" type="context">cõcludit</reg> melius <reg norm="autem" type="wordlist">aũt</reg></xhtml:td> + <xhtml:td xml:space="preserve">proponit, <reg norm="non" type="context">nõ</reg> concludit, melius <reg norm="autem" type="wordlist">aũt</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 ſuſtineantur</xhtml:td> + <xhtml:td xml:space="preserve">materiales ſuſtineantur</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">exiſtente, 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">ſufficere</xhtml:td> + <xhtml:td xml:space="preserve">ſufficeret</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">verſa</xhtml:td> + <xhtml:td xml:space="preserve">verſam</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">ſit</xhtml:td> + <xhtml:td xml:space="preserve">ſint</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">ſcindenda</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">oppeſitus</xhtml:td> + <xhtml:td xml:space="preserve">oppoſitum</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">ſestineret</xhtml:td> + <xhtml:td xml:space="preserve">ſustineret</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">ſuffieiant</xhtml:td> + <xhtml:td xml:space="preserve">ſufficiat</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"> ſubſeſquialter<unsure/> + </xhtml:td> + <xhtml:td xml:space="preserve">ſubſeſquialterum</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">philoſophi ſupra</xhtml:td> + <xhtml:td xml:space="preserve">philoſophi, proximè ſupra</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 menſe</xhtml:td> + <xhtml:td xml:space="preserve">contingeret vt menſe</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 conſiderans</xhtml:td> + <xhtml:td xml:space="preserve">regionis) in quo non conſiderauit</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 poſuerimus, 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 ipſos</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">phyloſophiæ</xhtml:td> + <xhtml:td xml:space="preserve">philoſophiæ</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">ſunt 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, creſcat</xhtml:td> + <xhtml:td xml:space="preserve">diffundatur, & creſcat</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 ſer</xhtml:td> + <xhtml:td xml:space="preserve">aliquando me ſer</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">ſoli, quod punctum</xhtml:td> + <xhtml:td xml:space="preserve">ſolis, 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 ſit futurum, vt</xhtml:td> + <xhtml:td xml:space="preserve">neceßarium ſit, 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 ſi velimus</xhtml:td> + <xhtml:td xml:space="preserve">ſi 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"> poteſt 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">ſit circulus</xhtml:td> + <xhtml:td xml:space="preserve">ſit 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 ſi 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">deſignabimus</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">ſi quæ</xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="ſique" type="simple">ſiq;</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">ſupradicta, minuta</xhtml:td> + <xhtml:td xml:space="preserve">ſupradicta 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 cœlo</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">ſit & ob id</xhtml:td> + <xhtml:td xml:space="preserve">ſit, 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> + <xhtml:td xml:space="preserve">in numero</xhtml:td> + <xhtml:td xml:space="preserve">numero</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">cum non videat ſi</xhtml:td> + <xhtml:td xml:space="preserve">cum non videat quod ſi</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">ſuo centro</xhtml:td> + <xhtml:td xml:space="preserve">ſuis centris</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">39</xhtml:td> + <xhtml:td xml:space="preserve">mille milliaria</xhtml:td> + <xhtml:td xml:space="preserve">mille milliariorum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">41</xhtml:td> + <xhtml:td xml:space="preserve">t<unsure/>recent a mille</xhtml:td> + <xhtml:td xml:space="preserve">tercenties mille</xhtml:td> + </xhtml:tr> + </xhtml:table> + <pb n="440" file="0440" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0440"/> + <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">257</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve">quem</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">257</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">quem</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">257</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <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 huiuſmodi ſitu, 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">ſupponuntur</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 ſit</xhtml:td> + <xhtml:td xml:space="preserve">a.K. ſit</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">ſemidiameter eße vnius</xhtml:td> + <xhtml:td xml:space="preserve">ſemidiameter 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 vbieſt <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 ſeu 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 ſub 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">eſt</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">ſatile</xhtml:td> + <xhtml:td xml:space="preserve">ſatilis</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">circunſcriptibilis</xhtml:td> + <xhtml:td xml:space="preserve">circunſcribentis</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="ſolidiſque" type="simple">ſolidiſq;</reg> + </xhtml:td> + <xhtml:td xml:space="preserve">anguli contingentiæ <reg norm="ſolidique" type="simple">ſolidiq;</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> + <xhtml:td xml:space="preserve">diſtincta, procedendo</xhtml:td> + <xhtml:td xml:space="preserve">diſtincta ſint, 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">eaſdem</xhtml:td> + <xhtml:td xml:space="preserve">eoſdem</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> + </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">quotieſcunque</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">ſi conſtitueremus</xhtml:td> + <xhtml:td xml:space="preserve">ſi nos conſtitueremus</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">ſimili ad</xhtml:td> + <xhtml:td xml:space="preserve">ſimili. 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">ipſi ellipſis</xhtml:td> + <xhtml:td xml:space="preserve">ipſius ellipſis</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> + <xhtml:td xml:space="preserve">vel remotiori</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">336</xhtml:td> + <xhtml:td xml:space="preserve">19</xhtml:td> + <xhtml:td xml:space="preserve">ſpeculi ſuperficiem</xhtml:td> + <xhtml:td xml:space="preserve">ſpeculi planam ſuperficiem</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">336</xhtml:td> + <xhtml:td xml:space="preserve">22</xhtml:td> + <xhtml:td xml:space="preserve">reflectit</xhtml:td> + <xhtml:td xml:space="preserve">reflectitur</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">337</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <xhtml:td xml:space="preserve">vnam tantummodo imaginem</xhtml:td> + <xhtml:td xml:space="preserve">vna tantummodo imago</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">337</xhtml:td> + <xhtml:td xml:space="preserve">12</xhtml:td> + <xhtml:td xml:space="preserve">ipſæ</xhtml:td> + <xhtml:td xml:space="preserve">ipſi</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">337</xhtml:td> + <xhtml:td xml:space="preserve">42</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">338</xhtml:td> + <xhtml:td xml:space="preserve">14</xhtml:td> + <xhtml:td xml:space="preserve">protracta</xhtml:td> + <xhtml:td xml:space="preserve">protracto</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">338</xhtml:td> + <xhtml:td xml:space="preserve">26</xhtml:td> + <xhtml:td xml:space="preserve">ratiotinare</xhtml:td> + <xhtml:td xml:space="preserve">apta</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">340</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">Alius modus</xhtml:td> + <xhtml:td xml:space="preserve">Alium modum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">341</xhtml:td> + <xhtml:td xml:space="preserve">9</xhtml:td> + <xhtml:td xml:space="preserve">æquale</xhtml:td> + <xhtml:td xml:space="preserve">æqualem</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">343</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">videbitur</xhtml:td> + <xhtml:td xml:space="preserve">videri</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">344</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve">hoc eſt</xhtml:td> + <xhtml:td xml:space="preserve">hoc est quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">344</xhtml:td> + <xhtml:td xml:space="preserve">@1</xhtml:td> + <xhtml:td xml:space="preserve">punctus</xhtml:td> + <xhtml:td xml:space="preserve">punctum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">345</xhtml:td> + <xhtml:td xml:space="preserve">6</xhtml:td> + <xhtml:td xml:space="preserve">ei</xhtml:td> + <xhtml:td xml:space="preserve">cum altitudine</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">345</xhtml:td> + <xhtml:td xml:space="preserve">19</xhtml:td> + <xhtml:td xml:space="preserve">ipſum repertum</xhtml:td> + <xhtml:td xml:space="preserve">ipſum, vt repertum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">347</xhtml:td> + <xhtml:td xml:space="preserve">16</xhtml:td> + <xhtml:td xml:space="preserve">reflectit</xhtml:td> + <xhtml:td xml:space="preserve">reflectitur</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">352</xhtml:td> + <xhtml:td xml:space="preserve">38</xhtml:td> + <xhtml:td xml:space="preserve">opinauerunt</xhtml:td> + <xhtml:td xml:space="preserve">opinati ſune</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <xhtml:td xml:space="preserve">parabolem</xhtml:td> + <xhtml:td xml:space="preserve">parabolam</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">15</xhtml:td> + <xhtml:td xml:space="preserve">hæc</xhtml:td> + <xhtml:td xml:space="preserve">hic</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">18</xhtml:td> + <xhtml:td xml:space="preserve">parallelam</xhtml:td> + <xhtml:td xml:space="preserve">parallela</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">356</xhtml:td> + <xhtml:td xml:space="preserve">15</xhtml:td> + <xhtml:td xml:space="preserve">muri, &</xhtml:td> + <xhtml:td xml:space="preserve">muri, in ori Zontali <var>.q.p.</var> ev<unsure/></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">356</xhtml:td> + <xhtml:td xml:space="preserve">33</xhtml:td> + <xhtml:td xml:space="preserve">mediante. cognoſeere</xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="mediante" type="context">mediãte</reg> cuius axis ad ori Zontem <lb/>erectus ſit <var>.o.n</var>: cognoſcere<unsure/></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">360</xhtml:td> + <xhtml:td xml:space="preserve">13</xhtml:td> + <xhtml:td xml:space="preserve">duplum</xhtml:td> + <xhtml:td xml:space="preserve">duplus</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">360</xhtml:td> + <xhtml:td xml:space="preserve">28</xhtml:td> + <xhtml:td xml:space="preserve">vnde iſte circulus</xhtml:td> + <xhtml:td xml:space="preserve">iſte circulus</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">361</xhtml:td> + <xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">364</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">diameter</xhtml:td> + <xhtml:td xml:space="preserve">diametrum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">367</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">prolung ando</xhtml:td> + <xhtml:td xml:space="preserve">produεendo</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">369</xhtml:td> + <xhtml:td xml:space="preserve">2</xhtml:td> + <xhtml:td xml:space="preserve">æqualem</xhtml:td> + <xhtml:td xml:space="preserve">æquale</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">370</xhtml:td> + <xhtml:td xml:space="preserve">4</xhtml:td> + <xhtml:td xml:space="preserve">ſinus</xhtml:td> + <xhtml:td xml:space="preserve">ſinum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">370</xhtml:td> + <xhtml:td xml:space="preserve">30</xhtml:td> + <xhtml:td xml:space="preserve">ſumma</xhtml:td> + <xhtml:td xml:space="preserve">ſummam</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">372</xhtml:td> + <xhtml:td xml:space="preserve">1</xhtml:td> + <xhtml:td xml:space="preserve">cæterarum</xhtml:td> + <xhtml:td xml:space="preserve">cæteris</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">372</xhtml:td> + <xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">qui qui</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">378</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">hæc</xhtml:td> + <xhtml:td xml:space="preserve">hæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">382</xhtml:td> + <xhtml:td xml:space="preserve">41</xhtml:td> + <xhtml:td xml:space="preserve">prolung atis</xhtml:td> + <xhtml:td xml:space="preserve">productis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">3</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">13</xhtml:td> + <xhtml:td xml:space="preserve">ginabis</xhtml:td> + <xhtml:td xml:space="preserve">ginaberis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">20</xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="triangulique" type="simple">trianguliq;</reg> + </xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="triangulaque" type="simple">triangulaq́;</reg></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383 <lb/>384</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">in prima parabola obliqua ex-<lb/>tremitatem inferiorem diametri <lb/>ſignabis charactere <var>.d.</var> in ſecunda <lb/>vero charactere <var>.r.</var></xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">384</xhtml:td> + <xhtml:td xml:space="preserve">29</xhtml:td> + <xhtml:td xml:space="preserve">maior proportio</xhtml:td> + <xhtml:td xml:space="preserve">maiorem proportionem</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">384</xhtml:td> + <xhtml:td xml:space="preserve">37</xhtml:td> + <xhtml:td xml:space="preserve">proportio halebit</xhtml:td> + <xhtml:td xml:space="preserve">proportionem habebis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">385</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.c.b.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.e.b.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">22</xhtml:td> + <xhtml:td xml:space="preserve">b.e.</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">29</xhtml:td> + <xhtml:td xml:space="preserve">poſitum</xhtml:td> + <xhtml:td xml:space="preserve">poſitus</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">31</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">33</xhtml:td> + <xhtml:td xml:space="preserve">& proportion alitate</xhtml:td> + <xhtml:td xml:space="preserve">& ex proportionalitate</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">34</xhtml:td> + <xhtml:td xml:space="preserve">antecedens in</xhtml:td> + <xhtml:td xml:space="preserve">antecedens <var>.A.B.</var> in</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">40</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">2</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.B.</var>)</xhtml:td> + <xhtml:td xml:space="preserve">ß)</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <xhtml:td xml:space="preserve">b. cum duplo <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.d.</var> cum duplo <var>.b.e.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">45</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve">b.c.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">45</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">46</xhtml:td> + <xhtml:td xml:space="preserve">ad <var>.e.b.</var></xhtml:td> + <xhtml:td xml:space="preserve">ad <var>.d.b.</var> cum <var>.e.b.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">390</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">diuiſa</xhtml:td> + <xhtml:td xml:space="preserve">diuiſas</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">392</xhtml:td> + <xhtml:td xml:space="preserve">3</xhtml:td> + <xhtml:td xml:space="preserve">ſit paraboles <var>.a.</var></xhtml:td> + <xhtml:td xml:space="preserve">ſit paraboles <var>.a.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">392</xhtml:td> + <xhtml:td xml:space="preserve">27</xhtml:td> + <xhtml:td xml:space="preserve">aliqua reliquarum</xhtml:td> + <xhtml:td xml:space="preserve">aliquis reliquorum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">393 <lb/>395 <lb/>397</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">in parabola vbi ſub <var>.g.</var><unsure/> eſt <var>.i.</var> pone <var>.r.</var> <lb/>& ſupra ſolidum minus dicatur <lb/>cubus minor (leantur <lb/>duo vero lineæ ſine characterib. de</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">18</xhtml:td> + <xhtml:td xml:space="preserve">vt <var>.y.</var> ad <var>.m.n.</var></xhtml:td> + <xhtml:td xml:space="preserve">vt p. ad <var>.m.n.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">33</xhtml:td> + <xhtml:td xml:space="preserve">x.o.</xhtml:td> + <xhtml:td xml:space="preserve">n.o.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">34</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.i.k.</var> ad <var>.i.k.</var> ad <var>f.g.</var></xhtml:td> + <xhtml:td xml:space="preserve">i.k. ad <var>.f.g.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">395</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">quintas</xhtml:td> + <xhtml:td xml:space="preserve">quintæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">396</xhtml:td> + <xhtml:td xml:space="preserve">1</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.d.b.</var></xhtml:td> + <xhtml:td xml:space="preserve">g.b.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">397</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">Vbirubrica dicit. Defenſio nostra <lb/>contra Antonium Bergam, & <lb/>Alexandrum Piccol.</xhtml:td> + <xhtml:td xml:space="preserve">dicatur. contra Anto. Bergam & <lb/>Alex Piccol atq, defenſio 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">ſemidiametrum</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">ſuis</xhtml:td> + <xhtml:td xml:space="preserve">ſuæ</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 ſupræ</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 ſors</xhtml:td> + <xhtml:td xml:space="preserve">qua ſorte</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 ſupra</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">fineſtræ</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 𐆑 (U+10191; D800+DD91). </s> + <s xml:id="s5187" xml:space="preserve">Das gleiche Zeichen innerhalb eines Wortes: vorher𐆑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 𐆒 (U+10192; D800+DD92) wird dagegen nicht normalisiert: vorher𐆒nachher. </s> + </p> + </div> + </div> + </text> +</echo> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/Benedetti_1585.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,15419 @@ +<?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 Philoſophi.</head> + <head xml:space="preserve"><emph style="it">DIVERSARVM SPECVLATIONVM <lb/>Mathematicarum, & Phyſicarum</emph> <lb/>Liber.</head> + <head xml:space="preserve">Quarum ſeriem ſequens 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 perſpectiuæ.</head> + <head xml:space="preserve">De Mechanicis.</head> + <head xml:space="preserve">Diſputationes de quibuſdam placitis Ariſt.</head> + <head xml:space="preserve">In quintum Euclidis librum.</head> + <head xml:space="preserve">Phyſica, & Mathematica reſponſa per Epiſtolas.</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 Celſitudinis, ac-<lb/>cerſitus ex vrbe Parmenſi 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 deſeruiendi, tam vebe-<lb/>menti cupiditate fui accenſus, vt ſub eius ditione quodſuper-<lb/>eßet vitæ agere conſtituerem. </s> + <s xml:space="preserve">Cuius in me benignitas, mea <lb/>in illum obſeruantia mirum in modum mutuo vſu, & conſue-<lb/>tudine eſt adaucta, vt idem Dux me ſecum dum ruſticaretur <lb/>eße vellet, ſæpè etiam ſecum pernoctare; </s> + <s xml:space="preserve">quo quidem tempo-<lb/>re de Matbematicis ſcientijs mecum agebat, in quibus perdi-<lb/>ſcendis mea opera vtebatur, quæſtiones, Arithmeticam, Geo<lb/>metriam, Opticen, Muſicam, aut Astrologiam ſpectantes <lb/>proponens. </s> + <s xml:space="preserve">Cui vt quod in me eßet ſatisfacerem, acrius <lb/>quàm anteainea studia (adquætamen ſemper fui propenſißi-<lb/>mus) incubui. </s> + <s xml:space="preserve"><reg norm="Illiusque" type="simple">Illiusq́ꝫ</reg> imitatione (vt ferècæteri Principum <lb/>studiaimitantur) non pauci aut præſentes, aut per litter as me <lb/>de his, atque illis Mathematicis quæstionibus conſuluerunt. <lb/></s> + <s xml:space="preserve">Cùmque ego nunquam laborem amicorum cauſa defugerim, <lb/>euenit vt post tot annorum curricula, mea ſcrinia ſcrutatus, <lb/>inuenerim tot abſolutas quæſtiones, vt ex eis corpus mediocre <lb/>effici poſſe videretur. </s> + <s xml:space="preserve">Quas, cùm rationibus in epiſtola ſub-<lb/>ſequenti allatis edere constituiſſem, non ſub cuiuſque alte-<lb/>rius nomine, & auſpicijs quam tuæ Celſitudinis 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 Celſitudine paternam in me fouendo, & augendo<unsure/> <lb/>benignit atem ineße ſemper ſum expertus, tum quòd tuæ Celſi-<lb/>tudinis interrog ationibus excitatus non pauca quæ hoc volu-<lb/>mine continentur, elucubraui. </s> + <s xml:space="preserve">Acceßit, quod ego ſemper in <lb/>his dedic ationibus ſpectandum put aui, tuam Celſitudinem 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/>ſarum Regum gloriam æquauit, & nos veluti in ſpem certam <lb/>fælicitatis buius ſæculi induxit, ſi verum eſt Platonis va-<lb/>ticinium, beat am eam futuram Rempublic am in qua <lb/>Principes Philoſophentur. </s> + <s xml:space="preserve">Tua igitur celſi-<lb/>tudo libellum tot ei nominibus debitum, <lb/>ea qua ſolet bumanitate accipe-<lb/>re nè grauetur. </s> + <s xml:space="preserve">Deus tuas <lb/>omnes cogitationes, <lb/>& conatus ad <lb/>fœlicißi-<lb/>mos <lb/>ſemper exitus perducat, <lb/><reg norm="teque" type="simple">teq́ꝫ</reg> diutißimè ſer-<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 diuerſis <lb/>diſciplinis contemplatus ſim, partim à præ-<lb/>ſtantibus viris patronis ac amicis meis exci-<lb/>tatus, quiſuper eis ſententiam meam exquire-<lb/>bant, partim, abingenito mihi deſiderio, ra-<lb/>tionem, & cauſam eorum percipiendi, com-<lb/>mittendum non putaui, quin qualiacunque <lb/>meaſcripta in illis ſcientijs, ſtudioſis impartirer, <lb/>non dubitans quin illis aliquid commodi atque vtilitatis allatura ſint, prę <lb/>ſertim cum in eiuſmodi quæſtionibus inueſtigandis atque perpendendis, <lb/>nemo ( quod ſciam ) hactenus elaborauerit. </s> + <s xml:space="preserve">Nihil enim his libris à me <lb/>traditum eſt, quod aut legiſſe, aut ab alijs audiuiſſe meminerim, nam ſi <lb/>aliena attigi, ea, aut cum aliqua differentia demonſtrationis, aut diluci-<lb/>dius ſcripſi, quod ſi 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 Ariſtoteles ipſe ſenſit facilè fieri poteſt, vt pluribus, eædem opinio-<lb/>nes in mentem veniant. </s> + <s xml:space="preserve">Immo multa ſcribenti euenire poteſt, vt cum <lb/>iamdiu aliquid ſcripſerit, iam oblitus, idem repetat, quod mihi etiam <lb/>nonnunquam accidit. </s> + <s xml:space="preserve">In his autemlibris non ſuſcepi munus integræ ali <lb/>cuius ſcientiæ tradendæ, ne, quæ abalijs iam tradita ſunt, ipſe inutiliter re <lb/>peterem, mihiq́ue viderer exalienis laboribus laudem voluiſſe comparare. <lb/></s> + <s xml:space="preserve">Singularum enim ſcientiarum volumina, iam ab alijs collecta, at-<lb/>que in ordinemſunt digeſta, & ſi pauciſſimi ſint libri quorum omnes <lb/>ſententiæ, omniaq́ue inuenta vnius ſint authoris, excipio Archime-<lb/>dis volumina. </s> + <s xml:space="preserve">Cumque multi ſint, qui vel vnam rem à ſe inuentam <lb/>in publicum proferre non dubitent, multo magis mihi qui multa ex-<lb/>cogitaui, & ſi inter ſe hætereogenea, atque vtcunque expreſſa, idem <lb/>licere ſum arbitratus. </s> + <s xml:space="preserve">In his autem meditandis, ex Arithmeticis autho-<lb/>ribus quos inſpexi, præcipuus fuit Nicolaus Tartalea, quippe quem fe-<lb/>rè omnia ab alijs ſcripta collegiſſe conſtat, nec alios ex præcipuis, quos le-<lb/>gere potui omittendos duxi, inter quos ſunt Hieronymus Cardanus, Mi-<lb/>chael Stifelius, Gemma Friſius, Ioannes Nouiomagus, Cuthebertus <lb/>Tonſtallus, <reg norm="cæterique" type="simple">cæteriq́;</reg> huiuſinodi. </s> + <s xml:space="preserve">Quorundam tamen volumina illorum <lb/>qui à Tartalea citantur, vt Leonardi Piſani, Proſdocimi, Ioannis Infor-<lb/>tunati, Fratris Lucæ, Petri Borgi, aliorumq́ue aliquot inſpiciendorum, + <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 nonnullę inueniantur <lb/>propoſitiones, quæ diſiunctam ab alijs habeant rationem, eæ non ſper-<lb/>nendæ tamen ſunt, viam fortaſſe alicui aperient vlterius progrediendi. <lb/></s> + <s xml:space="preserve">Quemadmodum enim, exempli gratia, ex ſub contraria coni ſectione, <lb/>ſumpta poſtea fuit diuina illa Planisferijdelineation, quæ ſub Ptolomæi no-<lb/>mine legitur, & ſicuti ex <ref id="ref-0001">penultima primi Euclidis</ref>, quam Pythagoras <lb/>excogitauit propè innumeræ pulchræ conſequentiæ in Aſtronomia, in <lb/>Architectura, in <reg norm="multisque" type="simple">multisq́;</reg> alijs ſcientijs deſumptæ ſunt, immo quemad-<lb/>modum ex ſingulis propoſitionibus à noſtris maioribus excogitatis mul-<lb/>ta egregia ſunt deducta, ita fortaſſe continget, vt ex mearum muentio-<lb/>num aliqua, <reg norm="nonnihil" type="context">nõnihil</reg> in poſterum vtilitatis deſumatur. </s> + <s xml:space="preserve">Si quid verò, hic in-<lb/>ueneris, quod tuo genio non arrideat, illa prudentiſſimi hominis ſen-<lb/>tentia in mentem veniat. </s> + <s xml:space="preserve"><emph style="it">Quot capita, tot ſententiæ</emph>, ac per raro con-<lb/>tingere, vt idem omnibus probari, atque placere queat, & perdifficulter <lb/>inueniri hominem cui placeant omnia quæ alteri ſatisfaciunt. </s> + <s xml:space="preserve">Nec te mo <lb/>ueat, quodhęc Theoremata ſiue excogitationes non videas ordine illo di-<lb/>ſpoſitas, quo collocari debere exiſtimaueris, tum in Arithmeticis, tum in <lb/>cæteris. </s> + <s xml:space="preserve">Cum enim in huiuſmodi rebus ordo non ſit neceſſarr<unsure/>us, vi-<lb/>ſum eſt mihi poſſe me, ſine repræhenſione, illum negligere, cum ſpe-<lb/>culationi, ſiue inuentioni preęcipuè adeo mihi incumbendum decreuerim <lb/>vtin collocatione operam ponere, & tempus abſumere operæpretium <lb/>non duxerim, quod idem in epiſtolarum collocatione feci, in quibus per-<lb/>ſonarum ad quas ſcribo nullus ferè graduum ordo ſeruatus eſt, nec tem-<lb/>poris, quo ſunt ſcriptæ, quæſitorum tantummodo ratione habita. </s> + <s xml:space="preserve">Nec <lb/>admirari quenquam velim, quod in ſpeculandis 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 eſt abſtractus, <emph style="it">quoniam oportet in-<lb/>telligentem phantaſmata ſpeculari</emph>, cum pręterea perſpicuum ſit, diſcretum <lb/>omne, ex continui diuiſione aliquo modo oriri, ſiue actu, ſiue potentia. <lb/></s> + <s xml:space="preserve">Deinde ſi forte meis in deinonſtrationibus tibi videbor aliquando bre-<lb/>uior, illud in cauſa fuiſſe ſcias, quod ibi ad viros ſcribebam in his diſcipli-<lb/>nis exercitatos, quibus ſatis fuit rem ſignificare. </s> + <s xml:space="preserve">Libuit autem mihi om-<lb/>nes voluminis Arithmetici propoſitiones potius vocabulo theorema-<lb/>tum appellare, quam problematum, quia pars earum ſpeculatiua tan-<lb/>tum mea eſt, & ſi ex varijs eiuſmodi propoſitionibus etiam operatiuam <lb/>adinuenerim. </s> + <s xml:space="preserve">Quoniam verò multis in locis accidit, vt veritatis iudi-<lb/>candæ cauſa neceſſe mihi fuerit quorundam ſententijs aduerſari 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">meq́;</reg> hoc nomine carptorem <reg norm="maledicumque" type="simple">maledicumq́;</reg> ha-<lb/>bere quod alienos errores aperiam, cum potius habenda ſit mihi gratia, <lb/>quod in ijs interdum laborans (quę Antiſthenes in diſciplinis magis ne-<lb/>ceſſaria eſſe dixit, <emph style="it">vt mala ſcilicet prius dediſcantur<sic comment="closing parenthesis in italics">)</sic></emph> falſas opiniones euel-<lb/>lere ſtudeam, <reg norm="veritatemque" type="simple">veritatemq́;</reg> oſtendere, quam omnis philoſophus, Ariſto-<lb/>telis exemplo, pluris quam cuiuſuis hominis authoritatem, aut gratiam <lb/>facere debet. </s> + <s xml:space="preserve">Cumq́ue in hoc volumine aliquid eiuſmodi legeris <lb/>te oratum volo, vt in iudicando, affectum omnem exuas, <lb/>Salluſtianum illud præ oculis habens. </s> + <s style="it" xml:space="preserve">Omnes qui dere-<lb/>bus <reg norm="dubijs" type="lig">dubijs</reg> conſultant, ab odio amicitia, ira, atque <lb/>miſericordia vacuos eſſe decet. </s> + <s xml:space="preserve">Hinc fiet, vt <lb/>non perſonæ (vt multiſolent) ſed <lb/>veritati, quę ſummo ſtudio di-<lb/>gniſſima eſt, ſemper po <lb/>tius faueas. </s> + <s xml:space="preserve">Vale <lb/>noſtrisq́ue <lb/>labo-<lb/>ribus vtere, ſi quem inde fructum, <lb/>ſicuti ſpero tuleris, illi præ-<lb/>cipuè habeas gratiam à <lb/>quo omnes fluunt <lb/>ſcientiæ.</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 philoſophi de nu<lb/>meris eorumq́ue effectibus excogitata poſteris tradide-<lb/>runt, quorum cum vix vllam rationem reddiderint, aut <lb/>certè per exiguam, occaſione diuerſorum problematum <lb/>mihi à Sereniſſimo Sabaudiæ Duce propoſitorum præbi-<lb/>ta, de ijs quæ ab antiquis propoſita fuerunt contemplanda <lb/>nonnulla occurrerunt, quæ poſteritati comendare non <lb/>inutile arbitratus fum, ne hæ meæ cogitationes intercide-<lb/>rent, & occaſionem præberem quamplurimis abſtruſa hęc <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 queſita, 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 cognoſci poſ-<lb/>ſet ſcientificè & ſpeculatiue (vt dicitur) productum ex duobus fractis numeris, <lb/>quolibet producentium minus eſſe. </s> + <s xml:space="preserve">Cui reſpondi, mente & cogitatione conci-<lb/>piendum eſſe fractos producentes cum fractis productis, non vnius eiuſdemq́ue 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 ſint <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 ſuperficiali <var>.d.b.</var> generato ex <lb/>vno in aliud totorum linearium, nam ſi conſtitueretur <var>.a.i.</var> octauum ipſius <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 ipſius <var>.<lb/>d.b</var>. </s> + <s xml:space="preserve">Quare <var>.d.b.</var> eſſet totum <reg norm="relatiuum" type="context">relatiuũ</reg> ipſius <var>.c.i.</var> non aliquod totum producentium. <lb/></s> + <s xml:space="preserve">Mirum itaque non eſt ſi productum <var>.c.i.</var> minus videatur fuis producentibus, cum <lb/>toto, diuerſæ naturæ à primis conferatur, fractum fiquidem ab integro eiuſdem <lb/>naturæ, linearis, ſuperficialis, aut corporeæ denominatur.</s> + </p> + <p> + <s xml:space="preserve">Quòd ſi amplioris cognitionis gratia ex ſcientiæ præceptis ſpeculari 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 ſit in ſuo integro <var>.d.b.</var> fracto <var>.a.i.</var> in <lb/>ſuo integro <var>.a.b.</var> aut fracto <var>.a.c.</var> in ſuo integro <var>.a.d.</var> conſideret is quo pacto pro-<lb/>portio <var>.c.i.</var> ad <var>.d.b.</var> minor ſit 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/>nifeſtum eſt ex <ref id="ref-0003">prima ſexti de quantitate <lb/>continua</ref>, aut <ref id="ref-0004">.18. ſeptimi Euclidis</ref> de diſcre <lb/> + <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> + ta, proportionem ipſius <var>.d.i.</var> ad <var>.d.b.</var> eſſe ſi-<lb/>cut <var>.a.i.</var> ad <var>.a.b.</var> & cum <var>.c.i.</var> minor ſit <var>.d.i.</var> <lb/>velut pars ſuo 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="eiuſdem" type="context">eiuſdẽ</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 eiſdem cauſis, medio <var>.c.b</var>. </s> + <s xml:space="preserve">Ex quibus pa-<lb/>tet ratio, cur fracti diuerſarum denomina-<lb/>tionum ad vnicam reducantur. </s> + <s xml:space="preserve">Cur etiam <lb/>numeros integros in partes fractis ſimiles <lb/>frangere liceat, quæ omnia ex ſubſequenti <lb/>figura facilè cognoſci 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> ſit ratio, cur hi, qui numeros, fractos diuerſarum denominationum col-<lb/>ligere volunt, & in ſummam redigere, multiplicent vnum ex numerantibus <lb/>per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum <lb/>vltimum productum, commune eſt denominans duorum priorum productorum, <lb/>quæ collecta in ſummam efficiunt quod quærebatur.</s> + </p> + <p> + <s xml:space="preserve">Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius <reg norm="eiuſdem- q́ue" type="context">eiuſdẽ-<lb/>q́ue</reg> magnitudinis quantitatis continuæ, linearum (verbigratia) <var>a.b.</var> et <var>.a.d.</var> <reg norm="æqualium" type="context">æqualiũ</reg> <lb/>in longitudine, <reg norm="quarum" type="context">quarũ</reg> <var>.a.b.</var> in quatuor partes diuidatur, et <var>.a.d.</var> in tres. </s> + <s xml:space="preserve">Quare ſi colli-<lb/>gere voluerimus duo tertia cum tribus quartis, multiplicabimus <var>.a.c.</var> duo tertia, <lb/>cum <var>.a.b.</var> diuiſa in 4. partes, produceturq́ue <var>.c.b.</var> octo partium ſuperficialium, de-<lb/>hinc multiplicando <var>.a.i.</var> tres quartas cum <var>.a.d.</var> diuiſa in .3. partes producetur <var>.i.d.</var> pri <lb/>mis ſingulis æqualis, nouem partium ſuper <lb/>ficialium, multiplicata deinde <var>a.b.</var> diui-<lb/> + <anchor type="figure" xlink:label="fig-0014-02a" xlink:href="fig-0014-02"/> + ſa in .4. partes per <var>.a.d.</var> in .3. diuiſa, produ-<lb/>cetur quadratum <var>.d.b.</var> in continuo, in 12. <lb/>partes diuiſum, quod erit totum commune <lb/>ſingulis productis, quorum primum erat <var>.c.<lb/>b</var>. </s> + <s xml:space="preserve">Quare <var>.c.b.</var> ita ſe habet ad totum <var>.d.b.</var> ſi-<lb/>cut <var>.a.c.</var> ad <var>.a.d.</var> ex prima ſexti in continuis, <lb/>aut .18. ſeptimi in diſcretis quantitatibus, <lb/>et <var>.d.i.</var> ad <var>.d.b.</var> ſicut <var>.a.i.</var> ad <var>.a.b.</var> ex eiſdem <lb/>propoſitionibus. </s> + <s xml:space="preserve">Collectis deinde parti-<lb/>bus producti <var>.c.b.</var> cum partibus producti <var>.<lb/>d.i.</var> manifeſtè depræhendetur eiuſmodi <lb/>ſummam componi ex partibus vnius totius <lb/>communis ſingulis 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 ſit fractus aliquis numerus reſpectu 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 ſi cauſam noſce vis, ſume <var>.o.i.</var> & <var>.o.u.</var> pro totis denominatoribus, tum <var>.o.e.</var> <lb/>& <var>.o.a.</var> pro numeratoribus (exempli cauſa) ſit <var>.o.i.</var> ſenarius <var>.o.u.</var> quaternarius <var>.o.e.</var> <lb/>quinarius <var>.o.a.</var> ternarius. </s> + <s xml:space="preserve">Si noſce vis quæ ſint tres quartę partes quinque ſextarum, <lb/>patet ex regulis practicis oriri quindecim vigeſimaſquartas. </s> + <s xml:space="preserve">Id quomodo fiat, ex <lb/>ſubſcripta ſigura depræhendetur, memores tamen eſſe oportet, quodlibet <reg norm="productum" type="context">productũ</reg> <lb/>conſiderari <reg norm="tanquam" type="context">tanquã</reg> ſuperficiem, producentia <reg norm="autem" type="context">autẽ</reg> tan-<lb/>quam lineas. </s> + <s xml:space="preserve">In hac igitur ſigura productum ex totis <lb/> + <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> + linearibus eſt <var>.u.i.</var> aggregatum ex .24. partibus, & <var>.u.e.</var> <lb/>productum aggregatum ex .20. </s> + <s xml:space="preserve">Quodita ſe habebit <lb/>ad productum totale <var>.u.i.</var> ſicut <var>.o.e.</var> ad <var>o.i.</var> ex prima <lb/>ſexti aut .18. ſeptimi, ita <var>.u.e.</var> erunt quinque ſextæ par <lb/>tes <var>.u.i.</var> quarum in propoſito exemplo, tres quartæ <lb/><reg norm="quæruntur" type="context">quærũtur</reg>. </s> + <s xml:space="preserve">Si <reg norm="itaque" type="simple">itaq;</reg> multiplicabitur <var>.o.e.</var> <reg norm="cum" type="context">cũ</reg> <var>.o.a.</var> orietur <lb/>productum <var>.a.e.</var> ita <reg norm="proportionatum" type="context">proportionatũ</reg> ad <var>.u.e.</var> ſicut <var>.o.a.</var> ad <lb/><var>o.u.</var> reperitur, ex prædictis rationibus. </s> + <s xml:space="preserve">Quòd ſi <reg norm="ſtatutum" type="context">ſtatutũ</reg> <lb/>eſt <var>.o.a.</var> tres quartas partes eſſe ipſius <var>.u.o.</var> <reg norm="etiam" type="context">etiã</reg> <var>.a.e.</var> tres <lb/>quartæ partes <reg norm="erunt" type="context">erũt</reg> <var>.u.e.</var> ſed <var>.u.e.</var> quinque ſextæ ſunt ip-<lb/>ſius <var>.u.i.</var> ex quo ſequitur bonum eſſe huiuſmodi 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, partianturq́ue 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 ſubſequenti figura, numerum integrorum <lb/>tanquam lineam <var>.a.e.</var> qui, verbigratia, ſit denarius, quorum vnuſquiſque ſit æqualis <lb/><var>a.i.</var> cogiteturq́ue productum ipſius <var>.a.e.</var> in <var>.a.i.</var> ſitq́ue <var>.u.e.</var> quod quidem erit dena-<lb/>rius ſuperficialis, conſtituta prius <var>.a.u.</var> æqualis <var>.a.i.</var> & <var>.a.o.</var> ſint duæ tertiæ <var>.a.u.</var> <reg norm="quarum" type="context">quarũ</reg> <lb/>duarum tertiarum productum in numerum <var>.a.e.</var> ſit <var>.o.e.</var> pariter <var>.u.i.</var> vnitas ſit ſuper-<lb/>ficialis prout <var>.a.i.</var> vnitas eſt linearis, quam <var>.u.i.</var> reſpicere debet productum <var>.o.e.</var> ex <lb/>quo integer ſuperficialis <var>.u.i.</var> erit tanquam ternarius, & productum <var>.o.i.</var> tanquam bi <lb/>narius, & quia quælibet pars è viginti ipſius <var>.o.e.</var> æqualis eſt tertiæ parti <var>.u.i.</var> vnita-<lb/>tis ſuperficialis; </s> + <s xml:space="preserve">ſi cupiamus ſcire quot integræ vnitates ſint in partibus <var>.o.e.</var> conſul-<lb/>tum eſt eaſdem diuidere per denominantem <var>.u.i.</var> compoſitum ex tribus partibus ſu <lb/>perficialibus, & cum tam linea <var>u.a.</var> quam ſuperficies <var>.u.i.</var> diuidatur in 3. partes <reg norm="aequa les" type="simple">ęqua<lb/>les</reg> noſce peroportunum eſt eiuſmodi partitionem numeri <var>.o.e.</var> fieri per numerum <lb/>ipſius <var>.u.i.</var> non <var>.u.a.</var> ex prædictis cauſis.</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 cauſa, ſpeculando inno-<lb/>teſcere poteſt, cuius rei gratia for-<lb/>metur ſequens figura <var>.e.o.a.u.n.</var> <lb/>eiuſmodi, vt <var>a.e.</var> ſit numerus li-<lb/>linearis integrorum, & <var>o.e.</var> produ-<lb/>ctum numerantis ipſorum <reg norm="fractorum" type="context">fractorũ</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">linearũ</reg> ſingulę ſtatuuntur æqua <lb/>les vnitati lineari, ſuperficies autem <lb/>parallelogramma <var>.u.n.</var> conſtituatur <lb/>æqualis magnitudinis ſuperficiei <var>.o.<lb/>e.</var> ex quo <var>.u.n.</var> erit nobis cognita ſu-<lb/>perficies. </s> + <s xml:space="preserve">Cognoſcetur pariter quan <lb/>titas partium <var>.a.u.</var> quam in propoſi-<lb/>to exemplo diximus eſſe trium par-<lb/>tium. </s> + <s xml:space="preserve">ex regula igitur de tribus, di-<lb/>cemus ſi <var>.u.a.</var> dat <var>.a.e.</var> ſine dubio <var>.o.<lb/>a.</var> dabit <var>.a.n.</var> numerum linearem. <lb/>quæ regula ex 15. ſexti in continuis, <lb/>& ex 20. ſeptimi in diſcretis, 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 ſpeculatione cognoſci poteſt hoc rectè fieri, mul-<lb/>tiplicantes enim has duas tertias per decem, debemus conſide-<lb/> + <anchor type="figure" xlink:label="fig-0016-02a" xlink:href="fig-0016-02"/> + rare quantitatem duarum <reg norm="tertiarum" type="context">tertiarũ</reg> decies produci, ex quo oriuntur <lb/>20. tertia, quandoquidem ſingulæ vnitates, </s> + <s xml:space="preserve">tunc pro duobus ter-<lb/>tijs ſumuntur, ſed c<unsure/>um quilibet integer tria fragmenta contineat, <lb/>ideo ex ratione partiendi quoties ternarius ingrediatur viginti, <lb/>ſtatim cognoſcemus 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 ipſum accideret ſi integri in eiuſmodi ſpecie fractorum diui-<lb/>derentur. quo facto hi multiplicandi eſſent cum numerante propo <lb/>ſito, & <reg norm="partiendum" type="context">partiendũ</reg> productum per quadratum denominantis.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s> + <s xml:space="preserve">Sit linea <var>.a.e.</var> conſtans ex <reg norm="quinque" type="simple">quinq;</reg> <lb/>integris numeris, quorum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> æqualis ſit <var>.a.u.</var> vel <var>.a.i.</var> & <var>.a.o.</var> <lb/>ſint duo tertia vnitatis integræ linearis. </s> + <s xml:space="preserve">cogitemus nunc hos <reg norm="quinque" type="simple">quinq;</reg> <lb/>integros diuidi in ſua <reg norm="fragmenta" type="context">fragmẽta</reg> linearia, quę in propoſito exemplo <lb/>erunt 15. multiplicatis iam 15. cum propoſitis, videlicet <var>a.o.</var> orie-<lb/>tur productum <var>.o.e.</var> triginta fragmentorum ſuperficialium, <reg norm="quorum" type="context">quorũ</reg> <lb/>in ſingulos integros ſuperficiales <reg norm="cadunt" type="context">cadũt</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, propoſitum con-<lb/>ſequemur.</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 ſpecies 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 cauſa) ſi 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 diuiſum per <lb/>productum ternarijin quaternarium, videlicet per duode <lb/>cim, quatuor integri proferentur cum ſeptem 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 ſubſequens figura in qua linea <var>a.i.</var> æqualis ſit li-<lb/>neæ <var>.u.a.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <reg norm="conſideretur" type="context">cõſideretur</reg> pro integro nume <lb/>ro: </s> + <s xml:space="preserve"><reg norm="cogiteturque" type="simple">cogiteturq́;</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">cũ</reg> <lb/>duobus tertijs, & <var>a.e.</var> æquipollens duobus integris & tri-<lb/>bus quartis. </s> + <s xml:space="preserve">Iam ſi hæ duæ lineæ in ſuos fractos redu-<lb/>cantur, multiplicata (vt in ſequenti figura apparet.) <var>a.o.</var> <reg norm="cum" type="context">cũ</reg> <lb/><var>a.e.</var> orietur productum <var>o.e.</var> fractorum ſuperficialium <lb/><reg norm="quinquagintaquinque" type="simple">quinquagintaquinq;</reg>, quorum integer ſuperficialis va-<lb/>let duodecim, ſcilicet <var>.u.i.</var> vt cuique manifeſtum eſt, 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 ſi fractiad vnam eandemq́ue denominationem reduceren-<lb/>tur, qui poſtmodum ſimul multiplicarentur, productumq́ue partiremur per qua-<lb/>dratum denominantis communis.</s> + </p> + <p> + <s xml:space="preserve">Exempli cauſa, ſint eadem quinque tertia, & vndecim quarta adinuicem multi-<lb/>plicanda, quæ ſi reducantur ad vnam & eandem denominationem quinarius <lb/>numerans vnius, multiplicabitur cum quaternario deno-<lb/>minante alterius, & vndenarius ſecundi 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 ſi multiplicentur vi <lb/>ginti cum trigintatribus, dabuntur 660. fracti, quorum in-<lb/>teger erit quadratum duodenarij, nempe 144. quibus qui-<lb/>dem 660. diuiſis per 144. proferentur quatuor integri & <lb/>ſeptem 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 ſit in ſubſcripta 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> ſit vndecim <reg norm="partium" type="context">partiũ</reg> <lb/>talium qualium <var>.a.i.</var> eſt quatuor, & <var>.a.o.</var> ſit quinque pro-<lb/>ut <var>.a.u.</var> eſt 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 ſuperficialium. </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">trigintatriũ</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> conſtabit ex duode-<lb/>cim partibus eiuſdem rationis cum reliquis duobus <lb/>productis, quod quadratum <var>.u.i.</var> vnitas eſt ſuperficia-<lb/>lis, & communis denominans duorum productorum. <lb/></s> + <s xml:space="preserve">quod ſi in præſentiarum cogitabimus lineam <var>.c.d.</var> tri-<lb/>gintatrium partium æqualium, et <var>.c.t.</var> duodecim ſimi-<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 ſuperficies <var>.f.d.</var> 660. fractorum <lb/>ſuperficialium, quorum vnitas integra ſuperficialis <lb/>erit quadratum <var>.n.t.</var> 144. partium cuiuſmodi <var>.f.d.</var> <lb/>partes habet .660. diuiſo itaque <var>.f.d.</var> per <var>.n.t.</var> pro-<lb/>poſitum conſequetur. </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 eſt 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 ſexti vel 18. ſeptimi, ſed vt <var>.f.d.</var> ad id <reg norm="quod" type="simple">ꝙ</reg> <lb/>fit ex <var>.f.c.</var> in <var>.c.t.</var> eſt vt <var>.c.d.</var> ad <var>.c.t.</var> & vt eius <reg norm="quod" type="simple">ꝙ</reg> fit ex <lb/><var>f.c.</var> in <var>.c.t.</var> ad <var>.n.t.</var> eſt vt <var>.f.c.</var> ad <var>.c.n.</var> ex dictis pro-<lb/>poſitionibus </s> + <s xml:space="preserve">quare ex æqua proportionalitate, eodem <lb/>modo diſcurrendo in figura <var>.o.a.e.</var> ita ſe 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 conſiderata fuerunt, apertè <lb/>ratio deprehenditur, cur productum, ſingulis producen <lb/>tibus ſemper minus ſit, cum producta ſint ſuperficialia <lb/>producentia verò ſemper linearia, omiſſis productis <lb/>corporeis, quæ omnia ad ſuperficialia 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 diuiſione, animaduertendum eſt, denominantes numeros <lb/>ſemper æquales inuicem eſſe debere, vnius ſcilicet ſpeciei, quòd ſi æquales non <lb/>fuerint, neceſſe eſt via multiplicationis ipſorum denominantium adinuicem effice-<lb/>re æquales vt ſint, ex quo productum oritur eiuſmodi, vt aptum ſit habere partes <lb/>fractorum, quæ deſiderabantur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponerentur diuidenda ſeptem 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 propoſita ori-<lb/>tur triginta duarum partium commune denominans, cuius duo numerantes ſunt 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 ſeptem octaua particularum vniformium, prout ope <lb/>primæ ſexti aut decimæoctauæ ſeptimi in ſubſcripta figura cognoſci poteſt.</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"/> + ſit <var>.u.i.</var> trigintaduarum particularum <lb/>fuperficialium fimilium & <reg norm="æqualium" type="context">æqualiũ</reg> ad-<lb/>inuicem. </s> + <s xml:space="preserve">fit deinde <var>.a.e.</var> ſeptem <reg norm="partium" type="context">partiũ</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 ſuperficialium vigintiocto <lb/>& productum <var>.a.o.</var> in <var>.a.i.</var> erit <var>.o.i.</var> par <lb/>ticularum <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg> vigintiquatuor <lb/>eiuſdem 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> ſeu diuidere vno numero alium numerum, eſt etiam quodammodo <lb/>eiuſmodi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis, <lb/>cuiuſmodi eſt vnitas in diuidente refpectu totius diuidentis, partem inquam numeri <lb/>diuiſibilis ſic ſe habentem ad totum numerum diuiſibilem ſicut vnitas ad totum di-<lb/>uidentem, quod ſimiliter ex regula de tribus præſtamus dicentes, ſi tantus numerus <lb/>diuidens dat <reg norm="vnitatem" type="context">vnitatẽ</reg>, quid dabit numerus diuifibilis, quemadmodum ex <ref id="ref-0006">.15. ſexti</ref> <lb/>ſeu <ref id="ref-0007">.20. ſeptimi</ref> licet ſpeculari, Idcircò quotieſcunque minorem numerum per <lb/>maiorem diuidimus, ſemper qui prouenit fractus eſt.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi cogitaremus lineam <var>.a.e.</var> diuiſam in octo partes æquales, qua <lb/>rum vna ſcilicet vnitas effet <var>.a.i.</var> & cupere-<lb/>mus eam diuidere in nouem partes, ac ſcire <lb/> + <anchor type="figure" xlink:label="fig-0019-02a" xlink:href="fig-0019-02"/> + quan a ſit nona illius pars; </s> + <s xml:space="preserve">manifeſtum eſſet, <lb/>nonam partem ipſius <var>.a.e.</var> minorem futuram <lb/>ipſa <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 innoteſcat, hoc <lb/>etiam modo licebit videre ſitlinea <var>.n.c.</var> no-<lb/>nupla ad <var>.a.i.</var> & parallela ad <var>.a.e.</var> dubium non <lb/>eſt quin <var>.n.c.</var> maior futura ſit ipſa <var>.a.e.</var> iam ſi <lb/>earum extrema congiungantur medijs duabus <lb/>lineis <var>.n.a.</var> et <var>.c.e.</var> quæ ſimul concurrant in <lb/>puncto <var>.o.</var> (quod eſt 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/>ipſius <var>.n.c.</var> quæ <var>.n.t.</var> æqualis erit <var>.a.i.</var> ex præſuppoſito. </s> + <s xml:space="preserve">ducatur deinde <var>.o.t.</var> quę <lb/>interſecet <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 eſt <var>.n.c.</var> neque enim dubium eſſe poteſt 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> ſint æquales inuicem quandoqui-<lb/> + <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> + dem vnaquæque earum ex triangulorum ſimi <lb/>litudine æqualis eſt proportioni <var>.o.n.</var> ad <var>.o.a</var>. <lb/></s> + <s xml:space="preserve">itaque <var>.n.t.</var> hoc eſt <var>.a.i.</var> tanto maior erit <var>.a.x.</var> <lb/>quanto <var>.n.c.</var> maior eſt <var>.a.e.</var> vnde ficut <var>.a.e.</var> con-<lb/>ſtat octo nonis ipſius <var>.n.c.</var> ita pars <var>.a.x.</var> ipſius <var>.<lb/>a.e.</var> octo nonis conſtabit ipſius <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 eſt præterea diuidere numerum <lb/>per numerum: </s> + <s xml:space="preserve">eſſe inuenire <reg norm="alterum" type="context">alterũ</reg> latus à quo <lb/>producitur, ſuppoſito ſemper quòd numerus <lb/>diuifibilis ſuperſicialis ſit, & rectangulus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponantur triginta diuidenda per quinarium, nihil aliud erit <lb/>hæc diuiſio, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-<lb/>cat triginta ſuperficies rectangulas, huiuſmodi verò eſt ſenarius, cuius ſingulæ vnita-<lb/>tes ſuperficiales erunt.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei gratia ſit ſubſcriptum rectangulum <var>.a.e.</var> triginta vnitatum <reg norm="ſuperſicialium" type="context">ſuperſicialiũ</reg>, <lb/>cuius latus <var>.e.n.</var> ſit quinque vnitatum. </s> + <s xml:space="preserve">hinc latus <var>.a.n.</var> erit ſex 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 ſex vnitatum. <lb/></s> + <s xml:space="preserve">Sin verò diuiſerimus 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 ſicut diuidentis ad vnita-<lb/>tem, ex prima ſexti, aut .18. vel .19. ſeptimi, & permutatim <lb/>ita ſe habebit diuiſibile ad diuidentem, ſicut 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 eſt, quam inuenire latus rectanguli, quod productum in <lb/>diuidente, numerum diuiſibilem compl at, ex quo numerus diuiſibilis ſuperficialis <lb/>eſt, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuſcemodi <lb/>numerum diuiſibilem. </s> + <s xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-<lb/>tem ex multiplicatione laterum ſiue linearum generatur ſuperficies, ex diuiſione po-<lb/>ſtea ipſius ſuperficiei inuenitur alterum latus. </s> + <s xml:space="preserve">quare mirum non eſt, ſi proueniens ex <lb/>vna diuiſione (via fractorum) ſit ſemper maius numero diuiſibili.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reſultat vnus integer nu <lb/>merus cum dimidio pro numero qui oritur. </s> + <s xml:space="preserve">Sit itaque dimidium ſuperſiciale diuiſi-<lb/>bile <var>.b.c.</var> cuius totum ſit <var>.b.p.</var> quadratum. </s> + <s xml:space="preserve">tertium verò lineare diuidens, <var>b.n.</var> cuius to-<lb/>tum lineare ſit <var>.b.d.</var> quærendum nobis eſt latus <var>.b.s.</var> quod cum latere <var>.b.n.</var> producat re <lb/>ctangulum <var>.n.s.</var> æquale dimidio ſuperſiciali propoſito <var>.b.c.</var> quod ſi ſiat, ex .15. ſexti, <lb/>aut .20. ſeptimi. erit eadem proportio <var>.b.n.</var> ad <var>.b.q.</var> quæ eſt <var>.q.c.</var> ad <var>.b.s.</var> dicemus itaque <lb/>ſi <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> ſed <var>.n.b.</var> eſt tertium lineare et <var>.b.q.</var> lineare <reg norm="in- tegrum" type="context">in-<lb/>tegrũ</reg>, & <var>b.s.</var> proueniens lineare. </s> + <s xml:space="preserve">& quia <var>.b.c.</var> dimidium ſuperficiale, 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> ſit ęqualis <var>.b.c.</var> & productum ex <var>.<lb/>b.n.</var> minori <var>.q.c.</var> neceſſe eſt, vt producatur in <var>.b.s.</var> maiore <var>.q.b.</var> quod <var>.q.b.</var> maius eſt <var>.q.c.</var> <lb/>quod quidem <var>.q.c.</var> ita appellatur ſicut <var>.b.c</var>. </s> + <s xml:space="preserve">quare mirum non eſt ſi proueniens per fra-<lb/>ctos numeros ex diuiſione, maior ſit numero diuiſibili.</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 manifeſte patet quamlibet <reg norm="diuiſionem" type="context">diuiſionẽ</reg> aut partitionem oriri ex regula de tri-<lb/>bus, quandoquidem ſinguli diuidentes æquipollent vni integro, & loco illius ſu-<lb/>muntur. </s> + <s xml:space="preserve">Perinde enim eſt diuidere centum per viginti, ac <reg norm="regulam" type="context">regulã</reg> obſeruare de tri-<lb/>bus <reg norm="dicentes" type="context">dicẽtes</reg>, ſi viginti æquipollent vni, quibus <reg norm="ęquiualebunt" type="context">ęquiualebũt</reg> <reg norm="centum" type="context">cẽtum</reg>? </s> + <s xml:space="preserve">Hoc autem ex ſub <lb/>ſequenti figura facile deprehendetur, in qua linea <var>.a.b.</var> ſignificat 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">centũ</reg>: </s> + <s xml:space="preserve">o.c. verò centum vnitates ſuperficiales, <lb/>et <var>.a.d.</var> <reg norm="quinque" type="simple">quinq;</reg> vnitates lineares, et <var>.d.b.</var> centum vnitates ſuperficiales, ex quo manife-<lb/>ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eſt, quam inueni <lb/>re <reg norm="productum" type="context">productũ</reg> ex duobus lateribus propoſitis, it a partiri nihil aliud eſt, quam da-<lb/>to vno latere inuenire aliud latus producti propoſiti.</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="quotieſcunque" type="simple">quotieſcunq;</reg> <reg norm="ratiocinantes" type="context">ratiocinãtes</reg> dicimus tantundem numeri, immediate produci <lb/>mus ſuperficiem, <reg norm="mediante" type="context">mediãte</reg> vnitate in huiuſmodi numero, qui numerus <reg norm="antequam" type="context">antequã</reg> pro-<lb/>ducatur in vnitatem, mente concipiendus eſt tanqua m linearis, tanquam linea in-<lb/>quam diuiſa in totidem particulas lineares, ſingulas continuas & æquales vnitati <lb/>propoſitæ. </s> + <s xml:space="preserve"><reg norm="Cum" type="context">Cũ</reg> verò productus fuerit numerus in vnitate ſuperficialis, erit ac ſi tot eſ-<lb/>ſent vnitates quadratæ, quod ſi ita non eſſet, nulla mentio facienda eſſet quo-<lb/>rumuis <reg norm="fractorum" type="context">fractorũ</reg>. </s> + <s xml:space="preserve">Ex <reg norm="eadem" type="context">eadẽ</reg> regula de tribus reduci poteſtad praxim <reg norm="tertium" type="context">tertiũ</reg> theorema.</s> + </p> + <p> + <s xml:space="preserve">Quare cupientes ſcire quæ ſint illæ partes, quæ ſunt tres quartę, ipſarum quin-<lb/>que ſextarum, dicemus ſi quatuor dant tria, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> ſextæ? </s> + <s xml:space="preserve">dabunt .15. <lb/>vigeſimas quartas, quæ quindecim ſunt tres quartæ ipſius .20. viginti <reg norm="autem" type="context">autẽ</reg> <reg norm="quinque" type="simple">quinq;</reg> ſex <lb/>tæ vigintiquatuor, quandoquidem nos numerum quęrimus, cui ita proportionentur <lb/><reg norm="quinque" type="simple">quinq;</reg> ſextæ alterius numeri, ſicut quatuor ad tria, vnde ſic ſe habent .20. ad .15. ſi-<lb/>cut .4. ad .3. ipſe autem .20. <reg norm="quinque" type="simple">quinq;</reg> ſextę partes ſunt vigintiquatuor, vt per ſe <reg norm="notum" type="context">notũ</reg> eſt.</s> + </p> + <p> + <s xml:space="preserve">Ex eadem regula de tribus, huiuſmodi quęſito reſponderi poteſt, ſi conſtituamus <lb/>prædictas <reg norm="quinque" type="simple">quinq;</reg> ſextas eſſe numerum, cuius tres quartæ quęrantur, dicentes, ſi vnus <lb/>integer dat tres quartas, quid dabunt <reg norm="quinque" type="simple">quinq;</reg> ſextæ? </s> + <s xml:space="preserve">quare ſequentes regulam de <lb/>tribus, dabuntur quindecim vigeſimæ quartæ. </s> + <s xml:space="preserve">Valet eadem regula de tribus; </s> + <s xml:space="preserve">vt quis <lb/>ſcire poſſit, quæ pars aut partes numeri propoſiti ſit aliquis numerus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſcire cupienti, quæ pars aut partes ipſius vigintiquatuor ſint ſex-<lb/>decim, conſtituentur .24. tanquam vnum totum, cuius pars aut partes ſint ſexdecim, <lb/>dicemus igitur ſi .24. dant ſexdecim, quid dabit vnum? </s> + <s xml:space="preserve">ſexdecim videlicet vigeſi-<lb/>masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ. <lb/></s> + <s xml:space="preserve">Eadem ratione qui ſcire uellet, quæ partes aut pars eſſent tres quartæ, octo no-<lb/>narum, diceret, ſi octo nonæ danttres quartas, quid dabit vnum? </s> + <s xml:space="preserve">prouenient .27. <lb/>trigeſimęſecundæ.</s> + </p> + <p> + <s xml:space="preserve">Subſeruit pariter ad <reg norm="ſciendum" type="context">ſciendũ</reg> <reg norm="naturam" type="context">naturã</reg> <reg norm="partium" type="context">partiũ</reg> numeri propoſiti. </s> + <s xml:space="preserve">Exempli cauſa, ſi quis <lb/>quærat, cuius numeri, duodecim ſint duæ tertiæ partes. </s> + <s xml:space="preserve">Dicet ſi 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 ſcilicet, <lb/></s> + <s xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita ſe <lb/>habeant duodecim, ſicut duo ad tria. </s> + <s xml:space="preserve">Ita etiam ſi quis quærat, cuius numeri duo <lb/>tertia ſint tres quintę, dicet, ſi 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> quęrere numerum ad quem ſic ſe <lb/>habeant duo tertia ſicut tria ad <reg norm="quinque" type="simple">quinq;</reg>, quod manifeſtum eſt per ſe.</s> + </p> + <p> + <s xml:space="preserve">Eadem ratione qui ſcire vellet, cuius numeri duæ ſeptimæ, eſſent octo integra-<lb/>rum cum duabus quintis, diceret, ſi duo dant ſeptem 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 ſi proponerentur vnde cim tertiædecimæ vnius totius, toto diui-<lb/>ſo in .13. partes, <reg norm="deſideraremusque" type="simple">deſideraremusq́;</reg> ſcire, quot partes totius <reg norm="eſsent" type="context">eſsẽt</reg> vndecim <reg norm="tertiaedeci- mæ" type="simple">tertiędeci-<lb/>mæ</reg>, toto in .4. partes diuiſo, diceremus ſi .13. dant .11. quid dabunt quatuor? </s> + <s xml:space="preserve">nem <lb/>pe <reg norm="dabunt" type="context">dabũt</reg> tres quartas <reg norm="cum" type="context">cũ</reg> <reg norm="quinque" type="simple">quinq;</reg> tertijsdecimis unius quartæ, hoc verò nihil aliud eſt <lb/>quam querere numerum, ad quem ſic ſe habeat totum in 4. partes diuiſum, ſicut <lb/>idem totum diuiſum in tredecim ſe habet ad undecim tertiasdecimas, Porrò ad <lb/>alia etiam multa hæc regula accommodata eſt.</s> + </p> + <p> + <s xml:space="preserve">Hæc enim <reg norm="non" type="context">nõ</reg> ſine propoſito dicta ſunt, ſed ut <reg norm="quiſque" type="simple">quiſq;</reg> videat cauſam ſimilium ope-<lb/>rationum, quæ à practicis circa fractos numeros ſcriptæ ſunt, omnem à diuina illa <lb/>regula de tribus originem trahere ut etiam in ſequentibus 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, ſemper æquale eſt numero <lb/>diuiſibili ſi queras ita accipe.</s> + </p> + <p> + <s xml:space="preserve">Sit numerus diuiſibilis <var>.b.</var> quod oritur ſit <var>.c.</var> diuidens <var>.d.</var> & vnitas diuidentis <var>.t.</var> cum <lb/>igitur, vt in præcedenti theoremate dictum <lb/>fuit, eadem ſit proportio <var>.b.</var> ad <var>.c.</var> quæ eſt <var>.d.</var> <lb/> + <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> + ad <var>.t.</var> manifeſte deprehenditur ex .20. ſepti <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 ipſum alia ratione contemplari licet.</s> + </p> + <p> + <s xml:space="preserve">Numerus diuiſibilis ſignificetur 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> ſit ſuperficies <var>.u.a</var>. </s> + <s xml:space="preserve">Dico ſuperficiem <var>.u.a.</var> componi <lb/>ex tot vnitatibus ſuperficialibus quot linearibus conſtat linea <var>.n.e.</var> nam ex ijs quæ <lb/>diuidendi ratione notauimus, <reg norm="conſtituitur" 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"/> + quę eſt <var>.a.e.</var> ad <var>.o.e</var>. </s> + <s xml:space="preserve">At ex prima ſexti aut <lb/>18. ſeptimi ſic ſe habet totale <reg norm="productum" type="context">productũ</reg> <var>.<lb/>u.a.</var> ad partiale <var>.u.o.</var> ſicut <var>.a.e.</var> ad <var>.o.e</var>. <lb/></s> + <s xml:space="preserve">quare ſic ſe habebit <var>.u.a.</var> ad <var>.u.o.</var> ſicut <var>.n.<lb/>e.</var> ad <var>.u.e.</var> ſed <var>.u.e.</var> et <var>.u.o.</var> numero non differunt, cum ſint vnius & eiuſdem ſpeciei, (ta-<lb/>met ſi numerus <var>.u.o.</var> ſit ſuperficialis 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 diuiſibilem per proueniens, oritur numerus diui-<lb/>dens?</s> + </p> + <p> + <s xml:space="preserve">Sit ſubſcriptus rectangulus <var>.o.e.</var> numerus diuiſi <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 ſi <var>.a.o.</var> diuidens fuerit <var>.a.e.</var> proue-<lb/>niens erit, ſi 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">HOcipſum, alia <reg norm="quoque" type="simple">quoq;</reg> uia licebit ſpeculari.</s> + </p> + <p> + <s xml:space="preserve">Sit linea <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> numerum diuiſibilem, et <var>.o.</var> primi prouenientis linea <var>.e.</var> pri <lb/>mi diuidentis <var>.u.</var> ſecundi prouenientis ideſt cum <var>.o.</var> pro diuidente ſumetur. </s> + <s xml:space="preserve">Iam ex <lb/>indicata definitione diuiſionis nono theoremate huius libri, dabitur proportio <var>.a.</var> <lb/>ad <var>.o.</var> prout datur <var>.e.</var> ad vnitatem ſignificatam li-<lb/>nea <var>.i.</var> & permutatim <var>.a.</var> ad <var>.e.</var> ſicut <var>.o.</var> ad <var>.i.</var> ſed <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0023-02a" xlink:href="fig-0023-02"/> + ad <var>.u.</var> ſic ſe habet prout <var>.o.</var> ad <var>.i.</var> ex eadem definitio-<lb/>ne diuiſionis, <reg norm="itaque" type="simple">itaq;</reg> ſic ſe habebit <var>.a.</var> ad <var>.u.</var> ſicut <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 cognoſcere cuius numeri quatuor quintæ par-<lb/>tes, ſint duæ tertię, aut quid ſimile, <reg norm="conſultiſſime" type="context">cõſultiſſime</reg> faciat, ſi ad unam <reg norm="eandemque" type="simple">eandemq;</reg> <lb/>denominationem reduxerit.</s> + </p> + <p> + <s xml:space="preserve">Prout in propoſito exemplo, <reg norm="cum" type="context">cũ</reg> <reg norm="denominans" type="context">denominãs</reg> <reg norm="communis" type="context">cõmunis</reg> ſit quindecim, cuius duæ ter <lb/>tiæ ſunt <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 ſit per quatuor quintas, ſcilicet duodecim, & productum diuidendum per <lb/>duas tertias, hoc eſt 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">numeratorũ</reg> ad vnam & eandem denominationem attinet, <lb/>ea de cauſa fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-<lb/>minis indiget, quo quartus à prędictis dependens, inueniri poſſit, quandoquidem <lb/>bini illi reſpectus, tribus terminis comprehendi <reg norm="poſsunt" type="context">poſsũt</reg>. </s> + <s xml:space="preserve">At quod ad multiplicatio-<lb/>nem ſpectat denominantis <reg norm="communis" type="context">cõmunis</reg> <reg norm="cum" type="context">cũ</reg> numerante denominantis in cogniti & diui-<lb/>ſionem producti per numerantem <reg norm="cognitum" type="context">cognitũ</reg> illę nihil aliud ſunt, quam <reg norm="quartum" type="context">quartũ</reg> <reg norm="terminum" type="context">terminũ</reg> <lb/>inuenire, ita proportionatum tertio, vt ſecundus primo.</s> + </p> + <p> + <s xml:space="preserve">Excmpli gratia, ſit <var>.a.</var> <reg norm="denotans" type="context">denotãs</reg> nume-<lb/>rantem denominantis cogniti, qui ſigni <lb/> + <anchor type="figure" xlink:label="fig-0023-03a" xlink:href="fig-0023-03"/> + ficetur linea <var>.o.</var> et <var>.e.</var> ſit denominantis in-<lb/>cogniti numerans, denotati linea <var>.u.</var> imò <lb/>verò & cogniti <var>.o.</var> nempe quatuor <lb/>quintæ, Iam ſi <var>.o.</var> cum <var>.e.</var> multiplicemus, & productum per <var>.a.</var> diuidemus dabitur <var>.u.</var> <lb/>ſic ſe habens ad <var>.e.</var> ſicut <var>.o.</var> ad <var>.a.</var> ex .20. ſeptimi.</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, ſint quatuor quintę partes, mul<lb/>tiplicandę eſſent duæ tertiæ per denominantem communem, & productum diui-<lb/>dendum per quatuor quintas ipſius de-<lb/>nominantis. </s> + <s xml:space="preserve">Ac ſi quis diceret ſi <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/>propoſito exemplo, terminus <var>.a.</var> loco <var>.e.</var> <lb/>duos ſortietur denominantes, cognitum <lb/>videlicet <var>.o.</var> et <var>.u.</var> incognitum quod po-<lb/>ſtea cognitum oritur ex regula de tribus, vt dictum eſt.</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 cognoſci poterit proportionem quantitatis cenſicæ cenſicæ ad <lb/>ſimilem quantitatem quadruplam eſſe ad eam, quæ eſt ſuarum 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="ſciendus" type="context">ſciẽdus</reg> eſt modus <reg norm="productionis" type="simple">ꝓductionis</reg> <reg norm="harum" type="context">harũ</reg> <reg norm="dignitatum" type="context">dignitatũ</reg> qui oritur ex produ-<lb/>ctione primæ radicis in ſeipſam, prout qui <reg norm="cubum" type="context">cubũ</reg> requirit, ducat radicé in ſuo quadra-<lb/>to, & orietur cubus, hæc poſtea ducta in cubum, <reg norm="quantitatem" type="context">quantitatẽ</reg> cenſicam <reg norm="cenſicam" type="context">cenſicã</reg>, et in <lb/>hanc, prædictam radicem, dabit quantitatem primam relatam. </s> + <s xml:space="preserve">Quod vbi ſciueri-<lb/>mus, meminiſſe oportet Euclidem decimaoctaua ſexti aut .11. octaui docere, pro-<lb/>portionem quadrati ad <reg norm="quadratum" type="context">quadratũ</reg>, duplam eſſe proportioni ſuarum radicum, & .36. <lb/>vndecimi aut .11. octaui, cubi ad <reg norm="cubum" type="context">cubũ</reg> triplam eſſe, ego verò nunc aſſero, cenſici cen <lb/>ſici 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 ſpeculationis gratia, detur linea <var>.d.</var> quæ cubum maiorem ſignificet. et <var>.b.</var> <lb/>minorem <var>.c.</var> verò ſit radixipſius <var>.d.</var> et <var>.e.</var> ipſius <var>.b.</var> ita ordinate adinuicem, vt in ſub-<lb/>ſcripta figura cernitur. </s> + <s xml:space="preserve">Iam <var>.c.</var> cum <var>.d.</var> producatur <reg norm="proueniatque" type="simple">proueniatq́;</reg> <var>.q.</var> cenſicum cenſi-<lb/>cum, tum producatur <var>.e.</var> cum <var>.b.</var> et dabitur <var>.p.</var> alterum cenſicum cenſicum. </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/>cauſa 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. ſexti, aut quinta octaui <lb/>depręhenditur. </s> + <s xml:space="preserve">Quare <reg norm="cum" type="context">cũ</reg> proportio <var>.d.</var> ad <var>.<lb/>b.</var> proportioni <var>.c.</var> ad <var>.e.</var> tripla ſit, 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, ſumptis ſemper <var>.d</var> et <var>.b.</var> pro duo-<lb/>bus cenſibus cenſuum, 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 ſit diuiſionis vnius radicis per alteram?</s> + </p> + <p> + <s xml:space="preserve">Sint exempli gratia duę lineæ <var>.b.q.</var> et <var>.f.g.</var> quæ ſignificent duas radices cuiuſuis <lb/>dignitatis; </s> + <s xml:space="preserve"><reg norm="demusque" type="simple">demusq́;</reg> eſſe radices duorum quadratorum, <reg norm="quadratumque" type="simple">quadratumq́;</reg> ipſius <var>b.q.</var> <lb/>per quadratum ipſius <var>.f.g.</var> diuidatur; </s> + <s xml:space="preserve">quadrataq́ue radix prouenientis ſit <var>.d.q.</var> <lb/>vnitas verò linearis ſit <var>.i.g</var>. </s> + <s xml:space="preserve">Dico ipſam <var>.d.q.</var> eſſe proueniens ex diuiſione <var>.b.q.</var> <lb/>per <var>.f.g</var>. </s> + <s xml:space="preserve">Patet enim ex definitione diuiſionis 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 ipſius <var>.d.q.</var> talem eſſe partem quadrati ipſius <var>.b.q.</var> qualis quadratum ipſius <var>.g.i.</var> <lb/>eſt quadrati ipſius <var>.f.g</var>. </s> + <s xml:space="preserve">Scimus pręterea ex .19. ſexti, aut vndecima octaui, propor-<lb/>tioné quadrati ipſius <var>.b.q.</var> ad <reg norm="quadratum" type="context">quadratũ</reg> ipſius <var>.d.q.</var> duplam eſſe proportioni <var>.b.q.</var> ad <var>.<lb/>d.q.</var> ſuarum radicum (cuborum enim tripla eſſet & cenſuum cenſuum, quadrupla, <lb/><reg norm="atque" type="simple">atq;</reg> ita deinceps ex præcedenti theoremate) Id ipſum dico de dignitatibus ipſius <var>.<lb/>f.g.</var> et <var>.i.g.</var> reſpectu radicum <var>.f.g.</var> et <var>.i.g</var>. </s> + <s xml:space="preserve">Vnde <lb/>cum proportio dignitatis ipſius <var>.b.q.</var> ad il-<lb/>lam <var>.d.q.</var> ęqualis ſit proportioni dignitatis <lb/> + <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/> + ipſius <var>.f.g.</var> ad illam <var>.g.i.</var> ex communi ſcien-<lb/>tia apertè cognoſcemus ſimplices propor-<lb/>tiones eſſe interſe æquales, nempe eam quę <lb/>eſt <var>.b.q.</var> ad <var>.d.q.</var> æqualem eſſe ei, quæ eſt <var>.f.<lb/>g.</var> ad <var>.i.g.</var> <reg norm="itaque" type="simple">itaq;</reg> ſequitur ex definitione diuiſionis <var>.d.q.</var> eſſe proueniens ex diuiſione <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, eſt quadrata radix, producti <lb/>ſuorum quadratorum ſimul?</s> + </p> + <p> + <s xml:space="preserve">In cuius rei gratiam, ſint duo quadrata <var>.d.a.</var> et <var>n.o.</var> coniuncta ſimul, prout in ſub-<lb/>ſcripta figura apparet, ita tamen vtangulus <var>.a.n.u.</var> ſitre <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">coniũgentur</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 ipſius <var>.a.n.</var> in <var>.n.u.</var> duarum videlicet radicum <lb/>quadratarum ſimul, dabitur ex prima ſexti, aut de-<lb/>cimaottaua ſeptimi, productum <var>.a.u.</var> medium propor <lb/>tionale inter quadratum <var>.a.d.</var> et <var>.u.c.</var> quod ſi cogi-<lb/>temus has tres ſuperficies, tres numeros eſſe, pate-<lb/>bit ex vigeſimaprima ſeptimi productum <var>.a.u.</var> in ſe-<lb/>ipſum, quadratum ſcilicet <var>.a.u.</var> æquale eſſe producto <var>.<lb/>a.d.</var> in <var>.u.c.</var> ex quo propoſiti euidentia conſequetur.</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 ipſum in cubis cognoſci poterit. <lb/></s> + <s xml:space="preserve">Sit cubus <var>.l.b.</var> & cubus <var>.o.p.</var> quorum productum ſit <var>.u.g.</var> quod aſſero eſle <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 demonſtra-<lb/>bo eſſe numeri æqualis numero <var>.m.q.</var> <lb/>qui <var>.m.q.</var> productum eſt ipſius <var>.m.e.</var> in <var>.e.<lb/>q.</var> radicum propoſitorum 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">productũ</reg> ſit 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, tantę 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">vocenturq́;</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 ſeptimi, eadem erit proportio <var>.a.g.</var> ad <var>.c.g.</var> <lb/>quæ eſt <var>.l.b.</var> ad <var>.l.x.</var> corporeum, ſed ex .25. vndecimi & prima ſexti, ita ſe habet <var>.a.K.</var> <lb/>ad <var>.K.c.</var> vnitatem linearé ſicut <var>.a.g.</var> ad <var>.c.g.</var> & ex <reg norm="eiſdem" type="context">eiſdẽ</reg> ita ſe habebit <var>.b.e.</var> ad <var>.e.x.</var> vnita-<lb/>tem linearem, ſicut <var>.l.b.</var> ad quadratum <var>.l.x.</var> corporeum. </s> + <s xml:space="preserve">Itaque ſic ſe habebit <var>.b.e.</var> ad <lb/>vnitatem linearem <var>.e.x.</var> videlicet <var>.K.c.</var> ſicut <var>.a.K.</var> ad ipſam <var>.K.c</var>. </s> + <s xml:space="preserve">Vnde ex nona quinti <var>.<lb/>a.K.</var> æqualis erit <var>.e.b.</var> & conſequenter æqualis <var>.m.e</var>. </s> + <s xml:space="preserve">Iam verò ſit <var>.u.g.</var> productum <var>.l.b.</var> <lb/>cubi, in cubum <var>.o.p.</var> vt ſupra dictum eſt, Hinc patebit ex quauis duarum propoſitio-<lb/>num, decimaoctaua, aut decimanona ſeptimi, eandem futuram proportionem <var>.u.g.</var> <lb/>ad <var>.a.g.</var> quæ eſt <var>.o.p.</var> ad <var>.x.p.</var> quadratum corporeum. </s> + <s xml:space="preserve">Quare ex poſtremis, dictis ratio-<lb/>nibus, eadem erit proportio <var>.u.K.</var> ad <var>.a.K.</var> quæ eſt <var>.o.e.</var> ad vnitatem linearem <var>.e.x.</var> at <lb/>ex dictis decimaoctaua & decimanona ſeptimi, ita ſe habet <reg norm="numerus" type="simple">numerꝰ</reg> <var>.m.q.</var> ad <reg norm="numerum" type="context">numerũ</reg> <lb/><reg norm="ſuperficialem" type="context">ſuperficialẽ</reg> <var>.m.e.</var> qui <reg norm="producitur" type="simple">ꝓducitur</reg> à lineari <var>.m.e.</var> in vnitaté <reg norm="linearem" type="context">linearẽ</reg> ipſius <var>.e.q.</var> ſicut nume <lb/>rus <var>.q.e.</var> ad ſuam vnitaté, ſed <reg norm="cum" type="context">cũ</reg> numerus <var>.a.K.</var> æqualis ſit numero <var>.m.e.</var> vt <reg norm="probatum" type="context">probatũ</reg> eſt <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 eſt 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 ſequitur <var>.u.g.</var> cubum eſſe, & <var>f.g.</var> radicem ipſius, 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 uniuerſum ſciri poſſit totum <reg norm="infinitum" type="context">infinitũ</reg> dignitatum, hoc eſt radicem <lb/>producti duarum dignitatum ſimilium, 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 ſit quadratum <var>.m.</var> radix cuius ſit <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">ꝓducto</reg> <var>.q.p.</var> in <var>.g.k.</var> <reg norm="quod" type="wordlist">qđ</reg> <reg norm="autem" type="context">autẽ</reg> ſit <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 ſuam vnitatem linearem, ex decimaoctaua, aut decima-<lb/>nona ſeptimi, hæc vero vnitas linearis ſit <var>.t.</var> cuius ſuperficialis ſit <var>.u.</var> vnitas ſcilicet to-<lb/>ties in ſeipſam multiplicata quoties propoſita dignitas patitur, tametſi in præſen <lb/>ti exemplo quadrata dignitas ſumatur. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex eiſdem propoſitionibus decimaocta <lb/>ua aut decimanona, ſic ſe habet <var>.m.</var> ad <var>.n.</var> ſicut <var>.i.</var> ad <var>.u</var>. </s> + <s xml:space="preserve">Scimus pręterea <reg norm="proportionem" type="context">proportionẽ</reg> <var>.<lb/>m.</var> ad <var>.n.</var> (eo quod in propoſito exemplo ſint quadrata) duplam eſſe proportioni <var>.b.<lb/>d.</var> ad <var>.q.p.</var> et ipſius <var>.i.</var> ad <var>.u.</var> pariter duplam proportioni <var>.g.k.</var> ad <var>.t.</var> iam autem dictum <lb/>fuit ſic ſe habere <var>.m.</var> ad <var>.n.</var> ſicut <var>.i.</var> ad <var>.u</var>. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.<lb/>b.d.</var> ſic ſe habebit ad <var>.q.p.</var> ſicut <var>.g.k.</var> ad <var>.t.</var> <lb/> + <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> + quandoquidem ſic ſe habeattotum ad <reg norm="to- tum" type="context">to-<lb/>tũ</reg>, ſicut pars ad <reg norm="partem" type="context">partẽ</reg>, <reg norm="dum" type="context">dũ</reg> ſimiles ſint, proba <lb/><reg norm="tum" type="context">tũ</reg> <reg norm="autem" type="context">autẽ</reg> eſt ſuperius ita ſe habere <var>.o.</var> ad <var>.q.p.</var> <lb/>ſicut <var>.g.k.</var> ad <var>.t.</var> <reg norm="itaque" type="simple">itaq;</reg> <var>.o.</var> ſic ſe habebit ad <var>.q.p.</var> <lb/>ſicut <var>.b.d.</var> ad <var>.q.p.</var> vnde <var>.o.</var> æqualis erit <var>.b.d.</var> <lb/>Hocipſum cęteris dignitatibus conueniet, <lb/>mutatis tantummodo proportionibus <var>.m.<lb/>n.</var> ad proportionem <var>.b.d</var>: <var>q.p.</var> ſic propor-<lb/>tionibus duarum dignitatum <var>.i.u.</var> ad pro-<lb/>portionem ſuarum 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 ſi quilibet numerus in duas partes inæquales diuiſus <lb/>fuerit, <reg norm="totumque" type="simple">totumq́</reg> diuiſum per <reg norm="vnam" type="context">vnã</reg> partium, & per eandem pars altera diuiſa fue-<lb/>rit: </s> + <s xml:space="preserve">differentia prouenientium ſemper vnitas erit. </s> + <s xml:space="preserve">quodquidem veriſſimum eſt.</s> + </p> + <p> + <s xml:space="preserve">Detur enim <var>.b.d.</var> propoſitus numerus in duas partes inæquales diuiſus <var>.b.c.</var> et <var>.c.d.</var> <lb/>& in primis <reg norm="totum" type="context">totũ</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">.ꝑ</reg> <var>.i.o.</var> ſigni-<lb/>ficetur, tum pars ipſa <var>.b.c.</var> <reg norm="per" type="simple punctuation">ꝑ.</reg> <reg norm="eandem" type="context">eãdem</reg> <var>.c.d.</var> diuidatur, <reg norm="ſitque" type="simple">ſitq́;</reg> <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a</var>. </s> + <s xml:space="preserve">Sanè ex defini-<lb/>tione diuiſionis, eadem erit proportio <var>.b.d.</var> ad <var>.e.o.</var> quæ eſt <var>.c.d.</var> ad <var>.i.o.</var> et ita <var>.b.c.</var> ad <var>.a.</var> <lb/>ſicut <var>.c.d.</var> ad <var>.i.o</var>. </s> + <s xml:space="preserve">Ex <ref id="ref-0009">.19. autem quinti</ref>, ita ſe habet <var>.b.c.</var> ad <var>.e.i.</var> ſicut <var>.b.d.</var> ad <var>.e.o.</var> at <var>.b.d.</var> <lb/>ad <var>.e.o.</var> ſic ſe habet ſicut <var>.c.d.</var> ad <var>.i.o.</var> hoc eſt ſicut <var>.b.c.</var> ad <var>.a</var>. </s> + <s xml:space="preserve">Quare ex .11. quinti ſic ſe <lb/>habebit <var>.b.c.</var> ad <var>.e.i.</var> ſicut .ad <var>.a.</var> ex quo ex .9. <reg norm="praedi cti" type="simple">prędi<lb/>cti</reg> <var>.a.</var> æqualis erit <var>.e.i.</var> ſed <var>.e.i.</var> minor eſt <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 ſequitur propoſitum verum eſ<lb/>ſe. </s> + <s xml:space="preserve">Quod ipſum pauciſſimis verbis ſic definiri <lb/>poteſt, ſi dixerimus, eiuſmodi diuidens .in par-<lb/>te diuiſibili, <reg norm="quam" type="context">quã</reg> in toto, ſemel minus ingredi, <lb/>quandoquidem altera pars eſt, 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">HOcipſum alia ratione contemplari po<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 diuiſus <var>.a.u.</var> et <var>.u.e.</var> totius autem diuidens ſit <var>.u.e.</var> & partis alterius <var>.a.u.</var> <lb/>totius verò <reg norm="proueniens" type="context">proueniẽs</reg> ſit <var>.a.c.</var> partis <reg norm="autem" type="context">autẽ</reg>, ſit <reg norm="proueniens" type="context">proueniẽs</reg> <var>.a.n.</var> tum differentia ſit <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, ſit <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 eſt <var>.u.e.</var> ad <var>.a.i.</var> ex definitione diuiſionis, et eandem <lb/>eſſe <var>.a.u.</var> ad <var>.a.n.</var> quæ eſt <var>.u.e.</var> ad <var>.a.i.</var> vnde ex .<lb/>11. quinti ſic ſe habebit <var>.a.e.</var> ad <var>.a.c.</var> ſicut <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. eiuſdem ſic ſe habe-<lb/>bit <var>.u.e.</var> ad <var>.n.c.</var> ſicut <var>.a.e.</var> ad <var>.a.c.</var> ſed. ſic ſe <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, ſic ſe habebit <var>.u.e.</var> ad <var>.n.c.</var> ſicut ad <var>.a.<lb/>i</var>. </s> + <s xml:space="preserve">Quare ex .9. eiuſdem <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 diuiſis adinuicem, <reg norm="multiplicatisque" type="simple">multiplicatisq́</reg> prouenien <lb/>tibus ſimul, productum, ſemper eſt vnitas ſuperficialis? </s> + <s xml:space="preserve">Nempe ex .20. ſeptimi, <lb/>quoniam vnitas linearis ſemper media proportionalis eſt inter bina prouenientia. <lb/></s> + <s xml:space="preserve">Quodita ſpecularilicet.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Significentur" type="context">Significẽtur</reg> duo propoſiti numeri per <var>.b.p.</var> et <var>.b.d.</var> mutuo diuiſi, proueniens au-<lb/>tem <var>.b.p.</var> per <var>.b.d.</var> diuiſum ſit <var>.b.n.</var> tum proueniens <var>.b.d.</var> diuiſum per <var>.b.p.</var> ſit <var>.b.a.</var> <lb/>et <var>.b.t.</var> ſit 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 diuiſionis, dabitur eadem proportio <var>.b.p.</var> ad <var>.b.n.</var> quæ eſt <var>.b.d.</var> <lb/>ad <var>.b.e.</var> et proportio <var>.b.d.</var> ad <var>.b.a.</var> quæ eſt <var>.b.p.</var> ad <var>.b.t</var>. </s> + <s xml:space="preserve">Sed cum ſic ſe habeat <var>.b.<lb/>p.</var> ad <var>.b.n.</var> ſicut <var>.b.d.</var> ad <var>.b.e.</var> permutando ſic ſe habebit <var>.b.p.</var> ad <var>.b.d.</var> ſicut <var>.b.n.</var> ad <var>.b.<lb/>e.</var> hoc eſt ad <var>.b.t.</var> et cum ſic ſe habeat <var>.b.d.</var> ad <var>.b.a.</var> ſicut <var>.b.p.</var> ad <var>.b.t</var>: permutando ſic ſe <lb/>habebit <var>.b.d.</var> ad <var>.b.p.</var> ſicut <var>.b.a.</var> ad <var>.b.t</var>. <lb/></s> + <s xml:space="preserve">Quare euerſim ſic ſe 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> ſicut <var>.b.t.</var> ad <var>.b.a.</var> ſed <var>.b.n.</var> ad <var>.b.t.</var> ſic <lb/>ſe 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/>quintiſic ſe habebit <var>.b.n.</var> ad <var>.b.t.</var> ſicut <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 eſt <var>.b.e.</var> et <var>.b.t.</var> idem omnino eſſe. </s> + <s xml:space="preserve">Quare ex .20. ſeptimi pro-<lb/>poſiti veritas innoteſcet.</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">IDipſum & hac altera uia patebit.</s> + </p> + <p> + <s xml:space="preserve">Duo illi numeri per <var>.o.</var> et <var>.u.</var> ſignificentur mutuo diuiſi, proueniens <reg norm="autem" type="context">autẽ</reg> <var>.o.</var> per <var>.<lb/>u.</var> ſit <var>.e.</var> et proueniens <var>.u.</var> per <var>.o.</var> ſit <var>.x.</var> vnitas uerò per <var>.i.</var> ſignificetur, quas tamen quanti-<lb/>tates ſubſcripto modo ad inuicem diſponi-<lb/>to. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> ex definitione diuiſionis, eadem erit <lb/> + <anchor type="figure" xlink:label="fig-0028-04a" xlink:href="fig-0028-04"/> + proportio <var>.o.</var> ad <var>.e.</var> quę eſt <var>.u.</var> ad <var>.i.</var> et <var>.o.</var> ad <var>.i.</var> quę <lb/>eſt <var>.u.</var> ad <var>.x</var>. </s> + <s xml:space="preserve">Quare ex æqualitate <reg norm="proportionum" type="context">proportionũ</reg> <var>.<lb/>c.</var> ad <var>.i.</var> ſic ſe habebit ſicut <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/>ſeptimi propoſitum concludetur. </s> + <s xml:space="preserve">Huiuſmodi rei cauſa etiam eſt, quod proueniens <lb/>diuiſionis vnius eſt numerator æqualis denominatori diuiſionis 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 diuiſis, <reg norm="sumptis" type="context">sũptis</reg> deinde prouenientibus ſimul et adinui <lb/>cem, & per hanc ſummam, diuiſa ſumma quadratorum dictorum <reg norm="propoſitorum" type="context">propoſitorũ</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 ſimul.</s> + </p> + <p> + <s xml:space="preserve">Sint exempli gratia propoſiti numeri .2. et .8. qui mutuo diuiſi in primis dent pro <lb/>uenientia quatuor integra, tum quartam partem pro altero proueniente, hæc colle-<lb/>cta dabunt ſummam quatuor integrorum et quartæ partis vnius, ſumma autem qua <lb/>dratorum binarij & octonarij erit .68. qui quidem numerus per quatuor & quar <lb/>tam partem vnius diuiſus 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 ſpeculatio, ſint duæ lineæ <var>.o.e.</var> et <var>.o.n.</var> quæ duos numeros pro-<lb/>poſitos ſignificent, inuicem ad angulum rectum <var>.o.</var> coniunctæ, quarum quadrata <lb/>ſint <var>.o.a.</var> et <var>.o.p.</var> ipſorum productum ſit <var>.n.e.</var> tum <var>.o.t.</var> ſit proueniens ex diuiſione <var>.o.e.</var> <lb/>per <var>.o.n</var>. </s> + <s xml:space="preserve">Hęc ſingulatim conſideremus (<reg norm="nam" type="context">nã</reg> ſi in partibus ſimplicibus quod dicimus ac <lb/>ciderit, id ipſum in compoſitis conſequenter eueniet) quamobrem ex definitione di <lb/>uiſionis dabitur eadem proportio <var>.o.e.</var> ad <var>.o.t.</var> quæ eft <var>.o.n.</var> ad vnitatem, quæ ſit <var>.o.<lb/>x</var>. </s> + <s xml:space="preserve">Nunc cogitemus <reg norm="ſuperficiem" type="context">ſuperficiẽ</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 diuiſione numeri quadrati <var>.o.a.</var> per <reg norm="numerum" type="context">numerũ</reg> <var>.o.t.</var> <reg norm="eritque" type="simple">eritq́</reg> <lb/><reg norm="eadem" type="context">eadẽ</reg> proportio <var>.c.t.</var> ad <var>.o.e.</var> quæ eſt <var>.o.e.</var> ad <var>.o.t.</var> ex ſecunda parte quintæ decimæ ſexti, <lb/>aut .20. ſeptimi. </s> + <s xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> <reg norm="autem" type="context">autẽ</reg> dictum eſt <var>.o.e.</var> ad <var>.o.t.</var> ſic ſe habere ſicut <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 ſic ſe habebit <var>.c.t.</var> ad <var>.o.e.</var> ſicut <var>.o.n.</var> ad <var>.o.x</var>. </s> + <s xml:space="preserve">Sed ex prima ſexti, aut .18. vel .<lb/>19. ſeptimi, ſic ſe habet <reg norm="productum" type="simple">ꝓductum</reg> <var>.n.e.</var> ad <var>.e.x.</var> ſicut <var>.o.n.</var> ad <var>.o.x</var>. </s> + <s xml:space="preserve">quare denuo ſic ſe ha-<lb/>bebit numerus <var>.c.t.</var> ad numerum <var>.o.e.</var> ſicut 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> ſpecie idem eſt, 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 ipſum de quadrato ipſius <var>.o.n.</var> videlicet <var>.p.o.</var> <lb/>dico. </s> + <s xml:space="preserve">Nam ſi proueniens <var>.o.n.</var> diuiſo per <var>.o.e.</var> ideſt <var>.<lb/>o.i.</var> proportionale reſpondens ad <var>.o.t.</var> cum <var>.o.t.</var> <lb/><reg norm="coniunctum" type="context">coniunctũ</reg> fuerit, et per <reg norm="hanc" type="context">hãc</reg> ſummam diuiſa ſumma <lb/>quadratorum <var>.o.a.</var> et <var>.o.p.</var> patet per ſe proueniens <lb/>futurum eiuſdem numeri <var>.c.t.</var> <reg norm="ipſumque" type="simple">ipſumq́</reg> <var>.c.t.</var> proue-<lb/>niens ſemper ſuturum.</s> + </p> + <p> + <s xml:space="preserve">Quo autem lucidius res hæc innoteſcat. </s> + <s xml:space="preserve">Cogi <lb/>temus proueniens quadrati <var>.o.p.</var> diuiſi ab <var>.o.i.</var> re-<lb/><reg norm="ſpondentisque" type="simple">ſpondentisq;</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 conſe-<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 compoſito 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/>poſito <var>.t.u.</var> <reg norm="quam" type="context">quã</reg> ex ſimplici <var>.o.c.</var> ex quo propoſiti ſe <lb/>ſe 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, ſed quod tamen citra al-<lb/>gebraticam effectionem, aut neſcierunt, aut noluerunt diſſoluere, quod nihi-<lb/>lominus facillimum eſt.</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 eiuſmodi partes diuidendum, vt ſumma qua-<lb/>dratorum dictarum partium, alteri numero poſsibili propoſito æqualis ſit, poſſi-<lb/>bili inquam, etenim ſi eiuſmodi numerus propoſitus, minor eſſet producto totius <lb/>primi in ſuum dimidium, eſſet huiuſmodi factum impoſſibile. </s> + <s xml:space="preserve">Quod nos exequi <lb/>cupientes, ſumamus primum <reg norm="numerum" type="context">numerũ</reg> propoſitum, quem in ſe ipſum multiplice-<lb/>mus. </s> + <s xml:space="preserve">ab hoc quadrato deducamus ſecundum numerum propoſitum, tum quod re-<lb/>manſerit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-<lb/>hi, accepta poſtea radice quadrata reſidui & dempta ex priori numero propoſito, <lb/></s> + <s xml:space="preserve">tunc dimidium reſidui vna pars erit ex duabus primi numeri quæſita.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia proponantur .20. diuidenda in duas eiuſmodi partes, vt ſumma <lb/>quadratorum ipſarum partium æqualis ſit .272. qui numerus maior eſt .200. maior <lb/>inquam dimidio quadrati .400. ipſorum .20. hic autem numerus .272. è quadra-<lb/>to .400. deducatur, <reg norm="remanebunt" type="context">remanebũt</reg> enim .128. quod duplicari iubeo, <reg norm="producentur" type="context">producẽtur</reg> <reg norm="ſiquidem" type="context">ſiquidẽ</reg> <num value="256">.<lb/>256.</num> quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem ſumi <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 propoſito .20. detra-<lb/>hetur, <reg norm="remanebitque" type="simple">remanebitq́</reg> .16. pro altera parte.</s> + </p> + <p> + <s xml:space="preserve">Cuius demonſtrationis cauſa, in primis cogitemus quadratum <var>.a.c.</var> cognitum nu-<lb/>meri <var>.a.b.</var> primò propoſiti, qui cogitetur diuiſus in duo quadrata <var>.d.e.</var> et <var>.e.b.</var> <reg norm="duo- que" type="simple">duo-<lb/>q́</reg> ſupplementa <var>.a.e.</var> et <var>.e.c.</var> numerus autem ſummæ duorum quadratorum <var>.d.e.<lb/>b.</var> pro ſecundo propoſito datur; </s> + <s xml:space="preserve">ex quo, ſumma duorum ſupplementorum <var>.a.e.c.</var> <lb/>conſequenter erit cognita, quę cum duplicata fuerit, & quatuor hæc ſupplementa<unsure/> <lb/>cogitatione accommodata, prout in <lb/>quadrato <var>.f.g.</var> apparet (<reg norm="quanuis" type="context">quãuis</reg> idipſum <lb/> + <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> + proueniret ſi modo Eucl. octaua <reg norm="ſecundi" type="context">ſecũdi</reg> <lb/>aptaretur) æquali quadrato <var>.a.c.</var> ita vt <lb/>cogitatis quatuor ſupplementis numeri <lb/>cogniti in quadrato <var>.f.g.</var> ex conſequen-<lb/>ti cognoſcetur 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 eſt) primo propoſiti, 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 ſupplementi, cum in <lb/>præcedenti theoremate dictum fuerit, <lb/>qua ratione manifeſtetur duplum ſupple-<lb/>menti ipſius.</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 ſubſcripta figura lineam <var>.<lb/>a.b.</var> tanquam primum numerum propoſi-<lb/>tum, & productum <var>.a.e.</var> ſupplemento <var>.a.e.</var> primæ præcedentis figuræ æquale ſit, <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 ſecundi Eucli. dabit quod propoſitum 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 eſt, cur ſubtracto duplo producti duorum numerorum ad inui-<lb/>cem <reg norm="multiplicatorum" type="context">multiplicatorũ</reg> ex ſumma ſuorum quadratorum, ſemper quod ſuper <lb/>eſt duorum numerorum quadratum differentiæ ſit?</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia ſi proponerentur duo numeri .16. et .4. duplum producti eorum <lb/>eſſet .128. quò detracto ex ſumma ſuorum 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 vtſciamus, duo numeri propoſiti, duabus lineis ſignificentur, maiore <var>.q.g.</var> <lb/>et minore <var>.g.p.</var> directè coniunctis, ſuper 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">dabũtur</reg> <var>.q.n.</var> et <var>.n.u.</var> ſingula 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 propoſi-<lb/>torum, quod ſatis <reg norm="ſuperque" type="simple">ſuperq́</reg> , probatur quarta ſecundi 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 ſecundi, 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 ipſo 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> ſed hæ duæ quantitates, ſunt par-<lb/>tes duorum quadratorum dictorum, & quæ ſuper <lb/>eſt <var>.m.e.</var> quadratum differentiæ vnius numeri pro-<lb/>poſiti ab altero, prout in ſubſcripta figura licebit cui <lb/>libet conſiderare. </s> + <s xml:space="preserve">Itaque veritas hæc manifeſta <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 propoſitis maiorem per minorem diuidunt, ſi <lb/>proueniens per maiorem numerum multiplicauerint, productum æquale <lb/>erit prouenienti ex diuiſione quadrati maioris numeri per minorem?</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia ſi proponantur duo numeri .20. et .4. <reg norm="ipſeque" type="simple">ipſeq́</reg> .20. per .4. diui-<lb/>datur, dabit quinque, tum .400. quadrato .20. diuiſo per prioré .4. dabit .100. <lb/>quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, ſint duo numeri, qui lineis <var>.x.u.</var> et <var>.x.s.</var> maiore <reg norm="atque" type="simple">atq;</reg> mi-<lb/>nore ſignificétur, tum <var>.u.x.</var> numerus per <var>.s.x.</var> di-<lb/>uidatur, ſitq́ue proueniens <var>.x.n.</var> poſtmodum qua-<lb/> + <anchor type="figure" xlink:label="fig-0031-02a" xlink:href="fig-0031-02"/> + dratum <var>.u.x.</var> ſit <var>.x.o.</var> et productum ex <var>.n.x.</var> in <var>.u.<lb/>x.</var> ſit <var>.x.e.</var> quod æquale eſſe dico prouenienti ex <lb/>diuiſione quadrati <var>.o.x.</var> per <var>.s.x.</var> quod ſit <var>.m</var>. </s> + <s xml:space="preserve">Patet <lb/>enim ex definitione diuiſionis, talem futuram pro-<lb/>portionem <var>.u.x.</var> ad <var>.n.x.</var> qualis eſt <var>.s.x.</var> ad vnitatem, <lb/>& quadratum <var>.o.x.</var> ad rectangulum <var>.e.x.</var> ita ſe 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, ſicut <var>.u.x.</var> ad <var>.n.x.</var> ex prima ſexti aut .18. vel .19. ſeptimi, </s> + <s xml:space="preserve">quare ex 11. <lb/>quinti ita ſe habebit <var>.o.x.</var> ad <var>.e.x.</var> ſicut <var>.s.x.</var> ad vnitatem; </s> + <s xml:space="preserve">ſed ſicut ſe habet <var>.s.x.</var> ad. <lb/>vnitatem, ita ſe habet pariter <var>.o.x.</var> ad <var>.m</var>. </s> + <s xml:space="preserve">vnde ex .11. prædicta ita ſe habebit <var>.o.<lb/>x.</var> ad <var>.m.</var> ſicut idipſum <var>.o.x.</var> ad <var>.e.x.</var> itaq́ue 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 propoſito aliquo numero in duas partes inæquales diuiſo, ſi rurſus per <lb/>quamlibet ipſarum diuidatur, prouenientia tantumdem coniuncta quantum <lb/>multiplicata efficiant.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſit denarius prop oſitus numerus, per binarium & octonarium <lb/>diuiſus, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta <lb/>crunt .6. cum quarta parte lineari, quæ ſi mul multiplicata, pariter erunt .6. cum <lb/>quarta parte ſuperficiali.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, totalis numerns, linea <var>.q.p.</var> ſignificetur, eius duæ <lb/>partes, per <var>.k.</var> maiorem et <var>.u.</var> minorem, ipſa vnitas per .t: proueniens ex diuiſio-<lb/>ne <var>.q.p.</var> per <var>.k.</var> ſit <var>.q.i.</var> proueniens autem ipſius <var>.q.p.</var> per <var>.u.</var> ſit <var>.q.f.</var> </s> + <s xml:space="preserve">quare ex defini-<lb/>tione diuiſionis ita ſe habebit <var>.q.p.</var> ad <var>.q.i.</var> ſicut <var>.k.</var> ad <var>.t.</var> et <var>.q.p.</var> ad <var>.q.f.</var> ſicut <var>.u.</var> ad <var>.t.</var> <lb/>hoc eſt <var>.q.f.</var> ad <var>.q.p.</var> ſicut <var>.t.</var> ad <var>.u.</var> vnde ex æqualitate <reg norm="proportionum" type="context">proportionũ</reg> ſic ſe habebit <var>.q.f.</var> <lb/>ad <var>.q.i.</var> ſicut <var>.k.</var> ad <var>.u.</var> et conuerſim. </s> + <s xml:space="preserve">Ad hæc in linea <var>.q.p.</var> vnitas, per lineam <var>.q.o.</var> ſigni-<lb/>ficetur, quo facto, dicamus, ſi <var>.q.p.</var> ad <var>.q.i.</var> ſic ſe habet vt <var>.k.</var> ad <var>.q.o.</var> itaque permu-<lb/>tando, ſic ſe habebit <var>.q.p.</var> ad <var>.k.</var> ſicut <var>.q.i.</var> ad <var>.q.o.</var> hoc eſt <var>.k.u.</var> ad <var>.k.</var> ſicut <var>.i.q.f.</var> ad <var>.<lb/>q.f.</var> (nam <var>.k.u.</var> partes ſunt integrales totius <var>.q.p.</var> et <var>.k.u.</var> ad <var>.k.</var> eſt ſicut <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> ſicut <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, ſit <var>.x.f.</var> quod probabo <lb/>æquale eſſe ſummæ <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 ſit <var>.q.o.</var> ſigneturq́ue 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 eſt <var>.q.x.</var> <lb/>ad <var>.q.s.</var> ex prima ſexti, aut .18. vel 19. <lb/>ſeptimi, hoc eſt, ſicut <var>.q.i.</var> ad <var>.q.o.</var> et <lb/>ex .11. quinti (vt dictum eſt) ſicut <var>.i.q.<lb/>f.</var> ad <var>.q.f.</var> ſed numerus <var>.s.f.</var> fuperficia-<lb/>lis tantus eſt, quantus linearis <var>.q.f</var>. <lb/></s> + <s xml:space="preserve">quare ex .9. quinti tantus erit (ſu-<lb/>perficialiter) numerus <var>.x.f.</var> quantus <lb/>(lineariter). <var>f.q.i.</var> quod erat pro-<lb/>poſitum.</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 diuiſo, ſi rurſus diuidatur per <lb/>ſingulas partes, ſumma duorum prouenientium per binarium, ſemper ma-<lb/>ior ſit ſumma prouenientium ex diuiſione vnius partis per alteram.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, ſi 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. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-<lb/>niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-<lb/>tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-<lb/>niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma eſſet <lb/>quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-<lb/>tium per binarium.</s> + </p> + <p> + <s xml:space="preserve">Cuius conſiderationis cauſa, propoſitus numerus linea <var>.q.p.</var> ſignificetur, eius duę <lb/>partes lineis <var>.q.x.</var> et <var>.x.p.</var> <reg norm="tum" type="context">tũ</reg> <var>.q.f.</var> ſit proueniens ex diuiſione totius <var>.q.p.</var> per <var>.x.p.</var> et <var>.<lb/>q.i.</var> ſit proueniens ex diuiſione eiuſdem <var>.q.p.</var> per <var>.q.x.</var> adhæc <var>.h.m.</var> ſit proueniens, <lb/>ex diuiſione <var>.q.x.</var> per <var>x.p.</var> et <var>.h.k.</var> proue-<lb/>niensex diuiſione <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 propoſitum.</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 eſt <lb/>proportionalis inter numerum <lb/> + <anchor type="figure" xlink:label="fig-0033-02a" xlink:href="fig-0033-02"/> + ſui 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 propoſitus, <lb/>qui linea <var>.a.u.</var> ſignificetur, cuiusqua-<lb/>dratum ſit <var>.u.n.</var> vnitas linearis ſit <var>.i.a.</var> <lb/>et ſuperficialis <var>.o.</var> patebit ex .18. ſexti <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> ſed <var>.i.a.</var> e<unsure/>t.o. eadem (ſpecie) <lb/>res <reg norm="sunt" type="context">sũt</reg>, tanta ſcilicet <var>.a.i.</var> quanta <var>.o.</var> vni <lb/> + <anchor type="figure" xlink:label="fig-0033-03a" xlink:href="fig-0033-03"/> + tas eſt, 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> ipſum quod diximus & alia ratione ſpeculari licebit.</s> + </p> + <p> + <s xml:space="preserve">Propoſitus numerus, nunc etiam per <var>.a.u.</var> ſignificetur, eius quadratum per <var>.<lb/>u.n.</var> vnitas linearis per <var>.a.i.</var> <reg norm="productumque" type="simple">productumq́;</reg> <var>.a.u.</var> in <var>.a.i.</var> terminetur, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.n.i</var>. </s> + <s xml:space="preserve">quare <lb/><var>n.i.</var> conſtabit numero íuperficiali æquali numero lineari <var>.a.u.</var> & ex prima fexti aut .<lb/>18. vel .19. ſeptimi, eadem erit proportio <var>.u.n.</var> ad <var>.i.n.</var> quæ eſt <var>.a.u.</var> ad <var>.a.i.</var> ſed nu-<lb/>merus <var>.a.u.</var> cum numero <var>.n.i.</var> idem ſpecie eſt. </s> + <s xml:space="preserve">Itaque medius eſt 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, & diuiſus, medius eſt propor-<lb/>tionalis inter productum multiplicationis, & proueniens diaiſionis.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi .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 eſt propor-<lb/>tionalis.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt ſpeculemur, proponatur numerus multiplicandus & diuidendus, qui ſi-<lb/>gnificetur linea <var>.u.e.</var> multiplicans autem & diuidens linea <var>.a.u.</var> multiplicationis <lb/>productum ſit <var>.e.a.</var> proueniens ex diuiſione ſit <var>.o.e</var>. </s> + <s xml:space="preserve">Nunc proueniens <var>.e.o.</var> per <reg norm="nu- merum" type="context">nu-<lb/>merũ</reg> <var>.a.u.</var> diuidentem multiplicetur, cuius multiplicationis productum ſit <var>.e.i.</var> <lb/>quare, eadem erit proportio numeri <var>.a.e.</var> <lb/>ad numerum <var>.e.i.</var> quæ eſt 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 ſextiaut .18. vel <lb/>19. ſeptimi. </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 ſit. </s> + <s xml:space="preserve">verum eſſe, quod propoſi-<lb/>tum fuit conſequetur.</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 propoſitum numerum ita multiplicare & diuidere cupiunt, vt pro <lb/>ductum multiplicationis, tam ſit multiplex prouenienti ex diuiſione, quam <lb/>quæritur, rectè ſumant aliquem numerum pro multiplicante & diuidente, qui ſit ra <lb/>dix quadrata denominantis quęſitę multiplicitatis.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-<lb/>ductum multiplicationis nonuplum ſit prouenienti ex diuiſione, nempè, vt pro-<lb/>ueniens, nona pars ſit eiuſmodi producti, </s> + <s xml:space="preserve">quare quadratam radicem ipſorum no-<lb/>uem, ideſt denominantis ſumunt, tria ſcilicet, multiplicant igitur & diuidunt <lb/>data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </s> + <s xml:space="preserve">& <lb/>propoſitum ſequitur.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, ſignificetur numerus propoſitus linea <var>.u.e.</var> multipli-<lb/>cans autem & diuidens linea <var>.u.a.</var> productum ſit <var>.e.a.</var> proueniens <var>.e.o.</var> quadratum <lb/>verò <var>.a.u.</var> ſit <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">terminenturq́;</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 eſt <var>.a.e.</var> ad <var>.u.e.</var> numerus enim <var>.e.i.</var> (quamuis ſuperficialis) idem eſt cum nume-<lb/>ro lineari <var>.u.e.</var> ſed <var>.a.e.</var> ad <var>.e.i.</var> ſic ſe habet ſicut <var>.a.u.</var> ad <var>.u.i.</var> ex prima ſexti aut .18. <lb/>vel .19. ſeptimi, (quod ipſum 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/>eſt <var>.x.u.</var> ęqualis erit <reg norm="proportioni" type="simple">ꝓportioni</reg> <var>.a.e.</var> ad <var>.u.e.</var> at trigeſimotertio & trigeſimoquarto theo <lb/>remate probatum eſt proportionem numeri <var>.a.x.</var> ad vnitatem, duplam eſſe propor-<lb/>tioni eiuſdem numeri <var>.a.x.</var> ad <var>.u.x.</var> ſequitur <lb/>igitur cum dimidia ſint æqualia, tota etiam <lb/>æqualia eſſe: </s> + <s xml:space="preserve">hoc eſt 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/>ſumitur numerus <var>.a.u.</var> eiuſmodi vt <reg norm="quadratum" type="context">quadratũ</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"/> + ipſius <var>.a.x.</var> tam ſit 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 ſummam colle-<lb/>cta, æqualia ſint numero propoſito, & ijſdem numeris multiplicatis ad-<lb/>inuicem, productum alteri numero propoſito ſit æquale, rectè ſumant dimidium <lb/>primi numeri propoſiti, cui ſumma quadratorum æquari debet, <reg norm="hocque" type="simple">hocq́;</reg> dimidium <lb/>in ſeipſum multiplicent, vnà etiam alterum numerum propoſitum in ſeipſum <lb/>multiplicent, quod quadratum detrahunt de primo, & reſidui quadratam radicem, <lb/>dimidio primi numeri propoſiti coniungunt, ex qua ſumma, quadratam radicem <lb/><reg norm="eruunt" type="context">eruũt</reg>, quæ duobus quæſitis numeris maior erit, cuius quadrato de primo numero <lb/>detracto, & exreliquo erutaradice quadrata, detur minor numerus, duorum <reg norm="quae- ſitorum" type="simple">quę-<lb/>ſitorum</reg>.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponerentur .34. pro primo numero cui æquari de-<lb/>beret ſumma duorum quadratorum, quorum radicum productum æquale eſſe de-<lb/>beret alteri numero, verbi gratia .15. iubet antiquorum regula, dimidium primi <lb/>numeri in ſeipſum multiplicari, cuius dimidij quadratum erit .289. è quo ſi detra-<lb/>has quadratum ſecundi numeri, nempe .225. remanebit .64. <reg norm="atque" type="simple">atq;</reg> huius ſi quadra-<lb/>tam radicem ſumas 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. ſcilicet, quorum <lb/>radices .5. et .3. eſſent ij numeri, qui quæruntur.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis gratia, cogitemus primum numerum, cui quadratorum fum <lb/>ma æquari debet, ſignificari linea <var>.a.n.</var> tum concipiamus quæſita quadrata ſignifi-<lb/>cari, <reg norm="coniungique" type="simple">coniungiq́</reg> modo ſubſcripto <var>.t.b.k.</var> ſecundum porrò numerum propoſitum, <lb/>ſignificari producto <var>.d.b</var>. </s> + <s xml:space="preserve">Iam nil ſupereſt 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> ſummæ quadratorum numerus detur, quadratum di-<lb/>midij <var>.o.a.</var> ſit <var>.s.a.</var> quod nobis erit cognitum; </s> + <s xml:space="preserve">ſit 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> ſit <reg norm="ſe- cundus" type="context">ſe-<lb/>cũdus</reg> numerus <lb/>propoſitus, quæ <lb/>minor erit <var>.a.s.</var> ex quinta ſecundi, aut ſeptima conſequentia poſt .16. noni Eucli-<lb/>dis. </s> + <s xml:space="preserve">Iam ſubtracta quantitate <var>.z.a.</var> è quadrato <var>.a.s.</var> cognoſcetur quadratum <var>.t.x.</var> <lb/>cuius radix æqualis erit <var>.o.u.</var> ex poſtremo adductis, Itaque cognoſcemus <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 cognoſcetur, & eorum radices conſequenter.</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 ipſum & alia ratione perfici poteſt, nempe, iuncta ſumma <var>.k.b</var>: <var>b.d</var>: ec<unsure/> <var>.<lb/>b.t.</var> alteri rectangulo æquali <var>.b.d.</var> quod ſit <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 conſequenter eius radicem <var>.d.k.</var> cognoſcemus, cuius <lb/>ope ac producti <var>.d.b.</var> cognoſcemus <var>.d.p.</var> et <var>.p.k.</var> prout ex theoremate quadrageſi-<lb/>moquinto huius libri patebit.</s> + </p> + <p> + <s xml:space="preserve">Michael Stifelius, vndecimo cap. tertij libri, problema eiuſmodi proponit, <lb/>quod tamen ipſe via algebræ diſsoluit.</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 propoſito æquetur, & quadratorum eorundem differentia alteri nu-<lb/>mero propoſito æqualis ſir. </s> + <s xml:space="preserve">Rectè dimidium ſecundi numeri propoſiti in ſeipſum <lb/>multiplicent, cui quidem numero differentia quadratorum æquari debet; </s> + <s xml:space="preserve">porrò <lb/>huic quadrato primi propoſiti numeri, cui æquandum eſt productum numerorum <lb/>quæſitorum, quadratum adiungant; </s> + <s xml:space="preserve">tum radicem quadratam huius ſummæ co-<lb/>pulet dimidio ſecundi numeri propoſiti, ei inquam, cui differentia quadratorum <lb/>æqualis eſſe debet, ex quo quadratum maius conſurgit, à quo, detracto ſecundo <lb/>numero, ſupereſt quadratum minus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponeretur primo loco numerus .8. cui æquandum eſt <lb/>productum numerorum quæſitorum, tum proponeretur numerus .12. cui, detra-<lb/>cto minore à maiore, differentia quadratorum vtriuſque quæſiti numeri æqualis <lb/>eſſe debet, oportet huius vltimi numeri .12. dimidium in ſeipſum multiplicare, <reg norm="fient- q́ue" type="context">fiẽt-<lb/>q́ue</reg> .36. quadratum dimidij, vnde in ſummam colligeremus quadratum primi <lb/>numeri .8. quod eſſet .64. quæ cum .36. efficerent .100. cuius centenarij radice, nem <lb/>pe .10. collecta in ſummam cum dimidio ſecundi numeri, nempe .6. daretur qua-<lb/>dratum maius, nempe .16. ex quo, detracto ſecundo numero, nempe .12. rema-<lb/>neret quadratum minus .4.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, maius quadratum <lb/> + <anchor type="figure" xlink:label="fig-0036-02a" xlink:href="fig-0036-02"/> + incognitum ſignificetur 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> ſua dimidia; </s> + <s xml:space="preserve">tunc cogite-<lb/>tur quadratum <var>.y.g.</var> ſuper <var>.b.g.</var> et <reg norm="parallelogram- mum" type="context">parallelogrã-<lb/>mum</reg> rectangu<unsure/>lum <var>.g.r.</var> deſignatum, & ita etiam <lb/>gnomon <var>.u.g.t.</var> prout ſexta ſecundi Euclidis pro <lb/>ponitur, ex quo quadratum <var>.b.i.</var> nempe <var>.u.t.</var> co-<lb/>gnitum erit, ſed gnomon æqualis eſt rectangulo <var>.g.r.</var> ex prædicta, aut ex .8. poſt .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">hocq́;</reg> rectangulum <var>.g.r.</var> quadratum eſt primi numeri propoſiti 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> cognoſcetur, <lb/>quare totum quadratum <var>.g.y.</var> <reg norm="eiusque" type="simple">eiusq́;</reg> radix <var>.b.g.</var> manifęſta erit, cui coniuncta <var>.q.b.</var> <lb/>data, maius quadratum <var>.q.g.</var> cognoſcetur, ex qua <var>.b.g.</var> detracta <var>.b.i.</var> data, cogno-<lb/>ſcetur <var>.i.g.</var> quadratum minus conſequenter, 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 idipſum definiri poteſt, prætermiſſa antiquorum via, <lb/>nempe multiplicatis in ſemetipſis primo & ſecundo, numeris propoſitis, qua-<lb/><reg norm="druplicatoque" type="simple">druplicatoq́;</reg> quadrato primi, qua ſumma coniuncta cum quadrato ſecundi nume-<lb/>ri, & ex hac altera ſumma eruta radice quadrata, ex qua detracto ſecundo nume-<lb/>ro, & è reliquo ſumpto dimidio, quod erit <reg norm="quadratum" type="context">quadratũ</reg> minus, quo detracto ex radi-<lb/>ce poſtremo iuncta, ſupererit quadrarum maius.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponeretur numerus .8. cui productum duorum numerorum <lb/>quæſitorum æquandum eſt, proponeretur idem .12. cui differentia quadratorum <lb/>duorum numerorum æqualis eſſe debet. </s> + <s xml:space="preserve">Iubeo primum numerum, nempe .8. in ſe <lb/>ipſum multiplicari, ex quo exurget .64. pro numero ſui quadrati, quod quadru-<lb/>plicari volo, <reg norm="eritque" type="simple">eritq́;</reg> productum .256. quod cenſeo <reg norm="coniungendum" type="context">coniũgendum</reg> cum quadrato ſe-<lb/>cundi numeri propoſiti, nempe .144. <reg norm="eritque" type="simple">eritq́;</reg> ſumma .400. ex quaſumetur radix, ſci <lb/>licet .20. & ex hac detrahetur ſecundus numerus .12. <reg norm="reſiduique" type="simple">reſiduiq́;</reg> dimidium, nempe <num value="4">.<lb/>4.</num> pro quadrato minore, quo in ſummam collecto cum, 12. dabit quadratum <lb/>maius .16.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, quadratum maius per lineam <var>.q.g.</var> minus per <var>.g.p.</var> ſi-<lb/>gnificetur: </s> + <s xml:space="preserve">ſuper integram autem <var>.q.p.</var> erigatur quadratum integrum <var>.d.p.</var> diuiſum, <lb/>vt quadratum <var>.f.g.</var> vigeſimiſeptimi theorematis huius libri, (idipſum accideret di-<lb/>uiſo quadrato modo octauæ ſecundi Euclidis) quæ quidem diuiſio, eſt via quatuor <lb/>productorum <var>.q.g.</var> in <var>.g.p.</var> è quibus vnum ſit <var>.g.r.</var> quod erit cognitum ex .19. theore <lb/>mate cum ſit <reg norm="quadratum" type="context">quadratũ</reg> primi numeri ppoſiti, ex quo illa quatuor cognita <reg norm="erunt" type="context">erũt</reg>. </s> + <s xml:space="preserve">Iam <lb/>verò ſi cogitemus <var>.q.p.</var> ſectam in puncto <var>.t.</var> ita vt <var>.q.t.</var> æqualis ſit <var>.p.g.</var> dabitur differen <lb/>tia <var>.t.g.</var> cognita, vt radix quadrati <var>.e.o.</var> cum ex præſup-<lb/>poſito <var>.r.n.</var> æqualis ſit <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> ipſius <var>.t.g.</var> cum quadruplo <var>.g.r</var>: cognitum erit <lb/>quadratum <var>.d.p.</var> ipſius <var>.q.p.</var> </s> + <s xml:space="preserve">quare cognoſcetur <var>.q.p.</var> de <lb/>quo numero detracta differétia quadratorum cognita <var>.<lb/>t.g.</var> ſupererit aggregatum <var>.p.g.</var> et <var>.q.t.</var> cognitum. </s> + <s xml:space="preserve">Qua-<lb/>re ex conſequenti, dimidium aggregati, nempe <var>.g.p.</var> <lb/>cognoſcetur, 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 eiuſmodi numeros inuenire, vt eorum maior mi-<lb/>norem, numero propoſito ſuperet, & productum vnius in alterum, alteri nu-<lb/>mero propoſito adęquetur, conſultiſsimum ſit dimidium primi numeri propoſiti, + <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 eſt, in ſeipſum <lb/>multiplicare, atque huic quadrato, ſecundum numerum propoſitum iungere, cui, <lb/>productum numerorum quæſitorum æquale eſſe debet, & ex hac ſumma eruere qua <lb/>dratam radicem, quæ coniuncta dimidio primi numeri propoſiti, 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, ſi proponeretur .12. cui differentia vnius numeri ab altero æqua-<lb/>ri deberet, tum proponeretur .64. cui productum multiplicationis duorum quæſi-<lb/>torum ſimul <reg norm="æquandum" type="context">æquãdum</reg> eſſet. </s> + <s xml:space="preserve">Dimidium primi numeri in ſeipſum multiplicaremus, <lb/><reg norm="proueniretque" type="simple">proueniretq́;</reg> <reg norm="quadratum" type="context">quadratũ</reg> .36. cui coniuncto ſecundo, nempe .64. totum eſſet .100. <lb/>ex quo detracta quadrata radice .10. etipſi coniuncto ſenario, 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 ſpeculatio hæc eſt. </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 ſiue cognitum ſit <var>.a.s</var>: conſide-<lb/>remus nunc <var>.e.i.</var> dimidium <var>.e.o.</var> datæ differentiæ, & ex compoſito <var>.a.i.</var> imaginetur <lb/>quadratum <var>.a.x.</var> in quo protracta ſit <var>.t.u.</var> æquidiſtans lateri <var>.a.i.</var> & tam ab ipſa <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 ſcilicet 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 ſe conſiderare, vnde ſequitur 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, ſi coniunctus fuerit quadrato <var>.e.t.</var> cognito <lb/>ex radice <var>.e.i.</var> cognita (vt dimidia toralis differentię <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 conſequenter quæ quidem ſpeculatio eadem eſt <lb/>quæ .6. ſecundi ſeu .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, hocipſum 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 propoſito numero, inuenturi ſunt duos numeros inter ſe <lb/>differentes, quorum quadratorum ſumma altero numero propoſito æqualis <lb/>ſit, rectè primum numerum propoſitum in ſeipſum multiplicant, quod quadratum <lb/>exſecundo numero <reg norm="detrahunt" type="context">detrahũt</reg>, & dimidium reſidui ſumunt, quod productum erit <lb/>multiplicationis duorum numerorum interſe, in reliquis præcedentis theorematis <lb/>ordinem ſequuntur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponeretur .12. tanquam numerus, cui differentia duorum <lb/>numerorum quæſitorum æquanda eſt, proponerentur præterea .272. quibus ſum-<lb/>ma quadratorum duorum numerorum quæſitorum æquari deberet, oporteret ſanè <lb/>primum numerum, nempe .12. in ſeipſum multiplicare, cuius <reg norm="quadratum" type="context">quadratũ</reg> hoc loco <lb/>eſſet .144. atque hoc detrahere ex ſecundo numero, ſupereſſet .128. ſumpto <lb/>deinde dimidio huiuſce numeri, népe .64. producto in quam duorum numerorum <lb/><reg norm="quæſitorum" type="context">quæſitorũ</reg>. </s> + <s xml:space="preserve">Cum hoc .64. proſtea et duodenario primo propoſito numero, præceden <lb/>tis theorematis ordinem ſequeremur.</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 ſpeculemus, conſideremus ſubſcriptam figuram, vigefiminoni theore-<lb/>matis figuræ ſimilem, 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">rũ</reg> quadrata <reg norm="erunt" type="context">erũt</reg> <var>.r.c.</var> et <var>.g.s.</var> <reg norm="quorum" type="context">quorũ</reg> <reg norm="summa" type="context">sũma</reg> <reg norm="iterum" type="context">iterũ</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">duorũ</reg> <lb/>numerorum primo propofita fit <var>.q.i.</var> eius verò qua-<lb/>dratum <var>.m.e.</var> quod cognitum eſt ex ſua radice <var>.q.i</var>. <lb/></s> + <s xml:space="preserve">quare gnomon <var>.e.n.m.</var> ſimul cum quadrato minori <var>.<lb/>g.s.</var> cognitus erit, quæ ſumma æqualis eſt duplo <var>.g.r.</var> <lb/>producto datorum numerorum. </s> + <s xml:space="preserve">Itaque & ipſa <var>.g.<lb/>r.</var> cognoſcetur, nunc ſi præcedentis theorematis ſpe-<lb/>culationem in reliquis conſuluerimus propoſitum <lb/>conſequemur.</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 idipſum conſequi poſſemus, non conſulto qua-<lb/>drageſimo theoremate. </s> + <s xml:space="preserve">Nam ſubtracto quadrato differentiæ, numeri primi <lb/>(<reg norm="inquam" type="context">inquã</reg>) propoſiti, ex <reg norm="summa" type="context">sũma</reg> duorum quadratorum, nempe ex ſecundo numero pro-<lb/>poſito colligendum eſſet reſiduum in ſummam cum prædicto ſecundo numero, & <lb/>ex ſumma hac deſumenda quadrata radix, quæ duorum numerorum ſumma erit, <lb/>de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius <lb/>dimidio addito primo numero propoſito, aut detracto minore inuento ex radice <lb/>poſtremo inuenta, dabitur numerus maior, qui quæritur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, cum ſuperfuerint .128. hæc ſi cum ſecundo numero <reg norm="nempe" type="context">nẽpe</reg> .272. <lb/>iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo <lb/>propoſito, nempe .12. ſupererunt .8. quorum <reg norm="dimidium" type="context">dimidiũ</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 reſiduum detractionis <lb/>quadrati <var>.m.e.</var> ex ſumma <reg norm="duorum" type="context">duorũ</reg> quadratorum <var>.r.c.</var> et <var>.g.s.</var> numerum præbet æqua-<lb/>lem duobus ſupplementis <var>.q.n.</var> et <var>.n.u.</var> ex .8. ſecundi Euclidis. qui coniunctus duo-<lb/>bus quadratis (quorum ſumma ſecundo propoſita fuit) cognitionem profert qua-<lb/>drati <var>.q.u.</var> & eius radicis <var>.q.p.</var> de qua, detracto primo dato numero, ſcilicet <var>.q.i.</var> ſu-<lb/>pereſt <var>.i.p.</var> cuius dimidium nempe <var>.g.p.</var> minor eſt numerus qui quęritur; </s> + <s xml:space="preserve">reſiduum <lb/>verò totius <var>.g.q.</var> maior ſcilicet.</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 ſumma æqualis propo-<lb/>fito alicui numero futura ſit, & ſumma quadratorum maior eorum produ-<lb/>cto per quantitatem alterius propoſiti numeri, rectè dimidium primi dati numeri in <lb/>ſeipſum multiplicant, quod quadratum ex <reg norm="ſecundo" type="context">ſecũdo</reg> dato numero detrahunt, ſumunt<lb/>q́ue tertię partis refidui quadratam radicem, quam dimidio primi numeri coniun-<lb/>gunt, ex quo maior numerus <reg norm="duorum" type="context">duorũ</reg> <reg norm="quæſitorum" type="context">quæſitorũ</reg> datur, quo ex toto primo detracto, ſu-<lb/>pererit minor.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, propoſito numero .20. cui æquanda eſt ſumma duorum nume-<lb/>rorum quæſitorum, <reg norm="datoque" type="simple">datoq́;</reg> ſecundo numero .208. qui ſemper 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 ſpeculatione huiuſmodi operis cognoſcetur, <reg norm="cuiæquanda" type="context">cuiæquãda</reg> <lb/>eſt <reg norm="differentia" type="context">differẽtia</reg> inter <reg norm="ſummam" type="context">ſummã</reg> <reg norm="quadratorum" type="context">quadratorũ</reg> <reg norm="duorum" type="context">duorũ</reg> qui <reg norm="quæruntur" type="context">quærũtur</reg> <reg norm="numerorum" type="context">numerorũ</reg>, ſimul <reg norm="cum" type="context">cũ</reg> pro <lb/>ducto <reg norm="eorum" type="context">eorũ</reg> radicum. </s> + <s xml:space="preserve">Dimidium numeri .20. in ſeipſum multiplicandum eſſet, qua-<lb/><reg norm="dratumque" type="simple">dratumq́;</reg> detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua <lb/>drata radix eſſet .6. quæ ſi 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 ſpeculationis cauſa, datus primus numerus ſignificetur linea <var>.g.h.</var> in qua <lb/>maior numerus incognitus ſit <var>.g.h.</var> minor verò <var>.b.h.</var> quorum quadrata ſint <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> ſit <var>.g.c.</var> <reg norm="cogitenturque" type="simple">cogitenturq́;</reg> duo <lb/>diametri <var>.q.h.</var> et <var>.g.p.</var> diuiſi per medium in puncto <var>.o.</var> per quod duę 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">vnumquodq́;</reg>, æquale erit quadrato <var>.<lb/>g.f.</var> dimidij ipſius <var>.g.h.</var> datę, </s> + <s xml:space="preserve">quare eorum <reg norm="vnumquodque" type="simple">vnumquodq́;</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 diſtan-<lb/>tem à <var>.g.k.</var> quantum <var>.y.l.</var> ab <var>.g.h.</var> diſtare 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 ſubtractis è <lb/>duobus quadratis ſuperius dictis <var>.a.t.y.x.</var> et <var>.b.l.</var> <lb/>producto <var>.y.b.</var> æqualibus, ſupererunt <var>.k.d.</var> et <var>.a.c.<lb/>x.</var> cognita, tanquam æqualia dato ſecundo nu-<lb/>mero, ſed <var>.k.d.</var> quadratum eſt medietatis <var>.g.f.</var> <lb/>cognitæ, cognoſcetur igitur reſiduum <var>.a.c.x.</var> vnà <lb/>etiam ſingulæ 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> ſingularum, <lb/>qua coniuncta dimidio <var>.g.f.</var> <reg norm="rurfusque" type="simple">rurfusq́;</reg> ab <reg norm="eodem" type="context">eodẽ</reg> de-<lb/>tracta, propoſitum conſequemur.</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 ſi quis cupiat numerum propoſitum in duas eiuſmodi partes diuidere, vt <lb/>quadratum maioris, quadratum minoris ſuperet quantitate alterius numeri <lb/>propoſiti, rectè primum numerum in ſeipſum multiplicabit, & ab eodem ſecun-<lb/>dum numerum detrahet, reſiduum 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, ſi proponantur .20. diuiſa in duas eiuſmodi partes, vt <reg norm="quadratum" type="context">quadratũ</reg> <lb/>maioris ſuperet quadratum minoris numero æquali ipſi .240. oportebit primum <lb/>numerum, qui quadratus cum fuerit, erit .400. in ſeipſum multiplicare, & ex hoc <lb/>quadrato ſecundum numerum nempe .240. detrahere, </s> + <s xml:space="preserve">tunc remanebunt .160. quę <lb/>diuiſa per .40. <reg norm="numerum" type="context">numerũ</reg> <reg norm="duplum" type="context">duplũ</reg> primo, dabuntur quatuor pro minori numero, à reſi-<lb/>duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s> + </p> + <p> + <s xml:space="preserve">Quod vt exactè conſideremus, primus numerus propoſitus ſignificetur 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">Poſtmodum eriga <lb/><gap extent="2"/>r quadratum <var>.q.e.</var> diuiſum diametro <var>.f.h.</var> <reg norm="ductisque" type="simple">ductisq́;</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 reſiduum 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 ſit ſecundo loco data, cogitemus <lb/>detrahi è toto quadrato cognito <var>.q.e.</var> ex quo <lb/>ſumma duorum ſupplementorum <var>.q.o.</var> et <var>.o.e.</var> <lb/>cognoſcetur, vnà cum quadratis <var>.u.n.</var> et <var>.p.a.</var> du <lb/>plo ſcilicet <var>.q.a.</var> quo diuiſo per duplum <var>.q.h.</var> aut <lb/>ſimplex <var>.q.a.</var> per <var>.q.h.</var> ſimplicem, 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 propoſitum in duas eiuſmodi partes, vt pro <lb/>ductum vnius in alteram, alteri numero propoſito æquetur, rectè dimidium <lb/>primi dati numeri in ſeipſum multiplicant, ex quo quadrato ſecundum datum nu-<lb/>merum detrahunt, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem ſumunt, qua coniuncta vni dimidio primi nu-<lb/>meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeſtabit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi 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/>ſeipſum multiplicaremus, cuius quadratum eſſet .289. de quo detracto ſecundo 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">detractoq́;</reg> ex .17. ſupereſſet .2. pars <lb/>inquam minor.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus propoſitus ſignificetur linea <var>.a.d.</var> cu-<lb/>ius dimidium <var>.c.d.</var> cognitum erit, vnà etiam eius quadratum <var>.c.f.</var> quo diuiſo per dia <lb/>metrum <var>.e.d.</var> ſupponantur partes ignotæ <lb/> + <anchor type="figure" xlink:label="fig-0041-02a" xlink:href="fig-0041-02"/> + ipſius <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 ſimi <lb/>li figuræ quintæ ſecundi 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 propoſitis tribus numeris, quorum prior in duas eiuſmodi partes diui-<lb/>dendus ſit, ut mutuò diuiſæ, & per ſummam prouenientium diuiſo ſecundo <lb/>numero, proueniens vltimum ſit æquale tertio numerorum propoſitorum. </s> + <s xml:space="preserve">Conſul <lb/>tiſsimum ſit ſecundum numerum per tertium diuidere, ex quo proueniens ſit ſum-<lb/>ma prouenientium è duabus partibus mutuò diuiſis, quam ſummam ſi quis velit di-<lb/>ſtinguere, rectè poſſit medio operationis <reg norm="pręcedentis" type="context">pręcedẽtis</reg> theorematis <reg norm="sumpta" type="context">sũpta</reg> vnitate ſuper <lb/>ficiali pro ſecundo numero diſtinctis poſtmodum 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, ſi ſumma vnius dictorum prouenientium cum vnitate dat primum numerum, <lb/>quid ipſa eadem vnitas dabit? </s> + <s xml:space="preserve">ex quo propoſitum oriatur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. ſecundus .34. tertius .8. <lb/>Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prębeant duo <lb/>prouenientia, quorum ſumma tanta ſit vt per eam diuiſo .34. proueniat numerus <lb/>æqualis tertio numero .8. </s> + <s xml:space="preserve">Quod vt præſtemus iubet regula ſecundum .34. per <reg norm="tertium" type="context">tertiũ</reg> <num value="8">.<lb/>8.</num> diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit ſumma pro <lb/>uenientium ex diuiſione duarum partium quæſitarum, quæ ſi diſtinguere volueri-<lb/>mus, præcedentis theorematis methodum ſequemur, vnitate ſuperficiali pro ſecun <lb/>do numero propoſito ſumpta, ac ſi diceremus, diuidatur .4. cum vna quarta parte <lb/>in duas eiuſmodi partes, vt productum vnius in alteram ſit vnitas ſuperficialis, cer-<lb/>tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemſeptem <lb/>quartæ lineares, verum cum neceſſe ſit, ex præcedenti theoremate, dimidium in <lb/>ſeipſum multiplicare, <reg norm="eſſetque" type="simple">eſſetq́;</reg> dimidium .8. quartarum partium cum octaua, com-<lb/>modius totum conſtituetur .34. octauarum, quarum dimidium, nempe decemſep-<lb/>tem octauæ, in ſeipſum multiplicatum erunt .289. ſexageſimæ quartæ vnius integri <lb/>ſuperficialis, quandoquidem <reg norm="integrum" type="context">integrũ</reg> ſuperficiale, cuius vnitas linearis in .8. partes <lb/>diuiditur eſt .64. vt ex primo theoremate huius libri depræhendi poteſt. </s> + <s xml:space="preserve">Nunc vni-<lb/>tate hac ſuperficiali, nempe .64. ex .289. detracta, ſupererit .225. cuius radix qua-<lb/>drata, ſcilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit <lb/>maius proueniens .32. <reg norm="detractaque" type="simple">detractaq́;</reg> ex altero dimidio, dabit proueniens minus .2. hoc <lb/>eſt pro maiore proueniente .32. octauas, & pro minore duas, quatuor ſcilicet inte-<lb/>gros pro maiore, & quartam partem vnius integri pro minore. </s> + <s xml:space="preserve">Nunc ſi ex regula <lb/>de tribus dixerimus, ſi .4. iuncta vni, nempe .5. dant .20. primum numerum, quid <lb/>dabunt .4. integra (proueniens inquam maius) <reg norm="dabunt" type="context">dabũt</reg> certè .16. partem maiorem. <lb/></s> + <s xml:space="preserve">Tum ſi dixerimus, ſi quarta pars coniuncta vnitati dat .20: </s> + <s xml:space="preserve">quid dabit quarta illa <lb/>pars (hoc eſt proueniens minus) dabit <reg norm="profectò" type="simple">ꝓfectò</reg> quatuor ſcilicet <reg norm="minorem" type="context">minorẽ</reg> partem, quod <lb/>ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-<lb/>ſcientes ijs vti ad inueniendas duas primi numeri partes.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis gratia, demus primum numerum ſignificari linea <var>.e.u.</var> cuius <lb/>partes <var>.e.a.</var> & <var>a.u.</var> ſint quæ quæruntur, alter verò numerus ſignificetur linea <var>.b.<lb/>d.</var> tertius linea <var>.g.f.</var> proueniens <reg norm="autem" type="wordlist">aũt</reg> diuiſionis <var>.e.a.</var> per <var>.a.u.</var> ſit <var>.n.t.</var> diuiſionis <reg norm="autem" type="wordlist">aũt</reg> <var>.a.u.</var> <lb/>per <var>.a.e.</var> ſit <var>.t.o.</var> ſumma erit <var>.n.t.o.</var> vnitas verò <var>.n.i.</var> et <var>.o.i</var>. </s> + <s xml:space="preserve">Iam ſi numerus <var>.f.g.</var> tertiò <lb/>propoſitus ex diuiſione ſecundi per <var>.o.t.n.</var> proferri debet. </s> + <s xml:space="preserve">Ex .13. theoremate patet, <lb/>quòd ſi <var>.b.d.</var> per <var>.g.f.</var> diuiſerimus, proferetur <var>.o.t.n.</var> qui cum fuerit inuentus, <reg norm="ſummam" type="context">ſummã</reg> <lb/>eſſe oportet <reg norm="duorum" type="context">duorũ</reg> <reg norm="prouenientium" type="context">prouenientiũ</reg>, ex diuiſione mutua <reg norm="duorum" type="context">duorũ</reg> numerorum, nempe <var>.<lb/>a.e.</var> per <var>.a.u.</var> et <var>.a.u.</var> per <var>.a.e.</var> deinde manifeſtum eſt ex .24. aut .25. theoremate <reg norm="eorum" type="context">eorũ</reg> <lb/>productum (multiplicatis prouenientibus adinuicem) vnitatem ſuperficialem 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/>ſolam vnitatem ſuperficialem <reg norm="contineat" type="context">cõtineat</reg>, quo <lb/> + <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> + facto, ſi, vt antedictum eſt, cogitauerimus <var>.n.<lb/>t.</var> <reg norm="proueniens" type="context">proueniẽs</reg> eſſe ex diuiſione <var>.e.a.</var> per <var>.a.u.</var> et <var>.<lb/>t.o.</var> proueniens ex diuiſione <var>.a.u.</var> per <var>.a.e.</var> pa-<lb/>tebit ex definitione diuiſionis, quod eadem <lb/>erit proportio <var>.a.e.</var> ad <var>.n.t.</var> quæ eſt <var>.a.u.</var> ad vni-<lb/>tatem <var>.n.i.</var> et <var>.a.u.</var> ad <var>.o.t.</var> eadem quæ eſt <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">permutandoq́;</reg> <var>.e.a.</var> ad <var>.a.u.</var> ſicut <var>.t.n.</var> ad <var>.n.i.</var> & componendo <var>.e.a.u.</var> <lb/>ad <var>a.u.</var> ſicut <var>.t.n.i.</var> ad <var>.n.i</var>: & euerſim <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. ſepti <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 ſit. </s> + <s xml:space="preserve">Non mirum tamen ſi huiuſmodi 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 diuiſis, ſi per ſummam prouenientium, produ-<lb/>ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-<lb/>tn<gap extent="2"/>m duorum numerorum æquale futurum ſit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit <num value="4">.<lb/>4.</num> integrorum cum quarta parte, qua ſumma multiplicata cum producto <reg norm="primorum" type="context">primorũ</reg> <lb/>numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-<lb/>torum duorum numerorum æquantur.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt conſideremus, duo numeri partibus <var>.a.e.</var> et <var>.e.i.</var> in linea <var>.a.i.</var> ſignificentur, <lb/>quorum productum ſit <var>.e.d.</var> & <reg norm="quadratum" type="context">quadratũ</reg> ipſius <var>.a.e.</var> ſit <var>.e.p</var>: ipſius verò <var>.e.i.</var> ſit <var>.e.q.</var> pro-<lb/>ueniens <reg norm="autem" type="wordlist">aũt</reg> ex diuiſione <var>.e.i.</var> per <var>.a.e.</var> ſit <var>.o.u.</var> proueniens <reg norm="autem" type="wordlist">aũt</reg> <var>.a.e.</var> per <var>.e.i.</var> ſit <var>.o.t.</var> quo-<lb/>rum ſumma ſit <var>.o.u.t.</var> tum productum <var>.e.d</var>: linea <var>.u.n.</var> ſignificetur ad angulum <reg norm="rectum" type="context">rectũ</reg> <lb/>coniuncta in puncto <var>.u.</var> extremo ipſius <var>.o.u.t.</var> productum <reg norm="autem" type="wordlist">aũt</reg> <var>.u.o.t.</var> in <var>.u.n.</var> ſit <var>.n.t</var>. </s> + <s xml:space="preserve">Iam <lb/>probandum nobis eſt <var>.n.t.</var> æqualem eſſe ſummæ duorum quadratorum <var>.q.e.p</var>. </s> + <s xml:space="preserve">Quod <lb/>ſingillatim probo, & aſſero productum <var>.o.n.</var> æquale eſſe quadrato <var>.q.e.</var> & <reg norm="productum" type="context">productũ</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 præſuppoſito ab <var>.e.a.</var> multiplicetur <lb/>& diuidatur, cuius multiplicationis produ-<lb/>ctum eſt <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"/> + diuiſione eſt <var>.o.u</var>: </s> + <s xml:space="preserve">quare ex dicto theorema-<lb/>te <var>.e.i.</var> media proportionalis eſt 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 eſt qua-<lb/>drato <var>.e.q.</var> ex .16. ſexti vel .20. ſeptimi. </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 eſt <var>.d.e.</var> nempe <var>o.s.</var> & <lb/>proueniens ex diuiſione <var>.o.t</var>: </s> + <s xml:space="preserve">inter quæ ex .<lb/>35. theoremate <var>.a.e.</var> media proportionalis <lb/>eſt. </s> + <s xml:space="preserve">Quare ex allatis propoſitionibus <reg norm="productum" type="context">productũ</reg> <var>.s.t.</var> æquale eſt quadrato <var>.e.p.</var> ſed <reg norm="totum" type="context">totũ</reg> <lb/>productum <var>.n.t.</var> ſumma eſt duorum productorum <var>.o.n.</var> et <var>.s.t.</var> ex prima ſecundi Eucli. <lb/></s> + <s xml:space="preserve">Itaque verum eſſe quod dictum eſt, conſequitur.</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 ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium, <lb/>per minorem diuidat, <reg norm="maioremque" type="simple">maioremq́;</reg> per proueniens multiplicet, productum, <lb/><reg norm="summæ" type="context">sũmæ</reg> ipſius maioris cum eodem proueniente æquale erit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-<lb/>to per proueniens, ipſo nempe .10: </s> + <s xml:space="preserve">datur productum .11. cum nona parte, tantum ſci + <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 ſumma eſt maioris cum proueniente.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, maior numerus ſignificetur <var>.a.i.</var> et minor linea <var>.a.o.</var> ex <lb/>quo ex præſupoſito <var>.o.i.</var> vnitas erit. </s> + <s xml:space="preserve">Sit autem proueniens ex diuiſione <var>.a.i.</var> per <var>.a.o.<lb/>a.e</var>: </s> + <s xml:space="preserve">quod <var>.e.a.</var> directè coniungatur ipſi <var>.a.i.</var> et productum <var>.a.i.</var> in <var>.a.e.</var> ſit <var>.u.i</var>. </s> + <s xml:space="preserve">Probabo <lb/>numerum ſuperficialem <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 ſi numerus diuiſibilis per pro-<lb/>ueniens diuidatur, proueniens futurus ſit numerus diuidens, </s> + <s xml:space="preserve">quare <var>.a.o.</var> erit pro-<lb/>ueniens ex diuiſione <var>.a.i.</var> per <var>.a.e.</var> & ex deſinitione diuiſionis ita ſe habebit <var>.e.a.</var> ad <var>.<lb/>a.i.</var> ſicut <var>.o.i.</var> ad <var>.o.a.</var> & componondo ita <var>.e.i.</var> ad <var>.a.i.</var> ſicut <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> ſed <var>.a.i.</var> non modò diuiſa <reg norm="nunc" type="context">nũc</reg> cogitatur ab <var>.e.a.</var> ex <lb/>quo ſit proueniens <var>.a.o.</var> ſed 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 eſt 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> ſicut <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 propoſitum.</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">IDipſtim etiam alia ratione conſiderari poteſt.</s> + </p> + <p> + <s xml:space="preserve">Linea <var>.u.a.</var> ſecetur in puncto <var>.t.</var> ita vt <var>.a.t.</var> æqualis ſit vnitati <var>.o.i.</var> & media paral <lb/>lela <var>.t.n.</var> terminetur productum <var>.t.i.</var> quod conſtabit æquali numero, quamuis ſuperfi-<lb/>ciali, numero <var>.a.i.</var> tametſi lineari. </s> + <s xml:space="preserve">Tumparallela ducatur à puncto <var>.o.</var> ipſi <var>.a.u.</var> termi <lb/><reg norm="neturque" type="simple">neturq́;</reg> productum <var>.o.u.</var> ex quo bina producta dabuntur <var>.u.o.</var> et <var>.t.i.</var> inter ſe æqualia <lb/>ex .15. ſexti aut .20. ſeptimi cum ita ſe habeat <var>.a.i.</var> ad <var>.a.u.</var> ſicut <var>.a.o.</var> ad <var>.a.t.</var> ſed <var>.a.i.</var> ad <var>.<lb/>a.o.</var> permutando ſic ſe habet ſicut <var>.a.u.</var> ad <var>.a.t.</var> & ex prima ſexti aut .18. vel .19. ſepti-<lb/>mi ſic ſe habet <var>.u.i.</var> ad <var>.u.o.</var> ſicut <var>.a.i.</var> ad <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0044-02a" xlink:href="fig-0044-02"/> + o. hoc eſt <var>.u.i.</var> ad <var>.t.i.</var> ope .11. quinti. </s> + <s xml:space="preserve">Iam <lb/>ex definitione diuiſionis ita ſe habet <var>.a.e.</var> <lb/>ad <var>.a.i.</var> ſicut <var>.o.i.</var> ad <var>.o.a.</var> & componendo <var>.<lb/>e.i.</var> ad <var>.a.i.</var> ſicut <var>.i.a.</var> ad <var>.o.a</var>. </s> + <s xml:space="preserve">Itaque ex præ-<lb/>dicta .11. ſic ſe habebit <var>.e.i.</var> ad <var>.i.a.</var> ſicut <var>.u.<lb/>i.</var> ad <var>.t.i.</var> ſed <var>.t.i.</var> numero conſtat 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 propoſitum in duas eiuſmodi partes, vt <reg norm="productum" type="context">productũ</reg> <lb/>vnius in alteram cum i pſarum differentia in ſummam collectum, æquale ſit <lb/>alicui alteri numero maiori primo. </s> + <s xml:space="preserve">Rectè primum ex ſecundo detrahunt, reſiduum <lb/>verò conſeruant, tum ex primo ſemper binarium deſumunt, <reg norm="dimidiumque" type="simple">dimidiumq́;</reg> conſer-<lb/>uant, alterum verò dimidium in ſeipſo multiplicant, & ex quadrato numerum con <lb/>ſeruatum eruunt, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem ex dimidio conſeruato, quod vltimum reſi-<lb/>duum propoſiti numeri quæſita pars minor eſt.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .20. ita <reg norm="diuidendus" type="context">diuidẽdus</reg>, vt <reg norm="productum" type="context">productũ</reg> vnius partis <lb/>in alteram, cum partium differentia collectum in ſummam, æquale ſit propoſito + <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 ſecundo, <lb/>nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi <lb/>narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi <lb/>dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81. <lb/>primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9. <lb/>tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua <lb/>dratum, ſic ſupererit .6. hoc eſt .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 eſt ſpeculatio. </s> + <s xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis <lb/>ſignificetur linea <var>.q.g.</var> maior vero linea <var>.x.</var> tum cogitemus <var>.q.g.</var> diuiſam, cuius maior <lb/>pars ſit <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> ſit autem produ-<lb/>ctum <var>.b.o</var>. </s> + <s xml:space="preserve">Oportet igitur, ut <var>.b.o.</var> ſimul cum differentia <var>.q.p.</var> æquale ſit numero <var>.x.</var> ſe-<lb/>cundò propoſito, qui notus eſt, </s> + <s xml:space="preserve">quare etiam ſumma producti <var>.b.o.</var> cum differentia <lb/><var>q.p.</var> cognita erit, ex qua detracto primo numero <var>.q.g.</var> reſiduum cognitum erit, nunc <lb/>igitur quodnam erit hoc reſiduum? </s> + <s xml:space="preserve">attendamus qua ratione ex ſumma <var>.b.o.</var> et <var>.q.p.</var> <lb/>detrahenda ſit <var>.q.g</var>. </s> + <s xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma <var>.q.p.</var> quę pars eſt <var>.q.g.</var> <lb/>ſupererit detrahenda <var>.p.g.</var> ex <var>.b.o.</var> pars inquam ipſius <var>.q.g.</var> quod fiet quotieſcunque <lb/>cogitauerimus <var>.q.o.</var> duabus vnitatibus diminutam, et per <var>.o.g.</var> multiplicatam, ſit au-<lb/>tem productum <var>.b.e.</var> nam cum <var>.o.g.</var> toties <var>.b.o.</var> ingrediatur, quot ſunt in <var>.q.o.</var> vnitates <lb/>ex prima ſexti aut .18. vel .19. ſeptimi, <reg norm="detrahendaque" type="simple">detrahendaq́;</reg> ſit <var>.p.g.</var> ex <var>.b.o.</var> quæ <var>.p.g.</var> dupla <lb/>eſt <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> ijſdem <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> cognoſcamus <lb/>ſimul cum <var>.q.i</var>: Sivoluerimus partes <var>.q.e.</var> <lb/>et <var>.e.i.</var> cognoſcere, vtemur .45. theorema-<lb/>te huius libri, & propoſitum obtinebimus, nam cognoſcemus <var>.e.i.</var> & ex conſequen-<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 eiuſmodi partes, quæ pro medio proportionali <lb/>alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni <lb/>hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra <lb/>to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod <lb/>tamen .45. theoremate fuit à nobis ſpeculatum.</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 poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-<lb/>dum per tertium multiplicetur, atque ſecundum hoc productum <reg norm="corporeum" type="context">corporeũ</reg>, <lb/>per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-<lb/>cundi in tertium.</s> + </p> + <p> + <s xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12. <reg norm="multiplicenturque" type="simple">multiplicenturq́;</reg> .10. <reg norm="cum" type="context">cũ</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. diuiſum dabit .132. numerum æqualem producto <lb/>ſecundi in tertium numerorum propoſitorum, ſcilicet .132.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt ſpeculemur, primus numerus ſignificetur line <var>a.o.u.</var> ſecundus <var>.e.o.</var> tertius <var>.<lb/>e.a.</var> productum verò <var>.o.u.</var> in <var>.o.e.</var> ſit <var>.o.i.</var> ipſius ve <lb/>rò <var>.o.i.</var> per <var>.e.a.</var> <reg norm="productum" type="context">productũ</reg> <reg norm="corporeum" type="context">corporeũ</reg> ſit <var>.i.c.</var> tum <lb/> + <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> + <reg norm="productum" type="context">productũ</reg> <var>.e.o.</var> in <var>.e.a.</var> ſit <var>.e.c</var>. </s> + <s xml:space="preserve">Dico <reg norm="nunc" type="context">nũc</reg> quod di-<lb/>uiſo numero corporeo <var>.i.c.</var> per <reg norm="primum" type="context">primũ</reg> <var>.o.u.</var> <reg norm="proue" type="simple">ꝓue</reg> <lb/>niens æquale erit numero producti <var>.e.c</var>. </s> + <s xml:space="preserve">Qua-<lb/>re in primis cogitandum eſt, 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/>ſequitur quòd diuiſo <var>.i.c.</var> per <var>o.u.</var> proueniens ſit <lb/><var>e.c.</var> corporeum, æquale nihilominus producto <var>.e.c.</var> ſuperficiali.</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 propoſitum numerum in tres partes ſic ſe habentes vt produ-<lb/>ctum primi in ſecundam, in tertia <reg norm="multiplicatum" type="context">multiplicatũ</reg>, præbeat numerum alteri nu-<lb/>mero propoſito æqualem. </s> + <s xml:space="preserve">Rectè ſecundum 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 ſumma cogni <lb/>ta erit, detracto numero diuidente ex primo dato, quam quidem ſi diſtinguere <lb/>quis voluerit, vtetur theoremate .45.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponitur numerus .20. in tres partes diuidendus, quæ ſic ſe <lb/>habeant, ut productum primæ in ſecundam in tertia multiplicatum det .90. itaque <lb/>ſumenda erit pro prima vna pars ipſius .20. quæcunque illa ſit, verbi gratia .2. qua <lb/>ſecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum <lb/>cæterarum partium inter ſe, quarum ſumma eſt .18. quam ſummam ſi diſtinguere <lb/>volueris in cęteris duabus partibus ſeparatis, vteris .45. theoremate, vt quàm citiſ-<lb/>ſimè quod cupis exequaris, erunt autem partes .3. et .15.</s> + </p> + <p> + <s xml:space="preserve">In cuius ſpeculationis gratiam nihil aliud occurrit, quàm quod præcedenti theo-<lb/>remate, & ſuperiore .45. allatum eſt.</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. eiuſmodi partes, vt quadratum vnius ſit æquale <lb/>producto reliquarum duarum inter ſe, idem omnino eſt cum 51. theoremate. <lb/></s> + <s xml:space="preserve">Nam qui ſumet quamlibet partem propoſiti numeri, quæ tertia parte maior tamen <lb/>non ſit, <reg norm="reſiduumque" type="simple">reſiduumq́</reg> in duas tales partes diuiſerit, vt prima ſumpta, media proportio <lb/>nalis ſit ex probatione .51. theoremate allata, propoſitum conſequetur.</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 ipſum alia ratione ab ea diuerſa <reg norm="quam" type="context">quã</reg> .51. theoremate adduximus, <reg norm="profici" type="simple">ꝓfici</reg> poteſt.</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, cuiuſcunque denique pro <lb/>portionalitatis, qui in ſummam colligantur, ac poſtmodum, regula de trib. dica-<lb/>mus. </s> + <s xml:space="preserve">Si ſumma hæc primo numero propoſito in tres partes diuidendo reſpondet, <lb/>cuireſpondebit vna ex tribus partibus huiuſcę <reg norm="summæ" type="context">sũmæ</reg>? </s> + <s xml:space="preserve">idem dereliquis duabus pa<unsure/>rti <lb/>bus dico.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .57. diuidendus in tres continuas partes <lb/>proportionales proportione ſeſquialtera, tres numeros in eiuſmodi proportio-<lb/>nalitate diſtinctos ſumemus, vt potè .4. 6. 9. qui in ſummam collecti dabunt <reg norm="ſum- mam" type="context">ſum-<lb/>mã</reg> .19. <reg norm="dicemusque" type="simple">dicemusq́;</reg> ſi .19. dant .4. quid <reg norm="dabunt" type="context">dabũt</reg> .57? </s> + <s xml:space="preserve">vnde proueniens vnius partis erit <num value="12">.<lb/>12</num>. </s> + <s xml:space="preserve">Tum ſi dicamus, ſi .19. dat .6. quid dabit .57? </s> + <s xml:space="preserve">nempe dabit .18. </s> + <s xml:space="preserve">Poſtremò, ſi <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 ſe.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſciamus, numerus propoſitus in tres quaſlibet partes diuidendus ſi-<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 ſe coniunctis denotentur. </s> + <s xml:space="preserve">Cogitemus pariter lineam <var>.d.a.</var> in <lb/>tres partes diuiſam <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 ſuarum partium, quæ eſt <var>.e.h.</var> ad reſponden <lb/>tem ipſius in <var>.a.d</var>: Verbi gratia reſpondentem <var>.a.b.</var> ipſi <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 ſe habebit <var>.a.d.</var> ad <var>.c.d.</var> ſicut <var>.e.h.</var> ad <var>.g.h</var>. </s> + <s xml:space="preserve">Nam cum ſic ſe habeat <var>.a.<lb/>b.</var> ad <var>.b.c.</var> ſicut <var>.e.f.</var> ad <var>.f.g.</var> ex præſuppoſito, permutando ſic ſe habebit <var>.a.b.</var> ad <var>.e.f.</var> ſi-<lb/>cut <var>.b.c.</var> ad <var>.f.g.</var> & eadem ratione ſic ſe habe-<lb/>bit <var>.c.d.</var> ad <var>.g.h.</var> ſicut <var>.b.c.</var> ad <var>.f.g.</var> & <reg norm="conſequen- ter" type="context">cõſequen-<lb/> + <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/> + ter</reg> ſicut <var>.a.b.</var> ad <var>.e.f.</var> ex quo ex .13. quinti ſic <lb/>ſe habebit tota <var>.a.d.</var> ad totam <var>.e.h.</var> ſicut <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 propoſitum manifeſtum erit, ipſum autem productum <var>.a.b.</var> in <var>.c.b.</var> <lb/>æquale erit quadrato <var>.b.c.</var> ex .15. fexti aut .20. ſeptimi.</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 propoſuerunt, quod ex <lb/>more ratione à me definietur, eſt autem eiuſmodi.</s> + </p> + <p> + <s xml:space="preserve">Quomodo propoſitus numerus in tres eiuſmodi partes diuidatur, vt <reg norm="quadratum" type="context">quadratũ</reg> <lb/>vnius æquale fit fummæ quadratorum reliquarum duarum partium.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt efficiamus tria quadrata ſeparata ſumamus, <reg norm="quorum" type="context">quorũ</reg> <reg norm="vnum" type="context">vnũ</reg> æquale ſit reliquis <lb/>duobus; </s> + <s xml:space="preserve"><reg norm="eorum" type="context">eorũ</reg> <reg norm="autem" type="context">autẽ</reg> radices in ſummam ſimul colligantur, tum regulam de tribus ſe <lb/>quemur, ratione præcedenti theoremate demonſtrata, & rectè vt infra docebimus, <lb/>quod autem dico de quadratis, etiam de cubis, & quibuſuis dignitatibus aſſero.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi numerus diuiſibilis proponatur .30. in tres eiuſmodi partes di <lb/>uidendus, vt quadratum vnius æquale ſit ſummæ quadratorum reliquarum duarum <lb/>partium, in primis radices trium quadratorum ſumemus, ſic quomodocunque ſe <lb/>habentes, vt maius ipſorum æquale ſit ſummæ reliquorum duorum, verbi gratia .25. <lb/>16. et .9. nempe .5. 4. et .3. quæ ſi colligantur in ſummam efficiunt .12. </s> + <s xml:space="preserve">Tum ex regu-<lb/>la de tribus dicemus, ſi .12. reſpondet .30: </s> + <s xml:space="preserve">cui, 5. radix maior reſpondebit? </s> + <s xml:space="preserve">nem-<lb/>pe .12. cum dimidio.</s> + </p> + <p> + <s xml:space="preserve">Deinde ſi dixerimus ſi .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 ſumma 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 ſummæ quadratorum duarum par <lb/>tium, nempe .100. cum .55.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt <reg norm="demonſtremus" type="context">demõſtremus</reg>, numerus diuiſibilis propoſitus ſignificetur linea <var>.a.d.</var> & ſum <lb/>ma radicum, noſtro modo ſumptarum, linea <var>.e.h.</var> quarum prima & maior ſit <var>.e.f.</var> ſe-<lb/>cunda <var>.f.g.</var> tertia <var>.g.h.</var> cogitemus etiam lineam <var>.a.d.</var> ea ratione diuiſam eſſe qua <var>.e.h.</var> <lb/>patebit cnim ex modo præcedentis theorematis vnamquanque partium <var>.a.d.</var> ita ſe <lb/>habituram ad ſuum totum ſicut ſe habent ſingulæ <var>.e.h.</var> ad ſuum. </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 ſic ſe habebit <var>.a.b.</var> ad <var>.b.c.</var> ſicut <var>.e.f.</var> ad <var>.f.g.</var> idem dico de reliquis. <lb/></s> + <s xml:space="preserve">Igitur ex .18. ſexti aut .11. octaui, eadem erit proportio quadrati <var>.a.b.</var> ad <reg norm="quadratum" type="context">quadratũ</reg> <var>.<lb/>b.c.</var> quæ quadrati <var>.e.f.</var> ad quadratum <var>.f.g.</var> tota enim ſunt æqualia, cum eorum partes <lb/>ſimiles inter ſe ſunt æquales. </s> + <s xml:space="preserve">Idem dico de proportione qu@drati <var>.a.b.</var> nempe ita <lb/>ſe habere ad <var>.c.d.</var> ſicut 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 ſummam quadratorum duarum partium <var>.b.c.</var> et <var>.c.d.</var> ſic ſe ha <lb/>bebit ut quadrati <var>.e.f.</var> ad ſummam 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"/> + eſt ſummæ quadratorum <var>.f.g.</var> et <var>.g.h.</var> igitur <lb/>ſic etiam ſe habebit quadratum <var>.a.b.</var> nempe <lb/>æquale quadratis <var>.b.c.</var> et <var>.c.g</var>. </s> + <s xml:space="preserve">Idipſum de cæ <lb/>teris dignitatibus dices, <reg norm="vterisque" type="simple">vterisq́;</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 eiuſ-<lb/>modi eſt.</s> + </p> + <p> + <s xml:space="preserve">An numerus aliquis in tres eiuſmodi partes di@idi poſſit, vt quadratum vnius æ-<lb/>quale ſit ſummæ quadratorum cæterarum duarum partium ſimul cum producto <lb/>vnius in alteram.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponatur numerus .50. vt iam dictum eſt diuidendus, repe <lb/>riendus erit alius quilibet numerus, qui tamen ſumma ſit trium radicum ſic ſe ha-<lb/>bentium, vt quadratum vnius æquale ſit ſummæ quadratorum duarum partium ſi-<lb/>mul cum producto vnius in alteram, eum autem qui primò occurrit ſumamus, utpo <lb/>tè .30. qui ſumma eſt numerorum .6. 10. 14. partium ſic ſe habentium, vt quadratum <lb/>ipſius .14. æquale ſit ſummæ quadratorum cæterarum partium ſimul cum produ-<lb/>cto vnius in alteram, agamusq́ue regula de tribus, ac dicamus, ſi .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. abſque @ractis, <lb/>ex quo quadratum primæ erit .544. cum .4. nonis, ſecundæ .277. cum ſeptem nonis, <lb/>tertiæ .100. & productum ſecundæ in tertiam .166. cum .6. nonis, quod productum, <lb/>cum quadratis ſecundæ & tertiæ collectum erit .544. cum .4. nonis.</s> + </p> + <p> + <s xml:space="preserve">Huius rei ſpeculatio eadem eſt, quę fuit præcedentis theorematis vſquequo no-<lb/>ueris eandem proportionem eſſe quadrati <var>.a.b.</var> ad ſummam quadratorum <var>.b.c.</var> et <var>.c.<lb/>d.</var> quæ quadrati <var>.e.f.</var> ad ſummam 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 ſummæ 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 eſt, è contrario, ſubſequentes figuræ cogitandæ erunt, qua-<lb/>rum <var>.i.</var> ſit quadratum <var>.a.b</var>: l. ſit 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. ſit productum <var>.b.c.</var> in <var>.c.d</var>: m. ſit 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 ſpeculatione præcedentis theorematis, eadem erit proportio <var>.n.<lb/>t.</var> ad <var>.o.u.</var> quæ eſt <var>.n.s.</var> ad <var>.o.r.</var> </s> + <s xml:space="preserve">quare pro-<lb/>ductum <var>.k.</var> ex definitione ſimile erit <lb/> + <anchor type="figure" xlink:label="fig-0049-01a" xlink:href="fig-0049-01"/> + producto <var>.m.</var> cum vtraque ſint 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. ſexti 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 ſic ſe ha-<lb/>bebit <var>.k.</var> ad <var>.i.</var> ſicut <var>.m.</var> ad <var>.l.</var> ſed <var>.x.p.</var> ad <var>.i.</var> <lb/>ſicſe habere probatum eſt vt <var>.y.q.</var> ad <var>.l</var>. <lb/></s> + <s xml:space="preserve">Quare ex eadem .24. quinti ſic ſe habe <lb/>bit <var>.x.p.k.</var> ad <var>.i.</var> ſicut <var>.y.q.m.</var> ad <var>.l.</var> ſed <var>.y.q.<lb/>m.</var> æqualis eſt <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 propoſuerunt, <lb/>nempe an aliquis numerus in .4. eiuſmodi partes diuidi poſſit, vt ſumma qua-<lb/>dratorum duarum partium dupla ſit ſummæ quadratorum reliquarum duarum.</s> + </p> + <p> + <s xml:space="preserve">Verum huius effectio & ſpeculatio non erit difficilis, <reg norm="cum" type="context">cũ</reg> ſit eadem quæ præmiſsis <lb/>proximè duobus theorematibus allata fuit, ſumpta nempe ſumma radicum quarun <lb/>cunque ſic ſe habentium, prout dictum fuit. </s> + <s xml:space="preserve">Verbigratia .44. cuius partes erunt. <lb/>16. 12. 14. 2. <reg norm="tunc" type="context">tũc</reg> progrediemur regula de tribus dicentes. </s> + <s xml:space="preserve">Si .44 numerum propoſi-<lb/>tum valet, quid .16. pars maior? </s> + <s xml:space="preserve">nempe valebit partem maiorem numeri propoſi-<lb/>ti reſpondentem .16. idem de cæteris dico.</s> + </p> + <p> + <s xml:space="preserve">Porrò ſpeculatio eadem eſt cum ſuperioribus.</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 propoſitum numerum in duas eiuſmodi partes, vt productum <lb/>radicum quadratarum ipſarum partium æquale ſit alteri numero propoſito, <lb/>cuius <reg norm="tamen" type="context">tamẽ</reg> quadratum maius <reg norm="non" type="context">nõ</reg> ſit quadrato dimidij primi numeri propoſiti. </s> + <s xml:space="preserve">Rectè <lb/>ſecundum numerum propoſitum in ſeipſum multiplicat, & <reg norm="eundem" type="context">eundẽ</reg> ex quadrato di-<lb/>midij primi detrahit, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> quadratam radicem ſubtrahit ex dimidio ipſius pri-<lb/>mi, ex quo datur minor pars quæſita, quaipſi dimidio coniuncta, maior pars ha-<lb/>betur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponatur numerus, 20. propoſito modo, in duas partes <lb/>eiuſmodi diuidendus, vt productum radicum æquale ſit (verbigratia) 8. </s> + <s xml:space="preserve">Dimi-<lb/>dium priminumeri in ſeipſum multiplicabimus, cuius quadratum erit .100. ex <lb/>quo quadratum ſecundi numeri, nempe .64. detrahemus, <reg norm="remanebitque" type="simple">remanebitq́;</reg> .36. cuius radi <lb/>ce quadrata coniuncta .10. dimidio inquam primi numeri propoſiti, dabitur nume <lb/>rus .16. pars maior, & ſubtracta 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 demonſtremus, primus nu-<lb/> + <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> + merus linea <var>.a.b.</var> ſignificetur, quam di-<lb/>uiſam 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 ſignificetur per <var>.d.</var> et <var>.e.</var> <lb/>productum autem radicum cognitum <var>.<lb/>f.</var> quandoquidem datum eſt, 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 ſit <var>.x.</var> <reg norm="itaque" type="simple">itaq;</reg> cognitum, quo facto, doctrinam .45. theorematis libri huius ſecuti, <lb/>propoſitum conſequemur.</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 ſummam ipſarum, ſemper <lb/>differentia ſit quadratorum ipſarum radicum.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gratia, quoslibet duos numeros pro radicibus ſumpſerimus, vt potè .3. et <num value="5">.<lb/>5.</num> quorum differentia eſt .2. certè ſi differentiam hanc per ſummam radicum ſcili-<lb/>cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eſt <lb/>ſuorum quadratorum, nempeinter .9. et .25.</s> + </p> + <p> + <s xml:space="preserve">Hoc vt ſpeculemur, duæ radices in linea <var>.n.i.</var> ſignificentur, quarum vna ſit <var>.n.c.</var> & <lb/>altera <var>.c.i.</var> ipſarum 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">ductaq́</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/>ipſi <var>.n.i.</var> & à puncto <var>.a.</var> quarta <var>.x.a.e.</var> parallela ipſi <var>.<lb/>o.</var> inueniemus <var>.b.n.</var> productum eſſe differentiæ <var>.n.<lb/>t.</var> in ſumma radicum <var>.n.i.</var> & cum <var>.d.o.</var> et <var>.a.o.</var> ſint <lb/>quadrata radicum prædictarum: </s> + <s xml:space="preserve">b.e. æquale erit <var>.<lb/>n.u.</var> cum vtrunque horum productorum æquale ſit <var>.<lb/>x.u.</var> ex quo gnomon <var>.e.d.u.</var> æqualis erit producto <var>.<lb/>b.n.</var> quod ſcire 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 propoſitum aliquem numerum diuiſuri in duas eiuſmodi partes, vt diffe-<lb/>rentia radicum quadratarum æqualis ſit alteri numero propoſito, cuius ta-<lb/>men quadratum dimidij primi quadratum non excedat. </s> + <s xml:space="preserve">Rectè ſecundum numerum <lb/>in ſeipſum multiplicant, productum verò ex primo numero detrahunt, <reg norm="rurſusque" type="simple">rurſusq́;</reg> di <lb/>midium reſidui quadrant, & quadratum hoc ex quadrato dimidij primi ſubtrahunt, <lb/>atque ita radice quadrata reſidui, dimidio primi coniuncta, pars maior datur, qua <lb/>ex ipſo dimidio detracta, pars minor relinquitur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, propoſito numero .20. ita ut propoſitum eſt, diuidendo, nem-<lb/>pe vt differentia radicum quadratarum dictarum partium æqualis ſit binario, bina-<lb/>rium hocin ſeipſum 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="ſupereritque" type="simple">ſupereritq́;</reg> numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-<lb/>cabimus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> numerus .64. qui cum ex quadrato dimidij primi detractus fue-<lb/>rit, nempe ex .100. & reſiduo .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 ſpeculationis cauſa, primus numerus <lb/>propoſitus ſigniſicetur linea <var>.x.y.</var> pro voto diui-<lb/> + <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> + ſa in puncto <var>.c.</var> et <var>.x.t.</var> productum ſit ipſius <var>.x.<lb/>c.</var> in <var>.c.y.</var> pariter etiam <var>.q.p.</var> ſit ſumma radicum <lb/>quadratarum, nempe <var>.q.g.</var> ipſius <var>.t.c.</var> et <var>.g.p.</var> ip-<lb/>ſius <var>.c.y</var>. </s> + <s xml:space="preserve">Tum ſuper <var>.q.p.</var> extruatur & diuidatur <lb/>quadratum <var>.q.u.</var> ea ratione qua .41. theoremate <lb/>aut .29. diuiſimus, in quo ſanè quadrato, quadra <lb/>tum ipſius <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 cognoſcimus productum <var>.g.r.</var> (vſi <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">cognitũ</reg>, ac proinde cum no <lb/>uerimus <var>.x.y.</var> ſi rationem ſequemur .45. theore <lb/>mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed <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 propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia <lb/><reg norm="ſuarum" type="context">ſuarũ</reg> <reg norm="radicum" type="context">radicũ</reg> <reg norm="quadratarum" type="context">quadratarũ</reg> æqualis ſit alteri numero propoſito. </s> + <s xml:space="preserve">Cuius <reg norm="tamen" type="wordlist">tamẽ</reg> <reg norm="qua- dratum" type="context">qua-<lb/>dratũ</reg> maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </s> + <s xml:space="preserve">Rectè <reg norm="etiam" type="context">etiã</reg> <lb/><reg norm="quadratum" type="context">quadratũ</reg> dimidij ſecundi numeri ex dimidio primi <reg norm="detrahunt" type="context">detrahũt</reg>, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> radicem per <lb/>ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum <lb/>pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi 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 ſecundi numeri eſſet .9. eo detra <lb/>cto ex dimidio primi, remaneret .16. cu <lb/>ius radix .4. ſcilicet per totum ſecundum <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="reſidum" type="context">reſidũ</reg> .50. hoc eſt .49. <lb/>radices autem erunt .1. et .7. differentes <lb/>inter ſe, numero ſenario.</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 ſciamus, duo numeri lineis <reg norm="ſi- gnificentur" type="context">ſi-<lb/>gnificẽtur</reg>, primus linea .b: <reg norm="ſecundus" type="context">ſecũdus</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/>ſi <var>.c.</var> æqualis & co gnita ſit <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> ſit <var>.a.i.</var> et <var>.t.i.</var> æqualis <var>.a.i.</var> et <var>.l.i.</var> pariter <lb/>ſecetur 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="ſupereritque" type="simple">ſupereritq́;</reg> quadratum <var>.q.u.</var> differentiæ <var>.a.h.</var> cognitum, hoc verò cogi-<lb/>temus diuiſum 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 cognoſcetur, & <reg norm="quadratum" type="context">quadratũ</reg> erit ipſius <var>.a.K.</var> aut ipſius <var>.K.h.</var> dimidij <var>.a.h</var>. </s> + <s xml:space="preserve">Quòd <lb/>ſi aliquod iſtorum quadratorum detrahere voluerimus, nempe <var>.n.r.</var> ex dimidio ſum <lb/>mæ <var>.b.</var> duorum quadratorum <var>.q.i.</var> et <var>.i.d.</var> cognitæ, hac via procedemus, primum con <lb/>ſiderabimus <var>.t.r.</var> coniunctam <var>.t.i.</var> quæ quantitates erunt ſumma dimidij <reg norm="duorum" type="context">duorũ</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">dimidiũ</reg> eſt quadrati <var>.t.l.</var> et <var>.t.i.</var> <reg norm="dimidium" type="context">dimidiũ</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 ipſius <var>.K.h.</var> nem <lb/>pe <var>.n.r</var>: </s> + <s xml:space="preserve">quare quod ſupereſt cognitum <lb/>erit nempe <var>.y.s.</var> cum <var>.n.i.</var> ſed <var>.y.m.</var> æqualis <lb/>eſt <var>.n.i.</var> et <var>.y.m.</var> cum <var>.y.s.</var> conſtituunt 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/>conſequenter <var>.p.s.</var> eius radix cognoſce-<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="eſtque" type="simple">eſtq́;</reg> produ-<lb/>ctum huiuſmodi ſemper 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> cognoſcetur, conſequen-<lb/>ter <var>.i.</var> @q. tanquam reſiduum ex <var>.b.</var> & eo-<lb/>rum radices quadratæ cognoſcentur <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 poteſt. </s> + <s xml:space="preserve">Secundus numerus in <reg norm="ſuum" type="context">ſuũ</reg> dimi <lb/><reg norm="dium" type="context">diũ</reg> multiplicetur, <reg norm="productum" type="context">productũ</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="ſicque" type="simple">ſicq́;</reg> huius ſummæ quadrata radix erit ſumma radicum quadratarum dictarum <lb/>partium, cui iuncto producto ex quadrageſimoquinto theoremate ſingulæ radices <lb/>proferentur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, primus numerus diuiſibilis erat .50. alter verò .6. </s> + <s xml:space="preserve">Iam ſi multi-<lb/>plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-<lb/>mi, nempe .25. detracto, ſupererit .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 ſcilicet .8. erit ſumma radicum duarum partium quæſitarum, <lb/>qua & producto .7. ex quadrag eſimoquinto theoremate dictæ radices diſtinguen, <lb/>tur, quarum vna erit .7. & altera <var>.I</var>.</s> + </p> + <p> + <s xml:space="preserve">Vtautem hocſpeculemur, præcedenti figura vti poterimus, in qua patet <var>.t.r.</var> pro <lb/>ductum eſſe ſecundi numeri <var>.c.</var> nempe <var>.a.h.</var> hoc eſt <var>.t.u.</var> in dimidio <var>.a.e.</var> ſcilicet <var>.p.t.</var> re-<lb/>ſiduum autem dimidij primi <var>.b.</var> eſſe <var>.t.i.</var> nempe <var>.a.i.</var> productum radicum, quod ſupple + <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 eſt quadrati <var>.q.d.</var> totalis. </s> + <s xml:space="preserve">Quare duplicato <var>.a.i.</var> & coniuncto <var>.b.</var> cognoſci-<lb/>mustotum <var>.q.d.</var> & conſequenter <var>.a.d.</var> ſuam radicem, hoc eſt ſummam duarum radi <lb/>cum <var>.a.g.</var> et <var>.g.d.</var> quæ medio <var>.a.i.</var> cognito, & quadrageſimoquinto theoremate ſingu-<lb/>læ cognoſcuntur.</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 propoſitum aliquem num erum in duas eiuſmodi partes diuiſuri, vt ſum-<lb/>ma radicum dictarum partium æqualis ſit alteri numero propoſito. </s> + <s xml:space="preserve">Rectè ſe-<lb/>cundum numerum in ſeipſum multiplicant, ex quo quadrato, primum datum nu-<lb/>merum detrahunt, <reg norm="rurſusque" type="simple">rurſusq́;</reg> reſiduum in ſeipſum multiplicant, & ex eo quadrato <lb/>quartam partem deſumunt, <reg norm="quam" type="context">quã</reg> ex quadrato dimidij primi numeri detrahunt, radi-<lb/>cemq́ue qua dratam reſidui cum iunxerint, & ex dimidio primi numeri detraxerint, <lb/>partes quæſitæ proferuntur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponeretur primus numerus .20. diuidendus et .6. ſecundus <lb/>pro ſumma radicum, hunc ſecundum .6. in ſeipſum multiplicabimus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> nu-<lb/>merus .36. ex quo quadrato primus numerus detrahetur, <reg norm="ſupereritque" type="simple">ſupereritq́;</reg> numerus .16. <lb/>qui quadratus dabit .256. cuius numeri quarta pars ſumetur, nempe .64. quæ ex qua <lb/>drato dimidij primi numeri detrahetur, nempe .100. <reg norm="ſupereritque" type="simple">ſupereritq́;</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 ſpeculatio, primus numerus diuiſibilis ſignificetur linea <var>.a.b.</var> diui-<lb/>ſa in puncto <var>.e.</var> in partes adhuc incognitas, et <var>.a.c.</var> ſit productum <var>.a.e.</var> in <var>.e.b.</var> item <var>.q.<lb/>p.</var> ſecundum numerum ſignificet, æqualem ſummæ radicum, quæ puncto <var>.n.</var> diſtin-<lb/>guantur. </s> + <s xml:space="preserve">Poſtmodum totum quadratum <var>.p.d.</var> erigatur (quod nobis eſt cognitum), <lb/>in duo quadrata diuiſum <var>.o.p.</var> et <var>.o.d.</var> quorum ſumma <var>.a.b.</var> cum detur, cognita rema-<lb/>net ſumma <reg norm="duorum" type="context">duorũ</reg> <reg norm="ſupplementorum" type="context">ſupplementorũ</reg> <var>.o.u.</var> et <var>.o.q.</var> qua quadrata <reg norm="cum" type="context">cũ</reg> fuerit dabit quadru <lb/><reg norm="plum" type="context">plũ</reg> quadrati <reg norm="ſupplementi" type="context">ſupplemẽti</reg> <var>.o.q.</var> <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quadruplum" type="context">quadruplũ</reg> producti <var>.a.c.</var> etenim <var>.a.c.</var> ex .19. theo <lb/>remate huius libri quadratum eft ipſius <var>.q.o.</var> <reg norm="ſicque" type="simple">ſicq́;</reg> poterant etiam veteres quadrare <lb/>dimidium differentiæ <var>.a.b.</var> ab <var>.p.d.</var> nempe quadrato tantummodo ſupplemento <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 ſingu <lb/>læ partes cognoſcentur.</s> + </p> + <p> + <s xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito ſupplemento <var>.<lb/>q.o.</var> diſtinguendæ radices <var>q.n.</var> et <var>.n.p.</var> ex .45. theoremate, quibus cognitis, eorum <lb/>etiam quadrata cognoſcuntur.</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 propoſito numero in tres qualeſcunque partes diuiſo, ſi prima in <lb/>tertiam multiplicetur, & huic producto, ſecundæ in primam productum <lb/>coniungatur, <reg norm="itemque" type="simple">itemq́;</reg> ſecundæ in tertiam, hæc ſumma duplicata æqualis ſit ſummæ <lb/>productorum ſingularum in cæteras duas.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponatur .20. diuiſus in tres partes nempe .12. 5. 3. multipli-<lb/>cato primo .12. per .3. tertiam partem dabitur .36. ſecunda verò multiplicata per re <lb/>liquas duas, hoc eſt .5. per .12. et .3. in primis dabitur .60. poſtea .15. <reg norm="quorum" type="context">quorũ</reg> <reg norm="trium" type="context">triũ</reg> pro <lb/>ductorum ſumma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-<lb/>citur ſummæ productorum ſingularum partium in reliquas duas, nempe ſummæ .60. <lb/>36. 60. 15. 36. 15. hoc eſt ipſis .222.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei per ſe patet ſpeculatio, cum in his ſex vltimis productis, ſingula 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 propoſito numero in .3. qualeſcunque partes diuiſo, ſi in reliquas duas ſin-<lb/>gulæ multiplicentur, & hæc producta cum ſumma ſuorum quadratorum con-<lb/>iungantur, tota ſumma hæc vltima æqualis erit quadrato totali propoſiti numeri.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi fuerit idem numerus .20. in .3. partes diuiſus .12. 5. 3. </s> + <s xml:space="preserve">Si .12. in <lb/>5. et .3. producatur, ſumma productorum erit .96. at .5. in .12. et .3. erit .75. poſtmo-<lb/>dum .3. in .12. et .5. erit .51. nempe in vniuerſum .222. quadratorum porrò ſumma <lb/>erit .178 quæ coniuncta .222. dabit .400. quadratum ipſius .20.</s> + </p> + <p> + <s xml:space="preserve">Erit autem huiuſce rei facillima ſpeculatio, ſi ſequentem figuram mente conce-<lb/>perimus, in qua <var>.a.b.</var> propoſitum numerum ſignificet, cuius partes diſtinctæ ſint me-<lb/>dio <var>.e.</var> et <var>.c</var>. </s> + <s xml:space="preserve">Ip ſum autem <var>.q.b.</var> ſit quadratum <lb/>totale parallelis <var>.e.s.</var> et <var>.c.x.</var> diuiſum, 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> compoſitum ex producto <var>.a.<lb/>e.</var> in ſemetipſam, nempe quadratum <var>.o.e.</var> & <lb/>ex producto eiuſdem <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 ſingularum 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 ſex 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 propoſuerunt, quod tamen à <lb/>nobis determinabitur.</s> + </p> + <p> + <s xml:space="preserve">Cur diuiſuri propoſitum numerum in duas eiuſmodi partes, vt mutuò diuiſis, & <lb/>per ſummam prouenientium diuiſa ſumma qua dratorum partium, oriatur proue-<lb/>niens alter numerus propoſitus.</s> + </p> + <p> + <s xml:space="preserve">Propoſito deinde tertio quolibet numero diuidendo per ſingulas 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 ſimul prouenientibus in ſummam collectis huius fummæ ad primum nume-<lb/>rum propoſitum proportio futura ſit ea quæ eſt tertij ad ſecundum. </s> + <s xml:space="preserve">Rectè dimidium <lb/>primi numeri in ſeipſum multiplicant, ex quo quadrato ſecundum numerum detra <lb/>hunt, tum reſidui radicem ſumunt, quam iungentes, & detrahentes ex dimidio <lb/>primi, partes quæſitas habent, cætera ex neceſsitate ſubſequuntur, 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/>ſtea mutuò diuiſis, & per ſummam prouenientium diuiſa ſumma quadratorum, <lb/>dent <reg norm="ſecundum" type="context">ſecundũ</reg> numerum propoſitum .36. nam reliqua conſequuntur. </s> + <s xml:space="preserve">Itaque .10. <lb/>dimidium primi in ſeipſum multiplicatur, & ex quadrato .100. eruitur numerus .36. <lb/>nempe ſecundus propoſitus reſidui porrò .64. quadrata radix .8. fumitur, quam con <lb/>iungimus & detrahimus ex dimidio primi ſcilicet .10. ex quo partes quæſitæ dabun <lb/>tur .18. et .2. quæ mutuo diuiſæ dabunt ſuorum prouenientium ſummam .9. cum no-<lb/>na parte, per quam diuidentes .328. ſummam quadratorum ipſarum partium, <lb/>exactè dabitur numerus .36. qui fuit ſecundò propoſitus. </s> + <s xml:space="preserve">Tum ſi per ſingu-<lb/>las iam inuentas partes quilibet numerus diuiſus fuerit, verbi gratia .72. ſumma pro <lb/>uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. ſer <lb/>uabit, quam tertius propoſitus .72. cum ſecundo .36.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſpeculemur, primus numerus ſignificetur linea <var>.n.e.</var> ita diuidendus à <lb/>puncto <var>.o.</var> vt diuiſa parte <var>.n.o.</var> per <var>.o.e.</var> et <var>.o.e.</var> per <var>.n.o.</var> & per ſummam prouenien-<lb/>tium diuiſa ſumma quadratorum <var>.n.o.</var> et <var>.o.e.</var> detur ſecundus 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 ſe futurum, nempe producto <var>.n.o.</var> in <var>.o.<lb/>e.</var> quod ſignificetur rectangulo <var>.n.e</var>. </s> + <s xml:space="preserve">Itaque datis <var>.n.e.</var> et <var>.q.K.</var> ſi .45. theorema conſu-<lb/>luerimus, partes <var>.n.o.</var> et <var>.o.e.</var> cognoſcemus.</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 ſi 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="ſummam" type="context">ſummã</reg> duorum <lb/>horum prouenientium, ſic primo nume-<lb/>ro <var>.n.e.</var> dato proportionatam eſſe, ſicut <lb/>tertius <var>.x.</var> <reg norm="ſecundo" type="context">ſecũdo</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 ſit <var>.o.b.</var> ex <lb/>quo <var>.q.d.</var> erit ſumma vltimò prouenien-<lb/>tium: </s> + <s xml:space="preserve">item producatur <var>.e.n.</var> donec <var>.n.u.</var> æ-<lb/>qualis ſit <var>.o.e.</var> <reg norm="termineturque" type="simple">termineturq́</reg> rectangulum <var>.<lb/>q.u.</var> quod tertio numero propoſito <var>.x.</var> vt <lb/>patet, æquale erit, </s> + <s xml:space="preserve">quare ex .15. ſexti aut .<lb/>20. ſeptimi 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> ſicut <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 eſt <var>.<lb/>b.o.</var> ad <var>.o.n.</var> nempe ſicut <var>.b.e.</var> ad <var>.e.n.</var> ſuperficialem, ex prima ſexti aut .18. vel .19. <lb/>ſeptimi, ſed rectangulum <var>.e.n.</var> conſtitutum fuit æquale numero <var>.q.K</var>. </s> + <s xml:space="preserve">itaque verum <lb/>eſt propoſitum.</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 diuiſo, <reg norm="productoque" type="simple">productoq́;</reg> duorum numerorum per pro-<lb/>ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua <lb/>dratum exiſtat.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo <lb/>bus numeris multiplicato, nempe .20. habe <lb/>bimus .100. quadratum numeri diuiſi.</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 ſint <var>.a.</var> et <var>.e.</var> por <lb/>rò <var>.a.</var> per <var>.e.</var> diuiſo detur <var>.u.</var> tum <var>.o.</var> produ-<lb/>ctum <var>.a.</var> in <var>.e.</var> eſſe conſtituatur, quo per <var>.u.</var> <lb/>multiplicato dabitur <var>.x.</var> quadratum <var>.a.</var> pro-<lb/>ptereà quòd <var>.a.</var> medium eſt proportionale <lb/>inter <var>.o.</var> et <var>.u.</var> ex .35. theoremate. </s> + <s xml:space="preserve">itaque <lb/>ex .16. ſexti aut .20. ſeptimi, propoſiti veri-<lb/>tas eluceſcet.</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 & diuiſo, ſi per horum duo-<lb/>rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-<lb/>timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden <lb/>dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336. <lb/>productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-<lb/>rimus, proueniens pariter erit .7.</s> + </p> + <p> + <s xml:space="preserve">In cuius <reg norm="gratiam" type="context">gratiã</reg> primus numerus ſignificetur linea <var>.q.b.</var> multiplicandus & diuiden-<lb/>dus numeris deſignatis per <var>.k.m.</var> et <var>.y.m.</var> productorum ſumma ſit <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> ſit <var>.f.<lb/>m</var>. </s> + <s xml:space="preserve">Dico quòd ſi <var>.k.z.</var> per <var>.f.m.</var> diuiſerimus proueni et <var>.a.e</var>. </s> + <s xml:space="preserve">Quod cum ſic fuerit, erit <lb/>quoque verum quòd diuiſa <var>.k.z.</var> per <var>.a.e.</var> proueniet <var>.f.m.</var> numerus ſcilicet æqualis <lb/>numero <var>.f.m.</var> ex .13. theoremate huius. </s> + <s xml:space="preserve">Itaque quotieſcunque probauero quòd di-<lb/>uiſa <var>.k.z.</var> per <var>.a.e.</var> proueniat numerus æqualis ipſi <var>.f.m.</var> propoſitum verum eſſe con <lb/>ſequetur. ex .13. theoremate. </s> + <s xml:space="preserve">Quòd ſi proueniens ex diuiſione <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 ipſum <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> ſuper 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. ſexti, aut, 20. ſeptimi, numerus autem <var>.k.p.</var> erit <lb/>proueniens, quod probandum eſt æquale eſſe <var>.f.m</var>.</s> + </p> + <p> + <s xml:space="preserve">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato 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 eſt ſic ſe habere <var>.k.g.</var> ad <var>.k.u.</var> ſicut <var>.k.p.</var> ad <var>.k.y.</var> ſufficiet igitur pro-<lb/>bare ſic ſe habere <var>.k.g.</var> ad <var>.k.u.</var> ſicut <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. ſatis erit igitur probare ita ſe habere <var>.q.b.</var> ad <var>.a.e.</var> ſicut <var>.f.m.</var> ad <var>.k.y</var>. </s> + <s xml:space="preserve">Scimus au-<lb/>tem quòd eadem eſt proportio <var>.q.b.</var> ad <var>.a.o.</var> quæ <var>.m.k.</var> ad vnitatem, quæ ſit <var>.x.</var> & quod <lb/>proportio <var>.o.e.</var> ad <var>.q.b.</var> eadem eſt, quæ <var>.x.</var> ad <var>.m.y.</var> ex definitione diuiſionis. </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 ſic ſe habebit <var>.k.y.</var> ad <var>.m.y.</var> ſicut <var>.e.a.</var> ad <var>.o.a.</var> & permutando <var>.k.y.</var> ad <var>.e.<lb/>a.</var> ſicut <var>.m.y.</var> ad <var>.o.a.</var> & ex .19. quinti ita <var>.k.m.</var> ad <var>.e.o.</var> ſicut <var>.k.y.</var> ad <var>.e.a.</var> & permutando <var>.<lb/>k.m.</var> ad <var>.k.y.</var> ſicut <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 ſit <var>.k.y.</var> <reg norm="produ- ctaque" type="simple">produ-<lb/>ctaq́;</reg> <var>.m.t.</var> done <var>c.t.s.</var> æqualis ſit vnitati <var>.x.</var> <reg norm="termineturque" type="simple">termineturq́;</reg> rectangulum <var>.s.i.</var> ex quo da-<lb/>bitur proportio numeri <var>.f.m.</var> ad numerum <var>.s.i.</var> compoſita ex <var>.m.t.</var> ad <var>.t.s.</var> et <var>.f.t.</var> ad <var>.t.i.</var> <lb/>ex .24. ſexti, aut quinta octaui, ſed ita etiam proportio <var>.q.b.</var> ad <var>.a.e.</var> componitur ex <lb/>eiſdem 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 eſt ad <reg norm="numerum" type="context">numerũ</reg> <lb/>ipſius <var>.k.y.</var> ęqualis eſt 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 eſt <var>.k.p.</var> ad <lb/><var>x.y.</var> ex quo ſequitur <var>.k.p.</var> conſtare numero ęquali <var>.f.m.</var> proueniens igitur ex diuiſione <lb/>numeri <var>.k.z.</var> per <var>.f.m.</var> æquale eſt numero ipſius <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 concluſio alia etiam via demonſtrari poteſt.</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> ſint <var>.k.m.</var> et <var>.m.y.</var> prouenientia ex diuiſione ſint <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 ſumma ſit <var>.a.e</var>: productum <lb/>autem <var>.b.a.</var> in <var>.k.m.</var> ſit <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> ſit <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">ueniensq́;</reg> ſit <var>.a.c.</var> cui, <var>a.c</var>: <reg norm="productum" type="context">productũ</reg> <var>.<lb/>a.s.</var> <reg norm="eandem" type="context context">eãdẽ</reg> <reg norm="proportionem" type="context">proportionẽ</reg> ſeruabit, <reg norm="quam" type="context">quã</reg> <lb/><var>k.y.</var> rectangulum ad vnitatem ex <lb/>definitione diuiſionis, hoc autem <lb/>proueniens <var>.a.c.</var> <reg norm="conſtare" type="context">cõſtare</reg> numero æ-<lb/>quali aſſero ſummæ <var>.a.e</var>. </s> + <s xml:space="preserve">Primum <lb/>enim ex dicta definitione diuiſio-<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 ſic ſe habet <var>.b.a.</var> <lb/>ad <var>.o.e.</var> ſicut <var>.k.m.</var> ad eandem vnita <lb/>tem. </s> + <s xml:space="preserve">Itaque vnitas hæc linearis ſi-<lb/>gnificetur per <var>.m.x.</var> in ſingulis late-<lb/>ribus <var>.k.m.</var> et <var>.m.y.</var> producentibus rectangulum <var>.k.y</var>: ſuperficialis autem vnitas ſit. + <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">cogiteturq́;</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 eſt <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> ſed ex prima ſexti aut .18. vel .19. ſeptimi, ſic ſe habet rectangu-<lb/>lum <var>.k.y.</var> ad <var>.x.y.</var> ſicut <var>.k.m.</var> ad <var>.m.x.</var> </s> + <s xml:space="preserve">quare ſicut <var>.b.a.</var> ad <var>.o.e.</var> ex .11. quinti, & eiuſdem <lb/>rectanguli <var>.k.y.</var> ad rectangulum <var>.k.x.</var> ſicut <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 ſcientia, ſic ſe habebit duplum rectanguli <var>.k.y.</var> ad ſummam <var>.y.x.</var> cum <var>.<lb/>k.x.</var> rectangulorum, ſicut duplum <var>.b.a.</var> ad ſummam <var>.a.o.e.</var> et proportio ſummæ re-<lb/>ctangulorum <var>.y.x.</var> et <var>.k.x.</var> duplo <var>.g.m.</var> ſicut duplum <var>.b.a.</var> ad <var>.a.o.e</var>. </s> + <s xml:space="preserve">Igitur ſumma 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 ſuperſicialis <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> ſicut dupli <var>.b.a.</var> ad <var>.a.e.</var> ex propoſitioni-<lb/>bus notatis, ſexti aut ſeptimi. </s> + <s xml:space="preserve">Quare etiam ſicut dupli rectanguli <var>.k.y.</var> ad <reg norm="ſummam" type="context">ſummã</reg> <lb/>rectangulorum <var>.y.x.</var> et <var>.k.x</var>. </s> + <s xml:space="preserve">Iam verò ſi conſtituatur <var>.e.c.</var> pro vnitate lineari ipſius <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/>eſt <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 hypotheſi, nempe quòd <var>.<lb/>e.r.</var> æqualis ſit numero <var>.k.m.y.</var> <lb/> + <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> + hoc eſt rectangulorum <var>.y.x.</var> et <var>.x.<lb/>k</var>. </s> + <s xml:space="preserve">Quamobrem <var>.a.r.</var> ex communi <lb/>ſcientia <reg norm="medium" type="context">mediũ</reg> proportionale erit <lb/>inter duplum <var>.a.s.</var> & duplum <var>.a.c.</var> <reg norm="ea demque" type="context simple">ea<lb/>dẽq́;</reg> <reg norm="proportio" type="simple">ꝓportio</reg> dupli prędicti <var>.a.s.</var> ad <lb/>duplum <var>.a.c.</var> ex æqualitate propor-<lb/>tionum ſimul collectarum, eadem <lb/>erit qùæ proportio dupli rectangu-<lb/>li <var>.k.y.</var> ad duplum <var>.m.g.</var> hoc eſt <var>.a.s.</var> <lb/>ſimplicis ad ſimplicem <var>.a.c.</var> quæ ſim <lb/>plicis rectanguli <var>.k.y.</var> ad ſimplicem <lb/>vnitatem <var>.g.m.</var> ſic enim ſe habet ſim <lb/>plex ad ſimplex, ſicut duplum ad <lb/>duplum. </s> + <s xml:space="preserve">Sed pariter ita ſe habet <var>.a.s.</var> ad <var>.a.</var> c<unsure/>. cogitato <var>.a.c.</var> tamquam proueniente <lb/>ex diuiſione <var>.a.s.</var> per rectangulum <var>.k.y.</var> vt conſtitutum eſt, ſicut <var>.k.y.</var> ad <var>.m.g.</var> ex defi-<lb/>nitione diuiſionis vt iam dictum eſt, </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 propoſitis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-<lb/>dis, ſi <reg norm="adinuicem" type="context">adinuicẽ</reg> diuiſi fuerint, <reg norm="duoque" type="simple">duoq́;</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 ſeruetur, ſi deinde ijdem numeri verſa vice mutuo diuiſi fue <lb/>rint, & inter ſe multiplicata prouenientia, <reg norm="productum" type="context">productũ</reg> hoc, primo ſeruato numero <lb/>æquale erit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia propoſitis his .4. numeris .20. 30. 5. 10. duo autem .20. ſcilicet <lb/>et .30. ſint 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 ſimul multiplicati <reg norm="proferent" type="context">proferẽt</reg> <num value="12">.<lb/>12.</num> tum .20. per .10. d iuiſo et .30. per .5. prouenientia erunt .2. 6. quæ inter ſe 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 ſi quæris, ſignificentur .4. numeri lineis, <var>a.e.o.u.</var> <reg norm="diuidaturque" type="simple">diuidaturq́;</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> diuiſo per <var>.o.</var> oriatur <var>.z.</var> & per <var>.u.</var> <lb/>proueniat <var>.f.</var> tum <var>.n.</var> ſit 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/>ſionis 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 ſe ha-<lb/>bet <var>.a.</var> ad <var>.y.</var> et <var>.e.</var> ad <var>.f.</var> ſicut <var>.u.</var> ad <var>.x.</var> ex <lb/>quo ſic ſe habebit <var>.s.</var> ad <var>.a.</var> ſicut <var>.z.</var> ad <lb/>e. et <var>.a.</var> ad. y, ſicut <var>.e.</var> ad <var>.f</var>. </s> + <s xml:space="preserve">Itaque ex <lb/>æqualitate proportionum ſic ſe ha-<lb/>bebit s. ad <var>.y.</var> ſicut <var>.z.</var> ad <var>.f</var>. </s> + <s xml:space="preserve">Igitur ex <lb/>15. ſexti aut .20. ſeptimi 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 eſt, nempe vt proponantur .4. <lb/>numeri qualeſcunque tandem, quorum duo diuiſibiles ſint, tertius diuiſor <lb/>vnius è duobus pro libito, <reg norm="quæramusque" type="simple">quæramusq́;</reg> alterius diuidentem, qui ſic ſe habeat vt pro <lb/>ductum duorum prouenientium quarto numero propoſito ſit æquale.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponuntur .4. numeri .20. 48. 5. 12. porrò .20. et .48. numeri <lb/>ſint diuiſibiles 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. eiuſmodi vt productum prouenientium æquale ſit .12. </s> + <s xml:space="preserve">Diuidam itaque <num value="20">.<lb/>20.</num> per .5. <reg norm="prouenietque" type="simple">prouenietq́;</reg> 4. quem per .48. multiplicabo, nempe per alterum diuiſibi-<lb/>lem, <reg norm="ſicque" type="simple">ſicq́;</reg> proueniet .192. quod productum per quartum numerum nempe .12. diui-<lb/>fum dabit .16. qui erit diuidens quæſitus, quo diuiſo .48. proueniet .3. ſecundum ſci <lb/>licet proueniens, quo per alterum hoc eſt .4. multiplicato producetur quartus nu-<lb/>merus .12.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſciamus, primus nume-<lb/>rus diuiſibilis ſignificetur <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> ſecundus 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> ſecundus diuidens la <lb/>tere <var>.e.u.</var> (hic autem eſt quem quæri-<lb/>mus) tum alterum proueniens ſigni <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 ſint 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 cognoſcetur, <lb/>cætera in ſubſcripta 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 eſt, nempe ſi duæ radices quadratæ in <lb/>ſummam collectæ fuerint, & ex dimidio eiuſmodi ſummæ detracta fuerit mi <lb/>nor radix, <reg norm="reſiduique" type="simple">reſiduiq́;</reg> quadratum duplicatum <reg norm="eique" type="simple">eiq́;</reg> ſummæ coniungatur du-<lb/>plum producti ipſius reſidui in dimidium ſummæ radicum, atque huic ſummæ du-<lb/>plum producti eiuſdem reſidui in radicem minorem coniunctum fuerit; </s> + <s xml:space="preserve">vltima hæc <lb/>ſumma differentia erit duorum quadratorum propoſitorum.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia duæ radices quadraræ ſint .5. et .11. harum ſumma erit .16. & dimi <lb/>dium .8. differentia minoris ab ipſo dimidio erit .3: duplum quadrati huius differen <lb/>tiæ erit .18: </s> + <s xml:space="preserve">duplum producti huius differentię in dimidium ſummę radicum erit .48. <lb/>item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium <lb/>ſumma erit .96. tantaq́ue erit differentia ſuorum quadratorum, quorum vnum <lb/>erit .25. alterum verò .121.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ſcientia, duæ quadratæ radices ſint <var>.h.o.</var> et <var>.o.d.</var> directæ inter ſe con-<lb/>iunctæ, quæ ſumma 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">ducanturq́;</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> ſimul cum du <lb/>plo producti <var>.o.e.</var> in <var>.o.h.</var> ex .12. ſecundi Eucli. </s> + <s xml:space="preserve">Sed ex .13. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> quadratum <var>.b.d.</var> <lb/>minus eſt 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/>eiuſmodi producti æquale eſt 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 eiuſdem, itaque duo quadrata ſcili-<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">cũ</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> ſummæ 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 ſi ex ſingulis duabus ſummis quadratorum demptum fuerit quadratum <var>.o.b.</var> <lb/>eadem producta & quadrata ipſius <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 ſumma duorum <reg norm="extremorum" type="context">extremorũ</reg> quatuor terminorum <reg norm="proportionalium" type="context">proportionaliũ</reg> arith-<lb/>meticè, æqualis eſt ſummæ duorum mediorum, vbi nota hac in re neceſſa-<lb/>rium non eſſe proportionalitatem continuam exiſtere.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea <lb/>dem eſſet primi ad ſecundum quæ tertij ad quartum, ſumma primi cum quarto eſſet <lb/>26. <reg norm="tantaque" type="simple">tantaq́;</reg> ſecundi cum tertio.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis cauſa, primus <reg norm="maiorque" type="simple">maiorq́;</reg> numerus ſignificetur linea <var>.e.o.</var> ſecun-<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> ſit <var>.i.o.</var> quæ æqualis <lb/>erit differentiæ <var>.r.c.</var> qua quartus à tertio ſuperatur ex hypotheſi. </s> + <s xml:space="preserve">Itaque aſſero ſum <lb/>mam <var>.e.o.</var> cum <var>.g.t.</var> nempe <var>.a.o.</var> æqualem eſſe ſummę <var>.q.s.</var> et <var>.u.c.</var> <reg norm="ſitque" type="simple">ſitq́;</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 tertiusq́ue terminus reperiuntur, eſt <lb/> + <anchor type="figure" xlink:label="fig-0061-01a" xlink:href="fig-0061-01"/> + enim ſecundus <var>.e.i.</var> tertius <var>.i.o.</var> et <var>.e.a.</var> quando-<lb/>quidem ex præſuppoſito <var>.e.i.</var> æqualis eſt <var>.s.q.</var> et <lb/><var>i.o.</var> æqualis <var>.r.c.</var> et <var>.a.e.</var> cum ſit æqualis <var>.g.t.</var> cui <lb/>pariter æqualis eſt <var>.r.u.</var> ex quo <var>.a.e.</var> æqualis <lb/>eſt <var>.u.r</var>. </s> + <s xml:space="preserve">Itaque illud ſequitur <var>.a.o.</var> ipſi <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 ſumma duorum terminorum extremorum imparium arithmeticæ pro-<lb/>portionalitatis ſemper duplo medij termini æqualis eſt.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſunt hitres termini proportionalitatis arithmeticæ .20. 15. 10 <lb/>ſumma duorum extremorum erit .30. quæ duplum eſt medij termini .15.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſpeculemur, tres termini, tribus lineis <var>.b.d</var>: <var>n.u.</var> et <var>.q.p.</var> <reg norm="ſignificentur" type="context">ſignificẽtur</reg>. </s> + <s xml:space="preserve">Di-<lb/>co nunc quòd ſumma <var>.b.d.</var> cum <var>.q.p.</var> nempe <var>.<lb/>h.d.</var> ſemper duplo <var>.n.u.</var> ſcilicet <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> ſit <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> ſed <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> manifeſtum 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">quorũ</reg> primum <lb/>eſt. </s> + <s xml:space="preserve">Datis tribus quantitatibus cognitis, ſi quis quartam inuenire voluerit, <lb/>quæ eiuſmodi ſit reſpectu tertiæ, qualis eſt ſecunda reſpectu primæ, ſecunda cum <lb/>tertia in ſummam colligenda erit, ex qua detracta prima, ſupererit quarta.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. ſi quartam inuenire vo <lb/>luerimus eiuſmodi proportionem cum tertia arithmeticè ſeruantem, quam ſecunda <lb/>cum prima, ſecundam cum tertia in ſummam colligemus, <reg norm="dabiturque" type="simple">dabiturq́;</reg> ſumma .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">Idipſum tamen proueniret ſi quis ex tertio termino differentiam primi atque ſe-<lb/>cundi detraheret; </s> + <s xml:space="preserve">hæc tamen via non tam vniuerſalis eſtqu àm illa. </s> + <s xml:space="preserve">N ſi quartus ter <lb/>minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-<lb/>mino in ſummam colligere oporteret.</s> + </p> + <p> + <s xml:space="preserve">Alterum problema eſt, quòd inuentis duobus terminis, ſi tertius requiratur, ſe-<lb/>cundus duplicandus erit, ex qua ſumma detracto primo, ſtatim 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 & ſecun-<lb/>di termini, eam detrahendo aut in ſummam cum ſecunda colligendo, attamen prior <lb/>ratio magis latè patet, ideſt vniuerſalior eſt.</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 ſi quis cupiat ſecundum 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">coniungetq́;</reg> vltimo termino, nempe quarto, ex qua ſum-<lb/>ma tertiam partem deſumet, quæ erit ſecundus terminus quęſitus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema <lb/>proponantur. </s> + <s xml:space="preserve">nempe .12. et .3. quorum ſecundus inueniendus ſit, ſumpto quolibet <lb/>pro primo, ſit autem .3. primus numerus, quartus verò .12. </s> + <s xml:space="preserve">quare duplicato 3. vtpo <lb/>tè primo, & coniuncto .12. quarto, ſumma erit .18. cuius eſt tertia pars .6. ſecundus <lb/>numerus ſcilicet ſumpto principio à minimo. </s> + <s xml:space="preserve">Idipſum euenit ſumpto principio à <lb/>maximo. </s> + <s xml:space="preserve">Nam ſi datur ſecundus à minimo aut à maximo, illico tertius datur diffe-<lb/>rentia inter hunc & primum, ſecundo coniuncta, aut ex eodem detracta.</s> + </p> + <p> + <s xml:space="preserve">Cuius ratio ſic demonſtratur, quatuor termini quatuor lineis <var>.m.g</var>: <var>q.p</var>: <var>u.n</var>: <var>c.t.</var> <lb/>ſignificentur, quorum <var>.m.g.</var> et <var>.c.t.</var> tantummodo cognoſcantur. </s> + <s xml:space="preserve"><reg norm="ſitque" type="simple">ſitq́;</reg> <var>.m.g.</var> primus ac <lb/>maior terminus: </s> + <s xml:space="preserve">k.g. verò ſit 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 ſumma totalis eſt, æqualem eſſe <var>.q.p</var>. </s> + <s xml:space="preserve">In primis enim <lb/>certi ſumus <var>.m.f.</var> in <var>.m.g.</var> reperiri æqualem <var>.q.p.</var> <reg norm="ſupereſtque" type="simple">ſupereſtq́;</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>: ſimul 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> ſimul 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">conſiderandum eſt an hætres quantitates <var>.f.g</var>: <var>K.d.</var> et <var>.b.K.</var> ſimul æquales ſint <var>.q.p.</var> <lb/>quod tamen per ſe manifeſtum eſt. </s> + <s xml:space="preserve">nam <var>.q.p.</var> ſuperat <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 differentię <var>.f.g.</var> ſuperat <var>.c.t.</var> ita <lb/>que <var>.f.g</var>: <var>k.d.</var> et <var>.K.b.</var> ipſi <var>.q.p.</var> ſunt <reg norm="ae- quales" type="simple">ę-<lb/>quales</reg>, ex quo ſequitur <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, ſumpto or-<lb/>dinis principio à minimo termino <lb/>rum, duplicetur <var>.c.t.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> duplum <lb/>hoc <var>.K.t.</var> cui <var>.k.b.</var> æqualis <var>.m.g.</var> con-<lb/>iungatur, quæſumma ſit <var>.b.t</var>. </s> + <s xml:space="preserve">Di-<lb/>co <var>.u.n.</var> tertiam eſſe partem ipſius. <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 ſemel reperitur <var>.u.n.</var> vnà cum <lb/>duabus differentijs, nempe <var>.i.g.</var> in <lb/>ipſa autem <var>.b.t</var>: <var>u.n.</var> ſignificetur pri <lb/>mo loco per <var>.r.K.</var> ex quo ſupererit <var>.b.r.</var> duabus differentijs prædictis æqualis, ſed ex <lb/>præſuppoſito <var>.u.n.</var> componitur ex <var>.o.u.</var> æquali <var>.c.t.</var> et <var>.o.n.</var> ęquali 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 ipſius <var>.b.t.</var> eſſe tertiam partem, quod erat <reg norm="propoſitum" type="context">propoſitũ</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 ſi quis velit ſecundum quinque continuorum proportionalium termi-<lb/>num inuenire, ſolis extremis cognitis. </s> + <s xml:space="preserve">Rectè <reg norm="vltimum" type="context">vltimũ</reg> triplo primi coniunget, <lb/>ex qua ſumma quartam partem detraher, quæ erit ſecundus terminus quæſitus. <lb/></s> + <s xml:space="preserve">Quod ipſum faciet qui inuenire vult ſecundum terminum ſenarij ſeptenarij, octo-<lb/>narij aut alterius cuiuſcunque, creſcente tamen multiplicatione primi, <reg norm="vltimoque" type="simple">vltimoq́;</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. ſi .18. primus erit, hoc eſt, ſi à genere maioris inæ-<lb/>qualitatis progrediemur, triplicabimus terminum .18. <reg norm="dabunturque" type="simple">dabunturq́;</reg> .54. cui numero <lb/>coniuncto quinto termino .10. dabitur numerus .64. cuius quarta pars erit .16. vtpo <lb/>tè ſecundus terminus gratia, aut ſecundi ſex terminorum, quadruplicandus eſſet pri <lb/>mus .18. deinde adiuncto vltimo, quinta pars ſummæ eſſet ſecundus terminus, <reg norm="atque" type="simple">atq;</reg> <lb/>ita deinceps.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis 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="ſigniſicentur" type="context">ſigniſicẽtur</reg>. <lb/></s> + <s xml:space="preserve">In primis ex genere maioris inæqualitatis, triplicabimus <var>.z.h.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> triplum hoc <var>.k.<lb/>h.</var> <reg norm="cuiconiungatur" type="context">cuicõiungatur</reg> <var>.b.k.</var> ęqualis 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 ſum-<lb/>mę <var>.b.h</var>. </s> + <s xml:space="preserve">Nam in <var>.k.h.</var> ſecundus terminus <var>.f.s.</var> ter cum tribus differentijs æqualibus <var>.n.h.</var> <lb/>reperitur. </s> + <s xml:space="preserve">Probandum nunc eſt tres has differentias <var>.n.h</var>: <var>a.c.</var> et <var>.d.k.</var> ſimul cum <var>.b.<lb/>K.</var> ęquales 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">dubiũ</reg> re <lb/>uocari <reg norm="non" type="context">nõ</reg> poteſt, <lb/>cum <var>.f.s.</var> ſuperet <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/>ſit <var>.x.a.</var> et <var>.a.b.</var> ſit <lb/>æqualis <var>.z.h.</var> & <reg norm="cum" type="context">cũ</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> ſu-<lb/>peret <var>.e.g.</var> quæ in <var>.<lb/>a.b.</var> ſint <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 ſumma <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 eſt vt quæ proportio (licet impropriè dicta) <lb/>eſt ipſius <var>.a.b.</var> ad <var>.c.d.</var> <reg norm="eadem" type="context">eadẽ</reg> ſit ipſius <var>.e.f.</var> ad <var>.g.h</var>. </s> + <s xml:space="preserve">Tunc permutando dico eandem pro <lb/>portionem fore ipſius <var>.a.b.</var> ad <var>.e.f.</var> quæ ipſius <var>.c.d.</var> ad <var>.g.h</var>.</s> + </p> + <p> + <s xml:space="preserve">Nam, ex hypotheſi, differentia qua <var>.a.b.</var> ſuperat <var>.c.d.</var> (quæ ſit <var>.m.b.</var>) æqualis eſt <lb/>differentiæ qua <var>.e.f.</var> ſuperat <var>.g.h.</var> (quæ ſit <var>.i.f.</var>) vnde <var>.a.m.</var> reſiduum ex <var>.a.b.</var> æquale erit <lb/><var>c.d.</var> & reſiduum <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 ſcientia. </s> + <s xml:space="preserve">Vnde <var>.K.m.</var> æqualis erit <var>.n.d.</var> hoc eſt ipſi <var>.g.h.</var> hoc eſt ipſi <var>e.i</var>. </s> + <s xml:space="preserve">Quare ex <lb/>communi conceptu <var>.b.K.</var> æqualis erit ipſi <var>.f.e.</var> ſed <var>.n.d.</var> æqualis eſt <var>.g.h.</var> vt dictum eſt. <lb/></s> + <s xml:space="preserve">Cum ergo <var>.b.K.</var> æqualis ſit <var>.e.f.</var> et <var>.d.n.</var> ipſi <var>.g.h.</var> et <var>.a.b.</var> maior ſit ipſa <var>.K.b.</var> per <var>.a.K.</var> æqua-<lb/>lem ipſi <var>.c.n.</var> per quam <var>c.n</var>: <var>d.c.</var> maior eſt ipſa <var>.d.n.</var> ſequitur verum eſſe <reg norm="propoſitum" type="context">propoſitũ</reg> hoc <lb/>eſt, quod eadem proportio ſit ipſius <var>.a.b.</var> ad <var>.e.f.</var> quæ <var>.c.d.</var> ad <var>.g.h.</var> arithmetice ſcilicet.</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 eiuſdem numeri diuiſi-<lb/>bilis, geometricè <reg norm="eandem" type="context">eandẽ</reg> inter ſe <reg norm="proportionem" type="context">proportionẽ</reg> ſeruant, <reg norm="quam" type="context">quã</reg> ipſimet <reg norm="diuidentes" type="context">diuidẽtes</reg>.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-<lb/>tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s> + </p> + <p> + <s xml:space="preserve">Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis <var>.u.x.</var> et <var>.n.e.</var> diuidentes <lb/>autem ſint <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/>ſo dabit <var>.x.o.</var> & diuiſo <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 ſub <lb/>ſcriptam figuram conſiderantibus, in qua, <lb/>ex .15. ſexti aut .20. ſeptimi, 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 prorſus <lb/>inter ſe proportionem ſeruabunt, <lb/>quam ipſi 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 aduerſo.</s> + </p> + <p> + <s xml:space="preserve">Cuius ratio ex .15. ſexti aut .20. ſeptimi dependet. </s> + <s xml:space="preserve">prout in ſubſcripto ordine fa-<lb/>cillimè deprehendi poteſt.</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 ſecta, & per ſingulas <lb/>ipſarum diuiſa, ſumma trium prouenientium quadrato medij prouenientis <lb/>æqualis eſt.</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="ipſeque" type="simple">ipſeq́;</reg> numerus .14. per ſingulas 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 ſecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-<lb/>nient .7. integri, qui in ſummam 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 ſpeculationis gratia, totalis numerus ſignificetur 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 ſint continuæ proportionales, quarum ſingulis, <lb/>numerum <var>.n.c.</var> diuiſum eſſe cogitemus, proueniens autem ex diuiſione <var>.n.c.</var> per <var>.n.<lb/>a.</var> ſit <var>.i.d.</var> quod verò prouenit ex diuiſione <var>.n.c.</var> per <var>.a.e.</var> ſit <var>.d.u.</var> proueniens quoque ex <lb/>diuiſione <var>.n.c.</var> per <var>.e.c.</var> ſit <var>.u.o.</var> quorum ſumma ſit <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 ſit <var>.o.u.</var> <reg norm="erigaturque" type="simple">erigaturq́;</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 ſit æqualis, <reg norm="terminenturque" type="simple">terminenturq́;</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 ſexti aut .20. ſeptimi, quandoquidem tria prouenientia <var>.o.u</var>: <var>u.d.</var> et <var>.d.i.</var> ex <lb/>pręcedenti theoremate ſunt inter ſe continua proportionalia, proportionalitate qua <lb/>partes <var>.n.c</var>. </s> + <s xml:space="preserve">Iam verò ſi probauero <var>.q.i.</var> productum, producto <var>.m.p.</var> æquale eſſe, pro-<lb/>poſitum quoque probatum erit. </s> + <s xml:space="preserve">Numerus enim producti <var>.q.i.</var> æqualis eſt numero. <lb/></s> + <s xml:space="preserve">ſummæ <var>.i.o</var>. </s> + <s xml:space="preserve">Habemus autem ex definitione diuiſionis ita ſe habere <var>.n.c.</var> ad <var>.i.d.</var> ſicut <var>.<lb/>n.a.</var> ad <var>.o.q</var>. </s> + <s xml:space="preserve">Itaque permutando ſic ſe habebit <var>.n.c.</var> ad <var>.n.a.</var> ſicut <var>.d.i.</var> hoc eſt <var>.m.o.</var> ad <var>.<lb/>o.q.</var> ſed ſicut ſe habet <var>.n.c.</var> ad <var>.n.a.</var> ita pariter ſe habet <var>.i.o.</var> ad <var>.o.u.</var> hoc eſt ad <var>.o.p</var>. </s> + <s xml:space="preserve">Ita-<lb/>que <var>.i.o.</var> ad <var>.o.p.</var> ſic ſe habebit ſicut <var>.m.o.</var> ad <var>.o.q.</var> ex quo ex .15. ſexti aut .20. ſeptimi <var>.<lb/>q.i.</var> æqualis erit <var>.m.p.</var> & conſequenter quadrato <var>.d.u</var>. </s> + <s xml:space="preserve">Vt autem lector minori labo-<lb/>re cognoſcere queat <var>.i.o.</var> ad <var>.o.u.</var> ſic ſe habere, vt <var>.n.c.</var> ad <var>.n.a.</var> ſciendum eſt quòd, ſic <lb/>ſe habet <var>.i.d.</var> ad <var>.d.u.</var> ut <var>.c.e.</var> ad <var>.e.a.</var> ex quo componendo ſic ſe habebit <var>.i.u.</var> ad <var>.d.u.</var> ſi-<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> ſed cum ex <reg norm="præ- cedenti" type="context">præ-<lb/>cedẽti</reg> theoremate ſic ſe habeat <var>.d.u.</var> <lb/>ad <var>.u.o.</var> ſicut <var>.e.a.</var> ad <var>.a.n.</var> permutando <lb/>ſic ſe habebit <var>.d.u.</var> ad <var>.a.e.</var> ſicut <var>.u.o.</var> ad <lb/><var>a.n.</var> ex quo ex .11. quinti ſic ſe habe-<lb/>bit <var>.i.u.</var> ad <var>.c.a.</var> prout <var>.o.u.</var> ad <var>.a.n.</var> per-<lb/>mutandoq́ue <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> ſicut <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 ſecta per-<lb/>q́ue ſingulas diuiſa, ſumma quatuor prouenientium æqualis ſit producto ſe-<lb/>cundi in tertium.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi triginta in quatuor partes proportionales ſecetur, hoc eſt. <lb/>16. 8. 4. 2. <reg norm="perque" type="simple">perq́;</reg> harum ſingulas idem numerus .30. diuidatur, primum proueniens <lb/>erit .1. cum ſeptem octauis partibus. </s> + <s xml:space="preserve">Secundum .3. cum tribus quartis, tertium .7. <lb/>cum dimidio, quartum .15. integri, quorum ſumma erit .28. cum octaua parte, tan <lb/><reg norm="tumque" type="simple">tumq́;</reg> erit productum ſecundi prouenientis in tertium.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſciamus, quantitas <var>.n.c.</var> in partes continuas proportionales quatuor ſe-<lb/>cetur <var>.n.a</var>: <var>a.t</var>: <var>t.e.</var> et <var>.e.c.</var> <reg norm="rurſusque" type="simple">rurſusq́;</reg> per ſingulas partes illa ipſa diuiſa, prouenientia <lb/>ſint <var>.i.d</var>: <var>d.x</var>: <var>x.u</var>: <var>u.o.</var> <reg norm="quorum" type="context">quorũ</reg> ſumma ſit <var>.i.o.</var> hanc <reg norm="ſummam" type="context">ſummã</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">quousq́;</reg> <var>.o.p.</var> æqua <lb/>lis ſit <var>.o.u.</var> <reg norm="erectamque" type="simple">erectamq́;</reg> <var>.m.o.</var> æqualem <var>.i.d.</var> perpendiculariter <var>.o.p.</var> & productam donec <var>.<lb/>o.q.</var> vnitati ſit æqualis. </s> + <s xml:space="preserve">Iam terminatis rectangulis <var>.m.p.</var> et <var>.i.q.</var> patebit ex .15. ſexti <lb/>aut .20. ſeptimi, productum <var>.m.p.</var> producto <var>.d.x.</var> in <var>.x.u.</var> æquale eſſe. </s> + <s xml:space="preserve">Ita quòd ſi pro-<lb/>bauero productum <var>.i.q.</var> producto <var>.m.p.</var> æquale eſſe, facile patebit propoſitum. </s> + <s xml:space="preserve">Cuius <lb/>gratia, ſequuti præcedentis theorematis ordinem, primum ex <reg norm="definitionem" type="context">definitionẽ</reg> diuiſionis, <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> ſic <lb/>ſe habebit vt <var>.i.d.</var> hoc eſt <var>.m.o.</var> ad <var>.o.q.</var> & ſi progrediamur eodem ordine, quo præ-<lb/>cedenti theoremate, ſumpto principio ab <var>.i.d.</var> et <var>.e.c.</var> verſus <var>.d.x.</var> et <var>.e.t.</var> gradatimq́ue <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 ſe 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. ſextiaut .20. ſeptimi <reg norm="produ- ctum" type="context">produ-<lb/>ctũ</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 quantaſuis partes diuiſa ſeruari poterit, cum huiuſmodi <reg norm="ſcientia" type="context">ſciẽtia</reg> in vni <lb/>uerſum 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, ſemper producto <lb/>rectanguli compræhenſi à maximo & medio in minimo termino æqualis ſit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, datis his tribus terminis continuis proportionalibus .9. 6. 4. ſi <lb/>ſumpſerimus productum maximi in medium nempe .54. quod per <reg norm="minimum" type="context">minimũ</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="ſignifi- centur" type="context">ſignifi-<lb/>cẽtur</reg>, cubus autem <var>.e.</var> ſignificetur figura <var>.d.n.</var> <reg norm="productumque" type="simple">productumq́</reg> <var>.a.</var> in <var>.e.</var> ſit <var>.b.n.</var> ipſius <reg norm="au- temmet" type="context">au-<lb/>tẽmet</reg> in <var>.i.</var> ſit <var>.p.o.</var> ita quod <var>.q.p.</var> aut <var>.b.o.</var> cum ſint <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpeciei, 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 abſolui corpus <var>.n.h.</var> ita ut <var>.b.<lb/>o.c.</var>ſit vnica recta linea, ex quo ex .25. <lb/>vndecimi proportio <var>.n.h.</var> ad <var>.n.k.</var> ea-<lb/>dem eſt quæ <var>.o.h.</var> ad <var>o.k.</var> ſed ſic ſe 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 ſexti aut .18. vel .19. ſe-<lb/>ptimi itaque <var>.n.h.</var> ad <var>.n.k.</var> ex .11. <lb/>quinti ſic ſe habebit. vt <var>.h.b.</var> ad <var>.b.k.</var> <lb/>ſed <var>.n.h.</var> ad <var>.n.d.</var> ex eiſdem ſic ſe 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 præſuppoſito. </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 propoſitum.</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 ſingulos tres termi <lb/>nos diuiſo, prouenientia, ſingulis dictis terminis ſint 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. diuiſum dabit .4: per .6: 6. per .4: 9.</s> + </p> + <p> + <s xml:space="preserve">Cuius gratia, ſint tres termini <reg norm="continui" type="context">cõtinui</reg> <reg norm="proportionales" type="simple">ꝓportionales</reg> <var>.a.o</var>: <var>o.c.</var> et <var>.c.q.</var> <reg norm="quadratum" type="context">quadratũ</reg> <reg norm="autem" type="context">autẽ</reg> <lb/>medij ſit <var>.e.c</var>. </s> + <s xml:space="preserve">Iam ſi applicetur <reg norm="rectangulum" type="context">rectangulũ</reg> <var>.a.d.</var> æquale quadrato <var>.e.c.</var> ipſi <var>.a.o.</var> & re-<lb/>ctangulum <var>.q.p.</var> æquale eidem quadrato <var>.e.c.</var> ipſi <var>.c.q.</var> ſi quadratum <var>.e.c.</var> per <var>.a.o.</var> diui <lb/>ſerimus, proueniens erit <var>.o.d.</var> <reg norm="diuiſoque" type="simple">diuiſoq́</reg> per <var>.c.q.</var> proueniens erit <var>.c.p.</var> quod ſi per ſuam <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 ſine dubio æquale eſt <var>.o.c.</var> ſed dico <var>.<lb/>o.d.</var> æqualem eſſe <var>.c.q</var>. </s> + <s xml:space="preserve">Nam ex .16. ſexti aut <lb/>20. ſeptimi eadem eſt proportio <var>.a.o.</var> ad <var>.o.<lb/>c.</var> quę <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 eſt <var>.c.q.</var> quandoqui <lb/>dem ex .11. ſic ſe habet <var>.o.c.</var> ad <var>.o.d.</var> ſicut <var>.o.<lb/>c.</var> ad <var>.c.q</var>. </s> + <s xml:space="preserve">Applicatis ijſdem rationibus ipſi <var>.<lb/>p.c.</var> probabimus <var>.c.p.</var> æqualem eſſe <var>.a.o.</var> cum <lb/><var>o.c.</var> media ſit 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 eſt <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 propoſitis tribus quantitatibus continuis proportionalibus proportione <lb/>aliarum duarum nobis datarum, multiplicata maiori poſtremarum dua-<lb/>rum in ſummam mediæ cum minima trium primarum, productum æqua-<lb/>le ſit producto minoris duarum in ſummam 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"/> + poſitorum .3. et .2. multiplicato .3. per .10. <reg norm="ſummam" type="context">ſummã</reg> .6. cum .4. dantur .30. quod pro-<lb/>ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</s> + </p> + <p> + <s xml:space="preserve">Quod vt cognoſcamus, tres quan <lb/> + <anchor type="figure" xlink:label="fig-0068-01a" xlink:href="fig-0068-01"/> + titates continuæ proportionales ſint <lb/><var>b.a.p.</var> proportione <var>.d.q.</var> productum <lb/>autem <var>.d.</var> in ſummam <var>.a.</var> cum <var>.p.</var> ſit <var>.f.t.</var> <lb/>& productum <var>.q.</var> in ſummam <var>.b.a.</var> ſit <var>.<lb/>K.h.</var> et <var>.K.n.</var> ſit 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> ipſi <var>.q.</var> et <var>.f.o.</var> ipſi <var>.d</var>. <lb/></s> + <s xml:space="preserve">quare ita ſe habebit <var>.K.n.</var> ad <var>.n.o.</var> ſicut <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 eſt <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 ſicut <var>.k.o.</var> ad <var>.o.t.</var> ex quo ex .15. ſexti aut .20. ſeptimi <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 ſingulis tribus quantitatibus continuis proportionalibus in <lb/>reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum 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 ſimul collecta efficiunt numerum .228 ſed <reg norm="tantum" type="context">tantũ</reg> eſt pro-<lb/>ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</s> + </p> + <p> + <s xml:space="preserve">Cuius <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg> cauſa, tres termini <reg norm="continui" type="context">cõtinui</reg> proportionales ſignificentur linea <var>.<lb/>b.e.</var> nempe <var>.b.d</var>: <var>d.c</var>: <var>c.e.</var> cuius duplum ſit <var>.u.e.</var> et <var>.b.f.</var> æqualis ſit <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>ſit <var>.u.s.</var> cui dico æqualem eſſe ſummam productorum <lb/>ſingulorum 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. ſexti aut .20. ſeptimi itemq́ue <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/>prędictas. </s> + <s xml:space="preserve">Quare ſex producta æquantur inter ſe, <reg norm="replentque" type="simple">replentq́</reg> productum <var>.u.s.</var> ex quo <lb/>verum eſt propoſitum.</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 cognoſci poſſit <reg norm="verum" type="context">verũ</reg> eſſe proportionem ſummæ quatuor quan-<lb/>titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-<lb/>dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-<lb/>es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma + <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 ſicſe habet .20. ad .8. nempe ad ſecundam, vt .30. <lb/>ad .12.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſciamus, quatuor prædictæ quantitates ſignificentur linea <var>.a.e.i.o.</var> pro-<lb/>babo ita ſe 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 ſic ſe 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 ſe habebit <var>.a.</var> ad <var>.i.</var> <lb/>vt <var>.e.</var> ad <var>.o.</var> & è conuerſo 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">permutandoq́</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 propoſitum. <lb/></s> + <s xml:space="preserve">Ex quo patet error antiquorum quiidipſum, accidere arbitrati ſunt in quantitatibus <lb/>diſcretæ proportionalitatis, quod tamen falſum eſt.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eſt quæ .4. <lb/>ad .2. </s> + <s xml:space="preserve">Sed à proportione .6. ad .4. frangitur, cum non ſit eadem quæ .12. ad .6. harum <lb/>autem ſumma erit .24. & ſumma ſecundæ cum tertia .10. ſed primæ cum tertia erit <lb/>16. ex quo .16. ad .6. non ſic ſe habebit vt .24. ad .10. <lb/>At in ſpeculatione quatuor quantitatum <var>.a.</var> <lb/> + <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> + <var>e.i.o.</var> ſi proportio <var>.e.</var> ad <var>.i.</var> non eſſet eadem <lb/>quæ <var>.a.</var> ad <var>.e.</var> minimè licuiſſet dicere ita ſe <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 quibuſlibet, productum duarum in tertiam, vna <lb/>ſemper <reg norm="eademque" type="simple">eademq́;</reg> ſit quantitas.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, proponuntur .15. 8. 2. ſi multiplicauerimus .15. per .8. tum produ <lb/>ctum per .2. tantum erit quantum ſi quis multiplicaret .8. per .2. & hoc per .15. et .15. <lb/>per .2. <reg norm="rurſusque" type="simple">rurſusq́;</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 ſignificentur <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> ſit æquale <lb/><var>m.f.</var> et <var>.u.t</var>: a: et <var>.u.c</var>: o. patebit manifeſtè <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> ſed idipſum 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, & idipſum <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, ſi prima in ſecundam multiplice-<lb/>tur & hoc productum in tertiam, <reg norm="rurſusque" type="simple">rurſusq́</reg> hoc alterum in quartam, vltimum <lb/>productum æquale ſit producto producti ſecundæ 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, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8. <lb/>per .5. & hoc .40. per .3. rurſus 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 ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis <var>.a.e.i.o.</var> <lb/>ſignifi cari, productum autem <var>.e.</var> in <var>.i.</var> eſſe <var>.m.f.</var> et <var>.r.s.</var> ſimiliter & productum <var>.a.</var> in <var>.o.</var> eſ-<lb/>ſe <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/>hocq́ue <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 eſt, in figura præcedentis theore matis ſigni-<lb/>ficetur linea <var>.n.u.</var> & linea <var>.r.u.</var> hu-<lb/>ius, nempe <var>.a.</var> ſignificetur 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/>ſignificabitur producto <var>.n.t.</var> quod <lb/><reg norm="productum" type="simple context">ꝓductũ</reg> <var>.u.s.</var> <reg norm="pręsens" type="context">pręsẽs</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="præsens" type="context">præsẽs</reg> <var>.o.</var> mul<lb/>tiplicatum, quod erat in præceden <lb/>ti <var>.u.c.</var> ſignificabitur 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/>ſentis oritur, ſed 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 eſt propoſitum.</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 quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-<lb/>cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-<lb/>rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi horum quatuor diſparium numerorum <reg norm="ſummam" type="context">ſummã</reg>, in dicta pro-<lb/>greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-<lb/>ma erit .16. numerus quadratus inquam. </s> + <s xml:space="preserve">Idem de cæteris.</s> + </p> + <p> + <s xml:space="preserve">Quamobrem animaduertendum eſt, vnitatem, tam ſumi pro ſui ipſius radicem, <lb/>quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate. <lb/></s> + <s xml:space="preserve">Nunc autem pro quadrato ſumamus per <var>.o.</var> ſignificato, <reg norm="cogitemusque" type="simple">cogitemusq́</reg> quadratum <var>.o.</var> <lb/>includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-<lb/>priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-<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> ſit <var>.o.u.c.e.</var> ita ex communi ſcientia quadrato <var>.o.</var> iun-<lb/>gitur gnomon <var>.e.c.u.</var> conſtans tribus vnitatibus, quare primus gnomon, numero im-<lb/>pari conſtat. </s> + <s xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-<lb/>ſparibus ſummam excreſcere, cum propius accedere <reg norm="quam" type="context">quã</reg> binario nequeant, ex quo <lb/>medio binario, ſibi inuicem ſuccedunt. </s> + <s xml:space="preserve">Dico igitur quòd quinario ternarium ſub <lb/>ſequente, coniuncto quadrato <var>.o.u.c.e.</var> profertur quadratum, quod in numeris, bi-<lb/>narij quadratum ſequitur, <reg norm="eritque" type="simple">eritq́;</reg> ternarij, <reg norm="quodque" type="simple">quodq́;</reg> ſignificetur 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 eſt <var>.e.c.u</var>. </s> + <s xml:space="preserve"><reg norm="Iam" type="context">Iã</reg> ſcimus 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">itemq́;</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 præſuppoſito, maior eſt 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> ſuperabit. </s> + <s xml:space="preserve">Qua-<lb/>re <var>.b.f.d.</var> erit impar immediatè ſequens ternarium, qui coniunctus quadrato <var>.o.c.</var> <lb/>quadratum ſubſequens componet. </s> + <s xml:space="preserve">Eadem ratione probabitur de quadrato <var>.o.n.</var> ſe <lb/>quenti <var>.o.f.</var> & gnomone <var>.i.n.a.</var> cum hic ordo ſpeculationis ſit vniuerſalis. </s> + <s xml:space="preserve">In <lb/>quo cernitur quemlibet gnomonem ſibi <reg norm="contiguum" type="context">contiguũ</reg> inferiorem ſemper duabus vni-<lb/>tat ibus excedere, cumque quadrata non niſi gnomonibus ſibi inuicem ſuccedant. <lb/></s> + <s xml:space="preserve">Sed <reg norm="cum" type="context">cũ</reg> primus <var>.e.c.u.</var> diſpar fuerit, <reg norm="proculdubio" type="simple">ꝓculdubio</reg> <reg norm="etiam" type="context">etiã</reg> <reg norm="neceſſarioque" type="simple">neceſſarioq́;</reg> cæteri diſpares <reg norm="erunt" type="context">erũt</reg>. <lb/></s> + <s xml:space="preserve">Ex qua ſpeculatione, oritur regula ab antiquis tradita <lb/>inueniendi vltimi numeri diſparis <reg norm="concurrentis" type="context">cõcurrentis</reg> ad <reg norm="compo ſitionem" type="context">cõpo<lb/> + <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/> + ſitionem</reg> alicuius quadrati. </s> + <s xml:space="preserve">Vt ſi quis ſeire deſideret nu-<lb/>merum vltimum diſparem, quo mediante quadratum <var>.<lb/>o.n.</var> conſtitutum fuit, quod aliud non eſt quam ſcire <lb/>quantus ſit numerus vltimi gnomonis <var>.i.n.a.</var> æqualis gno <lb/>moni <var>.i.o.a</var>. </s> + <s xml:space="preserve">Itaque vt ſciamus hunc gnomonem <var>.i.o.a.</var> <lb/>patet duplicandam eſſe radicem <var>.o.e.b.i.</var> <reg norm="dabiturque" type="simple punctuation">dabiturq́,</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 ſcire gnomonem .i.b.e.o.u.d.a. </s> + <s xml:space="preserve">Itaque <lb/>minor eſt vnitate duplo radicis, cum unitas <var>.o.</var> bis repe-<lb/>tatur, quæ tamen in gnomone ſemel tantum ſumebatur.</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 ſumma quadratorum, quorum radices ſunt in proportione ſeſquitertia <lb/>nempe .4. ad .3. quadrata ſit.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſumemus quadratum .3. ſcilicet 9. quod in ſummam cum qua-<lb/>drato .4. colligemus, nempè .16. <reg norm="eritque" type="simple">eritq́;</reg> quadratum .25. & ita quadratum .6. hoc eſt <num value="36">.<lb/>36.</num> collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-<lb/>dratum .9. hoceſt .81. coniunctum quadrato .12. nempè .144. producet quadra-<lb/>tum .225.</s> + </p> + <p> + <s xml:space="preserve">In cuius gratiam ſint duo quadrata ſubſcripta <var>.q.o.</var> et <var>.q.a.</var> quorum radices ſint <var>.q.</var> <lb/> + <anchor type="figure" xlink:label="fig-0071-02a" xlink:href="fig-0071-02"/> + g. et <var>.q.p.</var> hoc eſt <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 ſiue la-<lb/>tum <reg norm="quam" type="context">quã</reg> gnomon <var>.b.a.g.</var> nempè vt <var>.h.</var> ſit æqua <lb/>lis .g: g. verò differentia ſit qua <var>.q.g.</var> maior eſt <var>.<lb/>q.p.</var> <reg norm="huncque" type="simple">huncq́;</reg> gnomonem <var>.f.s.h.</var> dico ęqualem eſ <lb/>ſe quadrato <var>.q.o.</var> nam ex preſuppoſito <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 ſe æquales ſunt vnde <lb/>etiam quadratum <var>.q.o.</var> nouem partibus ſuper-<lb/>ficialibus quadratis conſtabit, quarum ſingula <lb/>rum radix æqualis erit <var>.g.</var> cumque præcedenti <lb/>theoremate didicerimus quemlibet gnomo-<lb/>nem quadrati immediatè ſequentis æquę 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 ſuperficialibus creſcere, <reg norm="quarum" type="context">quarũ</reg> <reg norm="ſingularum" type="context">ſingularũ</reg> radix æqualis eſt <var>.g.</var> ne <lb/>ceſſariò ſequitur gnomonem <var>.b.a.g.</var> duabus partibus aut vnitatibus gnomonem <var>.d.<lb/>o.p.</var> ſuperare, ita vt gnomon <var>.b.a.g.</var> ſeptem vnitatibus, aut partibus ſuperficialibus <lb/>quadratis conſtet. </s> + <s xml:space="preserve">Quare eadem ratione gnomon <var>.f.s.h.</var> conſtabit nouem ſimilibus. <lb/></s> + <s xml:space="preserve">Itaque æqualis erit quadrato <var>.q.o</var>. </s> + <s xml:space="preserve">Quamobrem verum eſt, 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 ſe continuæ proportionales erunt. <lb/></s> + <s xml:space="preserve">Idipſum dico ſi <var>.q.p.</var> fuerit .6. et <var>.q.g</var>: 8: </s> + <s xml:space="preserve">tunc enim ſingulæ 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 ſummam 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 ſit <var>.g</var>. </s> + <s xml:space="preserve">Idem occurret ſi <var>.q.g.</var> erit .12. et <var>.q.p.</var> <lb/>quinque, quod cum fuerit patebitex præce-<lb/>dentis theorematis ſpeculatione, gnomonem <lb/><var>f.s.h</var>: 25. vnitatibus conſtare, cogitatum am-<lb/>plitudinis ſimplicis vnitatis denominatæ in <var>.q.<lb/>p.</var> aut <var>.q.g.</var> non amplitudinis gnomonis <var>.b.a.g.</var> <lb/>qui ſeptem vnitatibus latus eſſet. </s> + <s xml:space="preserve">Cum igitur <var>.<lb/>q.p.</var> quinque vnitatibus linearibus conſtet ſcimus <var>.q.o</var>: 25. ſuperficialibus conſtare, <lb/>collecto itaque in ſummam quadrato <var>.q.o.</var> cum quadrato <var>.q.a.</var> cognoſcetur quadra-<lb/>tum <var>.q.s.</var> vnà etiam eius radix. </s> + <s xml:space="preserve">Eadem ratione, alia multa quadrata ſimilia 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 propoſito numero pari maiori binario, qui detrahi & in ſummam colli-<lb/>gi debeat ex altero numero quærendo, vt tam reſiduum quam ſumma ſint <lb/>quadrata numerorum integrornm. </s> + <s xml:space="preserve">Rectè dimidium propoſiti numeri in ſeipſum <lb/>multiplicamus, & quadrato huic addimus vnitatem, <reg norm="eritque" type="simple">eritq́;</reg> numerus quæfitus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-<lb/>ro inueſtigando, ut reſiduum detractionis, & ſumma ſint quadrati numeri. </s> + <s xml:space="preserve">Addi-<lb/>ta vnitate ipſi .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s> + </p> + <p> + <s xml:space="preserve">Cuius ſpeculationis gratia, ſubſcripta 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">cogitemusq́;</reg> quadratum <var>.g.p.</var> eſſe quadratum ſummæ, <var>K.n.</var> verò reſidui ſubtractio-<lb/>nis: <var>u.i.</var> <reg norm="autem" type="wordlist">aũt</reg> numerum <reg norm="inueſtigandum" type="context context">inueſtigãdũ</reg>, ex quo gnomon <var>.u.d.i.</var> cognoſcetur ita etiam et <var>.n.<lb/>o.K.</var> qui inter ſe ſunt 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">dimidiũ</reg> eſſe gnomonis <var>.<lb/>n.o.K.</var> ex unitatibus ſuperficialibus quarum una erit <var>.m.a</var>.</s> + </p> + <p> + <s xml:space="preserve">Cuius numeri quadratum ſit <var>.t.c.</var> vnde etiam cognitum & cum <var>.K.c.</var> ex communi <lb/>ſcientia ſit vnitas linearis, </s> + <s xml:space="preserve">propterea quod <var>.m.a.</var> eſt ſuperficialis hoc eſt quadrata, <lb/>quæ detracta ex <var>.q.c.</var> dimidio gnomonis <var>.n.o.K.</var> (quamuis lineari) ſupererit <var>.K.q.</var> co <lb/>gnita, numerorum integrorum (nota <var>q.K.i.</var> ſemper minor erit duabus vnitatibus li-<lb/>nearibus & maior vna ex dictis vnitatibus, ut ex te ipſo 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> ipſius quadratum numerorum integrorum cognoſcetur, cui addito gnomone <var type="gnomon">.<lb/>n.o.K.</var> cognoſcemus numerum <var>.u.i.</var> quæſitum.</s> + </p> + <p> + <s xml:space="preserve">Sed cum nobis hæc via, tenenda propoſitum non fuit, hoc eſt primo loco inue <lb/>niendi quadrati minoris <var>.n.K.</var> ideo ſupereſt probandum gnomonem <var>.t.o.c.</var> vnitati <reg norm="ae- qualem" type="simple">ę-<lb/>qualem</reg> eſſe, nempe quadratulo <var>.m.a.</var> quod patebit, ſi conſideremus nos ſumpſiſſe <lb/>rectangulum <var>.r.c.</var> pro dimidio gnomonis <var>.n.o.K</var>. </s> + <s xml:space="preserve">etenim ſi ſupplemento 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 ſupplementa <var>.m.t.</var> et <var>.m.c.</var> inter ſe <lb/>fint æqualia. </s> + <s xml:space="preserve">Quamobrem inuento quadrato <var>.t.c.</var> ex dimidio gnomonis cognito, <lb/>additur vnitas, gnomon ſcilicet <var>.t.o.c.</var> ex quo cognoſcitur numerus <var>.u.i.</var> quæſitus. <lb/></s> + <s xml:space="preserve">Quod autem quadratum <var>.g.p.</var> numeris integris conſtet, hac ratione probatur viſum <lb/>enim fuit ſupra quadratum <var>.n.K.</var> verè quadratum eſſe, & numeris integris conſtare, <lb/>pariter etiam <var>.t.c.</var> <reg norm="ſeque" type="simple">ſeq́;</reg> mutuo conſequi (nam <var>.K.c.</var> eſt vnitas linearis) ex quo gnomon <lb/><var>n.a.K.</var> numero diſpari conſtabit, ex ijs quæ .90. theoremate probata fuerunt. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <lb/>ex eodem theoremate neceſſe eſt gnomonem <var>.t.d.c.</var> etiam numero diſpari conſtare, <lb/>ita vt à numero <var>.n.a.K.</var> non niſi duabus vnitatibus differat, nempe vt <var>.c.p.</var> ſit vnitas li-<lb/>nearis, ſed ita reuera eſt, numerus enim <var>.u.d.i.</var> ex præſuppoſito par eſt, </s> + <s xml:space="preserve">quare nume <lb/>rus <var>.t.d.c.</var> diſpar erit, cum alterum vnitate ſuperet, videlicet gnomone <var>.t.o.c.</var> vnita <lb/>ri æquali, tum <var>.n.a.K.</var> minor eſt <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 ſe-<lb/>quitur <var>.g.p.</var> eſſe quadratum <reg norm="integrorum" type="context">integrorũ</reg> ex dicto theoremate ac con ſequens 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 ſe different. </s> + <s xml:space="preserve">Vnà etiam ſcienda eſt cauſa, cur numerus propoſitus 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">cũ</reg> ipſe <lb/>ſit futurus gnomon <var>.n.o.K.</var> nec poſſit minor eſſe <lb/>numero ternario, vt patet ex .90. theoremate, <lb/>idcirco ſequitur neceſſariò maiorem eſſe bina-<lb/>rio debere. </s> + <s xml:space="preserve">Quòd ſi diſpar numerus propone-<lb/>retur, nec forma operis nec ſpeculationis <reg norm="mutan- da" type="context">mutã-<lb/>da</reg> eſſet. </s> + <s xml:space="preserve">Non erit tamen neceſſarium vt ipſa <lb/>quadrata <var>.n.K.</var> et <var>.g.p.</var> numeris integris conſta-<lb/>rent. </s> + <s xml:space="preserve">Sæpius enim fractis <reg norm="componerentur" type="context">cõponerentur</reg>, quod <lb/>ex .90. theoremate facile erit ſpeculari nihilo-<lb/>minus fractis integris, <reg norm="ipſisque" type="simple">ipſisq́;</reg> collectis cum ſuis <lb/>fractis ſummæ 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 propoſitis duobus numeris altero pari, altero verò diſpari, duplo primi <lb/>minore per vnitatem, ſi alium inuenire numerum voluerimus, cui alterum iſto <lb/>rum coniunctum proferat quadratum, & altero detracto, quadratum ſuperſit. </s> + <s xml:space="preserve">Re-<lb/>ctè datos numeros in ſummam colligemus, quam ſummam in duas quam maximas <lb/>poterimus partes diuidemus, quarum vna pari, altera diſpari conſtet, tum vtran-<lb/>que in ſeipſam multiplicabimus, & quadrato minori, duorum numerorum propo-<lb/>ſitorum quemuis ademus, ex quo cupimus nobis quadratum minus ſupereſſe, & 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 ſit, ex quo proferri debeant bina qua-<lb/>drata. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> numeri illi in ſummam collecti dabunt .17. differentiam minoris quadra <lb/>ti & maioris. </s> + <s xml:space="preserve">I am ſi ex hoc .17. binas partes fecerimus, altera erit .8. altera .9. qui <lb/>bus in ſeipſis multiplicatis alterum quadratum erit .64. alterum .81. addito <reg norm="itaque" type="simple">itaq;</reg> ipſi. <lb/>64. 11. aut .6. pro libito, propoſitum numerum conſequemur. </s> + <s xml:space="preserve">cui addito .6. vel .11. <lb/>dabit nobis .81. vel ex ipſo 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 ſpeculatio <lb/>ex .90. dependet, quo demonſtratum fuit gnomonem proximè quadratum ſequen <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 ſi quis cupiat ſummam progreſſionis arithmeticæ quam citiſſimè cogno <lb/>ſcere. </s> + <s xml:space="preserve">Rectè coniunget vltimo termino vnitatem primum terminum, huius <lb/>poſtea vltimi termini dimidium cum numero terminorum multiplicabit, ex <lb/>quo multiplicationis productum, erit omnium propoſitorum terminorum ſumma, <lb/>aut eundem vltimum terminum iunctum primo, per dimidium numeri terminorum <lb/>multiplicabit. </s> + <s xml:space="preserve">Nam idipſum eueniet.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proponerentur .17. termini in prima progreſſione arithmeti-<lb/>ca naturali, vltimus eſſet .17. cui coniuncta vnitate primo termino ſumma erit .18. <lb/>cuius dimidium cum numero terminorum, nempe .17. multiplicatum cum fuerit, <lb/>oritur productum .153. </s> + <s xml:space="preserve">Idpſum eueniet, multiplicato dimidio numeri <reg norm="terminorum" type="context">terminorũ</reg> <lb/>per vltimum coniunctum vnitati primo termino.</s> + </p> + <p> + <s xml:space="preserve">Quod vt ſciamus, cogitemus terminos progreſſionis collocari, vt in figura ſub-<lb/>ſcripta <var>.a.o.n.</var> collocantur, <reg norm="tanquam" type="context">tanquã</reg> per gradus, ſumpto principio ab vnitate <var>.n.</var> tum <var>.<lb/>u.t.</var> atque ita gradatim. </s> + <s xml:space="preserve">Sic cogitato abſoluto parallelogrammo <var>.q.o.</var> ſciemus aper-<lb/>tè ſummam 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 duę <lb/>ambientes diametrum inter ſe ſunt æquales. </s> + <s xml:space="preserve">Sumpto <reg norm="itaque" type="simple">itaq;</reg> diametro cum altera di <lb/>ctarum duarum partium, patet ſumi pluſquam <reg norm="dimidium" type="context">dimidiũ</reg> totius <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>. </s> + <s xml:space="preserve">pro <lb/>tanta portione, quantum eſt dimidiam occupatam à diametro, qui <reg norm="cam" type="context">cã</reg><unsure/> ex diſcretis <lb/>reſpondentibus numero terminorum componatur, conſtat numero æquali eſſe di-<lb/>cto numero terminorum <var>.o.n</var>. </s> + <s xml:space="preserve">Iam ſi quis multiplicet <var>.a.o.</var> per dimidium <var>.o.n.</var> procul <lb/>dubio, ex prima ſexti aut .18. ſeptimi, orietur <reg norm="dimidium" type="context">dimidiũ</reg> numeri <reg norm="parallelogrammi" type="context">parallelogrãmi</reg> <var>.q.o.</var> <lb/>quod minus erit ſumma progreſſionis dimidio numeri diametri, aut quod idem eſt <lb/>dimidio <var>.o.n.</var> ſed hoc dimidium <var>.o.n.</var> æquale eſt producto dimidij vnitatis <var>.n.</var> in <var>.o.n.</var> <lb/>ex .20. ſeptimi, cum dimidium <var>.o.n.</var> ſit 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 ſumma quæſita <lb/>propoſitæ progreſſionis. </s> + <s xml:space="preserve">Idipſum accidet multiplicata ſumma <var>.o.a.</var> & vnitate <var>.n.</var> <reg norm="per" type="simple">ꝑ</reg> <lb/>dimidium <var>.o.n.</var> ex .20. ſeptimi, cum proportio totius ad totum eadem ſit, quæ dimi <lb/>dijad dimidium, ex cauſa permutationalitatis. </s> + <s xml:space="preserve">Patet etiam in progreſſionibus, <lb/>quæ ab vnitate initium ducunt, ſi fiat aſcenſus per binarium ſumma vltimi termini <lb/>cum primo ſemper duplam futuram eſſe numero terminorum, quod ſequentes 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 ſpeculari licebit, Diametros harum figurarum notaui literis ſiue <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, idipſum vt <lb/>monuimus accidit, hoc tamen notato, quòd ex conſequenti 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, ſingulæ partes diametri, vnitati ſui primi termini æ-<lb/>quales ſunt. </s> + <s xml:space="preserve">At in reliquis progreſſionibus, vt in figura patet, eadem eſt 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 eiuſmodi progreſſionibus accidit quoque <reg norm="parallelo- grammum" type="context">parallelo-<lb/>grãmum</reg> à diametro in tres partes diuidi, quarum vnam ipſe occupat, reliquæ ve-<lb/>ro inter ſe æquales ipſum ambiunt. </s> + <s xml:space="preserve">Ex quo illud etiam ſequitur, productum <var>.a.o.</var> in <lb/>dimidium <var>.o.n.</var> æquale eſſe dimidio <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, quod minus eſt ſumma progreſ-<lb/>ſionis dimidio diametri, quod dimidum ſi inuenire voluerimus, minimum <reg norm="terminum" type="context">terminũ</reg> <var>.<lb/>n.n.n.n.</var> per dimidium <var>.o.n.</var> multiplicabimus, & ex .18. aut .19. ſeptimi ipſum 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 eſt 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. ſeptimi. <lb/></s> + <s xml:space="preserve">idem dimidium diametri oriri, ſi 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 ſtatim ſummam propoſitæ progreſſionis <lb/>cognoſcere voluerit, <lb/> + <anchor type="figure" xlink:label="fig-0075-03a" xlink:href="fig-0075-03"/> + ſemper primum termi <lb/>num <var>.n.n.n.n.</var> cum <var>.a.o.</var> <lb/>coniunget, qua ſumma <lb/>per <reg norm="dimidium" type="context">dimidiũ</reg> <var>.o.n.</var> mul-<lb/>tiplicata, aut <var>.o.n.</var> per <lb/>dimidium dictæ ſum-<lb/>mæ, ex prædictis rationibus propofitum conſequemur.</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 ſi 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 eſttamen quoties primus terminus differentia <reg norm="terminorum" type="context">terminorũ</reg> <lb/>eſt, nempe aſcendens ipſorum ter minorum.</s> + </p> + <p> + <s xml:space="preserve">Cuius ratio manifeſtè ſpeculari poteſt in figura præcedentis theorematis. </s> + <s xml:space="preserve">Nam <lb/>diuiſa <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 diuiſionis. </s> + <s xml:space="preserve">At ſuperius dictum fuit ita ſe 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 ſequitur 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 eſt detrahere primum ex vltimo, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> per numerum aſcenden-<lb/>tem differentiam ſcilicet, partiri, <reg norm="proueniensque" type="simple">proueniensq́;</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 ſummam collectis ſimul <lb/>cum vnitatibus primi termini, <reg norm="totque" type="simple">totq́;</reg> funt termini, quot interualla ſimul 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 diuiſo, 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">refiduoq́;</reg> per interuallum, hoc eft numerum dif-<lb/>ferentiæ diuifo, proueniens erit numerus terminorum abſque primo quod vnus eft, <lb/>coni uncto quoque dicto prouenienti propoſitum conſequemur.</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">refiduumq́;</reg> per numerum terminorum excepto vno diuidet.</s> + </p> + <p> + <s xml:space="preserve">Huius theorematis ſpeculatio ex .13. theoremate manifeſta crit, nam in præce-<lb/>denti cap. numerus terminorum erat proueniens diuiſionis reſidui ſubtractionis 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 ſi quis maximum omnium terminorum dictæ progreffionis cognofcere <lb/>voluerit, dato primo vnà cum numero aſcendenti, <reg norm="numeroque" type="simple">numeroq́;</reg> terminorum. </s> + <s xml:space="preserve">Re-<lb/>ctè numerum afcendentem cum numero terminorum excepto vno multiplicabit, <lb/><reg norm="productoque" type="simple">productoq́;</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 eſt 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 ſummam 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">cũ</reg> toto fequente multiplicabant, <reg norm="productumque" type="simple">productumq́;</reg> ſumma quæſita erat.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi vltimus terminus eiuſmodi 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 ipſius nempe .3. & dimidio, cum numero ipſum terminum <reg norm="ſequenti" type="context">ſequẽti</reg>, nem <lb/>pè .8. ſumma dictorum terminorum erit .28.</s> + </p> + <p> + <s xml:space="preserve">Huius autem ſpeculatio ex .94. theoremate dependet, in quo facilè depræhen-<lb/>dere licet ex figura continuæ progreſſionis naturalis, numerum terminorum maxi-<lb/>mo termino ſemper æqualem eſſe; </s> + <s xml:space="preserve">ex quo <reg norm="tantum" type="context">tãtum</reg> eſt dimidium numeriterminorum, <lb/>quantum maximi dimidium, <reg norm="tantusque" type="simple">tantusq́;</reg> eſt vltimus terminus vnitati coniunctus, quan <lb/>tus numerus is, qui vltimum terminum conſequitur.</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 diſpar eſt ſcire cupientes, numerum integrorum proximè dimidium <lb/>maximi ſequentem ſumebant, quem per maximum multiplicabant, ex quo <lb/>ſumma quæſita oriebatur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi dimidium maximi fuiſſet .3. cum dimidio, fumebant quatuor, <lb/>& per maximum .7. multiplicabant, ex quo pariter proferebatur ſumma .28.</s> + </p> + <p> + <s xml:space="preserve">Cuius ratio ex .20. ſeptimi Euclidis oritur, cum eadem ſit proportio numeri fe-<lb/>quentis ma ximum ad numerum dimidium maximi ſequentem; </s> + <s xml:space="preserve">quæ maximi ad <reg norm="fuum" type="context">fuũ</reg> <lb/>dimidium, eſt 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 eſt à nonnullis, à veteribus obſeruatam fuiſſe hancregulam, qua ſci-<lb/>re poſſent ſummam alicuius progreſſionis arithmeticæ diſcontinuæ 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">dimidiũ</reg> vltimi termini per <lb/>pro ximum numerum dimidio dicto maiorem, ex quo <reg norm="inquiebant" type="context">inquiebãt</reg> ſemper productum <lb/>ſummæ quæſitæ æquale eſſe, <reg norm="ſubijciuntque" type="simple">ſubijciuntq́;</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 eſt, non quia ex ſe vnus ex producentibus numeris dimi-<lb/>dium termini maioris futurus ſit, alter uerò proximè ſequens dimidium, fed quia <lb/>vt dictum eſt .95. theoremate, eadem eſt proportio maximi termini ad numerum <lb/>terminorum, quæ minimi ad vnitatem. </s> + <s xml:space="preserve">Cumq́ue in præfenti exemplo minimum <lb/>ſit duplum vnitati in eiuſmodi caſu, numerus terminorum, dimidio maximi termini <lb/>æqualis eſt, qui terminorum numerus ex ſe, vt patet, vnus eſt ex producentibus, al-<lb/>ter verò producens numerus, eſt proximè dimidium ſequens, non exſe, fed quia nu <lb/>merus ſequens, dimidium eſt ſummæ maximi, & minimi, quæ per fe alter eſſe de-<lb/>bet producens numerus. </s> + <s xml:space="preserve">In cæteris enim progreſſionibus, quæ binario non creſcút <lb/>regulafal<unsure/>fa eſt, prout facilè patere poteſt ei, qui ex ſcientiæ legibus ope ſpeculatio-<lb/>nis .95. theorematis ſpeculatus 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 vſos fuiſſe dicunt, quo ſum-<lb/>mam ſcire poſſent progreſſionis diſcontinuæ, quænumero diſpari abſolui-<lb/>tur. </s> + <s xml:space="preserve">Ea autem eſt eiuſmodi. </s> + <s xml:space="preserve">Vltimum terminum in duas quam maximè poterant ma-<lb/>ximas partes diuidebant, quarum vna ſemper altera maior erat, banc autem maio-<lb/>rem in ſeipſam multiplicabant, at que quadratum hoc, ſummam 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 ſanè regula, non ſemper, etſi interdum vera ſit.</s> + </p> + <p> + <s xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creſcit per bina <lb/>rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè ſe-<lb/>quens, nempe è duabus partibus vltimi termini maior, æqualis ſit numero termino <lb/>rum, qui per ſe vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet; <lb/></s> + <s xml:space="preserve">alter vero producens, qui per ſe dimidium ſummæ primi & vltimi eſſe debet, per <lb/>accidens pars maior eſt duarum vltimi termini, & alteri producenti æqualis.</s> + </p> + <p> + <s xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuſmodi progreſſione dimidium <lb/>ſummæ vltimi termini cum primo, ſemper medium proportionale eſt inter eam <lb/>ſummam & dimidium numeri terminorum, etenim huiuſmodi ſumma numero ter-<lb/>minorum ſemper dupla eſt, prout .94. theoremate tradimus. </s> + <s xml:space="preserve">Itaque ex .20. ſeptimi, <lb/>quadratum partis maioris, producto ſummæ dictæ in numerum dimidij <reg norm="terminorum" type="context">terminorũ</reg> <lb/>æquale erit, quod productum per ſe ſummæ progreſſionis eſt æquale. </s> + <s xml:space="preserve">At in cæte-<lb/>ris eiuſmodi progreſſionibus fallit regula, vt ex ſupradictis facilè demonſtratur.</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 propoſitis, diſpoſitis nihilominus progreſſio-<lb/>ne, aut proportionalitate geometrica continua, ſi minimus ex maximo & exfe-<lb/>quenti minimum detrahatur, reſiduum maximi, eam proportionem ad fum-<lb/>mam reliquorum omnium terminorum retinebit, quam reſiduum ſecundi 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, ſi primum, hoc eſt minimum, ex ſecundo, & maximo detra <lb/>has, exſecundo ſupererit .9. ex maximo .189. quod ſi minimum per reſiduum maxi <lb/>mi multiplicaueris, hoc eſt .189. orietur .567. tum ſi huiuſmodi productum per .9. <lb/>( refiduum ſecundi ) diuiſeris, proueniet .63. quod proueniens æquale erit ſummæ <lb/>reliquorum omnium terminorum, maximo excepto. </s> + <s xml:space="preserve">Ex quo inferre licet ex .20. ſe <lb/>ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut ſi reſiduum ſecundi <lb/>per ſummam dictorum terminorum multiplicaueris produceturidem .567. </s> + <s xml:space="preserve">quare <lb/>ex .20. ſeptimi & cætera.</s> + </p> + <p> + <s xml:space="preserve">Quod vt <reg norm="ſcientificè" type="context">ſciẽtificè</reg> poſſimus, & in vniuerſum ſpeculari. </s> + <s xml:space="preserve">Quatuor termini propo-<lb/>ſiti, quatuor ſubſcriptis lineis <reg norm="ſignificentur" type="context">ſignificẽtur</reg> <var>.b.i</var>: <var>c.a</var>: <var>f.r</var>: <var>m.s.</var> (quod <reg norm="autem" type="wordlist">aũt</reg> de his quatuor di <lb/>co de <reg norm="centummillibus" type="context">centũmillibus</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="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.n.i.</var> <reg norm="idemque" type="simple">idemq́;</reg> <var>.m.s.</var> ex ſecundo termino <var>.f.r.</var> ſubtra-<lb/>hatur, <reg norm="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.o.r</var>. </s> + <s xml:space="preserve">Dico proportionem <var>.n.i.</var> ad ſummam 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 ſecundus <var>.f.r.</var> <reg norm="detrahatur" type="simple">detrahat̃</reg>, ex <reg norm="tertioque" type="simple">tertioq́;</reg> ſuperſit <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="ſuperſitque" type="simple">ſuperſitq́;</reg> <var>.d.i.</var> ſanè <lb/> + <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> + ſic ſe 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 ſe ſcire poteſt. </s> + <s xml:space="preserve">Quare ex <lb/>19. quinti ſic ſe 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> & ſeparando ſic <var>.a.t.</var> ad <var>.<lb/>a.c.</var> (hoc eſt <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 ſic ſe 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 eſt <var>.o.r.</var> ad <var>.m.s.</var> ex .11. quinti. </s> + <s xml:space="preserve">Itaque ex communi ſcientia ſic ſe 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 ſit <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 ſit <var>.o.r</var>. </s> + <s xml:space="preserve">Iam ſi ſic ſe 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> ſic <lb/>ſe 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 ſic <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 ſic <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 eſt. ut <var>.o.r.</var> ad <var>.m.s.</var> quod erat propoſitum.</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 deſideranti ſummam quorumcunque terminorum progreſſionis conti-<lb/>nuæ geometricæ cognoſcere. </s> + <s xml:space="preserve">Rectè minimus terminus ex maximo detrahen <lb/>dus eſt, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> per denominantem progreſſionis dempta vnitate diuidendum, <lb/><reg norm="prouenientique" type="simple">prouenientiq́;</reg> maximum terminum addendum, ex quo oritur ſumma quæſita.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi darentur quatuor termini continui proportionales .8. 12. 18. <lb/>27. primum hoc eſt minimum .8. ex vltimo .27. detraheremus: </s> + <s xml:space="preserve"><reg norm="remaneretque" type="simple">remaneretq́;</reg> .19. qui <lb/>per denominantem progreſſionis, dempta vnitate, diuideretur. </s> + <s xml:space="preserve">Quo loco animad <lb/>uertendum eſt, quamlibet <reg norm="denominationem" type="context">denominationẽ</reg> cuiuſcunque proportionis numerorum <lb/>ſupra vnitatem fieri, nam de proportionibus multiplicibus dubitandum non eſt, & <lb/>idipſum de ſuperparticularibus, & ſuperpartientibus eſt intelligendum, vt in præ-<lb/>ſenti proportio ſeſquialtera inter duos terminos cogitanda eſt, nempe inter vnum <lb/>& dimidium, atque vnum. </s> + <s xml:space="preserve">Seſquitertia autem inter vnum & tertiam partem, <lb/>& vnum. </s> + <s xml:space="preserve">Seſquiquinta inter vnum cum quinta parte, & vnum. </s> + <s xml:space="preserve">De ſuperpartien <lb/>tibus idem aſſero quod de proportione <reg norm="ſuperbipartiente" type="context">ſuperbipartiẽte</reg> tertias appellata, vt .5. <lb/>ad .3. quæ cogitanda eſſet inter vnum duas tertias, & vnum, ſuperbipartiens quar-<lb/>tas inter vnum tres quartas, & vnum, ita vt minor terminus, numerans ſcilicet, ſem <lb/>per ſit 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, ſupererit tantummodo dimi-<lb/>dium, quo diuiſo .19. proueniet .38. qui numerus æqualis erit ſummæ <reg norm="reliquorum" type="context">reliquorũ</reg> <lb/>omnium terminorum, cui coniuncto vltimo termino .27. dabitur ſumma quæſita .65.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius ſpeculatione, quatuor termini ſignificentur, 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="reſiduumque" type="simple">reſiduumq́;</reg> ſit <var>.n.i.</var> & ex <lb/>ſecundo <var>.f.r.</var> cuius reſiduum ſit <var>.o.r.</var> proportio verò progreſſionis ea ſit, quæ <var>.g.h.</var> ad <var>.<lb/>y.</var> quo vnitas repræſentatur (ex quo ſic ſe 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> ſuperſit <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> quę diui <lb/>datur in puncto <var>.x.</var> ita vt <var>.n.x.</var> æqualis <lb/>ſit vnitati <var>.y.</var> & in puncto <var>.u.</var> ita. vt <var>.n.<lb/>u.</var> æqualis ſit <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 ſic ſe habeat <var>.<lb/>f.r.</var> ad <var>.m.s.</var> hoc eſt ad <var>.f.o.</var> vt <var>.g.h.</var> ad <var>.y</var> <lb/>hoc eſt ad <var>.g.</var> permutando <reg norm="quoque" type="simple">quoq;</reg> ſic <lb/>ſe 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. eiuſdem <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 eſt ad <var>.m.s.</var> vt <var>.h.</var> ad <var>.g.</var> hoc eſt <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">rectangulũ</reg> <var>.u.z.</var> cogitetur ęquale rectangulo <var>.x.i.</var> cuius <var>.x.i.</var> <lb/>ſuperficialis 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 ſuperficialis <var>.u.z.</var> lineari <var>.n.i.</var> æqualis erit, qui <lb/>quidem numerus in figura rectangu-<lb/>la ſuperficialis cognitandus erit, cum <lb/> + <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> + diuidendus ſit per <var>.h.</var> hoc eſt per <var>.n.u.</var> <lb/>ex quo proueniens ex huiuſmodi di <lb/>uiſione 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. ſepti-<lb/>mi eadem eſt proportio <var>.n.i.</var> ad <var>.n.z.</var> <lb/>quæ <var>.n.u.</var> ad <var>.n.x.</var> hoc eſt <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 ſummæ trium terminorum <var>.c.a</var>: f. num <var>.s.</var> cui coniuncto <lb/>quarto termino <var>.b.i.</var> propoſitum 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">PRopoſuere veteres quæſita nonnulla de itineribus <reg norm="interque" type="simple">interq́;</reg> hoc vnum fuit. </s> + <s xml:space="preserve">Po-<lb/>namus duos iter agere per eandem viam quorum alter quatuor milliaria ſin-<lb/>gulis diebus conficiat, alter verò prima die milliare vnum, ſecunda duo, tertia tria, <lb/>atque ita ſingulis diebus milliare addit; </s> + <s xml:space="preserve">quærimus quot dierum ſpacio ſocium con <lb/>ſequetur.</s> + </p> + <p> + <s xml:space="preserve">Quamobrem numerus milliarium primi viatoris duplicatur, ſic ſunt .8. milliaria. <lb/></s> + <s xml:space="preserve">ex quo ſemper vnitas detrahicur, quæ in præſenti exemplo erit .7. <reg norm="totque" type="simple">totq́;</reg> dies erunt <lb/>quibus ſocius ſocium conſequetur, & milliarium numerum æqualem abſoluerit.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei facilis erit ſpeculatio, ſi ſubſcripta figura diligenter conſideretur, in <lb/>qua primus viator, die prima, quatuor milliaria linea <var>.q.d.</var> ſignificata conficit, at-<lb/>que illa ipſa die alter vnum tantum defignatum per <var>.d.</var> perficit, ita vt primus via-<lb/>tor tribus milliaribus ſocium anteceſſerit, altera verò die fecundus uiator cum duo <lb/>milliaria <reg norm="conficiat" type="context">cõficiat</reg>, excedètur à primo duobus milliaribus tantummodo, quę cum tri-<lb/>bus primæ diei quinque erunt; </s> + <s xml:space="preserve">tertia die ijſdem de cauſis primus ſex tantum millia-<lb/>ribus à ſecundo diſtabit, cum verò quarta die tot ſecundus quot primus milliaria <lb/>conficiat, primus à ſecundo amplius quam antea non diſtabit; </s> + <s xml:space="preserve">quinta verò cum ſe <lb/>cundus vnum milliare amplius quam primus conficiat. </s> + <s xml:space="preserve">propius accedit ad primum <lb/>vno ex ſex milliaribus, quibus anteà diſtabat, tum ſexta cum duobus primum ſupe-<lb/>ret, detrahet ex ſex milliaribus præteritæ diſtantiæ tria, ſeptima tandem illa ſex <lb/>detraxerit. </s> + <s xml:space="preserve">In quo conſiderandum eſt ſecundum 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">quarũ</reg> <lb/>duarum figurarum <var>.d.o.p.f.</var> pars <reg norm="communis" type="context">cõmunis</reg> eſſe reperitur, quæ quantitates ſi inuicem <lb/>æquales eſſe debent, neceſſe eſt ſeparatas partes <var>.u.q.n.</var> et <var>.t.i.c.</var> inter ſe æquales eſſe, <lb/>& quoniam quarta die (hoc eſt die ſic diſtante 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 abſoluat vnus <lb/> + <anchor type="figure" xlink:label="fig-0081-01a" xlink:href="fig-0081-01"/> + quot alter abſque vlla differentia, quæ ſigni-<lb/>ficetur per <var>.o.s.</var> neceſſe eſtitaque ex communi <lb/>conceptione tot dies eſſe poſt <var>.o.s.</var> quotante-<lb/>ceſſerant, vt exceſſus æqualis ſit defectui, qui <lb/>ſimul collecti, iuncta etiam <var>.o.s.</var> duplum erunt <lb/><var>d.s.</var> dempta vnitate, prout facilè in ſubſcripta <lb/>figura qui ſque per ſe ſcientificè poterit ſpecu <lb/>lari. </s> + <s xml:space="preserve">Quamobrem conſultum erit duplicare <lb/>numerum <var>.o.s.</var> & exduplo vnitatem detrahe-<lb/>re, quandoquidem dies ſupra <reg norm="infraque" type="simple">infraq́;</reg> <var>.o.s.</var> cum <lb/>die <var>.o.s.</var> minores ſunt duplo numeri <var>.d.s.</var> aut <var>.o.<lb/>s.</var> (quodidem eſt) 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="ſiſecundus" type="context">ſiſecũdus</reg> viator <reg norm="ordinem" type="context">ordinẽ</reg> <reg norm="ſecundæ" type="context">ſecũdæ</reg> progreſſionis arithmeticæ <reg norm="ſeruansiter" type="context">ſeruãsiter</reg> <lb/>agat, nempe ea quæ ab vno per binarium aſcendit, ſemper numerus dierum <lb/>æqualis erit numero milliarium diurnorum primi viatoris.</s> + </p> + <p> + <s xml:space="preserve">In cuius gratiam animaduertendum eſt numerus ne milliarium diurnorum pri-<lb/>mi viatoris par an impar ſit. </s> + <s xml:space="preserve">Etenim ſi par eſt, primus viator in fine ſingulorum die-<lb/>rum primæ medietatis numeri omnium dierum ſecundum antecedet numero diſpa <lb/>ri milliarium; </s> + <s xml:space="preserve">altero verò dimidio numero dierum, à ſecundo numero etiam diſpa <lb/>ri præteribitur, vt in ſequenti figura patet. </s> + <s xml:space="preserve">Nam prima die, ſecundus ex primo <lb/>milliare vnum ex numero pari, qui à primo conficitur detrahit; </s> + <s xml:space="preserve">ſecunda verò die <lb/>idem ſecundus, duo ſubtrahit milliaria exdiſpari, qui primo reliquus fuerat, <reg norm="ſicque" type="simple">ſicq́;</reg> <lb/>perpetuò diſpar remanet vſque ad vnitatem, ad quam cum peruenerint, nempe ad <lb/>illius diei exitum, quo primus ſecundum vnitate tantummodò ſuperat, manifeſtè <lb/>depræhendetur ſubſequente dieſecundum vnitate primum ſuperaturum, altera ve <lb/>rò tribus vnitatibus, prout penultima die ſecundus à primo tribus vnitatibus ſupera <lb/>batur. </s> + <s xml:space="preserve">Quare neceſſe erit, tot diebus ſecundum cum primo iter agere, inchoan-<lb/>do ab ea die, qua fecundus primam ſuperabit, quot egerat dum à primo ſuperare-<lb/>tur, vt ex communi conceptione, media figura <var>.A.</var> depræhendi poteſt. </s> + <s xml:space="preserve">Quod au-<lb/>tem ſingula dimidia dierum, dimi-<lb/> + <anchor type="figure" xlink:label="fig-0081-02a" xlink:href="fig-0081-02"/> + dia ſint numeri milliarium diurno-<lb/>rum primi; </s> + <s xml:space="preserve">patebit exſequenti fi-<lb/>gura, cogitato termino <var>.u.n.</var> vlti-<lb/>mo progreſſionis ſuperatę à primo <lb/>vſque ad vnitatem <var>.e.</var> quiterminus <lb/><var>u.n.</var> coniunctus primo <var>.o.</var> nempe <var>.e.</var> <lb/>ſemper <reg norm="duplum" type="context">duplũ</reg><unsure/> eſt 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 conſtat numero <lb/>milliarium diurnorum primi viatoris, ex quo ſequitur totum numerum dierum, quo <lb/>rum <var>.o.n.</var> dimidium eſt, 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 ſi numerus milliarium primi viatoris diſpar fuerit, ſecundum numero pari <lb/>ſemper ſuperabit, vt facile erit ſequentem figuram conſideranti intelligere, <lb/>ex quo illud ſequetur, futuram quandam diem, qua paria milliaria conficient. </s> + <s xml:space="preserve">Sit-<lb/>q́ue illa dies <var>.u.n.</var> ſequitur <reg norm="etiam" type="context">etiã</reg> tranſacta 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 ſuperetur, <reg norm="quanto" type="context">quãto</reg> ſecundum primus <lb/>ſuperauerat, vnde totalis numerus <var>.o.f.</var> mi <lb/>nor erit duplo <var>.o.n.</var> vnitate ex communi <lb/>conceptione, ſed ita etiam ſe habet termi-<lb/>nus <var>.u.n.</var> hoc eſt 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 propoſitum.</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 ſecundi viatoris, non ab vnitate ſed à binario inchoata, <lb/>per binarium quoque aſcenderet, <reg norm="numerusque" type="simple">numerusq́;</reg> milliarium diurnorum primi via <lb/>toris par eſſet, abſque dubio quadam die paria milliaria <reg norm="vterque" type="simple">vterq;</reg> conficeret, quę ſigni <lb/>ficetur <var>.u.n.</var> qua tranſacta, tot diebus vtrique <reg norm="ambulandum" type="context">ambulãdum</reg> erit, quot <reg norm="fuerunt" type="context">fuerũt</reg> <reg norm="dum" type="context">dũ</reg> primus <lb/>ſecundum ſuperaret, vt totidem alijs pri-<lb/>mus à ſecundo ſuperetur, in qua tamen <lb/> + <anchor type="figure" xlink:label="fig-0082-02a" xlink:href="fig-0082-02"/> + progreſſione terminus <var>.u.n.</var> ſemper duplus <lb/>eſt numero terminorum <var>.o.n.</var> ex .95. theo-<lb/>remate, <reg norm="totque" type="simple">totq́;</reg> ſunt infra <var>.u.n.</var> termini vſque <lb/>ad <var>.f.</var> quot ſupra. </s> + <s xml:space="preserve">ex quo illud ſequitur 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, ſi 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 ſi in eiuſmodi progreſſione numerus milliarium diurnorum primi viato-<lb/>ris diſpar fuerit, patet quòd primus ſecundum numero diſpari ſuperabit, 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"/> + ciſſimq́;</reg> primum ſecundus, 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, ſit itaque vltima <lb/>dies, qua primus ſecundum vnita <lb/>tem antecedit <var>.u.n.</var> qui terminus <lb/>duplus eſt numero terminorum <lb/><var>o.n.</var> & cum illa die primus ſecun-<lb/>dum milliario antecedat, ſequen + <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 ſcientia neceſſe <lb/>eſt ſecundum tot diebus <reg norm="cum" type="context">cũ</reg> primo iter agere quot ſunt <var>.o.n.</var> qui ſimul æquales erunt <var>.<lb/>u.n.</var> ſed <var>.u.n.</var> minor eft numero milliarium diurnorum primi vnitate <var>.e</var>. </s> + <s xml:space="preserve">Itaque rectè <lb/>ſequemur 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 ſecundi viatoris progreſſio per ternarium aſcenderet, ſumpto initio ab <lb/>ipſo ternario, animaduertendum eſt an numerus milliarium diurnorum primi, <lb/>ternario menſuretur necne, etenim ſi menſuretur, tandem aliquando paria millia-<lb/>ria conficient, quæ dies ſit <var>.u.n.</var> </s> + <s xml:space="preserve">quare ſub <lb/> + <anchor type="figure" xlink:label="fig-0083-01a" xlink:href="fig-0083-01"/> + <var>u.n.</var> totidem quot ſupra termini <reg norm="erunt" type="context">erũt</reg>, <lb/>& <reg norm="cum" type="context">cũ</reg> <var>.o.n.</var> tertia ſit 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 ſumendæ ſint duæ tertiæ partes <var>.u.n.</var> <lb/>ex quibus vnitas detrahatur ſuperſitq́ue <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/>aſcendente ſecundi, patet nullam futuram diem qua pari milliaria conficient, <lb/></s> + <s xml:space="preserve">quare illa vltima qua primus ſecundum antecedet, vno aut duobus milliaribus an-<lb/>tecedet in præſenti caſu. </s> + <s xml:space="preserve">Antecedat itaque duobus milliaribus, <reg norm="ſitque" type="simple">ſitq́;</reg> dies <var>.u.n.</var> & alte <lb/>ra <var>.t.i.</var> ſecundus primum vno milliari ſuperabit, ita quod ſub <var>.t.i.</var> non poterunt plu-<lb/>res integros dies iter agere, quam ambulauerunt ante diem <var>.u.n.</var> hoc eſt vſquequo <lb/>ſecundtis iunctus ſit primo, qui numerus dierum, tertia parte <var>.o.n.</var> ipſius <var>.u.n.</var> vnitate <lb/>minor erit, cum ex .95. theoremate <var>.o.n.</var> ſit tertia pars <var>.u.n.</var> ex quo numerus <var>.o.f.</var> ter-<lb/>minorum aut dierum intergrorum cognitus erit, qui ſi cum numero alcendente <lb/>cognoſcetur, ſtatim ex .99. theoremate deueniemus in cognitionem vltimi diei in <lb/>tegri <var>.s.f.</var> atque ita etiam totius ſummæ progreſſionis ex .95. theoremate. </s> + <s xml:space="preserve">Iam verò <lb/>cognito numero milliarium diurnorum primi, ſimul cum numero terminorum, aut <lb/>dierum conſequenter nouerimus rectanguli ſummam, hoc eſt productum à primo <lb/>viatore formatum, quarum duarum ſummarum in præſenti caſu ſemper ea, quæ <lb/>huiuſmodi producti eſt, maior erit, cum conſtitutum fuerit ſecundum viatorem à <lb/>primo ſuperari ipſa die <var>.u.n.</var> vno milliari amplius quam ſequente die <var>.t.i.</var> primus à ſe <lb/>cundo ſuperatur, tum pari gradu iter egerunt ſub <var>.t.i.</var> quo ſupra <var>.u.n.</var> ambulauerant. <lb/></s> + <s xml:space="preserve">Hoc animaduertendo, quòd ſi ſumma progreſſionis maior eſſet rectangulo, ex ea <lb/>ſumma neceſſe eſſet <reg norm="numerum" type="context">numerũ</reg> mil <lb/>liarium vltimi termini in ſumma <lb/> + <anchor type="figure" xlink:label="fig-0083-02a" xlink:href="fig-0083-02"/> + incluſi detrahere, & reſiduo ope-<lb/>rari. </s> + <s xml:space="preserve">Nunc verò ſummam pro-<lb/>greſſionis exſumma rectanguli à <lb/>primo viatore facti ſubtrahi de-<lb/>bet, <reg norm="reſiduumque" type="simple">reſiduumq́;</reg> ſeruari <reg norm="voceturque" type="simple">voceturq́;</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">primũ</reg> <reg norm="reſiduum" type="context">reſiduũ</reg>. </s> + <s xml:space="preserve">Ad hæc <reg norm="numerum" type="context">numerũ</reg> <reg norm="milliariorum" type="context">milliariorũ</reg>, quæ <reg norm="ſecundus" type="context">ſecũdus</reg> viator die <reg norm="ſeqennti" type="context">ſeqẽnti</reg> <var>.s.f.</var> confi <lb/>ciet <reg norm="ſumatur" type="simple">ſumat̃</reg>, ex quo numerus <reg norm="milliarium" type="context">milliariũ</reg> <reg norm="diurnorum" type="context">diurnorũ</reg> primi detrahatur, <reg norm="residuumque" type="context simple">reſiduũq́;</reg> pariter re <lb/>ſeruetur, <reg norm="voceturque" type="simple">voceturq́;</reg> ſecundum reſiduum, poſtmodum numerum milliarium primi <lb/>vnius diei multiplicetur per primum reſiduum ſeruatum, <reg norm="productumque" type="simple">productumq́;</reg> per <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>reſiduum diuidatur <var>.a.c.</var> <reg norm="proueniens" type="context">proueniẽs</reg> (<reg norm="quod" type="simple">ꝙ</reg> erit iter primi in ſequenti die) iungatur reſi-<lb/>ſiduo primo, tot enim erunt milliaria conficienda a ſecundo ſequenti die, vt <lb/>ſeſe conſequantur.</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 ſciamus quantam partem diei <reg norm="ſeqenntis" type="context">ſeqẽntis</reg>, ſingulos itinere agere oporteat, <lb/><reg norm="proueniens" type="context">proueniẽs</reg> per .24. horas multiplicetur (ſuppoſito 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">ꝓductũq́;</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="numerum" type="context">numerũ</reg> milliariorum vuius diei primi viatoris di <lb/>uidatur, ex quo dabitur <reg norm="quantitas" type="context">quãtitas</reg> <reg norm="horarum" type="context">horarũ</reg>, & pars horæ, qua <reg norm="cuique" type="simple">cuiq;</reg> illa die <reg norm="ambulandum" type="context">ambulandũ</reg> <lb/>eſt. </s> + <s xml:space="preserve">Idem accideret ſi primum reſiduum reſeruatum cum proueniente in ſummam <lb/>colligeretur, <reg norm="eaque" type="simple">eaq́;</reg> ſumma per .24. horas multiplicaretur, <reg norm="productumque" type="simple">productumq́;</reg> per <reg norm="nume- rum" type="context">nume-<lb/>rũ</reg> milliariorum ſequenti die à ſecundo conficiendorum diuideretur. </s> + <s xml:space="preserve">I dipſum quo-<lb/>que eueniret multiplicato primo reſiduo per .24. & producto per ſecundum reſi-<lb/>duum diuiſo.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, primus viator diurna milliaria vndecim conficit, ſecundus, pri-<lb/>ma die tria, ſecunda .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="ſupereritque" type="simple">ſupereritq́;</reg> .2. </s> + <s xml:space="preserve">quare: <var>u.n.</var> ab <var>.e.n.</var> duobus milliaribus ſu-<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">totq́;</reg> diebus ambulandum erit poſt <var>.t.i.</var> hoceſt .6. in vniuerſum inte-<lb/>gris. </s> + <s xml:space="preserve">Ad hęc multiplicato <var>.o.f.</var> hoc eſt .6. per <var>.x.o.</var> hoc eſt .3. habebimus <var>.s.f.</var> <reg norm="milliariorum" type="context">milliariorũ</reg> <lb/>18. <reg norm="tum" type="context">tũ</reg> <reg norm="coniuncto" type="context context">cõiũcto</reg> <var>.x.o.</var> primo termino hoc eſt .3. <reg norm="cum" type="context">cũ</reg> <var>.s.f.</var> hoc eſt .18. vltimo termino, habe <lb/>bimus .21. quo multiplicato <reg norm="cum" type="context">cũ</reg> dimidio <var>.o.f.</var> hoc eſt .3. habebimus totam ſummam <lb/>progreſſionis .63. ſex dierum integrorum ex .94. theoremate, tum multiplicato .11. <lb/>nempe numero <reg norm="milliariorum" type="context">milliariorũ</reg> diurnorum primi cum .6. hoc eſt cum <var>.o.f.</var> habebimus pa <lb/>rallelogrammum à primo ſex diebus integris confectum milliariorum .66. ex quo <lb/>detracta .63. ſumma inquam progreſſionis, ſupererit pro primo reſiduo .3. ſumptis <lb/>poſtea milliaribus .21. pro itinere, quod ſecundus die ſequenti <var>.s.f.</var> conficeret, & ex <lb/>ijs detracto numero <reg norm="milliariorum" type="context">milliariorũ</reg> diurnorum primi, nempe .11. ſecundum reſiduum <lb/>erit 10. quod pro diuidenti ſeruabitur. </s> + <s xml:space="preserve">Iam multiplicato .11. cum primo reſiduo <num value="3">.<lb/>3.</num> dabitur .33. qui diuiſus per .10. ſecundum reſiduum profert .3. cum tribus decimis, <lb/><reg norm="eritque" type="simple">eritq́</reg> iter à primo viatore ſequenti die conficiendum, hoc etiam ipſum proueniens <lb/>cum primo reſiduo .3. coniunctum, dat .6. cum tribus decimis, quod eſt iter ſecundi <lb/>viatoris illa ſequenti die. </s> + <s xml:space="preserve">Ad inueniendam autem quantitatem diei, qua vtrique <lb/>ambulandum eſt, perinde erit multiplicare proueniens .3. & tres decimas per .24. ho <lb/>ras, & productum per .11. dimidium iter primi viatoris partiri, ac multiplicare ſum <lb/>mam .6. & tres decimas cum .24. horis, <reg norm="productumque" type="simple">productumq́</reg> diuidere per .21. hoc eſt periter <lb/>ſecundi viatoris ſequentis diei, vtrinque enim ſemper ſeptem horæcum .12. minu <lb/>tis prouenient. </s> + <s xml:space="preserve">Idipſum accidet multiplicato per .24. horas primo reſiduo .3. pro-<lb/><reg norm="ductoque" type="simple">ductoq́;</reg> diuiſo per ſecundum reſiduum .10.</s> + </p> + <p> + <s xml:space="preserve">Quarum ſpeculationum gratia, totum iter parallelogrammi primi viatoris die-<lb/>rum integrorum fignificetur linea <var>.n.e.</var> ſumma verò progreſſionis ſecundi linea <var>.f.m.</var> <lb/>parallela <var>.n.e.</var> <reg norm="eritque" type="simple">eritq́;</reg> <var>.f.m.</var> minor <var>.n.e</var>. </s> + <s xml:space="preserve">Conſtituamus deinde à termino <var>.f.n.</var> (majoris <lb/><reg norm="intelligentię" type="context">intelligẽtię</reg> gratia) vtranque <reg norm="perpendiculariter" type="context">perpẽdiculariter</reg> duci, <reg norm="producatur" type="simple">ꝓducatur</reg> deinde <var>.n.e.</var> donec <var>.e.<lb/>d.</var> æqualis ſit itineri diurno primi viatoris, item etiam producatur <var>.f.m.</var> donec <var>.m.K.</var> <lb/>æqualis ſit itineri à ſecundo confecto ſequenti 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 poſito. </s> + <s xml:space="preserve">Poſtmodum <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 eſt propor-<lb/>tioni <var>.m.e.</var> ad <var>.e.b.</var> ex .2. ſexti, ſed cum <var>.m.k.</var> et <var>.g.k.</var> notæ ſint, pariter cognoſcetur <var>.m.<lb/>g.</var> ſecundum reſiduum, cum etiam notæ ſint <var>.n.e.</var> et <var>.n.a</var>. </s> + <s xml:space="preserve">Itaque cognoſcemus <var>.a.e.</var> hoc <lb/>eſt <var>.m.h.</var> cognitis verò <var>.m.g</var>: <var>g.k.</var> et <var>.m.h.</var> ex .15. ſexti aut .20. ſeptimi cognoſcetur <var>.h.<lb/>q.</var> erit igitur <var>.a.e.</var> aut quod idem eſt <var>.m.</var> hprimum reſiduum, et <var>.m.g.</var> ſecundum, et <var>.h.<lb/>q.</var> aut <var>.e.o.</var> proueniens, et <var>.n.o.</var> et <var>.f.q.</var> itinera vtriuſque viatoris inter ſe æqualia. <lb/></s> + <s xml:space="preserve">Nec verò prætermittenda eſt ſpeculatio vltimæ rationis inueniendæ quantitatis <lb/>diei, quæ conſtat ope diuiſionis producti <var>.m.h.</var> in .24. per <var>.m.g</var>. </s> + <s xml:space="preserve">Ea autem eiuſmodi <lb/>eſt. </s> + <s xml:space="preserve">Probatum fuit ſic ſe 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/>ſic ſe 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 ſic ſe habeat <var>.m.h.</var> ad <var>.h.q.</var> vt <var>.m.g.</var> ad <var>.g.k.</var> permutando ſic ſe 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> ſed <lb/><reg norm="cum" type="context">cũ</reg> <var>.m.k.</var> ſit motus toti diei reſpon <lb/>dens, ſecurè dicere poterimus, ſi <lb/><var>m.g.</var> talis eſt reſpectu horarum <num value="24">.<lb/>24.</num> ſignificatarum 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/>ſpondens: </s> + <s xml:space="preserve">quæ <lb/><reg norm="poſtmodum" type="context">poſtmodũ</reg> erit <var>.<lb/>m.q.</var> quæ, vt <reg norm="di- ctum" type="context">di-<lb/>ctũ</reg> fuit, talis eſt <lb/>reſpectu <var>.m.k.</var> <lb/>qualis <var>.m.h.</var> re-<lb/>ſpectu <var>.m.g</var>. </s> + <s xml:space="preserve">Reli <lb/>quę duæ ſpecula <lb/>tiones priorum <lb/><reg norm="modorum" type="context">modorũ</reg>, vna & <lb/>eadem eſt, <reg norm="facilisque" type="simple">facilisq́;</reg> per ſe mediocriter intelligenti. </s> + <s xml:space="preserve">Eodem modo reliquæ omnes <lb/>progreſſiones ſecundi viatoris <reg norm="cum" type="context">cũ</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 ſcilicet viatorem iter incipere diebus aliquot an-<lb/>tè ſecundum, primum tamen lentius, quàm ſecundum ambulare, & utrunque <lb/>eorum certa quædam milliaria conficere. </s> + <s xml:space="preserve">Iam ſiſcire voluerimus in quot diebus <lb/>ſeſe conſequentur, uulgaris regula iubet, inſpici quot milliaria primus ſolus 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 ſoli 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, ſi primus octo diebus antequam ſecundus iter arripuiſſet, con-<lb/><reg norm="feciſſetque" type="simple">feciſſetq́;</reg> fingulis diebus .20. milliaria, tum ſecundus .25. quotidie perfeciſſet, mul <lb/>tiplicandus eſſet numerus .8. cum .20. ex quo darentur .160. milliaria à primo ſolo <lb/><reg norm="ambulante" type="context">ambulãte</reg> confecta, quibus diuiſis 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 eſt. </s> + <s xml:space="preserve">Sint enim duo rectanguli <var>.a.n.</var> et <var>.u.i.</var> æquales inter <lb/>ſe, quibus motus itinerarium ſignificentur, quorum <var>.a.n.</var> ſit primi, et <var>.u.i.</var> ſecundi, præ <lb/>tereà. <var>a.c.</var> numerum milliarium diurnorum primi .et <var>.u.e.</var> ſecundi, 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/>gnoſcetur. </s> + <s xml:space="preserve">Tum <var>.c.o.</var> numerum <reg norm="dierum" type="context">dierũ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> + primi ſoli iter agentis denotet, <reg norm="cumque" type="context simple">cũq;</reg> <lb/>conſtituamus <var>.a.n.</var> æqualem eſſe <var>.u.i.<lb/>o.i.</var> ęqualis erit <var>.o.a.</var> atque <var>.o.a.</var> cogni <lb/>tus ex ſuis producentibus <var>.a.c.</var> et <var>.c.o.</var> <lb/>itaque <var>.o.i.</var> etiam cognitus, qui diui-<lb/>ſus per latus cognitum <var>.o.e.</var> dabit <var>.e.<lb/>i.</var> cognitum numerum ſcilicet dierum, quibus ſecundo ambulandum eſt, vt primum <lb/>conſequatur.</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 ſumpſi ordinem illius operationis, numeris mediantibus, ad <lb/>inueniendam exactam temporis quantitatem, ſeu interuallum, <reg norm="quod" type="simple">ꝙ</reg> tranſit, vel in <lb/>tercedit inter vnam mediocrem coniunctionem & aliam proximam <reg norm="ſequentem" type="context">ſequentẽ</reg> duo <lb/>rum planetarum, vt patet in epiſtola noſtra ad Illuſtrem Bernardum Trottum con-<lb/>tra Benedictum Altauillam repræhenſorem Ephemeridum. </s> + <s xml:space="preserve">Verum tamen eſt <reg norm="quod" type="simple">ꝙ</reg> <lb/>cum praxis huiuſmodi theorematis ſit multiplex, viſum fuit vnam proponere, quę <lb/>non ita perſpicua ſit, ſed ſubobſcura, non quòd aliquid voluerim latere illum ami <lb/>cum mihi dilectiſſimum, cui priuatim omnes modos prius oſtenderam, ſed vt cere-<lb/>brum illius mei aduerſarij in laberintum conijcerem inextricabilem vt feci, quam-<lb/>uis modus ille egregius etiam ſit, vt nunc oſtendam.</s> + </p> + <p> + <s xml:space="preserve">In dicta epiſtola igitur mente cogitaui medium motum tardioris planetæ, pu-<lb/>ta ſaturni, illius temporis quo velocior planeta, ſcilicet Iupiter, percurrit ſuo medio <lb/>motu totum zodiacum, incipiendo ambo eodem temporis puncto, nec non ab vna <lb/>eorum media coniunctione, hoc eſt ab eodem zodiaci puncto, in quo coniunctę fue <lb/>runt eorum lineæ mediorum motuum, vbi inueni vi regulæ de tribus, quòd Satur <lb/>nus ſpacio dierum vnius mediocris reuolutionis Iouis, qui ſunt .4328. progreditur <lb/>medio motu gra .145. min .4. hoc eſt min .8704. pofito quòd ipſe Saturnus perficiat <lb/>vnam mediam reuolutionem ſpacio 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/>ipſum ſpacio .4328. dierum, certus eram hos dies ſignificatos eſſe à linea <var>.a.u.</var> vel <var>.c.<lb/>o.</var> (æquales enim inuicem ſunt) in figura huiuſmodi theorematis, & quòd rectangu <lb/>lum <var>.a.o.</var> præbebat ſummam graduum .145. min .4. hoc eſt min .8704. et quòd <var>.a.c.</var> <lb/>vel <var>.o.u.</var> ſignificabat iter vnius diei ipſius Saturni, et <var>.u.e.</var> iter vnius diei Iouis. </s> + <s xml:space="preserve">Cogi-<lb/>temus nunc <var>.u.x.</var> ſignificari dies .30. & à puncto <var>.x.</var> productam eſſe <var>.x.f.</var> parallelam ipſi <lb/><var>u.o.e.</var> vnde certi erimus rectangulum <var>.e.x.</var> ſignificare iter Ionis ſpacio 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. & ſecundorum .43. et <var>.o.x.</var> minutorum .60. & ſecun. <lb/>20. vt in dicta epiſtola, vnde rectangulum <var>.o.f.</var> erit min .89. & ſecun .23. & quia re-<lb/>ctangulum <var>.o.i.</var> æquale eſt rectangulo <var>.a.o.</var> ergo <var>.o.i.</var> ſimiliter 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. ſecun <var>.o.</var> & tertiarum .40. videlicet tertiarum <num value="7240">.<lb/>7240.</num> ſuppoſito periodo totaliipſius Saturni dierum .10740. et <var>.u.e.</var> erit <reg norm="minutorum" type="context">minutorũ</reg>. <lb/>4. ſecun .59. & ter .27. vel circa hoc eſt tertiarum .17967. vnde <var>.o.e.</var> erit tertiarum <num value="10727">.<lb/>10727.</num> </s> + <s xml:space="preserve">Nuncſi dixerimus cum <var>.o.e.</var> tertiarum .10727. dat <var>.o.u.</var> vel <var>.a.c.</var> (nam tam <lb/>vna quam altera eſt tertiarum .7240.) quid dabit <var>.a.u.</var> vel <var>.o.c.</var> (quia tam vna quam <lb/>altera eſt 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/>ſius <var>.a.u.</var> dabunt totam <var>.a.t.</var> <reg norm="partium" type="context">partiũ</reg> .7249. quæ erunt tot dies, hoc eſt periodus quæſita.</s> + </p> + <p> + <s xml:space="preserve">Alia methodo ſimiliter poſſumus idem cognoſcere, ſcilicet dicendo ſi rectangu <lb/>lum <var>.f.o.</var> quod eſt minutorum .89. & ſecun .23. hoc eſt ſecundorum .5363. dat rectan <lb/>gulum <var>.o.x.</var> minutorum .60. & ſecun .20. hoc eſt ſecun .3620. quid dabit <var>.a.u.</var> partium <lb/>4328. vnde veniet <var>.u.t.</var> partium .2921. ſimiliter, eo quod eadem proportio eſt rectan <lb/>guli <var>.f.o.</var> ad <var>.o.x.</var> quæ <var>.e.o.</var> ad <var>.o.u.</var> ex prima ſexti, vel .18. 19. ſeptimi ſeu .15. quinti.</s> + </p> + <p> + <s xml:space="preserve">Poſſet etiam aliquis dicere ſi <var>.f.o.</var> dat <var>.o.x.</var> quid dabit <var>.o.a.</var> vnde veniet <var>.o.t.</var> quo <lb/>diuiſo per <var>.o.u.</var> daret <var>.u.<lb/>t.</var> quia ita ſe 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/>ſupra 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/>ſtola, aliam methodum <lb/>obſeruaui, quæ eſt multi <lb/>plicando minuta .8704. <lb/>per .30. <reg norm="productumque" type="simple">productumq́;</reg> di <lb/>uiſi per min .5363. quaſi <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 ſupradictis propoſitionibus veniet <var>.e.i.</var> & quia permu-<lb/>tando ita ſe habet <var>.o.f.</var> ad <var>.e.f.</var> vt <var>.o.i.</var> ad <var>.e.i.</var> ideo dixi, ſi min .89. cum ſecun .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 ſi duo iter agentes, eodem in-<lb/>ſtanti diuerſis è locis proficiſcantur, ita vt vnus locum vnde alter profectus <lb/>eſt petat, <reg norm="alterque" type="simple">alterq́;</reg> altero velocior ſit, quo loco <reg norm="quanue" type="context">quãue</reg> die ſibi inuicem occurrent.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, Patauio profectus quidam Taurinum petit, eodem inſtanti al-<lb/>ter Taurino Patauium, <reg norm="eſtque" type="simple">eſtq́;</reg> iter .400. milliarium, ille tamen vndecim diebus, hic <lb/>9. motu regulari & vniformi appellit. </s> + <s xml:space="preserve">Quærimus quot milliaria quiſque confece-<lb/>rit, <reg norm="quotque" type="simple">quotq́;</reg> diebus iter egerit, priuſquam ſibi occurrant.</s> + </p> + <p> + <s xml:space="preserve">Iubent nos veteres dies vtriuſque inuicem inter ſe multiplicare, <reg norm="eritque" type="simple">eritq́;</reg> produ-<lb/>ctum .99. item etiam in ſummam colligere, <reg norm="eritque" type="simple">eritq́;</reg> ſumma .20. per quam <reg norm="productum" type="context">productũ</reg>. <lb/>99.@diuiſerimus dabuntur dies .4. cum .19. vigeſimis vnius diei. </s> + <s xml:space="preserve">At pro milliaribus <lb/>vtriuſque, pro eo qui .11. diebus iter conficit, multiplicatis .400. per .4. et .19. vigeſi <lb/>mis, tum diuiſo 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 quiſque confecerit.</s> + </p> + <p> + <s xml:space="preserve">Dum autem hæc ſpecularer attentius, occurrit alius ſoluendi modus, quamuis pro <lb/>lixior. </s> + <s xml:space="preserve">Is <reg norm="autem" type="wordlist">aũt</reg> eſt eiuſmodi. </s> + <s xml:space="preserve"><reg norm="Accipiatur" type="simple">Accipiat̃</reg> medietas minoris numeri <reg norm="dierum" type="context">dierũ</reg>, <reg norm="nempe" type="context">nẽpe</reg> .4. <reg norm="cum" type="context">cũ</reg> dimi <lb/>dio, & per .400. multiplicetur, <reg norm="productumque" type="context simple">productũq́;</reg> per <reg norm="maiorem" type="context">maiorẽ</reg> numerum diuidemus ſcilicet <lb/>11. ex quo dabuntur .163. cum .7. vndecimis, quo proueniente è dimidio <reg norm="millia- riorum" type="context">millia-<lb/>riorũ</reg> itineris .200. detracto, & <reg norm="preſiduum" type="context">preſiduũ</reg> <reg norm="nempe" type="context">nẽpe</reg> .36. <reg norm="cum" type="context">cũ</reg> .4. vndecimis multiplicato pro <lb/><reg norm="ductoque" type="simple">ductoq́;</reg> diuiſo <reg norm="per" type="simple">ꝑ</reg> <reg norm="ſummam" type="context">ſummã</reg> dimidij itineris .200. <reg norm="cum" type="context">cũ</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">ꝑ</reg> .363. ct .7. vndecimas partes <reg norm="proueniet" type="simple">ꝓueniet</reg> .16. <reg norm="cum" type="context">cũ</reg> .4. vndecimis, quo <reg norm="coniuncto" type="context">cõiuncto</reg> pri <lb/>mo <reg norm="prouenienti" type="simple context">ꝓueniẽti</reg>, primus .180. milliaria <reg norm="confecerit" type="context">cõfecerit</reg>, quæ è .400. detracta ſupererunt .220. <lb/>pro itinere ſecundi, qui .9. diebus iter abſoluit. </s> + <s xml:space="preserve">Ad hæc ſi tempus ſcire velimus <lb/>eius, qui .11. diebus appellit, multiplicabimus .11. cum .180. <reg norm="productumque" type="simple">productumq́;</reg> per .400. <lb/>partiemur, <reg norm="prouenientque" type="simple">prouenientq́;</reg> paulominus, quam quinque dies, nempe .4. cum .22. horis <lb/>et .48. minutis, quod tempus vtrique viatori inſeruiet, quandoquidem idipſum pro <lb/>uenit multiplicato .220. per .9. <reg norm="productoque" type="simple">productoq́;</reg> per .400. diuiſo.</s> + </p> + <p> + <s xml:space="preserve">Huius autem, qui à me pręſcribitur modi, ſpeculatio talis eſt. </s> + <s xml:space="preserve">Duo termini duabus <lb/>rectis lineis æqualibus, & parallelis inter ſe <var>.b.p.</var> et <var>.d.q.</var> ſignificentur, quæ alijs dua-<lb/>bus <var>.b.d.</var> et <var>.q.p.</var> <reg norm="coniungantur" type="simple">coniungant̃</reg>, quę parallelæ & æquales erunt ex .33. primi, quibus ſigni <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/>ſumatur <reg norm="punctum" type="context">punctũ</reg> medium <var>.q.p.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.k.</var> & ab ipſo ad <var>.b.d.</var> ducatur <var>.k.i.</var> parallela <var>.d.q.</var> aut <lb/><var>b.p.</var> quod idem eſt, ex quo <var>.b.i.</var> æqualis erit <var>.p.k.</var> ex .34. primi, hoc eſt <var>.q.k.</var> <reg norm="certique" type="simple">certiq́;</reg> eri-<lb/>mus primum viatorem <var>.q.p.</var> in dimidio itineris <var>.q.k.</var> occurrere non potuiſſe viatori ip <lb/>ſius <var>.b.i.</var> quandoquidem eo tempore, quo is, qui ipſius <var>.q.p.</var> mouetur per <var>.q.k.</var> (cum ſit <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 eſt in <var>.k.</var> ex quo certi erimus eos inter <var>.c.</var> et <var>.i.</var> ſibi in-<lb/>uicem obuiaturos eſſe. </s> + <s xml:space="preserve">Cogito deinde rectam lineam ductam <var>.k.c.</var> & ut ſe habet <var>.i.<lb/>c.</var> ad <var>.c.b.</var> ita cogito ſe 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/>ſtum eſt, lineam <var>.k.c.</var> in puncto <var>.e.</var> interſecabit, à quo cum fuerit ducta <var>.e.o.n.</var> parallela <lb/><var>k.i.</var> habebimus <var>.o.n.</var> ea ſcilicet puncta, quibus occurrunt ſibijpſis, nam cum ſic ſe 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 ſimilitudine manifeſta <lb/>triangulorum, ex æqualitate proportionum ſic ſe 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> ſpatia ſint tempori-<lb/>bus æqualibus confecta, itaque ſpatia <var>.k.n.</var> et <var>.c.o.</var> ex communi ſcientia temporibus <lb/>æqualibus conficientur.</s> + </p> + <p> + <s xml:space="preserve">Quare rectè dicimus, ſi 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> cognoſcitur, quo ex dimidio itineris detracto, remanet <var>.c.i.</var> cogni <lb/>tus, ſed cum probauerimus <var>.q.k.</var> ad <var>.k.n.</var> hoc eſt <var>.i.o.</var> (cum ſint æquales inter ſe, ex .34 <lb/>primi) ita ſe habere. vt <var>.b.c.</var> ad <var>.c.o.</var> permutando ſic ſe 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 ſi ſumma <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/>ſcitur <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 ſi <var>.b.d.</var> tot diebus abſolui <lb/>tur, aut etiam <var>.q.p</var>: quo <var>.b.o.</var> aut <var>.q.n.</var> abſoluetur.</s> + </p> + <p> + <s xml:space="preserve">Vt autem ad ſpeculationem regulæ antiquorum deueniamus, cogitemus pri-<lb/>mum viatorem ipſius <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> abſoluit. </s> + <s xml:space="preserve">Is autem ad <var>.g.</var> pertingat, ex quo eadem pro-<lb/>portio ſpacij <var>.q.g.</var> ad <var>.q.p.</var> hoc eſt <var>.b.d.</var> dabitur, quæ temporis quo <var>.b.d.</var> abſoluitur 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> ſolum, qui per <var>.q.p.</var> mouetur (mo-<lb/>tus enim continui regulares & vniformes conſtituuntur) 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 ſe habere <var>.k.n.</var> ad <var>.c.o.</var> vt <var>.q.k.</var> ad <var>.b.c.</var> itaque ſic ſe 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 ſe habere <var>.q.k.</var> ad <var>.k.n.</var> vt <var>.b.c.</var> ad <var>.c.o.</var> ex quo <lb/>componendo ſic ſe 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 eſt <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 eſt <var>.b.d.</var> ad <var>.b.o.</var> vt ſumma <reg norm="dierum" type="context">dierũ</reg> vnius & <lb/>alterius viatoris ad <reg norm="minorem" type="context">minorẽ</reg> <reg norm="numerum" type="context">numerũ</reg> <reg norm="dierum" type="context">dierũ</reg> velocioris. </s> + <s xml:space="preserve">Breuiter <reg norm="itaque" type="simple">itaq;</reg> obtineremus in <lb/><reg norm="tentum" type="context">tentũ</reg> <reg norm="quando" type="wordlist">qñ</reg> diceremus ſi ſumma dierum, quibus iter agitur à viatoribus talis eſt (20) re-<lb/>ſpectu numeri dierum velocioris(9) qualis & cui <reg norm="reſpondebit" type="context">reſpõdebit</reg> totum ſpacium <var>.b.d</var>? </s> + <s xml:space="preserve">vn-<lb/>de dabitur ſpacium <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 poſtmodum diuidi productum per ſummam omnium <lb/>dierum, rectèid quidem fit. </s> + <s xml:space="preserve">Nam cum ſic ſe habeat <var>.b.d.</var> ad <var>.b.o.</var> vt ſumma omnium <lb/>dierum ad minorem quantitatem dierum velocioris ſcilicet. </s> + <s xml:space="preserve">Ideo temporis propor <lb/>tio à mobili per <var>.b.d.</var> abſumpti ad tempus mobilis per <var>.b.o.</var> eadem erit, quæ ſummæ <lb/>omnium dierum ad namerum dierum velocioris. </s> + <s xml:space="preserve">Quarerectè dicemus, ſi eiuſmodi <lb/>ſumma talem reſpectum habet ad minorem numerum dierum, quem numerum re-<lb/>ſpiciet dies ipſius <var>.b.d</var>? </s> + <s xml:space="preserve">ex quo proferentur dies reſpondentes ipſi <var>.b.o.</var> cætera iam <lb/>dicta fuerunt.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi verò ſpeculationis 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> ſit ſubſcripta <lb/>figura pars <reg norm="inquam" type="context">inquã</reg> pręcedentis, in qua <lb/><reg norm="conſtituamus" type="context simple">cõſtituamꝰ</reg> <var>.o.n.</var> <reg norm="locum" type="context">locũ</reg> <reg norm="eum" type="context">eũ</reg> eſſe quo ſibi <lb/>viatores obuient, ex quo ſpacium <var>.q.<lb/>n.</var> à ſuo viatore conficietur, eo ipſo <lb/>tempore, quo à ſuo ſpacium <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. proficiſcitur, & componendo <reg norm="eadem" type="context">eadẽ</reg> <lb/>erit proportio <var>.d.b.</var> ad <var>.b.o.</var> quæ ſum-<lb/>mæ dierum ad minorem numerum ipſorum, & eadem quæ dierum <var>.b.d.</var> ad dies <lb/>ipſius <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 ipſa itinera aliud quæritur peruenuſtè, in quo quæſito illud con<lb/>ſtituitur cognitum eſſe, nempe interuallum inter duo diuerſa loca, è quibus <lb/>duo viatores eodem inſtanti vt ſibi occurrant proficiſcuntur, <reg norm="certaque" type="simple">certaq́;</reg> milliaria ſin-<lb/>gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur <lb/>deinde quoto die ſibi occurrent. </s> + <s xml:space="preserve">Hoc autem fit diuiſo toto interuallo locorum per <lb/>ſummam milliariorum quam vterque quotidie abſoluit.</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, diſtant loca .100. milliaribus à ſe inuicem; </s> + <s xml:space="preserve">vnus autem viator <lb/>ſingulis diebus .15 milliaria, alter .10. conficit ſi ita que .15. cum .10. <reg norm="coniungamus" type="context">coniũgamus</reg>, <lb/>ſumma erit .25. per <reg norm="quam" type="context">quã</reg> diuiſis milliaribus .100. totius interualli proferetur .4. nume <lb/>rus quæſitus dierum quo viatoribus iter agendum erit prius quam ſibi obuient.</s> + </p> + <p> + <s xml:space="preserve">In cuius ſpeculationis gratiam totum iter ſignificetur 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> ſit occurſus ita vt eodem <lb/>tempore, alter ſpacium <var>.a.i.</var> alter <var>.u.i.</var> confecerit, ſpacij autem <var>.a.e.</var> tempus <lb/>per <var>.b.</var> ſignificetur & tempus ſpacij <var>.u.o.</var> per <var>.c.</var> quæ tempora erunt inter ſe <lb/>æqualia, porrò ſpacij <var>.a.i.</var> tempus per <var>.d.</var> & ſpacij <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 ipſius <var>.b.</var> ad iter ipſius <var>.c.</var> quæ <lb/>itineris <var>.d.</var> ad iter ipſius <var>.f.</var> & componendo itinerum ipſius <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"/> + ipſius <var>.f.</var> meritò itaque quęritur ſi 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> ſed <var>.c.</var> <lb/>ſignatum eſt pro vna die, </s> + <s xml:space="preserve">quare in pro <lb/>poſito exemplo <var>.f.</var> ſignificabit 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 ſummæ, rurſus eiuſdem ſummę <lb/>dimidium adiungere, tum quærimus, quoties noueratius totam eam ſummam ingre <lb/>diatur patefactis fractis ſi qui occurrant.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi quis cogitaſſet numerum .12. iubebant huic dimidium addi, <lb/>nempe .6. ex quo ſumma erat .18. iubebant, præterea dimidium huius ſummæ nem-<lb/>pe .9. toti ſummæ adiungi, quæ fuiſſet .27. adhæc quærebant ſibi patefieri quoties <num value="9">.<lb/>9.</num> ſummam prædictam ingrederetur, & ſi in prima aut ſecunda diuiſione aut <reg norm="etiam" type="context">etiã</reg> <lb/>vtraque, fracti reperirentur, ac quoties nouem vltimam ſummam ingrediebatur, <lb/>toties .4. multiplicabant. </s> + <s xml:space="preserve">Quod ſi in prima diuiſione fracti erant, vltimo produ-<lb/>cto addebant vnitatem; </s> + <s xml:space="preserve">ſin verò in ſecunda, binarium adiungebant, ex quo exa-<lb/>ctus numerus quæſitus proferebatur.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ratione ſit <var>.a.</var> numerus cogitatione compræhenſus et <var>.e.</var> ipſius <var>.a.</var> cum <lb/>eiuſdem medietate ſumma et <var>.i.</var> ipſius <var>.e.</var> cum eiuſdem medietate itidem ſumma, vn <lb/>de <var>.i.e.a.</var> tres numeri continui proportionales, in ſeſquialtera 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 ęqua <lb/>litatis proportionum ita ſe 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. ipſam <var>.i</var>. </s> + <s xml:space="preserve">Sed <lb/>quia ſępe contingit, vt in ſecunda diuiſione, aut in ambabus etiam diuiſionibus re <lb/>periantur numeri fracti, anima duertendum eſt numerum animo compræhenſum <var>.a.</var> <lb/>ſcilicet aut parem aut imparem ſemper futurum. </s> + <s xml:space="preserve">Si par eſt, aut multiplex erit ad <num value="4">.<lb/>4.</num> aut non. </s> + <s xml:space="preserve">Si priori modo ſe habebit in duabus diuiſionibus, nullus numerus fra-<lb/>ctus admittetur, ſed ſi ad .4. multiplex non erit, à multiplicibus per duo ſemper dif <lb/>feret, & ſi per medium diuidatur, eiuſdem medietas impar ſemper 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 ſumma par nunquam exiſtet, cuius medietatem aliquod medium ſemper <lb/>ingredietur, & hanc ob cauſam poſterior ſumma cum fracto ſemper erit, & nume-<lb/>rum deſumptum maiorem eſſe multiplici ad quatuor per duo ſignificabit.</s> + </p> + <p> + <s xml:space="preserve">At verò ſi inter impares reponatur, aut eorum erit qui ſuperant multiplicem <lb/>ipſius quatuor per vnum, ſeu per tria, quod hinc innoteſcet, nempe, quia ſi eorum <lb/>erit qui dictum multiplicem per vnum tantum vincunt, ſua medietate ipſi numero <lb/>addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior <lb/>ſumma in numerum parem ſemper euadet, vnde in poſteriori ſumma nullus nume-<lb/>rus fractus conſpicietur, & hanc ob <reg norm="causam" type="context">causã</reg> multiplici ipſius .4. vnitas ſemper addetur.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi numerus deſumptus, in ſerie eorum, qui multiplicem ipſius .4. pertria ſu-<lb/>perant, collocabitur, hinc compræhendetur, quia primæ ſummæ numerus cum <lb/>media vnitate ſemper impar erit, vnde ſecunda ſumma præter integras cum me-<lb/>dia vnitate nobis ſemper occur ret.</s> + </p> + <p> + <s xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiſione, & ſecunda numerus <lb/>aliquis fractus conſiſtat, eò tantum nobis inſeruit, quò deueniamus in cognitionem <lb/>an numerus animo conceptus multiplicem ipſius .4. per vnum, per duo, aut tria ſupe <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 poteſt, quin aliquis fra-<lb/>ctus in prima diuiſione, aut in ſecunda ſequatur: </s> + <s xml:space="preserve">vnde à numeris imparibus, qui mul <lb/>tiplicem ipſius .4. unitatis tantum exceſſu <reg norm="ſuperant" type="context">ſuperãt</reg>, poſterior ſumma <reg norm="cum" type="context">cũ</reg> quarta parte <lb/>vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipſius <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 ipſius .4. per duo cum medietate vnitatis præter <lb/>integros ſemper procedit. </s> + <s xml:space="preserve">Ita cum is qui numerum ſecum conſiderat, ſi in nume-<lb/> + <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> + ris fractis verſatus eſſet, qui eum in-<lb/>terrogat prudenter ſe gereret, ſi ſibi <lb/>declarari curaret, quis nam ex fractis <lb/>ſu per integros <reg norm="ſecundæ" type="context">ſecũdæ</reg> <reg norm="summæ" type="context">sũmæ</reg> remane <lb/>ret, quia <reg norm="per" type="simple">ꝑ</reg> quot quarta integros <reg norm="ſecun- dæ" type="context">ſecũ-<lb/>dæ</reg> ſummæ ſuperaret, per <reg norm="totidem" type="context">totidẽ</reg> inte <lb/>gros numerus mente conceptus multiplicem ipſius .4. ſuperaret.</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 ſi ali quis quemuis numerum animo compræhendat, eique <lb/>numero alium etiam quemlibet numerum propoſitum addat, & à tertia par <lb/>te huius ſummæ tertiam partem numeri imaginati detrah et, reſiduum ſecundi nu-<lb/>meri adiuncti, ideſt propoſiti, tertia pars erit.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia, ſi aliquis de numero denario cogitaſſet, <reg norm="huicque" type="simple">huicq́;</reg> .24. adderet, <lb/>vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-<lb/>rij cogitatione concepti, ideſt .3. cum tertia parte vnius, à tertia parte huius ſum mæ <lb/>ideſt ab vndecim & vna tertia parte remanerent .8. ideſt tertia pars numeri additi. <lb/></s> + <s xml:space="preserve">Id quod mihi inter iocos in honeſtorum 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 eſt <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> ſit tertia pars ipſius <var>.a.b.</var> prioris numeri im aginati, et. b <lb/>c. tertia pars ipſius, <var>b.d.</var> ſecundi numeri propoſiti, vnde coniunctum vnius harum ter <lb/>tiarum <reg norm="partium" type="context">partiũ</reg> <reg norm="cum" type="context">cũ</reg> alia ſit <var>.e.c.</var> quod quidem <var>.e.c.</var> eſſe tertiam partem ſummæ duorum <lb/>primorum ideſt <var>.a.d.</var> aſſero. </s> + <s xml:space="preserve">Iam manifeſtum eſt ipſius <var>.d.b.</var> ad <var>.b.c.</var> eſſe quemadmo <lb/>dum ipſius <var>.a.b.</var> ad <var>.b.e.</var> vnde viciſſim ipſius <var>.d.b.</var> ad <var>.b.a.</var> erit quemadmodum ipſius <var>.b.<lb/>c.</var> ad <var>.b.e.</var> & coniunctim ipſius <var>.d.a.</var> ad <var>.a.b.</var> quemadmodum ipſius <var>.c.e.</var> ad <var>.e.b.</var> & viciſ-<lb/>ſim ipſius <var>.d.a.</var> ad <var>.c.e.</var> quemadmodum ipſius <var>.b.a.</var> ad <var>.b.e.</var> ſed proportio ipſius <var>.b.a.</var> ad <var>.<lb/>b.e.</var> eſt tripla, ergo ea quæ eſt ipſius <var>.a.d.</var> ad <var>.e.c.</var> erit quoque tripla; </s> + <s xml:space="preserve">vnde ſumendo <var>.e.<lb/>c.</var> pro tertia parte ipſius <var>.a.d.</var> & ab ipſa <var>.e.c.</var> ſubtrahendo tertiam partem ipſius <var>.a.b.</var> <lb/>tertia pars ipſius <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, ſupponendo <var>.e.c.</var> tertiam partem ipſius <var>.a.d.</var> et <var>.e.b.</var> ipſius <var>.a.<lb/>b.</var> exiſter. </s> + <s xml:space="preserve">Dico <var>.b.c.</var> tertiam partem ipſius <var>.b.d.</var> futuram:</s> + <s xml:space="preserve">quia ſi totius <var>.a.d.</var> ad totum <lb/><var>e.c.</var> ita ſe habet, quemadmodum <var>.a.b.</var> à toto <var>.a.d.</var> diffecti atque diuulſi ad <var>.e.b.</var> à toto <var>.<lb/>e.c.</var> diſractum, 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> reſidui <var>.b.d.</var> totius <var>.a.d.</var> ad reſiduum <var>.b.c.</var> <lb/>totius <var>.e.c.</var> erit, vt totius <var>.a.d.</var> ad <reg norm="totum" type="context">totũ</reg> <var>.e.c.</var> at-<lb/>que hic quidem modus rem <reg norm="propoſitam" type="context">propoſitã</reg> ſpe-<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 reſolu-<lb/>ta, proptereà quòd <reg norm="non" type="context">nõ</reg> ſemper nobis ſuccurrit 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 ſecunda dupla ſit primę, & adhuc eam ſuperet tribus vnitatibus, tertia ve <lb/>rò æqualis ſit aggregato primæ cum ſecunda, & amplius ipſum aggregatum ſuperet <lb/>quinque vnitatibus.</s> + </p> + <p> + <s xml:space="preserve">Ad hoc autem quæſitum ſoluendum antiqui vtebantur regula falſi, quod reuera <lb/>breuiori modo poteſt ſolui, videlicet detra hendo illud ſecundum exceſſum, quin-<lb/>que ſcilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eſt .22. cum dimidia <lb/>vnitate, ſi addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-<lb/>tia parte quæſita ipſius numeri .50. deinde ſi 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 eſt .6. cum dimidia vnitate, prima pars, ex tri-<lb/>bus quæſita erit, quæ quidem ſi detraxerimus ex .19. cum dimidia vnitate, reli-<lb/>quum erit .13. cui <reg norm="cum" type="context">cũ</reg> additus fuerit primus exceſſus ideſt .3. </s> + <s xml:space="preserve">Iam propoſitum re-<lb/>ſultabit nobis .16. pro ſecunda parte quæſita.</s> + </p> + <p> + <s xml:space="preserve">Ratio verò huiuſmodi operationis talis eſt, ſit verbi gratia totalis numerus pro-<lb/>poſitus ſignificatus per lineam <var>.a.b.</var> cuius ſecundæ partis numerus datus ſignificetur <lb/>per lineam <var>.g.</var> & numerus tertiæ partis propoſitus 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 diuiſa in pun <lb/>cto <var>.e.</var> & ipſi <var>.e.f.</var> addita <var>.f.b.</var> tota <var>.e.b.</var> nobis cognita erit, quæ quidem tertia pars <lb/>quæſita ipſius <var>.a.b.</var> erit, proptereà quòd <var>.a.e.</var> (quæ æqualis eſt ipſi <var>.e.f.</var>) erit ſumma <lb/>primæ, & ſecundæ partis. </s> + <s xml:space="preserve">Detrahatur poſteà. g. ex <var>.e.a.</var> & remanebit <var>.d.a.</var> cuius ter <lb/>tia pars ſit <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 ſecundam 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> compoſita eſt ex <var>.<lb/>e.f.</var> æquali <var>.a.e.</var> hoc eſt æquali compoſi-<lb/>to ex prima, & ſe 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 ſint horæ, alius verò reſpondit tot eſſe, quot duæ tertiæ præteriti <lb/>temporis ſimul iuncta cum tribus quintis futuri temporis totius dieri naturalis effi-<lb/>ciunt. </s> + <s xml:space="preserve">Nunc quæritur quot ſint horę.</s> + </p> + <p> + <s xml:space="preserve">Antiqui, hoc etiam problema ſoluebant mediante regula falſi, ſed mihi alio mo <lb/>do ſoluendum 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">cũ</reg> quin-<lb/>que futuri temporis, vnde veniunt .14. vnitates, ex regula poftea de tribus ita dico <lb/>ſi ex .14. mihi prouenit .9. quid reſultabit ex .24. & prouenient mihi horæ .15. cum <lb/>tribus ſeptimis vnius horæ, hoc eſt minuta ferè .26.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius ratione, quinque vnitates, feu partes temporis futuri ſignificentur à <lb/>linea <var>.e.u.</var> quarum trium ſigniſicentur a linea <var>.e.i.</var> ſumpta deinde ſit linea <var>.e.o.</var> æqualis <lb/>lineæ <var>.e.i.</var> et <var>.e.a.</var> tripla ſit ad <var>.o.e.</var> vel ad <var>.e.i.</var> quod idem eſt, vnde <var>.a.e.</var> compoſita erit <lb/>ex <var>.a.o.</var> (hoc eſt ex duabus tertijs ip ſius <var>.a.e.</var>) & ex <var>o.e.</var> (hoc eſt ex. tribus quintis ip-<lb/>ſius <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. ſeptimi 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 ſocios nummos habere, quorum ſumma primi & <lb/>ſecundi cognita ſit, item ſumma primi & tertij cognita & ſumma ſecundi & <lb/>tertij item cognita, at que ex huiuſmodi tribus aggregatis veniunt in cognitionem <lb/>particularem vniuſcuiuſque illorum.</s> + </p> + <p> + <s xml:space="preserve">Gemafriſius ſoluit hoc problema ex regula ſalſi. </s> + <s xml:space="preserve">At ego tali ordine progredior. <lb/></s> + <s xml:space="preserve">Sit verbi gratia, ſumma primi cum ſecundo .50. & ſecundi cum tertio .70. & primi <lb/>cum tertio .60. harum trium ſummarum accipiantur duæ quæuis, vt puta .50. & .70 <lb/>quæ coniunctæ ſimul dabunt .120. à qua ſumma detrahatur reliqua, ideſt .60. & <lb/>reſtabit nobis .60. cuius medietas erit .30. hoc eſt numerus nummorum ſecundi <lb/>ſocij quo numero detracto à .70. hoc eſt à ſumma ſecundi cum tertio remanebit .40. <lb/>hoc eſt numerus tertij ſocij, & hic numerus deſumptus à .60. reſiduus erit nume-<lb/>rus primi ſocij.</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 conſideremus triangulum hic ſubnotatum <var>.a.b.c.</var> cuius <lb/>unumquodque latus ſignificet ſummam duorum ſociorum, vtputa latus <var>.a.b.</var> ſignifi-<lb/>cet ſummam primi cum ſecundo, latus verò <var>.b.c.</var> ſummam ſecundi cum tertio, la-<lb/>rus autem <var>.a.c.</var> ſummam primi cum tertio, et <var>.a.e.</var> ſeu <var>.a.o.</var> ſit numerus primi ſocij, et <var>.<lb/>e.b.</var> vel <var>.b.u.</var> ſit ſecundi ſocij, et <var>.c.u.</var> ſeu <var>.c.o.</var> ſit tertij, cum autem <var>.a.e.</var> æqualis ſit <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 ſuppoſito ſi <reg norm="dempta" type="context">dẽpta</reg> fuerit ſumma ſeu <lb/>latus <var>.a.c.</var> datum ex aggregato laterum <var>.<lb/>a.b.</var> cum <var>.b.c.</var> reliquarum ſummarum, 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> ſiue <var>.b.u.</var> nobis cognita erit, qua <lb/>detracta exſumma <var>.b.a.</var> relinquetur no <lb/>bis cognitus numerus <var>.a.e.</var> detracto ve-<lb/>ro numero <var>.a.e.</var> hoc eſt <var>.a.o.</var> ex <var>.a.c.</var> ſum-<lb/>ma, ſeu latus, aut <var>.b.u.</var> ex <var>.b.c.</var> remanebit <lb/><var>o.c.</var> ſeu <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="ſolum" type="context">ſolũ</reg> de tribus ſocijs, ſed <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"/> + ſint ſex ſocij <var>.a.b.c.d.e.f.</var> quorum ſumma per binos co-<lb/>gnita, vtputà ſumma numeri <var>.a.</var> cum <var>.b.</var> cognita nobis ſit, <lb/>& ſumma numeri <var>.b.</var> cum <var>.c.</var> & ſumma <var>.c.</var> cum <var>.d.</var> & ſum-<lb/>ma <var>.d.</var> cum <var>.e.</var> & ſumma <var>.e.</var> cum <var>.f.</var> neceſle eft etiam ſcire <lb/>ſummam duorum vno relicto, vtputa ſummam <var>.a.</var> cum <lb/>c. vt poſſimus triangulum <var>.a.b.c.</var> conſtituere. </s> + <s xml:space="preserve">Vnde ex <lb/>præmiffa, cognitus numerus nobis erit vniuſcuiuſque <var>.a.<lb/>b.c</var>. </s> + <s xml:space="preserve">Quapropter dempto numero <var>.c.</var> ex ſumma <var>.c.</var> cum <lb/>d. & numero <var>.d.</var> ex ſumma <var>.d.</var> cum <var>.e.</var> & numero <var>.e.</var> ex ſum <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 ſcribit Vitruuius, proderet, con-<lb/>templarer, mihi etiam viſum eſt, vt aliquem modum ſcientiſicum inueſtigarem, quo <lb/>proportio auri ad argentum, quod in aliquo propoſito corpore exipſis miſto cogni <lb/>ti ponderis cognoſci poſſet. </s> + <s xml:space="preserve">Et cum multos diuerſis temporibus excogitarim offi-<lb/>cio meo deeſſe nolui in ijſdem literarum monumentis mandandis, quorum hic <lb/>vnus erit: </s> + <s xml:space="preserve">propoſita nobis ſint tria corpora <var>.A.M.V.</var> æqualia inter ſe, ſed diuer-<lb/>ſarum ſpecierum materiei, vtputa quod <var>.A.</var> ſit argenteum, & omogeneum <var>.V.</var> ve-<lb/>rò aureum omogeneum, & M. mixtum exauro, & argento, ideſt heterogeneum, <lb/>cupimusergo ſcire <reg norm="iuſtam" type="context">iuſtã</reg> quantitatem auri & argenti, quæ eſt in ipſo corpore <var>.M.</var> <lb/>miſto. </s> + <s xml:space="preserve">Ita igitur faciamus. </s> + <s xml:space="preserve">Videamus primum quantum ſit pondus vniuſcuiuſque <lb/>ipſorum 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> miſti <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 ſeruan-<lb/>da, dematur etiam pondus <var>.M.</var> ex pondere <var>.V.</var> reliquum erit .18. pro ſecunda diffe-<lb/>rentia, etiam ſeruanda, multiplicetur poſteà pondus <var>.A.</var> per ſecundam 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 ipſius corpo-<lb/>ris miſti <var>.M.</var> quo etiam detracto ex pondere totali ipſius <var>.M.</var> reliquum erit quanti-<lb/>tas auri eius corporis, hoc eſt .180.</s> + </p> + <p> + <s xml:space="preserve">In cuius operationis ſpeculatione, aliquid natura ſua prius cognitum præcedere <lb/>oportet hoc eſt, quod omnia corpora omogenea eandem proportionem obtinent <lb/>inter quantitates, quam inter pondera. </s> + <s xml:space="preserve">Quo ſuppoſito 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>: ſed <var>.e.o.</var> ſignificet par-<lb/>tem argenti, et <var>.o.u.</var> partem auri in corpore miſto <var>.M.</var> vnde ex communi conceptu <lb/>habebimus <var>.o.e.</var> æqualem <var>.u.c.</var> cum ex hypotheſi <var>.e.u.</var> æqualis ſit <var>.o.c.</var> et <var>.a.o.</var> ſimiliter. <lb/></s> + <s xml:space="preserve">Significetur poſteà 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> ſit diffe-<lb/>rentia, qua <var>.f.g.</var> maior eſt .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 eſt <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> ſeu <lb/><var>f.g.</var> (quodidem eſt) 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/>ſeparando 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 eſt, vt differentia in <lb/>ter <var>.f.</var> g, et <var>.f</var>: cognita nobis eſt etiam <var>.f</var>: cognoſcimus itidem <var>.d.</var> vt differentiam inter <var>.<lb/>x.b.d.</var> et <var>.b.x.</var> quapropter cognoſcemus <var>.b.</var> ex .20. ſeptimi Eucli. & ſic <var>.x.</var> reſiduum. <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 propoſiti deuenire poſſumus in cognitio-<lb/>nem veræ quantitatis auri, & argenti confuſi in corona Hieronis conſtituen-<lb/>do primum duo corpora ſimplicia æqualia inter ſe, & coronæ hoc modo videlicet, <lb/>immergendo coronam, ſeu corpus miſtum in aliquod vas aqua plenum, & diligen-<lb/>ter colligere aquam, quæ ex eo effundetur, poſteà verò oportet, aliud vas inuenire <lb/>præciſæ capax illius a quæ collectæ, in quod demum infundatur tantum auri, & po-<lb/>ſteà tantum argenti, quantum ſieri poteſt, vnde vnumquodque horum duorum cor <lb/>porum ſimplicium æquale erit mixto, ſeu coronæ, & ſic quod dictum eſt 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 propoſito di-<lb/>ctum eſt, quædam theoremata præmittenda ſunt, videlicet quòd quotíeſcunque <lb/>fuerint tria corpora, quorum duo inuicem æqualia ſint in quantitate, ſed diuerſa- + <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 ſpecierum materiæ, tertium verò corpus maius, vel minus ſit in quantitate vtro-<lb/>que illorum, ſed eiuſdem 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 cuiuſuis æqualium.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſit <var>.b.</var> corpus aliquod aureum æquale corpori <var>.u.</var> argenteo, ſit <lb/>etiam corpus <var>.a.</var> argenteum maius corpore <var>.b.</var> vel <var>.u.</var> ſed ponderis eiuſdem, 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 eſt 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 ſit, 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. eiuſdem 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 eſt 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">QVotieſcunque nobis propoſita fuerint duo corpora cuiuſuis magnitudinis æ-<lb/>que ponderantia, ſed diuerſarum ſpecierum materiæ, cum ſcire volueri-<lb/>mus proportionem ponderum illarum ſpecierum inter ipſas hoc modo faciemus.</s> + </p> + <p> + <s xml:space="preserve">Sint exempli gratia, duo nobis propoſita corpora <var>.a.</var> et <var>.b.</var> (vt dictum eſt) quæ ſi <lb/>fuerint æqualium magnitudinum inter ſe, clarum erit quod quæritur, ſed inæqua-<lb/>lia erunt, immergatur <reg norm="unumquodque" type="simple">unumquodq;</reg> eorum in vas aqua plenum, & collecta ſit aqua <lb/>effuſa ab vnoquoque illorum, </s> + <s xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iſtarum aquarum æqualis magnitudi-<lb/>nis erit ſui corporis impellentis, & proportio ponderoſitatis illarum eadem erit, <lb/>quæ earum magnitudinum ex omogeneitate, quapropter ſi vnamquamque illarum <lb/>ponderabimus, habebimus propoſitum 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 ſcire voluerimus pondus alicuius magnitudinis aquæ æqualis alicui <lb/>corpori ponderoſo, breuiſſimus modus erit ponderando ipſum corpus tam in ae-<lb/>re, quàm in aqua, & quia ſemper leuius erit in aqua, </s> + <s xml:space="preserve">tunc differentia ponderum ip-<lb/>ſius corporis, erit pondus quæſitum, hoc eſt vnius corporis aquei æqualis magnitu-<lb/>dinis magnitudini corporis propoſiti ex <ref id="ref-0014">.7. propoſitione lib. Archimedis de inſi-<lb/>dentibus aquæ</ref>. </s> + </p> + <p> + <s xml:space="preserve">Quare ex præmiſſis quotieſcunque immerſa fuerint in aquam dicti vaſis duo cor <lb/>pora æquè ponderantia, ſed diuerſarum ſpecierum, vt dictum eſt, proportio pon-<lb/>deris aquæ maioris ad pondus aquæ minoris magnitudinis eadem ſemper erit, quæ <lb/>ponderis minoris corporis ad pondus alicuius corporis eidem æqualis, ſpeciei verò <lb/>maioris, vel eadem proportio ponderis alicuius corporis æqualis maiori, ſpeciei ve <lb/>rò minoris ad pondus ipſius maioris.</s> + </p> + <p> + <s xml:space="preserve">Vt puta ſit 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">aqueũ</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 eſt, 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> eſt vt <lb/>ponderis <var>.a.</var> ad pondus <var>.u.</var> ex corporum omogenei-<lb/>tate, & ex æqualitate magnitudinum corporum antecedentium & conſequentium.</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 quotieſcumque fuerint duo corpora aquea, quo-<lb/>rum vnum æqualis magnitudinis ſit alicui miſto, quod quidem miſtum graue <lb/>ſit tam in aere, quàm in aqua, alterum verò corpus aquem æqualis ſit magnitudi-<lb/>nis alicui corpoli ſimplici, quod quidem corpus ſimplex æqualis ponderis ſit dicto <lb/>corpori miſto. </s> + <s xml:space="preserve">Tunc proportio ponderis aquei, cuius magnitudo æquatur magni <lb/>tudini corporis miſti, ad pondus corporis aquei, cuius magnitudo æqualis eſt ma-<lb/>gnitudini corporis ſimplicis, eadem erit, quæ proportio ponderis alicuius corpo-<lb/>ris ſimplicis, cuius magnitudo æqualis ſit magnitudini corporis miſti ſuperius dicti, <lb/>ſed ſpeciei corporis ſimplicis iam dicti, ad pondus dicti miſti.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſit corpus aqueum <var>.e.</var> magnitudinis æqualis corpori <var>.m.</var> mixto, <lb/>corpus verò aqueum <var>.i.</var> æqualis magnitudinis ſit corpori ſimplici <var>.a.</var> quod quidem <lb/>corpus <var>.a.</var> æqualis ponderis ſit cum corpore <var>.m.</var> & corpus <var>.u.</var> ſit æqualis magnitudinis <lb/>cum corpore <var>.m.</var> ſed ſpeciei 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 eſt, quin <lb/>eadem proportio ſit magnitudinis <var>.e.</var> ad magnitudinem <var>.i.</var> quæ magnitudinis <var>.m.</var> ad <lb/>a. ſed <var>.m.</var> ad <var>.a.</var> eſt vt <var>.u.</var> ad <var>.a.</var> ex .7. quinti </s> + <s xml:space="preserve">quare <lb/>ex .11. eiuſdem 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 ipſius magnitudinibus loquendo, ſed propor-<lb/>tio ponderis <var>.u.</var> ad pondus <var>.a.</var> eadem eſt, 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 eſt quæ ponderis <var>.u.</var> ad pondus <var>.a.</var> ex .7. quinti, ergò ex .11. <lb/>eiuſdem 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 eſt propoſitum.</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 cognoſcendam proportionem duarum diuerſarum ſpecierum in <lb/>corpore miſto propoſito, tribus corporibus aqueis mediantibus, quæ <reg norm="quidem" type="context">quidẽ</reg> <lb/>corpora æqualium magnitudinum ſint alijs tribus corporibus vnius & eiuſdem pon <lb/>deris, quorum vnum ſit mixtum, reliqua verò duo ſimplicia, ſed ſpecierum 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/>ſitquinque librarum, quorum vnum ſit aureum, aliud argenteum, reliquum verò <lb/>mixtum ex ijs duobus metallis, vnde corpus aureum ſimplex minus erit, & argen <lb/>teum maius corpore mixto, quod nulli dubium eſt, ſit nunc pondus corporis aquei <lb/>ęqualis corpori aureo, <reg norm="librarum" type="context">librarũ</reg> .3. aquei verò ęqualis miſto, ſit <reg norm="librarum" type="context">librarũ</reg> 3. <reg norm="cum" type="context">cũ</reg> quarta par <lb/>te, aquei demum æqualisargenteo, librarum .4. cum dimidia, vnde exijs, quæ in præ <lb/>cedenti theoremate, & in .126. theoremate diximus, ſi imaginatione concipiemus <lb/>alia duo corpora ſimplicia, auri, & argenti, ſed æqualium magnitudinum mixto, <lb/>habebimus proportionem ponderis aurei ad pondus corporis mixti vt <reg norm="trium" type="context">triũ</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 eſt aurei ad mixtum, vt .13. ad .12. & mixti ad argenteum, vt .18. ad .13. <lb/>& aurei ad argenteum, vt .3. ad .2. ideſt, vt .18. ad .12.</s> + </p> + <p> + <s xml:space="preserve">Nunc inueniantur duo numeri ita inter ſe proportionati, vt .3. ad .2. habentes ta-<lb/>men inter ipſos numerum ita proportionatum ad maximum, vt .12. ſe habet ad <num value="13">.<lb/>13.</num> & ita proportionatum ad minimum, vt ſe habet .18. ad .13. quod hoc modo in-<lb/>ueniemus, multiplicabimus .18. per .12. & proueniet nobis .216. pro numero me-<lb/>dio, poſteà multiplicabimus .18. per .13. & proueniet .234. pro maximo, <reg norm="demum" type="context">demũ</reg> multi <lb/>plicando .12. per .13. proueniet .156. pro minimo, ita quod .234. correſpondebit <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, ſi ordinem theo <lb/>rematis .122. ſequemur, 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. ſed quia ſuppoſitum fuit corpus mixtum eſſe quinque librarum, propte-<lb/>reà dicemus. </s> + <s xml:space="preserve">Si .216. hoc eſt toti corpori mixto correſpondent quinque libræ tunc <lb/>parti .180. hoc eſt auro in ipſo corpore mixto, correſpondent libræ .4. cum duabus <lb/>vncijs, ex regula detribus, reſiduum verò quinque librarum, ideſt vnciæ decem, <lb/>correſpondent parti .36. hoc eſt argento in dicto corpore mixto.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi 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">cũ</reg> pon <lb/>dus mixti fuiſſet .120. librarum, </s> + <s xml:space="preserve">tunc aurum ipſius eſſet librarum .78. argentum ve-<lb/>rò librarum .42. ex eadem regula.</s> + </p> + <p> + <s xml:space="preserve">Pro quarum rerum ſpeculatione nil aliud oportet nunc dicere cum ſatis dictum à no <lb/>bis ſuperius fuerit, vno excepto, hoc eſt rationem reddere, qua motus fui ad inue <lb/>niendos illos .3. numeros ita inter ſe diſpoſitos, vt dictum eſt, quæ quidem ratio fuit, <lb/>vt haberemus .3. numeros ita inter ipſos ordinatè diſpoſitos, vt ſunt 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, reſpectu extremorum, vnde medius numerus in <lb/>ſuo vero ſitu inter .18. et .12. fuiſſent .16. <reg norm="cum" type="context">cũ</reg> .8. tertijs decimis, ſed vt <reg norm="fractorum" type="context">fractorũ</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 eſt .18. ad .12. ex .18. ſepti <lb/>mi, iuſſi etiam multiplicari .18. per .12. vt nobis prodiret .216. ad quem numerum, <lb/>numerus .234. ita ſe haberet, ut .13. ad .12. ex .19. ſeptimi, quod autem ita ſit 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. maniteſtum eſt exijſdem, 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">ſupponitur obſidio alicuius loci, vbi <lb/>alimento ad nutriendos .10000. homines ſufficiunt pro quinque menſibus tan-<lb/>tum, ſed quia eum locum obſidione non liberari putatur niſi .18. menſibus exactis, <lb/>quæritur, quot homines eo tempore illis alimentis nutriri poſſint, hoc eſt .18. <lb/>menſibus.</s> + </p> + <p> + <s xml:space="preserve">Præcipitregula, vt multiplicetur primus numerus, hoc eſt hominum .10000. cum <lb/>ſecundo, hoc eſt menſium quinque, productum verò diuidatur per .18. hoc eſt men-<lb/>ſium, </s> + <s xml:space="preserve">tunc proueniet .2777. cum .7. nonis.</s> + </p> + <p> + <s xml:space="preserve">Cuius operationis ratio eſt hæc, ſint exempli gratia duo hic ſubſcripta producta <lb/>ſuperficialia <var>.a.n.</var> et <var>.o.u.</var> inuicem æqualia, ſed tal@ figura delineata, vt proportio <var>.u.<lb/>x.</var> ad <var>.x.o.</var> ſit, vt .10000. ad quinque, & proportio <var>a.x.</var> ad <var>.x.o.</var> ſit vt .18. ad quinque, <lb/>ct <var>.x.n.</var> ſit nobis ignota, quæ quidem eſt illa, quæ indagatur, ita <reg norm="quod" type="simple">ꝙ</reg> vnumquodque <lb/>iſtorum productorum ſignificabit alimentum, et <var>.u.x.</var> ſignificabit numerum homi-<lb/>num .10000. qui quidem homines comederent totum alimentum <var>.u.o.</var> ſpacio tem-<lb/>poris <var>.x.o.</var> quinque menſium, proptereà quòd <var>u.o.</var> ſupponitur productum eſſe ab <var>.<lb/>u.x.</var> in <var>.x.o</var>. </s> + <s xml:space="preserve">Deinde <reg norm="ſupponendo" type="context">ſupponẽdo</reg> <var>.a.x.</var> tem <lb/> + <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> + pus eſſe .18. menſium, ergo <var>.x.n.</var> ſignifi-<lb/>cabit numerum hominum, qui eo tem-<lb/>poris ſpacio ali poſſunt, hoc eſt <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. ſexti, ſeu ex, 20. <lb/>ſeptimi proportio <var>.x.u.</var> ad <var>.x.n.</var> <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quę <var>.a.x.</var> ad <var>.x.o.</var> quapropter rectè factum <lb/>erit accipere <reg norm="productum" type="context">productũ</reg> <var>.u.o.</var> quodidem <lb/>eſt in quantitate, quod productum .2. n. & ipſum 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">QVotieſcunque nobis propoſitum fuerit inuenire tertium terminum, trium ter <lb/>minorum continuè proportionalium armonicæ proportionalitatis, quo-<lb/>tum duo nobis cogniti ſint, 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 ſint 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 ſumma <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 eſt 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. ſeptimi, proptereà quòd proportio <var>.q.o.</var> ad <var>.o.p.</var> hoc eſt ad <var>.<lb/>d.p.</var> eſt vt <var>.a.g.</var> ad <var>.g.n.</var> coniunctim cum diſiunctim it a ſit <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> (ideſt ad <var>.e.c.</var>) ita ſe <reg norm="hent" type="context">hẽt</reg> ut <var>.p.o.</var> (hoc eſt <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 ſi 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 ſi <var>.e.c.</var> tertius terminus nobis propoſitus eſſet ſimul cum <var>.a.g.</var> medio, & volue <lb/>rimus maiorem inuenire <var>.q.p.</var> ſcilicet, oportebit <var>.e.c.</var> ex <var>.a.g.</var> detrahere, differentiam <lb/>verò <var>.n.g.</var> ſimiliter 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 reſiduo <var>.c.t.</var> me-<lb/>diante diuidemus productum, <reg norm="quod" type="simple">ꝙ</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">ꝙ</reg> <lb/><reg norm="productum" type="context">productũ</reg> quod ſit ex <var>.e.t.</var> in <var>.d.p.</var> <lb/>æquale eſt producto quòd fit ex <var>.a.g.</var> in <var>.t.c.</var> per 20. ſeptimi Eucli. eo quòd <var>.a.g.</var> (id-<lb/>eſt <var>.q.d.</var>) ad <var>.d.p.</var> eſt ut <var>.e.t.</var> ad <var>.t.c.</var> diſiunctim, cum coniunctim ita ſit <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> eſt vt <var>.d.p.</var> ad <var>.t.c.</var> hoc eſt 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">Propoſiti enim cum nobis fue <lb/>rint duo termini <var>.c.e.</var> minimus et <var>.g.a.</var> medius, maximus verò quærendus ſit, de <lb/>trahatur differentia <var>.g.n.</var> ex <var>.e.c.</var> & per reſiduum <var>.e.t.</var> diuidatur productum <reg norm="quod" type="simple">ꝙ</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. ſeptimi 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">ꝙ</reg> naſcitur ex <var>.p.d.</var> (hoc eſt <var>.a.g.</var>) in <var>.e.c.</var> æquale eric producto <var>.e.t.</var> in <var>.q.p.</var> qua-<lb/>propter ſi diuiſerimus 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">cũ</reg> nobis propoſiti fuerint duo termini <var>.q.p.</var> maximus, et <var>.a.g.</var> medius, ſi <reg norm="mini- mum" type="context">mini-<lb/>mũ</reg> <var>.e.c.</var> <reg norm="voluerimus" type="simple">voluerimꝰ</reg> inuenire. </s> + <s xml:space="preserve">Termino <var>.q.p.</var> maximo, <reg norm="iungatur" type="context simple">iũgat̃</reg>. <var>p.o.</var> ęqualis, <var>p.d.</var> <reg norm="differentię" type="context">differẽtię</reg> <lb/>propoſitæ, diuidatur poſtea productum <reg norm="quod" type="simple">ꝙ</reg> ex <var>.q.p.</var> in <var>.a.g.</var> generatur per <var>.q.o.</var> prouen <lb/>tus autem ſit <var>.e.c.</var> qui quidem erit terminus quæſitus.</s> + </p> + <p> + <s xml:space="preserve">Cuius operationis ſpeculutio hæc erit, ſupponatur terminum <var>.e.c.</var> inuentum eſſe <lb/>vnde <var>.n.g.</var> differentia ſit 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">productũ</reg> quòd fit ex <var>.a.g.</var> <lb/>in <var>.q.p.</var> (ex .20. ſeptimi) æquale erit <lb/>producto <var>.q.o.</var> in <var>.a.n</var>. </s> + <s xml:space="preserve">Quare ſi diuiſum 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 ſit inuenire tertium <lb/>terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com <lb/>municantibus in tali proportionalitate, hoc eſt harmonica. </s> + <s xml:space="preserve">Vthoc oſtendatur.</s> + </p> + <p> + <s xml:space="preserve">Sint duo termini dati <var>.a.o.</var> et <var>.a.e.</var> inter ſe communicantes, tertius verò inuentus <lb/>ſit <var>.a.c.</var> qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem <lb/>eſſe cum primis datis.</s> + </p> + <p> + <s xml:space="preserve">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-<lb/>dem proportionem eſſe <var>.a.c.</var> ad <var>.a.o.</var> quæ eſt <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. eiuſdem <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/>eſt propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi datus fuerit maximus <var>.a.c.</var> cum medio <var>.a.e.</var> interſe 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 ſuppoſito). <var>a.c.</var> commu-<lb/>nicat cum <var>.e.c.</var> hoc eſt 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">erũt</reg> <lb/> + <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> + inter ſe 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> ſimiliter.</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 ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide <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, reſiduum verò <var>.q.x.</var> multiplicetur per <var>.b.r.</var> productum poſteà 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. ſeptimi <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 abſque aliqua <lb/>diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet. <lb/></s> + <s xml:space="preserve">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram <reg norm="proportionem" type="context">proportionẽ</reg> inuenian-<lb/>tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc <lb/>eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-<lb/>dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum + <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 eſt quod fit ex quinque in .3. quod erit .15. </s> + <s xml:space="preserve">Vt <lb/>autem medium terminum harmonicum inter iſtos habeamus, accipiatur <reg norm="duplum" type="context">duplũ</reg> pro-<lb/>ducti, quod fit ex primis minimis terminis, quod erit .12.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ſpeculatio eſt iſta: </s> + <s xml:space="preserve">ſignificentur duo termini datæ proportionis ab <var>.q.b.</var> <lb/>et <var>.b.r.</var> quorum ſumma erit <var>.q.r.</var> cuius quadratum ſit <var>.q.o.</var> ſit etiam imaginata <var>.b.e.</var> <lb/>parallela ad <var>.o.r</var>. </s> + <s xml:space="preserve"><reg norm="Sitque" type="simple">Sitq́;</reg> <var>.b.x.</var> æqualis <var>.b.r.</var> et <var>.q.u.</var> ſimiliter, & 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 huiuſmodi proportionalitatis.</s> + </p> + <p> + <s xml:space="preserve">Primum ergo certi ſcimus ex prima ſexti vel .18. ſeptimi eandem exiſtere pro-<lb/>portionem <var>.q.e.</var> ad <var>.b.o.</var> ſeu ad <var>.b.y.</var> quæ <var>.q.b.</var> ad <var>.b.r</var>: ſed <var>.u.y.</var> ad <var>.u.x.</var> eſt vt <var>.y.l.</var> ad <var>.l.x.</var> <lb/>hoc eſt vt <var>.q.b.</var> ad <var>.b.r.</var> ideſt vt <var>.q.e.</var> ad <var>.b.o.</var> & ſumma <var>.u.y.</var> cum <var>.u.x.</var> ideſt <var>.q.y.</var> minor eſt <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 eſt <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 ſpeculatione <reg norm="præcedentis" type="context">præcedẽtis</reg> theo <lb/>rematis, ſequitur <var>.u.y.</var> eſſe differentiam inter <reg norm="maximum" type="context">maximũ</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 eſt <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 ſunt 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">coniunctũ</reg> cum <var>.x.u.</var> medius <lb/>terminus erit, qui quidem (vt dictum eſt) duplus eſt 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 eſt. </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 vnuſquiſque dictorum extremorum, <lb/>deinde multiplicentur ipſi extremi interſe, vnde <lb/>habebimus tria producta eadem proportione inui <lb/>cem exiſtentia, 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 propoſitos 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 ideſt maius eſſet <lb/>7. cum dimidia vnitate, reliquum verò eſſet <lb/>quinque, productum poſteà quod ex ipſis extremis <lb/>prouenit, erit .6. quod quidem eſt harmonicè collo <lb/>catum inter .7. cum dimidia vnitate, & quinque.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ſpeculatio 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"/> + propoſiti <var>.a.e.</var> maior, et <var>.e.o.</var> minor, <reg norm="Sitque" type="simple">Sitq́;</reg> <var>.o.k.</var> medius arithmeticus inter dictos, vn-<lb/>de clarè patebit <var>.o.k.</var> eſſe dimidium ſummæ 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> ſit <reg norm="productum" type="context">productũ</reg> <lb/>quod fit ex <var>.e.o.</var> in <var>.o.k.</var> et <var>.n.m.</var> ſit productum quod ſit ex <var>.a.e.</var> in <var>.e.o.</var> quorum vnum-<lb/>quodque erit dimid ium vniuſcuiuſque producti præcedentis theorematis, <lb/>ex .18. et .19. ſeptimi Eucli. vnumquodque ſui relatiui. </s> + <s xml:space="preserve">Quare argumentando per <lb/>mutando à concluſionibus 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">harmonicũ</reg> inuenire cum quis voluesit inter duos <lb/>propoſitos terminos, ita faciendum erit, hoc eſt per ſummam datorum ex <lb/>tremorum diuidatur productum quod fit ex minimo termino in <reg norm="differentiam" type="context">differẽtiam</reg> dato-<lb/>rum, prouentus poſtea erit differentia inter maximum & med<unsure/>um quæſitum.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia, ſi nobis propoſiti fuerint hi duo termini .3. et .2. ſumma eo-<lb/>rum erit quinque, per quam cum diuiſerimus productum, quod naſcitur ex mini-<lb/>mo .2. in differentiam eorum, quæ eſt vnum, quod quidem erit .2. </s> + <s xml:space="preserve">tunc duæ quintæ <lb/>partes prouenient, quæ ſi demptæ fuerint ex maximo termino, reliquum erit .2. <reg norm="cum" type="context">cũ</reg> <lb/>3. quintis, hoc eſt 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 propoſitos, in-<lb/>ter quos deſideremus inuenire <var>.o.s.</var> medium ita illis <reg norm="relatum" type="context">relatũ</reg>, vt proportio exceſſus ip-<lb/>ſius ſupra <var>.x.c.</var> (qui ſit <var>.e.n.</var>) ad exceſ-<lb/>ſum <var>.u.d.</var> ſupra <var>.o.s.</var> (qui ſit <var>.n.d.</var>) ea-<lb/> + <anchor type="figure" xlink:label="fig-0103-01a" xlink:href="fig-0103-01"/> + dem ſit 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> & hæcſumma 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. ſed quia <var>.d.b</var>: <var>u.b.</var> et <var>.e.d.</var> quantitates no-<lb/>bis cognitę ſunt, ideò <var>.d.n.</var> ex .20. ſeptimi 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 diuerſis vnius <lb/>anni temporibus, </s> + <s xml:space="preserve">tunc in fine anni ſumma torius lucri datur cognita, ſed quæ-<lb/>ritur quantuni<unsure/> vnicuique illorum exipſa ſumma debeatur.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, primus in principio anni poſuit .100. aurcos, ſecundus verò .100 <lb/>diebus poſt primum poſuit .50. aureos tertius autem .200. diebus poſt primum po-<lb/>ſuit .25. aureos ſumma lucri poſtea in fine anni fuit aureorum .60.</s> + </p> + <p> + <s xml:space="preserve">Nunc vt ſciamus quantum huius ſummæ vniduique illorum proueniat, præcipit <lb/>regula, vt faciamus tria producta, quorum primum ſit ex numero dierum totius an-<lb/>ni in numerum aureorum primi, vnde tale productum in præſenti caſu erit .36500. <lb/>ſecundum verò ſit ex numero dierum à primo die in quo ipſe ſecundus poſuit uſque <lb/>ad finem anni, in numerum ipſorum nummorum, quod erit .13250. tertium autem <lb/>productum ex diebus tertij in numerum ſuorum aureorum, quod <reg norm="quidem" type="context">quidẽ</reg> erit .4125. <lb/>quæ producta ſimul 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"/> + ipſorum prochictorum per ſummam lucri hoc eſt per .60. vnde multiplicatio primi <lb/>producti erit .2190000. multiplicatio verò ſecundi producti erit .795000. tertij po <lb/>ſtca erit .247500. quarum multiplicationum vnaquæque diuidatur per ſummam <lb/>53875. productori<unsure/>t, & proueniet ex prima diuiſione .40. <reg norm="cum" type="context">cũ</reg> fractis .35000. vnius in-<lb/>tegri diuiſi in partes .53875. quod erit lucrum primi, prouentus autem ſecundæ di-<lb/>uiſionis erit .14. cum fractis .41050. vnius integri diuiſi in partes .53875. lucrum <reg norm="ſecu di." type="context">ſecũ<lb/>di</reg> </s> + <s xml:space="preserve">prouentus verò quartæ diuiſionis erit .4. cum fractis .32000. vnius integri, vt ſu <lb/>pra diuiſi in partes .53875. hoc eſt lucrum tertij.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ſpeculatio ex ſe in ſub ſcripta figura patet, vbi <var>.a.q.</var> ſignificat numerum <lb/>dierum totius anni pro primo mercatore <var>.q.n.</var> autem ſignificat numerum dierum ſe <lb/>cundi mercatoris <var>.e.q.</var> poſteà ſignificat numerum dierum tertij ſit etiam <var>.s.a.</var> pro nu-<lb/>mero denariorum primi, et <var>.o.n.</var> pro numero ſecundi, et <var>.e.t.</var> pro numero <lb/>tertij, productum autem <var>.q.s.</var> ſignificet valorem primi lucri, et <var>.q.o.</var> ſecundi, <lb/>et <var>.q.t.</var> tertij <var>.x.y.</var> autem ſignificet ſummam lucri omnium, et <var>.x.i.</var> ſignificet <lb/>partem primi, et <var>.i.p.</var> ſecundi, et <var>.p.y.</var> tertij. </s> + <s xml:space="preserve">vnde clarè patebit ex communi <lb/>ſcientia 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 ſi aliquis dice-<lb/>ret, ſi ex dicto aggregato, prouenit <lb/><var>x.y.</var> quid proueniet vnicuique <reg norm="illo- rum" type="context">illo-<lb/>rũ</reg> <reg norm="productorum" type="context">productorũ</reg>. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ſi numerus dena-<lb/>riorum <reg norm="ſecundi" type="context">ſecũdi</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">lucrũ</reg> <lb/>ſignificaretur à rectangulo <var>.q.b.</var> & ita <lb/>de tertio dico <reg norm="quod" type="simple">ꝙ</reg> ſignificaretur à <reg norm="re- ctangulo" type="context">re-<lb/>ctãgulo</reg> <var>.q.c.</var> vel ſi ſi<unsure/>antibus <reg norm="ijſdem" type="context">ijſdẽ</reg> <reg norm="denariorum" type="context">denariorũ</reg> quantitatibus <var>.n.o.</var> et <var>.e.t.</var> omnes ſuas pe-<lb/>cunias eodem tempore poſuiſſent, </s> + <s xml:space="preserve">tunc rectangula ſignificantia eorum lucra eſlent <lb/><var>q.s.q.d.</var> et <var>.q.f.</var> ſed cum nec eodem tempore, nec eandem quantitatem poſueruntr<unsure/>e <lb/>ctè eorum lucra ſignificantur à rectangulis <var>.q.s.q.o.</var> et <var>.q.t.</var> <reg norm="quod" type="simple">ꝙ</reg> ex prima .6. vel .18. aut <num value="19">.<lb/>19.</num> ſeptimi 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 propoſiti trianguli, cuius la-<lb/>r<unsure/>erum proportio nobis data ſit ſimul cum area ſuperſiciali ipſius trianguli, ſed quia <lb/>ipſe Tartalea vtiturregula algebræ, mihi viſum eſt breuiori methodo hoc idein fa <lb/>cere, & etiam vniuerſaliori via.</s> + </p> + <p> + <s xml:space="preserve">Supp onamus igitur duo triangula, quorum vnum <var>.u.n.i.</var> ſit nobis <reg norm="propoſitum" type="context">propoſitũ</reg>, & <lb/>cognitæ ſuperficiei, proportiones ſimiliter laterum <var>.i.n.</var> ad <var>.n.u</var>: et <var>.u.n.</var> ad <var>.u.i.</var> ſint no <lb/>bis datæ, <reg norm="alterum" type="context">alterũ</reg> verò <reg norm="triangulum" type="context">triangulũ</reg> ſit <var>.a.o.u.</var> à nobis tamen ita <reg norm="confectum" type="context">confectũ</reg>, v<unsure/>latera ſint in<lb/>er ſe proportionata eodem modo, quo latera prioris trianguli, ſed hæc nobis <reg norm="etiam" type="context">etiã</reg> <lb/>cognita ſint, <reg norm="quod" type="simple">ꝙ</reg> facillimum eſt. </s> + <s xml:space="preserve">Nunc vero ſi <reg norm="demptum" type="context">demptũ</reg> fuerit <reg norm="quadratum" type="context">quadratũ</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="ſupponendo" type="context">ſupponẽdo</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> conſequenter 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 eſt, ideo cognoſcemus <var>.e.u.</var> ſed <reg norm="cum" type="context">cũ</reg> <var>.e.u.</var> minor ſit <var>.a.u.</var> ex .18. & penultima primi, <lb/>ſi <reg norm="demptum" type="context">demptũ</reg> fuerit quadratum <var>.e.u.</var> ex quadrato <var>.a.u.</var> remanebit nobis <reg norm="cognitum" type="context">cognitũ</reg> <reg norm="quadra- tum" type="context">quadra-<lb/>tũ</reg> <var>.a.e.</var> & ſic nota erit nobis perpendicularis <var>.a.e.</var> ex penultima primi, quæ quidem <var>.<lb/>a.e.</var> ſi multiplicata fuerit in dimidium <var>.o.u.</var> dabit nobis <reg norm="ſuperficiem" type="context">ſuperficiẽ</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 ſimi <lb/>litudinem) eſt vt quadrati <var>.o.u.</var> ad quadratum <var>.n.i.</var> ex communi ſcientia cum vna-<lb/>quæque iſtarum proportionum dupla ſit proportioni <var>.o.u.</var> ad <var>.n.i.</var> ex .17. et .18. ſexti, <lb/>deinde cum nobis cognitæ ſint tres iſtarum quatuor quantitatum hoc eſt ſuperficies <lb/>trianguli <var>.a.o.u.</var> ſuperficies trianguli <var>.u.n.i.</var> & quadrati <var>.o.u.</var> </s> + <s xml:space="preserve">quare ex regula de tribus <lb/>cognoſcemus etiam quadratum <var>.n.i.</var> & ſic <var>.n.i.</var> latus primi trianguli, vnde reliqua la <lb/>tera illicò nobis innoteſcent exipſa regula de tribus, cum dixerimus, ſi <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/>ſcilicet inuenire <var>.x.</var> quantitatem me-<lb/> + <anchor type="figure" xlink:label="fig-0105-01a" xlink:href="fig-0105-01"/> + diam proportionalem inter duas ſu-<lb/>perficies triangulorum, vnde ſuper-<lb/>ficies trianguli <var>.i.a.u.o.</var> ad <var>.x.</var> ſe ha-<lb/>beret ut <var>.o.u.</var> ad <var>.i.n.</var> & ita ex regula <lb/>detribus cognoſcemus <var>.i.n</var>. </s> + <s xml:space="preserve">Multo <reg norm="tem" type="context">tẽ</reg> <lb/>pore poſtquàm hoc theorema conſtruxi, ipſum conſcriptum inueni in decimo <lb/>ſecundi libri Ioannis de monte Regio, ſatis tamen obſcurè 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 algebrę <lb/>quòd quælibet duo latera trianguli orthogonij, angulumrectum continentia, <lb/>ſint tertio longiora per diame-<lb/> + <anchor type="figure" xlink:label="fig-0105-02a" xlink:href="fig-0105-02"/> + trum circuli inſcriptibilis in ip-<lb/>ſo triangulo. </s> + <s xml:space="preserve">ſed hoc breuius <lb/>geometricè poteſt <reg norm="demonſtrari" type="context">demõſtrari</reg>, <lb/>quemadmodum in ſubſcripta <lb/>hic figura videre eſt, proptereà <lb/>quòd cum anguli <var>.A.o.u.</var> et <var>.n.</var> <lb/>omnes ſint recti et <var>.A.u.</var> æqualis <lb/><var>o.n.</var> et <var>.A.n.</var> ęqualis <var>.u.o.</var> ipſæ <var>.A.<lb/>u.</var> et <var>.A.n.</var> æquales erunt diame-<lb/>tro ipſius circuli. </s> + <s xml:space="preserve">Sed eædem <var>.<lb/>A.u.</var> et <var>.A.n.</var> ſunt ſuperfluum, quo <var>.A.B.</var> et <var>.A.C.</var> ſunt maiores <var>.B.C.</var> cum <var>.B.u.</var> et <var>.C.n.</var> <lb/>ſint 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 ſecundi libri nouæ ſcientiæ poterat ipſe Tartalea breuio <lb/>ri methodo abſque vlla operatione ipſius Algebræ inuenire <var>.A.H.</var> reſpectu <var>.A.<lb/>E.</var> eſſe vt .4. <reg norm="cum" type="context">cũ</reg> vno ſeptimo ad <reg norm="vnum" type="context">vnũ</reg>. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> ipſe ſupponit <var>.A.E.</var> <reg norm="decimam" type="context">decimã</reg> <reg norm="partem" type="context">partẽ</reg> eſſe ipſius + <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. ideſt .14. cum vno ſepti-<lb/>mo ferè. </s> + <s xml:space="preserve">quare <var>.A.L.</var> iuncta <var>.A.O.</var> erit .28. cum duobus ſeptimis. </s> + <s xml:space="preserve">ſed <var>.L.O.</var> ex ſuppoſi-<lb/>to erit .20. eo quòd <var>.L.I.</var> ęquatur ipſi <var>.A.I.</var> ſimiliter et <var>.I.O.</var> vt ipſe 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 ipſi <var>.L.O.</var> ex .<lb/>35. tertij ipſius Eucli. partium .8. <reg norm="cum" type="context">cũ</reg> duabus ſeptimis. cuius <reg norm="dimidium" type="context">dimidiũ</reg> hoc eſt <var>.A.H.</var> erit <lb/>4. cum una ſeptima, quod eſt propoſitum. </s> + <s xml:space="preserve">Reſpice figuram ipſius 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">QVadrageſimum nonum quæſitum ſimiliter poſſumus alio modo ſoluere, vt <lb/>putà cum vnumquodque latus rhombi ſimul cum area cognitum, ſeu datum <lb/>nobis ſit <reg norm="cognitum" type="context">cognitũ</reg> ſimiliter nobis erit quadratum lateris <var>.a.d.</var> hoc eſt ſumma 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/>ſit totalis ſuperficies rhombi, cognita etiam nobis erit eius medietas, hoc eſt produ-<lb/>ctum <var>.o.d.</var> in <var>.o.a.</var> vnde ex methodo .37. Theorematis cognoſcemus .a<unsure/><var>.o.</var> et <var>.o.d.</var> & ſic <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 huiuſmodi eſt. </s> + <s xml:space="preserve">Aliquis habet dolium mero plenum, ex quo <lb/>duas vrnas extrahit ipſius vini, ſed loco ipſius vini infundit duas vrnas aquæ. </s> + <s xml:space="preserve">Dein <lb/>de poſt aliquot dies extrahit iterum alias duas vrnas illius miſti, & iterum infundit <lb/>duas vrnas aquæ, & poſt 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 ipſius Tartaleæ bona eſt, cum ſupponat illas quatuor quantitates vini eſſe <lb/>inuicem continuas proportionales, vt putà primò totum vinum merum, poſteà re-<lb/>ſiduum pro ſecunda quantitate, deinde pro tertia in ſecunda, & pro quarta in ter-<lb/>tia extractione, hoc eſt quòd proportio totius vini meri ad vinum in prima ſit, vt hu <lb/>ius ad vinum in ſecunda, & vt huius ad vinum in tertia miſtione. </s> + <s xml:space="preserve">Sed quia ipſe <lb/>non probat hanc continuam proportionalitatem ex methodo ſcientifica, mihi <reg norm="visum" type="context">visũ</reg> <lb/>eſt hoc loco illam deſcribere.</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 ſupponamus quamlibet partem huius miſti omogeneam eſſe ſuo <lb/>toto, quapropter ſequetur 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 diuiſam, vt proportio <var>.i.e.</var> ad <var>.i.o.</var> eadem ſit 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 miſti hoc eſt primæ miſtionis, vnde cum eadem pro <lb/>portio ſit <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 eſt <var>.i.o.u.</var> (proptereà quòd <var>.i.o.</var> æqualis eſt <var>.a.e.</var> <lb/>vt reſidua 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/>ſto, & poſteà infuſa aqua vſque ad plenitudinem dolij, proportio ingredientium <lb/>huius ſecundi miſti erit ea, quæ eſt inter <var>.o.u.</var> et <var>.o.a.</var> eo quòd in prima miſtione pro-<lb/>proportio ingredientium erat ea, quæ eſt inter <var>.o.u.</var> et <var>.a.e.</var> vel inter <var>.a.e.</var> et <var>.o.u.</var> <lb/>vt demonſtrauimus. </s> + <s xml:space="preserve">Accipiamus ergo <var>.t.m.</var> huiuſmodi ſecundi mifti, magnitudi-<lb/>nis <var>.a.i.</var> vel <var>.e.o.</var> ſignificantis duas vrnas, & permutemus eum in tantam aquam, <lb/><reg norm="ſitque" type="simple">ſitq́;</reg> punctum <var>.o.</var> quod nobis diuidat <var>t.m.</var> in <var>.o.m.</var> et, <var>o.t.</var> partes ſimplices, tali propor <lb/>tione inuicem relatas, vt ſunt <var>.o.u.</var> et <var>.o.a.</var> vnde habebimus ex ſupradictis rationibus <lb/>eandem proportionem ipſius <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 eſt <var>.i.m.u.</var> (eo quod cum <var>.t.m.</var> æqualis ſit <var>.a.i.</var> per conſequens <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> ſed 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 ſupra diximus. </s> + <s xml:space="preserve">Ergo proportio <var>.i.m.u.</var> ad <var>.m.u.</var> erit dupla <lb/>ſimiliter proportioni <var>.i.o.u.</var> ad <var>.o.<lb/>u.</var> quapropter <var>.o.u.</var> erit media pro<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 eſt 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/>eſt vt intellectus acquieſcat, 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 eſt, ſed pulchrum etiam vide-<lb/>tur quærere proportionem ingredientium in ultima miſtione, cum cognita fue <lb/>rit nobis proportio continentiæ dolij ad capacitatis vrnæ ſimul <reg norm="cum" type="context">cũ</reg> numero vitium <lb/>extractionum & impletionum.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proportio <var>.a.u.</var> ad <var>.a.i.</var> cognita nobis fuerit, cognoſcemus etiam <lb/><var>e.i.</var> ex regula de tribus & per conſequens etiam <var>.i.o.</var> reſiduum ex <var>.e.o.</var> & ſimiliter ag-<lb/>gregatum <var>.a.i.</var> cum <var>.i.o.</var> & ſic <var>.o.u.</var> reſiduum totius, et <var>.o.t.</var> ſimiliter, eo quòd <var>.a.u.</var> ad <var>.a.<lb/>o.</var> eſt ut <var>.t.m.</var> ad <var>.o.t.</var> vnde cognoſcemus etiam <var>.o.m.</var> vt reſiduum <var>.t.m.</var> & ſimiliter ag-<lb/>gregatum <var>.a.o.</var> cum <var>.o.m.</var> hoc eſt <var>.a.m.</var> & etiam <var>.m.u.</var> reſiduum totius.</s> + </p> + <p> + <s xml:space="preserve">Cognoſcere autem proportionem totius dolij ad vrnam, vel ècontrà, cum cogni <lb/>ta nobis fuerit proportio ingredientium in vltima miſtione ſimul cum numero vi-<lb/>tium extractionum, & repletionum, quod ſcribit Tartalea, hoc etiam modo <lb/>poſſumus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proportio <var>.m.u.</var> ad <var>.m.a.</var> cognita nobis fuerit, illicò ſcie-<lb/>mus proportionem <var>.a.u.</var> ad <var>.m.u.</var> & cum ſciuerimus numerum vitium extractionum, <lb/>& impletionum illicò cognoſci-<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 eſt <var>.a.u.</var> ad <var>.i.u.</var> & ſimiliter ea, quæ eſt <var>.a.u.</var> ad <var>.a.i.</var> & è conuerſo ſimiliter.</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 ipſum, vt dictum eſt. </s> + <s xml:space="preserve">Nunc verò velim ſcire proportio-<lb/>nem vini ad a quam hac vltima vice. </s> + <s xml:space="preserve">Nunc igitur ſi procedemus iuxta doctrinam <lb/>primi exempli huius theorematis, obtinebimus quod quærebamus.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi diceret iuxta Tartaleæ quæſitum, hoc eſt 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="ſupradictum" type="context">ſupradictũ</reg> <lb/>eſt, ita vt vltima vice proportio vini ad aquam ſit ſeſquialtera. </s> + <s xml:space="preserve">Tunc ſi iuxta mo-<lb/>dum ſecundi exempli huius theorematis procedemus habebimus quod cupimus.</s> + </p> + <p> + <s xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eſt, habeo <reg norm="dolium" type="context">doliũ</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 ſcire quoties me hoc facere oporteat, ita vt poſtrema vi-<lb/>ce vinum ſe habeat ad aquam in proportione ſeſquialtera, vnde multoties accidet <lb/>vltimam extractionem, & impletionem mutilatam, ſeu imperfectam, euadere.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſi proportio vini ad aquam in vltima miſtione 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 ſupra dicto <lb/>proportio <var>.a.m.</var> ad <var>.m.u.</var> nobis in-<lb/>noteſceret, & proportio <var>.n.a.</var> ad <var>.<lb/>n.u.</var> nobis data eſt ſimul cum <reg norm="quan titate" type="context">quã<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 ſimiliter, etiam, et <var>.t.n.</var> reſiduum vaſis, quo metimur, vnde <lb/>neceſſe erit, quo<unsure/>d vltima vice vas contineret ſolum <var>.t.n.</var> reliqua uerò per ſe 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. ſuæ arithmeticæ cap .66. quæſtione .56.</ref> quam Car<lb/>danicam vocat, ita inquit.</s> + </p> + <p> + <s xml:space="preserve">Quidam perambulauit prima die certam quantitatem ſpatij, & ſecunda die, <reg norm="tan tò" type="context">tã<lb/>tò</reg> plus proportionaliter, quantò diameter eſt maior coſta, & tertia die tantò plus <lb/>ſecunda, quantò proportionaliter portio lineæ diuiſæ ſecundum proportionem ha <lb/>bentem medium, & duo extrema excedit minorem portionem, & quarta die in <lb/>proportione ad tertiam vt ſecunda ad primam, & quinta die proportionaliter tan-<lb/>tò plus quarta, quantò in tertia plus ſecunda, & 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 eſt, quàm ſi aliquis diceret, propono tibi, exempli gratia, <lb/>lineam <var>.a.l.</var> nouem partibus inuicem non æqualibus ita diuiſam <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> & ſic de cæteris vſque ad poſtremam <var>.k.l.</var> quæ quidem proportiones ſint <lb/>etiam inuicem diſſimiles, ſeu inæquales, do tibi etiam <reg norm="proportionem" type="context">proportionẽ</reg> totius lineæ <var>.a.l.</var> <lb/>ad <var>.a.b.</var> ſuam partem, quæ vt in propoſito exemplo nonupla eſt.</s> + </p> + <p> + <s xml:space="preserve">Quæro nunc quam proportionem habebit <var>.a.c.</var> ad <var>.a.b.</var> & ſic de cæteris partibus <lb/>eiuſdem ad eandem <var>.a.b</var>.</s> + </p> + <p> + <s xml:space="preserve">Quod quidem facillimum erit ſpeculari, nec non operari vnicuique, qui omnino <lb/>practicæ numerorum ignarus non fuerit, dum ab ordine ſcientifico non diſcedat.</s> + </p> + <p> + <s xml:space="preserve">Cum enim cognoſcimus proportionem <var>.a.c.</var> ad <var>.c.d.</var> conſequenter cognoſcemus <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> ſi <var>.c.d.</var> accipiemus, vt medium inter <var>.a.d.</var> et <var>.d.e.</var> cognoſcemus 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 poſteà. <lb/><var>d.e.</var> inter <var>.e.f.</var> et <var>.a.e.</var> innoteſcet ea, quæ eſt <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/>cognoſcemus proportionem totius <var>.a.l.</var> ad <var>.i.k.</var> & hac mediante, cam cognoſcemus, <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> & ſic gradatim, co <lb/>gnita nobis erit proportio totius <lb/>lineæ <var>.a.l.</var> ad ſuam partem <var>.a.c.</var> be-<lb/> + <anchor type="figure" xlink:label="fig-0109-01a" xlink:href="fig-0109-01"/> + neficio poſteà totius lineæ <var>.a.l.</var> co <lb/>gnoſcemus proportionem <var>a.c.</var> ad <lb/><var>a.b.</var> & ſic aliarum reſpectu lineæ <var>.a.b.</var> vt quærebatur, quæ quidem propoſitio, etſi car <lb/>danica uocetur leuiſſima tamen eſt.</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 ſumma colligendi, ſubtrahendi, <reg norm="multiplicandi" type="context">multiplicãdi</reg>, & di <lb/>uidendi proportiones ſcripſerint, nullus tamen (quod ſciam) perfectè, ac <lb/>ſcientificè ſpeculatus eſt has operationes, quapropter hanc rem cum ſilentio tranſi <lb/>re nolui, quin aliquid de ipſa conſcribam à ſumma dictarum proportionum in-<lb/>cohando.</s> + </p> + <p> + <s xml:space="preserve">Quotieſcunque igitur volunt duas proportiones inuicem aggregare, ſimul ea-<lb/>rum antecedentia multiplicant, & ſimiliter earum conſequentia. </s> + <s xml:space="preserve">Tunc proportio <lb/>terminata ab illis productis euadit in ſummam illarum duarum propoſitarum <lb/>proportionum.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia, ſi voluerimus colligere proportionem ſeſquialteram cum ſeſ-<lb/>quitertia, multiplicando .3. cum .4. antecedentia ſcilicet, pro ductum erit .12. poſteà <lb/>multiplicando .2. cum .3. conſequentia, tunc productum erit .6. </s> + <s xml:space="preserve">Proportio igitur, <lb/>quæ inter .12. et .6. reperitur. (quæ dupla eſt) eſt ſumma propoſitarum <reg norm="proportionum" type="context">proportionũ</reg>.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ſpeculatio erit huiuſmodi ſint <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 ſint eorum conſequentia, productum <lb/>autem antecedentium ſit <var>.a.g.</var> illud verò quod <reg norm="con" type="context">cõ</reg> <lb/>ſequentium ſit <var>.d.a.</var> vnde proportio <var>.a.g.</var> ad <var>.a.d.</var> <lb/>compoſita erit ex proportione <var>.x.</var> ad <var>.t.</var> & ex ea, <lb/>quæ eſt <var>.u.</var> ad <var>.n.</var> per .24. ſexti vel quintam octaui. <lb/></s> + <s xml:space="preserve">Patet igitur ratio rectè faciendi, vt ſuprà dictum <lb/>eſt.</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">QVotieſcunque deinde detrahere volunt vnam proportionem ex altera mul-<lb/>tiplicant antecedens vnius cum conſequenti alterius. </s> + <s xml:space="preserve">Tunc proportio, quę <lb/>inter talia duo producta incluſa reperitur, eſt reſiduum, ſeu differentia illarum dua-<lb/>rum proportionum datarum.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia, ſi aliquis vellet ex proportione dupla detrahere ſeſquialte-<lb/>ram, multiplicaret .2. antecedens duplæ cum .2. conſequenti ſeſquialteræ, quorum <lb/>productum eſſet .4. pro antecedenti reſiduę proportionis. </s> + <s xml:space="preserve">Deinde multiplicaret .3 <lb/>antecedens ſeſquialteræ cum .1. conſequenti duplæ, & productum eſſet .3. pro <reg norm="con- ſequenti" 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"/> + ſequenti</reg> reſiduę proportionis; </s> + <s xml:space="preserve">quæ quidem reſidua proportio eſſet vt .4. ad .3. hoc <lb/>eſt ſeſquitertia, & ſic de cæteris.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius ratione, ſit proportio <var>.x.</var> ad <var>.n.</var> ea quæ (exempli gratia) maior ſit, à <lb/>qua volumus demere proportionem <var>.t.</var> ad <var>.u.</var> minorem ſcilicet. </s> + <s xml:space="preserve">Nunc autem <lb/>productum <var>.x.</var> in <var>.u.</var> ſit <var>.a.g.</var> illud verò <var>.t.</var> in <var>.<lb/>n.</var> ſit <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 reſiduam 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> quę <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 ſexti, ſeu .18. vel .19. ſe-<lb/>ptimi, ſed proportio <var>.a.g.</var> ad <var>.a.b.</var> hoc eſt <var>.x.</var> ad <var>.<lb/>n.</var> componitur ex ea, quæ eſt <var>.a.g.</var> ad <var>.a.d.</var> & ea, <lb/>quæ eſt <var>.a.d.</var> ad <var>.a.b.</var> hoc eſt <var>.t.</var> ad <var>.u.</var> ergò ea, quę <lb/>eſt <var>.a.g.</var> ad <var>.a.d.</var> erit quàm 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, quotieſcunque aliquam proportionem dupli-<lb/>care volentes, quadramus terminos ipſius proportionis, vel ſi eam triplicare <lb/>voluerimus, cubamus ipſos terminos, vel ſi eam quadruplicare voluerimus <lb/>inuenimus cenſicos cenſicos terminorum ipſius proportionis, & ſic de ſingulis, in <ref id="ref-0016">.17 <lb/>Theo. huiuſmodi tractatus</ref> manifeſta eſt.</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">QVotieſcunque nobis propoſiti fuerint duo numeri ad libitum, deſideraremus<lb/>q́ue duas proportiones tali relatione inuicem refertas, quali ſunt hi duo pro <lb/>poſiti numeri inter ſe, ita faciendum erit.</s> + </p> + <p> + <s xml:space="preserve">Sciendum primo eſt proportionem maioris numeri propoſiti ad minorem ſem-<lb/>per eſſe alicuius ex quinque generum, hoc eſt aut erit generis multiplicis, aut ſu-<lb/>perparticularis, aut multiplicis ſuperparticularis, aut ſuper partientis, aut multi-<lb/>plicis ſuperpartientis.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem ſi erit ex genere multiplici, iam ab antiquis traditus eſt modus, <reg norm="quem" type="context">quẽ</reg> <lb/>ſequi debemus. </s> + <s xml:space="preserve">Cuius ſpeculatio à me inuenta patet .in .17. Theo. huius libri, vt <lb/>in præcedenti dixi.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi talis proportio datorum numerorum erit alicuius aliorum generum, ita <lb/>agemus, ſi fuerit ſuperparticularis.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia, ſeſquialtera, tunc ſumantur duo numeri inuicem inæquales, <lb/>quos à caſu volueris <var>.o.</var> et <var>.c.</var> qui quidem cubentur, & eorum cubi ſint <var>.a.</var> et <var>.e</var>. </s> + <s xml:space="preserve">Inuenia <lb/>tur poſteà. u. ita proportionatus ad <var>.o.</var> vt <var>.o.</var> eſt ad <var>.c.</var> ex regula de tribus, hoc eſt diui-<lb/>dendo quadratum ipſius <var>.o.</var> per <var>.c.</var> vnde nobis proueniat <var>.u.</var> & quia proportio <var>.a.</var> ad <var>.e.</var> <lb/>tripla eſt proportioni <var>.o.</var> ad <var>.c.</var> & proportio <var>.u.</var> ad <var>.c.</var> dupla eſt <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> ſeſquialtera erit proportioni <var>.u.</var> ad <var>.c</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi proportio numerorum propoſitorum fuerit ſeſquitertia, faciemus <var>.a.</var> et <var>.e.</var> <lb/>eſſe cenſica cenſica ipſius <var>.o.</var> et <var>.c</var>. </s> + <s xml:space="preserve">tunc ſumemus <var>.u.</var> conſequentem ad <var>.o.</var> vt dictum eſt, <lb/>deinde inueniremus <var>.i.</var> conſequens ad <var>.u.</var> ita ut <var>.u.</var> conſequens ipſius <var>.o</var>. </s> + <s xml:space="preserve">tunc habebi-<lb/>mus proportionem <var>.i.</var> ad <var>.c.</var> triplam, & eam quæ eſt <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 ſuperparticularibus.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi data proportio numerorum fuerit ex ſuper partientibus, vt exempli gra-<lb/>tia de quinque ad tria, efficiemus, vt <var>.a.</var> et <var>.e.</var> ſint prima relata ipſius <var>.o.</var> et <var>.c.</var> vnde <lb/>proportio <var>.a.</var> ad <var>.e.</var> ita ſe 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">vnũ</reg> & propor-<lb/>tio <var>.i.</var> ad <var>.c.</var> ut tria ad <reg norm="vnum" type="context">vnũ</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> ſe habebit, <lb/>vt quinque ad tria, & ſic 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 ſeruando, 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. ſexti et .20. ſeptimi manifeſtè pateat ratio, quare rectè fiatac <lb/>cipiendam radicem quadratam illius producti, quod fit ex duobus datis <lb/>terminis, vt medium proportionale geometricè inter ipſos habeamus: </s> + <s xml:space="preserve">nihilomi-<lb/>nus, quia per aliam methodum hoc idem ſcire 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 ſint, 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 eſt <var>.h.b.</var> <lb/>ad <var>.b.g.</var> ex prima .6. aut .18. vel .19. ſepti-<lb/>mi, ſed per .11. octaui ita eſt 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/>eſt 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 eſt diuidere vnam datam <lb/><reg norm="proportionem" type="context">proportionẽ</reg> per æqualia, hoc eſt in duas <lb/>æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poſteriori-<lb/>bus Theorematibus ad demonſtrandum priora illis, ſed hoc .148. dictum ſit 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">ꝙ</reg> ſi quis inuenire voluerit ſecundum terminum ex quatuor nume <lb/>ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-<lb/>modo nobis cogniti ſint, rectè factum ſit quadrare primum eorum, & hoc quadra-<lb/>tum poſteà per alium terminum cognitum multiplicare, cuius producti demum ac-<lb/>cipere radicem cubam pro ſecundo termino quæſito, hocloco videbimus.</s> + </p> + <p> + <s xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eſt, 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 ſint, <reg norm="ſitque" type="simple">ſitq́</reg> imagina <lb/>tione deſcriptus cubus <var>.a.q.</var> primi termini, <reg norm="cubusque" type="simple">cubusq́</reg> <var>.d.k.</var> ſecundi rermini, conſidere-<lb/>mus etiam baſim <var>.a.i.</var> quadratam ipſius cubi <var>.a.q.</var> hoc eſt præcedentem dignitatem ip <lb/>ſius cubi eiuſdem radicis, quæ quidem baſis <var>.a.i.</var> multiplicetur per quartum <reg norm="terminum" type="context">terminũ</reg> <lb/><var>g.h.</var> productum autem ſit <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, ſed per primam ſexti, vel .18. aut .19. ſeptimi ita eſt <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> ideſt <lb/> + <anchor type="figure" xlink:label="fig-0112-01a" xlink:href="fig-0112-01"/> + vt <var>.a.b.</var> ad <var>.g.h.</var> ſed vt eſt <var>.a.b.</var> ad <var>.g.h.</var> <lb/>ſic eſt <var>.a.q.</var> ad <var>.k.d.</var> per .36. vndecimi, <lb/>ſeu per .11. octaui, vnde per .11. quin <lb/>ti ſic 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. eiuſdem <var>.a.g.</var> ęqualis 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="ſecundo" type="context">ſecũdo</reg> termino <var>.c.d.</var> <lb/>id, quod nobis inſeruit ad inueniendam tertiam partem vnius propoſitæ 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 ſpeculatio iſta ita vniuerſalis 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, ſed eiuſdem <lb/>ſpeciei, 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 ipſius <var>.a.i.</var> multiplicatione in <var>.g.h.</var> reſultet <var>.a.<lb/>g.</var> vnde ex .18. vel .19. ſeptimi eadem proportio erit <var>.a.q.</var> ad <var>.a.g.</var> quæ <var>.a.b.</var> ad <var>.g.h.</var> ſed <lb/>eadem etiam eſt <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 huiuſmodi dignitatis ex quantitate <var>.a.g.</var> habebimus <var>.c.d.</var> ſecundum 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, ſi accipiat radices quadratas illorum datorum rer-<lb/>minorum, etſi voluerit tertiam partem, accipiat radices cubas: </s> + <s xml:space="preserve">ſi autem quartam, <lb/>accipereradices cenſicas cenſicas ipſorum, & ſic de ſingulis 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 diuiſerit per denominationem ipſorum fractorum.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem ſeſquiquar-<lb/>tam per duo tertia, multiplicabit prius ipſam proportionem per numeratorem .2. <lb/>& productum, erit proportio .25. ad .16. qua poſtea diuiſa 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 ſunt, cu <lb/>tam vna, quàm alia, ſit 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 eſt, quàm accipere duas tertias partes vnius <lb/>totius ſuperficialis, imaginemur igitur hoc totum <var>.a.b.</var> lineare diuiſum eſſe in tertias <lb/>partes mediantibus <var>.e.</var> et <var>.d</var>. </s> + <s xml:space="preserve">& tunc multiplicando ipſum per 2. tertias lineares produ-<lb/>ctum erit <var>.a.c.</var> ſex vnitatum ſuperficialium, quod quidem productum poſteà diuiſum <lb/>per .3. dabit <var>.d.c.</var> hoc eſt duas tertias ſuperficiales (quæ eſt tertia pars ipſius <var>.a.c.</var>) & <lb/>ęquales numero <var>.c.b.</var> duabus vnitatibus linearibus, ideſt duabus tertijs ipſius <var>.a.b</var>. </s> + <s xml:space="preserve">No <lb/>tandum etiam eſt, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà <lb/>etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eſt quàm <lb/>quædam operatio ipſius regulæ de tribus, vt eyempli gratia volo multiplicare .25. <lb/>per 20. hoc nihil aliud eſt niſi quærere alium numerum ita proportionatum ad .25. <lb/>vt 20. ſe habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo <lb/>per vnum exregula de tribus, prouentus eſt idem numerus ipſius producti, & propte <lb/>rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eſt <lb/>quàm quærere aliquem numerum ita proportionatum ad ipſum numerum datum, <lb/>vt ſe habet numerator ad denominatorem, exempli gratia ſi .24. aliquis voluerit mul <lb/>tiplicare per duo tertia hoc idem eſt vt ſi quæreret numerum ad quem .24. ita ſe <lb/>habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eſt quod aliud non eſt mulri-<lb/>plicare aliquam proportionem per fractos, quàm aliam proportionem quærere ad <lb/><reg norm="quam" type="context">quã</reg> data ſe habeat, vt denominator ſe <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="conſtituendo" type="context context">cõſtituẽdo</reg> <reg norm="denominatorem" type="context">denominatorẽ</reg> in primo loco, quilocus eſt diuiſoris, numerato <lb/><reg norm="rem" type="context">rẽ</reg> verò in <reg norm="ſecundo" type="context">ſecũdo</reg> loco, <reg norm="multiplicando" type="context">multiplicãdo</reg> poſteà pro <lb/>portionem per <reg norm="numeratorem" type="context">numeratorẽ</reg>, & <reg norm="productum" type="context">productũ</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 ſe habebit, vt denomi-<lb/>nator ſe <reg norm="hent" type="context">hẽt</reg> ad numeratorem ex ratione ipſius re <lb/>gulę 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 ſe ſatis patet, <lb/>cum idem ſit modus diuidendi quemhbet nume <lb/>rum integrum per fractos. </s> + <s xml:space="preserve">Quare, quæ vnius, <lb/>& alterius eſt 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 ſoluit .24. quæſitum ſi-<lb/>bi propoſitum à Hieronymo Cardano, via particulari & non generali. </s> + <s xml:space="preserve">Quæ-<lb/>ſitum autem tale eſt quamlibet propoſitam rectam lineam in duas partes ita diuide <lb/>re via Euclidis, ut cubus totius lineæ ad cubos partium ſe habeat in proportione <lb/>tripla.</s> + </p> + <p> + <s xml:space="preserve">Tartalea igitur inquit quòd vt ſatisfiat ſpeculatiuis ingenijs ſoluendum ſit huiuſ-<lb/>modi quæſitum, ſecando lineam propoſitam <var>.a.b.</var> in tres æquales partes, quarum vna <lb/>fit <var>.c.b.</var> vnde problema ſolutum erit.</s> + </p> + <p> + <s xml:space="preserve">Verum dicit, ſed hæc non eſt methodus generalis, proptereà, quod cum tale <lb/>problema alterius fuiſlet 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> diuiſibilis in puncto <var>.c.</var> ita vt cubum totius dictæ <var>.a.b.</var> lineæ ad <lb/>ſummam cuborum <reg norm="ſuarum" type="context">ſuarũ</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 ſe contemplari po-<lb/>teſt, conſtituendo punctum <var>.c.</var> in medio loco inter <var>.a.</var> et <var>.b.</var> vnde proportio totalis <lb/>cubi ad ſummam 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 propoſitum reuertamur, conſiderabimus cubum totalem ipſius <var>.a.b.</var> <lb/>eſſe vt .125. & ſummam partialium vt .65. quam detrahemus ex cubo totali & nobis <lb/>remanebit .60. pro ſumma trium ſolidorum inuicem æqualium, quorum longitu-<lb/>do vniuſcuiuſque erit tota linea <var>.a.b.</var> nobis cognita vt radix dati cubi totalis, quæ erit <lb/>in hoc exemplo quinque partium, latitudo verò vniuſcuiuſque dictorum <reg norm="ſolidorum" type="context">ſolidorũ</reg> <lb/>erit <var>.a.c.</var> pars maior ipſius <var>.a.b.</var> quæ quidem <var>.a.c.</var> adhuc nobis ignota eſt, profunditas <lb/>ſeu altitudo vniuſcuiuſque illorum ſolidorum, erit <var>.c.b.</var> pars reliqua ipſius <var>.a.b.</var> & <reg norm="etiam" type="context">etiã</reg> <lb/>nobis incognita, ſed quia ſumma horum trium ſolidorum nobis manifeſta ſuperius <lb/>fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuſcuiuſque illorum ſoli-<lb/>dorum, vt tertia pars totius ſummæ ipſorum quæ erit .20. in propoſito <reg norm="exemplo" type="context">exẽplo</reg>, dein <lb/>de cum vnumquodque illorum ſolidorum producatur à ſuperficie contenta ſeu pro<lb/> ducta ab <var>.c.a.</var> in <var>.c.b.</var> in tota linea <var>.a.b.</var> ſequitur quòd ſi diuiſerimus hoc ſolidum .20. <lb/>per lineam <var>.a.b.</var> quinque partium proueniet nobis cognita ſuperficies producta ab <var>.<lb/>a.c.</var> in <var>.c.b.</var> quatuor partium, ſed cum quadratum totius <var>.a.b.</var> nobis cognitum ſit, eo <lb/>quod <var>.a.b.</var> vt eius latus etiam cognitum eſt. </s> + <s xml:space="preserve">Tunc dictum quadratum erit .25. quod <lb/>quidem æquale eſt quadruplo illius quod fit ex <var>.a.c.</var> in <var>.c.b.</var> ſimul cum quadrato diffe <lb/>rentiæ inter <var>.a.c.</var> et <var>.c.b.</var> per .8. ſecundi 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 eſt, vt <lb/>16. eo quod ſimplum quod eſt .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 ſi hoc quadru-<lb/>plum .16. demptum fuerit ex totali <lb/>quadrato .25. reliquum erit .9. qua <lb/><reg norm="dratum" type="context">dratũ</reg> ſcilicet vnius partis <var>.a.c.</var> ipſius <lb/>hoc eſt illius partis, quæ differentia <lb/>eſt inter <var>a.c.</var> et <var>.c.b.</var> quæ quidem erit <num value="3">.<lb/>3.</num> partium quæ differentia cum ſub-<lb/>tracta fuerit ex <var>.a.b.</var> reliquum erit du <lb/>plum ipſius <var>.c.b.</var> duo ſcilicet. </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 ſuperficialium. <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 ſuam Arithmeticam re <lb/>cepiſſet, mox poſteà per literas petijt <reg norm="explicationem" type="context">explicationẽ</reg> regulæ falſi.</s> + </p> + <p> + <s xml:space="preserve">Similiter poſt inciſas omnes ſuperiorum Theorematum figu-<lb/>ras, <reg norm="opereque" type="simple">opereq́;</reg> Typographo commiſſo, amicus quidam omnium <lb/>ſcientiarum ornatiſſimus maxima neceſſitudine mecum coniun-<lb/>ctus monuit me, vt aliquid de regula falſi ſcribere vellem, cuius <lb/>ſuaſu hæc, quæ ſequuntur appendicis vice ponere libuit, nelector, quidpiam quod <lb/>ad hancrem pertinet iure merito à nobis deſiderare poſſet; </s> + <s xml:space="preserve">vt autem ad ipſam <reg norm="re- gulam" type="context">re-<lb/>gulã</reg> accedamus Ego ſicut, & in alijs multis, ita & in huiuſcæ regulę inuentione cum <lb/>ipſo Stifelio maximè conuenio, putans regulam falſi, ſeu falſarum poſitionum in-<lb/>uentam fuiſſe per paruos numeros in quæſtionibus facillimis & cognitis, eodem fer <lb/>mè modo, quo ipſe monſtrat illis duobus exemplis, quæ quamuis ipſe appellet theo <lb/>remata, nihilominus the oremata ego illa non vocarem, niſi adiuncta fuerit ſpecu-<lb/>latio ab ipſo præterita, & non experientia tantummodo, vt ipſe fecit. </s> + <s xml:space="preserve">Primum eius <lb/>exemplum eſt, quòd.</s> + </p> + <p> + <s xml:space="preserve">Quorumcumque <reg norm="duorum" type="context">duorũ</reg> numerorum differentia, ſi fuerit multiplicata in aggre <lb/>gatum eorum, producit ipſam differentiam, quæ eſt inter quadrata eorum.</s> + </p> + <p> + <s xml:space="preserve">Secundum verò exemplum eſt, quod.</s> + </p> + <p> + <s xml:space="preserve">Datis tribus numeris ſecundum progreſſionem arithmeticam diſpoſitis, facit mul <lb/>tiplicatio medij in ſe, <reg norm="quantum" type="context">quãtum</reg> multiplicatio extremorum inter ſe cum multiplicatio <lb/>ne differentiarum inter ſe.</s> + </p> + <p> + <s xml:space="preserve">Talia enim exempla ipſe aliter non probat niſi experientia in aliquibus numeris, <lb/>arbitratus ex eo inuentam eſſe regulam falſi, experientia tantummodo confirma-<lb/>tam, quod quidem etiam & ego credo. </s> + <s xml:space="preserve">At experientia in philoſophia mathema-<lb/>tica, aut <reg norm="nullam" type="context">nullã</reg> prorſus facit <reg norm="ſcientiam" type="context">ſcientiã</reg>, aut omnino ſuperfluus fuit Euclides in multis <lb/>ſuis propoſitionibus, & præcipuè in eius ſecundo libro, ſi ſufficeret experientia. </s> + <s xml:space="preserve">Id-<lb/>circo quo magis ad euidentiam ipſius veritatis, quam profiteor, deuenire poſſim, <lb/><reg norm="accipiam" type="context">accipiã</reg> primò primum exemplum <lb/>ipſius Stifelij hic ſuperius citatum, <lb/>& pro numero maiori, in prima hic <lb/> + <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> + ſubſcripta figura .AE. accipio <var>.a.i.</var> <lb/>cuius quadratum ſit <var>.a.c</var>: pro minori <lb/>vero numero capio <var>.a.e.</var> <reg norm="partem" type="context">partẽ</reg> ipſius <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 ipſius <var>.a.i</var>: & differen <lb/>tia ipſorum quadratorum erit gno-<lb/>mon <var>.e.c.o</var>: Nunc autem protraho <var>.<lb/>i.c.</var> latus quadrati maioris quouſque <lb/><var>c.n.</var> æqualis ſit <var>.a.e.</var> numero minori, <lb/><reg norm="perficioque" type="simple">perficioq́;</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, ſed 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 ipſi <var>.u.o.</var> reliquæ ſcilicet parti ipſius 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> ſed <var>.e.t.</var> ęquatur <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 ſit ipſi <var>.a.e.</var> erit etiam æqualis ipſi <var>.<lb/>o.t</var>. </s> + <s xml:space="preserve">quare <var>.u.n.</var> æqualis erit ipſi <var>.u.o.</var> <lb/>& tunc intellectus quieſcit, & <reg norm="abſque" type="simple">abſq;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> + aliqua alia experientia verè ſcientifi <lb/><reg norm="ceque" type="simple">ceq́;</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/>ipſam <reg norm="differentiam" type="context">differentiã</reg>, quæ eftinter qua-<lb/>drata eorum.</s> + </p> + <p> + <s xml:space="preserve">Hæcautem propoſitio à me ipſo <lb/>etiam in .60. </s> + <s xml:space="preserve">Theoremate huius libri <lb/>aliter demonftrata fuit.</s> + </p> + <p> + <s xml:space="preserve">DE ſpeculatione autem, etſcientia ſecundi exempli, in ſecunda hic ſubſcripta <lb/>figura <var>.ω.</var> cogitemus lineam <var>.u.a.</var> tribusin partibus arithmeticè diuiſam, qua <lb/>rum maxima ſit <var>.u.o.</var> media. ſit <var>.o.e.</var> minima verò ſit <var>.e.a.</var> multiplicatio autem mediæ <var>.<lb/>o.e.</var> in ſe ſit quadratum <var>.o.t.</var> abſcindatur 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 ſe ducta procreabit quadratum <var>.o.c.</var> quod erit productum ex differen <lb/>tijs ipſarum partium, & erit pars quadrati <var>.o.t.</var> ſuperius dicti, vt exſe 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> quouſque <var>.t.r.</var> æqualis ſit ipſi <var>.o.i</var>. </s> + <s xml:space="preserve">tunc <var>.e.r.</var> erit æqualis <var>.o.u.</var> quod etiam clarum <lb/>eſt. </s> + <s xml:space="preserve">Claudatur ergo rectangulum <var>.i.r.</var> quod erit æquale producto ipſius <var>.e.a.</var> in <var>.o.u.</var> <lb/>Nam <var>.e.i.</var> ſumpta fuit <lb/>æqualis <var>.e.a.</var> ſed 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">ꝓductum</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, ſcientifice-<lb/>q́ue poſſumus affirma <lb/>re, quòd. </s> + <s xml:space="preserve">Datis tribus <lb/>numeris <reg norm="ſecundum" type="context">ſecundũ</reg> pro <lb/>greffionem arithme-<lb/>ticam diſpofitis, fa-<lb/>cit multiplicatio me-<lb/>dij in ſe quantum mul <lb/>tiplicatio extremorum inter ſe, cum multiplicatione differentiarum inter ſe.</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 ſic de alijs huiuſmodi inuentionibus infero.</s> + </p> + <p> + <s xml:space="preserve">DIcturus igitur aliquid circa <reg norm="regulam" type="context">regulã</reg> falſi, videtur mihi nullam oportere facere <lb/>mentionem de origine huiuſcæ regulæ, cum in hoc Stifelius ſatisfecerit, ſed + <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">propriaq́;</reg> fundamenta huiuſmodi operationis oftendere, fu-<lb/>mendo eadem exempla propoſita abipſis practicis, & maximè à Nicolao Tartalea <lb/>viro accuratiffimo, qui vbicunque potuit ſpeculatus eſt cauiſas <reg norm="ipſarum" type="context">ipſarũ</reg> operationum, <lb/>etſi de huiuſmodi falſi regula circa finem cap .8. lib. 17. promittat poſtea loqui, nub-<lb/>libi tamen loquutus eft. </s> + <s xml:space="preserve">Monendum etiam cenſeo, me nihil de rationibus regulæ <lb/>falſi ſimplicis dicturum, cum ex ſeipſis ſatis appareant, quod non ita eſt de poſitio-<lb/>nibus duplis. </s> + <s xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipſius Tartaleæ, quo <reg norm="etiam" type="context">etiã</reg> <lb/>ipſe vtitur pro exemplo docendi gratia, ipſam regulam duplæ poſitionis, quod qui <lb/>dem problema aliter à me <reg norm="ſolutum" type="context">ſolutũ</reg> fuit in .118. </s> + <s xml:space="preserve">Theoremate huius mei lib. quod ſimi <lb/>liter ob hanc demum occaſionem mihi oblatam, alia etiam via, ſpeculatus ſumidem <lb/>poſſe fieri, quæ quidem via ſeu methodus generalis erit, & ita ſe habet.</s> + </p> + <p> + <s xml:space="preserve">Accipio enim propoſitum numerum diuiſibilem, à quo detraho ſummam <lb/>datorum numerorum, primo duplicato, eo quòd tam in ſecunda quam in <lb/>tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à <lb/>quo detraho ſummam dictorum numerorum, quæ eſt .11. nam tres, & tres, & <lb/>quinque ſunt vndecim, eo quòd primus ingreditur in ſecunda, & in tertia parte, <lb/>dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-<lb/>dus eſt pro ſumma trium partium ſimplicium adhuc incognitarum, à quo extrahen <lb/>da eſt prima, eo modo quo nunc proponam exregula de tribus, hoc eſt aggregan <lb/>do dictas partes ſimplices ſine aliqua additione vtcunque volueris (ſed commodius <lb/>erit in minimis numeris) iuxta propoſitum, quod quidem propoſitum eſt, vt ſecun <lb/>da pars dupla ſit primæ, tertia verò æqualis fit primæ & ſecundæ, quæ partes in di-<lb/>ctis minimis numeris, ita diſpoſitæ erunt .1. 2. 3. quarum ſumma erit .6. </s> + <s xml:space="preserve">Nunc ſi ex <lb/>regula de tribus dixerimus, cum hæc ſumma proueniat nobis ab vno, à quo proue-<lb/>niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoſito nume-<lb/>ro .39. cum ergo habuerimus primam <reg norm="partem" type="context">partẽ</reg>, reliquas poſteà illicò cognoſcemus.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi verò operationis ratio ex ſe manifeſta patet, eo quòd proportio ſum <lb/>mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ <lb/>ipſius .39. ad primam partem quæſitam huiuſmodi aggregati partium <reg norm="ſimplicium" type="context">ſimpliciũ</reg>, ſed <lb/>quia nemo adhuc, quod ſciam, ſatis animaduertit rationem modorum, qui ab anti-<lb/>quis obſeruati ſunt, qui quidem modi duo ſunt circa hoc Helcataym duplæ falſæ <lb/>pofitionis, igitur non prætermittam aliquid de hacreſpeculari, & primo de pri-<lb/>mo modo.</s> + </p> + <p> + <s xml:space="preserve">In primis igitur <reg norm="ſciendum" type="context">ſciendũ</reg> eft, <reg norm="quod" type="simple">ꝙ</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 ſimpli<lb/>cibus partibus, vt <reg norm="etiam" type="context">etiã</reg> <reg norm="median- tibus" type="simple">median-<lb/>tibꝰ</reg> <reg norm="compoſitis" type="context">cõpoſitis</reg>, ut in pręſenti <reg norm="exem" type="context">exẽ</reg> <lb/>plo pro primis pofitionibus ac-<lb/>ceperunt .10. et .8. pro ſecundis <lb/>verò compoſitis <reg norm="cum" type="context">cũ</reg> numero .3. <lb/><reg norm="inuenerunt" type="context">inuenerũt</reg> .23. et .19. pro tertijs <lb/><reg norm="autem" type="wordlist">aũt</reg> <reg norm="compoſitis" type="context">cõpoſitis</reg> <reg norm="cum" type="context">cũ</reg> <reg norm="quinque" type="simple">quinq;</reg>, notaue <lb/>runt .38. et .32. vnde prima ſum <lb/>marefultauit .71. ſecunda verò <lb/>59. ita <reg norm="quod" type="simple">ꝙ</reg> <reg norm="primus" type="simple">primꝰ</reg> error remanebat <lb/>21. <reg norm="ſecundus" type="context simple">ſecũdꝰ</reg> <reg norm="autem" type="wordlist">aũt</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 exſummis partium ſimplicium.</s> + </p> + <p> + <s xml:space="preserve">Vtexempli gratia, in figura <var>.B.</var> ſumma propoſita partium ſimplicium eſt .39. <lb/>vt diximus, eo quòd ab ipſo .50. detraxerimus .11. ſumma ſcilicet numerorum adij <lb/>ciendorum ad efficiendas partes compofitas, ſumma poſteà fimplicium partium <lb/>primæ poſitionis, erit .60. eo quòd prima pars erat .10. ſecunda autem ſimplex 20. <lb/>tertia verò fimplex .30. iuxta ordinem propoſiti. </s> + <s xml:space="preserve">Summa deinde ſimplicium <reg norm="partium" type="context">partiũ</reg> <lb/>fecundæ poſitionis effet .48. quia prima eius pars erat .8. ſecunda verò ſimplex .16. <lb/>tertia autem ſimplex .24. vnde prima ſumma excederet datam .39. per .21. differen-<lb/>tiæ, ſecunda verò per .9. vt ſupra vidimus de ſummis compoſitis à dato .50. compo-<lb/>fito, & hoc quidem mirandum non eft, quod ſcilicet tres ſummæ fimplicium par-<lb/>tium ſintinuicem inæqua-<lb/>les, ijſdem differentijs me-<lb/> + <anchor type="figure" xlink:label="fig-0118-01a" xlink:href="fig-0118-01"/> + diantibus, quibus <reg norm="differunt" type="context">differũt</reg> <lb/>dictæ tres ſummæ compofi <lb/>tæ, cum ab vnaquaque <reg norm="con" type="context">cõ</reg> <lb/><reg norm="poſitarum" type="context">poſitarũ</reg> ablatus fit nume-<lb/>rus .11. æqualiter, vnde ex <lb/>neceſſitate, permutando, <lb/><reg norm="earum" type="context">earũ</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 noſtri lib.</ref> ſummæ 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, ſed à fimplici-<lb/>bus per ſe, & 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 ſcilicet ſecundum 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 ſe-<lb/>cundæ poſitionis. <reg norm="&c." type="unresolved">&c.</reg> quòd benè ſe habet. </s> + <s xml:space="preserve">Vbi animaduertendum eſt, quod ille <lb/>numerus .12. non eft accipiendus per ſe vt differentia errorum hoc eft .21. et .9. nifi <lb/>peràccidens, fed benè perfe, vt <reg norm="differentia" type="context">differẽtia</reg> inter .60. er .48. ſimplices ſummas, quem <lb/>admodum .9. in hoc propoſito eft differentia per ſe inter .48. et .39 per accidens ve-<lb/>ro inter .59. et .50.</s> + </p> + <p> + <s xml:space="preserve">Cognoſcendum igitur eft mediante .24. quinti Eucli. quod eadem proportio <lb/>eft primæ ſummæ (ſimplicium dico) ad ſuam primam partem, quæ ſecundæ ſum-<lb/>mæ ad ſuam, & tertiæ ſummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-<lb/>co antiqui ſi multiplicauiffent tertiam ſummam fim plicem cum prima parte prioris <lb/>fummæ fimplicis, & productum diuififfent per primam ſummam, vnde prima pars <lb/>quæſita tertiæ ſummæ orta fuiffet, abſque ullo negotio ipfius plus velminus) </s> + <s xml:space="preserve">Quare <lb/>habebimus tres terminos antecedentes ab vna parte, & tres terminos conſequen-<lb/>tesab alia parte continentes vnam <reg norm="eandemque" type="simple">eandemq́;</reg> proportionem, vnde ex .19. quinti, <lb/>vel .12. ſeptimi eorum differentiæ proportionales erunt, hoc eft, <reg norm="quod" type="simple">ꝙ</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 eſt inter primam & fecundam ſummam, ad differen-<lb/>tiam quæ eſt inter primas earum partes, quæ illius differentiæ, quæ eſt inter ſecun-<lb/>dam & tertiam ſummam, ad differentiam, quæ eft inter primas illarum partes, ſed <lb/>harum .4. differentiarum, tres nobis cognitæ ſunt, ideft .12. 2. et .9. ergo ex regula de <lb/>tribus ab Eucli. in .20. ſeptì<unsure/>mi ſpeculata inueniebatur quarta differentia, quæ eft .1. <lb/>cum dimidio.</s> + </p> + <p> + <s xml:space="preserve">A compofitis ſummis idem etiam proueniet, ſed non vt ex proprijs caufis, & per <lb/>ſe, ſedper 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) ipſius .71. ad .59. quæ ipſius .60. ad .48. nec ea quæ ipſius .59. ad <num value="50">.<lb/>50.</num> eft quæ ipſius .48. ad .39: </s> + <s xml:space="preserve">Vnde non erit eadem proportio ipſius .71. ad .59. quæ <lb/>ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipſius .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 inęqualitatis.</s> + </p> + <p> + <s xml:space="preserve">A fimplicibus igitur ſummis pendet ratio huiuſmodi effectus.</s> + </p> + <p> + <s xml:space="preserve">Si vero prima pars fecundæ poſitionis effet .4. tunc ſecunda eius pars effet .8. & ter-<lb/>tia .12. quarum ſumma effet .24. (harum fimplicium partium ſeilicet) & minor vera <lb/>(39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum <lb/>partium effet .6. differentia vero primæpartis ſecundæ poſitionis, a prima parte quę <lb/>fita effet .2. cum dimidio. </s> + <s xml:space="preserve">Vnde in huiuſmodi 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 cognoſcatur differentia .36. <lb/>differentia ſcilicet ſimplicium ſummarum ipſarum poſitionum.</s> + </p> + <p> + <s xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eſt, quod multiplicabant <lb/>diametraliter errores cum primis partibus, hoc eſt primum errorem cum prima par <lb/>te, hoc eſt cum numero ſecundæ poſitionis, ſecundum vero errorem cum prima <lb/>parte, hoc eſt cum numero primæ poſitionis, differentiam poſteà vel aggregatum <lb/>horum duorum productorum diuidebant per differentiam vel aggregatum dicto-<lb/>rum errorum, proueniens poſteà erat prima pars quæſita numeri propoſiti. </s> + <s xml:space="preserve">Vn-<lb/>de oriebantur tria producta, quorum <reg norm="tertium" type="context">tertiũ</reg>, hoc eſt differentia, ſeu aggregatum il-<lb/>lorum conſtituebatur ex differentia feuaggregato errorum, & ex numero quæ-<lb/>fito.</s> + </p> + <p> + <s xml:space="preserve">Vtin præfenti exemplo, primus error eſt .21. qui multiplicatus cum prima par-<lb/>te ſecundæ poſitionis, quæ eſt .8. producit .168. <reg norm="ſecundus" type="context">ſecũdus</reg> verò error eſt .9. qui multi-<lb/>plicatus cum prima parte primę poſitionis producit .90. differentia autem horum <lb/>productorum eſt .78. quæ diuifa per differentiam errorum, quæ eſt 12. dabit .6. <reg norm="cum" type="context">cũ</reg> di <lb/>midio, pro prima parte quæſita dati numeri diuiſibilis, qui erat .50.</s> + </p> + <p> + <s xml:space="preserve">Hæc omnia rectè ſe habent. </s> + <s xml:space="preserve">Sed, vt ſupra dixi diuiſor non eft per ſe differentia <lb/>errorum, neque etiam differentia per ſe ſummarum compoſitarum, fed bene fim-<lb/>plicium.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ſpeculatione, accipiendæ ſunt ſummæ ſimplices, quarum differen-<lb/>tiæ per ſe vtiles ſunt in huiuſmodi operatione; </s> + <s xml:space="preserve">& quia etiam rationes veritatis ex <lb/>iſtis, & non ex illis fluunt; </s> + <s xml:space="preserve">quamuis tam vnæ, quam aliæ ſint eædem in quantitate, <lb/>ideſt 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">Diſponantur igitur huiuſmo-<lb/> + <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> + di numeri tali ordine, vt fim-<lb/>plex ſumma, quæ ab vna reli-<lb/>quarum ſuperatur, & aliam ſupe-<lb/>rat, medium locum teneat; </s> + <s xml:space="preserve">@t <lb/>in propoſito exemplo ſumma <lb/>mediocris eft .48. quę à ſumma <num value="60">.<lb/>60.</num> ſuperatur, & ſuperat ſum-<lb/>mam .39. locata igitur fit hęc .48. <lb/>inter illas, ſuæ verò primæ partes <lb/>fimiliter conftitutæ ſint ſupra di-<lb/>ctas ſummas, cum ſuis <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 ipſius plus ſimpli-<lb/>citer: </s> + <s xml:space="preserve">figura verò <var>.D.</var> pro exem-<lb/>plo ipſius 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 ſolum pro hypoteſi, <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">Demonſtrandum nobis nunc relinquetur, maximum productum æquale effere-<lb/>liquis duobus; </s> + <s xml:space="preserve">hoc eſt productum .168. æquale effe productis .90. et .78. quorum <lb/>duorum productorum alterum .90. ſcilicet, generatur à differentia .9. quæ eft ſe-<lb/>cundę, & tertię ſummæ, in primum numerum antecedentem, qui eſt .10. alterum vc-<lb/>ro productum .78. ſcilicet, generatur à differentia .12. quę eſt primę, & ſecundę, ſum <lb/>mę in tertium numerum antecedentem, qui eſt .6. cum dimidio, maximum vero <lb/>productum .168. ſcilicet generatur à differentia maxima .21. quę eft primę, & tertię <lb/>ſummę (& ſemper ęqualis prioribus duabus differentijs .12. et .9.) in ſecundum nu-<lb/>merum antecedentem, qui eſt .8.</s> + </p> + <p> + <s xml:space="preserve">Conſtituantur igitur duo producta fimul iuncta ęqualia duobus .90. et .78. <lb/>lateralibus ſupra vnam aliquam rectam lineam <var>.q.p.</var> <reg norm="fitque" type="simple">fitq́;</reg> productum <var>.f.g.</var> ęquale <num value="90">.<lb/>90.</num> productum verò <var>.g.n.</var> ęquale .78. fit etiam baſis <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 ſubſcripta videre licet, et <var>.q.p.</var> erit .21. <lb/>Cogitemus nunc differentiam <var>.d.i.</var> diuiſam eſſe in puncto <var>.e.</var> ita vt eadem proportio <lb/>ſit ipſius <var>.d.e.</var> ad <var>.e.i.</var> quæ ipſius <var>.q.g.</var> ad <var>.g.p.</var> hoc eſt 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 ipſorum antecedentium numerorum, deinde à puncto <var>.e.</var> ducatur imagina-<lb/>tione <var>.u.e.o.</var> æ quidiſtans ipſi <var>.q.p.</var> & producatur <var>.q.n.</var> vſque ad <var>.u.</var> vnde ita ſe 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. ſexti vel .20. ſeptimi <var>.n.e.</var> <lb/>rectangulum æquale crit ipſi <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>: ſed cum <var>.g.i.</var> ſit 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 eſt igitur quod dictum fuit, hoc eſt <reg norm="quod" type="simple">ꝙ</reg> maximum productum ęquale ſit 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 poſteà quo volueris horum altero productorum ex maximo, <reg norm="diuiſoque" type="simple">diuiſoq́;</reg> <lb/>reliquo per differentiam conſequentium, ipſi diametraliter oppoſitam, pro <lb/>ueniet tibi numerus antecedens <reg norm="correſpondensque" type="simple">correſpondensq́;</reg> illi.</s> + </p> + <p> + <s xml:space="preserve">Animaduertendum tamen eſt, quòd ſi in figura à me ita ordinata, ſumma ſim-<lb/>plex propoſita medium locum occuparet, vt in figura <var>.D.</var> arithmetica videri poteſt; <lb/></s> + <s xml:space="preserve">tunc vt habeatur eius productum, addenda ſimul erunt circunſtantia producta .eo <lb/><reg norm="quod" type="simple">ꝙ</reg> eius ſecundum latus eſſet antecedens medio loco conſtitutum, & prima pars <reg norm="quae- ſita" type="simple">quę-<lb/>ſita</reg> numeri propoſiti: </s> + <s xml:space="preserve">in qua figura <var>.D.</var> manifeſtè patet ratio, quare colligendi ſint <lb/>tam errores, quam producta, dum eorum alterum eſt plus, reliquum verò minus.</s> + </p> + <p> + <s xml:space="preserve">Speculatio figurę <var>.D.</var> arithmeticę videbitur in figura <var>.D.</var> geometrica, eodem fe <lb/>rè modo quo fecimus in figuris <var>.C.</var> mutatis mutandis, reſpectu ipſius plus, & minus.</s> + </p> + <p> + <s xml:space="preserve">Collectio namque <reg norm="errorum" type="context">errorũ</reg> ſimiliter accidentalis eſt, eo quod eſſentialis numerus <lb/>diuiſor per ſe, eſt maxima differentia ſummarum ſimplicium, vt in dicta figura <var>.D.</var> <lb/>cerni poteſt.</s> + </p> + <p> + <s xml:space="preserve">Sed vt ſuperius dixi, nunc etiam repeto, quòd rectè hoc loco multiplicabatur <lb/>ſumma ſimplex propoſita, cum prima par <lb/>te primę poſitionis, vt productum diuide <lb/>retur per primam ſimplicem ſummam, <lb/> + <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> + vnde proueniret nobis pars prima <reg norm="quaeſi- ta" type="simple">quęſi-<lb/>ta</reg> noſtri numeri propoſiti, ex regula de <lb/>tribus, vnica poſitione.</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 ſit .100. in quinque partes, tales <lb/>verò, <reg norm="quod" type="simple">ꝙ</reg> ſecunda duplo maior ſit prima <lb/>cum .2. ſimul, tertia autem æqualis ſit pri-<lb/>mæ & ſecundæ cum .3. vnitatibus iunctis, <lb/>quarta poſteà maior ſit prima ſecunda, & <lb/>tertia per .4. vnitates, quinta demum ſu-<lb/>peret reliquas omnes per quinque vnita <lb/>tes, vt in figura <var>.E.</var> videre eſt, quæ quidem <lb/>partes compoſitæ (ſumpta vnitate pro <lb/>prima) ita diſpoſitæ erunt .1. 4. 8. 17. 35. <lb/>quarum ſumma erit .65. ſimplices autem <lb/>cum diſpoſitæ fuerint erunt .1. 2. 3. 6. 12. <lb/>quarum ſumma erit .24. dempta igitur <lb/>cum fuerit hæc ſimplex ſumma .24. à com <lb/>poſita .65. reſiduum erit .41. hoc eſt ſum-<lb/>ma numerorum propoſitorum cum ſuis <lb/>iterationibus in ipſis partibus, quod cum <lb/>per ſe clariſſimum ſit, ſuperſluum eſt <reg norm="ipsam" type="context">ipsã</reg> <lb/>ſummam annatomizare per ſingulas par-<lb/>tes, niſi quis habuerit eius cerebrum à fi-<lb/>gura Omega <reg norm="terminatum" type="context">terminatũ</reg>, cui tamen poſ-<lb/>ſemus dicere dictam ſummam .41. in .4. <lb/>partes diuidi, cuius prima eſſet .2. pro ad <lb/>ditione ad <reg norm="ſecundam" type="context">ſecũdam</reg> partem ſimplicium, + <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"/> + ſecunda verò eſſet .5. pro additione ad tertiam partem ſimplicium, tertia autem eſ-<lb/>ſet .11. pro additione ad quartam partem ſimplicium, quarta demum eſſet .23. pro <lb/>additione quintæ partis ſimplicium, quarum partium .2. 5. 11. 23. ſumma eſt .41. vt <lb/>diximus. </s> + <s xml:space="preserve">Hæc igitur ſumma .41. ſubducenda eſt à numero .100. propoſito, vnde re-<lb/>linquetur .59. pro ſumma partium ſimplicium numeri propoſiti, quarum prima erit <lb/>2. cum vndecim vigeſimisquartis ex diuiſione huiuſmodi .59. per .24. ſummam par-<lb/>tium ſimplicium ex viregulæ de tribus, dicendo ſi .24. prouenit nobis ab .1. prima <lb/>partium ſimplicium, à quo proueniet nobis .59? </s> + <s xml:space="preserve">vnde proueniet à .2. cum vndecim <lb/>vigeſimisquartis pro prima parte quæſita, ſecunda verò iuxta propoſitum, erit .6. <lb/>cum .22. vigeſimisquartis, tertia autem .12. cum nouem vigeſimisquartis, quarta po<lb/>ſteà .25. cum .18. vigeſimisquartis, quinta demum erit .52. cum .12. vigeſimisquartis, <lb/>quarum omnium ſumma erit .100.</s> + </p> + <p> + <s xml:space="preserve">STifelius in primo exemplo regulæ falſi, ita inquit.</s> + </p> + <p> + <s xml:space="preserve">Quæratur numerus, à cuius dimidio ſubtractæ partes tertia, & quarta relin-<lb/>quatur .300.</s> + </p> + <p> + <s xml:space="preserve">Ipſe enim ſupponit .300. pro reſiduo cognito alterius numeri incogniti, deinde <lb/>accipit .24. pro prima poſitione numeri cogniti, à cuius medietate abſcindit tertiam <lb/>& quartam partem ipſius medietatis, vnde remanet .5. qui quidem numerus .5. ex <num value="22">.<lb/>22.</num> quinti vel .15. ſeptimiſe ha-<lb/>bebit ad .24. vt .300. ad <reg norm="numerum" type="context">numerũ</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 diuiſerit 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">Conſideremus igitur <reg norm="meam" type="context">meã</reg> di-<lb/>ſpoſitionem numerorum huiuſ-<lb/>modi exempli, in figura hic ſup-<lb/>poſita <var>.F.</var> in qua videre licebit <lb/>quo pacto ipſe etiam Stifelius ac <lb/>cipiat diuiſorem .5. vt <reg norm="differentiam" type="context">differentiã</reg> <lb/>errorum & non ut differentiam <lb/>duorum conſequentium .5. et .10 <lb/>ſicuti eſt re uera, ut diuiſor dico, <lb/>ex rationibus à me hic ſupra ad-<lb/>ductis, quamuis vna & eadem ſit <lb/>quantitas neceſſariò ut patet.</s> + </p> + <p> + <s xml:space="preserve">ACcipiamus adhuc aliud exemplum à Tartalea propoſitione .9. <reg norm="datum" type="context">datũ</reg>, & <reg norm="oppoſitum" type="context">oppoſitũ</reg> <lb/>priori; </s> + <s xml:space="preserve">nam ſicut in illo numerus ſimplex habebatur per ſubtractionem ſum-<lb/>mæ numerorum adijciendorum, in hoc fitèconuerſo, hoc eſt per additionem nu-<lb/>merorum ſubtrahendorum.</s> + </p> + <p> + <s xml:space="preserve">Problema igitur ita ſe habet. </s> + <s xml:space="preserve">Fuit quidam mercator qui habebat aliquot au-<lb/>reos, cuius quantitas poſteà quærenda erit, hic enim fecit duo itinera, ut aliquod <lb/>dictis aureis mediantibus lucrum faceret, in primo autem itinere duplicauit nume-<lb/>rum ſuorum aureorum, ex quibus poſteà conſumpſit .4. pro aliquibus expenſis, 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"/> + ſecundo itinere iterum duplicauit ſuos aureos, ex quibus etiam poſtea conſumpſit <num value="8">.<lb/>8.</num> numeratis poſteà pecunijs reperit tantummodo .24. aureos in eius marſupio, <reg norm="quae ritur" type="simple">quę<lb/>ritur</reg> nunc quot habebat aureos in principio primi itineris.</s> + </p> + <p> + <s xml:space="preserve">Intali caſu, cum ipſe quolibet itinere duplicabat eius pecuniam, nulli dubium <lb/>eſt quòd in fine ſecundi itineris ipſe habuiſſet pecuniam ſuam quadruplicatam, ſi <lb/>ex ipſa nihil detractum fuiſſet, ſed quia in fine primi itineris conſumpſit .4. aureos, <lb/>quibus alios .4. lucratus eſſet in ſecundo itinere, poſteà conſumpſit iterum <num value="8">.<lb/>8.</num> aureos, ita <reg norm="quod" type="simple">ꝙ</reg> ex quadruplo ſuæ primæ pecuniæ, rectè dici poteſt, quod conſum-<lb/>pſerit .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 ſi addiderimus .16. ipſi .24. habebimus .40. pro quadruplo eius <lb/>prioris pecuniæ. </s> + <s xml:space="preserve">Rectè igitur dici poteſt, ſi .4. prouenit ab vno, à quo numero pro <lb/>ueniet .40.</s> + </p> + <p> + <s xml:space="preserve">Videamus igitur nunc quo pacto hoc reſpondeat cum methodo antiquorum. <lb/></s> + <s xml:space="preserve">Ego enim inueni duas poſitiones ſcriptas à Tartalea pro prima pecunia hoc eſt .12. <lb/>et .14. ſed à .12. pro primo errore reperi .8. more antiquo à .14. verò pro ſecundo er-<lb/>rore proueniebat .16. producta autem horum numerorum diametraliter, ſunt .112. <lb/>et .192. quorum differentia eſt .80. pro tertio producto, quo diuiſo per differen-<lb/>tiam <reg norm="errorum" type="context">errorũ</reg> .8. ſcilicet, præbetnobis .10. pro pecunia quæſita, vt etiam ego inueni.</s> + </p> + <p> + <s xml:space="preserve">Sed hoc mihi viſum eſt ſubtilius examinare mea methodo mediante, vtin figu-<lb/>ra <var>.G.</var> videre eſt, prius enim ſuo loco poſuitria producta dicta, deinde duas poſitio <lb/>nes .12. et .14. & quia ſciebam productum .112. oriri à multiplicatione .14. cum .8. <lb/>ideo poſui talem numerum .8. ſuo loco diametraliter oppoſito ei producto .112. <lb/>& quia ſciebam etiam productum .192. naſci ex .12. et .16. ideo ſuo loco poſui hunc <lb/>numerum .16. qui eſt maxima differentia inter duos conſequentes ( ita à me ſupra <lb/>nominatos) à qua differentia dempta priori .8. iam inuenta, reliqua .8. mihi daba-<lb/>tur, <reg norm="quam" type="context">quã</reg> ſuo loco notaui, ſuo <reg norm="etiam" type="context">etiã</reg> <lb/>loco ſcripſi .2. differentiam inter <lb/>12. et .14. antecedentium. </s> + <s xml:space="preserve">ſed <lb/> + <anchor type="figure" xlink:label="fig-0124-01a" xlink:href="fig-0124-01"/> + quia ſciebam eandem proportio <lb/>nem eſſe inter hanc differentiam <lb/>& <reg norm="differentiam" type="context">differentiã</reg> .8. huic <reg norm="ſuppoſitam" type="context">ſuppoſitã</reg>, <lb/>quæ reperitur inter .12. <reg norm="antecedem" type="context">antecedẽ</reg> <lb/>tem, & <reg norm="ſuum" type="context">ſuũ</reg> conſequentem; </s> + <s xml:space="preserve">ideo <lb/>poſui .48. pro dicto conſequenti, <lb/>diuiſi poſtea productum .80. per <num value="8">.<lb/>8.</num> differentiam ei diametraliter <lb/>oppoſitam, vnde prouenit mihi <lb/>10. cui ita proportionatus eſt <reg norm="ſuus" type="simple">ſuꝰ</reg> <lb/>numerus conſequens .40. vt .48. <lb/>ad .12. et .56. ad .14. exijſdem ra-<lb/>tionibus à meſupra dictis. <lb/></s> + <s xml:space="preserve">In tali igitur figura videntur nu-<lb/>merinaturaliter <reg norm="correſpondentes" type="context">correſpondẽtes</reg> <lb/>ipſis poſitionibus, & hac metho-<lb/>do poſſumus inuenire tales numeros conſequentes in omnibus alijs exemplis à no-<lb/>ſtris maioribus ſcriptis.</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 ſit quarta pars totius vaſis cum oper <lb/>culo, pars autem media ſine operculo, ſit quinta pars ipſius 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 eſt pes .4. cum .14. decimisnonis ta-<lb/>lium partium, ſeu librarum, qualium operculus eſt .18. </s> + <s xml:space="preserve">Videamus igitur & nos ex <lb/>noſtra figura, quo pacto hoc reſpondeat veritati.</s> + </p> + <p> + <s xml:space="preserve">Inuenta enim ſunt tria producta, <reg norm="iam" type="context">iã</reg> orta ex dicta methodo .10. 100. 90. quæ ſuis <lb/>locis notaui, vt in ſigura <var>.H.</var> ſubſcripſi etiam duas illorum poſitiones .5. et .10. cum <lb/>ſua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat <num value="1">.<lb/>1.</num> quod ſuo loco notaui, ſimiliter quia .100. productum, pro vno eius laterum erat <num value="5">.<lb/>5.</num> reliquum autem .20. ſuo loco poſui, & quia differentia inter .20. et .1. duo latera, <lb/>quę eſt .19. æqualis eſt ei, quæ inter duo conſequentia duarum poſitionum, etiam <lb/>ſuo loco ipſam conſtitui, ſed quia hæc differentia eſt vnum laterum producti .90. er <lb/>go reliquum latus <reg norm="quæſitum" type="context">quæſitũ</reg> erit .4. cum .14. decimisnonis, rectè igitur operatur. <lb/></s> + <s xml:space="preserve">ſed cum eadem proportio ſit inter differentiam .5. ſuperiorem, et .19. inferiorem, <lb/>quæ eſt vnius <reg norm="antecedentis" type="context">antecedẽtis</reg> ad ſuum <lb/>conſequens, </s> + <s xml:space="preserve">quare .10. <reg norm="antecedens" type="context">antecedẽs</reg> <lb/>habebit pro ſuo conſequenti .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 ſi .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">Huiuſmodi autem rei ratio ita <lb/>ſe <reg norm="hent" type="context">hẽt</reg>, eſto linea <var>.a.e.u.</var> cuius pars <lb/>a. ſit quarta reliquarum <var>.e.u.</var> iuncta <lb/>rum, ſed <var>.e.</var> ſit quinta ipſius <var>.a</var>. </s> + <s xml:space="preserve">Tunc <lb/>clarum erit quod <var>.e.</var> erit vigeſima <lb/>dictarum <var>.e.u.</var> </s> + <s xml:space="preserve">quare erit decima-<lb/>nona ipſius <var>.u.</var> ſed <reg norm="cum" type="context">cũ</reg> u. <reg norm="sumpta" type="context">sũpta</reg> ſit vt <num value="18">.<lb/>18.</num> rectè igitur dici poteſt, ſi .u: ut <num value="19">.<lb/>19.</num> prouenit ab <var>.a.</var> ut quinque, à <lb/>quot ipſius <var>.a.</var> proueniet <var>.u.</var> ut .18.</s> + </p> + <p> + <s xml:space="preserve">Quis enim non uidet quod diui <lb/>ſa cum fuerit <var>.u.</var> in partes .19. quod <lb/>quinque illarum æquabuntur ipſi <var>.<lb/>a.</var> cum quælibet fuerit æqualis <var>.e.</var> <lb/>quintæ parti ipſius <var>.a</var>.</s> + </p> + <p> + <s xml:space="preserve">HAc igitur mea numerorum diſpoſitione mediante reperiuntur ipſi numeri in <lb/>feriores naturaliter conſequentes, correſpondentesq́ue ipſis ſuperioribus an <lb/>tecedentibus; </s> + <s xml:space="preserve">quamuis multoties <reg norm="contingere" type="context">cõtingere</reg> poſſit, ut generationes ſeu com-<lb/>poſitiones ipſorum ignorentur: </s> + <s xml:space="preserve">& quia tam à differentijs errorum, quam ab illis, <lb/>quę ſunt inter ueros conſequentes numeros ( propter eorum æqualitatem ) elicitur <lb/>ipſa ueritas, proptereà rectè antiqui illis vſi ſunt, quamuis ſint potius ſenſum <lb/>ſequuti, uel experientiam, quam rationem: </s> + <s xml:space="preserve">quæ quidem ratio pendet ab ipſis na-<lb/>turalibus numeris conſequentibus ( ut ſupra uidimus ) etſi incognitis ut plurimum, <lb/>quod ſi ipſos inuenire primò nobis datum fuiſſet, unica <reg norm="tantummodo" type="context">tantũmodo</reg> poſitio ſuffice- + <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 ipſa regula de tribus, vt <reg norm="iam" type="context">iã</reg> ſępius <reg norm="dictum" type="context">dictũ</reg> eſt, quod <reg norm="etiam" type="context">etiã</reg> clarè patet ex di-<lb/>uerſis problematibus .17. lib. ipſius Tartaleæ, vt ex primo, quod aſſumpſimus pro <lb/>noſtro etiam primo exemplo, ex .9. 15. 16. 17. 18. 19. 20. 27. 28. 29. 30. 33. & ex <lb/>alijs multis, vbi facillimè inue nitur conſequens ipſius poſitionis, qui quidem nume-<lb/>rus eſt diuiſor producti ipſius numeri propoſiti in numerum poſitionis, vnde poſteà <lb/>prouenit <reg norm="ſecundum" type="context">ſecundũ</reg> latus huiuſmodi producti, hoc eſt 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 manifeſtè pateant <reg norm="eorum" type="context">eorũ</reg> rationes abſque regula falſi, cuius <lb/>regulæ rationes non ita promptè ipſi intellectui ſe offerunt, vt ſupra vidimus.</s> + </p> + <p> + <s xml:space="preserve">Accipiamus pro exemplo .21. problema ipſius Tartalæ in dicto .17. libr. vbi ſup-<lb/>ponit vnum hædum diuiſum in .4. partes, quarum quælibet vendebatur eodem pre <lb/>cio, interiora vero .6. denarijs minus quam quælibet dictarum partium, ſumma <lb/>autem omnium iſtorum denariorum fuit .127. quæritur nunc precium cuiuſque <lb/>partis.</s> + </p> + <p> + <s xml:space="preserve">Tale enim problema hoc etiam alio breuiori modo poteſt ſolui, vt rationes ma-<lb/>gis pateant, quam ex regula falſi.</s> + </p> + <p> + <s xml:space="preserve">Nam ſi illi numero .127. denariorum, additus fuerit numerus .6. ſumma erit .133. <lb/>qua diuiſa per quinque, illico proueniet .26. cum tribusquintis pro precio vniuſcu-<lb/>iuſque 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">preciũ</reg> <reg norm="vniuſcuiuſque" type="simple">vniuſcuiuſq;</reg> illorum.</s> + </p> + <p> + <s xml:space="preserve">Hoc etiam problema, hac alia methodo ſolui poteſt, dicendo exregula de tribus, <lb/>ſi ex .20. vtrorunque qui ea vendit, vult .44. quid volet ex .12? </s> + <s xml:space="preserve"><reg norm="manifeſtum" type="context">manifeſtũ</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, ſed 12. cum .28. pomis valebant .36. ergo .16. poma ſola valebunt .9. <lb/>cum tribus quintis, hoc enim clarè ex ſe patet; </s> + <s xml:space="preserve">quare cum dixerimus, ſi .16. poma ſo <lb/>la valent .9. cum tribusquintis, vnum valebit <var>.o.</var> cum tribusquintis, ſed 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/>ſcilicet vnius pomi, reliquum .1. cum tribusquintis, erit precium vnius pyri.</s> + </p> + <p> + <s xml:space="preserve">Idem etiam dico de .28. problemate, vbi ſupponit quod quidam comparaſſet <lb/>quatuor petias, vt vulgo dicitur, panni pro ducatis .96. quarum primæ precium ob-<lb/>litus ſit, ſed memoria tenet pro ſecunda ſoluiſſe .6. plusquam pro prima, & pro ter-<lb/>tia ſoluiſſe .8. plus quam pro ſecunda, & pro quarta ſoluiſſe .10. plus quam pro ter-<lb/>tia, quæritur nunc quantum fuerit precium vniuſcuiuſque 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 ſolui, vt in <lb/>ſubſcripta figura <var>.I.</var> videri <lb/>poteſt, <reg norm="addendo" type="context">addẽdo</reg> ſimul omnes <lb/>exceſſus. </s> + <s xml:space="preserve">Nam exceſſus <reg norm="ſecum" type="context">ſecũ</reg> <lb/>dæ ſupra primam eſt .6. ſed <lb/>cum exceſſus tertiæ ſupra ſe <lb/>cundam ſit .8. ergo exceſſus <lb/>tertiæ ſupra 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"/> + ſed exceſſus quartæ ſupra tertiam eſt .10. vnde ſupra ſecundam erit .18. & ſupra pri-<lb/>mam erit .24. quæ omnia ſimul addita erunt .44. & in qualibet harum trium remane-<lb/>bit una pars æqualis primæ quantitati, </s> + <s xml:space="preserve">quare ſi ex .96. detractus fuerit numerus .44. <lb/>reliquus 52. erit quadruplus primæ, </s> + <s xml:space="preserve">quare prima pars valebit .13. ſecunda .19. ter-<lb/>tia .27. & quarta .37. quarum omnium ſumma eſt .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 poſitionibus autem Tartaleæ in noſtra figura <var>.K.</var> digeſtis, videre poſſumus <lb/>quo pacto <reg norm="colligantur" type="context">colligãtur</reg> huiuſ <lb/>modi <reg norm="conſequentes" type="context">conſequẽtes</reg> numeri ſimpli-<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 compoſi <lb/>tæ .9. 15. 23. 33. ex quarum ſumma <lb/>80. ſubtrahitur .36. ſumma ſim-<lb/>plex ex ſimplicibus partibus .9. 9. <lb/>9. 9. & <reg norm="reſiduum" type="context">reſiduũ</reg> quod eſt .44. ſubdu <lb/>citur ex .96. ſumma compoſita & <lb/>propoſita, vnde remanet .52. pro <lb/>ſumma ſimplici, ex numero dato, <lb/>cuius proportio ad .13. eadem eſt <lb/>quæ .36 ad .9. & proptereà ſuper-<lb/>flua eſt ſecunda poſitio, <reg norm="quando" type="context">quãdo</reg> ſci <lb/>mus inuenire tales duos numeros <lb/>conſequentes, vt in hoc exemplo <lb/>ſunt .36. et .52. quia ex regula de <lb/>tribus poſteà elicitur veritas quæ-<lb/>ſita. </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 poſitiones primi <reg norm="exem" type="context">exẽ</reg> <lb/>pli Tonſtalli hoc eſt .33. et .31. vt in figuris hic ſubiectis <var>.P.Q.</var> facile quis poteſt vi-<lb/>dere, vbi in figura P. videbit nume-<lb/>ros compoſitos, in figura verò <var>.Q.</var> cer <lb/> + <anchor type="figure" xlink:label="fig-0127-02a" xlink:href="fig-0127-02"/> + net numeros ſimplices, à quibus pro <lb/>ueniunt rationes per ſe huiaſmodi <lb/>operationis, in figura autem <var>.R.</var> vide <lb/>bitur meus ordo, & iſtæ tres figuræ ſi <lb/>miles <reg norm="erunt" type="context">erũt</reg> tribus illis primis <var>.A.B.C.</var> <lb/>ita quòd cum quis illas intellexerit, il <lb/>lico etiam iſtas cognoſcet, 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/>ſimplicium <reg norm="numerorum" type="context">numerorũ</reg>, à quibus fluit <lb/>tota ratio (vt ſupra dixi) huiuſcemo <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 poteſt dici de .15. problemate (ſicut de alijs multis) vbi ponit tres <lb/>homines habentes .40. aureos quorum primus habet duas quintas partes ſecun-<lb/>di, <reg norm="ſecundus" type="context">ſecũdus</reg> verò <reg norm="quinque" type="simple">quinq;</reg> octauas tertij, <reg norm="quæritur" type="simple">quærit̃</reg> <reg norm="nunc" type="context">nũc</reg> quot ducatos habeat vnuſquiſque.</s> + </p> + <p> + <s xml:space="preserve">Quis non videt quæſo, <reg norm="quod" type="simple">ꝙ</reg> omnes partes erunt .15. quare cum dixerimus ſi .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. ſimiliter aſſero, vbi ponit <reg norm="alium" type="context">aliũ</reg> emiſſe tria fruſta panni pro ducatis .48. <lb/>quarum ſecundam habuit pro dimidio precio primæ, tertiam autem pro quarta <lb/>parte ipſius ſecundæ, </s> + <s xml:space="preserve">quare omnes partes erunt .13. quapropter precium tertiæ pe-<lb/>tiæ erit tertiadecima pars ipſius .48. hoc eſt .3. cum .9. tertijs decimis.</s> + </p> + <p> + <s xml:space="preserve">Adhuc duo exempla videtur mihi proponere, quorum primum eſt .38. eiuſdem <lb/>lib. vbi ſupponitur operarium quendam velle perficere opus quoddam ſpacio die-<lb/>rum .36. tali pacto, quod qualibet die, in qua ipſe operaturus ſit lucretur ſolidos <num value="16">.<lb/>16.</num> qualibet verò die, in qua nihil agat perdat ſolidos .24. </s> + <s xml:space="preserve">Tunc accidit, vt exacto <lb/>termino <reg norm="perfectoque" type="simple">perfectoq́;</reg> opere, tantum lucratus ſit, quantum perdiderit. </s> + <s xml:space="preserve">Quæritur <reg norm="nunc" type="context">nũc</reg> <lb/>quot fuerint dies lucri, quotúe perditionis.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi problematis operatio breuiſſima abſque vlla falſa poſitione ita erit, <lb/>hoc eſt diuidendo productum .36. in .24. per .40. ideſt per aggregatum ipſius .24. <reg norm="cum" type="context">cũ</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 ſe ſatis 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/>& vigeſimaſepti <lb/>mi ex regula de <lb/>tribus reperiun <lb/>tur partes ipſius <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 ſe inuicem habentes, vt .24. et .16. quæ ſunt .21. cum tribus quintis, et .14. <reg norm="cum" type="context">cũ</reg> dua <lb/>bus quintis, ex quo ſequitur, vt quod fit ex .21. cum tribus quintis, in 16. ęquale ſit 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 eſt .39. quod quidem à ſuperiori non differt, niſi quod <lb/>in fine operationis, operarius dictus lucratus eſt ſolidos .60: </s> + <s xml:space="preserve">quęritur <reg norm="nunc" type="context">nũc</reg> vt ſu-<lb/>pra, quot fuerunt dies lucri, & quot perditionis.</s> + </p> + <p> + <s xml:space="preserve">Hoc etiam abſque vlla falſa poſitione dicto citius poteſt ſolui, hoc modo, <reg norm="diuidem" type="context">diuidẽ</reg> <lb/>do ſcilicet illos .60 ſolidos per .40. ideſt per aggregatum .24. cum .16. proueniens <lb/>autem, quod erit .1. cum dimidio, adde ad latus ſuperius inuentum, hoc eſt .21. cum <lb/>tribus quintis, & ſunima erit .23. cum decima parte pro numero dierum lucri, dein-<lb/>de idem prouentum deme ex alio latere ſuperius 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 ſpeculatione cogitemus in figura <var>.N.</var> duo dicta producta inuicem <lb/>æqualia <var>.o.b.</var> et <var>.n.c.</var> exiſtente 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 ſi mente concepta fuerit recta <var>.e.<lb/>a.t.</var> æquidiſtans <var>.o.c.</var> ita vt rectangulum <var>.o.e.</var> ſit .60. </s> + <s xml:space="preserve">tunc rectangulum, ſeu productum <lb/><var>b.t.</var> ſuperabit rectangulum ſeu productum <var>.n.e.</var> per idem .60. ex communi conceptu, <lb/>eo quòd ex producto <var>.n.c.</var> ſublatum eſt productum <var>.a.c.</var> 24. & producto <var>.o.b.</var> additum <lb/>eſt productum <var>e.a.</var> 16. rectè igitur feci cum diuiſerim .60 per .40. vnde prouenit mi <lb/>hi <var>.u.a.</var> ideſt .1. cum dimidio, quod additum ipſi <var>.b.u.</var> compoſuit <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 ſi idem operator perdidiſſet .60. tunc cogitaremus parallelam dictam <var>.e.a.</var> t <lb/>ſuperius ductam eſſe ita vt ſecaret <var>.b.u.</var> & non <var>.u.n.</var> vnde adderet .24. ipſi 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 ſi aliquid notatu di-<lb/>gnum aduerterim, hoc eſt quod ſæpe accidere poterit ut caſus impoſſibiles <lb/>proponantur. </s> + <s xml:space="preserve">Quemadmodum ſi aliquis diceret, cupio mihi ueſtimentum 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 ſit albus, rubeus uerò alter, deinde albus ſit pretij .40. ſolidorum uniuſcuiuſ-<lb/>que cubiti, rubeusuerò precij .50. uellemq́ue omnes cubitos eſſe .8. nec plus <lb/>nec minus. </s> + <s xml:space="preserve">Vellem etiam ſoluere ſolidos 450. neque minus.</s> + </p> + <p> + <s xml:space="preserve">Hic igitur caſus impoſſibilis eſt, eo quòd .8. cubiti totius rubei eſſent precij ſo-<lb/>lidorum .400. tantummodo, unde ex alio panno albo minoris precij ſumere ali-<lb/>quid non poſſumus.</s> + </p> + <p> + <s xml:space="preserve">Idem etiam eueniret ſi uoluiſſet ſoluere ſolidos .320. neque plus, eo quòd .8. cu-<lb/>biti illius minoris precij, hoc eſt .40. ſolidorum, eſsent ualoris .320. ſolidorum tan <lb/>tummodo, quare pro alio panno nullus eſset locus. </s> + <s xml:space="preserve">Animaduertendum igitur erit <lb/>quod numerus poſſibilis ad ſoluendum tale quæſitum erit inter .400. et .320. & non <lb/>extra iſtos terminos, vt vnicuique patere poteſt.</s> + </p> + <p> + <s xml:space="preserve">Similiter idem in hoc alio caſu accidere poterit, ut ſi quis diceret.</s> + </p> + <p> + <s xml:space="preserve">Emi quinque petias panni pro aureis .55. pretium tamen primæ oblitus ſum, ſed <lb/>memoria teneo, quòd ſecunda altioris pretij erat quam ipſa prima per .4. & ter-<lb/>tia precioſior ſecunda per .7. et quarta carior tertia per .9. quinta verò ſuperabat <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 ſolutum eſſet, ſuperat aureos .55 quòd <lb/>quidem nullo pacto fieri poteſt, vt veri ſint ſupra dicti exceſsus, ſi verus eſt 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. ſed .22. cum .20. et .11. et .4. dant .57. qui numerus maior <lb/>eſt 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 ſciam) veras <reg norm="internasque" type="simple">internasq́</reg> cauſas <lb/>operationis perſpectiuæ perſectè docuerit, operæpre-<lb/>cium exiſtimaui <reg norm="aliquam" type="context">aliquã</reg> de ijs diſputationem ſuſcipere.</s> + </p> + <p> + <s xml:space="preserve">Multi enim <reg norm="eorum" type="context">eorũ</reg>, qui huiuſmodi operationis regulas <lb/>præſcribunt, cum eius effectuum veras cauſas igno-<lb/>rent, varios <reg norm="diuerſosque" type="simple">diuerſosq́;</reg> errores committunt, vt exempli <lb/>gratia in ſubſcripta figura ſuperficiali <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> ipſi <var>.q.d.</var> à puncto <var>.B.</var> interſecationis lineæ-<lb/>o.l. cum latere <var>.i.d.</var> trianguli, & (idem) indifferenter, ean-<lb/>dem quoque à puncto <var>.Z.</var> interſecationis ipſius <var>.o.l.</var> cum perpendiculari <var>.x.i.</var> ducunt. <lb/></s> + <s xml:space="preserve">neſcientes hunc ſolum eſſe verum modum, n onitem alium, quia ſi alius, talis eſſet, <lb/>hic, verus non exiſteret, nam ſi vellent ſeſe excuſ are, quòd ducendo dictam paralle-<lb/>lam à puncto <var>.B.</var> hoc fiat præſupponendo planum ipſius <var>.i.d.q.</var> verſus rectangulum <var>.<lb/>q.a.</var> orizontale inclinatum, ſecundum angulum <var>.i.d.q.</var> hæc excuſatio accipien-<lb/>da non eſſet, quia horum conſenſu, præſupponendo planum <var>.i.d.q.</var> inclinatum, <lb/>anguli inferiores rectanguli degradati, non tam acuti, quam ſunt duo <var>.i.d.q.</var> et <var>.i.q.<lb/>d.</var> eſſe deberent, quod facilè eorum ratione innoteſcet, quæ de figura corporea <var>.<lb/>A.</var> hîc ſubſcripta mox proponam, præter id, quòd volentes deinde aſpicere qua-<lb/>dratum degradatum, oporteret huiuſmodi planum reſpectu oculi ita collocare, <lb/>quemadmodum ſe habet linea <var>.i.d.</var> reſpectu <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/>ſis <var>.o.l.</var> et <var>.x.i.</var> perpendiculari ipſi <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 ſpeculatione imaginemur in figura corporea .A: <var>q.a.</var> eſſe figuram re-<lb/>ctangulam <reg norm="orizontalemque" type="simple">orizontalemq́;</reg> ad degradandam ſuper 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, ſit autem oculus reſpicientis <var>.o.</var> cuius altitudo <var>.o.p.</var> ab orizonte, qui <reg norm="quidem" type="context">quidẽ</reg> <lb/>conſpicit 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> ſit verò primum ita <lb/>collocatus pes <var>.p.</var> eius qui reſpicit, vt linea <var>.p.l.</var> perpendicularis ipſi <var>.u.a.</var> lateri re-<lb/>ctanguli, medio loco poſita ſit, inter <var>.a.n.</var> et <var>.u.s</var>. </s> + <s xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</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 ſuperficie-<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> poſitæ erunt. </s> + <s xml:space="preserve">Imaginemur etiam lineam <var>.p.s.n.</var> perpendicula-<lb/>rem ipſi <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 conſtituet, cum ex .28. duæ <var>.u.q.s.</var> et <var>.a.d.n.</var> ſint parallelæ ipſi <var>.p.l.</var> et <var>.s.n.</var> ipſi <var>.u.<lb/>a.</var> & quia ſupponitur <var>.o.p.</var> perpendicularis plano orizontali, Angulus ergò <var>.o.p.l.</var> re-<lb/>ctus erit ex ſecunda 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 ſuperficiei, in qua ſunt duæ lineæ <var>.o.p.</var> et <var>.s.n.</var> ex .4. <lb/>11. perpendicularis erit, & ſuperficies orizontalis <var>.a.s.</var> perpendicularis erit cum dicta <lb/><var>o.s.n.</var> ex .18. eiuſdem 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 ſcientia, in ſuperficiebus duorum triangulorum <var>.o.u.q.</var> et <var>.<lb/>o.a.d.</var> erunt, ſi noluerimus cogere aduerſarium ad confitendum duas lineas rectas in-<lb/>cludere ſuperficiem, quemadmodum cogere-<lb/> + <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> + tur facere, ſi opinaretur duas alias rectas per <lb/>eadem puncta <var>.o.s.n.</var> tranſire, quæſunt in di-<lb/>ctis ſuperficiebus. </s> + <s xml:space="preserve">Vnde <var>.o.s.</var> et <var>.o.n.</var> communes <lb/>erunt ſectiones duarum dictarum <reg norm="ſuperficierum" type="context">ſuperficierũ</reg> <lb/>cum ſuperficie <var>.o.s.n</var>. </s> + <s xml:space="preserve">Imaginemur nunc has <lb/>duas ſuperficies <var>.o.u.</var> et <var>.o.a.</var> quarum commu-<lb/>nis ſectio ſit <var>.o.t.</var> (quæ erit linea recta ex .3. lib. <lb/>II.) quæ erunt perpendiculares ſuperficiei <var>.o.s.<lb/>n.</var> ex .4. et .14. iam dictis. </s> + <s xml:space="preserve">& ex .19. eiuſdem <lb/><var>o.t.</var> perpendicularis eidem ſuperficiei <var>.o.s.n.</var> <lb/>erit, & ex .6. eiuſdem hæc linea <var>.o.t.</var> duabus <var>.u.<lb/>q.s.</var> et <var>.a.d.n.</var> parallela exiſter, & ex .9. eiuſdem <lb/>hæc linea <var>.o.t.</var> duabus <var>.u.q.s.</var> et <var>.a.d.n.</var> parallela <lb/>exiſtet, & ex eadem .9. erit parallela ipſi <var>.p.l.</var> <lb/>Imaginemur nunc planum, ſuper quod deſide <lb/>remus videre quadrangulum orizontale, quod <lb/>planum, exempli gratia, ſit primo, vt iam dixi-<lb/>mus, locatum in linea <var>.q.d.</var> ad angulos rectos <lb/>cum plano orizontali, cuius communes ſectio <lb/>nes cum ſuperficiebus <var>.s.t.</var> et <var>.n.t.</var> viſionis la-<lb/>terum <var>.u.q.</var> et <var>.a.d.</var> ſint <var>.i.q.</var> et <var>.i.d.</var> & com-<lb/>munis ſectio trianguli <var>.o.u.a.</var> ideſt viſionis <lb/>lateris <var>.a.u.</var> cum dicto plano, ſit <var>.r.e</var>. </s> + <s xml:space="preserve">Vnde ex <lb/>communi ſcientia rectangulum orizontale, <lb/>oculo <var>.o.</var> ſeipſum patefaciet in plano <var>.i.q.d.</var> ſe- + <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 ſectio ſuperficiei <var>.p.t.</var> <lb/>cum dicto plano, ſit <var>.i.x.</var> quæ <var>.i.x.</var> perpendicularis erit <var>.s.a.</var> ſuperficiei orizontali ex <lb/>19. lib. 11. quia <var>.p.t.</var> eſt etiam orizonti perpendicularis ex .18. eiuſdem, cum <var>.o.p.</var> ei-<lb/>dem perpendicularis exiſtat. </s> + <s xml:space="preserve">Vnde <var>.i.x.</var> erit altitudo trianguli <var>.i.q.d.</var> & æqualis ipſi <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> ſectio ſuperficiei triangularis <var>.o.a.u.</var> <reg norm="cum" type="context">cũ</reg> <lb/>ſuperficie <var>.p.t.</var> quæ <var>.o.l.</var> ſecando lineam <var>.e.r.</var> in puncto <var>.Z.</var> nobis oſtendet quantum di-<lb/>ſtare ſeu eminens eſſe debeat latus <var>.e.r.</var> in plano ab <var>.q.d.</var> medio ipſius <var>.z.x</var>. </s> + <s xml:space="preserve">Et quia <lb/>præſuppoſuimus <var>.p.l.</var> in eodem medio, inter <var>.u.s.</var> et <var>.a.n.</var> ideo <var>.x.q.</var> ęqualis erit <var>.x.d.</var> <lb/>& ex .4. lib. primi <var>.i.q.</var> ipſi <var>.i.d.</var> et <var>.e.r.</var> parallela ipſi <var>.q.d.</var> ex .6. lib. 11. cum ipſa quoque <lb/>ſit perpendicularis ſuperficiei <var>.p.t.</var> ex <ref id="ref-0020">.19. eiuſdem</ref>. </s> + <s xml:space="preserve">Hucuſque igitur in figura cor-<lb/>porea <var>.A.</var> prodeunt in lucem omnes cauſæ effectuum figuræ ſuperficialis <var>.A.</var> ideſt vn <lb/>de fiat, vt in ipſa figura ſuperficiali, triangulum <var>.o.p.l.</var> tale conſurgat, & quid ſignifi-<lb/>cet <var>.o.</var> et <var>.o.p.</var> et <var>.p.l.</var> et <var>.o.l.</var> & quam ob cauſam tale quoque formetur triangulum <var>.i.<lb/>q.d.</var> atque in tantam altitudinem, quantam obtinet <var>.o.p.</var> & quid ſint latera <var>.i.q.</var> et <var>.i.<lb/>d.</var> & quare erigatur <var>.x.i.</var> parallela ipſi <var>.p.o.</var> ab eadem <var>.p.o.</var> tanto ſpatio diſtans, & qua <lb/>ratione producatur à puncto <var>.Z.</var> ipſa <var>.Z.r.e.</var> parallela ipſi <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 obſeruandum eſt, quòd ſi planum ipſius <var>.i.q.d.</var> in figura corporea aliquan-<lb/>tulum inclinatum eſſet orizontem verſus, anguli <var>.i.q.d.</var> et <var>.i.d.q.</var> maiores exiſterent, <lb/>quàm cum idem eſt ipſi orizonti perpendiculare, quemadmodum clarè demonſtra-<lb/>tum fuit in .39. primi Vitelionis.</s> + </p> + <p> + <s xml:space="preserve">Non igitur rectè fit ſi in figura ſuperficiali ducatur à puncto <var>.B.</var> parallela ipſi <var>.q.d.</var> <lb/>abſque 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 ſubſequentes <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 ſumere ſemper <var>.p.x.</var> figuræ ſuperfi-<lb/>cialis æqualem ei, quæ eſt corporeæ, & eidem ſuperficiali, adiungere <var>.x.d.</var> æqualem <lb/>ei, quæ eſt corporeæ, & compoſito <var>.p.d.</var> ex dictis duabus lineis, in figura ſuperficiali, <lb/>addere <var>.d.q.</var> æqualem ei, quæ eſt 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 diſtans 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"/> + ſteà. <var>x.i.</var> erigetur æqualis lineæ <var>.p.o.</var> & ſuis terminis <lb/>concludetur triangulum <var>.i.q.d.</var> & id quod remanet. <lb/></s> + <s xml:space="preserve">Vnde ſi longius diſerendo progrediare, patebit ex <num value="4">.<lb/>4.</num> primi <var>.i.d.</var> ſuperficialem, 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 eſt, & vniuerſalis, licet in figura ſuperficiali <var>.A.</var> ſupe-<lb/>rius poſita, ſecundum communem antiquorum conſuetudinem exemplum <lb/>dederim, effectus enim idem eſt. </s> + <s xml:space="preserve">Sed ſi quis vellet conſiderare dictam figuram ſu-<lb/>perficialem <var>.A.</var> ſecundum eum modum, quem de figura <var>.B.</var> ſuperficiali pręſcripſi, id <lb/>poterit in ſubſcripta figura <var>.C.</var> ſpeculari in hunc modum. </s> + <s xml:space="preserve">Accipiet enim <var>.p.x.</var> ſuper <lb/>ficialem, æqualem corporeæ, & quia in ipſa corporea <var>.A.</var> ſuppoſita fuit linea <var>.p.l.</var> <lb/>ideſt punctum <var>.x.</var> inter duas <var>.u.s.</var> et <var>.a.n.</var> ſecabimus <var>.p.x.</var> ſuperficialem in puncto <var>.d.</var> ita, <lb/>vt <var>.d.x.</var> ſuperficialis, æqualis ſit corporeæ, & ipſi ſuperficiali <var>.p.x.</var> addetur <var>.x.q.</var> æqualis <lb/>corporeæ. </s> + <s xml:space="preserve">vnde <var>.q.d.</var> ſuperfi-<lb/>cialis æqualis erit corporeę,<lb/> + <anchor type="figure" xlink:label="fig-0134-03a" xlink:href="fig-0134-03"/> + et <var>.p.x.</var> ſuperficiali addetur <var>.<lb/>x.l.</var> æqualis corporeæ. </s> + <s xml:space="preserve">De <lb/>ijs poſtea quæ dicenda ſuper-<lb/>ſunt, iam ſatis ſuperq́ue 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 ſe-<lb/>quentes, in errorem non in-<lb/>cidemus, imò efficietur qua -<lb/>drilaterum <var>.q.r.</var> ſuperficiale, <lb/>ſimile, & 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 eſt punctum perſpectiuæ, vt practici dicere ſo <lb/>lent) quid ſit, hac via & ratione ſub noſtram 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 eſt, quàm punctum, varijs ſectionibus commune, & huiuſmodi punctum, ocu <lb/>lus non eſt, quemadmodum multi Pictores, Sculptores, Architecti, & Perſpectiui <lb/>ignari, ipſum punctum, oculum appellando, falsò crediderunt, quaſi <reg norm="punctum" type="context">punctũ</reg> <var>.i.</var> per-<lb/>ſpectiuæ oculus eſſet.</s> + </p> + <p> + <s xml:space="preserve">In ſupradictis igitur figuris manifeſte eluceſcit cauſa diminutionis obiectorum, <lb/>& altitudinis trianguli æqualis ei, quæ eſt oculi à plano orizontali, vt etiam diſtantię <var>.<lb/>p.l.p.x.</var> & cuiuſuis tandem rei. </s> + <s xml:space="preserve">Sed vt huius effectus ſcientia magis in vniuerſum pa-<lb/>retur. </s> + <s xml:space="preserve">Volo duas hic ſubſcriptas figuras <var>.D.</var> corpoream, & <var>.D.</var> ſuperficialem à vo-<lb/>bis conſiderari, in quarum corporea, linea <var>.p.l.</var> ſit 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> diſiunctum ſit à rectangulo ſuper-<lb/>ficiali, ideſt, vt ſeparatum exiſtat à linea <var>.q.d.</var> latere ipſius rectanguli, & ſit etiam obli <lb/>quum, reſpectu ipſius rectanguli, ideſt vt communis ſectio dicti plani cum ſuperficie <lb/><var>a.s.</var> orizontalis ipſi <var>.u.a.</var> parallela non ſit, ſed ſit obliqua, ſi tamen idem planuni per-<lb/>pendiculare dictæ ſuperficiei orizontali <var>.a.s.</var> erit: </s> + <s xml:space="preserve">& dicta communis ſectio exprima<lb/>tur characteribus .q.ω.α.d.x. nunc in figura corporea habebimus figuram <var>.e.r.c.m.</var> <lb/>in plano, quod viſualem pyramidem ſecat, medio cuius figuræ <var>.e.r.c.m.</var> oculus po-<lb/>ſitus in <var>.o.</var> rectangulum orizontale conſpicit. </s> + <s xml:space="preserve">Volentes vero nunc in figura <var>.D.</var> ſuper-<lb/>ficiali eam deſcribere, faciem us <var>.p.x.</var> ſuperficialem, æqualem corporeæ, eiq́ue <lb/>addemus <var>.x.l.</var> æqualem corporeæ, aut ſumemus <var>.p.l.</var> eidem corporeæ <reg norm="aequa- lem" type="simple">ęqua-<lb/>lem</reg>, quam ſecabimus in puncto <var>.x.</var> eodem planè modo, quo corporea reperi-<lb/>tur diuiſa; </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> ſimiles & 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 ſecta fuit corporea, & ducemus lineam <var>.d.i.</var> vnde habebi-<lb/>mus triangulos <var>.i.d.q.</var> et <var>.i.d.x.</var> ſimiles 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"/> + cuſque habebimus ſitus duorum laterum figurę <lb/>rectanguli degradati, ideſt ſitus ipſius <var>.e.m.</var> et <var>.r.<lb/>c.</var> etiam ſi adhuc neſciatur in qua parte ipſius <var>.i.<lb/>q.</var> & ipſius <var>.i.d.</var> eſſe <reg norm="debeant" type="context">debeãt</reg>. </s> + <s xml:space="preserve">Quod ſi ſcire volue <lb/>rimus <reg norm="ſecabitur" type="simple">ſecabit̃</reg>. <var>p.l.</var> in <reg norm="puncto" type="context">pũcto</reg> <var>.g.</var> ſimilis corporeæ, <lb/>ſi in ipſa tamen corporea prius protraxerimus <lb/>lineam <var>.q.d.</var> latus rectanguli vſque ad <var>.p.l.</var> in pun <lb/>cto <var>.g</var>. </s> + <s xml:space="preserve">Ducetur deinde linea <var>.o.g.</var> ſuperficialis, <lb/>quæ ſecabit lineam <var>.i.x.</var> in puncto <var>.f.</var> linea vero <var>.<lb/>o.l.</var> in puncto <var>.z.</var> punctis ſitis in <var>.i.x.</var> ſuperficiali, <lb/>pręcisè vt in corporea, <reg norm="quemadmodum" type="context">quemadmodũ</reg> quilibet <lb/>ex ſe facilè cognoſcere poteſt. </s> + <s xml:space="preserve">Deinde in cor <lb/>porea, in ſuperficie 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 ſuperficie <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 ſectiones dictorum <reg norm="duorum" type="context">duorũ</reg> <lb/>triangulorum cum plano trianguli <var>.i.q.x.</var> ipſi quoque plano ex .19. eiuſdem lib. erunt <lb/>perpendiculares. </s> + <s xml:space="preserve">Nuncautem ſecetur <var>.q.x.</var> ſuperficialis in punctis <var>.ω.</var> et .α. eadem ra-<lb/>tione; </s> + <s xml:space="preserve">qua corporea ſecta ſuit à duabus <var>.p.q.</var> et <var>.p.u.</var> à quibus punctis <var>.ω.</var> et. α. ſuperficia <lb/>libus ductæ ſint duæ <var>ω.m.</var> et .α.e. perpendiculares vſque ad latus <var>.i.q.</var> in punctis <var>.m.</var> et <var>.<lb/>e.</var> quę ſitum habebunt in <var>.i.q.</var> ſuperficiali pręcisè, vt in corporea, ex .26. primi, du-<lb/>cendo deinde in ſuperficiali duas <var>.m.f.</var> et <var>.e.Z.</var> eæ æquales erunt corporeis ex .4. pri-<lb/>mi, & ſic anguli <var>.i.e.z.</var> et <var>.i.m.f.</var> & eę duę lineę <var>.e.z.</var> et <var>.m.f.</var> fectę 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, ſed ita quo-<lb/>que ſe habent duę <var>.e.m.</var> et <var>.r.c.</var> ſi verum eſt <reg norm="quod" type="simple">ꝙ</reg> dif <lb/>ferentię rerum æqualium ſint adinuicem etiam <lb/>æquales. </s> + <s xml:space="preserve">Hac ratione igitur habebimus figu-<lb/>ram quadrilateram <var>.m.e.r.c.</var> ſuperficialem om <lb/>ninò ſinlilem, & ęqualem corporeæ. </s> + <s xml:space="preserve">Is tamen <lb/>modus prolixus eſt, & arduus, quam ob cau-<lb/>ſam neque ego vnquam <reg norm="eum" type="context">eũ</reg> viui accommo-<lb/>darem, neque alijs, vt eodem vterentur ſua-<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 ſciendum, quòd qui ſciuerit vnum ſolum punctum locare in perſpe <lb/>ctiua, eo modo quem nunc proponam, facilè quoque ſciet ſupra quoduis <reg norm="planum" type="context">planũ</reg> <lb/>(quod tamen ſit perpendiculare orizonti) quamlibet rem locare. </s> + <s xml:space="preserve">Quam ob cauſam <lb/>imaginemur hic ſubſcriptas duas figuras <var>.E.</var> <reg norm="corpoream" type="context">corporeã</reg>, & E. ſuperficialem, & 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 ſuppo-<lb/>nebatur in figura <var>.A.</var> corporea. </s> + <s xml:space="preserve">Imaginemur ergo in ipſa figura <var>.E.</var> corporea radi-<lb/>um viſualem <var>.o.b.</var> qui ſectus ſit à noſtro plano in <var>.k.</var> quod quidem <var>.k.</var> quærendum eſt <lb/>in triangulo <var>.i.q.d.</var> ipſius plani. </s> + <s xml:space="preserve">Volo ob hanc igitur rem, vt à puncto <var>.b.</var> in figura <var>.E.</var> <lb/>ſuperficiali ducatur <var>.b.c.</var> ad rectos <reg norm="cum" type="context">cũ</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 ipſi <var>.q.d.</var> quę ab ipſa <var>.x.l.</var> in puncto <var>.m.</var> erit diuiſa, & hęc <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> ſecta 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 diuiſa. </s> + <s xml:space="preserve">Atque id erit quod nobis <lb/>inquirendum propoſueramus.</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="ſpeculationem" type="context">ſpeculationẽ</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 eſ <lb/>ſe vſque ad <var>.y.</var> lineæ <var>.s.n.</var> & imaginatione ſit com <lb/><reg norm="præhenſa" 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> ſimul cum linea <var>.o.b.</var> erunt <lb/>in vna eademq́ue ſuperficie plana, quam cha-<lb/>racteribus <var>.y.</var> R<unsure/>. notemus .et <var>.i.c.</var> eius erit ſe-<lb/>ctio communis cum plano, in quo quæritur <reg norm="pun- ctum" type="context">pũ-<lb/>ctum</reg>, et <var>.f.k.</var> ipſius plani cum triangulo <var>.o.b.m.</var> <lb/>erit ſectio communis, & parallela ipſi <var>.q.d.</var> ex <ref id="ref-0022">.<lb/>6. lib. 11.</ref> quia <var>.k.f.</var> perpendicularis eſt ſuperfi-<lb/>ciei <var>.p.t.</var> ex .19. eiuſdem cum triangulus <var>.o.<lb/>b.m.</var> eidem ſuperficiei <var>.p.t.</var> ex .18. eiuſdem <lb/>perpendicularis exiſtat. </s> + <s xml:space="preserve">Vnde perſpicuè pa-<lb/>tet ratio quare protracta<unsure/> ſit parallela <var>.b.c.</var> et <lb/>quare ducta ſit <var>.i.c.</var> et coniuncta <var>.x.m.</var> cum <var>.x.<lb/>p.</var> directè, & quare ducta ſit <var>.o.m.</var> et <var>.f.k</var>. </s> + <s xml:space="preserve">Lau-<lb/>do igitur vt ſemper præſupponatur <var>.p.x.</var> perpen <lb/>dicularis baſi ipſius plani & præſupponatur, (vt <lb/>rem totam vnò verbo complectar) ſuperficies <var>.<lb/>p.t.</var> perpendicularis plano, & orizonti. </s> + <s xml:space="preserve">Quod <lb/>reliquum eſt, neceſſariv<unsure/>m non eſt, niſi ad ſpe-<lb/>culandum. </s> + <s xml:space="preserve">Neceſſariæ ergo non ſunt aliæli-<lb/>neæ, quàm.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 ſit ipſi <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 cauſas, 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 noſtro plano perpendiculari orizonti, & ita <reg norm="locatum" type="context">locatũ</reg><unsure/>, <lb/>vt poſtremo diximus, inueniamus; </s> + <s xml:space="preserve">duas hîc ſubſcriptas figuras conſiderabimus <var>.<lb/>G.</var> corpoream, & G. ſuperficialem, ſimiles duabus <var>.E.E.</var> proximè præcedentibus, <lb/>in quarum corporea ſit linea <var>.b.M.</var> altitudinis perpendicularis orizonti. </s> + <s xml:space="preserve">Quare ſi <lb/>deſiderabis inuenire in noſtro plano ſitum puncti <var>.M.</var> ideſt punctum radij <var>.o.M.</var> vi-<lb/>ſualis in quo ipſe radius à plano eſt diuiſus, quod ſit <var>.R.</var> quamuis extra <reg norm="triangulum" type="context">triangulũ</reg> <lb/><var>i.q.d.</var> tibi imaginatione confige ductam eſſe lineam <var>.p.b.</var> quæ erit ſectio commu-<lb/>nis orizontis cum ſuperficie <var>.o.p.b.M.</var> quæ ſuperficies erit perpendicularis ipſi ori-<lb/>zonti ex .18. lib 11. </s> + <s xml:space="preserve">Quòd autemnon minus <var>.o.p.</var> quàm.M.b. ſit in vna eademq́ue <lb/>ſuperficie dubitandum non eſt, quia ſi 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/>demq́ue ſuperficie conſtantem, 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 ſuperficie ſit ex .2. lib. 11. ſe-<lb/>quetur ſuperficiem <var>.o.p.b.M.</var> planam eſſe, & vnicam, cuius communis ſectio cum no-<lb/>ſtro plano ſit. θ.K.R. quæ perpendicularis orizonti exiſtet ex .19. lib. 11. eritq́ue pa-<lb/>rallela ipſi <var>.i.x.</var> ex .6. eiuſdem. </s> + <s xml:space="preserve">Imaginare nunc erectam eſſe <var>.m.T.</var> æqualem ipſi <var>.<lb/>b.M.</var> orizonti perpendicularem, quæ extenſa erit in ſuperficie <var>.p.t.</var> quod ex ſe ad <lb/>conſiderandum admodum facilè, clarumq́ue exiſtit, 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> ęqualis <var>.b.M.</var> parallela eſt <lb/>ipſi <var>.b.M.</var> ex .6. lib. 11. præter hæc <var>.b.m.</var> parallela eſt ipſi <var>.q.d.</var> quia ſic fuit ducta <lb/>ſuperius, vnde <var>.M.T.</var> parallela erit ipſi <var>.q.d.</var> ex .9. vndecimi, & obid perpendi-<lb/>cularis erit ſuperficiei <var>.b.t.</var> ex .8. eiuſdem. </s> + <s xml:space="preserve">Nunc ſit <var>.R.V.</var> communis ſectio trian-<lb/>guli <var>.o.M.T.</var> cum noſtro plano, vnde <var>.R.V.</var> perpendicularis erit ſuperficiei <var>.p.t.</var> <lb/>ex .19. lib. 11. quam ob cauſam parallela erit ipſi <var>.q.d.</var> ex .6. aut ex .9. eiuſdem <lb/>quia ex .6. dicta, parallela eſt ipſi <var>.M.T</var>. </s> + <s xml:space="preserve">Atſi <var>.R.V.</var> parallela eſt ipſi <var>.q.d.</var> <lb/>etiam <var>.f.K.</var> probatum iam fuit parallelam eſſe eidem, ergo <var>.R.V.</var> parallela erit <lb/>ipſi <var>.K.f.</var> ex .30. primi, </s> + <s xml:space="preserve">Vnde ex .34. æqualis erit ipſi <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> ſupra figuram <var>.E.</var> ſuperficialem, & erigamus <var>.m.T.</var> perpendi-<lb/>cularem ipſi <var>.m.p.</var> ſed æqualem perfectæ altitudini, & ducamus <var>.T.o.</var> vt ſecet li-<lb/>neam <var>.i.x.</var> in puncto <var>.V.</var> ab ipſo ducentes <var>.V.R.</var> parallelam ipſi <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 ipſi <var>.i.a.</var> habebimus altitudinem <var>.k.R.</var> quam quærebamus in <lb/>noſtro plano. </s> + <s xml:space="preserve">Quod cum ſui 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 ſcien-<lb/>tia erudito relinquo. </s> + <s xml:space="preserve">ideſt, vt probetur <var>.k.R.</var> <reg norm="ſu- perficialem" type="context">ſu-<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="ſuper- fluitatem" type="context">ſuper-<lb/>fluitatẽ</reg> linearum, & hoc <reg norm="accommodantes" type="context">accõmodantes</reg> vt in figura <var>.<lb/>F.</var> diligenter conſideretur 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 ſit 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 abſcindatur in puncto <var>.s.</var> ita quòd <var>.c.s.</var> æqualis ſit alti <lb/>tudini perfectæ, deinde coniungatur rectà. s. cum <var>.i</var>. </s> + <s xml:space="preserve">Tunc ſi ab <var>.k.</var> vſque 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"/> + neę <var>.c.s.</var> hæc <var>.R.k.</var> erit altitudo <lb/>quæſita ſeu 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 eſt quin <var>.f.V.</var> ſit æqualis alti-<lb/>tudini quęſitæ ſeu degradatę, quo <lb/><reg norm="tieſcunque" type="simple">tieſcunq;</reg> ergo <reg norm="probauerimus" type="simple">ꝓbauerimus</reg> <var>.k.R.</var> <lb/>æqualem eſſe lineæ <var>.f.V.</var> habebi-<lb/>mus propoſitum. </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 ſimilitudine <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/>ſed <var>.c.s.</var> ſumpta 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">ꝗnti</reg>, & ex .11. eiuſ-<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. eiuſdem <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 philoſophis obſeruatus, eſt etiam vtilis, <reg norm="compendioſaque" type="simple">compendioſaq́;</reg> via <lb/>progreditur, cuius ſpeculationem, in ſubſcripta figura, quadam ex parte <reg norm="ſecun- dum" type="context">ſecũ-<lb/>dum</reg> morem antiquum, quadam etiam ex parte ſecundum ingenij mei vires <reg norm="conſtru- cta" type="context">cõſtru-<lb/>cta</reg>, cognoſcemus. </s> + <s xml:space="preserve">In qua ego diuiſi <var>.x.i.</var> in puncto <var>.s.</var> ab <var>.x.</var> ita eleuato, quanta eſt <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 ipſius <var>.M.T.</var> et <var>I.s.</var> duxi ſupponendo eſſe <var>.I.</var> <reg norm="punctum" type="context">punctũ</reg> pſpectiuæ <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>antiquos, ideſt angulum ſupremum trianguli antiquorum à punctoq́ue <var>.k.</var> meo duxi <lb/><var>k.f.</var> parallelam ipſi <var>.c.m.p.</var> vſque ad <var>.i.x.</var> in puncto <var>.f.</var> & à puncto à communi ipſis <var>.k.f.</var> <lb/>et <var>.i.x.</var> vſque ad <var>.I.s.</var> duxi quoque <var>.A.B.</var> parallelam ipſi <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 conſiderans modum, quem ego de figuris <var>.G.H.</var> antecedentibus <lb/>præſcripſi, videndum eſt, an punctum <var>.B.</var> tribus lineis <var>.A.B.I.s.</var> et <var>.R.V.</var> quarum hęc vl <lb/>tima à me iam ducta fuit, commune exiſtat, ideſt vtrum <var>.A.B.</var> æqualis exiſtat ipſi <var>.K.<lb/>R.</var> quam ſecundum modum à me adinuentum, reuera ſcimus eſſe deſideratam altitu <lb/>dinem in perſpectiua. </s> + <s xml:space="preserve">Quod tunc à nobis probatum erit, quando rationibus clarè <lb/>patebit ipſam <var>.A.B.</var> æqualem eſſe ipſi <var>.f.V</var>. </s> + <s xml:space="preserve">Quamobrem ducamus <var>.I.f.</var> vſque ad <var>.ω.</var> <lb/>lineæ <var>.c.p.</var> vnde ratione ſimilitudinis triangulorum manifeſtè intelligemus, eandem <lb/>proportionem eſſe ipſius <var>.m.T.</var> ad <var>.f.V.</var> quæ eſt <var>.m.o.</var> ad <var>.f.o.</var> & eius, quæ eſt <var>.m.o.</var> ad <var>.f.<lb/>o.</var> quæ eſt <var>.ω.I.</var> ad <var>.f.I.</var> & eius, quæ eſt <var>.ω.I.</var> ad <var>.f.I.</var> quæ eſt <var>.x.I.</var> ad <var>.A.I.</var> & eius, quæ eſt <var>.x.<lb/>I.</var> ad <var>.A.I.</var> quæ eſt <var>.x.s.</var> ad <var>.A.B.</var> ideſt vt eius, quæ eſt <var>.m.T.</var> ad <var>.A.B.</var> ſed 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 ſequitur <var>.A.B.</var> æqualem eſſe <var>.f.V.</var> ex .9. quinti Eucli. <reg norm="atque" type="simple">atq;</reg> etiam <lb/>ipſi <var>.k.R.</var> quod à nobis propoſitum eſt 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 ſermonem de figuris ſu-<lb/>perficialibus orizontalibus, ſeu de plantis, <lb/>pulcherrimum quendam modum, quem ego ad <lb/>locandum quodlibet punctum in perſpectiua, <lb/>(degradatum cum fuerit <reg norm="parallelogrammum" type="context">parallelogrãmum</reg> quod <lb/>dam rectangulum, in noſtro plano perpendicula <lb/>ri orizonti, quemadmodum in ſuperioribus figu-<lb/>ris <var>.A.</var> demonſtrauimus) conſideraui, ſilentio <lb/>haud prætereundum eſſe.</s> + </p> + <p> + <s xml:space="preserve">Sit igitur in ſubſcripta hîc figura <var>.K.</var> in paralle <lb/><reg norm="logrammo" type="context">logrãmo</reg> perfecto <reg norm="punctum" type="context">pũctum</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> vſque 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> ipſorum <lb/>laterum, ideſt vt non ſecent 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="correſpondet" type="context">correſpõdet</reg> <var>.<lb/>u.</var> perfecti, mediante linea <var>.e.f.</var> quæ erit <var>.u.f.</var> degra <lb/>dita in noſtro plano. </s> + <s xml:space="preserve">Deinde ſumatur punctum <var>.<lb/>n.</var> in linea <var>.q.d.</var> tam diſtans à <var>.q.</var> quàm.g. diſtat ab <var>.<lb/>u.</var> ducaturq́ue linea <var>.i.n.</var> quæ lineam <var>.e.r.</var> in puncto <lb/>c. diuidet, quod exijs, quæ ſuperius iam diximus <lb/>ad ipſum <var>.g.</var> referetur. </s> + <s xml:space="preserve">Ducendo poſtea lineam oc <lb/>cultam <var>.q.c.</var> patebit eam correſpondere lineæ <var>.q.g.</var> quæ ſecans lineam <var>.e.f.</var> in puncto <var>.<lb/>t.</var> hoc, communi ſcientiæ ratione, reſpondebit ipſi <var>.b.</var> vt omnes cognoſcent.</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 ſi 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 ſit vna diagonalis quam <lb/>volueris puta <var>.q.a.</var> deinde à puncto <var>.b.</var> ad reli-<lb/>quum angulum oppoſiti lateris ducta ſit recta <var>.<lb/>b.d.</var> ita quod à diagonali ſecetur in puncto <var>.ω.</var> <lb/>per quod punctum demum à reliquo angulo la <lb/>teris <var>.q.u.</var> ducta ſit <var>.u.ω.</var> vſque ad latus <var>.q.d.</var> in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.f.</var> quo facto, ita faciendum erit in rectangu <lb/>lo degradato, hoc eſt ducenda erit diagonalis <var>.<lb/>q.r.</var> quę correſpondet diagonali <var>.q.a.</var> perfecti <lb/>deinde <var>.f.e.</var> quæ correſpondet rectæ <var>.f.u.</var> perfe-<lb/>cti, quæ etiam interſecabitur à diagonali <var>.q.r.</var> <lb/>in puncto <var>.o.</var> correſpondens <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> vſque ad <var>.t.</var> <lb/>in latere <var>.q.e.</var> hoc <reg norm="punctum" type="context">punctũ</reg> <var>.t.</var> correſpondebit 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 ſi loco diametri <var>.q.a.</var> ſumpta <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 correſpondens ipſi <var>.f.</var> in figura de-<lb/>gradata erit in latere ſupremo <var>.e.r.</var> correſpon-<lb/>dens lateri <var>.u.a.</var> & ita ducenda erit diameter <var>.d.<lb/>e.</var> correſpondens diametro <var>.d.u.</var> et <var>.q.f.</var> ſurſum <lb/>verſus correſpondens <var>.q.f.</var> imum verſus deinde <var>.<lb/>r.o.</var> reſpondens <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> ſuperficiali perſpectiuæ facillimum modum locandi quoduis <lb/>punctum in perſpectiua 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, quàm latitudinis protenſa, quæ quidem latitudo erat ad duos circiter pedes <lb/>deſignata 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 ſuperficiem eiuſdem tabulæ excederent, vt <reg norm="vnum" type="context">vnũ</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 poſtea, 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 poſteà. <var>o.i.</var> parallela ipſi <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 extenſam, quanta conſtat <var>.A.B.</var> <reg norm="conſtitui" type="context">cõſtitui</reg> etiam, vt <reg norm="quoddam" type="context">quoddã</reg> angulum re <lb/>ctum materiale tantæ magnitudinis, quanta nobis vſui eſſe poterat ſuper eadem ta-<lb/>bula; </s> + <s xml:space="preserve">necnon regula quædam materialis neceſſariæ longitudinis <reg norm="ſtatuerentur" type="simple">ſtatuerent̃</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/>ideſt <var>.p.E.</var> ſuper eadem tabula adeò diſtantem 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, ſignaui. </s> + <s xml:space="preserve">Hæc autem diſtantia, 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 ſignificabat. </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/>quotieſcunque volebam in perſpectiua aliquid delineare. </s> + <s xml:space="preserve">Quod cum facere deſi-<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 conſtitaere <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.b.</var> <reg norm="ſumebam" type="context">ſumebã</reg> <reg norm="angulum" type="context">angulũ</reg> <reg norm="rectum" type="context">rectũ</reg> <reg norm="materialem" type="context">materialẽ</reg>, ſeu <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> tranſire faciebam, & vbi hoc latus <reg norm="lineam" type="context">lineã</reg> <lb/><var>E.p.</var> diuidebat, punctum .c ſignabam per quod efficiebatur, vt regula <var>.i.M.</var> tranſiret, <lb/>quieſceretq́ue aliquantulum aliquo modo in huiuſmodi ſitu, opera deinde circini <lb/>interuallum <var>.b.c.</var> ſumebam, & in <var>.p.E.</var> à puncto <var>.x.</var> verſus <var>.E.</var> punctum <var>.m.</var> <reg norm="ſignabam" type="context">ſignabã</reg>: </s> + <s xml:space="preserve">per <lb/>quod faciebam, vt tranſiret 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> tranſibat, quod quidem latus regulam <var>.i.M.</var> in puncto <var>.k.</var> <lb/>(quod ſtatim ſuper folio papyri ſignabatur) interſecabat, atque hoc erat <reg norm="punctum" type="context">punctũ</reg>, <lb/>quod quærebam, puncto <var>.b.</var> correſpondens. </s> + <s xml:space="preserve">Huiuſmodi effectus rationes ab ijs, quæ <lb/>ſuperius dixi eliciuntur. </s> + <s xml:space="preserve">Atque hæc ad baſes rerum, vt in ſubſcripta figura eluceſcit, <lb/>ſpectabant.</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, vſus ſum 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 ſeſe offerebat occaſio. </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, ſeu normæ regulæ <var>.B.C.</var> anniteretur, aliud verò <lb/>per <var>.m.</var> in rectitudine cuius ſignabam <var>m.T.</var> interuallum æquale altitudini perfecti, <lb/>ideſt punctum <var>.T.</var> æqualiter diſtans ab <var>.m.</var> tranſire faciebam, deinde regulam <var>.o.Q.</var> <lb/>per punctum <var>.T.</var> tranſire quoque faciebam: </s> + <s xml:space="preserve">& notabam interſectionem ipſius 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, aliudq́ue per punctum <var>.k.</var> tranſire faciebam, & in huiuſmodi <lb/>rectitudine à puncto k. ſignabam quandam menſuram æ qualem lineæ <var>.f.V.</var> quę 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, ideſt ba <lb/>ſis eiuſdem perfecti tranſire faciebam. </s> + <s xml:space="preserve">Et in huiuſmodi rectitudine ſignabam <var>.c.s.</var> <lb/>æquale interuallum altitudini perfecti, ideſt punctum <var>.s.</var> ita diſtans à <var>.c.</var> efficiendo de <lb/>inde, vt latus anguli recti, lateri <var>.B.C.</var> tabulæ anniteretur, aliudq́ue per punctum <var>.k.</var> <lb/>tranſire faciens ſignabam <var>.k.R.</var> indeterminatam. </s> + <s xml:space="preserve">Faciens deinde tranſire regulam <lb/><var>i.M.</var> per punctum <var>.s.</var> notabam punctum <var>.R.</var> interſectionis eiuſdem 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, expeditiorq́ue 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 poſtquam de perſpectiuis inter nos ſermonem <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 ſatis laudatus Daniel Barba-<lb/>rus ſe accepiſſe profitetur à Ioanne Zamberto patritio Veneto, qui ad verbum om <lb/>nia deſumpſerat a Ioanne Cuſino Pariſienſe. </s> + <s xml:space="preserve">Nec parum mirabar peritiſſimum il-<lb/>lum Cuſinum, quod in capite quarto ſecundæ partis perſpectiuæ, vt quodpiam pla-<lb/>num quadrilaterum in quadratam figuram redigeret, ſuper vnam datam <reg norm="lineam" type="context">lineã</reg> qua-<lb/>dratam compoſuiſſe. </s> + <s xml:space="preserve">Non animaduertens diſtantiam aut interuallum <var>.b.c.</var> degra-<lb/>datum ín linea <var>.b.f.</var> (quod eſt <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 ſubſcriptæ hic figuræ facilè depræhendi poteſt. </s> + <s xml:space="preserve">Vbi <var>.b.c.</var> latitudo eſſe po-<lb/>teſt, 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">latũ</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> ſcientificè intelligi poteſt. </s> + <s xml:space="preserve">Hinc, ad inueniendum perfectum alicuius pla-<lb/>ni degradati, non ſufficere degradationem ſolum interualli inter duos terminos <lb/>ſolos, ideſt <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">ꝙ</reg> non ſunt omnia quadrata. </s> + <s xml:space="preserve">Ad <reg norm="inquirendum" type="context">inquirendũ</reg> <lb/>igitur perfectum alicuius plani <reg norm="parallelogrammi" type="context">parallelogrãmi</reg>, alicuius propoſiti degradati, oportet <lb/>vniuerſam degradationem tam latitudinis, <reg norm="quam" type="simple">ꝗ̃</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 ſubſcripta hic figura, <reg norm="volendo" type="context">volẽdo</reg> inuenire perfectum paral <lb/>lelogrammum degradati <var>.b.h.l.m.</var> dando diſtantiam 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 vſque ad planiſitum in quo degradatio facta ſit: </s> + <s xml:space="preserve">ſtatim altitudo <var>.A.</var> oculi à pe <lb/>de, quæ tanta ſemper eſſe debet quanta eſt altitudo trianguli <var>.b.n.m.</var> qui clauditur, <lb/>protrahendo <var>.m.l.</var> et <var>.b.h.</var> vſque ad concurſum 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> & vſque 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> vſque ad <var>.c.</var> ipſius <var>.d.b.m.</var> produ <lb/>ctæ ducere <var>.A.E.c.</var> atque deinde protrahere lineam <var>.o.b.</var> vſque ad <var>.T.</var> ita vt <var>.b.T.</var> ęqua <lb/>lis ſit ipſi <var>.b.c.</var> & ad ipſam à puncto <var>.m.</var> ducere parallelam <var>.m.R.</var> & à puncto <var>.T.</var> ducere <var>.<lb/>T.R.</var> parallelam ipſi <var>.b.m</var>. </s> + <s xml:space="preserve">Vnde ex .34. primi Eucli <var>.m.R.</var> æqualis erit ipſi <var>.b.T.</var> et <var>.R.<lb/>T.</var> ipſi <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 eſt hic quod magis nos commoueat, quia cum ex linea <var>.b.c.</var> quadratum <var>.b.g.</var> pro <lb/>duxerit, vult eum poſtea degradare. </s> + <s xml:space="preserve">Quod vt faciat (hanc figuram videbis in cap <num value="4">.<lb/>4.</num> ſecundæ partis Danielis Barbari) oculum <var>.A.</var> in eadem ſuperficie extenſa, <lb/>quadrati <var>.b.g.</var> collocat. </s> + <s xml:space="preserve">Quod rectè fieti non poteſt, quia oculum hoc mo-<lb/>do locantes, viſualesq́ue radios beneficio vnius plani ſituati in <var>.b.f.</var> ſecantes in ipſo <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, & ſi <lb/>natura ſua ſit omnibus notum, ponit <reg norm="tantum" type="wordlist/context">tñ</reg> id ipſum Vitelio pro quinta propoſitione <lb/>quarti libri de perſpectiua. </s> + <s xml:space="preserve">Præter hæc, credit latera <var>.b.d.</var> et <var>.c.e.</var> quadrati degra-<lb/>dati ſemper videri mediantibus angulis <var>.b.A.c.</var> et <var>.f.A.g.</var> quod fieri <reg norm="non" type="context">nõ</reg> poteſt, quem-<lb/>a dmodum ex mea figura corporea <var>.A.</var> facilè cognoſcere 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 ſuperfi-<lb/>ciem <var>.s.a.</var> exiſtunt videntur, vnde ſi quis imaginaretur in puncto <var>.p.</var> oculum eſſe, & <lb/>ab ipſo 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 diuerſis igitur an-<lb/>gulis, pro maiori parte, deteguntur latera, à partibus quadrati tam degradati, quàm <lb/>perfecti, quæ non ſunt anguli <var>.b.A.c.</var> et <var>.f.A.G</var>. </s> + <s xml:space="preserve">Quod vero idem poſtea dicat eam <lb/>proportionem eſſe ab <var>.b.E.</var> ad <var>.f.h.</var> & ſimul ad <var>.c.g.</var> quæ eſt 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 conſiderandum relinquo; </s> + <s xml:space="preserve">cum rationabilis degrada-<lb/>tio eſſe debeat, qua ratione neceſſarium ſit, vt diſtantiæ reſq́ue, in vna & <reg norm="eadem" type="context">eadẽ</reg> pro-<lb/>portione cum altitudine oculi ad rem degradatam exiſtant? </s> + <s xml:space="preserve">Cum poſtea degrada-<lb/>uerit <reg norm="quadratum" type="context">quadratũ</reg>, is ſcriptor, in figura <var>.d.b.c.e.</var> eum bene & ex perſpectiuæ optimis <lb/>legibus degradatum fuiſſe probare nititur; </s> + <s xml:space="preserve">ſolum probans <var>.d.e.</var> æqualem eſſe ipſi <var>.<lb/>E.h.</var> <reg norm="qua" type="wordlist">q̃</reg> <var>.E.h.</var> <reg norm="ſecundum" type="context">ſecundũ</reg> ipſum eſt degra datio lateris <var>.c.g.</var> & <reg norm="cum" type="context">cũ</reg> ſuperius dixerit, ſetria <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 ſi <var>.c.g.</var> <lb/>perfecti, degradatum eſt in <var>.E.h</var>: et <var>.d.e.</var> rectè protracta exiſtit, cum ſit æqua-<lb/>lis ipſi <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/>ſa ipſis <var>.b.E.</var> et <var>.f.h.</var> quæ, ex ipſo, ſunt 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 reſpondere, <reg norm="quod" type="simple">ꝙ</reg> hoc pacto nulla ſuperficies 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 ſunt degradata, <reg norm="eiusque" type="simple">eiusq́;</reg> <reg norm="proportionalitates" type="simple">ꝓportionalitates</reg> ma <lb/>lè intellectæ nil probant. </s> + <s xml:space="preserve">quia ſi 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 exiſterent, quarum quælibet maior eſt <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 ſcribit eam eſſe rationem, aut comparationem ab <var>.A.d.</var> ad <var>.b.<lb/>E.</var> quæ eſt ab <var>.d.c.</var> ad <var>.b.c.</var> eandemq́ue eſſe ab <var>.E.h.</var> ad <var>.c.g.</var> quæ eſt ab <var>.A.E.</var> ad <var>.A.c.</var> nil <lb/>probet; </s> + <s xml:space="preserve">nec ſimilitudinem triangulorum, nec aliquam propoſitionem Eucli. citans. <lb/></s> + <s xml:space="preserve">In quo excuſari non poteſt, quòd non ſoleat Euclidem, aut alium quemuis autorem <lb/>citare, cum vel in ipſo operis principio capite .3. primæ partis, A pollonium <reg norm="Pergeum" type="context">Pergeũ</reg> <lb/><reg norm="Euclidemque" type="simple">Euclidemq́;</reg>, & ſi etiam præter rem, citet. </s> + <s xml:space="preserve">Deinde <reg norm="quum" type="context">quũ</reg> idem probare vult <var>.d.e.</var> æqua <lb/>lem eſſe ipſi <var>.E.h.</var> eandem inquit eſſe proportionem <var>.a.b.</var> ad <var>.a.d.</var> quæ eſt ipſius <var>.A.c.</var> <lb/>ad <var>.A.E.</var> quod & ſi verum ſit, hic tamen modus ratiocinandi nullo ordine nititur, <lb/>quia rectius dixiſſet pro clariori intelligentia ipſius <var>.a.c.</var> ad <var>.a.e.</var> eandem proportio-<lb/>nem eſſe, quæ eſt <var>.A.c.</var> ad <var>.A.E.</var> propter ſimilitudinem, quæ inter duos triangulos <var>.A.<lb/>c.a.</var> et <var>.E.c.e.</var> intercedit, cum <var>.E.e.</var> ſupponatur parallela ipſi <var>.A.a.</var> quod etiam vt de-<lb/>monſtraretur longiori oratione ei opus fuiſſet ſi voluiſſet intellectum eorum, qui pa <lb/>rum ſunt exercitati, perduci ad <reg norm="cognoſcendum" type="context">cognoſcendũ</reg> idem planè futurum de <var>.a.c.</var> ad <var>.a.e.</var> vt <lb/>eſt ipſius <var>.A.c.</var> ad <var>.A.E.</var> in hunc modum, ideſt 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/>ſint parallelæ. </s> + <s xml:space="preserve">Vnde ex .4. ſexti. 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 ipſius <var>.E.c.</var> ad <var>.e.c.</var> & ex .19. eiuſdem de <var>.A.E.</var> ad <var>.a.<lb/>e.</var> vt ipſius <var>.A.c.</var> ad <var>.a.c.</var> & ex .16. iam dicta de <var>.A.E.</var> ad <var>.A.c.</var> vt ipſius <var>.a.e.</var> ad <var>.a.c.</var> ideſt <lb/>ipſius <var>.A.c.</var> ad <var>.A.E.</var> vt eſt ipſius <var>.a.c.</var> ad <var>.a.e</var>: Aut hoc alio modo, qui breuior eſt pro-<lb/>cedendum, incipiendo ſcilicet à ſecunda ſexti Eucli. dicendo <reg norm="quod" type="simple">ꝙ</reg> exiſtente <var>.E.e.</var> paral <lb/>lela ipſi <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 ſtatim 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 ſupradictis duabus propoſitionibus ideſt .29. primi, & 4. ſexti, cogno-<lb/>ſcitur idem planè eſſe de <var>.b.c.</var> ad <var>.d.e.</var> quod ipſius <var>.a.c.</var> ad <var>.a.e.</var> & ex eiſdem idem eſſe <lb/>de <var>.c.g.</var> ad <var>.E.h.</var> quod ipſius <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> ſed cum ex ſuppoſito <var>.c.g.</var> ſit æqualis ipſi <var>.c.b.</var> idem <lb/>erit de <var>.c.g.</var> ad <var>.e.d.</var> quod ipſius <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 eiuſdem <var>.c.g.</var> ad <var>.e.d.</var> ex .9. igitur eiuſdem <var>.d.e.</var> æqualis erit ipſi <var>.E.<lb/>h.</var> atque hic verus eſt modus ducendi intellectum parum exercitatum in cognicio-<lb/>nis campum. </s> + <s xml:space="preserve">quem quidem mihi obſeruandum proponerem ſi onus ſcribendi ſu-<lb/>ſciperem ijs, qui in ſcientijs parum verſati ſunt, 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 ipſo adducta propter quam <var>.E.</var> repreſentatur oculo al-<lb/>tius quam <var>.b.</var> nempe eo quod <var>.A.</var> ſuperſtet ipſi <var>.E.</var> nihil valet, quia ſi inferius eſſet, <lb/>idem contingeret, ſed hoc euenit eo quod <var>.E.</var> altius eſt ipſo <var>.b</var>. </s> + <s xml:space="preserve">Idem dico de <var>.h.</var> <lb/>vbi ſimiliter decipitur. </s> + <s xml:space="preserve">Idem etiam in .7. cap. fallitur in ſecundo modo, quem oſten <lb/>dit pro ſecundo quadrato aliquo degradato à parallelogrammo degradato magis <lb/>longo quàm lato, cum ducat parallelam <var>.l.m.</var> ad <var>.b.c.</var> à puncto <var>.l.</var> interſection is ipſius <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 innoteſcit.</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 proceſsit, quia <reg norm="cum" type="context">cũ</reg> <lb/>angulus .2. trianguli perfecti magis diſtet à plano ſuper quod degradari debet <lb/>triangulum, quàm latus .1. 3. oppoſitum dicto angulo .2. & per confequens longère <lb/>motior ſit ab oculo, ipſe in degradato, <reg norm="eum" type="context">eũ</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 eſt, <reg norm="admittendum" type="context">admittendũ</reg>. <lb/></s> + <s xml:space="preserve">Quod verò in ſecunda parte ab eo traditur, ideſt alius quidam modus quem de <reg norm="tranſ" type="context">trãſ</reg> <lb/>ferendis punctis à perfecto in degradato proponit, non eſt modus vniuerſalis; </s> + <s xml:space="preserve">quia <lb/>ſi 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 ſit in ſubſcripta 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 ſit parallela duobus lateribus <var>.A.B.</var> et <var>.C.D.</var> in cuius quolibet puncto <var>.<lb/>Q.</var> ſit infimus terminus altitudinis oculi, & in <var>.<lb/> + <anchor type="figure" xlink:label="fig-0148-01a" xlink:href="fig-0148-01"/> + T.</var> ad perpendiculum ipſius <var>.Q.</var> ſit verus ſitus <lb/>eiuſdem, tantum eleuatus à <var>.Q.</var> quanta eſt <lb/>medietas ipſius <var>.D.B.</var> ſitq́ue figura corpo-<lb/>rea finita ſimilis meæ <var>.A.</var> vnde <var>.Q.T.</var> æqualis <lb/>erit ipſi <var>.Q.æ.</var> & planum perpendiculare <reg norm="orizon- ti" type="context">orizõ-<lb/>ti</reg>, ſuper quod punctum <var>.k.</var> perfecti duci debet <lb/>ſit <var>.R.D.B.</var> ſintq́ue ductæ per imaginationem <lb/>lineæ <var>.T.K</var>: <var>Q.K.</var> et ſit <var>.K.N.</var> perpendicularis la-<lb/>teri <var>.C.D.</var> à quo puncto <var>.N.</var> imaginatione ſit <reg norm="con" type="context">cõ</reg> <lb/>præhenſa linea <var>.N.Q.</var> at que hæ tres lineæ ſectæ <lb/>ſint à plano in punctis <var>.c.i.</var> et .2. quorum <reg norm="punctum" type="context">punctũ</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 ſecti <reg norm="erunt" type="context">erũt</reg> <lb/>à plano <var>.R.B.D.</var> quorum communes ſectiones <lb/>erunt .1. 2. et <var>.D.c.</var> & quia <var>.N.K.D.i.</var> et <var>.æ.Q.</var> <lb/>inuicem ſunt parallelæ, ſequitur eandem pro-<lb/>portionem futuram ipſius <var>.Q.K.</var> ad <var>.K.i.</var> quæ eſt <lb/>ipſius <var>.æ.N.</var> ad <var>.N.D.</var> imaginatione concipien <lb/>do a puncto <var>.K.</var> vſque ad <var>.æ.Q.</var> quandam paral-<lb/>lelam ipſi <var>.N.æ.</var> quemadmo dum ex te ipſo intel <lb/>ligere potes. </s> + <s xml:space="preserve">Sed ratione ſimilitudinis trian-<lb/>gulorum ita ſe 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 ipſius <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. eiuſdem <lb/>de <var>.Q.T.</var> ad <var>.Q.æ.</var> quod de .1. 2. ad <var>.c.D.</var> & exiſtente <var>.æ.Q.</var> ex ſuppoſito æquali ipſi.</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 huiuſmodi regula tunc bona redditur, quando <var>T.Q.</var> æqualis eſt ipſi .œ.<lb/>Q. ideſt medietati ipſius <var>.D.B.</var> at verò ſi æqualis non eſſet hoc minime ſequeretur, <lb/>vt facilè patet. </s> + <s xml:space="preserve">Quòd verò .2. <var>R.Z.</var> &. ſint benè diſpoſita, dubitandum non eſt, quia <lb/>punctum <var>.i.</var> meæ hic ſubſcriptæ figuræ, quod coreſpondet K. eius ſiguræ adeò diſtat <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 manifeſtè cognoſci poteſt, ab eo, quod in <lb/>ſuperius poſitis figuris corporeis dixi. </s> + <s xml:space="preserve">Huiuſmodi modus ducendi res in perſpectiua, <lb/>non ſolum à Gallis, ſed à Germanis etiam in vſum reducitur. </s> + <s xml:space="preserve">Sed quia ad hæc <reg norm="vſque" type="simple">vſq;</reg> <lb/>tempora eiuſdem perfectionis ratio, quam ego ſuperius propoſui, <reg norm="nondum" type="context">nõdum</reg> in lucem <lb/>emerſit, factum fuit, vt <reg norm="errorum" type="context">errorũ</reg> laqueis irretirentur, ſumentes <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/>vniuerſalem eſſe putet, labitur in errorem, cum debuiſſet longitudinem ipſius <var>.T.Q.</var> <lb/>debere eſſe æqualem medietati ipſius <var>.D.B.</var> proferre. </s> + <s xml:space="preserve">Aſſerit deinde diſtantiam ip-<lb/>ſius <var>.T.Q.</var> à latere <var>.B.D.</var> æ qualem eſſe debere lateri <var>.C.D.</var> quod neceſſarium non eſt, <lb/>quia in quibuslibet diſtantijs, iuſta operatio fieri poteſt, quemadmodum in ſubſcri-<lb/>pta hîc figura facile patet, ideſt, quòd quibuſcunque modis <var>.c.D.</var> æqualis remaneat <lb/>ipſi .1. 2. & ſic interualla, quæ <reg norm="per" type="simple">ꝑ</reg> tranſuerſum aguntur <reg norm="vſque" type="simple">vſq;</reg> ad <reg norm="medium" type="context">mediũ</reg> trianguli <var>.D.R.B.</var> <lb/>Neque etiam probandus eſt auctor ille, cum pro oculo, ſuum <var>.T.</var> loco <var>.Q.</var> à me poſi-<unsure/><lb/>ti, ponit, cum is locus ſit verus ſitus pedis eius quireſpicit, & non oculi. </s> + <s xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> <lb/>Auctor iſte, modo vniuerſali intelligat, vt iam diximus, <reg norm="conſideretur" type="context">cõſideretur</reg> figura tertij mo <lb/>di primi cap. tertiæ partis, in qua ſuum oculum (vt ita dicam) ponit in <var>.o.</var> altius ſeu <lb/>diſtans à rectitudine lateris <var>.c.d.</var> plus quam ſit 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/>ſam, quæ me in alijs meis literis impulit ad <reg norm="dicendum" type="context">dicendũ</reg>, <reg norm="angulum" type="context">angulũ</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 cauſam imagineris in ſubſcripta hîc figura duo triangula <var>.q.o.u.</var> et <var>.q.p.u.</var> <lb/>quorum <var>.q.p.u.</var> perpendiculariter ſit ſuper ſuperficie trianguli <var>.q.o.p.</var> collocatum, <lb/>præcisè vt in mea figura corporea <var>.A.</var> ſuperficies verò trianguli <var>.q.o.p.</var> ſit exempli-<lb/>gratia <var>.V.M.</var> & trian-<lb/>guli <var>.u.o.p.</var> ſit <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> ſe-<lb/>ctio ſit <var>.V.p.o.x.</var> non <lb/>eſt enim <reg norm="dubitandum" type="context">dubitãdum</reg> <lb/>quin triangulum <var>.q.<lb/>p.u.</var> ſit perpendicula-<lb/>re triangulo <var>.q.o.p.</var> <lb/><reg norm="cum" type="context">cũ</reg> hoc ex .18. lib. 11. <lb/>Eucli. perpendicula-<lb/>re ſit ſuperficiei <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 ſuperficiei <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"/> + eſt angulum <var>.p.q.u.</var> obtuſum eſſe; </s> + <s xml:space="preserve">Imaginemur ergo circa triang ulum <var>.p.q.u.</var> circun-<lb/>ſcriptum eſſe circulum, cuius portio <var>.p.q.u.</var> minor erit medietate eiuſdem medij cir-<lb/>culi, vt iam ex <ref id="ref-0023">30. Eucli. lib. tertij</ref> nouiſti. </s> + <s xml:space="preserve">nunc imaginemur dictum circulum circum <lb/>lineam <var>.q.u.</var> loco axis verſus <var>.x.</var> moueri, vnde girus eiuſdem, per quem tranſibat linea <lb/><var>V.x.</var> remouebitur ab eadem linea non nihil cum motus erit à primo ſitu vſquequò <lb/>ad ſecandam 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 ſi <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 ſi <lb/>idem <reg norm="punctum" type="context">punctũ</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">ꝗ-<lb/>dẽ</reg> re tu ipſe 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">ꝙ</reg> <lb/>hæc Ioannis Cuſini di <lb/>cta ad hæc vſque tempora tanto in prætio ſint habita, vt ab excellentibus ſcriptori-<lb/>bus quaſi ſi proprij eorum ingenij partus eſſent, de verboad verbum vt theſauros, in <lb/>fuis <reg norm="ipſorummet" type="context">ipſorũmet</reg> libris reſcripta fuerint, quemadmodum iam omnes admonui in mea <lb/>gnomonica Orontium, Munſterum, <reg norm="aliosque" type="simple">aliosq́;</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 perſpectiua tertíum corpus regu <lb/>lare, <reg norm="quod" type="simple">ꝙ</reg> octo triangulis æquilateribus eſt term inatum, ſcire deſideras, hoc <reg norm="vnum" type="context">vnũ</reg> <lb/>eſt caput: </s> + <s xml:space="preserve">vnde fiat, aut quomodo probetur quaſlibet duas facies oppoſitas eiuſ-<lb/>dem corporis octoaedri <reg norm="inuicem" type="context">inuicẽ</reg> æquidiſtantes eſſe. </s> + <s xml:space="preserve">Quamobrem ſit hîc <reg norm="ſubſcriptum" type="context">ſubſcriptũ</reg> <lb/><reg norm="octoaedrum" type="context">octoaedrũ</reg>, cuius diameter vna ſit <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> æquidiſtantes eſſe contendo <lb/>ſint 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 ſint 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 ęquales erunt <reg norm="ſumendo" type="context">ſumẽdo</reg> eas vt <lb/>baſes <reg norm="triangulorum" type="context">triangulorũ</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="æquidiſtabunt" type="simple context">æꝗdiſtabũt</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">ꝗa</reg> ſi imaginabimur dia <lb/>metrum <var>.b.q.</var> tunc ex .4. aut ex .8. eiuſdem lib. <lb/>habebimus angulos <var>.a.b.q.</var> et <var>.u.q.b.</var> æquales <lb/>inuicem; </s> + <s xml:space="preserve">ſed ob eaſdem rationes <var>.p.l.</var> paralle-<lb/>la eſt ipſi <var>.d.k.</var> vnde ex 15. lib. 11. facies <var>.b.p.l.</var> <lb/>parallela fit, aut æquidiſtans ipſi <var>.q.d.k.</var> ideſt <lb/>primum propoſitum.</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 diſtantiæ, aut interualli ſimul cum ſitu, in fa-<lb/>cie <var>.q.d.k.</var> quem latus <var>.p.l.</var> perpendiculariter reſpicit. </s> + <s xml:space="preserve">Imaginemur à puncto <var>.u.</var> ſuper <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 conſtet, <reg norm="quodquidem" type="context">quodquidẽ</reg> triangulum, medietas eſt qua-<lb/>drilateri, ſeu. rumbi <var>.q.a.b.u.</var> cui vnaquæque dictarum quatuor facierum perpendi-<lb/>cularis exiſtit ex .4. ct .18. lib. 11. & ob id linea <var>.u.o.</var> extenſa in ſuperficie 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. eiuſdem lib <var>.o.u.</var> perpendicularis erit faciei <var>.b.p.l</var>. </s> + <s xml:space="preserve">Ha bebimus ergo ſitum in fa-<lb/>cie <var>.q.d.k.</var> qui reſpicietur 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> ſit latus exagoni æquilateris circumſcrip tibilis ab eodem circu <lb/>lo, qui vnam ex faciebus triangularibus æquilateribus propoſiti corporis circunſcri-<lb/>bere pot eſt, ita oſtenditur. ſit <reg norm="comprehenſum" type="context">cõprehenſum</reg> imaginatione, triangulum <var>.a.q.u.</var> ſepara <lb/>tim, cuius latus <var>.a.u.</var> æquale eſt vni ex lateribus <reg norm="triangulorum" type="context">triangulorũ</reg> eiuſdem 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 ſit perpendicularis <var>.u.o.</var> ab vna <reg norm="extremitatum" type="context">extremitatũ</reg> <lb/>lateris maioris, ad vnum ex minoribus lateribus, quę perpendicularis intra triangu-<lb/>lum cadet, quia dictum triangulum oxigonium eſt. </s> + <s xml:space="preserve">quod autem attinet ad duos angu <lb/>los <var>.a.</var> et <var>.u.</var> cum æquales ſint ex quinta lib. primi; </s> + <s xml:space="preserve">17. nos certiores facit; </s> + <s xml:space="preserve">quod verò an<lb/>gulus <var>.q.</var> ſit <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">ꝗa</reg><var>.a.u.</var> minor eſt diametro <reg norm="ſphae ræ" type="simple">ſphę<lb/> + <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> + ræ</reg> datum corpus circumſcribentis, cum <var>.q.</var> <lb/>dictæ ſphęrę ſuperficiem 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> ęqualem eſſe lateri <lb/>exagoni dicti, ſatis erit probare <var>.a.q.</var> ſeſqui <lb/>alteram eſſe ad <var>.a.o.</var> quia ſi in ſubſcripto <lb/>hîc circulo ducemus duas ſemidiametros <var>.<lb/>n.p.</var> et <var>.n.l.</var> ad. angulos <reg norm="trianguli" type="context">triãguli</reg> ęquilateri <var>.p.</var> <lb/>et <var>.l.</var> & cum quodlibet laterum ipſius exago <lb/>ni, ęquale ſit ſemidiametro 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. eiuſ <lb/>dem <var>.o.n.</var> ęqualiserit ipſi <var>.o.q.</var> ideſt <var>.q.a.</var> ſeſ <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> ſeſquialteram eſſe ad <var>.a.o.</var> eſt <reg norm="quoque" type="simple">quoq;</reg> <lb/>ſciendum primò omne latus trianguli ęquilateri in potentia ſeſquitertium eſſe ad <lb/>perpendicularem eiuſdem trianguli, quod vndecima lib. 14. Eucli. breuiter demon <lb/>ſtratum eſt.</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 poſteriore loco dixi <lb/>mus, & quia quadratum ipſius <var>.q.a.</var> eſt tantò minus aliorum duorum quadratorum <lb/>ſumma, quantum eſt duplum producti ipſius <var>.q.a.</var> in <var>.a.o.</var> ex .13. ſecundi, ſed alia duo <lb/>quadrata ſimul collecta faciunt .21. à quo numero ſubtrahendo quadratum ipſius <var>.a.<lb/>q.</var> ideſt nouem, remanebit numerus .12. pro duplo producti ipſius <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/>eſt ſimplex <lb/>productum <lb/>ipſius <var>.q.a.</var> <lb/><reg norm="in" type="wordlist">ĩ</reg> <var>a.o.</var> erit 6. <lb/>Sed <reg norm="quia" type="simple">ꝗa</reg> qua <lb/>dratum ip-<lb/>ſius <var>.q.a.</var> eſt <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/>ipſius <var>.q.a.</var> <lb/>in <var>.a.o.</var> pro <lb/>latere <var>.a.o.</var> <lb/>conſurgent <lb/>duo, cum er <lb/>go <var>.a.o.</var> ſint <lb/>duo tertia <lb/>ipſius <var>.a.q.</var> <lb/>certi <reg norm="erimus" type="simple">erimꝰ</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="DEſiderantes" type="context">DEſiderãtes</reg> ſcire 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 perſpectiuę Danielis <lb/>Barbari, ſeu Zamberti, eſſe veram <reg norm="altitudinem" type="context">altitudinẽ</reg> cor-<lb/>poris octoaedri, <reg norm="primum" type="context">primũ</reg> ſcire debemus <reg norm="quod" type="simple">ꝙ</reg> <reg norm="exiſtente" type="context">exiſtẽte</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, quę ſuperius <reg norm="iam" type="context">iã</reg> <lb/>diximus, facile percipi poteſt; </s> + <s xml:space="preserve">ex penultima primi <var>.<lb/>b.l.</var> in potentia, ſeſquioctaua erit ad <var>.k.l.</var> ipſa 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">cũ</reg> eſſet <var>.h.<lb/>l.</var> exiſteret in potentia tripla ad <var>.h.k.</var> & ſeſquialtera <lb/>ad <var>.l.k.</var> & ſeſquitertia ad <var>.l.b.</var> & ſic ad <var>.h.b.</var> vnde <var>.l.h.</var> <lb/>æqualis eſſet vni ex lateribus <reg norm="trianguli" type="context">triãguli</reg> ęquilateri di-<lb/>cti corporis. </s> + <s xml:space="preserve">Ex rationibus igitur ſuperius hîc poſi-<lb/>tis <var>.l.k.</var> erit altitudo dicta, id eſt diſtantia inter duas <lb/>facies inuicem oppoſitas, 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">aliũ</reg> <reg norm="non" type="context">nõ</reg> <reg norm="paruum" type="context">paruũ</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">cũ</reg> ſint acuti, <reg norm="nam" type="context">nã</reg> <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> minor eſt angulo cubi ſolido.</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 ſcitißimè, de mechn<unsure/>-<lb/>nicis, at cum natura <reg norm="vſusque" type="simple">vſusq;</reg> aliquid ſemper vel nouum, vel <lb/>Latens in apertum emittere ſoleant, nec ingenui aut grati ſit <lb/>animi, posteris inuidere, ſi quid ei contigerit comperuiße prius <lb/>tenebris inuolutum: </s> + <s xml:space="preserve">cum tam multa ipſe ex aliorum diligentia <lb/>ſit conſequut us. </s> + <s xml:space="preserve">Paucula <reg norm="quædam" type="context">quædã</reg> futùra, vt reor, non ingrata his <lb/>qui in biſce mechanicis verſantur, nuſquam ante bac tentata, <lb/>aut ſatis exastè explicata in medium proferre volui: </s> + <s xml:space="preserve">quo vel iuuandi deſiderium, vel <lb/>ſaltem non ocioſi 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 ſitus brachiorum libra.</head> + <head xml:space="preserve">CAP.I.</head> + <p> + <s xml:space="preserve"><emph style="sc">OMne</emph> pondus poſitum 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 diuerſa ratione ſitus ipſius brachij. </s> + <s xml:space="preserve">ſit 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> ſit pondus, & <var>.C.O.</var> linea inclinationis, ſeuicineris<unsure/> <var>.C.</var> verſus cen-<lb/>trum mundi, cum qua <var>.B.C.</var> angulum rectum conſtituat in puncto <var>.C</var>. </s> + <s xml:space="preserve">Exiſtente <lb/>igitur in huiuſmodi ſitu brachio <var>.B.C.</var> dico pondus <var>.C.</var> grauius futurum, quam <lb/>in alio quolibet ſitu. </s> + <s xml:space="preserve">quia ſupra centrum <var>.B.</var> omninò non quieſcet, quemadmodum <lb/>in quouis alio ſitu faceret. </s> + <s xml:space="preserve">Ad quod intelligendum, ſit dictum brachium, in ſitu <var>.B.<lb/>F.</var> cum eodem pondere in puncto <var>.F.</var> & linea itineris ſeu inclinationis dicti ponderis <lb/>ſit <var>.F.u.M.</var> per quam lineam dictum pondus progredi non poteſt, niſi 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 ſupra cen <lb/>trum <var>.B.</var> mediante brachio <var>.B.F.</var> nititur. <lb/></s> + <s xml:space="preserve">Eſt quidem verum, quòd pondus <var>.C.</var> nec <lb/>ipſum etiam per lineam <var>.C.O.</var> proficiſce-<lb/>tur, quia iter extremitatis brachij eſt cir-<lb/>cularis, & <var>.C.O.</var> in vno <reg norm="quodam" type="context">quodã</reg> puncto eſt <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 ſuæ inclinatio-<lb/>nis (ponamus <var>.F.u.M.</var>) propinqua erit di <lb/>cto centro <var>.B.</var> quod ſequenti cap. proba-<lb/>bo, vt exempli gratia, ſit <var>.F.</var> ſuper <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> vndeſe- + <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 eſt <var>.A.F.</var> & <lb/>minus ſupra 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-ſupra dictum <lb/>centrum <var>.B.</var> quieſcet, & hac ratione grauius quoque erit, & quanto magis vicinum <lb/>erit ipſi <var>.A.</var> à dicto <var>.F.</var> tantò magis ſuper centrum <var>.B.</var> quoque quieſcet, vnde <reg norm="tantò" type="context">tãtò</reg> quo-<lb/>que leuius exiſtet. </s> + <s xml:space="preserve">Idem dico de omni ſitu brachij per girum inferiorem <var>.C.Q.</var> vbi <lb/>pondus pendebit à centro <var>.B.</var> dictum centrum attrahendo, quemadmodum ſuperius <lb/>illud impellebat. </s> + <s xml:space="preserve">Hæc verò omnia cap. ſequenti 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 diuerſo ſitu 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> poſitam inter centrum & lineam <var>.F.u.M.</var> inclinatio-<lb/>nis, quam pondus ab extremitate <var>.F.</var> liberum verſus mundi <reg norm="centrum" type="context">centrũ</reg> conficeret. </s> + <s xml:space="preserve">Quod <lb/>vt facilius intelligamus imaginemur <reg norm="alterum" type="context">alterũ</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 eſt <var>.B.D.</var> cla-<lb/>rè cognoſcetur ex .6. lib. primi de ponderibus Archimedis, quòd ſi in puncto <var>.u.</var> col-<lb/>locatum erit pondus ipſius <var>.C.</var> libra nihil penitus à ſitu orizontali dimouebitur. </s> + <s xml:space="preserve">Sed <lb/>perinde eſt quòd pondus <var>.F.</var> æquale <var>.C.</var> ſit in extremo <var>.F.</var> in ſitu brachij <var>.B.F.</var> <reg norm="quam" type="context">quã</reg> vt ſit <lb/>in puncto <var>.u.</var> in ſitu ipſius <var>.B.u.</var> orizontali. </s> + <s xml:space="preserve">Ad cuius rei euidentiam imaginemur <reg norm="filum" type="context">filũ</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 ſi 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 eſt ſitu ipſius <var>.C.</var> quantò <var>.u.<lb/>B.</var> minus eſt ipſo <var>.B.C</var>. </s> + <s xml:space="preserve">Idem aſſero ſi brachium eſſet in ſitu <var>.e.B.</var> quod facilè cogno-<lb/>ſcere poterimus, ſi imaginemur filum appenſum ipſi <var>.u.</var> brachij <var>.B.C.</var> & vſque ad <var>.e.</var> <lb/><reg norm="perpendicularem" type="context">perpendicularẽ</reg>, in quo extremo <reg norm="appensum" type="context">appensũ</reg> eſſet pondus æquale ponderi <var>.C.</var> & <reg norm="liberum" type="context">liberũ</reg> <lb/>ab <var>.e.</var> brachij <var>.B.e.</var> vnde libra orizontalis manebit. </s> + <s xml:space="preserve">Sed ſi brachium <var>.B.e.</var> conſolida-<lb/>tum fuiſſet in tali ſitu cum orizontali <var>.B.D.</var> <lb/> + <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> + & <reg norm="appenſo" 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="aſcenderet" type="context">aſcẽderet</reg>, <reg norm="neque" type="simple">neq;</reg> deſcenderet. </s> + <s xml:space="preserve">quia tantum <lb/>eſt quod ipſum ſit appenſum filo, <reg norm="quod" type="simple">ꝙ</reg> pendet <lb/>ab <var>.u.</var> quantum quòd ab ipſo liberum <reg norm="appem" type="context">appẽ</reg> <lb/>nſum fuiſſet <var>.e.</var> brachij <var>.B.e.</var> & hoc procede <lb/>ret ab eo quòd partim pendereta centro <var>.<lb/>B.</var> & ſi <reg norm="brachium" type="context">brachiũ</reg> eſſet in ſitu <var>.B.Q.</var> totum <reg norm="pon" type="context">põ</reg> <lb/>dus centro <var>.B.</var> remaneret appenſum, <reg norm="quem- admodum" type="context">quem-<lb/>admodũ</reg> in ſitu <var>.B.A.</var> <reg norm="totum" type="context">totũ</reg> dicto centro an-<lb/>niteretur. </s> + <s xml:space="preserve">vnde fit vt hoc modo pondus <lb/>magis aut minus ſit 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 eſt cauſa proxima, & per ſe, <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">demq́;</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"/> + ſtat. </s> + <s xml:space="preserve">Et quamuis appellem latus <var>.B.C.</var> orizontale, ſupponens illud angulum rectum <lb/>cum <var>.C.O.</var> facere, vnde angulus <var>.C.B.Q.</var> fit vt minor ſit recto, ob quantitatem vnius <lb/>anguli ęqualis ei, quem duæ <var>.C.O.</var> et <var>.B.Q.</var> in centro regionis <reg norm="elementaris" type="context">elemẽtaris</reg> <reg norm="conſtituunt" type="context">conſtituũt</reg>, <lb/>hoc tamen nihil refert, cum dictus angulus inſenſibilis ſit magnitudinis. </s> + <s xml:space="preserve">Ab iſtis au-<lb/>tem rationibus elicere poſſumus, quod ſi 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> & ſi dictum <var>.u.</var> erit propius <var>.B.</var> quam puncto <var>.C.</var> pendebit ab ipſo, aut nitetur <lb/>ipſi amplius <reg norm="quam" type="context">quã</reg> exmedietate, & ſi magis verſus <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 cuiuſlibet ponderis, aut uirtus mouens re-<lb/>ſpectu alterius quantitatis cognoſcatur 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 hucuſque ſunt dicta, facilè intelligi poteſt, <reg norm="quod" type="simple">ꝙ</reg> quantitas <var>.B.u.</var> <lb/>quæ ferè perpendicularis eſt à centro <var>.B.</var> ad lineam <var>.F.u.</var> inclinationis, ea eſt, <lb/> + <anchor type="handwritten" xlink:label="hd-0155-01a" xlink:href="hd-0155-01"/> + quæ nos ducit in cognitionem quantitatis virtutis ipſius <var>.F.</var> in huiuſmodi ſitu, conſti <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/>ſint appenſa duo pondera, aut duæ virtutes mouentes <var>.e.</var> et <var>.c.</var> ita tamen <reg norm="quod" type="simple">ꝙ</reg> linea incli-<lb/>nationis <var>.e.</var> ideſt <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> ideſt <var>.a.c.</var> faciat angulum acutum, aut obtuſum 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. ſecetur deinde imaginatione <var>o.a.</var> in puncto <var>.i.</var> ita ut <lb/><var>o.i.</var> æqualis. </s> + <s xml:space="preserve">ſit <var>.o.t.</var> & puncto <var>.i.</var> appenſum ſit pondus æquale ipſi <var>.c.</var> cuius inclinationis <lb/>linea parallela ſit lineæ inclinationis ponderis <var>.e.</var> ſupponendo tamen pondus aut vir <lb/>tutem <var>.c.</var> ea ratione maiorem eſſe ea, quæ eſt <var>.e.</var> qua <var>.b.o.</var> maior eſt <var>.o.t.</var> abſque dubio <lb/>ex .6. lib. primi Archi. de ponderibus <var>.b.o.i.</var> non mouebitur ſitu, ſed ſi loco <var>.o.i.</var> imagi <lb/>nabimur <var>.o.t.</var> conſolidatam cum <var>.o.b.</var> & per lineam <var>.t.c.</var> attractam virtute <var>.c.</var> ſimiliter <lb/>quoque continget ut <var>b.o.</var> t; </s> + <s xml:space="preserve">communi quadam ſcientia, non moueatur ſi tu. </s> + <s xml:space="preserve">Eſt ergo <lb/> + <anchor type="handwritten" xlink:label="hd-0155-02a" xlink:href="hd-0155-02"/> + quod propoſuimus verum quantitatem alicuius ponderis reſpectu ad eam, quæ eſt <lb/>alterius debere depræhendi à perpendicularibus, quæ à centro libræ ad lineas incli <lb/>nationis exiliunt. </s> + <s xml:space="preserve">Hinc autem innoteſcit 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 ſequetur, 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 exſupradictis cauſis omnes staterarum & <lb/>uectium cauſæ dependeant.</head> + <head xml:space="preserve">CAP. IIII.</head> + <p> + <s xml:space="preserve">VIs brachij longioris alicuius ſtateræ, aut vectis, maior breuioris, ab ijs, quæ in ſu <lb/>perioribus capitibus diximus, ideſt <reg norm="quod" type="simple">ꝙ</reg> nitatur <reg norm="pendeatuem" type="context">pendeatuẽ</reg> magis aut minus à <lb/>centro pondus in extremitate brachij maioris poſitum, oboritur. </s> + <s xml:space="preserve">Quamobrem illud <lb/>à nobis primò eſt cognoſcendum, ſtateras, aut vectes, puras mathematicas li-<lb/>neas non eſſe, ſed naturales, hincque exiſtere corpora cum materia coniuncta. </s> + <s xml:space="preserve">Nunc <lb/>igitur imaginemur <var>.n.s.</var> eam ſuperficiem eſſe, quæ ſecundum longitudinem axem ſta <lb/>teræ ſcindit. </s> + <s xml:space="preserve">& ſupponamus ipſius 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 ſit, quanta eſt ſpiſſitu-<lb/>do, aut craſſities ipſius ſtateræ à ſuperiori latere ad inferius, ad faciliorem intelligen-<lb/>tiam, ſupponendo <var>.n.s.</var> <reg norm="parallelogrammam" type="context">parallelogrãmam</reg>. </s> + <s xml:space="preserve">Poſitis igitur duobus ponderibus æquali-<lb/> + <anchor type="handwritten" xlink:label="hd-0156-01a" xlink:href="hd-0156-01"/> + bus in extremitatibus brachiorum, experientia innoteſcit, <reg norm="quod" type="simple">ꝙ</reg> pondus ad <var>.u.s.</var> appen-<lb/>ſum, viol entiam faciet ponderi appenſo ad <var>.n.x.</var> ſed nos volumus inueſtigare <reg norm="causam" type="context">causã</reg> <lb/>huius effectus, quæ à nemine vnquam literarum monumentis, <reg norm="quod" type="simple">ꝙ</reg> ſciam, conſignata <lb/> + <anchor type="handwritten" xlink:label="hd-0156-02a" xlink:href="hd-0156-02"/> + fuit. </s> + <s xml:space="preserve">Iam diximus ſtateram, aut vectem materialem eſſe & <var>.n.s.</var> eius ſuperficiem me-<lb/>diam, ſupponendo <var>.i.</var> eſſe centrum quo nititur dicta ſtatera aut vectis; </s> + <s xml:space="preserve">Cum hocer-<lb/>go ita ſe habeat, ſint <var>.u.s.</var> et <var>.n.x.</var> lineæ inclinationum ponderum, & imaginemur, <reg norm="quod" type="simple">ꝙ</reg> <lb/>dicta pondera pendeant à punctis <var>.u.</var> et <var>.n.</var> vt reuera pendent, etiam ſi appenſa eſſent <lb/>ſub <var>.s.</var> et <var>.x.</var> quia punctum <var>.u.</var> & punctum <var>.n.</var> ita coniuncta ſunt cum <var>.s.</var> et <var>.x.</var> ut qui <reg norm="vnum" type="context">vnũ</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> quę <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, ſi quis ipſi <lb/>e. pondus ipſius <var>.u.</var> (<reg norm="quod" type="simple">ꝙ</reg> æquale eſt ponderi <var>.n.</var>) appenderet, id eandem planè vim habe <lb/>ret, quam pondus ipſius <var>.n.</var> habet, & ſtateram neque ſurſum, neque deorſum moue-<lb/>ret, quia ambo pondera ad centrum <var>.i.</var> mediantibus lineis <var>.e.i.</var> et <var>.n.i.</var> exęquo annite-<lb/>rentur, ſed dicto pondere poſito 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 diſtans à centro <var>.i.</var> <lb/> + <anchor type="handwritten" xlink:label="hd-0156-03a" xlink:href="hd-0156-03"/> + quàm linea <var>.e.t.</var> vnde huiuſmodi pondus magis quoque liberum à centro <var>.i.</var> reſultat. <lb/></s> + <s xml:space="preserve">magisque ponderoſum, quam cum erat in <var>.e.</var> ratione eorum, quæ primo & ſecundo <lb/>capitibus diximus, & ob hanc cauſam ſuperat pondus poſitum in <var>.n</var>. </s> + <s xml:space="preserve">Sed ſi centrum <lb/>fuerit .in <var>.o.</var> imaginabimur duas lineas <var>.o.s.</var> et <var>.o.x.</var> & ſupponemus quòd pondera po-<lb/>ſita ſint in <var>.s.</var> et <var>.x.</var> vnde exiſtente magis orizontali linea <var>.o.s.</var> quam erit <var>.o.x.</var> & linea <lb/><var>u.s.</var> inclinationis longius diſtante à centro <var>.o.</var> quàm 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 ſuperius dixi-<lb/>mus, inueniemus eundem effectum verum eſſe. </s> + <s xml:space="preserve">In ſtateris, rectè & propriè appella <lb/>ri poteſt <var>.x.i.s.</var> aut <var>.n.o.u.</var> orizontalis, ſed in omni vectium ſpecie, hoc <reg norm="tantum" type="context">tãtum</reg> per quan <lb/>dam ſimilitudinem dicetur. </s> + <s xml:space="preserve">Idem contemplari licet ſupponendo centrum in medio <lb/>inter <var>.o.</var> et <var>.i.</var> quod vnuſquiſque ex ſe abſque 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 quibuſdam rebus animaduerſione 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/>ſunt neceſſaria. </s> + <s xml:space="preserve">Quod autem quærimus, in eo conſiſtit, quòd aliqui vectes <lb/>adhibeantur ad opus, quorum centrum, quod Græci <reg norm="hypomochlion" type="context">hypomochliõ</reg> appellant vnum <lb/>eſt ex extremis ipſius vectis, & pondus, quod ſurſum eleuari debet, inter ipſa-<lb/>met extrema iacet, propinquum tamen hypomochlio, vt exempli gratia, ſi vectis <lb/>eſſet infraſcripta 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">ꝙ</reg> cum eleuari debeat <var>.n.</var> oportebit quoque opera manus ele-<lb/>uari <var>.u</var>. </s> + <s xml:space="preserve">Nunc conſiderandum eſt quomodo pondus <var>.n.</var> annitatur ad <var>.u</var>. </s> + <s xml:space="preserve">Hanc ob cau <lb/>ſam 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> verſus mundi cen <lb/>trum ſit poſita, 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 ſuperius conſtante, vt <var>.n.</var> ad inferius (re-<lb/>mota tamen grauitate materiæ vectis) huiuſmodi virtus, totum pondus ipſius <var>.n.</var> com <lb/>muni quadam ſcientiæ notione ſuſtinebit. </s> + <s xml:space="preserve">& ſi <reg norm="pondus" type="context">põdus</reg> ipſius <var>.n.</var> eſſet in <var>.x.</var> è directo ſu-<lb/>per <var>.o.</var> totum pondus ſuper hypomochlio ſe haberet, & tanta virtus ipſius hypomo-<lb/>chlij ſufficeret ad reſiſtendum pro ſuſtinendo, quanta eſt grauitas ipſius ponderis, <lb/>ſed ipſum iterum ponamus in <var>.n.</var> ibi clarum erit, quòd ſi alia virtus à parte inſeriori <lb/>ad ſuperiorem vectis non opponitur, excepto tamen hypomochlio, oportebit virtu <lb/>te cuiuſdam partis ponderis <var>.n.</var> (abſque conſideratione tamen, vt iam dixi, ponderis <lb/>materiæ vectis) vt vectis à parte <var>.s.u.</var> deprimatur, & dixi vnius cuiuſdam partis pon-<lb/>deris <var>.n.</var> quia alia <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ponderis pars annititur ipſi 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 ſeſe huiuſ-<lb/>modi reſiſtentia, vt vectis non deprimatur, clarum erit communi ſcientia, <reg norm="quod" type="simple">ꝙ</reg> virtus <lb/>ponderis <var>.n.</var> diuiſa erit per medium æqualiter, cuius vna medietas ſuper <var>.o.</var> quieſcet, <lb/>& alia ſuper <var>.t.</var> mediantibus duabus lineis <var>.n.o.</var> et <var>.n.t</var>. </s> + <s xml:space="preserve">Imaginemur nunc reſiſtentiam <lb/>t. ablatam eſſe, <reg norm="poſitamque" type="simple">poſitamq;</reg> in <var>.e.</var> clarum quoque erit, <reg norm="quod" type="simple">ꝙ</reg> maior pars ponderis <var>.n.</var> ipſi <var>.e.</var> <lb/>annitetur beneficio lineæ <var>.n.e.</var> quàm ipſi <var>.o.</var> cum linea <var>.n.i.</var> inclinationis ipſi <var>.e.</var> ſit pro <lb/>pinquior quam <var>.o.</var> quia omnis reſiſtentia aut in <var>.i.</var> aut in <var>.e.</var> aut in <var>.t.</var> aut in <var>.u.</var> eſt loco <lb/>centri, quemadmodum eſt <var>.o.</var> & alter alterius opera iuuatur. </s> + <s xml:space="preserve">Si verò eadem reſiſten <lb/>tia poſita erit in <var>.u.</var> clarum quoque erit, <reg norm="quod" type="simple">ꝙ</reg> minor pars ponderis <var>.n.</var> annitetur ipſi <var>.u.</var> <reg norm="quam" type="context">quã</reg> <lb/>ipſi <var>.o.</var> cum dicta <var>.n.i.</var> à centro <var>.u.</var> longius quam à centro <var>.o.</var> diſter, & 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="ſecundum" type="context">ſecũdum</reg> propor <lb/>tionem angulorum <var>.u.n.i.</var> et <lb/><var>o.n.i.</var> ſed ſecundum propor <lb/>tionem <var>.u.i.</var> ad <var>.i.o.</var> quod cla <lb/>rè compræhendi poteſt 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 conuerſo, ideſt, vt quemadmodum nunc ſupponuntur <var>.o.</var> et <var>.u.</var> eſſe duo <lb/>centra quibus <reg norm="ſuſtinetur" type="simple">ſuſtinet̃</reg> pondus <var>.e.</var> ipſius <var>.n.</var> imaginemur <var>.n.</var> eſſe quoddam centrum à <lb/>quo pendeant duo pondera <var>.o.</var> et <var>.u.</var> ſic inuicem proportionata, ut ſunt <var>.u.i.</var> et <var>.i.o.</var> <lb/>certe horum ponderum cauſa ſtatera <var>.o.s.</var> quam vectem appellabamus à nulla parte <lb/>inclinabitur. </s> + <s xml:space="preserve">Redeuntes nunc ad propoſitum, dicemus <reg norm="quod" type="simple">ꝙ</reg> annitente pondere ipſius <var>.<lb/>n.</var> minus ad <var>.u.</var> quam ad <var>.o.</var> ideſt ad <var>.t.</var> minori vi opus erit in <var>.u.</var> quàm in <var>.t.</var> ad attollen-<lb/>dum pondus ipſius <var>.n.</var> & ſic per conſequens quantò longius erit punctum <var>.u.</var> ab <var>.t.</var> tan <lb/>tò minori quoque vi egebit, & conſequenter quando vis, aut reſiſtentia in <var>.u.</var> ita pro <lb/>portionata erit illi, quæ eſt ipſius <var>.o.</var> vt eſt <var>.o.i.</var> ad <var>.i.u.</var> vectis non mouebitur. </s> + <s xml:space="preserve">Sed quan <lb/>do erit proportio maior, reſiſtentiæ ipſius <var>.u.</var> ad eam, quæ eſt ipſius <var>.o.</var> ea, quæ eſt <var>.o.<lb/>i.</var> ad <var>.i.u.</var> </s> + <s xml:space="preserve">tunc vectis à par-<lb/>teipſius <var>.u.s.</var> eleuabitur, ſi <lb/> + <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> + vero proportio minor eſſet <lb/>quàm.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 cuiuſdam uis adauctæ.</head> + <head xml:space="preserve">CAP. VI.</head> + <p> + <s xml:space="preserve">QVibuſdam in locis vtuntur <reg norm="quidam" type="context">quidã</reg> <reg norm="quodam" type="context">quodã</reg> <reg norm="inſtrumento" type="context">inſtrumẽto</reg> piſtorio ad <reg norm="ſubigendam" type="context context">ſubigẽdã</reg> pa-<lb/>ſtam, 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 deſcriptione imaginemur planum, in quo ſedet ille, qui voluit paſtam, & in quo <lb/>ipſa paſta eſt repoſita <var>.T.S.D.</var> & triangulum <var>.T.A.S.</var> immobile perpendiculare-<lb/>q́ue ſuperficiei dicti plani, angulo autem <var>.A.</var> coniunctum lignum <var>.A.E.</var> vt ſemidiame <lb/>trum mobilem, & æqualem perpendiculari ipſius trianguli, </s> + <s xml:space="preserve">unde <var>.A.</var> loco centri erit <lb/>et <var>.D.O.</var> ſit ſemidiameter, qui paſtam contundit, & ab eius extremo <var>.O.</var> (quod <var>.O.</var> <lb/>quando <var>.D.O.</var> orizontalis eſt, in baſi dicti trianguli reperitur) veniat lignum <var>.O.V.</var> <lb/>quod cum <var>.A.V.</var> ſit æquale perpendiculari imaginatæ ab angulo <var>.A.</var> baſi <var>.T.S.</var> <reg norm="deno- datum" type="context">deno-<lb/>datũ</reg> <reg norm="tantum" type="wordlist/context">tñ</reg> utvulgo <reg norm="dicitur" type="simple">dicit̃</reg> ſeu flexile in <var>.O.</var> & in <var>.V.</var> vt elleuare <reg norm="atque" type="simple">atq;</reg> deprimere ſemidiame <lb/>trum <var>.D.O.</var> poſſit, et <var>.V.O.</var> ſit æqualis <var>.A.V.</var> et <var>.V.</var> medium ſit 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> ſunt deinde duo ligna <reg norm="perpendicularia" type="context">perpẽdicularia</reg> ab <var>.A.</var> ad baſim <lb/>fixa, & immobilia inter ſe adeò diſtantia, vt inter ipſa <reg norm="pertranſeant" type="context context">pertrãſeãt</reg> <var>.O.V.</var> et <var>.D.O.</var> ſupra <lb/>& infra, ne deuiet ſemidiametrum <var>.D.O</var>. </s> + <s xml:space="preserve">In extremitate deinde ipſius <var>.E.</var> ſit lignum <lb/>quoddam tenue, vt digitus polex, ad angulos rectos cum <var>.A.E.</var> quod ab aliquo, qui <lb/>antedictam machinam ſtet, manibus teneatur, qui quidem homo idipſum lignum, <lb/>ideſt ſemidiametrum <var>.A.E.</var> à ſuperficie trianguli dicti, ad ſe trahendo, & deinde ver <lb/>ſus eundem triangulum impellendo, vim quandam maximam mediante ſemidia <lb/>metro <var>.D.O.</var> ſuper paſtam excitat.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei contemplatione volo vt ſecundam hanc ſubſcriptam 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> baſem trianguli <var>.a.u.o.</var> cui <var>.o.t.</var> perpendicularis dictæ baſi <var>.<lb/>u.a.</var> addatur. </s> + <s xml:space="preserve"><reg norm="Hucuſque" type="simple">Hucuſq;</reg> igitur <var>.u.o.</var> æqualis erit <var>.o.x.</var> & ipſi <var>.o.a.</var> imaginemur etiam <var>.a.o.</var> <lb/>vſque ad <var>.b.</var> ita productam vt <var>.o.b.</var> æqualis ſit <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 verſus <var>.u.</var> vnde linea eius inclinationis ſit ſemper <var>.a.u.</var> ſupponamus etiam <var>.a.o.b.</var> <lb/>eſſe <reg norm="libram" type="context">librã</reg>, aut ſtateram, aut vectem, & <var>.o.</var> eius centrum, vnde vis, aut virtus ipſius <var>.a.</var> <lb/>proportionalis erit ipſi <var>.o.t.</var> reſpectu virtutis, aut vis imaginatæ in <var>.b.</var> inclinationis <lb/>perpendicularis ipſi <var>.b.a.</var> quæ quidem virtus, aut vis in <var>.b.</var> proportionalis erit ipſi <var>.b.<lb/>o.</var> ex tertio capite huius tractatus; </s> + <s xml:space="preserve">Si ergo fuiſſet poſita in <var>.b.</var> virtus quædarn ad an-<lb/>gulum rectum, trahens lineam <var>.b.o.</var> tam proportionatam virtuti perpendiculari ip-<lb/>ſius <var>.a.</var> quam eſt <var>.o.t.</var> proportionata ipſi <var>.o.b.</var> ſtatera <var>.b.o.a.</var> non moueretur, ſed quæuis <lb/>portio maior in <var>.b.</var> ſuperaret <var>.a.</var> cum autem fuerit <var>.o.x.</var> æqualis ipſi <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 ſcientia, 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 ſuperare debet reſiſtentiam <lb/>in <var>.a.</var> quæ ipſi <var>.u.</var> contraponitur, debet ha-<lb/>bere aliquantulum maioris proportionis <lb/>ad reſiſtentiam, quæ in <var>.a.</var> angulum re-<lb/>ctum efficeret cum <var>.a.o.</var> ea, quæ eſt <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 quibuſdam erroribus Nicolai Tartaleæ circa pondera <lb/>corporum & eorum motus, quorum aliqui deſumpti <lb/>fuerunt à fordano ſcriptore quodam antiquo.</head> + <head xml:space="preserve">CAP. VII.</head> + <p> + <s xml:space="preserve">CVm magis amici veritatis eſſe debeamus quàm cuiuſquam hominis, quemad-<lb/>modum Ariſto. ſcribit, detegam hoc loco quoſdam errores Nicolai Tartaleę <lb/>de ponderibus corporum, & velocitatibus motuum localium. </s> + <s xml:space="preserve">Et primum decipitur <lb/>is in .8. lib. ſuarum diuerſarum inuentionum in ſecunda propoſitione, cum non ani-<lb/>maduerterit quanti momenti ſint extrinſecæ reſiſtentiæ.</s> + </p> + <p> + <s xml:space="preserve">Subiectum quoque tertiæ propoſitionis eſt malè demonſtratum, quia idem pla-<lb/>nè ex eius demonſtratione iam dicta corporibus hætereogeneis, aut figura diuerſis <lb/>contingeret, quod ad velocitates attinet.</s> + </p> + <p> + <s xml:space="preserve">In quarta propoſitione, quod ad <reg norm="diſputandum" type="context context">diſputãdũ</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="ſequitur" type="simple">ſequit̃</reg>, quod Archimedes in .6. propoſitione lib. primi de <reg norm="ponderibus" type="context">põderibus</reg> <reg norm="probauit" type="simple">ꝓbauit</reg>.</s> + </p> + <handwritten/> + <p> + <s xml:space="preserve">Sed in ſecunda parte quintę propoſitionis non uidet <reg norm="quod" type="simple">ꝙ</reg> uigore ſitus eo modo, quo <lb/>ipſe diſputat, nulla elicitur ponderis differentia. </s> + <s xml:space="preserve">quia ſi corpus <var>.B.</var> deſcendere debet <lb/>per arcum <var>.i.l.</var> corpus <var>.A.</var> aſcendere debet per arcum <var>.u.s.</var> æqualem, & ſimilem. eadem <lb/>quoque ratione ſituatum, vt eſt arcus <var>.i.l.</var> vnde vt eſt facilè corpori <var>.B.</var> deſcendere <lb/>per arcum <var>.i.l.</var> difficile ita erit corpori <var>.A.</var> aſcendere per arcum <var>.u.s</var>. </s> + <s xml:space="preserve">Hęc autem qnin <lb/>ta propoſitio Tartaleæ eſt ſecuuda quæſtio à Iordano propoſita.</s> + </p> + <p> + <s xml:space="preserve">Quòd autem ad primum corollarium dictæ propoſitionis attinet, verum ille qui <lb/>dem ſcribit, eius tamen effectus cauſa & à Iordano prius, & ab ipſo poſtea citata, na-<lb/>tura ſua vera non eſt. </s> + <s xml:space="preserve">quia vera cauſa per ſe ab eo oritur, <reg norm="quod" type="simple">ꝙ</reg> à centro libræ dependeat <lb/>vt primo cap. huius tractatus oſtendi. </s> + <s xml:space="preserve">Secundum verò corollarium falſum eſſe, ijs ra <lb/>tionibus quas nunc ſubiungam, patebit. </s> + <s xml:space="preserve">Imaginemur <var>.u.</var> pro centro regionis ele-<lb/>mentaris, & libram <var>.b.o.a.</var> obliquam reſpectu ad <var>.u.</var> & brachijs æqualibus <reg norm="conſtantem" type="context">conſtãtem</reg>, <lb/>& pondera in <var>.a.</var> et in <var>.b.</var> etiam æqualia. </s> + <s xml:space="preserve">lineæ autem inclinationum ſint <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 ipſius <var>.a.</var> in huiuſmodi ſitu 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">ꝙ</reg> tertio cap. hu-<lb/>iustractatus probaui, ſed linea <var>.o.t.</var> maior eſt linea <var>.o.e.</var> quod ſic probo. </s> + <s xml:space="preserve">Imaginemur <lb/>triangulum <var>.u.a.b.</var> circunſcriptum eſſe à circulo <var>.u.a.n.b.</var> cuius <var>.c.</var> ſit centrum, <reg norm="quod" type="simple">ꝙ</reg> erit <lb/>extra lineam <var>.u.o.</var> cum ſupponatur <var>.a.o.b.</var> obliquam eſſe reſpectu 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> vſque ad circunferentiam, quæ perpendicula-<lb/>ris erit ipſi <var>.a.b.</var> ex tertia lib. 3. Eucli. </s> + <s xml:space="preserve">ſi poſteà 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> ſed ſi imaginabimur <var>.u.o.</var> ad circunferentiam vſque <lb/>productam, clarum erit <reg norm="quod" type="simple">ꝙ</reg> arcum <var>.s.b.</var> ſecaret in puncto <var>.n.</var> vnde arcus <var>.n.b.</var> minor erit <lb/>arcu <var>.n.a.</var> & ſic 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> ſit diameter, cuius circunferentia <lb/>per duo puncta <var>.e.</var> et <var>.t.</var> <reg norm="prætergradiatur" type="simple">prætergradiat̃</reg>, cum in ipſis ſint angulirecti, quod quilibet ex <lb/>ſeratio cinando colligere poteſt, ſi .30. lib. 3. in mentem reuocauerit. </s> + <s xml:space="preserve">Sed cum angu-<lb/>lus <var>.o.u.t.</var> ſit 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 ipſius <var>.o.e.</var> ex conuerſo .27. lib. 3. quod eſt propoſitum. </s> + <s xml:space="preserve">Pon-<lb/> + <anchor type="handwritten" xlink:label="hd-0160-01a" xlink:href="hd-0160-01"/> + dusigitur ipſius <var>.a.</var> in huiuſmodi ſitu, pondere ipſius <var>.b.</var> grauius erit. </s> + <s xml:space="preserve">Quod è directo ijs <lb/>repugnat quæ Tartalea in 2. parte quinræ propoſitionis ediſerit, & per conſequens <lb/>2. corollarij falſitatem oſtendit, vt eam quoque, quæ in 6. propoſitione latet. </s> + <s xml:space="preserve">quia <reg norm="cum" type="context">cũ</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/>dusipſius <var>.b.</var> eadem ſit cum <lb/>ea quę eſt <var>.o.t.</var> ad <var>.o.e.</var> ſub co <lb/><reg norm="gnitionem" type="context">gnitionẽ</reg> noſtram cadere po <lb/>teſt, primum cognoſcendo <lb/>angulos obliquitatis librę, <lb/>ideſt angulos <var>.b.o.u.</var> et <var>.a.o.<lb/>u.</var> quia oportet ſemper ſup-<lb/> + <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> + ponere ſitum aliquem no-<lb/>tum. </s> + <s xml:space="preserve">Si nobis deinde co-<lb/>gnita erit proportio ipſius <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 conſe-<lb/>quens ipſius <var>.o.a.t.</var> eius reſi-<lb/>dui, vnde poſtea 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. propoſitione, & Iordanus in ſecunda parte. <lb/></s> + <s xml:space="preserve">ſecundæ propoſitionis ſcribunt, maximum quoque errorem inſe continet. <lb/></s> + <s xml:space="preserve">Dicunt enim <reg norm="angulum" type="context">angulũ</reg> <lb/><var>h.a.f.</var> differentem ab <lb/>angulo <var>.d.b.f.</var> alia ra-<lb/>tione non eſſe quàm <lb/>per angulum conta-<lb/>ctus <reg norm="duorum" type="context">duorũ</reg> <reg norm="circulorum" type="context">circulorũ</reg>, <lb/>vt in ſua figura ſcribit <lb/>Tartalea; </s> + <s xml:space="preserve">id quod fal-<lb/>ſiſſimum eſt. </s> + <s xml:space="preserve"><reg norm="Quam" type="context">Quã</reg> ob <lb/>cauſam in ſubſcripta <lb/>figura ſit 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">centrũ</reg> regionis ele <lb/>mentaris, et <var>.A.u.</var> et <var>.B.<lb/>u.</var> lineæ <reg norm="inclinationum" type="context">inclinationũ</reg>. <lb/></s> + <s xml:space="preserve">Imaginemur deinde <lb/>lineam <var>.B.K.</var> <reg norm="parallelam" type="context">parallelã</reg> <lb/>ipſi <var>.A.u.</var> quæ gyrum <var>.<lb/>B.F.A.</var> in puncto <var>.K.</var> <lb/>communi ſcientiæ <reg norm="prae cepto" type="simple">prę<lb/>cepto</reg> ſcindet, & habe <lb/>bimus angulum <var>.K.B.<lb/>Z.</var> æqualem angulo <var>.<lb/>H.A.F.</var> ideſt <var>.u.A.F.</var> <lb/>(quia <var>.H.u.</var> et <var>.D.</var> <reg norm="unum" type="context">unũ</reg> <lb/>ſunt) 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 ſit <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">nũc</reg> <lb/>comparatio eſt inter <lb/>angulum <var>.D.B.F.</var> & an <lb/>gulum <var>.K.B.Z.</var> miſtili-<lb/>neos, qui quidem duo <lb/>anguli, <reg norm="communem" type="context">cõmunem</reg> ha-<lb/>bent angulum miſtili <lb/>neum <var>.K.B.F.</var> quapro-<lb/> + <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> + pter ſi angulus <var>.K.B.Z.</var> <lb/>miſtilineus maior eſt <lb/>angulo <var>.D.B.F.</var> miſti-<lb/>lineo per angulum <var>.<lb/>K.B.Z.</var> contingentiæ, <lb/>circulorum ergo angu <lb/>lus miſtilineus com-<lb/>munis <var>.K.B.F.</var> æqualis <lb/>erit miſtilineo, angu-<lb/>lo <var>.D.B.F.</var> pars vide-<lb/>licet ſui toto. </s> + <s xml:space="preserve">Omnis <lb/>autem error in quem <lb/>Tartalea, <reg norm="Iordanusque" type="simple">Iordanusq;</reg> <lb/>lapſi fuerunt ab eo, <reg norm="quod" type="simple">ꝙ</reg> <lb/> + <anchor type="handwritten" xlink:label="hd-0162-0a" xlink:href="hd-0162-01"/> + lineas inclinationum <lb/>pro parallelis viciſſim <lb/>ſumpſerunt, 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 propoſitio Tartaleæ, quæ eſt <reg norm="quinta" type="simple">ꝗnta</reg> quæſtio Iordani mihi <reg norm="videtur" type="simple">videt̃</reg> excipien-<lb/>da riſu, cum pondus ipſius <var>.A.</var> ponderi ipſius <var>.B.</var> exiſtens æquale, grauius ſit pondere <lb/>eiuſdem <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 ſcilicet ipſe putet <lb/>lineas <var>.A.E.</var> et <var>.B.D.</var> figuræ ab eo confictæ ſibi inuicem eſſe parallelas, quæ etiam ſi <lb/>æquidiſtantes eſſent (vnde angulus <var>.E.A.G.</var> minor eſſet angulo <var>.D.B.F.</var>) non eam ta <lb/>men ob cauſam huiuſmodiangulorum differentia cauſa eſſet differentiæ <reg norm="grauitatum" type="context">grauitatũ</reg> <lb/>ipſorum <var>.A.</var> et <var>.B.</var> ob ea quæ cap .4. huius tractatus poſui.</s> + </p> + <p> + <s xml:space="preserve">Octaua autem propoſitio, quæ eſt .6. quæſtio Iordani Iongè melius demonſtratur <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">Tartalęa citat, qui <lb/>neque etiam probat nonam .10. 11. 12. et .13. cum ad pręcedentes probandas mini <lb/>mè acceſſerit.</s> + </p> + <p> + <s xml:space="preserve">Quartadecima verò, quæ eſt .10. quęſtio Iordani, duas ob cauſas eſt falſa, quarum <lb/>vna eſt, <reg norm="quod" type="simple">ꝙ</reg> (ſupponendo <var>.A.D.E.G.B.</var> eſſe vnum brachium librę et <var>.A.</var> punctum <reg norm="centri" type="context">cẽtri</reg> <lb/>eiuſdem, et <var>.D.</var> pondus ęquale 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> ſibi <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="non" type="context">nõ</reg> ſunt ęquales; </s> + <s xml:space="preserve"><reg norm="cum" type="context">cũ</reg> ille angulus ſit intrinſecus, hic <lb/>verò extrinſecus & oppoſitus dicto intrinſeco <reg norm="vnius" type="simple">vniꝰ</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 vſque ad centrum regionis elementaris, vnde dictus angu-<lb/>lus <var>.M.E.G.</var> maior eſt alio, ex .16 lib. primi Eucli. </s> + <s xml:space="preserve">Qua ratione fit, vt hanc ob cauſam <lb/>E. grauius ſit ipſo <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 eſtratio, qua dictum <var>.E.</var> grauius fit ipſo <var>.D.</var> quę quidem eſt <lb/>maior diſtantia à centro <var>.A.</var> libræ, per ſimiles 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, quę eſt .11. quęſtio Iordani, cuius Autho-<lb/>ris opuſculum opera Traiani Bibliopolę Venetijs è tenebris in lucem emerſit.</s> + </p> + </div> + <div type="section" level="3" n="9"> + <head style="it" xml:space="preserve">Quòdſummaratione ſtateræper æqualia interualla <lb/>ſint diuiſæ.</head> + <head xml:space="preserve">CAP. IX.</head> + <p> + <s xml:space="preserve">MAgna cum ratione <reg norm="diuiduntur" type="context">diuidũtur</reg> ſtateræ per interualla ęqualia, in libras, aut in <lb/>vncias, aut quoquo alio modo. </s> + <s xml:space="preserve">Nam ſit ſtatera exempli gratia <var>.a.b.</var> <lb/>& punctum, <reg norm="quod" type="simple">ꝙ</reg> eam ſuſtinet ſit <var>.c.</var> & vas illud, <reg norm="quod" type="simple">ꝙ</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">cũ</reg> <lb/>eo, <reg norm="quod" type="simple">ꝙ</reg> eſt dicti vaſis <var>.f.</var> ab altera parte, ſint cauſę, quibus ſtatera <var>.a.b.c.</var> ſtet orizonta-<lb/>lis. cui ſic orizontali manenti imaginemur ad punctum <var>.a.</var> adiunctum eſſe pondus, <lb/>veluti vnius librę. & ad punctum <var>.d.</var> tam diſtanti à <var>.c.</var> ut eſt <var>.a.</var> ab ipſo <var>.c.</var> aliud quoque <lb/>pondus vnius libræ <reg norm="additum" type="context">additũ</reg> eſſe, vnde <reg norm="coni" type="context">cõi</reg> <reg norm="quadam" type="context">quadã</reg> ſcientia ſtatera, non mouebitur ſitu. <lb/></s> + <s xml:space="preserve"><reg norm="quia" type="simple">ꝗa</reg> exiſtentibus duobus hiſce 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> abſque dubio <var>.a.d.</var> non mutaret ſitum, ſed <var>.d.b.</var> et, f. in ſitu, in <lb/>quo reperiuntur, à centro paribus viribus prędita ſunt. </s> + <s xml:space="preserve">Addendo igitur <var>.d.b.</var> ipſi <var>.d.</var> <lb/>et <var>.f.</var> ipſi .a: ſumma earum, æqualibus quoque viribus conſtabunt. ex communi ſen-<lb/>tentia, quæ habet ſi ęqualibus addas ęqualia, tota quoque fient ęqualia. </s> + <s xml:space="preserve">Si verò <lb/>ponderi ipſius <var>.a.</var> aliud adderetur eidem ęquale, haberemus in <var>.a.</var> duplum pon-<lb/>dus ei <reg norm="quod" type="simple">ꝙ</reg> eſt ipſius <var>.d.</var> ſed volentes vt ſolum cum pondere ipſius <var>.d.</var> ſtatera ſtet orizon <lb/>talis, ſi dictum pondus ipſius <var>.d.</var> longè diſtabit à centro <var>.c.</var> per duplum ipſius <var>.c.a.</var> ideſt <lb/>ipſius <var>.c.d.</var> id <reg norm="quod" type="simple">ꝙ</reg> volumus aſſeque-<lb/>mur, beneficio ſupradictarum ra <lb/> + <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> + tionum, adiuti opera ſextę lib. pri<lb/>mi de <reg norm="ponderibus" type="context">põderibus</reg> Archimedis. </s> + <s xml:space="preserve">Et <lb/>ſi quis aliud <reg norm="quoque" type="simple">quoq;</reg> pondus adiun <lb/>geret ipſi <var>.a.</var> æquale illi priori, ad <lb/><reg norm="efficiendum" type="context">efficiẽdum</reg>, vt ſtatera ſemper ori <lb/>zontalis maneret, oporteret, vt <reg norm="pondus" type="context">põdus</reg> ipſius <var>.d.</var> ab <var>.c.</var> longè diſtaret, ita vt huiuſmodi <lb/>diſtantia tripla eſſet primæ, & ſic per quoſdam quaſi 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">ARiſtoteles 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 falſum <lb/>eſt: </s> + <s xml:space="preserve">quia huiuſmodi linea partes nullas ſecundum latitudinem habet, (vt ipſe etiam <lb/>confirmat) ſed eſt idem conuexum circuli: </s> + <s xml:space="preserve">linea verò quæ terminus eſt ſuperficiei <lb/>ambientis, & amplectentis circulum eſt eadem concauitas dictæ ſuperficiei eun-<lb/>dem circulum ambientis, quæ nullam conuexitatem habet. </s> + <s xml:space="preserve">& hæ duæ ſunt lineæ, <lb/>quarum vna diuerſa eſt 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 Ariſtoteles ſcribit de duplici reſpectu motus vnius puncti ſecun <lb/>dum vnam datam pro portionem, non ſufficit, ille enim ſic ait.</s> + </p> + <p> + <s xml:space="preserve">Sit proportio ſecundum quam latum fertur, quam habet <var>.A.B.</var> ad <var>.A.C.</var> et <var>.A.</var> qui <lb/>dem feratur verſus .B: <var>A.B.</var> verò ſubterferatur verſus <var>.M.C.</var> latum autem ſit <var>.A.</var> <reg norm="quidem" type="context">quidẽ</reg> <lb/>ad <var>.D.</var> vbi autem eſt <var>.A.B.</var> verſus <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 eſt & <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 eſt 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 reſpondeo, punctum <var>.A.</var> quod mouetur in linea <var>.A.M.</var> ab <var>.A.</var> verſus <var>.M.</var> vſque <lb/>ad <var>.F.</var> non moueriab aliqua proportione determinata magis quàm ab alia: </s> + <s xml:space="preserve">vnde <reg norm="non" type="context">nõ</reg> <lb/>ſolum poſſumus imaginari dictum punctum <var>.A.</var> moueri ab <var>.A.</var> vſque ad <var>.F.</var> eiuſdem <lb/>velocitatis ſub alia quadam proportione, ſed etiam ſub alia, quæ iam datæ contraria <lb/>ſit, vt eſt proportio ipſius <var>.A.C.</var> ad <var>.A.B.</var> <reg norm="imaginantes" type="context">imaginãtes</reg> moueri <var>.A.</var> verſus <var>.C.</var> et <var>.A.C.</var> ver <lb/>ſus <var>.B.M.</var> delatam. </s> + <s xml:space="preserve">Dico etiam idem <var>.A.</var> moueri vſque ad <var>.F.</var> ſecundum proportio-<lb/>nem ipſius <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">cũ</reg> linea <var>.A.F.</var> face-<lb/>re <reg norm="angulum" type="context">angulũ</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">triangulũ</reg> <var>.A.H.F.</var> ęqui <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="ipſius" type="simple">ipſiꝰ</reg> <lb/><var>A.H.</var> ad <var>.F.H.</var> quę <reg norm="enipſius" type="context">ẽipſius</reg> <lb/><var>A.O.</var> ad <var>.O.P.</var> punctum <lb/>igitur <var>.A.</var> vſque ad <var>.F.</var> mouetur ſecundum proportionem etiam ipſius <var>.A.O.</var> ad <var>.O.P.</var> <lb/>Huiuſmodi igitur conſideratio, ab Ariſtotele facta, nullius eſt 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 cauſam non attulerit.</head> + <head xml:space="preserve">CAP. XI.</head> + <p> + <s xml:space="preserve">QVærens Ariſtoteles vnde fiat, vt eæ libræ, quæ brachia habent alijs longiora, <lb/>ſint exactiores cæteris, ait hoc euenire ratione maioris velocitatis extremo <lb/>rum earundem. </s> + <s xml:space="preserve">Quod verum non eſt; </s> + <s xml:space="preserve">quia hîc effectus nil aliud eſt, quam clarius pro <lb/>ponere ob omnium oculos obliquitatem brachiorum à linea orizontali, & oſtende-<lb/>re etiam facilius à dicto orizontali ſitu exire brachia iam dicta. </s> + <s xml:space="preserve">Quæ quidem per ſe <lb/>neque à velocitate, neque à tarditate motus, ſed à ratione vectis, & à ma-<lb/>iori interuallo inter ſecundum ſitum extremorum à primo proficiſcuntur. </s> + <s xml:space="preserve">Vt exem-<lb/>pli gratia, imaginemur magnam libram <var>.A.B.</var> orizontalem, cuius centrum ſit <var>.E.</var> et <lb/>pondus <var>.B.</var> maius ſit pondere ipſius <var>.A.</var> vnde conceditur, quòd ob hanc rationem di-<lb/>cta libra ſitum mutabit, qui ſecundus ſitus ſit 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 & ſecundus ſitus ſit in <var>.h.f.</var> ita tamen vt anguli circa <var>.e.</var> æquales <lb/>ſint ijs, qui ſunt circa <var>.E.</var> ideſt <var>.b.e.f.</var> ſit ęqualis <var>.B.E.F</var>. </s> + <s xml:space="preserve">Nunc dico ſitum <var>.H.F.</var> <reg norm="exa- ctiorem" type="context">exa-<lb/>ctiorẽ</reg> futurum & clariorem ſitu <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 eſt ipſo <var>.b.f.</var> in qua <var>.B.E.</var> maius eſt <var>.b.e.</var> <lb/>quod autem interuallum <var>.B.F.</var> breuiori, aut longiori temporis ſpacio quam <var>.b.f.</var> ſit fa <lb/>ctum, nil planè refert. </s> + <s xml:space="preserve">Ratione vectis deinde, dico <reg norm="quod" type="simple">ꝙ</reg> ſi ſupponemus duas libras pa-<lb/>res <reg norm="æqualesque" type="simple">æqualesq́;</reg> in omni alio reſpectu, præter quàm in brachiorum longitudine, pon-<lb/>dus <var>.B.</var> maiorem vim habebit ad deprimendum brachium <var>.E.B.</var> quàm pondus <var>.b.</var> quia <lb/>libræ materiales, cum ſuſtineantur ab <var>.E.e.</var> & non à puncto mathematico, ſed <lb/>à linea, aut ſuperficie naturali in materia exiſtente. </s> + <s xml:space="preserve">vnde aliqua reſiſtentia ipſi mo-<lb/>tui brachiorum oritur, & hanc ob cauſam, ſupponendo hanc reſiſtentiam æqualem <lb/>tam in <var>.E.</var> quàm in <var>.e.</var> clarum erit ob ea, quæ in cap .4. huius tractatus oſtendi <var>.B.</var> cum <lb/>minus dependeat ab <var>.E.</var> aut minus quoque eidem <var>.E.</var> annitatur, ponderoſum magis <lb/>futurum, quam <var>.b.</var> & hac de cauſa mouebit ad partem inferiorem, maiori cum agilita <lb/>te, brachium <var>.E.B.</var> multo magis etiam illud ipſum deprimet, ideſt maiorem etiam an <lb/>gulum <var>.B.E.F.</var> quàm 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 cauſa ſecundæ, & tertiæ quæstionis mechanicæ <lb/>ab Ariſtotele nonperſpecta.</head> + <head xml:space="preserve">CAP. XII.</head> + <p> + <s xml:space="preserve">ARiſtoteles in ſecunda quæſtionum mechanicarum quærens illius rationem ſic <lb/>ſcribit.</s> + </p> + <p> + <s xml:space="preserve">Cur ſiquidem ſurſum fuerit ſpartum quando deorſum lato pondere quiſpiam id <lb/>admouet rurſus aſcendit libra: </s> + <s xml:space="preserve">ſi autem deorſum conſtitutum fuerit non aſcendit, <lb/>ſed manet? </s> + <s xml:space="preserve">an quia ſurſum quidem ſparto exiſtente plus libræ extra perpendiculum <lb/>ſi (ſpartum enim eſt perpendiculum) quare neceſſe eſt deorſum ferriid, quod plus <lb/>eſt, quare & cætera.</s> + </p> + <p> + <s xml:space="preserve">Sed vera cauſa, vnde fiat, vt ſi ſpartum fuerit ſurſum, & brachium vnum <lb/>ipfius libræ deprimendo, & idem liberum deinde permittendo, ad ſitum ori-<lb/>zontalem redeat, non ſolum eſt maior quantitas ponderis brachiorum quæ iam præ <lb/>tergreſſa eſt vltra verticalem lineam, ſed etiam eſt 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, ſit <var>.A.B.</var> libra in ſitu orizontali, cuius <lb/>ſpartum ſit <var>.E.</var> ſuper ipſam. </s> + <s xml:space="preserve">& deprimentes brachium ipſius <var>.A.</var> vſque ad <var>.F.</var> eius ſitus <lb/>ſit in <var>.F.H.</var> vnde medium <reg norm="punctum" type="context">pũctum</reg> <var>.G.</var> prætergreſſum erit lineam verticalem <var>.V.Z.</var> ver <lb/>ſus <var>.B.</var> quæ <var>.V.Z.</var> ſecabit brachium <var>.F.G.</var> in puncto <var>.D.</var> vnde <var>.D.H.</var> longius erit ipſo <var>.<lb/>F.D</var>. </s> + <s xml:space="preserve">Nunc nobis ſupponendum eſt id, <lb/>quod veriſſimum exiſtit, dictam ſcilicet li <lb/> + <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> + bram in ſitu <var>.F.H.</var> <reg norm="etiam" type="context">etiã</reg> ſi ſuſtineatur à pun-<lb/>cto <var>.E.</var> idem tamen futurum ac ſi ſuſtenta-<lb/>retur in puncto <var>.D.</var> vnde ſequitur, quod <lb/>pondus appenſum ex ipſa <var>.H.</var> ita grauius <lb/>reddatur, ipſo <var>.F.</var> in eadem propor-<lb/>tione, quæ maioreſt <var>.D.H.</var> ipſo <var>.D.F.</var> ob <lb/>rationes quas in primis huius tractatus ca-<lb/>pitibus poſui, vt etiam ſi <var>.D.H.</var> quodmate <lb/>riale eſſe ſupponitur, nullam planſe4; </s> + <s xml:space="preserve">graui-<lb/>tatem haberet, <reg norm="ſolustamem" type="context">ſolustamẽ</reg> exceſſus vis pon <lb/>deris in <var>.H.</var> poſiti, longè maior pondere in <lb/>F. collocato pro maiorilongitudine ipſius <lb/><var>D.H.</var> ſufficiat. ad præſtandum vt libra ad <lb/>ſitum 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 ſecunda deinde huius quęſtionis par <lb/> + <anchor type="handwritten" xlink:label="hd-0166-01a" xlink:href="hd-0166-01"/> + te, in qua ſcribit libram in ſitu, in quo poſi <lb/>ta eſt, firmam manere, toto cęlo aberrat, quia <reg norm="neceſſarium" type="context">neceſſariũ</reg> eſt, vt omninò cadat, eò <reg norm="uſque" type="simple">uſq;</reg> <lb/>quò ſpartum ſurſum remaneat: </s> + <s xml:space="preserve">ablato tamen omni impedimento, quod nulla eget <lb/>probatione, cum natura ſua 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">Cauſa, deinde, vera tertiæ quæſtionis non eſt ea, quam Ariſtoteles ponit, ſed hu-<lb/>iuſmodi effectus ab eo, quod capitibus .4. et .5. huius tractatus propoſui 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 Ariſtotelisratio in 6. quæſtione poſit a non ſit admittenda.</head> + <head xml:space="preserve">CAP. XIII.</head> + <p> + <s xml:space="preserve">VOlens Ariſtoteles rationem proponere, vnde fiat, vt nauis velocius moueatur <lb/>cum antennam altiorem quàm cum depræſſiorem habet, id ad vectis ratio-<lb/>nem refert, quod verum <reg norm="non" type="context">nõ</reg> eſt. </s> + <s xml:space="preserve">Huiuſmodi enim ratione nauis tardius potius, quàm <lb/>velocius ferri deberet, quia quantò altius eſt velum, vi venti impulſum, <reg norm="tantò" type="context">tãtò</reg> magis <lb/>proram ipſius nauis in aquam demergit. </s> + <s xml:space="preserve">Sed huiuſmodi effectus à maiori potius <lb/>quantitate venti quam recipit, quàm ab alia aliqua cauſa oritur, quia ventus liberius <lb/><reg norm="vehementiusque" type="simple">vehementiusq́;</reg> in altiore parte, quàm in depræſſione vagatur & perflat.</s> + </p> + </div> + <div type="section" level="3" n="14"> + <head style="it" xml:space="preserve">Quòdrationes ab Ariſtotele de octaua quæstione confictæ <lb/>ſufficient es non ſint.</head> + <head xml:space="preserve">CAP. XIIII.</head> + <p> + <s xml:space="preserve">RAtiones etiam ab Ariſtotele propoſitæ pro indaganda octauæ quæſtionis ve-<lb/>ritate, in qua quærit vnde fiat, vt corpora rotundæ figuræ, ad <reg norm="voluendum" type="context">voluendũ</reg> ſint <lb/>faciliora reliquis, quarum reuolutionum corporum tres ſpecies aſſignat, <reg norm="quarum" type="context">quarũ</reg> vna <lb/>eſt, vt rotarum <reg norm="curruum" type="context">curruũ</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 vaſorum a figulis fabricatorum, <reg norm="ſufficientes" type="context">ſufficiẽtes</reg> <reg norm="non" type="context">nõ</reg> ſunt.</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 ſolo quodam puncto contingat, quemadmodum <lb/>probat Theodoſius in .3. lib. primi & Vitellio in .71. lib. primi, & <reg norm="ducendo" type="context">ducẽdo</reg> per <reg norm="centrum" type="context">centrũ</reg> <lb/>ſphæræ lineam vſque ad punctum contactus, ipſa erit perpendicularis plano contin-<lb/>genti ſphęram dictam, vt probat <reg norm="idem" type="context">idẽ</reg> Thęodoſius 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 eſt omnem inclinationem ponderoſam huiuſ <lb/>modi corporis homogęnei totam hanc lineam æqualiter omni ex parte circundare; <lb/></s> + <s xml:space="preserve">cuius quidem rei exemplum in carta deſcribere poſſumus mediante figura circulari <lb/>hîc ſubſcripta <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 ipſi <var>.b.d.</var> ex .17. lib. 3. Eucli. & <reg norm="tantum" type="context">tantũ</reg> ponderis habebimus à parte <var>.a.u.e.</var> quan <lb/>tum ab ipſa <var>.a.n.e</var>. </s> + <s xml:space="preserve">Nuncigitur ſi imaginabimur ductum eſſe centrum verſus <var>.u.</var> per <lb/>lineam <var>.o.u.</var> parallelam ipſi <var>.a.d.</var> clarum nobis <lb/>erit, <reg norm="quod" type="simple">ꝙ</reg> <reg norm="abſque" type="simple">abſq;</reg> vlla difficultate aut reſiſtentia <reg norm="idem" type="context">idẽ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> + ducemus, quia huiuſmodi centrum ab inferiori <lb/>parte ad ſuperiorem, nunquam mutabit ſitum <lb/>reſpectu <reg norm="diſtantiæ" type="context">diſtãtiæ</reg> ſeu interualli, quę inter ipſum <lb/>lineamq́ue <var>.a.d.</var> intercedit, <reg norm="quod" type="simple">ꝙ</reg> quidem centrum <lb/>in ſe colligittotum pondus figurę <var>.a.n.e.u.</var> & be <lb/>neficio lineæ <var>.e.o.a.</var> illud ipſum 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 verſus ipſum <var>.d.</var> aut verſus <lb/>b. <reg norm="dirigatur" type="simple">dirigat̃</reg>; </s> + <s xml:space="preserve">ita vt <reg norm="cum" type="context">cũ</reg> non oporteat vt huius figuræ <lb/><reg norm="pondus" type="context">põdus</reg>, vna vice, magis eleuetur, quàm alia, ſed <lb/>ſemper ęqualiter ſuper lineam <var>.b.a.d.</var> quieſcat.</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> ſemper diuiſum à linea <var>.a.o.e.</var> per medium, ſequitur 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, ſi imaginabimur pentagonum <var>.K.i.h.f.l.</var> quie <lb/>ſcere <reg norm="ſuper" type="simple">ſuꝑ</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">primũ</reg> <reg norm="totum" type="context">totũ</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> poſteà centrum <var>.o.</var> (ponamus.) verſus <var>.l.</var> dubium non eſt, 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 diſtet, voluens ſe per <reg norm="arcum" type="context">arcũ</reg> vnum <lb/>circuli, <reg norm="qui" type="simple">ꝗ</reg> <reg norm="pro" type="simple">ꝓ</reg> ſuo ſemidiametro habeat <var>.o.K.</var> quę maior eſt ipſa <var>.o.a.</var> ex .18. li. primi Eu <lb/>cli. vnde ſi à puncto <var>.K.</var> imaginabimur lineam <var>.K.c.</var> reſpicientem centrum regionis <lb/>elementaris, dubium non eſt, quin ſi velimus transferre <reg norm="centrum" type="context">cẽtrum</reg> hoc à priori ſitu <reg norm="vſque" type="simple">vſq;</reg> <lb/>ad dictam lineam, oporteat addere pondus parti ipſius <var>.l.</var> quæ à linea <var>.K.c.</var> fuit ſecta, <lb/>aut aliquid de ipſo pondere partis centri detrahere. </s> + <s xml:space="preserve">quod quibuſuis modis fiat, ar-<lb/>duum certè eſt 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">ꝙ</reg> perfectè in medio ipſius ponderis eſt, reperiatur ſemper in linea per-<lb/>pendiculari ipſi plano, in quo animaduertendum eſt, <reg norm="quod" type="simple">ꝙ</reg> etiam ſi ipſum planum ap-<lb/>pellem; </s> + <s xml:space="preserve">pro plano tamen perfecto intelligi nolo, ſed pro ſuperficie perfectè <reg norm="ſphaeri- ca" type="simple">ſphęri-<lb/>ca</reg> circa centrum à corporibus grauibus expetitum; </s> + <s xml:space="preserve">nam ratione magnæ amplitudi-<lb/>nis huiuſmodi ſuperficiei, nullam differentiam notatu dignam à perfecto aliquo pla <lb/>no exigui interualli ad curuitatem eiuſdem ſuperficiei imaginari poterimus. </s> + <s xml:space="preserve">Sed ut <lb/>redeamus ad ſermonem de reuolutione figuræ rotundæ ſuſceptum, <reg norm="clarum" type="context">clarũ</reg> igitur erit <lb/>quamlibet minimam vim (vt ita dicam) quę trahat, aut impellat centrum <var>.o.</var> verſus <var>.u.</var> <lb/>huiuſmodi figuram reuoluturam, cuius media pars ad trahendum, aut impellendum <lb/>punctum <var>.e.</var> ſufficiere; </s> + <s xml:space="preserve">Imaginemur autem <reg norm="quod" type="simple">ꝙ</reg> li <lb/>nea <var>.n.o.u.</var> eſſet libra <reg norm="quędam" type="context">quędã</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> poſita, & vis, quę trahere cen <lb/>trum deberet, diuiſa eſſet per medium, cuius <lb/>medietas appenſa eſſet extremitati <var>.u.</var> diame-<lb/>tri <var>.n.o.u.</var> <reg norm="clarum" type="context">clarũ</reg> erit, <reg norm="quod" type="simple">ꝙ</reg> abſque vlla difficultate <lb/>reuolueret figuram ſuper lineam <var>.b.a.d.</var> verſus <var>.<lb/>d.</var> quia huius vis, aut pondus <reg norm="nullum" type="context">nullũ</reg> contra pon <lb/>dus haberet vltra centrum <var>.o.</var> uerſus <var>.n.</var> <reg norm="quod" type="simple">ꝙ</reg> cen-<lb/>trum <var>.o.</var> perpetuo quieſcit <reg norm="ſuper" type="simple">ſuꝑ</reg>. a. in linea <var>.e.o.<lb/>a.</var> per medium diuidente ſemper totum pon-<lb/>dus figurę ſuppoſitę. </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">ipsũ</reg> ver <lb/>ſus <var>.u.</var> <reg norm="trahens" type="context">trahẽs</reg> <lb/>per lineam <lb/><reg norm="parallelam" type="context">parallelã</reg> ip <lb/>ſi <var>.a.d.</var> <reg norm="dictam" type="context">dictã</reg> <lb/>figuram re-<lb/>uolueret. </s> + <s xml:space="preserve">Et <lb/>ſi linea qua <lb/>dictum cen <lb/>trum trahi-<lb/>tur ab ipſo + <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 æquediſtaret, ſed ſurſum traheret ſuper <var>.u.</var> aut ſubter, aliquid de ſua vi vir <lb/><reg norm="tuteque" type="simple">tuteq́;</reg> amitteret, & tantò plus, quantò inclinata magis eſſet verſus <var>.a.o.e.</var> & tandem <lb/>cum eſſet vnita cum <var>.a.o.e.</var> aut ad ſuperius, aut ad inferius quantalibet ui, etiam ſi in-<lb/>finita, figuram extra ſitum primæ lineæ <var>.a.o.e.</var> non moueret, ſed ſi ſurſum traheret ſe <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 ſpecies, tribus reuolutionum modis, abſque axis mutatione conſta <lb/>re poteſt, ideſt 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 ſi priorem <lb/>modum conſiderabimus, vt mediante fune quælibet figura, quæ circularis non ſit, <lb/>voluatur, ſupponamus 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">laterumq́;</reg> diſparium) plures inæqualitates, quæ reuolutionem eiuſdem fi-<lb/>guræ irregularem efficient; </s> + <s xml:space="preserve">quarum vna erit, quod duæ partes funis, ideſt <var>.u.q.</var> et <var>.i.p.</var> <lb/>non erunt in vna <reg norm="eademque" type="simple">eademq́;</reg> inter ſe diſtantia ſemper, quod facile intellectu erit, ſi ima <lb/>ginabimur ductas eſſe lineas <var>.a.i</var>: <var>u.i</var>: et <var>.i.c.t.</var> ſi funis duo pondera habebit alterum <lb/>altero maius, ſuis extremis appenſa, 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> eiuſdem funis, <reg norm="mundi" type="context">mũdi</reg> centrum, dum firmæ ma <lb/>nebunt, reſpicient; </s> + <s xml:space="preserve">ſed 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, eiuſdem figuræ mundi <reg norm="quoque" type="simple">quoq;</reg> cen <lb/>trum reſpiciet, vt in <lb/> + <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> + figura <var>.A.</var> <reg norm="ſicque" type="simple">ſicq́;</reg> etiam <lb/>linea <var>.i.c.t.</var> (pro <reg norm="exem- plo" type="context">exẽ-<lb/>plo</reg>) erit menſura di-<lb/>ſtantiæ funium inter <lb/>ipſas, & deinde <reg norm="circum" type="context">circũ</reg> <lb/>uoluendo etiam di-<lb/>ſtabuntinter ſe 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">ĩnotuit</reg> <reg norm="exem" type="context">exẽ</reg> <lb/>plo, & ſic etiam ali-<lb/>quando erunt magis <lb/><reg norm="diſtantes" type="context">diſtãtes</reg>, quàm linea <lb/><var>t.i.</var> & minus quàm.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 ſunt 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, ſi <lb/>extrema <var>.q.</var> et <var>.p.</var> eſſent quoque ſemper in vna <reg norm="eademque" type="context simple">eadẽq́;</reg> <lb/>diſtantia; </s> + <s xml:space="preserve">neque à corpore <reg norm="ponderoſo" type="context">põderoſo</reg> eſſent attracta, <lb/>quia aliæ partes ipſius <var>.u.q.</var> et <var>.i.p.</var> ex ſupradictis ratio-<lb/>nibus vnam <reg norm="eademque" type="simple">eademq́;</reg> diſtantiam <reg norm="non" type="context">nõ</reg> ſemper <reg norm="ſeruarent" type="context">ſeruarẽt</reg>. <lb/></s> + <s xml:space="preserve">vnde fieret vt cum diuerſis 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> ſemidiametros <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/>ſemper traherent ope ſeu virtute anguli æqualis ipſi <var>.<lb/>c.i.p</var>. </s> + <s xml:space="preserve">Hæc autem inę qualitas communis eſt 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, quàm diſparis numeri. </s> + <s xml:space="preserve">Sed aliam quandam maiorem <lb/>inęqualitatem habent hæ figuræ numeri diſparis, quæ eſt, quòd <reg norm="quando" type="context">quãdo</reg> linea <var>.t.i.</var> tam <lb/> <var>.u.q.</var> quàm ipſi <var>.i.p.</var> <lb/> + <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> + <reg norm="perpendicularis" type="simple context">ꝑpẽdicularis</reg> fuerit, <lb/>ideſt <reg norm="quando" type="context">quãdo</reg> <var>.t.i.</var> cum <lb/>dictis partibus funis <lb/>angulos rectos con-<lb/>ſtituerit, <reg norm="tunc" type="context">tũc</reg> ratione <lb/><reg norm="longitudinis" type="context">lõgitudinis</reg> ipſius <var>.c.<lb/>i.</var> maioris quam <var>.t.<lb/>c.</var> (quia cum ſit <var>.c.i.</var> <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg> ipſi <var>.c.a.</var> et <var>.c.a.</var> <lb/>maior ipſa <var>.c.t</var>: <var>c.i.</var> <lb/>etiam maior ſit ipſa <var>.<lb/>c.t.</var>) pondus aut vis <lb/>ipſius <var>.p.</var> ſuperabit <reg norm="eam" type="context">eã</reg> <lb/>quæ eſt ipſius <var>.q.</var> ſed <lb/>quando <var>.t.</var> erit in oppoſita parte, et <var>.i.</var> in ea, quæ eſt <lb/>ipſius .t: q. <reg norm="eandem" type="context">eãdem</reg> ob cauſam ſuperabit <var>.p.</var> & ſic 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="aſcendentem" type="context">aſcendẽtem</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 inęqualitatem habent figuræ pares, quæ <lb/>etiam in imparibus cernitur, etſi aliquantulum di-<lb/>uerſa; </s> + <s xml:space="preserve">quæ ab eo oritur, quod funes ſit modò ma-<lb/>gis, modo minus propinquę centro; </s> + <s xml:space="preserve">quæ inæqualis <lb/>diſtantia, maiorem <reg norm="minoremque" type="simple">minoremq́;</reg> vim ſuper dictum <lb/>centrum ob rationes in ſecunda parte cap. decimi <lb/>huius tractatus propoſitas, 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 diſparibus accommodari poteſt.</s> + </p> + <p> + <s xml:space="preserve">Secundus modus eſt earum rotarum, in quibus aliquod animal incedit, quæ ſi cir-<lb/>culares non eſſent, tantò difficilius voluerentur, quantò pauciores angulos haberent. <lb/></s> + <s xml:space="preserve">quod cum per ſe pateat, non demonſtrabo. </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 eſt 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, quàm <reg norm="violentię" type="context">violẽtię</reg>, quam anguli aeri faciunt, eum <lb/>expellendo, vt ipſi occupent locum, quem ipſe <reg norm="aer" type="conjecture">aér</reg> implebat. </s> + <s xml:space="preserve">Quod nullo modo po <lb/>teſt euenire circulari figuræ.</s> + </p> + <p> + <s xml:space="preserve">Nunc nobis ad dicendum reſtat de ſpecie reuolutionis rotarum, quæ parallelæ <lb/>ſunt orizonri, quibus accidit poſſe volui primo <reg norm="tertioque" type="simple">tertioq́;</reg> modo ſecundę ſpeciei, & ob <lb/>id ſi circulares non erunt, eadem ſubibunt incommoda, de quibus in ſecunda illa ſpe <lb/>cie loquuti ſumus. </s> + <s xml:space="preserve">ſed circulares rotæ huius tertiæ ſpeciei ad reuoluendum erunt re-<lb/>liquis eò faciliores, <reg norm="quod" type="simple">ꝙ</reg> vno <reg norm="ſolum" type="context">ſolũ</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 ſpecie formari poteſt problema, vnde fiat, vt quieſcens huiuſ-<lb/>modi rota parallela orizonti ſuper vnum punctum, & quantò fieri poteſt exiſtens <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg>, ſi eam circunuoluamus maiore qua poterimus ui, & <reg norm="eandem" type="context">eãdem</reg> poſtea dimitten-<lb/>tes non perpetuò circunuoluatur.</s> + </p> + <p> + <s xml:space="preserve">Hoc quidem, quatuor fit ob cauſas. quarum prima eſt, quia huiuſmodi motus, eius <lb/>rotæ non ſit naturalis. </s> + <s xml:space="preserve">ſecunda eſt, quia etiamſi rota ſuper punctum mathematicum <lb/>quieſceret, oporteret tamen vt ſuperius <reg norm="alterum" type="context">alterũ</reg> haberet polum, qui ipſam <reg norm="orizontalem" type="context">orizontalẽ</reg> <lb/>teneret, qui quidem munimento aliquo corporeo indigeret; </s> + <s xml:space="preserve">vnde fricatio quędam <lb/>conſequeretur, ex qua reſiſtentia prodiret.</s> + </p> + <p> + <s xml:space="preserve">Tertia eſt, quia aer contiguus eam perpetuò aſtringit, <reg norm="hocque" type="simple">hocq́;</reg> modo eius motui <lb/>reſiſtit.</s> + </p> + <p> + <s xml:space="preserve">Quarta eſt, quia quęlibet pars corporea, quę à ſe mouetur, impetu eidem à quali-<lb/>bet extrinſeca virtute mouente impręſſo, habet naturalem inclinationem ad rectum <lb/>iter, non autem curuum, vnde ſi à dicta rota particula aliqua ſuę circunferentiæ <reg norm="diſium" type="context">diſiũ</reg> <lb/>geretur, abſque dubio per aliquod temporis ſpatium pars ſeparata recto itinere fer <lb/>retur per aerem, vt exemplo à fundis, quibus iaciuntur lapides, ſumpto, cognoſce <lb/>re poſsumus, in quibus, impetus motus impręſſus naturali quadam propenſione <lb/>rectum iter peragit, cum euibratus lapis, per lineam rectam contiguam giro, quem <lb/>primo faciebat, in puncto, in quo dimiſſus fuit, rectum iter inſtituat, vt rationi con-<lb/>ſentaneum eſt.</s> + </p> + <p> + <s xml:space="preserve">Eadem, quoque ratione fit, vt quantò maior eſt aliqua rota, tantò maiorem quo <lb/>que impetum, & impreſſionem motus eius circunferentiæ partesrecipiant, vnde <reg norm="ſae pe" type="simple">ſę<lb/>pe</reg> euenit, vt dum eam ſiſtere volumus, id <reg norm="cum" type="context">cũ</reg> labore & cum diſſicultate agamus ; </s> + <s xml:space="preserve">quia <lb/>quantò maior eſt diameter vnius circuli, tantò minus curua eſt eiuſdem circunferen <lb/>tia, & tantò propius accedit angulum eiuſdem circunferentiæ ad quantitatem duo-<lb/>rum angulorum rectorum rectilineorum, ideſt circunferentia ad rectitudinem linea <lb/>rem. </s> + <s xml:space="preserve">Vnde earundem partium dictæ circunferentiæ motus ad inclinationem ſibi à <lb/>natura tributam, quæ eſt 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 queſtionis <lb/>admittendanon ſit.</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, accerſiri debet.</s> + </p> + </div> + <div type="section" level="3" n="16"> + <head style="it" xml:space="preserve">Quod Aristotelis rationes de decima queſtione <lb/>ſint reijciende.</head> + <head xml:space="preserve">CAP. XVI.</head> + <p> + <s xml:space="preserve">ARiſtotelis rationes, vnde fiat, vt facilius moueantur libræ vacuæ, quàm plenè <lb/>ad propoſitam diſputationem non pertinent; </s> + <s xml:space="preserve">quia ſemper ineunda eſt ratio <lb/>proportionis virtutis mouentis ſuper mobile; </s> + <s xml:space="preserve">quod ipſe 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 ſolum <lb/>ponderis obtineat, & ſit libra <var>.n.i.u.</var> æqualis priori, quæ pro ſingula 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">Ariſtoteles admiratur, quòd addendo ipſi <var>.e.</var> mediam pon <lb/>deris vnciam, brachium <var>.i.e.</var> velocius cadat, quàm <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> ipſi <var>.u.</var> <lb/>brachij <var>.i.u</var>. </s> + <s xml:space="preserve">Quod à duabus cauſis proficiſcitur, quarum prior eſt, magna differentia <lb/>proportionis vnius libræ ad medietatem vnius vnciæ, ad proportionem vnius vnciæ <lb/>ad ipſam medietatem, quia ſi pondus adiectum extremo <var>.u.</var> dimidiæ eſſet libræ , & <lb/>cum eadem tarditate brachium moueret, optimo iure in admirationem poſſet Ari-<lb/>ſtoteles duci. </s> + <s xml:space="preserve">Sed hoc fieri non poſſet, quia ipſum deprimeret cum eadem quaſi ve <lb/>locitate, qua media vncia brachium <var>.i.e</var>. </s> + <s xml:space="preserve">Dixi autem quaſi, quia nonnihil diſcrimi-<lb/>nis intercederet, quod proficiſcitur à ſecunda ratione. </s> + <s xml:space="preserve">Et hæc, reſiſtentia eſt , quæ <lb/>oritur à ſparto, quia quantò maius pondus continet libra, tantò magis præmit ſpar <lb/>tum in loco, in quo ſuſtinetur; </s> + <s xml:space="preserve">vnde maior reſiſtentia in circunuolutione <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpar <lb/>ti, in loco, in quo quieſcit, exoritur, quia ipſum eſt 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> deſcenderet, oporteret, vt propor-<lb/>tio dimidiæ librę adiectæ ponderi ipſius <var>.u.</var> <lb/> + <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> + quod eſt vnius libræ, vim ſuam haberet, <lb/>quæ excederet reſiſtentiam ſui ſparti (me-<lb/>dio brachiorum maiorum ijs qui ſunt <var>.a.i.<lb/>e.</var>) ita proportionatam, vt proportionata <lb/>eſt vis dimidiæ vnciæ ipſi e. iunctæ, reſiſten <lb/>tiæ ſui ſparti. </s> + <s xml:space="preserve">Huiuſmodi rationes cum ro-<lb/>tis grauioribus <reg norm="leuioribusque" type="simple">leuioribusq;</reg>, & ijs, quę à cor <lb/>poribus quibuſlibet 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 cauſa .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, quàm fieret manu, quod corpus liberatum deinde cum <lb/>fuerit à funda, natura duce, iter <reg norm="fuum" type="context">fuũ</reg> à puncto, à quo proſilijt, per lineam contiguam <lb/>giro, quem poſtremò faciebat, ſuſcipit. </s> + <s xml:space="preserve"><reg norm="Dubitandumque" type="simple">Dubitandumq́</reg> non eſt, quin dicta funda <lb/>maior impetus motus dicto corpori imprimi poſſit, <reg norm="cum" type="context">cũ</reg> ex multis circumactibus, ma-<lb/>ior ſemper impetus dicto corpori accedat. </s> + <s xml:space="preserve">Manus autem eiuſdem corporis motus, <lb/>dum illud ipſum circunuoluitur (pace Ariſtotelis dixerim) centrum non eſt, neque <lb/>funis eſt ſemidiameter. </s> + <s xml:space="preserve">Immo manus quam maximè fieri poteſt in orbem cietur; <lb/></s> + <s xml:space="preserve">qui quidem motus in orbem, vt circumagatur etiam ipſum corpus, cogit, quod qui-<lb/>dem corpus, naturali quadam inclinatione, exiguo quodam impetu iam incępto, <lb/>vellet recta iter peragere, vt in ſubſcripta figura patet, in qua <var>.e.</var> ſignificat 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 eſt, impręſſum illum impetum, continuò paulatim decreſcere <lb/>vnde ſtatim inclinatio grauitatis eiuſdem corporis ſubingreditur, quæ ſeſe miſcens <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, ſed citò fiat curua, cum dictum corpus <var>.a.</var> duabus virtutibus moueatur, qua-<lb/>rum vna eſt, 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 ſimul & ſemel moueri <lb/>poſſe. </s> + <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> eſt <reg norm="ſilentio" type="context">ſilẽtio</reg> <reg norm="prætereundus" type="context">prætereũdus</reg> <lb/>hac in re <reg norm="quidam" type="simple context">ꝗdã</reg> notatu <reg norm="dignus" type="simple">dignꝰ</reg> effectus, <lb/>qui eiuſmodi eſt, <reg norm="quod" type="simple">ꝙ</reg> quanto magis <lb/>creſcit impetus in corpore <var>.a.</var> cauſa <lb/>tus ab augumento velocitatis giri <lb/>ipſius <var>.e.</var> <reg norm="tantò" type="context">tãtò</reg> magis oportet, vt ſen-<lb/>tiat ſe trahi manus à dicto corpore <lb/>a. mediante fune, quia quantò ma-<lb/>ior impetus motus ipſi <var>.a.</var> eſt impręſ <lb/>ſus, 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 eſt referenda. </s> + <s xml:space="preserve">Imaginari debemus <lb/>axem cylindrici iugi, hypomochlion eſſe. </s> + <s xml:space="preserve">Quod reſtat, illud ipſum totum de <lb/>pendet à .4. <reg norm="quintoque" type="simple">quintoq́;</reg> cap. huius tractatus. </s> + <s xml:space="preserve">Vna tamen differentia inter hanc machi-<lb/>nam, <reg norm="vectemque" type="simple">vectemq́;</reg> reperitur, quæ eſt, <reg norm="quod" type="simple">ꝙ</reg> iugum aliquam reſiſtentiam pro coniunctione <lb/>calcata in loco, in quo voluitur, magis quàm hypomochlion vecti efficiat.</s> + </p> + </div> + <div type="section" level="3" n="19"> + <head style="it" xml:space="preserve">De decimaquart a queſtione.</head> + <head xml:space="preserve">CAP. XIX.</head> + <p> + <s xml:space="preserve">RAtiones etiam decimæquartæ quæſtionis dependent ab ijs, quæ ſunt vectis, vt <lb/>exempli gratia ſit 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">ꝙ</reg> tenentes marlus longè à medio, in locis <lb/>a. et <var>.c.</var> facilius <reg norm="minorique" type="simple">minoriq́;</reg> cum labore illum frangemus, quàm ſi eaſdem vicinas me-<lb/>dio eiuſdem 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> ſunt <reg norm="cum" type="context">cũ</reg> ijs, quæ <lb/>primis huius tractatus capitibus propoſitæ 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 manifeſtè perſpiciemus eorum, quæ iam diximus <lb/>ratione, <reg norm="quod" type="simple">ꝙ</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, quàm loco <var>.a.</var> et <var>.c.</var> <reg norm="duarum" type="context">duarũ</reg> <reg norm="linearum" type="context">linearũ</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">habebũt</reg> <lb/> + <anchor type="figure" xlink:label="fig-0173-02a" xlink:href="fig-0173-02"/> + in <var>.a.</var> et <var>.c.</var> <lb/>quàm 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. queſtionis.</head> + <head xml:space="preserve">CAP. XX.</head> + <p> + <s xml:space="preserve">DEcimaſeptima quæſtio ab Ariſtotele haud benè percepta fuit, quia is non ac-<lb/>commodat partes vectis ſuis 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 ſint <var>.o.</var> & pondera, <lb/>quæ ſunt attollenda ſint <var>.a.</var> et <var>.e.</var> inter ſe æqualia, & diſtantię ſint <var>.a.o.</var> et <var>.e.o.</var> ſibi <reg norm="inuicem" type="context">inuicẽ</reg> <lb/>æquales, ſed <var>.o.n.</var> æqualis ſit ipſi <var>.o.u</var>: clarum erit, <reg norm="quod" type="simple">ꝙ</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 ſupponuntur æ qua <lb/> + <anchor type="handwritten" xlink:label="hd-0174-01a" xlink:href="hd-0174-01"/> + lia, clarum quoque erit, commu-<lb/>ni ſcientia, tantam virtutem in <lb/> + <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> + n. quanta ſufficiet ad <reg norm="attollendum" type="context">attollendũ</reg> <lb/>a. in <var>.u.</var> <reg norm="quoque" type="simple">quoq;</reg> ſuffecturam ad ele-<lb/>uandum <var>.e.</var> quia <reg norm="cum" type="context">cũ</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 ſuo 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">quintoq́;</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 ſatis ſu <lb/><reg norm="perque" type="simple">perq́;</reg> dixi in dicto capit .5. conuenire poſ-<lb/>ſunt.</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 ſit aliqua pars ligni cindenda ſe-<lb/>cundum venulas ſuas <var>.d.e.f.g.</var> & ſit cuneus <lb/><var>a.b.c.</var> qui vi mallei <var>.P.</var> vſque 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, poſt quam infigitur cuneus ſe <lb/>cundum venas, longíor erit parte <var>.x.b.t.</var> cu <lb/>nei, quæ ingreſſa eſt. </s> + <s xml:space="preserve">Oportet nunc ima-<lb/>ginari duos vectes ſimiles ſupradictæ <var>.u.e.<lb/>o.</var> in hunc modum, vt puncta <var>i.r.</var> lìni ſint <lb/>loco <var>.u.</var> extremi <reg norm="ipſius" type="simple">ipſiꝰ</reg> vectis, et <var>.t.x.</var> loco vir <lb/>tutis applicatæ ipſi <var>.u.</var> & reſiſtentia circa <lb/>punctum <var>.m.</var> loco ponderis <var>.e.</var> vectis <var>.o.e.u.</var> <lb/>dicti, & pars <var>.K.</var> quaſi immediata poſt <var>.m.</var> <lb/>verſus extremitatem <var>.f.e.</var> ligni, ſit 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 & intrinſeca cauſa trocble arum.</head> + <head xml:space="preserve">CAP. XXI.</head> + <p> + <s xml:space="preserve">PRo intelligenda vera, & intrinſeca ratione, vnde fiat ut multitudo rotularum in <lb/>trochleis cauſa ſit, ut exigua vis ſurſum moueat, aut attollat <reg norm="pondera" type="context">põdera</reg> magna. </s> + <s xml:space="preserve">Ima <lb/>ginemur duas hîc ſubſcriptas trochlæas explicatas tranſuerſaliter in hunc modum, <lb/>ideſt ſit <reg norm="paruum" type="context">paruũ</reg> <reg norm="tignum" type="context">tignũ</reg> <var>.a.b.</var> fixum & <reg norm="parallelum" type="context">parallelũ</reg> orizonti. cui ſint rotulæ appenſe ab infe <lb/>riori parte ad ſuperiorem <reg norm="huicque" type="simple">huicq́;</reg> è regione <reg norm="oppoſitus" type="simple">oppoſitꝰ</reg> ſit aliud <reg norm="tignum" type="context">tignũ</reg> <var>.c.d.</var> quod moueri <lb/>poſſit ab imo ad ſumum, ſuper quod totidem ſint rotulæ aut radij, <reg norm="cum" type="context">cũ</reg> annexa poſtea <lb/>fuerit funis puncto <var>.b.</var> fixo, eam faciendo pertranſire per rotulas tam à parte ſupe-<lb/>riore, quam ab inferiore; </s> + <s xml:space="preserve">& appenſum deinde cum erit paruo illi tigno <var>.c.d.</var> mobili <lb/>pondus <var>.E.</var> ducendo poſtmodum extremum <var>.f.</var> funis tranſeuntis per rotulas, idem pla <lb/>nè fiet quod à trochlęis ſimul unitis fieri ſolet. </s> + <s xml:space="preserve">Cuius quidem effectus ratio ſub no-<lb/>ſtram cognitionem cadet facilius in huiuſmodi figura. </s> + <s xml:space="preserve">Imaginemur ſeparatim ſta-<lb/>teram <var>.g.h.</var> cuius <reg norm="centrum" type="context">cẽtrum</reg> ſit <var>.K.</var> ita ſitum, ut brachium <var>.g.k.</var> ſit duplum ad brachium <var>.K.<lb/>h.</var> ſupponendo igitur in puncto <var>.g.</var> pondus, aut virtutem mouentem unius libræ, & in <lb/>h. duarum librarum, <reg norm="abſque" type="simple">abſq;</reg> dubio hæ duæ uirtutes in huiuſmodi diſtantijs à centro <lb/> + <anchor type="handwritten" xlink:label="hd-0175-01a" xlink:href="hd-0175-01"/> + ęquales <reg norm="inuicem" type="context">inuicẽ</reg> <reg norm="erunt" type="context">erũt</reg>, ob rationes prioribus capitibus iam allatas, & ſtatera orizontalis <lb/>manebit. </s> + <s xml:space="preserve">Vnde clarum erit, <reg norm="quod" type="simple">ꝙ</reg> quæuis etiam exigua virtus adiuncta ipſi <var>.g.</var> mouebit <lb/>ſtateram extra orizontalem ſitum. </s> + <s xml:space="preserve">Nunc ſi puncto <var>.i.</var> ex æquo medio inter <var>.g.</var> et <var>.K.</var> <lb/>applicata erit virtus ipſius <var>.h.</var> non amplius conſiderato brachio <var>.K.h.</var> inclinante uirtu-<lb/>te ipſius <var>.i.</var> eandem partem verſus, in quam inclinabat, quando erat in <var>.h.</var> ſed uirtus ip <lb/>ſius <var>.g.</var> inclinet contrario modo, <reg norm="diuerſoque" type="simple">diuerſoq́;</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 ſunt dicta <var>.g.h.</var> ſemper <lb/>in eodem ſitu abſque motu manſuram, <reg norm="hancque" type="simple">hancq́;</reg> ſtateram appellabimus mobilem, & <lb/>primam. </s> + <s xml:space="preserve">Imaginemur nunc à puncto <var>.e.</var> fixo deſcendere 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">& ſit <var>.n.l.m.</var> diameter vnius ex rotulis ſuperioribus alterius parui <lb/>tigni defixi à parte inclinationis ipſius <var>.g.</var> & parallela diametro <var>.g.K.</var> cuius diametri <lb/>centrum fixum ſit <var>.l.</var> & ſit coniunctum <var>.g.</var> punctum, à fune cum puncto <var>.m.</var> quæ <reg norm="tam" type="context">tã</reg> per-<lb/>pendicularis ſit primo diametro <var>.g.i.K.</var> quàm ſecundo <var>.n.m.</var> ideſt 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> ſint recti. </s> + <s xml:space="preserve">Imaginemur <reg norm="quoque" type="simple">quoq;</reg> virtutem ipſius <var>.g.</var> applicatam eſſe extremo <var>.<lb/>n.</var> cum inclinatione tamen contraria, ideſt ad inferiorem partem, quæ quidem virtus <lb/>communi quodam conceptu eandem poſſidebit vim ſuſtentandi 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 ſuperiorem partem, <lb/>& ſic etiam diameter <var>.n.l.m.</var> non magis ab una, quàm ab alia parte declinabit, quia <lb/>cum quædam virtus in <var>.n.</var> reperiatur æqualis medietati uirtutis ipſius <var>.i.</var> quæ uirtus ip <lb/>ſius <var>.i.</var> uim habet deprimendi ipſum <var>.g.</var> ideſt <var>.m.</var> pro dimidia ſui ipſius parte, ſequitur <var>.<lb/>n.m.</var> debere immobilem permanere. </s> + <s xml:space="preserve">Nunc ſi alia diameter rotulæ mobilis erit de-<lb/>ſumpta, quæ ſit <var>.p.q.o.</var> cuius centrum ſit <var>.q.</var> in ſitu parallelo ipſi <var>.n.l.m.</var> & ſic collocata, <lb/>vt coniungendo <var>.o.</var> cum <var>.n.</var> anguli <var>.m.n.o.</var> et <var>.n.o.p.</var> ſint recti: </s> + <s xml:space="preserve">ſi imaginati fuerimus <reg norm="tranſ" type="context">trãſ</reg> <lb/>latum eſſe <reg norm="pondusipſius" type="simple">pondusipſiꝰ</reg> <var>.n.</var> in <var>.o.</var> <reg norm="cum" type="context">cũ</reg> <reg norm="eadem" type="context">eadẽ</reg> inclinatione ad depræſſiorem partem, illud ip <lb/>ſum, ac ſi eſſet in <var>.n.</var> communi conceptu, ſine alicuius diametri mutatione præſtabit. <lb/></s> + <s xml:space="preserve">Et ſi centrum <var>.q.</var> fixum eſſet, & extremo <var>.p.</var> appoſitum fuiſſet pondus ipſius <var>.o.</var> cum in <lb/>clinatione ad ſuperiorem partem, idem etiam planè pręſtaret, etiam ſi nullum ullius <lb/>diametri ſitum, communi ſcientia, mutaret, cum extremum <var>.m.</var> deorſum ſit ductum <lb/>à. g. uirtute dimidiæ partis ipſius <var>.i.</var> & ab alia huic ſimili <var>.m.</var> quoque deorſum ſit tra-<lb/>ctum ab .o: quod quidem <var>.o.</var> deorſum eſt alteratum, ob inclinationem ad ſuperius <lb/>à uirtute poſita in <var>.p.</var> ſupponendo centrum <var>.q.</var> fixum. </s> + <s xml:space="preserve">Sed ſi loco centri fixi, imagina <lb/>bimur in <var>.q.</var> pondus aliquod æquale ipſi <var>.i.</var> quod duplum erit in uirtute ad eam, quæ <lb/>eſt ipſius <var>.p.</var> & ipſius quoque .g: ſequetur <reg norm="etiam" type="context">etiã</reg> eadem immobilitas horum trium dia-<lb/>metrorum. </s> + <s xml:space="preserve">Quia cum ſit huiuſmodi pondus ſeu virtus in <var>.q.</var> cum inclinatione con-<lb/>traria virtuti in <var>.p.</var> quæ æquipollet dimidiæ parti ipſius <var>.q.</var> & ſic ei quæ eſt ipſius <var>.o.</var> ſi-<lb/>militer quia <var>.o.</var> tractum eſt ſupra ab <var>.n.</var> virtute ipſius <var>.g.</var> quod <var>.m.</var> deorſum trudit; </s> + <s xml:space="preserve">idcir <lb/>co quanta erit vis quam habebit virtus in <var>.q.</var> ferendi deorſum diametrum <var>.p.o.</var> tanta <lb/>quoque virtutes ipſorum <var>.p.</var> et <var>.o.</var> æquales, & æqualiter diſtantes à <var>.q.</var> ipſum ad ſupe-<lb/>riorem partem inclinabunt. </s> + <s xml:space="preserve">Quamobrem nec aſcender, nec deſcendet, nec locum <lb/>mutabit. </s> + <s xml:space="preserve">Supponamus nunc quartum diametrum rotulæ <var>.s.t.r.</var> quæ ſit ſecunda rotu <lb/>larum fixarum, parallela ipſi <var>.p.o.</var> & in eo ſitu, quo coniungendo extrema <var>.r.p.</var> anguli <lb/><var>o.p.r.</var> et <var>.p.r.s.</var> ſint recti, & imaginemur virtutem ipſius <var>.p.</var> reperiri in <var>.s.</var> cum inclinatio <lb/>ne tamen contraria, ideſt deorſum verſus, ex his <reg norm="idem" type="context">idẽ</reg> quoque planè ſequetur, ideſt <reg norm="quod" type="simple">ꝙ</reg> <lb/>nulla <reg norm="harum" type="context">harũ</reg> quatuor diametrorum mouebitur. </s> + <s xml:space="preserve">quia eundem <reg norm="effectum" type="context">effectũ</reg> <reg norm="cum" type="context">cũ</reg> inclinatione <lb/>deorſum verſus efficeret dicta virtus in <var>.s.</var> quem in <var>.p.</var> cum inclinatione ſurſum verſus. <lb/></s> + <s xml:space="preserve">et iam dictum eſt virtutem ipſius <var>.g.</var> dimidium virtutis ipſius <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 ipſius <var>.r</var>. </s> + <s xml:space="preserve">Hucuſque <reg norm="ſcien- tificè" type="context">ſciẽ-<lb/>tificè</reg> nouimus pondus, aut virtutem ipſius <var>.s.</var> quæ eſt dimidium <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.i.</var> ſuſtinere uim <lb/>ipſorum <var>.i.</var> et <var>.q.</var> nam quater tantum, quanta ipſamet virtus ipſius <var>.s.</var> eſſe conſpicitur. <lb/></s> + <s xml:space="preserve">Et ſi adiunctę nobis eſſent duæ aliæ diametri cum ijſdem planè conditionibus <reg norm="ijſdem" type="context">ijſdẽ</reg> <lb/>rationibus vtentes, cognoſceremus quod eadem medietas ipſius <var>.i.</var> ſexies tantum <reg norm="pon" type="context">põ</reg> <lb/>deris, quanta ipſa exiſteret, ſeſtineret. </s> + <s xml:space="preserve">Vnde <reg norm="manifeſtum" type="context">manifeſtũ</reg> euadit, <reg norm="quod" type="simple">ꝙ</reg> eidem medietati <lb/>ipſius <var>.i.</var> in <var>.s.</var> nonnihil virtutis addendo, dictæ diametri, illicò <reg norm="mouerentur" type="context">mouerẽtur</reg> ſitu. </s> + <s xml:space="preserve">Et quia <lb/>rotulæ in quolibet puncto, aliquam diametrum habent, neceſſariò ſequitur <reg norm="quod" type="simple">ꝙ</reg> infe-<lb/>riores ad ſuperiores accedere debeant. </s> + <s xml:space="preserve">Attamen ſi forte extremum immobile ip-<lb/>ſius funis non pendet à puncto <var>.e.</var> trochleæ ſuperioris, ſed alligatum fuerit ad <reg norm="medium" type="context">mediũ</reg> <lb/>inferioris trochleæ ut ad punctum <var>.i.</var> ope unius trochleę ſuperioris immobilis vt in fi <lb/>gura <var>.A.</var> videre licet, clarè patebit <reg norm="quod" type="simple">ꝙ</reg> à tribus virtutibus æqualibus pondus in <var>.i.</var> <reg norm="poſitum" type="context">poſitũ</reg> <lb/>ſuſtinebitur: </s> + <s xml:space="preserve">hoc eſt à <var>.g.</var> ab <var>.i.</var> & ab <var>.k.</var> <reg norm="quarum" type="context">quarũ</reg> vnaquęque tertia pars erit ipſius <var>.i.</var> in con <lb/>c<unsure/>ontrariam <reg norm="partem" type="context">partẽ</reg>, hoc eſt tertia pars reſiſtentiæ. </s> + <s xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</reg> ex æquo inter ſe <reg norm="diſtant" type="context">diſtãt</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 reſiſtentię, 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="ſuffieiant" type="context">ſuffieiãt</reg> reſiſtentiæ in <var>.o.</var> et <var>.p.</var> <lb/>ipſum ſuſtinere, idcirco ipſum pondus <var>.q.</var> ſubſeſquialter erit <reg norm="ponderi" type="context">põderi</reg> in <var>.i.</var> poſiti. </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 ſi adhuc. duo diametri vnus in-<lb/>ferior, alter verò ſuperior 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 ſeptima pars trium ponderum <var>.i.q.</var> & tertij additi, ex <lb/> + <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> + ſupradictis rationibus. </s> + <s xml:space="preserve">Et quia virtus ſuſti <lb/> + <anchor type="figure" xlink:label="fig-0177-02a" xlink:href="fig-0177-02"/> + nens totale pondus trochleæ inferiori ap-<lb/>penſum in tot diuiditur partes æquales, <lb/>quot ſunt diametri orbiculorum trochleæ <lb/>inferioris, quando extremum immobile fu <lb/>nis alligatum fuerit trochleę ſuperiori, 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 ſemper ſeſqui <lb/>altera erit vnicuique aliorum <reg norm="diametrorum" type="context">diametrorũ</reg> <lb/>ideò virtus reſiſtentię alterius extremi mo <lb/>bilis funis, puta <var>.s.</var> ſubmultiplex erit totalis <lb/>ponderis, eo modo quo diximus, cuius vir <lb/>tus, ſeu grauitas diuiditur ſeu diſtrubuitur <lb/>diametris inferioris trochleæ vt dictum eſt.</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 cauſa .24. quæſtionis.</head> + <head xml:space="preserve">CAP. XXII.</head> + <p> + <s xml:space="preserve">VEra cauſa effectus, qui vigeſimaquarta quæſtione exprimitur, adhuc à nemine <lb/>(quod ſciam) animaduerſa fuit, licet non ſit admodum ardua vel obſcura. </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> ſimul coniunctos, vt ſi ip <lb/>ſorum vnus feratur in orbem, alius quoque circumagatur, eo modo, quo curruum ro <lb/>tæ voluuntur. </s> + <s xml:space="preserve">Et imaginemur primò ſuper lineam <var>.f.i.</var> reuolui maiorem, & quando <lb/>idem circulus erit in <var>.l.</var> dictam lineam <var>.f.i.</var> tangere circunferentiam eiuſdem 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 ipſi <var>.f.l.</var> quia ex .17. tertii, anguli <var>.f.</var> et <var>.g.</var> <lb/>ſunt æquales, vnde ex .28. primi <var>.f.l.</var> et <var>.g.K.</var> ſunt parallelæ. </s> + <s xml:space="preserve">& ſic erunt <var>.k.l.</var> cum <var>.f.g.</var> ex <lb/>eadem ſupradicta. </s> + <s xml:space="preserve">Ratio autem, qua arcus <var>.g.b.</var> tranſierit lineam <var>.g.K.</var> maiorem ipſa, <lb/>eſt, quia dum mouetur, quodlibet punctum ipſius <var>.g.b.</var> virtute reuolutionis ipſius <var>.f.c.</var> <lb/>omne punctum eiuſdem arcus <var>.g.b.</var> vlterius verſus <var>.K.</var> quam ſi moueretur virtute re-<lb/>uolutionis ipſius <var>.g.b.</var> ſuper 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 ſitu 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 ſint 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/>ſam 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 ipſo <var>.b.</var> et <var>.g.</var> fa-<lb/>ciet iter <var>.g.n.</var> quę itinera alijs multò breuiora ſunt, quia breuioribus cruribus reuolu-<lb/>ta ſunt dicta puncta; </s> + <s xml:space="preserve">& ſic dico de reliquis omnibus punctis inter <var>.g.</var> et <var>.b.</var> & in hoc ca <lb/>ſu 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 ſolum non erunt delata anteà, ſed potius à primo ſitu retrorſum <lb/>erunt repulſa. </s> + <s xml:space="preserve">Vnde non eſt, quòd in tantam admirationem ducamur ſi dum reuol <lb/>uitur circulus maior, arcus <var>.g.b.</var> circuli minoris, totam lineam <var>.g.K.</var> tranſire 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 ſeſe 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 ſolum mouetur ver <lb/>ſus <var>.m.</var> quòd circa centrum <var>.a.</var> feratur, cum ipſum etiam centrum moueatur verſus <var>.m.</var> <lb/>ſed quia pręter hoc deferantur quoque à circulo maiori verſus <var>.m.</var> vſque 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 verſus <var>.i.</var> virtute reuolutionis circuli minoris, <lb/>& alter ex eo, <reg norm="quod" type="simple">ꝙ</reg> dictus circulus maior circa centrum <var>.a.</var> voluatur, vnde omne <reg norm="punctum" type="context">punctũ</reg> <lb/>contactus circuli maioris cum recta <var>.f.i.</var> tetrorſum pellitur verſus <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 cauſa .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 ſedet ( non tamen Turcarum more ) ſi velit <lb/>ſeſe in pedes erigere, calcaneos retrahit, vt efficiat angulum acutum, cum <reg norm="fae- moribus" type="simple">fę-<lb/>moribus</reg> coxis à parte inferiori, & ventrem inclinat, ad conſtituendum etiam angu <lb/>lum acutum in ſuperiori parte, ea eſt; </s> + <s xml:space="preserve">vt totius corporis pondus, ex ęquo, ideſt ab <lb/>oppoſitis partibus circundet lineam rectam, quæ tranſit per locum, in quo conquie <lb/>ſcunt pedes verſus mundi centrum. </s> + <s xml:space="preserve">ideſt, ut edatur ęquilibrium ponderis ipſius cor-<lb/>poris circum lineam illam, quę ſub pedibus inſeruit pro ſparto. </s> + <s xml:space="preserve">Vnde aperiendo, <lb/>deinde dictos duos angulos circa dictam <reg norm="lineam" type="context">lineã</reg>, abſque vlla difficultate erigitur cor-<lb/>pus, & abſque 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, quę reperiuntur in vorticibus aquarum, ſemper verſus <lb/>medium ipſarum vertiginum vniuntur, inde promanat, quod media <lb/>vertiginum ſemper depreſſiora ſunt. </s> + <s xml:space="preserve">vnde quòd dicta corpora ad medium acce-<lb/>dant, nihil aliud eſt, quàm ipſa corpora ſuo pondere grauitateq́ue deſcendere, figu <lb/>ra enim vorticibus eſt quaſi conica, & concaua cum angulo deorſum, & gyro baſis <lb/>ſurſum. </s> + <s xml:space="preserve">Atque hæc vera eſt huius effectus cauſa, & non ea quam Ariſtoteles ponit, <lb/>à<unsure/> quo aliarum omnium quæſtionum, quas ego omiſi rationes ſunt benè propoſitæ.</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 eſt certè Ariſtotelis amplitudo at que authoritas, vt dif-<lb/>ficillimum ac periculoſum ſit quidpiam ſcribere contra quam <lb/>ipſe docuerit, & mihi præſertim, cui ſemper viſa est viri <lb/>illius ſapientia admirabilis. </s> + <s xml:space="preserve">Veruntamen studio veritatis im-<lb/>pulſus, cuius ipſe amore in ſeipſum ſiviueret excitaretur, in me <lb/>dium <reg norm="quædam" type="context">quædã</reg> proferre non dubitaui, in quibus me inconcußa mathematicæ philoſophiæ <lb/>baſis, cui ſemper inſiſto, ab eo dißentire coegit.</s> + </p> + <div type="section" level="3" n="1"> + <head style="it" xml:space="preserve">Qualiter & ubi Ariſtoteles de uelocitate motuum natura-<lb/>lium localium aliter tractauerit quam nos <lb/>ſentiamus.</head> + <head xml:space="preserve">CAP.I.</head> + <p> + <s xml:space="preserve">VOlens Ariſtoreles probare vacuum non eſſe in rerum natura .8. cap. lib. 4. phy<lb/>ſicorum ait, idem corpus per varia <reg norm="diuerſaque" type="simple">diuerſaq́;</reg> media, vt per <reg norm="aerem" type="context">aerẽ</reg>, & per <reg norm="aquam" type="context">aquã</reg> <lb/>ſi moueretur, proportionem velocitatis eiuſdem corporis per aerem, ei, quæ per <lb/>aquam fit, vnam <reg norm="eandemque" type="simple">eandemq́;</reg> futuram cum ea, quæ eſt ſubtilitatis aereę ad ſubtilitatem <lb/>aquæ. </s> + <s xml:space="preserve">In poſtrema autem parte eiuſdem capitis ſic ſcribit: </s> + <s xml:space="preserve">Nam cum ea quę ma-<lb/>iorem vel ponderis velleuitatis pręſtantiam habent, ſi ſimili ſigura ſint, <reg norm="ſpacium" type="context">ſpaciũ</reg> par, <lb/>& æquale, maiore celeritate conficere cernamus, ea quam magnitudines inter ſe ha <lb/>bent, proportione: </s> + <s xml:space="preserve">profectò idem etiam perinane fieret. </s> + <s xml:space="preserve">Aliam quoque rationem <lb/>proponit phyloſophus .2. cap. ſexti phyſicorum ſcribens eademmet proportione, <lb/>qua tempus diuiditur, magnitudinem etiam diuidi. </s> + <s xml:space="preserve">Sexto autem cap. primi de cœ-<lb/>lo ſcribit, tempora eandem proportionem habere, quam habentè conuerſo ponde-<lb/>ra; </s> + <s xml:space="preserve">vt ſi media pars vnius ponderis, vnius horæ ſpatio moueretur, vniuerſum pondus <lb/>in media hora moueretur. </s> + <s xml:space="preserve">Secundo cap. lib. 3. de cœlo duobus in locis apertè com<lb/>monſtrat velocitatem corporis minoris, maiori corpori comparatam, in eadem exi-<lb/>ſtere proportione, in qua dicta corpora adinuicem relata exiſtunt. </s> + <s xml:space="preserve">Quinto cap. eiuſ<lb/>dem lib. idem affirmat, exemplo ab igne deſumpto. </s> + <s xml:space="preserve">Ex alijs etiam plurimis locis <lb/>cognoſci poteſt, ſenſiſſe Ariſtotelem duo corpora eadem ſpecie, & figura prædita <lb/>eandem planè proportionem in ſuorum motuum velocitatibus, quam in ſuis ma-<lb/>gnitudinibus habent, retinere. </s> + <s xml:space="preserve">Alij quoque permulti eandem opinionem retinue <lb/>runt, & <reg norm="omnium" type="context">omniũ</reg> poſtremus Nicolaus Tartalea, ſecunda propoſitione vigeſiminoni <lb/>quæſiti octaui libri, vbi profitetur ſe demonſtratiuè probare hanc propoſitio-<lb/>nem veram exiſtere; </s> + <s xml:space="preserve"><reg norm="neque" type="simple">neq;</reg> videt quàm magna reſiſtentiarum ſit differentia, quætam <lb/>ex diuerſitate figurarum, quàm ex magnitudinum varietate exoriri poteſt; </s> + <s xml:space="preserve">quas qui <lb/>dem diuerſitates ne conſiderat 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 ſupponenda ut conſtet cur circa uelocit atem motuum <lb/>natur alium localium ab Ariſtotelis placitis <lb/>recedamus.</head> + <head xml:space="preserve">CAP. II.</head> + <p> + <s xml:space="preserve">CVM ſuſceperimus prouinciam probandi quod Ariſtoteles circa motus <lb/>locales naturales deceptus fuerit, ſunt quædam primo veriſſima & obie-<lb/>cta intellectus perſe cognita pręſupponenda, ac primum quælibet duo corpora, <lb/>grauia, aut leuia, area æquali, <reg norm="ſimilique" type="simple">ſimiliq́</reg> figura, ſed ex materia diuerſa conſtantia, <reg norm="eodem q́ue" type="context">eodẽ<lb/>q́ue</reg> modo ſitum habentia, eandem proportionem velocitatis inter ſuos motus loca <lb/>les naturales, ut inter ſuamet pondera aut leuitates in vno <reg norm="eodemque" type="simple">eodemq́;</reg> medio, ſeruatu-<lb/>ra. </s> + <s xml:space="preserve">Quod quidem natura ſua notiſſimum eſt ſi conſiderabimus non aliunde maio-<lb/>rem tarditatem, aut velocitatem gigni, quàm à .4. cauſis (dummodo medium vnifor <lb/>mè ſit & quietum) ideſt à maiori aut minori pondere aut leuitate; </s> + <s xml:space="preserve">à diuerſa figura; <lb/></s> + <s xml:space="preserve">à ſitu ciuſdem figuræ diuerſo, reſpectu lineę 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 eiuſdem figu-<lb/>ræ ſitum, motum fore proportionatum virtuti mouenti, quæ erit pondus aut leuitas. <lb/></s> + <s xml:space="preserve">Quod autem de qualitate, de quantitate & ſitu eiuſdem figuræ dico, reſpectu reſi-<lb/>ftentiæ ipſius medii dico: </s> + <s xml:space="preserve">Quia diffimilitudo aut inęqualitas figurarum, aut ſitus di-<lb/>uerſus non <reg norm="parum" type="context">parũ</reg> alterat dictorum corporum motus, cum figura parua facilius diui-<lb/>dat continuitatem medij, quam magna; </s> + <s xml:space="preserve">vt etiam cęlerius idem facit acuta, quàm ob <lb/>tufa; </s> + <s xml:space="preserve">& illa quæ cum angulo, qui antecedat mouebitur velocius quàm illa, quæ ſecus. <lb/></s> + <s xml:space="preserve">Quotieſcunque igitur duo corpora vnam <reg norm="eandemque" type="simple">eandemq́;</reg> reſiſtentiam ipſorum ſuperfi-<lb/>ciebus, aut habebunt aut recipient, eorum motus inter ſeipſos eodem planè modo <lb/>proportionati conſurgent, quo erunt ipſorum virtutes mouentes: </s> + <s xml:space="preserve">& è conuerſo, quo <lb/>tieſcunque duo corpora vnam <reg norm="eandemque" type="simple">eandemq́;</reg> grauitatem, aut leuitatem, & diuerſas reſi <lb/>ſtentias habebunt, eorum motus inter ſeipſos eandem <reg norm="proportionem" type="context">proportionẽ</reg> ſortientur, <reg norm="quam" type="context">quã</reg> <lb/>habebunteorum reſiſtentiæ conuerſo modo; </s> + <s xml:space="preserve">quæ quidem reſiſtentiæ inter ſeipſas, <lb/>eandem proportionem quàm ipſarum ſuperficies habebunt, aut in qualitate ſola fi <lb/>guræ, aut in quantitate ſola, aut in ſitu, aut in aliquibus ex dictis rebus, eo tamen mo <lb/>do, quiſuperius poſitus fuit, vt ſcilicet corpus illud quod alteri comparatum, æqua-<lb/>lis erit ponderis, aut leuitatis, ſed minoris reſiſtentiæ, exiſtet velocius altero, in <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportione, cuius ſuperficies reſiſtentiam ſuſcipit minorem ea quæ alterius eſt cor-<lb/>poris, ratione facilioris diuiſionis continuitatis aeris, aut aquæ; </s> + <s xml:space="preserve">Vt exempli gratia, <lb/>ſi proportio ſuperficiei corporis maioris ſuperficiei minoris ſeſquitertia eſſet, pro-<lb/>portio velocitatis dicti corporis maioris, velocitati corporis minoris, eſſet ſubſeſqui <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 exiſtit.</s> + </p> + <p> + <s xml:space="preserve">Aliud quoque ſupponendum eſt, velocitatem ſcilicet motus naturalis alicuius <lb/>corporis grauis, in diuerſis medijs, propor-<lb/>tionatam eſſe ponderi eiuſdem corporis in <lb/>ijſdem medijs; </s> + <s xml:space="preserve">Vt exempli gratia, ſi pondus <lb/> + <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> + totale alicuius corporis grauis ſignificatum <lb/>crit ab <var>.a.i.</var> quo corpore poſito 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 denſo, quàm ipſum ſit, (quia in medio ſe denſiore ſi poner etur, non graue <lb/>e<gap extent="2"/>et, ſed leue, quemadmodum Archimedes oſtendit) illud medium ſubtrahat par-<lb/>tem <var>.e.i.</var> vnde pars <var>.a.e.</var> eiuſdem ponderis libera manear; </s> + <s xml:space="preserve">& poſito deinde eodem cor <lb/>pore in aliquo alio medio denſiore, minus tamen denſo quam ipſum ſit corpus, <lb/>hoc medium ſubtrahat partem <var>.u.i.</var> dicti ponderis, vnde pars <var>.a.u.</var> ei uſdem <lb/>ponderis remanebit. </s> + <s xml:space="preserve">Dico proportionem velocitatis eiuſdem corporis per <reg norm="medium" type="context">mediũ</reg> <lb/>minus denſum, ad velocitatem eiuſdem per medium magis denſum futuram vt <var>.a.e.</var> <lb/>ad <var>.a.u.</var> vt eſt etiam rationi conſonum, magis quàm ſi dicamus huiuſmodi velocitates <lb/>eſſe, vt <var>.u.i.</var> ad <var>.e.i.</var> cum velocitates à virtutibus mouentibus ſolum (cum figura vna, <lb/><reg norm="eademque" type="simple">eademq́;</reg> in qualitate, quantitate <reg norm="ſituque" type="simple">ſituq́</reg> erit) proportionentur. </s> + <s xml:space="preserve">Quæne<unsure/>nc diximus, <lb/>planè ſimilia ſuntijs, quæ ſupra ſcripſimus, quia idem eſt dicere, proportionem velo <lb/>citatum, duorum corporum hetereogeneorum, ſed ſimilium figura, & magnitudine <lb/>æ qualium, in vno ſolo medio, æqualem eſſe <lb/>proportioni ponderum ipſorum, vt ſi dicam?</s> + <s xml:space="preserve"><unsure/> <lb/>proportionem velocitatum vnius ſolum cor-<lb/> + <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> + poris per diuerſa media eandem eſſe cum ea. <lb/>quæ eſt <reg norm="ponderum" type="context">ponderũ</reg> dicti corporis in iſidem 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 diuerſis medijs habere cum denſitate eorum.</head> + <head xml:space="preserve">CAP. III.</head> + <p> + <s xml:space="preserve">POſſibile eſt in rerum natura corpus aliquod huiuſmodi denſitate præditum re-<lb/>periri, vt velocitas eius motus naturalis per aerem, velocitati per aquamita pro <lb/>portionata exiſtat, vt eſt <reg norm="denſitas" type="context">dẽſitas</reg> aquæ denſitati aeris. </s> + <s xml:space="preserve">Denſitas aquæ notetur (exem-<lb/>pli gratia) per <var>.u.i.</var> & ea, quæaeris eſt per <var>.e.i.</var> & pondus alicuius corporis in aere per <lb/><var>e.a.</var> & pondus eiuſdem corporis in aqua per <var>.u.a.</var> ita tamen, quod eadem proportio <lb/>ſit <var>.e.a.</var> ad <var>.u.a.</var> vt <var>.u.i.</var> ad <var>.e.i.</var> vnde per vltimam ſuppoſitionem <reg norm="præcedentis" type="context">præcedẽtis</reg> capitis, pro <lb/>portio velocitatis prædicti corporis per aerem, <lb/>proportioni eiuſdem 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">Oſcitanter ab Ariſtotele nonnibil prolatum cap 8. <lb/>lib. 4 Phyſicorum.</head> + <head xml:space="preserve">CAP. IIII.</head> + <p> + <s xml:space="preserve">EX ſupradictis patet in vniuerſum non eſſe verum quod Ariſto .8. cap .4. lib. phy<lb/>ſicorum ſcribit, velocitates ſcilicet motuum alicuius corporis per diuerſa me-<lb/>dia, proportionatas eſſe denſitatibus eorundem mediorum. </s> + <s xml:space="preserve">Quocirca, ſit propor-<lb/>tio <var>.u.i.</var> ad <var>.e.i.</var> vt <reg norm="denſitatis" type="context">dẽſitatis</reg> aquę ad <reg norm="aeream" type="context">aereã</reg> <reg norm="denſitatem" type="context">dẽſitatem</reg> .et <var>.e.a.</var> ad <var>.u.a.</var> vt ponderis alicuius <lb/>corporis in aere ad pondus eiuſdem in aqua, ita tamen vt maior aut minor propor-<lb/>tio ſit <var>.e.a.</var> ad <var>.u.a.</var> quam <var>.u.i.</var> ad <var>.e.i.</var> vnde exiſtente 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/>cauſam nimis diſſentaneum eſt rationi, opi-<lb/>nari proportionem velocitatis omnium cor <lb/>porum grauium per aerem vnam <reg norm="eandemque" type="simple">eandemq́;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0183-01a" xlink:href="fig-0183-01"/> + eſſe cum velocitate eorundem per aquam, <lb/>quemadmodum Ariſtoteles ſenſit.</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 denſitate dupla ad aerem, & aliquod <lb/>graue corpus in aqua duplum ad denſitatem ipſius aquæ, vnde dictum corpus in <lb/>denſitate ad aerem quadruplum erit; </s> + <s xml:space="preserve">quam ob cauſam, mediam ſui ponderis tota-<lb/>lis partem in aqua, & in aere quartam partem, ex .7. lib. de inſidentibus aquæ ab Ar-<lb/>chimede conſcripto, amitteret. </s> + <s xml:space="preserve">Moueretur igitur in aqua virtute illius mediæ partis <lb/><reg norm="ponderis" type="context">põderis</reg> ſui, in aere <reg norm="autem" type="wordlist">aũt</reg> uirtute <reg norm="trium" type="context">triũ</reg> <reg norm="quartarum" type="context">quartarũ</reg>; </s> + <s xml:space="preserve">vnde proportio facultatis <reg norm="mouentis" type="context">mouẽtis</reg> dicti <lb/>corporis in aere ad facultatem mouentem eiuſde m in aqua ſeſquialtera erit. </s> + <s xml:space="preserve"><reg norm="hocque" type="simple">hocq́;</reg> <lb/>corpus appelletur <var>.A</var>. </s> + <s xml:space="preserve">Sit aliud quoque corpus, quod <var>.B.</var> nominetur, ſimile figura, & <lb/>magnitudine corporea corpori <var>.A.</var> ſed <reg norm="denſitate" type="context">dẽſitate</reg>, in proportione ſeſquialtera ad <reg norm="aquam" type="context">aquã</reg>, <lb/>& denſius 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 ſeſquialtera, vnde etiam velocius erit ipſo <var>.B.</var> in aere <lb/>in eadem proportione, ſed 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 Ariſtoteli concedam corpus <var>.B.</var> in aere, quam in aqua velocius futurum in ea-<lb/>dem proportione, in qua, aqua eſt <reg norm="denſior" type="context">dẽſior</reg> aere, ex Euclidis vndecima propoſitione <lb/>lib. quinti. </s> + <s xml:space="preserve">Sed præter hæc omnia, ſi corpus <var>.A.</var> eſſet etiam velocius in aere, <reg norm="quam" type="context">quã</reg> in <lb/>aqua, in eadem proportione, ſequeretur ex .16. dicti lib. quinti proportionem velo-<lb/>citatis <var>.A.</var> in aqua ad <reg norm="velocitatem" type="context">velocitatẽ</reg> ipſius <var>.B.</var> in aqua etiam ſeſquialteram eſſe. </s> + <s xml:space="preserve">Sed cum <lb/>corpus <var>.A.</var> in denſitate ad aquam <reg norm="duplum" type="context">duplũ</reg> ſit, & corpus <var>.B.</var> <reg norm="ſeſquialterum" type="context">ſeſquialterũ</reg> ad ipſam <reg norm="aquam" type="context">aquã</reg>, <lb/>ſequetur <reg norm="proportionem" type="context">proportionẽ</reg> ponderis ipſius <var>.A.</var> ad <reg norm="pondus" type="context">põdus</reg> ipſius <var>.B.</var> in aqua eſſe in propor-<lb/>tione dupla; </s> + <s xml:space="preserve">Vnde ex primo ſuppoſito capitis ſecundi proportio velocitatis <var>.A.</var> ad <lb/>velocitatem <var>.B.</var> in aqua dupla erit, non ſeſquialtera. </s> + <s xml:space="preserve">Si ergo proportio velocitatis <var>.<lb/>A.</var> ad eam quæ eſt <var>.B.</var> in aqua dupla eſt, & ea, quæeſt <var>.B.</var> in aere, ad eam, quæ eſt ipſius <lb/>per aquam eſt etiam dupla (vnde ea quę eſt <var>.A.</var> per aquam ęqualis erit ei, quæ eſt <var>.B.</var> <lb/>peraerem, ex .9. lib. quinti) & cum ea, quæ eſt <var>.A.</var> ſit ei, quæ eſt <var>.B.</var> per aerem ſeſqui-<lb/>altera, erit ergo ea, quæ eſt <var>.A.</var> per aerem, ei, quæ eſt ſuimet ipſius per aquam ſeſqui <lb/>altera, non autem dupla, ex .7. eiuſdem libr. quinti. </s> + <s xml:space="preserve">Hiſce rationibus accedimus ad <lb/>confirmandam veritatem vltimi ſuppoſiti cap .2. proportionem videlicet velocita <lb/>tis <reg norm="motus" type="simple">motꝰ</reg> naturalis in diuerſis medijs <reg norm="alicuius" type="simple">alicuiꝰ</reg> corporis <reg norm="ponderoſi" type="context">põderoſi</reg> in ipſis medijs eſſe ean <lb/>dem cum ea, quæeſt 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/>ctuoſa, aut pinguia non ſunt, ut ſunt <lb/>oleum, lac, aut huiuſmodialia, 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 eiuſdem corporis in diuerſis medijs pro <lb/>portiones eorum mediorum denſit atum non ſeruant. 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 diuerſa media, vnde <lb/>proportiones velocitatum inæquales exiſtunt. </s> + <s xml:space="preserve">Vt exempli gratia, ſi fue-<lb/>rit corpus <var>.A.</var> cuius pondus totale ſit <var>.o.a.</var> quod in aqua diminutum ſit ratione partis <var>.<lb/>e.o.</var> ita vt ei ſolum relinquatur pondus <var>.a.e.</var> & in aeie adempta ſit ei pars <var>.i.o.</var> vnde ſo <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 denſum, quàm aer, quemadmodum aer minus denſus eſt, aqua, in quo, cor <lb/>pus <var>.A.</var> ammittat partem <var>.t.o.</var> ponderis ſui, vnde ex .7. lib. de inſidentibus aquæ Ar-<lb/>chimedis, eadem proportio erit <var>.e.o.</var> ad <var>.i.o.</var> quæ eſt <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> eſt <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 eſt <var>.i.o.</var> ad <var>.t.o</var>. </s> + <s xml:space="preserve">Cum ſit ergo proportio <var>.a.i.</var> <lb/>ad <var>.a.e.</var> ut <var>.e.o.</var> ad <var>.i.o.</var> erit diſiunctim <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. quin<lb/>ti erit <var>.a.e.</var> æqualis <var>.i.o.</var> ſed cum ita ſehabeat <var>.e.o.</var> ad <var>.i.o.</var> vt <var>.i.o.</var> ad <var>.t.o.</var> ita quoque <lb/>ſe 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/>ęqualis ſit ipſi <var>.i.o.</var> non poterit eſſe proportio <var>.t.a.</var> ad <var>.i.a.</var> vt eſt <var>.o.i.</var> ad <var>.t.o.</var> quia ſi <lb/>hoc eſſet, eſſet etiam diſiunctim proportio <var>.i.t.</var> ad <var>.i.a.</var> vt eſt <var>.i.t.</var> ad <var>.t.o.</var> & ex ſupradicta <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> ideſt minorem <var>.a.e.</var> æqualem eſſe <var>.a.i.</var> quæ maior eſt <var>.a.e</var>. </s> + <s xml:space="preserve">Oſtenſiuè tamen idem <lb/>hoc modo probari poteſt, vt exiſtente <var>.i.o.</var> ęquali ipſi <var>.a.e.</var> per conſequens <reg norm="quoque" type="simple">quoq;</reg> erit <lb/>minor ipſa <var>.a.i.</var> cum <var>.a.e.</var> pars ſit ipſius <var>a.i</var>. </s> + <s xml:space="preserve"><reg norm="Pereandem" type="context">Pereãdem</reg> tamen rationem <var>.o.t.</var> minoreſt <var>.<lb/>o.i</var>. </s> + <s xml:space="preserve">Tanto magis igitur minor erit <var>.t.o.</var> ipſa <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="eiuſdem" type="context">eiuſdẽ</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, quàm.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> quàm.t.a. ad <var>.i.a.</var> ita ergo ſe habebunt ipſorum 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 eiuſdem figur æ et materiæ ſed inæqualis <lb/>magnitudinis, in ſuis motibus natur alibus uelocit atis, in eo <lb/>dem medio, proportionem longè diuerſam ſeruatura <lb/>eße quam Aristoteliuiſum fuerit.</head> + <head xml:space="preserve">CAP. VII.</head> + <p> + <s xml:space="preserve">ESt mihi nunc probandum <reg norm="quod" type="simple">ꝙ</reg> in uno <reg norm="eodemque" type="simple">eodemq́;</reg> mcdio duo corpora inæqualia, ſed <lb/>ſimili figura & materia, mouebuntur naturali motu, diuerſa 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, quàm Ariſtoteles præſcripſit.</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 ſit, & per conſequens in eadem quoque proportione grauius ipſo <var>.o.</var> in <lb/>qua eſt maius, communi omnium ſententia.</s> + </p> + <p> + <s xml:space="preserve">Scribit ergo Ariſtoteles proportionem velocitatis corporis <var>.a.</var> ad eam, quæ eſt <lb/>corporis <var>.o.</var> (naturaliterſe vnoquoque mouente) eandem futuram, quæ eſt 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 eſt, <lb/>ſed eandem grauitatem obtinere, quæ communicata eſt corpori <var>.o.</var> quod ex quauis <lb/>materia conſter. </s> + <s xml:space="preserve">Hinc ex primo ſuppoſito ſecundi capitis certi erimus proportio-<lb/>nem velocitatis corporis <var>.a.</var> ſi comparetur cum velocitate corporis <var>.u.</var> futuram, vt <reg norm="eam" type="context">eã</reg>, <lb/>quæ eſt ponderis corporis <var>.a.</var> ad pondus ipſius corporis <var>.u</var>. </s> + <s xml:space="preserve">Ex .9. igitur lib. quinti Eu-<lb/>cli. cogitur fateri Ariſtoteles velocitatem corporis <var>.o.</var> eſſe vnam <reg norm="eandemque" type="simple">eandemq́;</reg> in ſpe-<lb/>cie, quæ eſt corporis <var>.u</var>. </s> + <s xml:space="preserve">Quod primo ſuppoſito cap. ſecundi huius lib. planè repugna<lb/>ret. </s> + <s xml:space="preserve">Igitur hæc Ariſtotelis opinio falſa eſt. </s> + <s xml:space="preserve">Idem quoque probaretur mediante cor <lb/>pore <var>.i.</var> æquali magnitudine, <reg norm="ſimilique" type="simple">ſimiliq́;</reg> figura cum corpore <var>.o.</var> prædito, ſed, quòd ad <lb/>quantitatem attinet, æquali corpori <var>.a.</var> vnde ex primo ſuppoſito cap. ſecundi huius li <lb/>bri in eadem pro <lb/>portione <reg norm="velocius" type="simple">velociꝰ</reg> <lb/>eſſet corpore <var>.o.</var> <lb/> + <anchor type="figure" xlink:label="fig-0185-01a" xlink:href="fig-0185-01"/> + in qua grauius eſt. <lb/>ex .9. igitur quin-<lb/>ti cogitur Ariſto-<lb/>teles affirmare <reg norm="tam" type="context">tã</reg> <lb/>velox eſſe corpus <lb/>a. <reg norm="quam" type="context">quã</reg> eſt corpus <lb/>i. vnde idem pla-<lb/>nè inconueniens emergit ex ſecundo ſuppoſito cap. ſecundi 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 eiuſdem materia in diuerſis <lb/>medijs eandem uelocitatis proportionem <lb/>retinebunt.</head> + <head xml:space="preserve">CAP. VIII.</head> + <p> + <s xml:space="preserve">QVælibet duo corpora inæqualia ſimili tamen figura & eadem materia con-<lb/>ſtantia, naturaliter ſe per diuerſa media mouentia, vnam <reg norm="eandemque" type="simple">eandemq́;</reg> ſem-<lb/>per proportionem velocitatum ſeruant.</s> + </p> + <p> + <s xml:space="preserve">Sint duo corpora <var>.A.</var> et <var>.B.</var> ſibi inuicem inæqualia quorum <var>.A.</var> ſit maius, ſed ſimile <lb/>figura & idem materia, <lb/>cuius pondus totaleſit <var>.<lb/> + <anchor type="figure" xlink:label="fig-0185-02a" xlink:href="fig-0185-02"/> + x.o.</var> & pondus totaleip <lb/>ſius <var>.B.</var> ſit <var>.u.s</var>. </s> + <s xml:space="preserve">Imagine-<lb/>mur quoque corpus <var>.A.</var> <lb/>poſitum 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 correſpondet <var>.e.o.</var> æquale eſt ipſi <var>.A.</var> & corpus <lb/>aqueum, cui correſpondet <var>.c.s.</var> æquale eſt i pſi <var>.B.</var> vt eſt ab Archimede <reg norm="probatum" type="context">probatũ</reg>: </s> + <s xml:space="preserve">com <lb/>muni quadam ſcientiæ ratione, ſequitur eandem proportionem futuram <var>.o.x.</var> ad <var>.e.o.</var> <lb/>quæ eſt <var>.u.s.</var> ad <var>.c.s.</var> ob <reg norm="eaſdemque" type="simple">eaſdemq́;</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/>quintí 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, ſi ſubtra <lb/> + <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> + hantur proportiones @reſi <lb/><reg norm="ſtentiarum" type="context">ſtẽtiarum</reg> extrinſecus <reg norm="ad- uenentium" type="context">ad-<lb/>uenẽtium</reg>, proportiones <lb/>quæ remanebunt, exter-<lb/>tio communi axiomate <lb/>ab Eucli. in principio pri<lb/>mi lib. poſito, ad inuicem <lb/>erunt æquales, ſecundum quas eorundem corporum ſunt 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 diſeruerit de proportionibus mo-<lb/>tuum in uacuo.</head> + <head xml:space="preserve">CAP. IX.</head> + <p> + <s xml:space="preserve">CVm verò Ariſtoteles circa finem cap .8. lib. 4. phyſicorum ſubiungit quod ea-<lb/>dem proportione dicta corpora mouerentur in vacuo, vt in pleno, id pace <reg norm="eius" type="simple">eiꝰ</reg> <lb/><reg norm="dictum" type="context">dictũ</reg> ſit planè <reg norm="erroneum" type="context">erroneũ</reg> eſt. </s> + <s xml:space="preserve">quia in pleno dictis corporibus ſubtrahitur proportio reſi <lb/>ſtentiarum extrinſecarum à proportione ponderum, vt velocitatum proportio re-<lb/>maneat, quę nulla eſſet, ſi dictarum reſiſtentiarum proportio, ponderum propor-<lb/>tioni æqualis eſſet, & hanc ob cauſam diuerſam velocitatum proportionem in va-<lb/>cuo haberent ab ea, quæ eſt in pleno.</s> + </p> + </div> + <div type="section" level="3" n="10"> + <head style="it" xml:space="preserve">Quòd in uacuo corpor a eiuſdem materiæ æquali uelocita-<lb/>te mouerentur.</head> + <head xml:space="preserve">CAP.X.</head> + <p> + <s xml:space="preserve">QVòd ſupradicta 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> ſit dimidia pars ipſius <var>.o.</var> ſint alia <lb/>quoque duo corpora <var>.a.</var> et <var>.e.</var> omogenea primis, quorum quodlibet æquale ſit ipſi <var>.g.</var> <lb/>& imaginatione compręhendamus ambo poſita in extremitatibus alicuius lineæ, cu <lb/>ius medium ſit <var>.i.</var> clarum erit, tantum pondus habiturum, punctum <var>.i.</var> quantum <reg norm="centrum" type="context">centrũ</reg> <lb/>ipſius <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 ipſius .<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 ſuam 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 eſt .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 eiuſdem materiæ & figuræ, ſireſiſten-<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 præſcripſimus, poſſet oſtendi, ſi duo cor-<lb/>pora <var>.o.</var> et <var>.g.</var> ſuas reſiſtentias, ita ad inuicem proportionatas haberent, utſunt <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 ipſius <var>.o.</var> cum à tanto pondere <lb/>i. motum eſſet; </s> + <s xml:space="preserve">quanto centrum ipſius <var>.o.</var> atquetan <lb/> + <anchor type="figure" xlink:label="fig-0187-01a" xlink:href="fig-0187-01"/> + tam reſiſtentiam duo corpora <var>.a.</var> et <var>.e.</var> <reg norm="quanta" type="context">quãta</reg> ipſum <lb/>o. ſolum haberet ex hypotheſi, dicta tamen corpo <lb/>ra <var>.a.</var> et <var>.e.</var> tam ſeparata, 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 demonſir atur eſſe proportio ponder is corpor is den <lb/>ſioris ad pondus minus denſi in <reg norm="medijs" type="lig">medijs</reg> <reg norm="denſioribus" type="context">dẽſioribus</reg>, quam <lb/>ſit eorundem corporum in medio minus denſo, nec <lb/>corporum ponder a ſeruare proportionem <lb/>denſitatis mediorum.</head> + <head xml:space="preserve">CAP. XII.</head> + <p> + <s xml:space="preserve">PRopoſita nobis cum fuerint duo corpora <var>.A.</var> et <var>.B.</var> area corporea æqualia, quo-<lb/>rum <var>.A.</var> denſius ſit ipſo <var>.B.</var> probabo in medio magis denſo, maiorem proportio <lb/>nem futuram ponderis ipſius <var>.A.</var> ad pondus <var>.B.</var> quàm in medio minus denſo.</s> + </p> + <p> + <s xml:space="preserve">Sit igitur <var>.p.g.</var> pondus totale ipſius corporis <var>.A.</var> et <var>.q.k.</var> ipſius corporis <var>.B.</var> vnde <var>.p.g.</var> <lb/>maius erit ipſo <var>.q.k</var>. </s> + <s xml:space="preserve">Sit quoque <var>.o.g.</var> pondus, quod medium magis denſum ſubtra-<lb/>hit à pondere <var>.p.g.</var> et <var>.n.k.</var> ſit pondus, quod idem medium ſubtrahit à pondere <var>.q.k.</var> et <lb/><var>f.g.</var> ſit pondus, quod medium minus denſum ſubtrahit à <var>.p.g.</var> et <var>.i.k.</var> illud, quodid@m <lb/><reg norm="medium" type="context">mediũ</reg> ſubtrahit ab <var>.q.k.</var> vnde <var>.o.g.</var> æquale erit <var>.n.k.</var> et <var>.f.g.</var> ipſi <var>.i.k.</var> quia quod ad <reg norm="aream" type="context">areã</reg> <lb/>attinet, corpora ſupponuntur æqualia, vnde proportio <var>.p.f.</var> ad <var>.q.i.</var> maior erit ea, quæ <lb/>eſt <var>.o.f.</var> ad <var>.n.i.</var> communi <lb/> + <anchor type="figure" xlink:label="fig-0187-02a" xlink:href="fig-0187-02"/> + ſcientiæ notione, quia ſi <lb/>ſcinderet <reg norm="aliquis" type="simple">aliꝗs</reg>.p.f. in pun <lb/>cto <var>.c.</var> ita. vt <var>.c.f.</var> æquale eſ-<lb/>ſet ipſi <var>.q.i.</var> proportio <var>.c.f.</var> <lb/>ad <var>.q.i.</var> eſſet vt ea, quæ eſt <var>.<lb/>o.f.</var> ad <var>.n.i.</var> (hoc eſt 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"/> + ſed proportio <var>.p.f.</var> ad <var>.q.i.</var> maior eſſet ea, quæ eſt <var>.c.f.</var> ad <var>.q.i.</var> ex. octaua lib. quinti, vn-<lb/>de ex .12. eiuſdem lib. maior eſſet <var>.p.f.</var> ad <var>.q.i.</var> quàm.o.f. ad <var>.n.i.</var> ex .33. igitur eiuſdem, <lb/>maior erit proportio <var>.p.o.</var> ad <var>.q.n.</var> quàm.p.f. ad <var>.q.i</var>. </s> + <s xml:space="preserve">Sic quoque ſe habebunt ad inui <lb/>cem velocitates, quod eſt propoſitum. </s> + <s xml:space="preserve">Cum autem proportio <var>.p.o.</var> ad <var>.q.n.</var> maior ſit, <lb/>quàm.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 euerſim maior erit proportio <var>.q.i.</var> ad <var>.q.n.</var> quàm.p.f. ad <var>.p.o.</var> vnde ſi proportio <lb/><var>p.f.</var> ad <var>.p.o.</var> eſſet ac ea, quæ eſt <var>.o.g.</var> ad <var>.f.g.</var> non eſſet <var>.q.i.</var> ad <var>.q.n.</var> ut eſt <var>.o.g.</var> ad <var>.f.g.</var> aut <lb/>vt <var>.n.k.</var> ad <var>.i.k.</var> quodidem <lb/>eſt, de quibus quidem re-<lb/> + <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> + bus, exemplis propoſitis <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 ſe-<lb/>quentibus pondera, ſequi <lb/>tur proportionem veloci-<lb/>citatum duorum corporum hetereogeneorum eandem non eſſe per diuerſa media, <lb/>contra id, quod ſequeretur ſi Ariſtotelis opinionem .8. cap. lib. 4. phyſicorum re-<lb/>ciperemus.</s> + </p> + </div> + <div type="section" level="3" n="13"> + <head style="it" xml:space="preserve">Longe aliter ueritatem ſe habere quam Aristoteles <lb/>doceat in fine libri ſeptimi phyſicorum.</head> + <head xml:space="preserve">CAP. XIII.</head> + <p> + <s xml:space="preserve">NOn tam facile eſt aſſignare proportionem velocitatum duorum corporum na <lb/>turalium, quam Ariſtoteles vltimo cap. lib. 7. phyſicorum putauit.</s> + </p> + <p> + <s xml:space="preserve">Quamobrem ſint duo corpora <var>.B.</var> et <var>.D.</var> materia <reg norm="magnitudineque" type="simple">magnitudineq́;</reg> diuerſa, pondere <lb/>tamen, & figura ſimilia, & proportio reſiſtentiarum, quas recipiunt à medio <reg norm="dum" type="context">dũ</reg> mo-<lb/>uentur, ſit. ut <var>.o.i.</var> ad <var>.a.e.</var> denotentur deinde velocitates totales abſque vlla reſiſten-<lb/>tia ab <var>.a.u.</var> et <var>.o.c.</var> quæ æquales erunt ad inuicem per communem ſcientiam ex ſup-<lb/>poſito, ſint alia deinde duo corpora <var>.V.</var> et <var>.M.</var> eodem modo ſe habentia ut prima <var>.B.</var> <lb/>et <var>.D.</var> in eodem medio, ſed ex diuerſa materia ab ea, quæ eſt illorum duorum corpo <lb/>rum, magnitudine tamen & figura ijſdem ſimilia: </s> + <s xml:space="preserve">ſignificentur quoque eo-<lb/>rundem reſiſtentiæ per <var>.t.s.</var> et <var>.n.r.</var> & eorundem velocitates à nulla ex reſiſtentijs 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> ipſi <var>.o.i.</var> et <var>.n.x.</var> ipſi <var>.t.g</var>: <var>n.x.</var> ta-<lb/>men et <var>.t.g.</var> non erunt ęqualia <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 ipſi <var>.B.</var> cum applicata <lb/>erit reſiſtentia <var>.a.e.</var> dicto corpori <var>.B.</var> quæ diminutam facit totam <var>.a.u.</var> per <var>.a.e.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.i.c.</var> <lb/>ea, quę remanet ipſi <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 eſt ex <var>.t.g.</var> corporis <var>.M</var>. </s> + <s xml:space="preserve">Vnde communi omnium <reg norm="conſenſu" type="context">cõſenſu</reg> aſſequemur <var>.e.u.</var> ma <lb/>iorem futuram <var>.r.x.</var> et <var>.i.c.</var> ipſa <var>.s.g</var>. </s> + <s xml:space="preserve">Scindatur deinde <var>.a.m.</var> ad ęqualitatem <var>.n.x.</var> et <var>.o.z.</var> <lb/>ipſius <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 ipſa <var>.z.i.</var> maior igitur erit proportio <var>.z.c.</var> <lb/>ad <var>.z.i.</var> quàm.m.u. ad <var>.m.e.</var> (quia <var>.z.c.</var> ad <var>.z.i.</var> ita ſe habet vt <var>.m.u.</var> ad <var>.i.z.</var> ex .7. lib. quin-<lb/>ti, ſed <var>.m.u.</var> ad <var>.i.z.</var> maior eſt quam ad <var>.m.e.</var> ex .8. dicti lib. vnde ex .12. eiuſdem <var>.z.c.</var> ad <lb/>ad <var>.z.i.</var> maior erit, quàm.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> ideſt <var>.s.</var> g <lb/>ad <var>.r.x.</var> quod Ariſtoteli in mentem non venerat. </s> + <s xml:space="preserve">Alijs quoque modis idem proba-<lb/>ri poteſt, vt ſi 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 eſt ratio vt ad <var>.z.c.</var> ex .7. quinti, ſed proportio <var>.e.m.</var> ad <var>.<lb/>z.c.</var> maior eſt quam <var>.i.z.</var> ad <var>.z.c.</var> ex .8. eiuſdem, ergo ea, quæ eſt <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 eſt <var>.e.u.</var> ad <var>.m.u.</var> maior erit illa, <lb/>quæ eſt <var>.i.c.</var> ad <var>.z.c.</var> & <reg norm="permutando" type="context">permutãdo</reg>, quam ea, quæ eſt <var>.e.u.</var> ad <var>.i.c.</var> ea, quæ eſt <var>.m.u.</var> ad <var>.z.c.</var> <lb/>& ex .33. quinti, ea, quæ eſt <var>.e.m.</var> ad <var>.i.z.</var> maior erit ea, quæ eſt <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 ſequatur ex ſupradistis.</head> + <head xml:space="preserve">CAP. XIIII.</head> + <p> + <s xml:space="preserve">EX præcedenti capite manifeſtè depræhenditur, in vniuerſum Ariſtotelis opi-<lb/>nionem veram non eſſe in prima parte vltimi capitis. lib. 7. phyſicorum; </s> + <s xml:space="preserve">quia <lb/>in eo loco ſupponens ipſe corpus <var>.B.</var> pręcedentis capitis eſſe dimidiam partem ipſius <lb/>D. quantum ad aream corpoream ſpectat (ſunt tamen pondere ad inuicem æqualia) <lb/>ait <var>.B.</var> futurum duplo velocius ipſo <var>.D</var>. </s> + <s xml:space="preserve">Ego verò præcedenti capite accepi <var>.e.u.</var> pro <lb/>velocitate reſidua corporis <var>.B.</var> (ſubtracta ea tamen parte, quam ei reſiſtentia adimit, <lb/>quæ erat <var>.e.a.</var>) et <var>.i.c.</var> pro ea, quæ eſt corporis <var>.D.</var> et <var>.r.x.</var> pro ea, quæ eſt corporis <var>.V.</var> et <var>.<lb/>s.g.</var> pro ea, quæ eſt corporis <var>.M</var>. </s> + <s xml:space="preserve">Dicat nunc Ariſtoteles, quę nam harum duarum pro <lb/>portionum dupla erit? </s> + <s xml:space="preserve">quia ſi earum aliqua talis erit, alia nullo modo eſſe poterit, <lb/>vt iam oſtendi, etiamſi duo corpora <var>.V.</var> et <var>.M.</var> eaſdem conditiones habeant, quas <var>.B.</var> <lb/>et <var>.D</var>. </s> + <s xml:space="preserve">Ratio autem, quæ Ariſtotelem induxerit ad illud credendum, nulla alia eſſe <lb/>potuit, quàm quod putarit reſiſtentias proportionatas eſſe magnitudinibus corpo-<lb/>reis, ideſt quemadmodum <var>.B.</var> erat corporaliter dimidia pars ipſius <var>.D.</var> ſic etiam habe <lb/>ret medietatem eius reſiſtentiæ, quam habuiſſet corpus <var>.D</var>. </s> + <s xml:space="preserve">Quod etſi verum eſſet, <lb/>non tamen ſequeretur neceſſariò in quibuſlibet corporibus futuram velocitatum <lb/>proportionem eandem, quæ reſiſtentiarum eſt, vt ſuperiore capite oſtendimus.</s> + </p> + </div> + <div type="section" level="3" n="15"> + <head style="it" xml:space="preserve">Numrestè ſenſerit Philoſophus reſistentias proportionales <lb/>eße cum corporibus mobilibus.</head> + <head xml:space="preserve">CAP. XV.</head> + <p> + <s xml:space="preserve">QVòd Ariſtoteles crediderit reſiſtentias proportionatas eſſe corporibus, erra-<lb/>uit. </s> + <s xml:space="preserve">Si ſuperficies ijſdem proportionatæ eſſent, dubium non eſt, quin <lb/>reſiſtentiæ quoque ipſæ, ijſdem proportionatæ exiſterent, ſupponendo eas ſimiles <lb/>ſitu, dum eadem corpora mouerentur. </s> + <s xml:space="preserve">Sed eadem proportio non eſt inter ſuperfi- + <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">Ariſtoteles igitur in eo defecit. </s> + <s xml:space="preserve">Quòd <reg norm="autem" type="context">autẽ</reg> inter <lb/>ſuperficies non eadem ſit proportio, quæ inter corpora extat, ſi primo ad ſphęricas <lb/>mentem verterimus, intelligemus proportionem eam, quæ inter duas ſphæras repe <lb/>ritur triplam ſemper exiſtere ei, quæ eſt inter ipſarum diametros ex vltima .12. libr. <lb/>Euclid. </s> + <s xml:space="preserve">Eſt autem proportio, quæ eſt inter ſuperficies ſphęricas ęqualis ei, quæ eſt <lb/>ipſorum circulorum maiorum ex .16. lib. quinti, cum ex .31. primi de ſphæra & cy-<lb/>lindro Archimedis, omnis ſphærica ſuperficies quadrupla, ſit maiori circulo ipſius <lb/>ſphęræ, ſed proportio, quæ eſt inter dictos circulos, eſt dupla ei, quæ eſt inter <reg norm="eorun- dem" type="context context">eorũ-<lb/>dẽ</reg> diametros ex .2. lib. 12. Euc. </s> + <s xml:space="preserve">ergo <reg norm="proportio" type="simple">ꝓportio</reg>, quæ eſt inter corpora, ſeſquialtera erit <lb/>ei, quæ eſt ſuperficierum, & non æqualis, ut Ariſtoteles putauit. </s> + <s xml:space="preserve">Idem de corporibus <lb/>ſimilibus à ſuperficiebus planis terminatis dico, ratiocinando mediante .36. lib. 11. <lb/>et .18. ſexti, vnde cognoſcemus proportionem corporum, proportioni laterum, tri-<lb/>plam futuram, & ſuperficierum proportionem, laterum proportioni duplam. </s> + <s xml:space="preserve">Quare <lb/>corporum proportio, ei, quæ ſuperficierum eſt, ſeſquialtera erit, ita ut ſi velocitates <lb/>extitiſſent ad inuicem proportionatæ, vt ſuperficies, proportio velocitatis corporis <var>.<lb/>B.</var> ei, quæ eſt corporis <var>.C.</var> fuiſſet ſubſeſquialtera proportioni corporum, & non æqua <lb/>lis eidem.</s> + </p> + </div> + <div type="section" level="3" n="16"> + <head style="it" xml:space="preserve">Fdipſum aliter demonſtr atur.</head> + <head xml:space="preserve">CAP. XVI.</head> + <p> + <s xml:space="preserve">ALio quoque modo probari poteſt non eſſe in vniuerſum verum id, quod Ari-<lb/>ſtoteles in prima parte capitis vltimi lib. 7. phyſicorum ait, ſic ſcribens.</s> + </p> + <p> + <s xml:space="preserve">Si <var>.A.</var> quidem ſit id quod mouet <var>.B.</var> verò id quod mouetur, et <var>.C.</var> ſit longitudo per <lb/>quam, et <var>.D.</var> tempus in quo eſt motum, in tempore nimirum ęquali, potentia æqua-<lb/>lis <var>.A.</var> dimidium ipſius <var>.B.</var> per duplum mouebit ipſius <var>.C.</var> per ipſum autem <var>.C.</var> in dimi <lb/>dio temporis <var>.D.</var> ſic enim erit rationis ſimilitudo.</s> + </p> + <p> + <s xml:space="preserve">Sit ergo corpus <var>.o.</var> ſeptimi capitis pondere æquali corpori <var>.u.</var> eiuſdem capitis, ſed <lb/>area corporea minusipſo <var>.u.</var> pro medietate. </s> + <s xml:space="preserve">Simile tamen figura. </s> + <s xml:space="preserve">Imaginemur <reg norm="nunc" type="context">nũc</reg> <lb/>tertium aliud corpus omogeneum ipſi <var>.u.</var> quod ſit <var>.i.</var> magnitudine & figura ſimile ipſi <lb/>o. vnde minor erit ipſo <var>.u.</var> pro media parte, & hanc ob cauſam ipſum <var>.u.</var> erit duplo ma <lb/>gis graue, quàm ipſum <var>.i.</var> & per conſequens ipſum quoque <var>.o.</var> duplo grauius erit <reg norm="quam" type="context">quã</reg> <lb/>ſit ipſum <var>.i.</var> ex .7. libr. quinti Euclidis. </s> + <s xml:space="preserve">Ipſum ergo corpus <var>.o.</var> duplo velocius erit, <lb/>quàm ipſum <var>.i.</var> ex primo ſuppoſito cap .2. huius lib. </s> + <s xml:space="preserve">Vnde ex .9. quinti, velocitas ipſius <lb/>i. æqualis eſſet ei, quæ eſt ipſius u. cum Ariſtoteles ſcribat <var>.o.</var> quoque futurum duplo <lb/>velocius ipſo <var>.u.</var> <reg norm="quod" type="simple">ꝙ</reg> cap .7. huius lib. falſum eſſe demonſtraui.</s> + </p> + </div> + <div type="section" level="3" n="17"> + <head style="it" xml:space="preserve">De alio Aristo. lapſu.</head> + <head xml:space="preserve">CAP. XVII.</head> + <p> + <s xml:space="preserve">SCribit Ariſtoteles in ultimo cap. lib. 7. phyſicorum in hunc modum. <lb/></s> + <s xml:space="preserve">Si duo quædam ſeorſum per tantum ſpatium tanto tempore duo ſeorſum pon <lb/>dera mouent, & compoſita per longitudinem æqualem, <reg norm="ęqualiuem" type="context">ęqualiuẽ</reg> in tempore, com-<lb/>poſitum ex ponderibus <reg norm="vtriſque" type="simple">vtriſq;</reg> mouebunt, eſt 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 vniuerſum nec etiam poteſt eſſe verum in pleno, quia cap .14. iam pro-<lb/>baui, non eandem proportionem eſſe inter ſuperſicies corporum, & ipſa corpora.</s> + </p> + </div> + <div type="section" level="3" n="18"> + <head style="it" xml:space="preserve">Quomodo dignoſcatur proportio uelocitatis duorum ſimilium <lb/>corporum omogeniorum inaqualium.</head> + <head xml:space="preserve">CAP. XVIII.</head> + <p> + <s xml:space="preserve">ETiam ſi reperire in qua proportione motus naturaliter moueantur duo corpo-<lb/>ra, figura & materia ſimilia, inęqualia tamen ad inuicem, non facile ſit, oſten-<lb/>dam tamen qua ratione id conſequi poſſimus.</s> + </p> + <p> + <s xml:space="preserve">Proponantur nobis, exempli gratia, duo corpora <var>.a.</var> et <var>.o.</var> ſphęrica, inęqualia inui-<lb/>cem, omogenea tamen materia, quorum <var>.a.</var> maius ſit; </s> + <s xml:space="preserve">ſi voluerimus inuenire in qua <lb/>nam velocitatis proportione naturaliter mouerentur. </s> + <s xml:space="preserve">Volo vt inquiratur corpus <var>.i.</var> <lb/>ſphęricum, alia tamen & diuerſa materia conſtans, ſed pondere ęquale corpori <var>.o.</var> & <lb/>ſuperſicie tam proportionata ſuperficiei corp oris <var>.a.</var> quàm eſt ea, quæ eſt ſui ponde-<lb/>ris ad pondus ipſius <var>.a</var>. </s> + <s xml:space="preserve">Hoc facto, indagetur, quænam erit proportio inter ſu-<lb/>perficies corporum <var>.i.</var> et <var>.o.</var> quę ſemper dupla eſt, vel ſubdupla ei quæ eſt diametro-<lb/>rum; </s> + <s xml:space="preserve">ut iam cap .15. dixi, & hęc proportio ſuperficierum ſphęricarum <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.o.</var> et <var>.i.</var> ſub <lb/>trahatur ab æqualitate, quod igitur remanebit, erit proportio <reg norm="velocitatum" type="context">velocitatũ</reg> inter duo <lb/>corpora <var>.o.</var> et <var>.i.</var> ideſt inter <var>.o.</var> et <var>.a.</var> vt exempli gratia, ſi proportio ſuperficiei <var>.o.</var> ſuperfi <lb/>ciei ipſius <var>.i.</var> ſeſquitertiα<unsure/> eſſet, ſub <lb/>trahendo eam ab ęqualitate, rema-<lb/> + <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> + neret <reg norm="proportio" type="simple">ꝓportio</reg> ſubſeſquitertia, vnde <lb/>velocitas corporis maioris ( quod in <lb/>pręſenti loco ſupponitur eſſe <var>.o.</var>) ei, <lb/>quę eſt corporis minoris, quale eſt <lb/>corpus <var>.i.</var> ſubſeſquitertia eſſet; </s> + <s xml:space="preserve">aut <lb/>dicamus quòd <var>.i.</var> eſſet velocius ipſo <lb/>o. in proportione ſeſquitertia ex ſe <lb/>cundo ſuppoſito ſecundi capitis huius libri. </s> + <s xml:space="preserve">Sed <var>.i.</var> tam velox eſt quam ipſum <var>.a.</var> ex <num value="11">.<lb/>11.</num> cap. ergo proportio velocitatis ipſius <var>.a.</var> ſeſquitertia erit ei. quæ eſt ipſius <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 ſit inanis ab Ariſtotele ſuſcepta demonſtratio quod <lb/>uacuum non detur.</head> + <head xml:space="preserve">CAP. XIX.</head> + <p> + <s xml:space="preserve">EX ijs, quæ ſuperius <reg norm="demonſtrauimus" type="context">demõſtrauimus</reg> facilè cognoſci poteſt irritam eſſc eam ratio <lb/>nem, quam Ariſtoteles .8. cap. lib. 4. phyſicorum ad deſtruendum vacuum, <reg norm="con" type="context">cõ</reg> <lb/>finxit. </s> + <s xml:space="preserve">Vtigitur idem facilius oſtendamus, compræhendamus imaginatione infini-<lb/>ta media corporea, quorum vnum altero rarius ſit, in qua placuerit nobis ex propor <lb/>tionibus, incipiendo ab uno, imaginemur etiam corpus <var>.Q.</var> denſius primo medio, cu-<lb/>ius corporis, totalis grauitas ſit <var>.a.b.</var> & poſitum in ipſo medio, amittat partem <var>.e.b.</var> ip-<lb/>ſius grauitatis, & in ſecundo medio amittat <var>.i.b.</var> & ſic 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 remanſuram ſuam totalem grauitatem <var>.a.b.</var> in quolibet <lb/>ex-dictis medijs. </s> + <s xml:space="preserve">Nunc ſi quærat à me Ariſtoteles proportionem velocitatis corpo-<lb/>ris <var>.Q.</var> per vacuum ad velocitatem dicti corporis per plenum, ego ei proponam pro-<lb/>portionem ipſius <var>.a.b.</var> ad <var>.a.e.</var> exempli gratia, dicens, <reg norm="quod" type="simple">ꝙ</reg> <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> <var>.a.b.</var> maius eſt <lb/>ip ſo <var>.a.e.</var> ſic etiam corpus <var>.Q.</var> velocius erit in vacuo, quàm in pleno, dicti autem ple-<lb/>ni denſitatem appellabimus <var>.e.b</var>. </s> + <s xml:space="preserve">Ariſtoteles dicet nunc, <reg norm="quod" type="simple">ꝙ</reg> aliud quoddam medium <lb/>in eadem proportione ſubtilius ipſo <var>.e.b.</var> deſumatur; </s> + <s xml:space="preserve">quemadmodum <var>.a.e.</var> minus eſt <lb/>ipſo <var>.a.b.</var> ſit ergo iſtud <var>.i.b.</var> in quo Ariſtoteles credit corpus Q. futurum tam velox ut <lb/>in vacuo, in quo aberrat, <reg norm="quia" type="simple">ꝗa</reg> proportio velocitatis corporis <var>.Q.</var> in medio <var>.i.b.</var> ad velo <lb/>citatem eiuſdem in medio <lb/><var>e.b.</var> ita ſe hàbebit, 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 ſuppoſito ca <lb/>pit .2. huius libr. quæ minor <lb/>eſſet ea, quæ eſt <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 ſatis dilucidè Ariſtotelem de loco ratiocinatum fuiße.</head> + <head xml:space="preserve">CAP. XX.</head> + <p> + <s xml:space="preserve">QVæ Ariſtoteles de loco ſcribit multas in ſe continent difficultates. </s> + <s xml:space="preserve">Primum, <lb/>cap .4. lib. 4. phyſicorum ait, omne corpus eſſe in ſuo proprio loco, ſupponen <lb/>do vnum centrum pro loco grauium, et unam circunferentiam pro loco leuium cor <lb/>porum. </s> + <s xml:space="preserve">Sed quomodo punctum poteſt eſſe locus ipſius corporis, cum omni dimen <lb/>ſione <reg norm="capacitateque" type="simple">capacitateq́;</reg> ſit denudatum? </s> + <s xml:space="preserve">vnde ſi <reg norm="centrum" type="context">centrũ</reg> locus eſſet corporum grauium, om <lb/>nia dicta corpora grauia, extra proprium locum exiſterent, quia nullum ex iis eſt, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſit in centro. </s> + <s xml:space="preserve">Adde quod neque hoc cum loci definitione ab ipſo poſita conſentiret <lb/>cum ipſe dicat in eodem cap. locum eſſe ſuperſiciem quandam, & non interuallum, <lb/>licet huiuſmodi definitio falſa appareat primo ex <reg norm="inconuenienti" type="context">incõuenienti</reg> falſo, quod ipſe hinc <lb/>ſequuturum dicit, ideſt, quod ſi locus interuallum eſſet, infinita loca exiſterent, quod <lb/>reuera nec ob hanc cauſam inconueniens exiſtit, quia eodem planè modo quo ali-<lb/>quod corpus poteſt eſſe infinita corpora, (quod ipſe diceret in potentia) ſic etiam in <lb/>teruallum aliquod poſſet eſſe infinita interualla. </s> + <s xml:space="preserve">Cum autem dicat ſuperficies cor-<lb/>poris ambientis eſſe locum eius corporis, quod continetur, cogitur dicere lineam, <lb/>quæ circundat ſuperficiem, ſuperficiei locum eſſe, & puncta ipſius lineæ, quod reue <lb/>ra abſurdum eſt. </s> + <s xml:space="preserve">Locus corporis eſt interuallum illud eadem magnitudine & figu-<lb/>ra, qua corpus ipſum pręditum eſt, quod ſi non eſſet, ſed eſſet ſuperficies, quemad-<lb/>modum Ariſtoteles voluit, maximum inconueniens ſequeretur, ſcilicet æquales lo-<lb/>cos capere inęqualia corpora, aut corpora æqualia, locos inęquales occupare, quod <lb/>ſcitu facillimum eſt, cum Theon ſuper Ptolomęi Almageſtum iam probarit ſphæ-<lb/>ricam ſuperficiem maius interuallum corporeum continere, quàm aliam <reg norm="quanuis" type="context">quãuis</reg> ſu-<lb/>perficiem dictæ ſphęricæ 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, quę tamen corpora apta eſſent <lb/>ad occupandos locos ęquales, quamuis Ariſtoteles dicat, locum, neque maiorem ne <lb/>que minorem eſſe debere locato. </s> + <s xml:space="preserve">Sed interualla corporea ęqualia à quauis figura <lb/>terminata, continebunt ſemper corpora ęqualia. </s> + <s xml:space="preserve">Corporeum igitur interuallum eſt + <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 adęquatus, cum corpus in interuallum ſuperſiciale non intret, <lb/>quam @is interuallum corporeum ingrediatur. </s> + <s xml:space="preserve">Et hoc modo <reg norm="nullum" type="context">nullũ</reg> eſt corpus, quod <lb/>in m@ do aut extra mundum ( dicat autem Ariſtoteles quicquid voluerit ) locum <lb/>ſuum non habeat.</s> + </p> + </div> + <div type="section" level="3" n="21"> + <head style="it" xml:space="preserve">V<unsure/>trum bene Aristoteles ſenſerit de infinito.</head> + <head xml:space="preserve">CAP. XXI.</head> + <p> + <s xml:space="preserve">TRactans Ariſtoteles in fine quinti cap. lib. 3. phyſicorum de infinito ait, impoſ<lb/>ſibile cum ſit inuenire locum infinitum, & omne corpus in loco cum ſit, impoſ <lb/>ſibile quoque eſſe in rerum natura aliquod: </s> + <s xml:space="preserve">infinitum corpus reperiri. </s> + <s xml:space="preserve">Omittamus <lb/>quòd cum Ariſtoteles debuerit beneficio loci deſtruere infinitum, ordine peruerſo <lb/>de infinito prius, quàm de loco diſputationem inſtituat; </s> + <s xml:space="preserve">ſed dicamus ipſum intelli-<lb/>gere de infinito corporeo, & cum probauerimus corporis locum eſſe corporeum in <lb/>teruallum, non autem ſuperficiem, neque opus ſit in definitione interualli mentio <lb/>nem aliquam facere terminorum, vnde ipſum infinitum eſſe poteſt, neque aliqua ra <lb/>tione de hac re dubitari poteſt; </s> + <s xml:space="preserve">hoc modo nullum inconueniens ſequeretur, quòd <lb/>extra cęlum reperiri poſſit corpus aliquod infinitum, quamuis, id ipſe nulla euiden-<lb/>ti ratione inductus perneget. </s> + <s xml:space="preserve">Senſit quoque, abſque eo, <reg norm="quod" type="simple">ꝙ</reg> aliquam rationem propo <lb/>nat, aliquid extra cœlum reperiri quemadmodum apparet ex fine cap .9. lib. primi <lb/>de cœlo, cum etiam ait cap .8. lib. 8. phyſicorum, infinitas partes alicuius continui eſ-<lb/>ſe ſolum in potentia, non item in actu, hoc non eſt illico concedendum, quia ſi omne <lb/>totum continuum, & re ipſa exiſtens, in actu eſt, omnis quoque eius pars erit in actu, <lb/>quia ſtultum eſſet credere, ea quæ actu ſunt, ex ijs, quæ potentia exiſtunt, componi. <lb/></s> + <s xml:space="preserve">Neque etiam dicendum eſt continuationem earundem partium efficere, vt poten-<lb/>tia ſint 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 eſt, quin ante <reg norm="diuiſionem" type="context">diuiſionẽ</reg>, <lb/>medietas <var>.a.e.</var> tam in actu (licet coniuncta cum alia <var>.e.u.</var>) reperiretur, quàm totum .2. <lb/>u. licet à ſenſu diſtincta non eſſet. </s> + <s xml:space="preserve">Idem affirmo de medietate <var>.a.e.</var> ideſt de quarta <lb/>parte totius <var>.a.u.</var> & pariter de octaua, de milleſima, & de quauis, ita vt eſſentia actua <lb/>lis infiniti hoc modo tutò concedi poſſit, <reg norm="cum" type="context">cũ</reg> ita ſit in natura. </s> + <s xml:space="preserve">Sed peius etiam ſenſit <lb/>Ariſtoteles eodem loco capitis quinti lib. 3. phyſicorum, 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 conſiderat eadem etiam ratione intelligi poſſe <lb/>aliquam quantitatem <reg norm="infinitorum" type="context">infinitorũ</reg> cubitorum, & in quantitatis definitione nullam eſ-<lb/>ſe neceſſitatem terminorum, vt exempli gratia in definitione numeri, non eſt neceſ <lb/>ſitas alicuius determinati numeri, quia multitudo, non minus infinita, quàm finita, <lb/>intelligi poteſt. </s> + <s xml:space="preserve">Vbi poſteà cap .8. libr .4. phyſicorum ait nullam eſſe differentiam <lb/>inter infinitum, & vacuum, reuera nihil abſurdius 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 Ariſtotele adductatemporis definitio.</head> + <head xml:space="preserve">CAP. XXII.</head> + <p> + <s xml:space="preserve">CVM ſenſerit Ariſtoteles <reg norm="tempus" type="context">tẽpus</reg> abſque motu eſſe <reg norm="non" type="context">nõ</reg> poſſe, ea tamen ab <reg norm="inui- cem" type="context">inui-<lb/>cẽ</reg> ſeparans, <reg norm="volens" type="context">volẽs</reg> definire <reg norm="tempus" type="context">tẽpus</reg> ait, <reg norm="ipsum" type="context">ipsũ</reg> eſſe <reg norm="motus" type="simple">motꝰ</reg> <reg norm="menſuram" type="context">menſurã</reg> <reg norm="numerumque" type="context simple">numerũq́;</reg>. </s> + <s xml:space="preserve">Quæ <lb/>quidem definitio, natura ſua non eſt bona, quia tempus, neque numerus eſt, neque <lb/><reg norm="etiam" type="context">etiã</reg> eſt <reg norm="menſura" type="context">mẽſura</reg> motus <reg norm="per" type="simple">ꝑ</reg> ſe, ſed <reg norm="tantum" type="wordlist">tm̃</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="accidens" type="context">accidẽs</reg>, quia nihil eſt, <reg norm="quod" type="wordlist">qđ</reg> numeret aut menſuret <lb/>aliud, quod non ſit <reg norm="eiuſdem" type="context">eiuſdẽ</reg> ſpeciei <reg norm="cum" type="context">cũ</reg> illo quod <reg norm="menſuratur" type="context">mẽſuratur</reg>, aut numero <reg norm="circunſcribitur" type="simple">circunſcribit̃</reg>, <lb/>vt <reg norm="exempli" type="context">exẽpli</reg> gratia, nulla <reg norm="vnquam" type="context">vnquã</reg> ſuperficies <reg norm="per" type="simple">ꝑ</reg> ſe numerabit aut <reg norm="menſurabit" type="context">mẽſurabit</reg> <reg norm="lineam" type="context">lineã</reg>, aut cor-<lb/>pus; </s> + <s xml:space="preserve">neclinea ſuperficiem <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 ſuperfi-<lb/>ciem; </s> + <s xml:space="preserve">Sed linea lineam menſurabit; </s> + <s xml:space="preserve">ſuperficies ſuperficiem; </s> + <s xml:space="preserve">& corpus corpus; </s> + <s xml:space="preserve"><reg norm="etianſi" type="context">etiãſi</reg> <lb/>tam vna ex iis quantitatibus quàm altera ſit continua. </s> + <s xml:space="preserve">Cum verò motus non ſit tem <lb/>pus, neque tempus ſit motus, ſed inter ſe maximè differant, ſequetur ex iis, alterum <lb/>nullo modo per ſe eſſe menſuram alterius, niſi per accidens. </s> + <s xml:space="preserve">Et ſi alicui videtur, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ad ſignificandam aliquam quantitatem motus, dicere huiuſmodi operationem dua-<lb/>rum horarum, aut duorum dierum, aut duorum annorum ſpatio completam eſſe, ſit <lb/>ponere tantum tempus: </s> + <s xml:space="preserve">animaduertere debet hoc ſimpliciter non eſſe verum, quia <lb/>horarum, dierum. </s> + <s xml:space="preserve">& annorum interualla, imaginatione <reg norm="concipiuntur" type="context">concipiũtur</reg> vt motus corpo-<lb/>rum cęleſtium, ſine quibus, neque anni, neque dies, neque horę exiſterent, <reg norm="etiam" type="context">etiã</reg> ſi om <lb/>nis motus ſit (vt ita dicam) locatus in tempore, ut corpus in loco, vnde motus motu, <lb/>& tempus tempore, non autem aliud ab alio menſuratur. </s> + <s xml:space="preserve">Tempus ex neceſſita-<lb/>te (phyloſophicè tamen loquendo) res eſt æterna, motus non item, quia diuerſis mo <lb/>dis terminari poteſt & ceſſare, & interim dum ceſſabit quieſcet corpus, quod <lb/>primo mouebatur. </s> + <s xml:space="preserve">nihilominus tamen, tempus continuabit curſum ſuum. </s> + <s xml:space="preserve">Tempus <lb/>igitur potius locus motus erit dicendum, quàm numerus aut menſura eius, & tale eſt, <lb/>vt conſumatum uideatur à continuò quodam fluxu vnius inſtantis, quemadmodum <lb/>iam dixi in .38. capite meę gnomonicæ, & cum dico ab vno inſtanti, vnum in ſpecie, <lb/>& non in numero intelligo, quod à ſenſibus noſtris percipi non poteſt, <reg norm="neque" type="simple">neq;</reg> etiam <lb/>notari, quia nouum ſemper inſtans nobis occurrit. </s> + <s xml:space="preserve">& ſi aliquis aliquod <reg norm="exemplum" type="context">exemplũ</reg> (lar <lb/>go modo) incompræhenſibilitatis ipſius inſtantis deſideraret, imaginetur rotam ali <lb/>quam albam, in qua ſit nigrum aliquod punctum ſenſibile, aut è contra rotam <reg norm="nigram" type="context">nigrã</reg> <lb/>imaginetur, in qua ſit punctum album, quæ rota velociſſimè moueatur; </s> + <s xml:space="preserve">huiuſmodi <lb/>punctum, nullo modo aſſignari poterit, magis ab una parte quàm ab altera; </s> + <s xml:space="preserve">immo ſe <lb/>ſe nobis offeret ſemper in forma lineæ circularis. </s> + <s xml:space="preserve">poſſumus aliquo modo etiam ſu-<lb/>mere exemplum à ſono, quia omnis chorda cuiuſlibet inſtrumenti muſici, dum ſo-<lb/>nus editur, tremit, unde huiuſmodi ſonus, appellari poteſt aggregatum aliquod ex <lb/>innumerabilibus ſonis. </s> + <s xml:space="preserve">eodem modo ſe habet ſonus, quem ędunt campanę, & omnia <lb/>inſtrumenta tam naturalia, quàm artificialia, quæ quantò velocius <reg norm="tremunt" type="context">tremũt</reg>, tanto acu <lb/><reg norm="tiorem" type="context">tiorẽ</reg> generant ſonum, & quantò tardius, tantò grauiorem. </s> + <s xml:space="preserve">Neque eſt quòd in ad-<lb/>mirationem ducamur, quòd ſenſui unum aliquod continuum appareat id, quod di-<lb/>ſcretorum eſt multitudo ( non putet tamen aliquis me negare continuitatem ſucceſ <lb/>ſiuam ipſius temporis) quod clare cognoſci poteſt à niue, aut à chryſtallo, 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 inſpiciendum offerunt, quod nihil aliud eſt, quàm innumerabilis quædam <lb/>multitudo minutorum reflexorum. </s> + <s xml:space="preserve"><reg norm="Idem" type="context">Idẽ</reg> dico de ſputo, & qualibet ſpuma, & 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 ſunt corpuſcula à quibus vt à ſpeculis reflectitur lumen, tantò magis ag <lb/><reg norm="gregatum" type="context">gregatũ</reg> illud <reg norm="album" type="context">albũ</reg> apparet. </s> + <s xml:space="preserve">Hæc <reg norm="autem" type="context">autẽ</reg> exempla <reg norm="cum" type="context">cũ</reg> ſint, nec non largo modo ſumpta, <lb/><reg norm="mirum" type="context">mirũ</reg> non erit ſi claudicare <reg norm="videbuntur" type="context simple">videbũt̃</reg>. </s> + <s xml:space="preserve">Sed ut ad <reg norm="motum" type="context">motũ</reg>, & <reg norm="tempus" type="context">tẽpus</reg> reuertamur ( quæ ſunt <lb/><reg norm="continua" type="context">cõtinua</reg> ſucceſſiua) Ariſtoteles in definiendo tempore, non reduxit in mentem, quod <lb/>ſcribit decimo metaphyſicę et .4. cap. ſecundo. libr. de cęlo, omnia videlicet, ab eo, <lb/>quod minimum eſt in ſuo genere, menſurari, & ex ſeipſo in phyſicorum libris, tem-<lb/>pus non eſt de genere motus; </s> + <s xml:space="preserve">ergo eius ipſius rationum ui, tempus non erit menſura <lb/>motus, ſed motus quidem poteſt menſurare motum, videlicet velocior minus velo-<lb/>cem, & breuior longiorem; </s> + <s xml:space="preserve">& <reg norm="numerus" type="simple">numerꝰ</reg> <reg norm="menſuratur" type="simple">menſurat̃</reg> numero, & tempus tempore in quan <lb/>tum longum eſt, aut breue, non in quantum velox, aut tardum; </s> + <s xml:space="preserve">Nullum autem in-<lb/>conueniens ſequetur ſumendo tempus tam ptoportionale motui, quam locus cor-<lb/>pori, quia motus decem milliarium, quæ aliquis vnius horæ ſpatio conficiat, erit pro <lb/>portionalis corpori denſo, & motus vnius milliaris eadem hora peracti, proportiona <lb/>lis erit corpori raro; </s> + <s xml:space="preserve">& quemadmodum corpus denſum occupat minus interuallum <lb/>loci, contra quam fiat in corpore raro: </s> + <s xml:space="preserve">ſic etiam motus velox breuiori temporis ſpa-<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/>Ariſtotele.</head> + <head xml:space="preserve">CAP. XXIII.</head> + <p> + <s xml:space="preserve">ARiſtoteles .8. capi .8. phyſicorum ait impoſſibile eſſe aliquid per <reg norm="lineam" type="context">lineã</reg> rectam <lb/>nunc vno modo, nunc altero, ideſt eundo, & redeundo per dictam lineam in <lb/>extremis abſque quiete moueri. </s> + <s xml:space="preserve">Id quod contrà poſſibile eſſe dico. </s> + <s xml:space="preserve">Pro ſpecula-<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="ſiniſtram" type="context">ſiniſtrã</reg> ferri; </s> + <s xml:space="preserve">& imaginemur <reg norm="punctum" type="context">pũctum</reg> <var>.b.</var> extra ipſum, ubi <lb/>magis nobis videbitur, à quo ducantur duæ lineæ recte <var>.b.u.</var> et <var>.b.n.</var> contiguæ ipſi 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 ſit <var>.<lb/>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> conſtituamus in quali <lb/> + <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> + bet parte, ſumemus etiam punctum <var>.a.</var> circun-<lb/>ferentiæ dicti circuli, à quo vſque ad <var>.b.</var> lineam <var>.<lb/>b.a.</var> imaginemur <reg norm="fixam" type="context">fixã</reg> in <var>.b.</var> ſed quod remanat mo <lb/>bile, ſecundum 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> verſus <var>.b.<lb/>n.</var> proficiſcetur, & aliquando ab <var>.b.n.</var> verſus <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 quieſcet, quia in inſtanti reuertetur, quia <lb/><var>b.u.</var> et <var>.b.n.</var> in puncto, <reg norm="dictum" type="context">dictũ</reg> circulum tangunt, & <lb/>dicta <var>.b.a.</var> interſecabit ſemper aliquam ex dictis <lb/><var>u.n.</var> aut <var>.c.d.</var> aut <var>.e.f.</var> aut <var>.g.h.</var> quod interſectionis <lb/>punctum ſit <var>.t</var>. </s> + <s xml:space="preserve">Imaginemur nunc quod <reg norm="ſecundum" type="context context">ſecũdũ</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 quieſcet, etiam ſi ſit in quouis ex <lb/>tremo. </s> + <s xml:space="preserve">Ariſtotelis igitur opinio, tuta non eſt.</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 grauisſimus an bene ſenſerit de motibus corporum <lb/>uiolentis & natur alibus.</head> + <head xml:space="preserve">CAP. XXIIII.</head> + <p> + <s xml:space="preserve">ARiſtoteles in fine .8. phyſicorum ſentit corpus per vim motum, & ſeparatum à <lb/>primo mouente, moueri, aut motum eſſe per aliquod tempus ab aere, aut ab <lb/>aqua, quæ ipſum <reg norm="ſequuntur" type="context">ſequũtur</reg>. quod fieri non poteſt; </s> + <s xml:space="preserve">quia imo aer, qui in locum defer-<lb/>tum à corpore ſubintrat ad fugandum vacuum, non ſolum hoc corpus non impellit, <lb/>ſed potius id cohibet à motu, quia aer per vim à corpore ducitur retrò, & diuiſus à <lb/>parte anteriori à dicto corpore, reſiſtit ſimiliter, & quantum dictus aer in dicta parte <lb/>condenſatur, tantum in poſteriori rarefit, vnde per vim ſeſe 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/>ipſum ab aere rapitur. </s> + <s xml:space="preserve">Huiuſmodi autem rarefactio aeris, naturalis non eſt, ſed vio <lb/>lenta; </s> + <s xml:space="preserve">& hanc ob cauſam reſiſtit, & ad ſe trahit, ſed non ſufferente natura, vt inter <reg norm="vnum" type="context">vnũ</reg> <lb/>& aliud ex dictis corporibus reperiatur vacuum; </s> + <s xml:space="preserve">iccirco funt hæc ſemper contigua, <lb/>& mobile corpus aerem deſerere cum nequeat, eius velocitas impeditur. </s> + <s xml:space="preserve">Huiuſmo <lb/>di igitur corporis ſeparatim à primo mouente velo citas oritur à quadam naturali im <lb/>pręſſione, ex impetuofitate recepta à dicto mobili, quæ impręsſio & impetuoſitas, <lb/>in motibus rectis naturalibus continuò creſcit, cum perpetuò inſe cauſam <reg norm="mouentem" type="context context">mouẽtẽ</reg>, <lb/>ideſt propenſionem eundi ad locum ei à natura asſignatum habeat. </s> + <s xml:space="preserve">Ariſto .8. cap. <lb/>primi lib. de cœlo, dicere non deberet <reg norm="qui" type="simple">ꝗ</reg> quantò propius accedit corpus ad <reg norm="terminum" type="context">terminũ</reg> <lb/>ad quem, tantò magis ſit velox; </s> + <s xml:space="preserve">ſed potius, <reg norm="qui" type="simple">ꝗ</reg> quantò longius diſtat à termino à quò <lb/>tantò velocius exiſtit. </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 eſt inclinatio ad locum ſuum eundi, extra quem per vim confiftit. <lb/></s> + <s xml:space="preserve">Neque etiam rectè ſeripſit Ariſto .9. cap. lib. 8. phyficorum et .2. lib. primi de cœlo <lb/>eſſe aliquem motum ex recto & circulari mixtum, <reg norm="qui" type="simple">ꝗ</reg> omninò imposſibile eſt.</s> + </p> + </div> + <div type="section" level="3" n="25"> + <head style="it" xml:space="preserve">Motum rectum & natur alem non eſſe primo & per ſe <lb/>quicquid Ariſtoteli uiſum ſit.</head> + <head xml:space="preserve">CAP. XXV.</head> + <p> + <s xml:space="preserve">MOtus rectus corporum naturalium ſurſum, aut dcorfum, non eft naturalis pri <lb/>mò & per ſe, quia motus naturalis perpetuus eſt, aut vt melius dicam, inceſ-<lb/>fabilis, & alius eſſe non poteft quàm circularis, <reg norm="nullaque" type="simple">nullaq́;</reg> pars cum fuo toto coniun-<lb/>cta, alium motum naturalem habere poteft, quàm eum, qui eft totius. </s> + <s xml:space="preserve">fi autem à ſuo <lb/>toto diuulfa atque difiuncta fit, <reg norm="libereque" type="simple">libereq́;</reg> vagetur, ſpontè, & quàm breuisſima poteft <lb/>via, ad locum, <reg norm="ſuitotius" type="simple">ſuitotiꝰ</reg> à natura ſtatutum proficiſcitur. </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 ſuum locum, vbi contra naturam ſuam reperitur. </s> + <s xml:space="preserve">Vnde hu-<lb/>iuſmodi motus, partim & non omninò, naturalis eft. </s> + <s xml:space="preserve">Is autem proprius eſt & natura <lb/>lis motus, qui dicti corporis eſſentiam conſeruat. </s> + <s xml:space="preserve">hoc autem non præſtat hic rectus, <lb/>cum deſtruat, ergò hic motus primò & per ſe 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 cęlo fic ſcribit.</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 ſe caufam nec grauitatis, nec leuitatis corporum <lb/>naturalium nofce, quæ eft denfitas auto raritas corporis grauis, aut leuis, maior denſi-<lb/>tate, aut raritate medij permeabilis, in quo reperitur.</s> + </p> + <p> + <s xml:space="preserve">Exemplum <reg norm="qui" type="simple">ꝗ</reg> ipſe de vtre inflato proponit, debuiſſet ſaltem ei oculos ad verita-<lb/>tem, quæ clarisſimè fulget, inſpiciendum aperire. </s> + <s xml:space="preserve">Verisſimum eſt, vtrem inflatum <lb/>plus ponderis habere quàm 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, quàm ſi eidem liberè vagari permit-<lb/>teretur, vnde violenter, quodam modo, con denfata eft, & quia corpus denfum in <lb/>minus denfo, femper deſcendit, & minus denſum in magis denſo aſcendit. </s> + <s xml:space="preserve">Hanc ob <lb/>caufam vter inflatus plenus corpore magis denſo, quàm eft medium quod eum cir-<lb/>cundat, deſcendit, 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, & ſpbæra.</head> + <head xml:space="preserve">CAP. XXVII.</head> + <p> + <s xml:space="preserve">CVm Ariftoteles fenſerit circulum eſſe figurarum ſuperficialium <reg norm="primam" type="context">primã</reg>, & <reg norm="ſphae ram" type="simple context">ſphę<lb/>rã</reg> eſſe <reg norm="primam" type="context">primã</reg> <reg norm="corporearum" type="context">corporearũ</reg> <reg norm="prope" type="wordlist">ꝑꝑ</reg> <reg norm="earum" type="context">earũ</reg> periferias, decipitur. </s> + <s xml:space="preserve">Sunt enim vltimæ, <lb/>non primæ. </s> + <s xml:space="preserve">Sunt quidem (in quò rectè ſentit) perfectè, licet rationem huius rei non <lb/>nouerit. </s> + <s xml:space="preserve">Nam <reg norm="centrum" type="context">centrũ</reg> cuiuſlibet rei, eiuſdem rei <reg norm="principium" type="context">principiũ</reg> eft, & eę figurę, quæ ipſum <lb/>æqualiter circundant, poſſunt appellari perſectæ, ſiue ſint ſuperficiales, ſiue corpo-<lb/>reæ, & ècontrà illæ, quæ contrario modo ſe habent, imperfectæ. </s> + <s xml:space="preserve">Quòd autem per-<lb/>ſectum eſt, licet natura fit primum, eſt tamen vltimum generatione. </s> + <s xml:space="preserve">Sed quando <lb/>Ariftoteles duas dictas figuras pronuntiauit primas, vt perfectas, prioritate ſcilicet <lb/>ea, quæ oritur à perfectione, verum dixit; </s> + <s xml:space="preserve">fed quando de figuris ſuperficialibus 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 ſeu elipſi, 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/>ſtrat, quia aliæ figuræ, à lineis curuis terminatę, eandem conditionem fortiuntur. <lb/></s> + <s xml:space="preserve">Circulus ſphę<reg norm="raque" type="simple">raq́;</reg>, non ex vno ſolo angulo recto conſtant, 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 cęlo, etiam fi triangulus ex duobus angulis rectis conſurgat, ſed ſunt <lb/>figurę infinitorum angulorum rectorum, & hanc ob cauſam à me dicuntur vltimæ & <lb/>perfectę, quia infinito nihil addi poteſt. </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">folidorũ</reg> <lb/>ſphęræ, minor eft quadruplo infinito per .4. infinita angulorum ſolidorum <reg norm="contingen- tiæ" type="context">cõtingen-<lb/>tiæ</reg>, quæ .4. infinita, minora ſunt quouis angulo ſolido acuto terminato à tribus pla-<lb/>nis. </s> + <s xml:space="preserve">Triangulus inter figuras planas ſuperſiciales eft primus, & circulus vltimus; </s> + <s xml:space="preserve">& <lb/>pyramis quadrilatera, inter corpora eft prima, & ſphęra vltima.</s> + </p> + </div> + <div type="section" level="3" n="28"> + <head style="it" xml:space="preserve">Occultam fuiße grauisſimo Stagirit & canſam ſcintilla-<lb/>tionis ſtellarum.</head> + <head xml:space="preserve">CAP. XXVIII.</head> + <p> + <s xml:space="preserve">VBi Ariſtoteles ait ſcintillationem ſtellarum ſieriratione aſpectus @oſtri ob, ma <lb/>ximam diſtantiam, maximum errorem committit, vt etiam facid quum putat <lb/>vifionem fieri extramittendo, contra id, quod alio loco, immo contra veritatem ip <lb/>ſam afferuit. </s> + <s xml:space="preserve">Scintillatio ergo ſtellarum, neque aſpectus noſtri ratione, neque ali-<lb/>cuius mutationis earundem ſtellarum, ſed ab inæqualitate motus corporum diapha <lb/>norum mediorum naſcitur, <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg> clarè cernitur, quòd fi inter aliquod obie <lb/>ctum, & nos, aliquis ſumus, qui aſcendat, intercefferit, videbimus obiectum illud qua <lb/>ſi tremere. </s> + <s xml:space="preserve">Hoc autem tantò magis fiet, quantò magis diſtabit obiectum ab ipſo fu <lb/>mo; </s> + <s xml:space="preserve">vnde admirationi locus non erit, fi ftellas fixas magis ſcintillare, quam errantes <lb/>cernamus. </s> + <s xml:space="preserve">Lumen ſtellæ ad oculum noſtrum 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, quàm in propinquis ſtel <lb/>lis apparet, quemadmodum ab exemplo de fumo allato, & etiam ab aliquibus vi-<lb/>tris ex ſuperficie non plana, ſed irregulari conſtantibus, quilibet cognoſcere poteft.</s> + </p> + </div> + <div type="section" level="3" n="29"> + <head style="it" xml:space="preserve">Daricontinuum infinitum motum ſuper rectam at que <lb/>finitam lineam.</head> + <head xml:space="preserve">CAP. XXIX.</head> + <p> + <s xml:space="preserve">OMnes hactenus ſenſerunt imposfibile eſſe dari per <reg norm="imaginationem" type="context">imaginationẽ</reg> motum con-<lb/>tinuum & <reg norm="perpetuum" type="context">perpetuũ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> + ſuper 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">quarũ</reg> <lb/><var>b.a.</var> fit <reg norm="infinita" type="context">ĩfinita</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 ſolis calorem à motu localι ipſius corporis ſolaris, <lb/>ut Ariſtoteli placuit.</head> + <head xml:space="preserve">CAP. XXX.</head> + <p> + <s xml:space="preserve">JD nullo planè modo eſt admittendum quod Ariftoteles credidit calorem folis à <lb/>motu locali ipſiuſmet 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, ſatis eſt eſſicere, & c. ſed <lb/>cap .7. lib. 2. de cælo fic ſeribit, Caliditas autem ab ipſis, <reg norm="lumenque" type="simple">lumenq́;</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 cœleſtium fit caufa ca <lb/>loris, ſed eriam luminis, paulò autem poſt dicit, ſuperiorum autem corporum vnum <lb/>quodque fertur in ſphæra, vt ipſa quidem non igniantur. </s> + <s xml:space="preserve">Opinio profecto abfur-<lb/>da. </s> + <s xml:space="preserve">Nam cùm corpus ſolate fixum fit in ſpisfitudine ſui orbis deferentis, fe-<lb/>cundum communem opinionem, non mouetur per fe, ſed accidentaliter, cum ſei-<lb/>licet fertur à dicto ſuo orbe, vnde fieri poteſt, 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/>ſam, rationi <reg norm="confentaneum" type="context">cõfentaneum</reg> non eſſet credere, quòd tantùm caloris gigneretur. </s> + <s xml:space="preserve">Quod <lb/>tamen fi posſibile eſſet, quid ergo fricatio ſuperficierum orbis ſui, cum iis, quæ funt <lb/>deferentium augis efficeret? </s> + <s xml:space="preserve"><reg norm="Quando" type="context">Quãdo</reg> tamen hoc fieret, vt ſcilicet 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">Mercuriusq́;</reg>, magis calefacere quam fol de <reg norm="berent" type="context">berẽt</reg>, <lb/>cum ita ſint veloces vt ipſe Sol, & eodem magis propinqua terræ. </s> + <s xml:space="preserve">Verum Ari-<lb/>ſtotelis <reg norm="temporibus" type="context">tẽporibus</reg>, <reg norm="nullum" type="context">nullũ</reg> <reg norm="alium" type="context">aliũ</reg> <reg norm="planetam" type="context">planetã</reg> quam folem <reg norm="putabant" type="context">putabãt</reg> philofophi ſupra <reg norm="Lunam" type="context">Lunã</reg> eſ-<lb/>ſe. </s> + <s xml:space="preserve">Atque etiam <reg norm="contigeret" type="context">cõtigeret</reg> menfe Decembri, quam Iunio, magis inualeſceret calor, <lb/>cum huiuſmodi menſe ſolad nos propius accedat, quam menfe Iunii. per differen-<lb/>tiam maiorem diametro regionis elementaris, (nam folaris eccentricitas maior eft <lb/>ſemidiametro <reg norm="elementaris" type="context">elemẽtaris</reg> regionis) non confiderans Ariftoteles differentiam ca-<lb/>loris, quæ naſcitur ex eo, <reg norm="qui" type="simple">ꝗ</reg> Sol aut altius ſupra orizontem excurrat, aut infra <reg norm="eundem" type="context">eundẽ</reg> <lb/>feratur; </s> + <s xml:space="preserve">neque eam, quę à longitudine, aut breuitate diei proficiſcitur. </s> + <s xml:space="preserve">Sed quia Ari <lb/>ſtoteles eodem cap tertio Metheororum intelligit de motu rapto, ideſt diurno, ſiue <lb/>dicamus vniuerfali, hinc ſequi deberet, <reg norm="qui" type="simple">ꝗ</reg> Sol maiorem caloris uim menſe Martij & <lb/>Septembris, quàm aliis menfibus, profunderet, quia in iiſdem temporibus, ſol virtu <lb/>te huiuſmodi motus velocior exiftat, quàm alio quolibet tempore anni, cum tunc <lb/>per æquatorem circuũoluatur. </s> + <s xml:space="preserve">Multa quoque alia incommoda ſequerentur ſi Ari <lb/>ſtorelis rationes admitteremus. </s> + <s xml:space="preserve">Sed clarè uidemus, mediante refl exione aut refra-<lb/>ctione radiorum folarium, <reg norm="qui" type="simple">ꝗ</reg> vniente ſeſe 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 ipſius ſolis, & ita in lumine ſedem habet, ut fi ſol <lb/>quieſceret, neque in orbe ſuo circumager etur, infęliciſſima eſſet ea regio, in cuiu; <lb/></s> + <s xml:space="preserve">Zenith ipſe reperiretur.</s> + </p> + </div> + <div type="section" level="3" n="31"> + <head style="it" xml:space="preserve">Vnde caloris ſolis prode at incrementum & state, et byeme <lb/>decrementum.</head> + <head xml:space="preserve">CAP. XXXI.</head> + <p> + <s xml:space="preserve">CVm capite ſuperiore oſtenderim calorem ſolis non aliunde, quàm à lumine <lb/>prouenire, oſtendam nunc ex ordine, ex quot, <reg norm="quibusque" type="simple">quibusq́;</reg> cauſis oriatur magna <lb/>differentia eius caloris æſtatis ad hyemem, quarum nonnullæ ab antiquis obſerua-<lb/>tæ fuerunt, aliæ autem à nemine, quod ſciam. </s> + <s xml:space="preserve">Sunt autem quinque ad minus eæ cau <lb/>ſæ, quarum vna eft diuturna folis mora, tempore æſtatis ſupra orizontem, quæ cau-<lb/>ſa ab antiquis pofita, & citata fuit. </s> + <s xml:space="preserve">Aliam quoque huius rei cauſam iidem antiqui <lb/>dicebant eſſe propinquitatem ſolis noftro Zenith, ſed hæc cauſa immediata non <lb/>eſt, quia ab ea tres caufæ immediatæ naſcuntur; </s> + <s xml:space="preserve">quarum vna eft maior unio radij re <lb/>flexi cum radio incidenti; </s> + <s xml:space="preserve">ſecunda maior quantitas luminis in ſuperficie terrę; <lb/></s> + <s xml:space="preserve">tertia, minor <reg norm="reſiſtentia" type="context">reſiſtẽtia</reg> vaporum ipſi radio luminoſo 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 reſlexi cum incidente, quiſque, uel ſaltem mediocriter <lb/>in cathoptricę cognitione verſatus hoc verum eſſe cognoſcet. </s> + <s xml:space="preserve">Vt hoc tamen in-<lb/>noteſcat facilius. </s> + <s xml:space="preserve">Imaginemur <var>.q.p.</var> et <var>.b.d.</var> eſſe duas particulas ęquales ſuperficiei <lb/>ipfius terræ, ſuper quas cadant duo radii luminofi ſolis <var>.e.q.</var> et <var>.n.d.</var> quorum <var>.e.q.</var> fit ad <lb/>modum obliquus, et <var>.n.d.</var> quaſi perpendicularis, vnde radii reſlexi <var>.p.a.</var> et <var>.b.u.</var> aſcen <lb/>dent cum angulis æqualibus eis, qui funt ſuorum cadentium, cum omnis angulus re-<lb/>flexionis femper æqualis ſit angulo ſuæ incidentię, 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 ſint angulis <var>.b.</var> et <var>.d.</var> trianguli <var>.b.i.d.</var> ex ſuppoſito. </s> + <s xml:space="preserve">Quòd uero attinet ad ma <lb/>iorem quantitatem luminis ſuper terræ ſuperſiciem; </s> + <s xml:space="preserve">Imaginemur radium <var>.a.q.</var> cuius <lb/>reſpectu etiam imagine mur duos ſuperficiei terræ ſitus, 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 eſt ex ſuppoſito, unde <var>.q.o.</var> <lb/>minor erit <var>.q.p.</var> ex .18. primi Euclidis. </s> + <s xml:space="preserve">hinc fit, vt ſuper <var>.q.o.</var> cadat vniuerſum lumen, <lb/>quod ſuper <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 ſuperficies <var>.q.u.</var> ipſius terræ, quàm.q.o. quia maius lumen in ſe maio-<lb/>rem calorem includit: </s> + <s xml:space="preserve">quod manifeſtè apparet in radiorum vnione mediante refle-<lb/>xione, aut refractione. </s> + <s xml:space="preserve">Sed quod attinet ad minorem refiſtentiam vaporum ad ip-<lb/>ſum radium luminoſum, etfi primo capite meæ Gnomonicæ leuiter id attigerim, ni <lb/>hilominus tamen, & idem ipſum hoc loco proponam. </s> + <s xml:space="preserve">Denotetur, exempli gratia, <lb/>ſuperficies terræ ab <var>.o.g.</var> et ea, quæ eft vaporum ab <var>.n.a.</var> ſupponatur etiam ſol 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 ſit Zenith <reg norm="puncti" type="context">pũcti</reg> <var>.o.</var> & <reg norm="etiam" type="context">etiã</reg> in <var>.p.</var> ipſi <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 ipſis vaporibus offuſcatæ, ſed <var>.o.n.</var> breuior eſt <var>.o.a.</var> ex .7. lib. 3. Eucli. </s> + <s xml:space="preserve">mino-<lb/>rem ergo reſiſtentiam habebit à vaporibus ſol in Zenith, quàm extra cundem com -<lb/>morans, & quantò longius erit idem ab ipſo Zenith, tanto maiorem reſiſtentiam 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 ſenſus expers à ſono offendi, præterquam <lb/>Aristoteles crediderit.</head> + <head xml:space="preserve">CAP. XXXII.</head> + <p> + <s xml:space="preserve">POſſe ſonum corpus aliquod, quod ſenſu ſit deſtitutum, vt Ariſtoteles .9. cap. li-<lb/>br .2. de cælo putauit, offendere, eſt falſum.</s> + </p> + <p> + <s xml:space="preserve">Corpus enim non niſi à corpore poteſt lædi, non ergo à ſono, cum ſonus corpus <lb/>non ſit. </s> + <s xml:space="preserve">Sed aer, & ignis, cum è contra ſint corpora, hoc facilè præſtare poſſunt im-<lb/>plendo aliquem locum velociter ad excludendum vacuum; </s> + <s xml:space="preserve">vnde generatur ſonus. <lb/></s> + <s xml:space="preserve">Quod hucuſque à nemine animaduerſum 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 ſonitu corporum cælestium non <lb/>fuiſſe ab Aristotele ſublatam.</head> + <head xml:space="preserve">CAP. XXXIII.</head> + <p> + <s xml:space="preserve">SEnſerunt Pythagorici orbes cæleſtes dum circunuoluuntur, non autem corpora <lb/>ſtellarum ſolum, æd ere ſonu. </s> + <s xml:space="preserve">Quibus dum Ariſtoteles contradicere cogitat, <lb/>maximè fauet. </s> + <s xml:space="preserve">Eatamen opinio è phyloſophorum ſcholis eſt explodenda, quia aut <lb/>orbes ſunt ſibi ipſis contigui, aut inuicem diſtantes: </s> + <s xml:space="preserve">ſi ab inuicem diſtant (quod <lb/>nemo adhuc conceſſit, quia hac ratione vacuum introduceretur) clarum eſt, quod <lb/>cum ſe minime tangant, ſonum edere nequeunt: </s> + <s xml:space="preserve">Si verò eorum vnus alteri ſit conti <lb/>guus, <reg norm="neque" type="simple">neq;</reg> etiam ab ipſis ſonus reſultare poterit, quia pro certo putandum eſt, <reg norm="ipſorum" type="context">ipſorũ</reg> <lb/>ſuperficies tam politas eſſe, ac lenas, vt nihil omnino aſperitatis, aut inæqualitatis <lb/>contineant. </s> + <s xml:space="preserve">Vt exempli gratia, ſi aliquis duo ſpecula plana inuicem confricaret, nul <lb/>lum planè ſonum audiret, ſed ſi hoc faceret cum duabus ſuperficiebus a ſperis, <reg norm="ſonum" type="context">ſonũ</reg> <lb/>perſentiret, & tanto maiorem; </s> + <s xml:space="preserve">quantò aſperiores eſſent dictæ ſuperficies, & qui vult <lb/>vtarcus lirę, ex corda ſonum eliciat, colophonia dictum arcum illinet, vt aſperiorem <lb/>reddat. </s> + <s xml:space="preserve">Neceſſarium quoque eſt vt tremat ſiue trepidet corpus, quod <reg norm="ſonum" type="context">ſonũ</reg> edere <lb/>debet; </s> + <s xml:space="preserve">Neque etiam abſque aere ſonus efficipotelt, quia aer ſonat ingrediendo <lb/>velociter ad implendum locum, vt non remaneat vacuus. </s> + <s xml:space="preserve">Sed ſupponendo in æche <lb/>rea regione neque aerem eſſe, neque corpus aliquod fluidum, clarè patebit orbes <lb/>cœleſtes ex ſeſe nullum emittere ſonum. </s> + <s xml:space="preserve">Idem affirmo de fricatione ſuperficiei con <lb/>cauæ infimi orbis lunaris cum conuexa materiæ à dicto orbe contentæ, ſuperioribus <lb/>rationibus fultus, vt etiam experientia à corpore aliquo fluido, quod in alio velociſ <lb/>ſimè moueretur deſumpta fretus, cuius corporis ſuperficies tamen lenis eſſet, à quo <lb/>ſonus 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 ſuum axem velociſſimè moueatur, nullum ſo-<lb/>num efficiet, quia nullam aeris partem extra ſuum <reg norm="locum" type="context">locũ</reg> impellit dum mouetur non <lb/>ſecundum totum, ſed ſecundum ſuas partes, quarum quælibet abſque reſiſtentia im-<lb/>mediatè ſubintrat locum alterius, abſque temporis interpoſitione. </s> + <s xml:space="preserve">nec huiuſmodi <lb/>locum aliquo modo eadem materia dicti corporis, quod circunuoluitur: </s> + <s xml:space="preserve">deſtitutum <lb/>dimittat. </s> + <s xml:space="preserve">Sed ſi Pythagorici de alia quadam harmoniæ ſpecie ab ea, quæ eſt ſono-<lb/>rum, vt à diuerſis velocitatibus motuum, aut à diuerſis magnitudinibus aut diſtantiis, <lb/>aut ſtellarum influxibus intellexiſſent, rectè ſenſiſſent exparte, non autem omnino, <lb/>quia ea harmoniam efficere nequeunt, quæ ad <reg norm="inuicem" type="context">inuicẽ</reg> ſecundum interualla harmoni-<lb/>ca proportionata non ſunt, vt ſunt dupla, ſeſquial tera, ſeſquitertia, ſeſquiquarta, ſeſ-<lb/>quiquinta, ſupertripartientia quintas, <reg norm="ſuperbipartientia" type="context">ſuperbipartiẽtia</reg> tertias, & quę ab ijs <reg norm="dependent" type="context">dependẽt</reg> <lb/>ideſt coniuncta ſunt cum duplis; </s> + <s xml:space="preserve">de conſonantijs loquendo. de diſſonantiis idem di <lb/>co, quæ harmonicis inſeruiunt modulationibus, vt <reg norm="ſeſquioctauum" type="context">ſeſquioctauũ</reg>, <reg norm="ſeſquinonum" type="context">ſeſquinonũ</reg>, ſeſqui <lb/><reg norm="quintundecimum" type="context context">quintũdecimũ</reg>, <reg norm="ſequiuigeſimunquartum" type="context context">ſequiuigeſimũquartũ</reg>, <reg norm="ſeſquioctuogeſimum" type="context">ſeſquioctuogeſimũ</reg>, & ſuperbipartiens vigeſi <lb/>masquintas. </s> + <s xml:space="preserve"><reg norm="Verum" type="context">Verũ</reg> quidem eſt nonnulla harmonica interualla in aſpectibus <reg norm="comperta" type="context">cõperta</reg> <lb/>fuiſſe, vt Prolomeus oſtendit, & alii quoque aſſerunt. </s> + <s xml:space="preserve">ineſt tamen huic rei nonnihil <lb/>difficultatis. </s> + <s xml:space="preserve">vt exempli gratia, ſi ſubtrahamus diateſſaron extra diapaſon, remanet <lb/>diapente, & ſi à diapente ſubtrahamus ſemiditonum, 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"/> + conſonantiæ, eum habent reſpectum ad inuicem, quem habent diateſſaron, & dia <lb/>pente, quia quemadmodum ſemiditonum, & ditonum ſimul coniuncta, compo <lb/>nunt diapente, ſic diateſſaron, & diapente ſimul vnita componunt diapaſon; </s> + <s xml:space="preserve">& <reg norm="quem" type="context">quẽ</reg> <lb/>admodum terminus, qui diuidit diapaſon in <reg norm="diateſſaron" type="context">diateſſarõ</reg>, & diapente, eſt mediator ha@ <lb/>monicus inter extrema diapaſon diuiſi, ſic etiam terminus, qui diuidit <reg norm="diapente" type="context">diapẽte</reg> in ſe-<lb/>miditonum, & ditonum, mediator eſt harmonicus inter extrema ipſius diapente diui <lb/>ſi) ſubtrahendo deinde à diapaſon ſemiditonum remanet exachordum maius, & ab <lb/>eodem diapaſon ſubtrahendo ditonum remanet exachordum minus, quę <reg norm="quidem" type="context">quidẽ</reg> <reg norm="non" type="context">nõ</reg> <lb/>accidunt aſpectuum circulo, quia ſubtrahendo aſpectum quadratum ab oppoſito, <lb/>remanet aliud quadratum, & ſubtrahendo ſextilem à trino remanet quoque alius <lb/>ſextilis. </s> + <s xml:space="preserve">Quòd autem attinet ad motus, ad magnitudines, ad diſtantias, & ad influ-<lb/>xus, nihil eſt, quod hiſce proportionibus conueniat, ſed quia hæc omnia <reg norm="dependent" type="context">depẽdent</reg> <lb/>ab <reg norm="infinita" type="context">ĩfinita</reg>, & diuina <reg norm="prouidentia" type="simple context">ꝓuidẽtia</reg> Dei, neceſſariò fit vt iſtæ velocitates, eæ magnitudines, <lb/>diſtantiæ, & influxus, talem ordinem, & reſpectum inter ſeipſa, & vniuerſum <reg norm="habeant" type="context">habeãt</reg>, <lb/>qualis perfectiſſimus ſit.</s> + </p> + </div> + <div type="section" level="3" n="34"> + <head style="it" xml:space="preserve">Deraro et denſo nonnulla, minus diligenter à Peripateticis <lb/>perpenſa.</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 diſputauerunt, quia hoc naſciturà condenſatione hali <lb/>tus, quę ab ambiente frigore fit. </s> + <s xml:space="preserve">quia halitus is abore, aut naſo animalis <reg norm="exiens" type="context">exiẽs</reg> <lb/>non eſt purus aer attractus primò, ſed mixtus eſt cum quodam vapore excrementi-<lb/>tio, & ſubtili, quo ſemper ab ea parte <reg norm="euacuatur" type="simple">euacuat̃</reg> corpus, qui ſtatim ab aere frigido cir-<lb/>cundatur, & denſatur, quam ob cauſam ab ipſo ea luminis pars reflectitur, quæ eum <lb/>penetrare non poteſt, quod in hypocauſtis, <reg norm="huiuſmodique" type="simple">huiuſmodiq́;</reg> calidis locis non fit. </s> + <s xml:space="preserve">Idem <lb/>exemplo ab aqua ſtatim à ciſternis, aut profundis puteis in hyeme extracta compro <lb/>bari poteſt, quia tunc temporis, huiuſmodi aqua, cum magis calida ſit, quàm fri-<lb/>gida, emittit vaporem, qui facillimè videtur, ob rationem iam dictam, quod <lb/>in æſtate non cernitur in aqua, etſi ea magis calida eſſet, quam ea, quæ in hyeme <lb/>hauritur.</s> + </p> + <p> + <s xml:space="preserve">Ratio autem, quam ab antiperiſtaſi deſumptam citarunt iidem ad inquirendum, <lb/>cur aqua ſubterranea magis calida, aut minus frigida, hyberno tempore, quàm ea, <lb/>quæ eſt ſupra terram ſit, vana eſt, quia hoc non aliunde fit, quàm ab eo, <reg norm="quod" type="simple">ꝙ</reg> terræ por-<lb/>ri à frigoris ſiccitate ſint clauſi, vnde vapores & exalationes non tam facilè exire poſ <lb/>ſunt. </s> + <s xml:space="preserve">quamobrem calefiunt ſubterraneæ partes. </s> + <s xml:space="preserve">Fimum, fœnum, frumentum hac in <lb/>re ſunt nobis exemplo, in quibus ſępiſſimè viſum eſt ignem accendi.</s> + </p> + <p> + <s xml:space="preserve">Priore illa quoque ratione de antiperiſtaſi dicta, volunt philoſophi maiorem ca-<lb/>liditatem hyęme, quàm ęſtate in animalium ſtomacho contineri, non animaduerten <lb/>tes ſiccitatem, frigiditatis partes ſuperficiales corporis, <reg norm="reſtringentem" type="context">reſtringẽtem</reg>, ſanguinem ver <lb/>ſus originem ſuam impellere, qui in eo loco copioſior cum ſit, eas partes tunc tem-<lb/>poris calefacit magis.</s> + </p> + <p> + <s xml:space="preserve">Neque etiam ijdem nouerunt cauſam, vnde fiat, ut in æſtate impleto vaſe vitreo, <lb/>aut argenteo, aut ex materia non poroſa conſtante, aqua frigida, vas ſudet, 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 niſi in calidis locis euenit, quem ſudorem, dicebantipſi, eſſe <lb/>eandem aquam, quæ per porros vaſis exiret, quod falſiſſimum eſt, quia ſi per porros <lb/>aqua frigida exiret multò magis exiret calida, cum ſubtilior ſit, & ad penetrandum <lb/>aptior. </s> + <s xml:space="preserve">Sed hoc non aliunde oritur, quàm à condenſatione aeris vas circundantis, <lb/>cauſata à frigiditate vaſis refrigerati ab aqua, quemadmodum tempore hyberno <lb/>clarè videmus mane ſuperficies interiores vitri feneſtrarum ſudare, quia <reg norm="extrinſecum" type="context">extrinſecũ</reg> <lb/>frigus refrigerando vitrum, intrinſecum aerem ſibi contiguum congelat.</s> + </p> + <p> + <s xml:space="preserve">Neque ſilentio inuoluendum eſt, nec Ariſtotelem, neque alium ex ſuis fautoribus <lb/>animaduertiſſe denſum, & rarum eſſe cauſam ventorum. </s> + <s xml:space="preserve"><reg norm="Rarum" type="context">Rarũ</reg> autem & denſum, me <lb/>diante calore & frigore fit, & ſi à partibus, in omogeneis, licet <reg norm="argumentari" type="context">argumẽtari</reg>, de toto <lb/>deducat conſequentiam qui velit, obſeruans in calidis æſtatis diebus, dum aliqua nu <lb/>becula ad Solem cooperiendum incedit, ibi ſtatim agitationem aeris ſentiri; </s> + <s xml:space="preserve">ea verò <lb/>nubecula prætergreſſa cum fuerit, & in ea parte, aer ad priſtinam raritatem cauſa-<lb/>tam à calore Solis redierit, quieſcit; </s> + <s xml:space="preserve">huiuſmodi autem aeris agitatio, à nulla certè ex <lb/>halatione proficiſcitur, ſed à motu ſolum locali, quem dum condenſatur, facit. </s> + <s xml:space="preserve">Om <lb/>ne denſum natura ſua frigidum eſt; </s> + <s xml:space="preserve">omne rarum calidum, & è conuerſo. </s> + <s xml:space="preserve">Et frigida <lb/>aura, quæ à flabellis cauſatur, non ſolum à nouo aere qui nos tangit, ſed etiam à den-<lb/>ſo, quod in agitatione eiuſdem aeris fit, naſcitur.</s> + </p> + <p> + <s xml:space="preserve">Cum autem de raritate & denſitate diſputationem ſuſceperim, non ſine ratione <lb/>mihi <reg norm="videtur" type="simple">videt̃</reg> <reg norm="illorum" type="context">illorũ</reg> <reg norm="opinionem" type="context">opinionẽ</reg> <reg norm="explodendam" type="context context">explodẽdã</reg> eſſe, qui Lunę maculas <reg norm="non" type="context">nõ</reg> aliud eſſe dixerunt, <lb/>quàm aliquas partes rariores aliis eiuſdem Lunæ partibus, non obſeruantes rarum, & <lb/>denſum, proportionabilia lumini, quod ab huiuſmodi corporibus reflectitur, non eſ<lb/>ſe. </s> + <s xml:space="preserve">quia corpus aliquod rarum aliquando aptum erit ad reflectendum maius lumen, <lb/>quàm corpus minus rarum ut manifeſtè 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, ideſt, eas Lunę partes, den <lb/>ſiores eſſe; </s> + <s xml:space="preserve">neque unquam aliquis fuit qui de diaphano, aut opaco mentionem fece <lb/>rit, quia melius eſt credere, eas partes diaphanas, ſiue perſpicuas magis eſſe, quàm a-<lb/>lias, quę per aliquod ſpatium, ſolis radio ingreſſum permittant, & alię partes <reg norm="cum" type="context">cũ</reg> ſint <lb/>opacæ ipſum à ſuperficie reflectant. </s> + <s xml:space="preserve">diuerſa tamen ratione à ſpeculo, cum in pleni-<lb/>lunio tota ferè Lunę pars illuminata cernatur, quamuis dictum lumen extenſiuè & in <lb/>tenſiuè ſit minus eo, quod ipſa in nouilunio recipit. </s> + <s xml:space="preserve">Indignum autem mihi videtur <lb/>ijs reſpondere, qui dixerunt huiuſmodi maculas, terræ vmbras exiſtere, cum craſſiſſi-<lb/>mæ ignorantiæ tenebris ſint circunfuſi, vt <reg norm="etiam" type="context">etiã</reg> fuit Cornelius Agrippa, qui primo de <lb/>occulta philoſophia dicens ſe noſſe modum <reg norm="quendam" type="context">quendã</reg> naturalem à Pythagora inuen-<lb/>tum, quo in Luna id totum, quod ipſe ſuper ſpeculum ſcripſiſſet, videretur. </s> + <s xml:space="preserve">oſtendit <lb/>manifeſtè ſe ignorare luminum <reg norm="vmbrarumque" type="simple">vmbrarumq́;</reg> naturam. </s> + <s xml:space="preserve">quia nulla vnquam vmbra ge <lb/>nerari poteſt à corpore, quod aut opacum non ſit, 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">adũ-<lb/>brare</reg> poteſt, niſi opacum illud in linea recta ſitum obtineat, quæ inter lucidum & il <lb/>luminatum extenditur. </s> + <s xml:space="preserve">Neque etiam reſpondebimus ijs, qui ſentiunt quotieſcun-<lb/>que nulla eſſet terra, ſed totus hic globus eſſet aqua, toties non futuram eclipſim lu-<lb/>narem, ratione diaphaneitatis aquæ. </s> + <s xml:space="preserve">Quod <reg norm="falſiſſimum" type="context">falſiſſimũ</reg> eſt, quia omne corpus <reg norm="ſphae- ricum" type="simple">ſphę-<lb/>ricum</reg> quantumuis diaphanum ſit, dummodo ſit denſius aere, luminoſos radios re-<lb/>frangit, & eos ad inuicem interſecare facit, qui deinde vltra interſectionis <reg norm="punctum" type="context">punctũ</reg> di-<lb/>ſgregantur, ita vt amittant illuminationis actum. </s> + <s xml:space="preserve">Adde <reg norm="quod" type="simple">ꝙ</reg> etſi huiuſmodi corpus <lb/>aqueum, ſphęricum non eſſet, ſed cubicum, illud ſuper <reg norm="aliquam" type="context">aliquã</reg> ex eius ſuperficiebus <lb/>ad angulos rectos radius ſolaris 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 ſit denſius aere acquirit, nec totus radius vnquam dictum corpus <lb/>ingr editur, cum ab eius ſuperficie magna pars reflectatur. </s> + <s xml:space="preserve">Reſiſtit ergo huiuſmo-<lb/>di corpus lumini, & quantò magis ſpiſſum aut profundum exiſtit, tantò, validius reſi <lb/>ſtit. </s> + <s xml:space="preserve">Habemus huius rei teſtes, piſcatores vnionum, in ijs mundi partibus, quæ pau-<lb/>cis ab hinc annis Hiſpanorum opera nobis innotuerunt, qui aſfirmanu ad maris <reg norm="vſque" type="simple">vſq;</reg> <lb/>fundum lumen Solis non peruenire.</s> + </p> + <p> + <s xml:space="preserve">Immediata ratio, cur nebulę in ijs locis in quibus <reg norm="conſpiciuntur" type="context">cõſpiciuntur</reg> permaneant, & <reg norm="uunc" type="context">uũc</reg> <lb/>altiores, nunc vero depręſſiores cernantur, non ea eſt, quam Ariſtoteles cap .3. lib. 1. <lb/>metheororum proponit, ſed inde oritur, quòd ſint eædem denſiores ea parte aeris, <lb/>quæ ipſis ſupereminet, & rariores e a, quæ ijſdem ſubiacet. </s> + <s xml:space="preserve">Quòd autem alicuius cor <lb/>poris denſitas maior ea, quæ eſt medij, in quo reperitur, cauſa ſit, vt ipſum corpus de <lb/>ſcendat, & maior raritas eiuſdem corporis, ea, quæ eſt quoque me dij, efficiat, vt di-<lb/>ctum corpus aſcendat, iam Archimedes in lib. de inſidentibus aquæ docuit.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Rectiſſimem" type="context">Rectiſſimẽ</reg> inſtituit natura, vt corpora denſiora verſus loca anguſtiora, & minora <lb/>(intelligendo ea loca orbicularis figuræ) quæ ad centrum propius accedunt, & rario <lb/>ra ad ampliora loca, & maius ſpatium occupantia, ſeſe reciperent. </s> + <s xml:space="preserve">tum quia eadem <lb/>quantitas materiæ condenſatæ, eget minori loco quam rarefacta, <reg norm="cum" type="context">cũ</reg> etiani, quia <reg norm="cum" type="context">cũ</reg> <lb/>corpus denſum non ita ad velocitatem motus localis, vt rarum, idoneum ſit, ad eas <lb/>partes accedat, quæ motibus tardioribus magis ſunt aptæ, corpora <reg norm="autem" type="context">autẽ</reg> rara ad eas, <lb/>quæ velocioribus motibus ſunt aptiores ſeſe transferant. </s> + <s xml:space="preserve">præterquam <reg norm="quod" type="simple">ꝙ</reg> reuera ap-<lb/>pareat pro maiori parte, corpus magis denſum, minus diaphanum; </s> + <s xml:space="preserve">aut magis <reg norm="opacum" type="context">opacũ</reg> <lb/>futurum, quàm rarum, licet ſæpiſſimè videamus contrarium, vt ſuperius innuimus. <lb/></s> + <s xml:space="preserve">eſt tamen naturale <reg norm="proportionatumque" type="simple">proportionatumq́;</reg> magis opacum denſo, & diaphanum raro, <lb/>quàm è contra. </s> + <s xml:space="preserve">Quamobrem ſumma ratione inducta natura voluit, vt corpora ma <lb/>gis opaca, aut minus diaphana, magis vicina centro colligantur, vt ſpatium, quòd re <lb/>manet, abſque vllo impedimento à radijs ſolaribus penetrari poſſit. </s> + <s xml:space="preserve">Tres autem eæ <lb/>cauſæ, quas hoc loco poſui, propriæ ſunt, immediatæ, & per ſe, ex quibus fit, vt corpo <lb/>ra denſiora deſcendant, & rariora aſcendant in mediis minus denſis, aut minus raris <lb/>dictorum corporum, quæ à nemine, <reg norm="quod" type="simple">ꝙ</reg> ſciam, hucuſque propoſitæ fuerunt.</s> + </p> + <p> + <s xml:space="preserve">Qui autem aſſerunt cucurbitæ, quam apponunt chirurgi, effectum ex eo naſci, <reg norm="quod" type="simple">ꝙ</reg> <lb/>calidi ſit attrahere, valdè aberrant à vero quia hoc, non niſi à raro, & à denſo imme-<lb/>diatè, à calido & frigido cauſatis efficitur, quia aer in cucurbita rarefactus à calore <lb/>& per conſequens dilatatus, ſtatim vt à dicto calore deſeritur, iterum condenſatur & <lb/>tantò citius, quantò aer ambiens frigidior exiſtet, & quia eadem materia cum con-<lb/>denſata fuerit minorem ſemper occupat locum, reſtringens igitur ſeſe in cucurbi-<lb/>ta aer dum condenſatur, neceſſariò fit, ne ulla, ſcilicet pars vacua remaneat, <reg norm="quod" type="simple">ꝙ</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 ſit corpus experiri poſſumus, ſi <reg norm="eam" type="context">eã</reg> ad ignem pri-<lb/>mò calefactam, deinde <reg norm="cum" type="context">cũ</reg> ore in amplo aliquo cyatho, aut alio vaſe. </s> + <s xml:space="preserve">vino, aut aqua <lb/>pleno vbi videbimus huiuſmodi liquorem ſtatim ſurſum ferri, quia dum calefit am-<lb/>phora, rarefit quoque aer qui in ea continetur, & quia rateſcit 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 remanſerit, iterum condenſatur ob defectum caloris, reſtringitur, <lb/><reg norm="minorique" type="simple">minoriq́;</reg> indiget loco; </s> + <s xml:space="preserve">Quod cum ita ſe habeat, neceſſarium eſt, 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 ſi corpus admodum non erit fluxile, aut humidum, ita vt ingre <lb/>di amphoram poſſit ita amphorę 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">cũ</reg> admiratione <reg norm="videmus" type="simple">videmꝰ</reg> fragile vas <reg norm="vitreum" type="context">vitreũ</reg> <reg norm="magnum" type="context">magnũ</reg>, & graue <reg norm="lapideum" type="context">lapideũ</reg> corpus eleuare. <lb/></s> + <s xml:space="preserve">Sed vt ad denſum & ad rarum redeamus, mihi videtur frigidum eſſe conſequentem <lb/>qualitatem denſi, & calidum rari, quia quæuis res dum calefit, rarefit, & quælibet <lb/>materia dum refrigeratur, ſimul condenſatur. </s> + <s xml:space="preserve">Qua ratione fit, vt terra frigidior <lb/>ſit aqua, & ignis calidior ſit aere.</s> + </p> + <p> + <s xml:space="preserve">Nec propriè locutus eſt Ariſtoteles .9. & .10. capite primi lib. & ſecundo ſecundi <lb/>metheororum cum dixerit <reg norm="calorem" type="context">calorẽ</reg> Solis eum eſſe, qui ſurſum humores, <reg norm="vaporesque" type="simple">vaporesq́;</reg> eue <lb/>hat, quia Sol nil aliud facit, quam calefacere, cuius caloris ratione, ea materia rarefit, <lb/>& ob rarefactionem leuior facta aſcendit, non quia ſurſum à Sole feratur.</s> + </p> + <p> + <s xml:space="preserve">Quę ſubſequuntur, cum raro ac denſo ſimbolum habere videntur. </s> + <s xml:space="preserve">cum raro, ſcili-<lb/>cet calidum, humidum, leue, ſublime, diaphanum, lumen, clarum, lux, <reg norm="album" type="context">albũ</reg>, dies, mo-<lb/>tus, velox, ſimplex, diſgregatum, molle, lene, acutum, ſubtile, coctum, <reg norm="ſpaciosum" type="context">ſpaciosũ</reg>, <lb/>dulce, voluptas, audacia, lætitia, liberalitas, veritas, induſtria, amor, miſericordia, hu-<lb/>manitas, ſanitas, vita, & iis ſimilia. </s> + <s xml:space="preserve">Cum denſo verò frigidum, ſiccum, graue, imum, <lb/>opacum, vmbra, obſcurum, tenebræ, nigrum, nox, quies, tardum, mixtum, congrega <lb/>tum, durum, aſperum, ob tuſum, craſſum, crudum, anguſtum, amarum, dolor, cimor, <lb/><reg norm="melancholia" type="context">melãcholia</reg>, auaritia, mendacium, inertia, odium, crudelitas, feritas, infirmitas, mors, <lb/>& ijs ſimilia.</s> + </p> + <p> + <s xml:space="preserve">Verum eſt quod ea ratio, qua Ariſtoteles ait aerem humidum eſſe, parui eſt mo-<lb/>menti, quia ſimiliter deigne inferri poſſet, qui facilius à termino alieno, <reg norm="quam" type="context">quã</reg> aer, aut <lb/>aqua terminari poteſt.</s> + </p> + </div> + <div type="section" level="3" n="35"> + <head style="it" xml:space="preserve">Motum rectum curuo poſſe comparari etiam diſentiente <lb/>Ariſtotele.</head> + <head xml:space="preserve">CAP. XXXV.</head> + <p> + <s xml:space="preserve">SEd vt ad <reg norm="Ariſtotelem" type="context">Ariſtotelẽ</reg> redeamus, rectè dicere non poteſt motum rectum ad <reg norm="curuum" type="context">curuũ</reg> <lb/>comparabilem non eſſe .4. cap. lib. 7. phyſicorum, 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 ſemi-<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 ſuis lateribus circundantibus angulum rectum æquale ſemidiametro dicti <lb/>circuli, aliud quoque latus ipſum angulum rectum circundans, ex neceſſitate, <reg norm="circun- ferentiæ" type="context">circũ-<lb/>ferentiæ</reg> dicti circuli habebit æquale. </s> + <s xml:space="preserve">Poteſt igitur dari vna quædam recta linea <reg norm="ae- qualis" type="simple">ę-<lb/>qualis</reg> circulari contra Ariſtotelis opinionem, qui non benè reuocauit in mentem, <lb/>quod ſcripſit de relatiuis, cum dixit quadraturam circuli poſſe quidem dari, etſi <reg norm="tunc" type="context">tũc</reg> <lb/><reg norm="temporis" type="wordlist">tꝑis</reg> de ea <reg norm="non" type="context">nõ</reg> <reg norm="haberetur" type="simple">haberet̃</reg> ſcientia. </s> + <s xml:space="preserve">Si <reg norm="igitur" type="simple">igit̃</reg> dicta quadratura dari <reg norm="pont" type="context">põt</reg>, poteſt <reg norm="etiam" type="context">etiã</reg> dari vna <lb/>recta linea ęqualis circunferentiæ <reg norm="eiuſdem" type="context">eiuſdẽ</reg> circuli, ob rationes <reg norm="iam" type="context">iã</reg> dictas. </s> + <s xml:space="preserve">Sed ſi Ariſt. <lb/>dixiſſet, circularem corporum cęleſtium motum, comparabilem non eſſerecto cor-<lb/>porum elementarium, verum dixiſſet, non quia eorum alter circularis, alter ve-<lb/>rò ſit rectus, ſed quia cœleſtis regularis ſit, neque modò tardus, modò velox, <lb/>ſed vnam ſemper & eandem velocitatem retinens, <reg norm="motus" type="simple">motꝰ</reg> <reg norm="autem" type="wordlist">aũt</reg>, qui eſt <reg norm="corporum" type="context">corporũ</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">ꝙ</reg> nunquam fuit neque ſit futurus aliquis <reg norm="horum" type="context">horũ</reg> <lb/>rectorum, qui naturales dicuntur, qui tam velociter moueatur, ut motus cœli, quia ſi <lb/>voluerimus conſiderare motum diurnum .24. horarum, ſecundum opinionem com-<lb/>munem, reperiemus calculando, Lunam in quadraturis cum Sole, dum inuenitur in <lb/>æquatore, ſingulis horarum minutis moueri per .500. milliaria Italica vel circa, & in <lb/>coniunctionibus, & oppoſitionibus ipſius Solis .1000. vel circa, & Solem tempore <reg norm="ae- quinoctiorum" type="simple context">ę-<lb/>quinoctiorũ</reg> .18000. & <reg norm="Saturnum" type="context">Saturnũ</reg> circa æquatoris <reg norm="ſitum" type="context">ſitũ</reg> .260000. & <reg norm="amplius" type="simple">ampliꝰ</reg> de ſtellis <reg norm="autem" type="wordlist">aũt</reg> <lb/>fixis circa æquatorem poſitis quiuis cogitet; </s> + <s xml:space="preserve">quod reuera diffi cillimum quibuſdam <lb/>videbitur, quod quidem non occurrit <reg norm="ſecundum" type="context">ſecũdum</reg> pulcherrimam Ariſtarchi ſamij opi-<lb/>nionem, diuinitus à Nicolao Copernico expreſſam, contra quam nil planè valent <lb/>rationes ab Ariſtotele; </s> + <s xml:space="preserve">neque etiam à Ptolomeo propoſitę. </s> + <s xml:space="preserve">Motu verò proprio, quo <lb/>libet horę 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> eſt, aut op <lb/>poſita 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, & ſi <reg norm="moueretur" type="simple">moueret̃</reg> motu recto hoc <reg norm="mon" type="context">mõ</reg>, <lb/>& velocius etiam corpore cęleſti, non obſeruans <reg norm="tamen" type="wordlist">tamẽ</reg> uniformitatem, ut dictum cœ <lb/>leſte facit, cum eodem nullo modo comparari poſſet, quia rectus dictus naturalis, <lb/>ſuam ſemper velocitatem adauget, ob continuam impreſſionem, quam recipit à cau <lb/>ſa perpetuò coniuncta cum ipſo corpore, quę eſt propenſio illa naturalis eundi bre-<lb/>uiori quadam via ad locum ſuum, ita vt etiam ſi dictum corpus elementare à motu <lb/>tardiore ad velociorem, ſuperare poſſet <reg norm="motum" type="context">motũ</reg> alicuius corporis cęleſtis, ij duo motus <lb/>interſecarent ſeſe in vno ſolo <reg norm="puncto" type="context">pũcto</reg>, quod diuidi <reg norm="diſtribuique" type="simple">diſtribuiq́;</reg> in partes nequiret, ideſt <lb/>non niſi in vno ſolo temporis inſtanti redderentur æquales, vt ita dicam. </s> + <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> <reg norm="ſolum" type="context">ſolũ</reg> <lb/>loquor de circulari cœleſti cum recto elementari, ſed de qualibet alia motuum ſpe-<lb/>cie, ſiue ſint ambo recti, ſiue ambo curui, quando aliquis eorum irregularis erit.</s> + </p> + </div> + <div type="section" level="3" n="36"> + <head style="it" xml:space="preserve">Minus ſufficienter exploſam fuiſſe ab Ariſtotele opinionem cre-<lb/>dentium plures mundos exiſtere.</head> + <head xml:space="preserve">CAP. XXXVI.</head> + <p> + <s xml:space="preserve">MAior ratio, qua Ariſtoteles eorum opinionem, qui plures eſſe mundos dixe <lb/>runt, refutare nititur, in eo conſiſtit, quod is credat partes terræ, quæ alijs <lb/>mundis aſſignarentur, ad huius mundi centrum inclinationem habere, & ſic ignem <lb/>illorum, propenſionem habiturum ad circunferentiam huius.</s> + </p> + <p> + <s xml:space="preserve">Quæ certè ratio tam debilis eſt, vt per ſe cadat, non conſideransipſe, quòd ſi <lb/>eſſent dicti mundi, eorum quilibet ſuum proprium centrum, <reg norm="ſuamque" type="simple">ſuamq́;</reg> propriam cir-<lb/>cunferentiam haberet, <reg norm="terrasque" type="simple">terrasq́;</reg> & ignes haberent inclinationem ad centra circunfe-<lb/><reg norm="rentiasque" type="simple">rentiasq́;</reg> ſuorum mundorum, abſque eo, <reg norm="quod" type="simple">ꝙ</reg> vna terra, alterius centrum appeteret; </s> + <s xml:space="preserve">vt <lb/>exempli gratia, ſi doctiſſimi Ariſtarchi opinio eſt vera, rationi <reg norm="quoque" type="simple">quoq;</reg> conſentaneum <lb/>erit maximè, vt quod Lunæ contingit, cuilibet <reg norm="etiam" type="context">etiã</reg> ex aliis quinque planetis eue-<lb/>niat, ideſt, vt quemadmodum Luna ſuorum epicyclorum ope <reg norm="circum" type="context">circũ</reg> terram voluitur, <lb/>quaſi per circunferentiam alterius cuiuſdam epicycli, in quo terra ſit inſtar centri <lb/>naturalis (ideſt ſit 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 ſui 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, ſitum habens, voluantur; </s> + <s xml:space="preserve">quod quidem corpus, & aliquem quoque ha-<lb/>beat motum circa ſuum axem, ſit opacum, ijs conditionibus, quæ terræ ſunt ſimi-<lb/>les, præditum exiſtat, & in dicto epyciclo ſint res ſimiles iſtis lunaribus.</s> + </p> + </div> + <div type="section" level="3" n="37"> + <head style="it" xml:space="preserve">Anrectè loquutus ſit Phyloſopbus de extenſione luminis <lb/>per uacuum.</head> + <head xml:space="preserve">CAP. XXXVII.</head> + <p> + <s xml:space="preserve">ARriſtoteles ſecundo lib. de anima ſentit <reg norm="quod" type="simple">ꝙ</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 veriſimile <reg norm="non" type="context">nõ</reg> eſt; </s> + <s xml:space="preserve"><reg norm="quia" type="simple">ꝗa</reg> <reg norm="quemadmo dum" type="wordlist">quẽadmo<lb/>dum</reg> quantò rarius eſt aliquod corpus, tanto aptius eſt vt diaphanum exiſtat; </s> + <s xml:space="preserve">& <reg norm="quan- tò" type="context">quã-<lb/>tò</reg> rarius eſt dictum corpus, tantò minorem quantitatem materiæ contineat; </s> + <s xml:space="preserve">ſic <reg norm="quam" type="context">quã</reg> <lb/>tò magis diaphanum eſt, cum ex perexigua materia conſtet, tantò magis liber tran-<lb/>ſitus luminis patet; </s> + <s xml:space="preserve">Vnde quantò minor quantitas materiæ erit in dicto ſpatio, tan <lb/>tò nitidius pertranſibit lumen. </s> + <s xml:space="preserve">Sequitur ergo, quòd vbi nulla eſſet materia, totum <lb/>lumen libere tranſiret. </s> + <s xml:space="preserve">Color cęruleus quem videmus in profunditate aquæ, & ae-<lb/>ris, color eſt a quæ & aeris, qui denotat reſiſtentiam factam ab aere & ab aqua ipſi lu <lb/>mini; </s> + <s xml:space="preserve">Quod quidem lumen ubi corpus aliquod non eſſet, minime reflecteretur, ſed <lb/>abſque vllo impedimento rectà tranſiret.</s> + </p> + </div> + <div type="section" level="3" n="38"> + <head style="it" xml:space="preserve">An rectè phyloſophiœ<unsure/> penus Ariſtoteles ſenſerit de loco im-<lb/>pellendo à pyramide.</head> + <head xml:space="preserve">CAP. XXXVIII.</head> + <p> + <s xml:space="preserve">ARiſtoteles .8. cap. lib. 3. de cœlo, diſputans contra antiquos de elementorum <lb/>figuris, ait pyramidem implere poſſe locum corporeum. </s> + <s xml:space="preserve">quod verum non <lb/>eſt. </s> + <s xml:space="preserve">Cubus quidem id facit ab .8. enim cubis perfectè impletur locus, ſed non <lb/>item .12. pyramides, ut Ariſtoteles ſenſit (ideſt ſex ſuper aliquam exagonam figu-<lb/>ram ſuperficialem & ſexſub eadem) id præſtant, cum potius maius vacuum rema-<lb/>neatad quamlibet partium ſupra, & infra, quam plenum. </s> + <s xml:space="preserve">Rectius Ariſtoteles <lb/>egiſſet, ſi probaſſet ratione immobilitatis conuenire pyramidem terræ, quam cu-<lb/>bum. </s> + <s xml:space="preserve">quamuis, de horum corporum altero, ſit ſtultum 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 eſt illud, quod ita ſe habet, vt multis rationibus probari poteſt, quarum vna <lb/>hæc nobis ſufficiet. </s> + <s xml:space="preserve">Scimus iam ex communi conceptu corpus ſphęricum eſſe ma-<lb/>gis volubile, <reg norm="inſtabileque" type="simple">inſtabileq́;</reg>, quàm alia ſint. </s> + <s xml:space="preserve">Illud ergo corpus, cuius figura ad ſphæri-<lb/>cam magis accedet, ad uoluendum, & ad mouendum facilius erit quouis alio, quod <lb/>æqualis ſit quantitatis, & ſibi omogeneum materia, vt exempli gratia corpus .20. ba <lb/>ſium ad voluendum, & ad mouendum promptius erit eo, quod ex .12. conſtat, & id, <lb/>quod eſt .12. eo, quod eſt .8. & id, quod eſt .8. eo, quod eſt .6. & id, quod eſt .6. vt <lb/>cubus eſt, eo, quod eſt .4. cuiuſmodi eſt pyramidale. </s> + <s xml:space="preserve">Huc accedit, quòd pyrami-<lb/>dale corpus aliam conditionem habet, quàm 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 ſit, cubicum autem econtrà ſit alterabile vndequaque, <reg norm="ſuaque" type="simple">ſuaq́;</reg> quadrata in <reg norm="rhum- bos" type="context">rhũ-<lb/>bos</reg> mutare poſſit, iiſdem exiſtentibus lateribus.</s> + </p> + </div> + <div type="section" level="3" n="39"> + <head style="it" xml:space="preserve">Examinatur quam ualida ſit ratio Aristotelis de <lb/>inalterabilitate Cœli.</head> + <head xml:space="preserve">CAP. XXXIX.</head> + <p> + <s xml:space="preserve">ARriſtoteles textu .22. primi lib. de Cœlo ita inquit.</s> + </p> + <p> + <s xml:space="preserve">Accidit autem, & hoc per ſenſum ſufficienter, quo ad humanam dixiſſe fi-<lb/>dem, & omni pręterito tempore ſecundum traditam inuicem memoriam, nihil vi-<lb/>detur tranſmutatum neque ſecundum totum vltimum cęlum, neque ſecundum par-<lb/>tem ipſius propriam vllam.</s> + </p> + <p> + <s xml:space="preserve">Hoc autem in loco Ariſto. non conſiderauit, <reg norm="quod" type="simple">ꝙ</reg> ſimiliter de terra dici poſſet, quan <lb/>do ipſa ita eminus proſpiceretur, imo abſque dubio putandum eſt, <reg norm="quod" type="simple">ꝙ</reg> ſi terra luce So <lb/>lis prædita eſſet, & aliquis ipſam ab octauo orbe vellet videre, nullo pacto cerne-<lb/>ret, cum ſidera illa quæ primæ magnitudinis vocantur, & quæ pluſquam centies ma <lb/>iora ip<unsure/>ſa terra putantur non niſi 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 ſint. <lb/></s> + <s xml:space="preserve">Animaduertimus tamen permultos ſumma <reg norm="cum" type="context">cũ</reg> <lb/>difficultate <reg norm="eorum" type="context">eorũ</reg> demonstr ationes percipere. </s> + <s xml:space="preserve">Prœ<unsure/>-<lb/>cipuè ubi quint a, aut ſeptima deffinitiones eiuſ-<lb/>dem libri neceſſariœ<unsure/> ſunt. </s> + <s xml:space="preserve">Illœ enim adeo obſcurœ <lb/>uidentur, ut longè facilius admißuri ſint hœc no-<lb/>ſtra poſtulat at anquam clarior a. </s> + <s xml:space="preserve">At que etiam tanquam intellectui <lb/>commodiora, quam ſit illud quintum <reg norm="idemque" type="simple">idemq́ꝫ</reg> ultimum postulatum <lb/>eiuſdem in primo libro poſitum, de line a duas alias ſecante. </s> + <s xml:space="preserve">Quan-<lb/>doquidem <reg norm="ijs" type="lig">ijs</reg> noſtris postulatis admißis, ſequentia 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 compoſita ex æquali numero partium æqualium, ſunt inuicem <lb/>æqualia.</s> + </p> + <p> + <s xml:space="preserve">Vtſi quis diceret omnes proportiones quæ <reg norm="compoſitæ" type="context">cõpoſitæ</reg> ſunt ex æquali numero alia-<lb/>rum proportionum inuicem æqualium, ſunt etiam inuicem æquales, quod Eucli-<lb/>des conatur demonſtrare 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> ſi à totis æqualibus detractæ fuerint æquales partes, quæ remanent erunt <lb/>partes inuicem æquales.</s> + </p> + <p> + <s xml:space="preserve">Et è conuerſo ſi æqualibus æqualia addas compoſita erunt inuicem æqualia.</s> + </p> + <p> + <s xml:space="preserve">Quod in ipſis proportionibus hoc loco ſemper intelligendum eſt.</s> + </p> + </div> + <div type="section" level="4" n="3"> + <head xml:space="preserve">TERTIVM.</head> + <head style="it" xml:space="preserve">Quę est εuclidis ſeptima propoſitio.</head> + <p> + <s xml:space="preserve"><emph style="sc">Qvod</emph> ſi fuerint plures termini æquales inuicem, ratio ſeu proportio vnius ip-<lb/>ſorum ad alium tertium terminum maiorem, minoremúe, ſed eiuſdem generis, erit <lb/>cadem quæ cuiuſuis alterius termini ad eundem tertium. </s> + <s xml:space="preserve">Et è conuerſo, quæ fuerit <lb/>proportio tertij termini ad vnum prædictorum æqualium, eadem erit, ſpecie, 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 propoſitio.</head> + <p> + <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius plurium terminorum collatorum cum ali <lb/>quo tertio eiuſdem generis, eadem fuerit cum ea quæ eſt cuiuſuis alterius dictorum <lb/>terminorum cum eodem tertio, aut proportio dicti tertij, cum aliquo dictorum, ea-<lb/>dem fuerit cum ea quæ ipſius eſt ad aliquem alium eorundem <reg norm="terminorum" type="context">terminorũ</reg>, tunc eiuſ-<lb/>modi termini, æquales erunt inter ſe.</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 propoſitio.</head> + <p> + <s xml:space="preserve"><emph style="sc">Qvoties</emph> plures erunt termini, quorum vnus fuerit maior altero, ſi compa-<lb/>rentur alicui tertio eiuſdem generis, proportio maioris adtertium illum, maior erit <lb/>ea, quæ eſt minoris ad prædictum tertium, & proportio illius tertij ad maiorem, mi-<lb/>nor erit ea quæ eiuſdem 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 propoſitio.</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, eiuſmodi 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 propoſitio.</head> + <p> + <s xml:space="preserve"><emph style="sc">Proportiones</emph>, quarum vnaquęque cum aliqua tertia æqualis eſt, ipſæ quo-<lb/>que inter ſe ſunt æquales. </s> + <s xml:space="preserve">Vtillud, Quæ vni & eidem ſunt æqualia, ſibi inuicem <lb/>ſunt 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 propoſitio.</head> + <p> + <s xml:space="preserve"><emph style="sc">Qvotiescvnqve</emph> proportio vnius ex pluribus antecedentibus cum ſuo ex <lb/>pluribus conſequentibus, æqualis fuerit ei cuiuſuis alterius dictorum <reg norm="antecedentium" type="context">antecedentiũ</reg>, <lb/>cum ſuo plurium <reg norm="conſequentium" type="context">cõſequentium</reg>, proportio totius aggregati antecedentium cum to-<lb/>to aggregato conſequentium, dictæ primę proportioni ęqualis erit, nempe illius an <lb/>tecedentis ad ſuum conſequens.</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 propoſitio.</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 ſe, 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 (ſiue coniuncti ſiue di-<lb/>ſiuncti ſint) æquales ſinguli 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="ſuum" type="context">ſuũ</reg> ter-<lb/>tium, æqualis erit proportioni aggregati reliquorum terminorum ad ſuum tertium, <lb/>& è conuerſo, ita ſe habebit primus tertius terminus ad ſuos multos terminos, ſicut <lb/>ſe habet ſecundus tertius terminus ad ſuos ſimul ſumptos.</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, ſem <lb/>per multiplex eſt ad ſingulas partiales proportiones, ex quibus ipſum 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 diuiſa fuerit, ex iis partibus componitur, in <lb/>quas diuiditur.</s> + </p> + <p style="it"> + <s xml:space="preserve">Cum enim bæ præpoſitiones ſint ita conſpicuæ ipſi intellectui, ut <reg norm="abſque" type="simple">abſq;</reg> dubio inter obie <lb/>ct a ipſius intellectus connumerari poſſint, nullus ſanæ 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, ſecundum, & tertium theorema quinti Euclidis ab ipſo ſatis exactè de <lb/>monſtratur, ſtudioſus itaque autorem conſulat.</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 ſic <reg norm="demonſtrarem" type="context">demonſtrarẽ</reg>. <lb/></s> + <s xml:space="preserve">ſit, verbi gratia, proportio <var>.a.</var> ad <var>.b.</var> <lb/>quæ eſt <var>.c.</var> ad <var>.d.</var> ſumptis 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 eſt <var>.f.</var> ad <var>.h</var>. </s> + <s xml:space="preserve">Habemus enim ex .10 <lb/>poſtulato præmiſſo, eandem futuram <lb/>proportionem <var>.e.</var> ad <var>.a.</var> quæ eſt <var>.f.</var> ad <var>.c.</var> <lb/>& ita <var>.b.</var> ad <var>.g.</var> quæ eſt <var>.d.</var> ad <var>.h.</var> ex præ-<lb/>ſuppoſito verò <reg norm="cum" type="context">cũ</reg> ſic ſe habeat <var>.a.</var> ad <lb/>b. ſicut <var>.c.</var> ad <var>.d.</var> erit ex primo poſtula-<lb/>to <reg norm="eadem" type="context">eadẽ</reg> proportio <var>.e.</var> ad <var>.g.</var> quæ eſt <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 ſunt <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> & ſimiliter 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 ſint de obiectis intelligibilibus, ſine vllo medio, <lb/>ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti <lb/>Euclid. fecimus tertium Poſtulatum, .8. quintum, .9. quartum, .10. ſextum, .11. ſepti<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 demonſtrabitur, mutatis tantum <lb/>theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur <lb/>tertium .5. 4. et .6. poſtulatum à me propoſitum.</s> + </p> + </div> + <div type="section" level="4" n="6"> + <head xml:space="preserve">THEOR. XV.</head> + <p> + <s xml:space="preserve">QVintumdecimum Theorema ſic demonſtrabo; </s> + <s xml:space="preserve">Sit, exempli gratia, a. termi-<lb/>nus antecedens. et <var>.b.</var> conſequens, qui-<lb/>bus duo multiplices ſumantur <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/>ſtè patet quamlibet partem ipſius <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/>ſtulato propoſitum eluceſcet.</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 ſic demonſtrabitur. </s> + <s xml:space="preserve">Sit, exempli cauſa, eadem pro <lb/>portio <var>.a.</var> ad <var>.b.</var> quæ eſt <var>.c.</var> ad <var>.d</var>. </s> + <s xml:space="preserve">Dico <reg norm="quod" type="simple">ꝙ</reg> ita ſe habebit <var>.a.</var> ad <var>.c.</var> ſicut <var>.b.</var> ad <var>.d</var>. </s> + <s xml:space="preserve">Cogi-<lb/>temus itaque alterum iſtorum 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 ipſius <var>.a.</var> ad <var>.d.</var> componetur ex ea quę <lb/>eſt <var>.a.</var> ad <var>.b.</var> & ea quæ eſt <var>.b.</var> ad <var>.d.</var> ex .12. poſtulato. </s> + <s xml:space="preserve">Et ex eodem, illa ipſa proportio <var>.<lb/>a.</var> ad <var>.d.</var> pariter componetur ex ea quæ eſt <var>.a.</var> ad <var>.c.</var> & ea quæ eſt <var>.c.</var> ad <var>.d.</var> ſumpto <var>.c.</var> pro <lb/>medio termino. </s> + <s xml:space="preserve">Ex quo ſequitur, 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 ſi duas proportiones æquales ſubtraxerimus, eam videlicet quæ eſt <var>.a.</var> ad <var>.b.</var> & il <lb/>lam quæ eſt <var>.c.</var> ad <var>.d.</var> ſupererunt duæ proportiones <lb/>inter ſe 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 ſecundi po <lb/>ſtulati diuiſim.</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 idipſum <reg norm="demonſtrari" type="context">demõſtrari</reg> poteſt, <lb/>ſumpto <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 eſt <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"/> + poſtulato. </s> + <s xml:space="preserve">Sed cum proportio <var>.a.</var> ad <var>.b.</var> ęqualis ſit <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 ſecunda <lb/>parte .2. poſtulati compoſitè, & ſic habebimus pro <lb/>poſitum, ita quòd quotieſcunque <reg norm="dabuntur" type="context">dabũtur</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">DEcimiſeptimi theorematis hæc eſt demonſtratio. </s> + <s xml:space="preserve">Ita ſe ha beat <var>a.c.b.</var> ad <var>.c.<lb/>b.</var> ſicut ſe habet <var>.d.f.e.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">Probo ita ſe habere <var>.a.c.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.d.<lb/>f.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">Cogitemus itaque alterum terminum ſcilicet <var>.n.f.</var> qui ſic ſe habeat. ad <var>.f.e.</var> <lb/>ſicut ſe habet <var>.a.c.</var> ad <var>.c.b</var>. </s> + <s xml:space="preserve">Quare ex præcedenti theoremate ita ſe habebit <var>.a.c.</var> ad <var>.n.<lb/>f.</var> ſicut ſe habet <var>.c.b.</var> ad <var>.f.e.</var> & ex .8 poſtulato ita ſe habebit <var>.a.c.b.</var> ad <var>.n.f.e.</var> ſicut ſe ha-<lb/>bet <var>.c.b.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">Sed cum ex præſuppoſito ita ſe habeat <var>.a.c.b.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.<lb/>d.f.e.</var> ad <var>.f.e.</var> ideo ex præcedenti theoremate ita ſe habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> ſicut ſe ha <lb/>bet <var>.c.b.</var> ad <var>.f.e.</var> demonſtratum autem eſt ita ſe habere <var>.c.b.</var> ad <var>.f.e.</var> ſicut ſe habet <var>.a.c.b.</var> <lb/>ad <var>.n.f.e</var>. </s> + <s xml:space="preserve">Quare ex .7. poſtulato 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. poſtulato <var>.d.f.e.</var> æqualis erit <var>.n.f.e</var>. </s> + <s xml:space="preserve">Itaque ex 3. poſtulato 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> ęqualis 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 ſecunda par-<lb/>te tertij axiomatis præmiſſi. </s> + <s xml:space="preserve">Igitur ita <lb/>ſe habebit <var>.a.c.</var> ad <var>.d.f.</var> ſicut <var>.c.b.</var> ad <var>.f.e.</var> ex <lb/>7. poſtulato. </s> + <s xml:space="preserve">& ſic ex præcedenti theo-<lb/>remate ita ſe habebit <var>.a.c.</var> ad <var>.c.b.</var> ſicut <var>.d.f.</var> ad <var>.f.e.</var> quod erat propoſitum: </s> + <s xml:space="preserve">Quotieſ-<lb/>cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiſim 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 demonſtrari poteſt. </s> + <s xml:space="preserve">Detur proportio <var>.a.c.</var> ad <var>.c.b.</var> ſi-<lb/>milis ei quæ eſt <var>.d.f.</var> ad <var>.f.e.</var> probo ita ſe habere <var>.a.c.b.</var> ad <var>.c.b.</var> ſicut ſe habet <var>.d.f.<lb/>e.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">In primis notum eſt ex .16. theoremate ita ſe habiturum, <var>a.c.</var> ad <var>.d.f.</var> ſi <lb/>cut <var>.c.b.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">Quare ex .8. poſtulato ita <lb/>ſe habebit <var>.a.c.b.</var> ad <var>.d.f.e.</var> ſicut <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 ſe habebit <var>.<lb/>a.c.b.</var> ad <var>.c.b.</var> ſicut <var>.d.f.e.</var> ad <var>.f.e</var>. </s> + <s xml:space="preserve">Quod erat <lb/>propoſitum. </s> + <s xml:space="preserve">Quotieſcunque igitur .4. <lb/>quantitates dabuntur vnius <reg norm="eiuſdemque" type="simple">eiuſdemq́;</reg> generis diſiunctim 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. ſatis quidem apud Euclidem demonſtratur: </s> + <s xml:space="preserve">eius tamentertia <lb/>pars commodius hac ratione demonſtrari poterit (nempe) quod cum ſit 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 eſt <var>.c.</var> ad <var>.d.</var> probabo ita ſe habituram proportionem <var>.b.</var> ad <var>.a.</var> ſicut <lb/>ſe habet <var>.d.</var> ad <var>.c.</var> hoc argumento: </s> + <s xml:space="preserve">ſi <var>.a.</var> ad <var>.b.</var> ita ſe <lb/>habet ſicut <var>.c.</var> ad <var>.d.</var> ex .16. theoremate ita ſe ha <lb/> + <anchor type="figure" xlink:label="fig-0215-01a" xlink:href="fig-0215-01"/> + bebit <var>.a.</var> ad <var>.c</var>, ſicut <var>.b.</var> ad <var>.d</var>. </s> + <s xml:space="preserve">Quare ſic ſe habebit <lb/>b. ad <var>.d.</var> ſicut <var>.a.</var> ad <var>.c</var>. </s> + <s xml:space="preserve">Itaque ex eodem .16. ita ſe <lb/>ſe habebit <var>.b.</var> ad <var>.a.</var> ſicut <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 demonſtratum fuerit, poteſt ni-<lb/>hilominus & hac via demonſtrari. </s> + <s xml:space="preserve">Sic ſe habeat proportio <var>.a.</var> ad <var>.b.</var> ſicut ſe <lb/>habet <var>.c.</var> ad <var>.d.</var> & proportio <var>.b.</var> ad <var>.e.</var> ſicut <var>.d.</var> ad <var>.<lb/>f</var>. </s> + <s xml:space="preserve">Dico <reg norm="quod" type="simple">ꝙ</reg> ſi <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> & ſi <var>.a.</var> minus fuerit .e: c. <reg norm="quoque" type="simple">quoq;</reg> minus erit <lb/>f. ſin verò ęquale, <reg norm="ent" type="context">ẽt</reg> æquale erit. </s> + <s xml:space="preserve">Nam ex pri <lb/>mo poſtulato certi ſumus ita ſe habere pro <lb/><reg norm="portionem" type="context">portionẽ</reg> <var>.a.</var> ad <var>.e.</var> ſicut ſe habet proportio <var>.c.</var> ad <lb/>p. </s> + <s xml:space="preserve">Quare ex .12. theor <reg norm="propoſitum" type="simple context">ꝓpoſitũ</reg> <reg norm="manifeſtum" type="context">manifeſtũ</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">VIgeſimum primum theorema, ſatis apud Eucli. probatum, nihilominus præ-<lb/>ſcripto nunc modo demonſtrari 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 poſtulatum collegimus. <lb/></s> + <s xml:space="preserve">Sequentia verò cum exactè apud Eucli. demonſtrentur non eſt 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, Philoſophi Mathematici.</head> + <div type="preface" level="3" n="1"> + <head xml:space="preserve">Ad Lectorem.</head> + <p> + <s xml:space="preserve">VT Nilmagis virtutis eſt proprium, quàm <lb/>agitari, & inceßabili motu prodeße. </s> + <s xml:space="preserve">Ac velu <lb/>ti fulgidum ſydus ante oculos <reg norm="ſpectantium" type="context">ſpectantiũ</reg> com <lb/>micare. </s> + <s xml:space="preserve">Ita mihi mathematicis <reg norm="ijsque" type="simple">ijsq;</reg> maxi <lb/>mè philoſophicis ſpeculationibus dedito, ſapiſ-<lb/>ſimè, ut in principium ſummorum aulis, & <lb/>amplißimis ciuitatibus degenti, ubi multa ſem <lb/>per Nobilium mir a curioſitate, ſciendi deſiderio, & conferendicu <lb/>piditate referta, uerſantur ingenia, contigit, modo ab his, modo ab <lb/>illis, aut uerbis tentari, aut literis prouocari ad diſſerendum, de <lb/>his, in quorum ſtudijs uerſamur. </s> + <s xml:space="preserve">Quarum concertationum & re <lb/>ſponſionum, quoniam non omnino indigna exiſtimaui, quæmemoriæ<unsure/> <lb/>comendarentur, partem aliquam apud me conſeruaui. </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 ſcientia ipſa quo magis diffundetur, creſcat; </s> + <s xml:space="preserve">& quicquid <lb/>ualeo, ſine inuidia in communem utilitatem conferam. </s> + <s xml:space="preserve">Tum ut ui-<lb/>rorum præctantiβimorum, qui me ſuis interrogationibus excitaue <lb/>runt, quantum in me erit, gratitudinis ergo, nomina reddam im-<lb/>mortalia, & eorum exemplo alios, ocio ſordidiore abiecto, quod ſolet <lb/>ourialium præcipuè excelſa ingenia corrumpere, ad ſciſcit andum <lb/>conferendum, & diſſerendum, derebus <reg norm="ſerijs" type="lig">ſerijs</reg>, & quæuſui 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/>conſule, & 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, Philoſophi <lb/>Mathematici.</head> + <head style="it" xml:space="preserve">AD SERENISS. CMANVELEM PHILIB. <lb/>Allobrogum & ſubalpinarum gentium Ducem <lb/>Inuictiβimum.</head> + <head xml:space="preserve">EPISTOLA.</head> + <p> + <s xml:space="preserve"><emph style="sc">MIrvm</emph>, Quàm lectione epiſtolæſeu (vt vocant) Breuis <var>.S.D.<lb/>N</var>. </s> + <s xml:space="preserve">Gregorij XIII. Pont. Max. quod ad me nuper tua Celſitu <lb/>do miſit ex Nicea, vt meam de ea re ſententiam proferrem, <lb/>delectatus ſim; </s> + <s xml:space="preserve">ex quo, non tantum recta illius mens ac verè <lb/>ſancta cogitatio, ſed etiam aperta <reg norm="maximaque" type="simple">maximaq́;</reg>, ſi ad exitum per <lb/><reg norm="ducatur" type="simple">ducat̃</reg>, imo ſummè neceſſaria vniuerſo orbi vtilitas percipi <lb/>poteſt; </s> + <s xml:space="preserve">qua de re memini cum Celſitudine tua aliquando ſer-<lb/>monem habuiſſe. </s> + <s xml:space="preserve">Vidi præterea cum ipſo breui tranſ-<lb/>miſſum compendium Domini Aloiſij Lilij: </s> + <s xml:space="preserve">cuius mihi ſententia perplacet, de corre <lb/>ctione eius diei, qui 134. quoque anno præter, neceſſitatem, gignitur. </s> + <s xml:space="preserve">qui ſanè dies <lb/>perpetuæ retrogradationis ingreſſus Solis in Zodiaci ſigna, cauſa fuit. </s> + <s xml:space="preserve">quod ita per-<lb/>ſpicuè patebit. </s> + <s xml:space="preserve">Cum Numa Pompilius anni curſum correxit <reg norm="emendauitque" type="simple">emendauitq́;</reg>, ea ſanè <lb/>mente id videtur præſtitiſſe, vt principium Ianuarij primi menſis anni, præcisè in ip <lb/>ſo hyemalis ſoltitij puncto collocaretur. </s> + <s xml:space="preserve">quod hac tempeſtate, dictam ob cauſam <lb/>adeò retroceſſit, vt circa vndecimam diem Decembris eſſe reperiatur. </s> + <s xml:space="preserve">quod ſi cen-<lb/>teſimo trigeſimo quarto quoque anno detractus dies vnus fuiſſet, nihil erroris pror-<lb/>ſus accidiſſet. </s> + <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dies hic (vt alias Celſit. tuæ ſignificaui) 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. ſecundorum ferè .16. (anni æqualis <lb/>ſiue medij) fallimur quarto quoque anno in minutis .42. <reg norm="ſecundis" type="context">ſecũdis</reg> propè .56. amplius <lb/>quàm par ſit minutis ſcilicet .10. ſecundis ferè .44. ſingulis annis; </s> + <s xml:space="preserve">qui numerus .134. <lb/>multiplicatus, diem penè horarum .24. conſtituit; </s> + <s xml:space="preserve">penè inquam, quia minutum <reg norm="vnum" type="context">vnũ</reg> <lb/>deeſſet <reg norm="tantummodo" type="wordlist">tm̃modo</reg>, & ſecunda .44. ſi <reg norm="decem" type="context">decẽ</reg> illa minuta, & .44. ſe cunda annua, exquiſita eſ <lb/>ſent 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/>eiuſmodi eſſet emendatio, <reg norm="aptaque" type="simple">aptaq́;</reg> eius ratio à D. </s> + <s xml:space="preserve">Lilio oſtenditur, prout etiam Pe-<lb/>trus Pitatus Veronenſis tradidit, in eo, quem de vera anni quantitate tractatu con-<lb/>ſcripſit, nempe vt tribus primis centeſimis annis, centeſimus quiſque annus commu-<lb/>nis ſit, quartus ſubſequens centeſimus intercalaris: </s> + <s xml:space="preserve">quod ſanè fierineceſſe eſt. </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 centeſimis <reg norm="communibus" type="context">cõmunibus</reg>, tres quartas diei partes plus æquo detraxerimus, <lb/>non enim centeſimo quoque anno, ſed centeſimo trigeſimo quarto, dictus dies de-<lb/>trahi debet, poſtquam tres integros dies, qui quadringentis detrahendi erant, tre-<lb/>centorum annorum ſpacio detraxerimus; </s> + <s xml:space="preserve">ſitq́ue 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 ſuo loco reſtituta erunt. </s> + <s xml:space="preserve">Idcirco dictus iam <lb/>quadringenteſimus annus intercalaris & non communis conſtituendus erit, non alia <lb/>de cauſa, quam vt biſſexti ordinem ſequamur.</s> + </p> + <p> + <s xml:space="preserve">Is verò modus, qui à D. </s> + <s xml:space="preserve">Lilio traditus eſt, de ratione inueniendi ſingulis menſibus <lb/>Nouilunij diem, interdum fallere nos poſſet vno die; </s> + <s xml:space="preserve">prout Ianuario proximè <lb/>lapſo accidit; </s> + <s xml:space="preserve">quo ex præſcripto modo nouilunij, dies nonus illius menſis eſſe debuiſ <lb/>ſet, qui fuit tamen dies ſeptimus, ſexta decima hora cum dimidia poſt meridiem. </s> + <s xml:space="preserve">Ne <lb/>que etiam tutum eſt, via integrorum dierum, nulla habita horarum aut minutorum <lb/>ratione, nec minus ea, quæ à Pitato tradita eſt, mediorum ſeu æqualium <reg norm="motuum" type="context">motuũ</reg> pro <lb/>gredi: </s> + <s xml:space="preserve">At cenſerem potius veros motus ſequendos eſſe ex calculis exactarum tabu-<lb/>larum, quales Prutenicas eſſe iudico; </s> + <s xml:space="preserve">Et cum ſolius Paſchæ cauſa laboremus hac in <lb/>re, pleniluniorum verorum, in multos annos tabulas formarem, quæ æquinoctia ver <lb/>nalia ſequuntur, cum aſſignatione diei Paſchatis præcisè, prout fecit Pitatus; </s> + <s xml:space="preserve">non <lb/>via tamen æqualium pleniluniorum ſed verorum. </s> + <s xml:space="preserve">Porrò quod ad Paſchatis cele-<lb/>brationem attinet, rationi conſentaneum eſt, concilij Niceni decretum ea de re ſer <lb/>uari, prima ſcilicet dominica die poſt primum plenilunium, quod æquinoctium ver-<lb/>nale ſequitur; </s> + <s xml:space="preserve">hoc tamen animaduerſo, ſi dictum plenilunium primum poſt æquino-<lb/>ctium contingens, <reg norm="diem" type="context">diẽ</reg> dominicum ſortiretur; </s> + <s xml:space="preserve">nulla ratione tali die Paſcha celebran-<lb/>dum eſſe; </s> + <s xml:space="preserve">verum ſubſequenti, ne cum Hębreis conſentiat Eccleſia Chriſti: </s> + <s xml:space="preserve">quæ fuit <lb/>cauſa, vt in decreto concilij Niceni ſtatutum ſit, à quartadecima, in vigeſimam 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">eiꝰ</reg> <reg norm="ſententia" type="context">ſentẽtia</reg> in ſuis tabulis die Paſchate declarata, huiuſce anni Paſca <reg norm="celebrandum" type="context">celebrandũ</reg> <lb/>fuerit .23. </s> + <s xml:space="preserve">Martij, ipſomet de plenilunij non tantum æqualis, ſed veri.</s> + </p> + <p> + <s xml:space="preserve">Dies autem Paſchatum elapſorum, 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 poſt nouilunium Februarij, carnis priuij diem eſſe; </s> + <s xml:space="preserve">non autem <reg norm="cum" type="context">cũ</reg> <lb/><reg norm="sanctione" type="context">sãctione</reg> Patrum concilij Niceni, qua ſtatuerunt à vigeſima prima Martii dirigen-<lb/>dum eſſe Paſchatis diem, vt potè qui ſibi perſuaſerunt, circa eum diem <reg norm="æquinoctium" type="context">æquinoctiũ</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/>ſcha ex huiuſmodi ſuppoſitione concilij, poſt vigeſimam primam lunę celebretur, <lb/>cum ſeruata regula concilij non fuerit. </s> + <s xml:space="preserve">Prout <reg norm="manifeſtum" type="context">manifeſtũ</reg> eſt de Paſchate anni .1566. <lb/>celebrato .14. </s> + <s xml:space="preserve">Aprilis (quę fuit .24. lunę) quod .7. dicti menſis <reg norm="celebrandum" type="context">celebrãdum</reg> erat. </s> + <s xml:space="preserve">Tum <lb/>anno .1569. 10. </s> + <s xml:space="preserve">Aprilis ſolenne fuit Paſca, quod tertia eiuſdem eſſe debuerat. </s> + <s xml:space="preserve">Anno <lb/>deinde 1572. 6. Aprilis, dies fuit Paſchatis, quæ .30. Martij futura erat, anno vero <lb/>1575. in tertiam Aprilis Paſcha incidit, caſurum in .27. Martii.</s> + </p> + <p> + <s xml:space="preserve">Cum igirur (vt ex diplomate ad Celſit. tuam miſſo patet) <var>S.D.N.</var> mens ſit <reg norm="atque" type="simple">atq;</reg> vo <lb/>luntas, ut quiſque liberè in medium proferat quid hac dere ſentiat: </s> + <s xml:space="preserve">quædam mihi <lb/>non omnino præmittenda occurrunt, quæ tantis cœptis 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 Paſcha cætera-<lb/>q́ue feſta mobilia ab illo manantia, vt decreto concilij Niceni ſancitum eſt, ſcilicet <lb/>vt ipſum Paſcha celebretur prima dominica poſt 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 ſequitur; </s> + <s xml:space="preserve">verum etiam quò anni principium emendetur, <lb/>ſcilicet vt ad ſuum verum principium reuocetur annus. </s> + <s xml:space="preserve">Nempè ad diem hyemalis <lb/>ſolſtitij, quæ prima Ianuarij dies eſſe debet.</s> + </p> + <p> + <s xml:space="preserve">Deinde, tot dierum menſes conſtituantur, quot hac noſtra tempeſtate, ſol in ipſis <lb/>Zodiaci ſignis verſatur. </s> + <s xml:space="preserve">Poſtremò, quædam feſta immobilia in alios dies <reg norm="transferam" type="context">transferã</reg> <lb/>tur, <reg norm="celebrenturque" type="simple">celebrenturq́;</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 feſtis mobilibus quam immobilibus agat, imo etiam planè <lb/>æquum ſit, vt habeatur vtrorunque ratio, quò ſtatutis 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">cenſerem <lb/>ex eo anno, quem corrigere voluerimus, non modò dies .10. eſſe detrahendos, <reg norm="verum" type="context">verũ</reg> <lb/>etiam vnum & uiginti, illo ipſo anno; </s> + <s xml:space="preserve"><reg norm="idque" type="simple">idq́;</reg> duplici via; </s> + <s xml:space="preserve">aut partiendo menſes, 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 menſes ſint die. <lb/></s> + <s xml:space="preserve">rum duorum ſpatio ſolito breuiores, alter menſis vno deficiat: </s> + <s xml:space="preserve">aut conſtituto <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> ſequitur, ſit & <lb/>primus Ianuarij, & dies ſolſtitij ob quam cauſam exiſtimarem conſultiſſimum eiuſ <lb/>modi annum eſſe mileſimum quingenteſimum ſeptuageſimum nonum. </s> + <s xml:space="preserve">Quo quam <lb/>primum <var>.S.D.N</var>. </s> + <s xml:space="preserve">Pontifex max. ſuis temporibus huius correctionis manifeſtos effe-<lb/>ctus experiri & perpendere, atque diſpoſitionem anni non ſolum principio, ſed <reg norm="cae teris" type="simple">cę<lb/>teris</reg> partibus ſuis in vniuerſum tam concinnè <reg norm="apteque" type="simple">apteq́;</reg> reſpondere, & aſtrorum moti-<lb/>bus, & Eccleſiæ ſacroſanctæ ſanctionibus, ſe authore lætari poſſit.</s> + </p> + <p> + <s xml:space="preserve">Omnino <reg norm="itaque" type="simple">itaq;</reg> iudico detrahendos eſſe vnum & viginti dies elapſi erroris: </s> + <s xml:space="preserve">non de <lb/>cem tantum, quo hyemmalis conuerſio ad initium Ianuarij reuocetur; </s> + <s xml:space="preserve">idq́ue ne à <lb/>communi opinione de ipſo anni principio veritas diſcrepet, quæ principium Ianua-<lb/>rij, anni principium arbitratur. </s> + <s xml:space="preserve">etenim cum credant omnes <reg norm="annum" type="context">annũ</reg> à Ianuario inchoa-<lb/>ri, veritas autem ipſa ſic ſe habeat, vt nobis ſeptentrionalibus tunc inchoet annus, <lb/>cum ad nos Sol accedere incipit, aut dies augetur; </s> + <s xml:space="preserve">non conuenit principia eiuſmo-<lb/>di ſeparata & diſcrepantia eſſe. </s> + <s xml:space="preserve">Et hanc fuiſſe Numæ Pompilio mentem credibile <lb/>eſt, qui ad annum Romuli decem menſium, Ianuarium & Februarium addidit, vt <lb/>principium Ianuarij principium eſſet anni: </s> + <s xml:space="preserve">cuius rei argumentum eſſe poteſt, quod <lb/>C. </s> + <s xml:space="preserve">Iulij Cæſaris temporibus (qui multis annis poſt Numam fuit) <reg norm="atque" type="simple">atq;</reg> vti Pont. </s> + <s xml:space="preserve">Max. <lb/></s> + <s xml:space="preserve">corrigendorum feſtorum curam ſuſcepit hyemale ſolſtitium per aliquot dies retro-<lb/>ceſſerat; </s> + <s xml:space="preserve">nec mirum tamen eſſet, ſi Numæ temporibus, exactè prima Ianuarij die non <lb/>fuiſſet hyemale ſolſtitium, adhuc pubeſcente in Italia Aſtronomia.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi autem correctio dierum .21. poſt .2300. annos à Numa, quæ ſit per-<lb/>petuo ſeruitura, media emendatione ea, quæ de tribus centeſimis annis communi-<lb/>bus, & quarto intercalari, ſuperius propoſita fuit, non repudianda ei videatur, qui <lb/>ſciet, 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 eſt hyemale ſolſtitium, reduceretur.</s> + </p> + <p> + <s xml:space="preserve">Alio item argumento cuique patere poteſt, priſcos Romanos ſtatuiſſe annum ab <lb/>hyemali ſolſtitio initium ſumere, vt inquit Ouidius primo Faſtorum.</s> + </p> + <p> + <s xml:space="preserve">Bruma noui prima eſt, <reg norm="veterisque" type="simple">veterisq́;</reg> nouiſſima Solis.</s> + </p> + <p> + <s xml:space="preserve">Principium capiunt Phębus, & annus idem.</s> + </p> + <p> + <s xml:space="preserve">co quod diem naturalem à medio noctis inchoarent, ab eo puncto ſcilicet, quo Sol <lb/>ad noſtrum hemiſpherium 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, reſpondeat puncto meridiani ſub terra, in quo Sol ſemel in die reperitur: <lb/></s> + <s xml:space="preserve">Quòd apertè norunt hi, qui ſub polo boreali conſtituti ſunt. </s> + <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> facilè diſcerne <lb/>re poſſumus, diem ſcilicet & annum, quaſi ſibi ad inuicem medio ſuarum partium <lb/>reſpondere; </s> + <s xml:space="preserve">ſolſtitium inquam hyemale, mediæ nocti, æſtiuum meridiei, æquino-<lb/>ctium vernale ortui Solis, autumnale occaſui. </s> + <s xml:space="preserve">Quam tamen ſimilitudinem, multò <lb/>quam nos manifeſtius deprehendunt, hi qui (ut diximus) ſub polo borcali verſantur.</s> + </p> + <p> + <s xml:space="preserve">Quod ſi quis dubitet hac ratione correcto anno, quo nam pacto ad calculos coe-<lb/>leſtes motus medijs tabulis aſtronomicis hactenus in lucem æditis redigi poſſint, id <lb/>facilimum ſanè erit, exempli gratia; </s> + <s xml:space="preserve">aliquis planetę ſitum, aut alicuius ſtellę fixæ, quo <lb/>cunque die menſis anni correcti inuenire cupit, detrahat ex huiuſmodi <reg norm="tempore" type="context">tẽpore</reg> dies <num value="21">.<lb/>21.</num> ab Aera Chriſti, cum reſiduo ſupputet ſtellam, cuius ſitum ſcire deſiderat; </s> + <s xml:space="preserve">ſum-<lb/>pta quacunque tabula, ſupputatio erit exacta: </s> + <s xml:space="preserve">Cuius ratio cuilibet manifeſta erit, <lb/>qui ſciet annum vt potè .1579. dierum .344. tantummodo conſtitutum fuiſſe. </s> + <s xml:space="preserve">Nam <lb/>in ijſdem locis cœli prima die Ianuarij correcti, erunt ſtellæ quibus eſſe ſolebant .11 <lb/>Decembris præcedentis anni ex ſupputatione tabularum: </s> + <s xml:space="preserve">atque ita deinceps. </s> + <s xml:space="preserve">Alia <lb/>præterea via idem perfici poſſet inuentione omnium motuum cęleſtium ipſo princi <lb/>pio anni .1580. correcti: </s> + <s xml:space="preserve">hoc ſtatuto, vt hi motus radices eſſent Aeræ <var>S.D.N.</var> Grego <lb/>rij XIII. quod ſi alio <reg norm="tempore" type="context">tẽpore</reg> quiſpiam motus cęleſtes ad calculos redigere voluerit, <lb/>ſupputabit ab Aera huiuſmodi, quæ anno .1580. principium habuerit: </s> + <s xml:space="preserve">Quæ vt nobi <lb/>lius nomen ſortiatur, <reg norm="idque" type="simple">idq́;</reg> merito ex nomine Gregorij. XIII. Pont. Max. appelletur; <lb/></s> + <s xml:space="preserve">exemplo antiquarum, quæ ex Principum nominibus ſunt 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 ſummæ, quæ ex huiuſmodi Aera Gregoriana ex tabu-<lb/>lis colligetur, ipſiuſmet Aeræ radices addantur, vt exactus calculus habeatur. </s> + <s xml:space="preserve">Et <lb/>hæc ſit primæ ſententiæ noſtræ explicatio.</s> + </p> + <p> + <s xml:space="preserve">Altera erit numerum dierum menſium anni alia ratione quam nunc ſe habeat, or <lb/>dinandum eſſe: </s> + <s xml:space="preserve">nempe vt Ianuarius, Nouember <reg norm="atque" type="simple">atq;</reg> December dies .29. ſinguli con <lb/>tineant, Februarius, Martius, & October .30. </s> + <s xml:space="preserve">Aprilis, Maius, Auguſtus, & September <lb/>dies .31. Iunius, ac Iulius .32. atque id hac potiſſimum de cauſa, vt Sol unum <reg norm="quodque" type="simple">quodq́;</reg> <lb/>ſignum calendis menſrum ingredi poſſit. </s> + <s xml:space="preserve">Nam detractis (ut dictum eſt) diebus .21. <lb/>& reuocato ingreſſu Solis in principium Capricorni ad principium Ianuarij, in quo <lb/>ſigno hac noſtra tempeſtate, Sol, dies propè .29. & quartam vnam verſatur: </s> + <s xml:space="preserve">ſi Ianua <lb/>rius .29. dies continebit, exactis hiſce diebus, ingredietur Aquarium circa princi-<lb/>pium Februarii; </s> + <s xml:space="preserve">hæret autem hoc noſtro ſæculo in Aquario Sol dies propè .29. cum <lb/>dimidio; </s> + <s xml:space="preserve">quare ſi Februarius erit .30. dierum, elapſis ipſis diebus, Sol ingredietur pi-<lb/>ſces circa principium Martii: </s> + <s xml:space="preserve">& ſic de cæteris.</s> + </p> + <p> + <s xml:space="preserve">Quamobrem ſi generali correctione annus emendandus erit, pulcherrimè acci-<lb/>det, ſi menſes anni cum duodecim partibus cœleſtibus, itineris annui Solis, concor-<lb/>dauerint; </s> + <s xml:space="preserve"><reg norm="eiſque" type="simple">eiſq́;</reg> aptè reſponderint. </s> + <s xml:space="preserve">Qua ex re, varię vtilitates promanabunt, pręſertim <lb/>Nautis, Agricolis, Medicis, & alijs qui vera principia, & interualla temporum per <lb/>ſpecta habebunt: </s> + <s xml:space="preserve">terminos item & interualla incrementi & diminutionis dierum & <lb/>noctium, & eorundem æqualitatis. </s> + <s xml:space="preserve">Exempli cauſa, ſcient 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 huiuſmodi conuerſione Solis ad nos dependent. </s> + <s xml:space="preserve">Pariter <lb/>ſcient omnes primam diem Iulij, non tantum æqualiter annum diuidere, ſed 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, quę Solis conuerſionem <lb/>ad auſtrales ſequuntur.</s> + </p> + <p> + <s xml:space="preserve">Neconon intelliget vnuſquiſque primam diem Aprilis, <reg norm="primamque" type="simple">primamq́;</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 ſuperat, & alia huiuſmodi, quæ ab æquinoctijs <reg norm="dependent" type="context context">depẽdẽt</reg>.</s> + </p> + <p> + <s xml:space="preserve">Si vero quiſpiam obijciat, modum hunc noſtrum & 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">tũc</reg> <lb/>Sol hærebit in ſigno Sagittarij .32. diebus, totidem in Capricorno, in Geminis vero <lb/>29. totidem in Cancro; </s> + <s xml:space="preserve">ex quo ſequetur prioribus <reg norm="contrarius" type="context">cõtrarius</reg> effectus; </s> + <s xml:space="preserve">huic ego reſpon <lb/>debo, tale <reg norm="quidpiam" type="context">quidpiã</reg> non euenturum, niſi exactis ab hoc anno annis .240@0. quod ſi <lb/>mundus poſthac totidem annis, quot fuit antehac, perdurauerit, punctus augis non <lb/>amplius à ſitu præſenti, quàm .45. gradibus diſtabit. </s> + <s xml:space="preserve">Verum demus <reg norm="modum" type="context">modũ</reg> noſtrum <lb/>& regulam in annos ter, aut quater mille ſubſeruire poſſe, nec amplius, certè hoc <lb/>toto tempore nullius momenti penè erit, quæ accidere poterit mutatio, tametſi ela-<lb/>pſis quatuor millibus annorum Februarius eſſe debebit .29. dierum. Aprilis & No-<lb/>uember .30. Iunius & October .31. Auguſtus .32. in aliis verò menſibus nihil mutan-<lb/>dum erit. </s> + <s xml:space="preserve">Ecce quam ſit nullius momenti mutatio.</s> + </p> + <p> + <s xml:space="preserve">Quæ ſi Iulij Cæſaris temporibus fuiſſent animaduerſa nunquam omiſſa fuiſſent, <lb/>ſed ſcientiæ Aſtronomicæ nondum (vt ita dicam) confirmata ætas, cum alibi, <reg norm="tum" type="context">tũ</reg> maxi <lb/>mè in Italia, quo minus hæc aut ſcirentur aut ſtatuerentur impediebat.</s> + </p> + <p> + <s xml:space="preserve">Tertia ratio eſt, vt non <reg norm="ſolum" type="context">ſolũ</reg> feſta mobilia, verum <reg norm="ent" type="context">ẽt</reg> immobilia ad meliorem regu <lb/>lam (ut dictum eſt) reuocentur, ſi ſuis temporibus celebranda erunt. </s> + <s xml:space="preserve">Quorum <reg norm="primum" type="context">primũ</reg> <lb/>eſt Natiuitas Domini, & quæ ab ea pendent; </s> + <s xml:space="preserve">nempe Circuncifio, Epiphania, Purifi-<lb/>catio, Annunciatio, & Natiuitas Io. Baptiſtæ. </s> + <s xml:space="preserve">ita vt dies Natalis Domini celebretur <lb/>prima die anni, cum Dei filius naſci voluerit circa verum principium anni, quod à <lb/>ſolſtitio hyemali initium ducit, & in ipſo principio diei naturalis ex <reg norm="Romanorum" type="context">Romanorũ</reg> ſen <lb/>tentia, media ſcilicet nocte, tanquam qui ſummæ lætitiæ principium, poſt longos & <lb/>graues filiorum Adæ mærores, eſſet allaturus. </s> + <s xml:space="preserve">Nec forſan Ianuarij nomini, à vete-<lb/>ribus Iano bifronti dicati hæc mutatio non conueniret, cum in ip ſo ſeruatore, duæ <lb/>veluti frontes & formæ vnitæ ſint, duæ ſcilicet naturæ diuina & humana. </s> + <s xml:space="preserve">Hac ratio <lb/>ne abuſus tolletur, natus ex diuerſis moribus Tabulariorum, quorum alij monumen-<lb/>ta, ſeu quæ uocant Inſtrumenta, à die Natiuitatis Domini incohant, alij à Circunci <lb/>ſione, alij à Calendis Martij, nonnulli à Paſchate; </s> + <s xml:space="preserve">quæ varietas innumerabiles lites <lb/>affert & abuſus propè infinitos, ob dubiam & ancipitem ſcripturam. </s> + <s xml:space="preserve">Indictionum <lb/>præterea ordini, hic noſter modus nihil officiet; </s> + <s xml:space="preserve">celebrato Natali celebrabitur Cir-<lb/>cunciſio octaua Ianuarij. </s> + <s xml:space="preserve">Epiphania .13. eiuſdem. </s> + <s xml:space="preserve">Purificatio .11. Februarij quæ erit <lb/>40: </s> + <s xml:space="preserve">dies à Natiuitate ſeruatoris. </s> + <s xml:space="preserve">Prima Aprilis Annunciatio Virginis ſolennis erit, <lb/>ipſo nempè die æquinoctij, natiuitas Diui Io: </s> + <s xml:space="preserve">Baptiſtę celebrabitur Prima Iulij die <lb/>quæ erit ſolſtitij æſtiui, cum illa diminutionem capit. vtrectè Diuus Auguſtinus il-<lb/>la verba Io: </s> + <s xml:space="preserve">Baptiſtę interpretatus fuerit. </s> + <s xml:space="preserve">Illum opportet creſcere, me autem minui: <lb/></s> + <s xml:space="preserve">in quibus ſic tantus Doctor philoſophatur, vt tempus etiam natiuitatis ſerui & do-<lb/>mini præclare notet dicens, natus eſt ſeruus cum decreſcunt dies, natus eſt Dominus <lb/>cùm creſcere incipiunt.</s> + </p> + <p> + <s xml:space="preserve">Inſignes etiam Theologi admonuerunt habendam rationem eſſe nonnullorum <lb/>feſtorum, vt Diui Antonij, diuorum Fabiani & Sebaſtiani, & aliorum ſanctorum, + <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 hęc <var>.S.D.N.</var> curæ erunt, ut in aptiſ <lb/>ſima tempora transferantur.</s> + </p> + <p> + <s xml:space="preserve">Admonuerunt præterea transferendos eſſe dies feſtos Beati Stephani, Ioannis, <lb/>& Innocentium, vt quemadmodum factum eſt hactenus, diem natalis proximè ſe <lb/>quantur, ob multorum Doctorum, non recentium modo, ſed etiam antiquorum ob <lb/>ſeruantiam; </s> + <s xml:space="preserve">qui ſuis omelijs & concionibus multa piè, de myſteriis ſucceſſionis Feſto <lb/>rum huiuſimodi 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/>ſe in ſigno Virginis ſol verſabitur, ita Eccleſia Der in cęlebrandi tantæ Virginis ma <lb/>tris Deilaudibus occupetur.</s> + </p> + <p> + <s xml:space="preserve">Atque hęc ſunt Serenisſime Princeps, quę longa & attenta cogitatione à me exa <lb/>minata, atque perpenſa fuerunt; </s> + <s xml:space="preserve">quæſitam diligenter & accuratè expendentur ab <lb/>his, quorum intereſt, quam mihi apta & rationi conſentanea, ac vera penitus, imo <lb/>(quod me magis afficit) etiam tibi viſa fuerunt; </s> + <s xml:space="preserve">non dubito quin placitura ſint; </s> + <s xml:space="preserve">& vo <lb/>tis ſummi Pont. aliqua ex parte ſatisfactura. </s> + <s xml:space="preserve">eò magis quòd te iubente, & cogitata à <lb/>me, & ſcripta fuerint. </s> + <s xml:space="preserve">Vale Princeps Sereniſſime, & qua ſoles hylaritate cętera no <lb/>ſtra, etiam has breues vigilias ſuſcipe & foue. </s> + <s xml:space="preserve">Dat. Auguſtæ Taurinorum Kal. <lb/></s> + <s xml:space="preserve">Aprilis. <num value="1578">MDLXXVIII</num>. </s> + </p> + <p> + <s xml:space="preserve">T. Celſitudinis.</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 celſitudine tua nobis proponitur non ſolum poſſibile eſt, ſed <lb/>facile etiam ad ſoluendum, hoc eſt quod circulus talis inueniatur, qui poſſit cir <lb/>cunſcribere, ſeu capere quadrilaterum ex quatuor datis rectis lineis terminatum, vel <lb/>ſic, datis quatuor rectis lineis ex quibus quadrilaterum poſſit eftici, tale efficiatur vt <lb/>circa ipſum, circulus poſſit circunſcribi.</s> + </p> + <p> + <s xml:space="preserve">Sint igitur .4. lineæ propoſitæ <var>.b.d</var>: <var>q.b</var>: <var>a.q</var>: et <var>.a.d.</var> ex <reg norm="quibus" type="simple">quibꝰ</reg> poſſibile ſit quadrilate <lb/>rum conſtitui, tale vero conſtituatur, vt aliquis circulus poſſit ipſum circunſcribere. <lb/></s> + <s xml:space="preserve">imaginemur autem hoc factum eſſe, quod quidem quadrilaterum ſit <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 ſint <var>.q.d.</var> et <var>.a.b.</var> quæ ſe inuicem interſecent in puncto <var>.o.</var> vnde cum anguli <lb/>contra ſe poſiti circa <var>.o.</var> æquales inuicem ſint 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> ſimilis erit triangulo <var>.d.o.b.</var> et <var>.q.o.b.</var> ſimilis trian-<lb/>gulo <var>.a.o.d.</var> ex definitione. </s> + <s xml:space="preserve">Vnde eadem proportio erit ipſius <var>.q.o.</var> ad <var>.b.o.</var> quæ ipſius <lb/><var>q.a.</var> ad <var>.b.d.</var> & ipſius <var>.b.o.</var> ad <var>.o.d.</var> eadem quæ <var>.q.b.</var> ad <var>.a.d.</var> & ipſius <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 compoſita ex <lb/>ea quæ eſt <var>.q.o.</var> ad <var>.o.b.</var> ex <var>.o.b.</var> ad <var>.o.d.</var> quæ nobis cognitę ſunt, mediante <lb/>proportione ipſius <var>.q.a.</var> ad <var>.b.d.</var> & ipſius <var>.q.b.</var> ad <var>.a.d.</var> proportio ſimiliter ipſius <var>.b.o.</var> <lb/>ad <var>.o.a.</var> nobis cognita erit, vt compoſita ex proportione ipſius <var>.b.o.</var> ad <var>.o.q.</var> & <lb/>ipſius <var>.o.q.</var> ad <var>.o.a.</var> cognitis, mediante proportione ipſius <var>.b.d.</var> ad <var>.q.a.</var> & ipſius <var>.q.b.</var> ad <lb/><var>a.d.</var> cum <reg norm="autem" type="context">autẽ</reg> proportio ipſius <var>.q.o.</var> ad <var>.o.b.</var> nobis cognita ſit, </s> + <s xml:space="preserve">tunc nobis cognita erit <lb/>proportio ipſius <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> eſt vt <var>.a.o.</var> ad <var>.o.d.</var> ex ſimilitudine, <lb/></s> + <s xml:space="preserve">quare proportio compoſiti ex primo, & quarto terminorum ad compoſitum ex .2. & <lb/>tertio, cognita erit. </s> + <s xml:space="preserve">ſed quod fit ex <var>.q.d.</var> in <var>.a.b.</var> cognitum nobis eſt, vt æquale duobus <lb/>productis, hoc eſt ex <var>.q.a.</var> in <var>.d.b.</var> & ex <var>.q.b.</var> in <var>.d.a.</var> ex ſecunda primi Almageſti. </s> + <s xml:space="preserve">quæ <lb/>producta nobis cognita ſunt, cum nobis data ſint eorum latera. </s> + <s xml:space="preserve">Quapropter facta <lb/>cum fuerit figura quadrilatera rectangula ſimilis alicui alterirectangulæ figuræ pro <lb/>ductæ à duobus lateribus inuicem ita proportionatis, vt ſe habet <var>.q.d.</var> ad <var>.a.b.</var> æqua-<lb/>lis tamen duobus productis, hoc eſt 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. ſexti Eucli quæ quidem figura, exempli gratia, ſit <var>.u.t.</var> eius verò latera <lb/>ſint <var>.u.n.</var> et <var>.n.t.</var> Hæc enim dico æqualia eſſe <var>.q.d.</var> et <var>.b.a.</var> hoc eſt <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> ſed, <var>s.n.</var> æqualis ſit <var>.q.d.</var> et <var>.n.r.</var> æqua-<lb/>lis <var>.a.b.</var> <reg norm="ſintque" type="simple">ſintq́;</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">eademq́</reg> proportio erit <var>.u.n.</var> ad <var>.n.t.</var> quę <var>.s.n.</var> ad <var>.n.r.</var> eo <reg norm="quod" type="simple">ꝙ</reg> ita fa-<lb/>ctum fuit, cum autem ita ſit <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> ſed quia ita eſt <var>.u.n.</var> ad <var>.n.r.</var> vt <var>.s.n.</var> ad <var>.n.t.</var> ex 15. ſexti, </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 ſequitur <var>.n.t.</var> et <var>.n.r.</var> inuicem æquales eſſe, et <var>.u.n.</var> ſimiliter <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> ipſius quadrilateri, difficile <lb/>non erit eius angulos inuenire, eo <reg norm="quod" type="simple">ꝙ</reg> mediante <var>.a.b.</var> cognita, ſimul 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 circunſcriptibilis cuiuſuis di-<lb/>ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo <lb/>dem circulo circunſcriptum erit, ex communi ſcientia.</s> + </p> + <p> + <s xml:space="preserve">SEd vt ipſa operatio facilior fiat, Sint eędem 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">quadrilaterũ</reg> effici. </s> + <s xml:space="preserve">Videatur deinde primò quas volumus oppoſitas ſibi <lb/>inuicem eſſe, ponatur ergò ut <var>.q.a.</var> et <var>.b.d.</var> velimus oppoſitas inuicem facere, et <var>.q.b.</var> <lb/>cum <var>.a.d.</var> ſimiliter, accipiemus nunc <var>.K.</var> cuiuſuis magnitudinis, cui comparetur <var>.e.</var> <lb/>ita proportionata, vt <var>.q.b.</var> eſt ipſi <var>.a.d.</var> ex doctrina .10. ſexti 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 ſimiliter <var>.h.</var> ita pro-<lb/>portionata ad <var>.e.</var> et <var>.g.</var> ad <var>.k.</var> vt <var>.b.d.</var> eſt ad <var>.q.a.</var> vel <var>.g.</var> ad <var>.h.</var> vt <var>.a.d.</var> ipſi <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 compoſitum ſit <var>.g.e.</var> & <lb/>ita duæ <var>.K.</var> et <var>.h.</var> ex quibus ſit <var>.K.h</var>. </s> + <s xml:space="preserve">Nunc ex iſtis 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/>ſit ex <var>.q.a.</var> in <var>.b.d.</var> reliquum verò ſit ex <var>.q.b.</var> in <var>.a.d.</var> quæ quidem ſint <var>.f.m</var>.</s> + </p> + <p> + <s xml:space="preserve">Quo facto deſignetur rectangulum <var>.u.t.</var> ex .25. ſexti, quod æquale ſit duobus re-<lb/>ctangulis <var>.f.</var> et <var>.m.</var> ſimile tamen <var>.Z.</var> cuius rectanguli vnum latus correſpondet <var>.e.g.</var> reli-<lb/>quum verò <var>.K.h.</var> in proportione, ſed in æqualitate, vnum correſpondet <var>.q.d.</var> <reg norm="reliquum" type="context">reliquũ</reg> <lb/>vero <var>.a.b.</var> diametris ipſius quadrilateri.</s> + </p> + <p> + <s xml:space="preserve">Accipiatur nunc latus illud quod correſpondet <var>.K.h.</var> hoc eſt ipſi <var>.a.b.</var> maius ſcili-<lb/>cet, & ſimul cum <var>.b.d.</var> et <var>.a.d.</var> formetur <reg norm="triangulum" type="context context">triãgulũ</reg> <var>.a.b.d.</var> ex .22. primi Eucli. circa quod <lb/>circunſcribatur circulus ex .5. quarti. </s> + <s xml:space="preserve">& inuentum erit quod quęrebamus.</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/>abſque correctione ſemper nauigari <lb/>non poſſe.</head> + <head style="it" xml:space="preserve">Vbi not antur Petri <reg norm="Nonij" type="lig">Nonij</reg> lapſus in correctione erroris nauis. <lb/>Et alij Petri Medinæ errores.</head> + <head xml:space="preserve">ILLVSTRISSIMO ANDREAE PROVANAE <lb/>Leinici Domino, Fruzaſci comiti, Aequiti Torquato, inthimo Sere-<lb/>niſsimi Sabaudiæ Ducis Conſiliario, <reg norm="eiuſque" type="simple">eiuſq́;</reg>, & ſacræ religio-<lb/>nis ſanctorum Mauritij, & Lazari Claſsi Præfecto.</head> + <p> + <s xml:space="preserve"><emph style="sc">INter</emph> Eximiastuas virtutes, reinauticæ peritia Illuſtris emicat <lb/>merito ad te ſcribendum duxi, quod ad eam facultatem perti-<lb/>nens excogitaui, ſimul cum quibuſdam alijs inſtrumentis, vt non-<lb/>nihil commodi attuliſſe videar maritimis negotijs, & aliqua ex <lb/>parte animi mei erga te propenſionem indicauiſſe.</s> + </p> + <p> + <s xml:space="preserve">PEr vnum <reg norm="eum" type="context">eũ</reg><reg norm="demque" type="simple">demq́;</reg> parallelum in primis <reg norm="abſque" type="simple">abſq;</reg> aliqua corre <lb/>ctione ſemper nauigari poſſe, omninò nego. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, <reg norm="verum" type="context">verũ</reg> eſtid quod Petrus Nonius in <lb/>initio ſui operis oſtendit, ideſt nauim verſus æquatorem ſemper declinare: </s> + <s xml:space="preserve">qui <reg norm="cum" type="context">cũ</reg> <lb/>corrigit errorem, fallitur, cum ipſe, eandem nauim, parallelam æquatori in vno ver <lb/>ticali ipſi æquatori propinquiori, & non in primo parallelo dirigit, itaque exiſtimat <lb/>in fine itineris, vbi deſcribit punctum <var>.o.</var> eam in eodem parallelo priori repe-<lb/>riri debere, quod <reg norm="verum" type="context">verũ</reg> <reg norm="non" type="context">nõ</reg> eſt, quia ea correctio efficit, vt motus nauis <reg norm="effectum" type="context">effectũ</reg> <reg norm="cuiuſdam" type="context">cuiuſdã</reg> <lb/><reg norm="deſcenſus" type="context">deſcẽſus</reg> <reg norm="per" type="simple">ꝑ</reg> ſcaligradum pręſtet, in quo à gradu in <reg norm="gradum" type="context">gradũ</reg> fiat deſcenſus, ſed ſi per gra <lb/>dus <reg norm="tantum" type="wordlist">tm̃</reg> aſcenderet <reg norm="quantum" type="context context">quãtũ</reg> <reg norm="deſcendit" type="context">deſcẽdit</reg>, <reg norm="dubium" type="context">dubiũ</reg> <reg norm="non" type="context">nõ</reg> eſt quin in fine ita eſſet ſe habitura <reg norm="quem" type="context">quẽ</reg> <lb/>ad modum in medio & in principio, cum verò ſemper deſcendat, abſque vlla aſcen-<lb/>ſione, neceſſariò ſic ſemper procedens, remota cum eſſent impedimenta terræ, ſub <lb/>æquatore reperiretur, ſub quo perpetuò circuiret globum.</s> + </p> + <p> + <s xml:space="preserve">Idem ſub quolibet meridiano præſtare poteſt, ideſt vno <reg norm="eodemque" type="simple">eodemq́;</reg> vento circun-<lb/>uerti: </s> + <s xml:space="preserve">ſed per alios circulos quam per hos duos (ſiue circulus magnus ſiue paruus) id <lb/>nunquam perfectè efficere poteſt, de parallelis iam manifeſtum eſt, cum impetus na <lb/>turalis corporum, quæ mota ſunt ſint ſemper in ſuperficiebus circulorum maiorum, <lb/>quorum circunferentię cum circunferentijs minorum, præter quam per vnum quod <lb/>dam punctum quando adinuicem contiguæ ſunt, aut per duo ideſt cum ſe ſe interſe-<lb/>cant non communicant, ita quod ad efficiendum, vt triremis aliqua, aut nauis, per <lb/>aliquem ex parallelis ad æquatorem moueatur, neceſſario ſit 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 conſtet virtus mouens remis, aut velis. </s> + <s xml:space="preserve">Sed per quemlibet <reg norm="alium" type="context">aliũ</reg> <lb/>circulum maiorem, qui non ſit aut æquator aut aliquis ex meridianis, eſt penitus im <lb/>poſſibile. </s> + <s xml:space="preserve">ideſt vt vnius venti vi nauis impellatur. </s> + <s xml:space="preserve">Quod vt clarè pateat, ſit orizon <var>.<lb/>a.c.b.d.</var> & æquator <var>.c.q.t.d.</var> & vnus meridianorum ſit <var>.a.r.n.t.b.</var> in quo <var>.n.</var> ſit Zenit ſub <lb/>quo primum nauis reperiatur et <var>.r.</var> ſit polus ſeptentrionalis. </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> conſtituat angulum <var>.a.n.p.</var> ſeu <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 ipſius orizontis cum azimut, erit medio in loco inter <var>.b.</var> et <var>.c.</var> & ideo quarta <var>.n.f.</var> ip <lb/>ſius azimut ſecabit quartam <var>.c.t.</var> ipſius 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 ſimul 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 ſi voluerimus.</s> + </p> + <p> + <s xml:space="preserve">Ponamus nunc nauem à puncto <var>.n.</var> diſcedere ſeu iter facere verſus <var>.u.</var> punctum, & <lb/> + <anchor type="figure" xlink:label="fig-0227-01a" xlink:href="fig-0227-01"/> + in ipſo <var>.u.</var> reperiri, iam in hoc ſitu ha-<lb/>bebimus angulum huius ſecundi me-<lb/>ridiani <var>.r.u.p.</var> qui quidem in hoc caſu <lb/>minor eſſet angulo <var>.r.n.p.</var> extrinſeco <lb/>trianguli <var>.r.u.n.</var> ex conuerſo ſecundæ <lb/>partis .48. <reg norm="propoſitionis" type="simple">ꝓpoſitionis</reg> tertij lib. de <reg norm="trian gulis" type="context">triã<lb/>gulis</reg> Monteregij, ſeu ex .13. primi Me <lb/>nelai, cuius anguli <var>.u.</var> arcus orizontalis <lb/>ſit <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. verſus <var>.q.</var> diuerſus ab illo qui <lb/>prius ab <var>.n.</var> verſus <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 eſt quod aliquo modo fieri <lb/>non poteſt, vt nauis ab aliquo loco ad <lb/>alium, breuiſſimo interuallo ire poſſit <lb/>ideſt per gyrum circuli maioris ſphæræ vno tantummodo <reg norm="vento" type="context">vẽto</reg> eam impellente, præ <lb/>ter quam in ęquatore, ſeu in aliquo quouis meridianorum, nos autem ire per gyrum <lb/>alicuius paralleli dementia eſſet, niſi neceſſitas cogeret.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi demonſtrationis ope, quantum decipiatur Petrus Medina cap .6. lib. <lb/>3. cognoſcitur, vbi ſic ſcribit; </s> + <s xml:space="preserve">Vbicunque locorum reperiatur homo, aliquem circu-<lb/>lum qui vniuerſum ambiat imaginatione ſibi confingens, per totum eum circulum <lb/>vno <reg norm="eodemque" type="simple">eodemq́;</reg> vento nauigatio ſuſcipitur. </s> + <s xml:space="preserve">Ex hac etiam demonſtratione, quàm fal <lb/>ſa ſit charta maritima patet, cuius beneficio exiſtimant nautę ſe per breuiſſimum iter <lb/>a loco ad locum vehi etiamſi dicti loci non ſint ambo in æquatore, aut in aliquo me <lb/>ridiano, ſed extra dictos circulos vnico tantum vento impellente & ſi in paruis æquo <lb/>fibus hic error parum depræhenditur, forte tamen in magno Oceano clarè pateret. <lb/></s> + <s xml:space="preserve">In ſuperius igitur dicta demonſtratione iam oſtendi, quod ſi velimus vehi ab vno lo <lb/>co ad alium beneficio alicuius circuli maioris, præter duos iam dictos, hoc fieri non <lb/>poteſt vno <reg norm="eodemque" type="simple">eodemq́;</reg> <reg norm="vento" type="context">vẽto</reg> impellente. </s> + <s xml:space="preserve">Vnde ſequitur, omnia ea interualla quæ vno <lb/><reg norm="eodemque" type="simple">eodemq́</reg> vento tranſibimus 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 eſ-<lb/>ſe, planiſphęrium Prolomei & Iordani citat, non animaduertit quam diuerſo mo-<lb/>do a charta maritima huiuſmodi inſtrumentum ſit fabricatum, cum exceptis orizon <lb/>te recto, & meridiano in dicto inſtrumento quilibet alius circulus ſit circulus, ſiue ſit <lb/>almicantarat, ſiue azimur, ſiue æquator, ſiue tropicus, ſiue zodiacus, ſiue alius quiuis <lb/>circulus, eum in charta maritima ne vna quidem ſit linea, quę non ſit 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 poſitæ meæ demonſtrationis ope, deuenimus in cognitionem magnitu <lb/>dinis arcus <var>.n.q.</var> cognoſcimus etiam angulum <var>.n.q.t.</var> vnde nobis <reg norm="manifeſtum" type="context">manifeſtũ</reg> eſſet quo <lb/>vento oporteret iter facere. cum à puncto <var>.q.</var> nauis aliqua diſceſſura eſſet, in eodem <lb/>azimut propoſito. </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 ſupponitur, ſimul 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 cognoſcendam diſtantiam <lb/>vnius meridiani ab alio præſcribit, in genere eſt falſus, etiam ſi is ab antiquis eum de <lb/>ſumat, qui, hic non viderunt quam magna inter meridianos differentia ſit interuallo <lb/>rum eorum quæ ſunt vicina polis & eorum quæ ſunt circa æquatorem.</s> + </p> + <p> + <s xml:space="preserve">Falſus eſt etiam modus ab eo traditus ad cognoſcendos gradus longitudinis per <lb/>medium itineris cogniti in quouis parallelo extra æquatorem facti, & hoc cap .14. li <lb/>bri tertij eiuſdem, & primo cap. lib. 4. <reg norm="continetur" type="context">cõtinetur</reg>, vbi .17. leucas cum dimidia cuilibet <lb/>gradui tam paralleli quàm meridiani aſſignat.</s> + </p> + <p> + <s xml:space="preserve">Falſum eſt etiam quod ab eo aſſeritur, Solem, cum reperiretur in æquatore, circa <lb/>eos qui ſub ipſo æquatore habitant, vnius diei <reg norm="noctisque" type="simple">noctisq́;</reg> ſpatio per omnes uentos cir-<lb/>cunuolui. </s> + <s xml:space="preserve">quia illis æquator idem eſt cum verticali, qui duos tantum rhumbos pro-<lb/>ducit, ideſt orientis, & occidentis: </s> + <s xml:space="preserve">hic verò error, in ſecundo cap. lib. 6. habetur.</s> + </p> + <p> + <s xml:space="preserve">Falſum eſt etiam quod profert Solem ijs qui habitant ſphæram obliquam, qua-<lb/>libet hora tertia, regulariter ab vno rhumbo ad alium ex præcipuis ideſt ab vno azi <lb/>mut ad alium progredi, quemadmodum eadem cap .2. lib. 6. et .7. cap. ſeptimi libr. <lb/>ſcribit. </s> + <s xml:space="preserve">Huius autem rei falſitas ita facile depræhendetur, ponamus hemiſphęrium <lb/>orientale, verbi gratia, cuius meridianus ſit <var>.p.z.b.</var> æquator <reg norm="autem" type="wordlist">aũt</reg> <var>.e.m.</var> vnus verò paral <lb/>lelorum ſeptentrionalium ſit <var>.c.a.</var> in quo Solem exiſtere ponamus, orizon autem ſit <var>.<lb/>b.m.</var> zenit vero <var>.z.</var> polus arcticus <var>.p.</var> ſit poſtea azimut <var>.z.q.</var> à meridiano diſtans per gra <lb/>dus .45. qui quidem azimut in hoc hemiſphęrio erit rhumbus illius venti, quem uul-<lb/>go Itali Sirocum dicunt, et <var>.z.m.</var> ſit azimut verticalis qui in hoc hemiſphærio erit <reg norm="ron" type="context">rõ</reg> <lb/>bus venti orientalis, ita <reg norm="quod" type="simple">ꝙ</reg> ſecundum Medinam à rhumbo <var>.z.m.</var> ad <var>.z.q</var>. </s> + <s xml:space="preserve">Solabſoluet <lb/>ſpatium <reg norm="temporis" type="context">tẽporis</reg> trium horarum, & aliud <lb/>æquale temporis ſpatium abſoluet à <reg norm="rhum" type="context">rhũ</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 ipſo 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">nũc</reg> Solem reperiri in ęqua <lb/>tore, vbi per ipſum Medinam arcus <var>.u.m.</var> <lb/>ſimiliter eſſet graduum .45. & ſic <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. ſed cum <lb/>arcus <var>.e.u.</var> ſit graduum .45. ex ſuppoſito <lb/>ipſius Medinæ, ſequeretur arcum <var>.e.f.</var> <lb/>æqualem eſſe arcui <var>.e.u.</var> pars igitur æqua <lb/>lis erit ſuo 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 diſtantijs Solis à meridiano ſcribit, hanc opinionem Petrià Medina & <lb/>corum qui idem ei perſuaſerunt falſam eſſe demonſtrat.</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">Falſum eſt 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> inſularum quæ azore dicuntur, pyxidem <lb/>verſus eum ventum qui vulgò Græcus dicitur, & ab occidentali verſus eum qui Ma <lb/>giſter dicitur, vergere, huius rei nulla eſt ratio.</s> + </p> + <p> + <s xml:space="preserve">Ego enim huiuſmodi rationem reperiri poſſe contendo, quæ talis eſt, quia pars <lb/>roſæ (ut vocant) à magnete tacta, ad aliquod punctum, aut ſitum globi terrę, in eo-<lb/>dem meridiano inſularum, quæ Azore dicuntur, vltra ſitum poli arctici in terra diri-<lb/>geretur, ita vt ſitus dicti poli in terra eſſet in dicto meridiano, inter locum qui ab in <lb/>dice roſæ aut pyxidis reſpiceretur, & dictas inſulas, id quod ſuperius ſcripto meridia <lb/>no facile cognoſci poteſt, ſumendo pro inſulis ſitum <var>.e.</var> in meridiano et <var>.z.</var> pro polo, <lb/>et <var>.p.</var> pro loco qui à pyxide ſit viſus, <reg norm="imaginando" type="context">imaginãdo</reg> deinde pyxidem in <var>.f.</var> magis orienta <lb/>li quam eſt <var>.e.</var> clarum eſt lineam quæ reſpicit (ponamus) <var>f.p.</var> verſus Græcum & ab alia <lb/>parte verſus Magiſtrum 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, ſeu nauigatoriæ chartę, quemadmodum aliquoties inter nos ſermonem <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">efficeretq́;</reg> vt nauis ſuper aquę globum, beneficio circulorum maiorum, quam <lb/>optimè poſſet, ideſt 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">ꝙ</reg> mihi ex <lb/>animi voto ſucceſſurum putaui, beneficio <reg norm="quinque" type="simple">quinq;</reg> circulorum circundantium <reg norm="aliquem" type="context">aliquẽ</reg> <lb/>globum <reg norm="terreſtrem" type="context">terreſtrẽ</reg> & maritimum, quales ij ſunt qui in inferiori Germania à Gerardo <lb/>Mercatore ſtruuntur, qui vno pede cum dimidio diametri conſtet, ideſtſeſquipede.</s> + </p> + <p> + <s xml:space="preserve">Sit ergo, exempli gratia, huiuſmodi globus <var>.a.b.d.</var> circa quem duo circuli, aut cir<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 ſe globi polos mediantibus extremitatibus axis mun-<lb/>di contineat, qui quidem poli à punctis ſuarum interſectionum per <reg norm="quartam" type="context">quartã</reg> ex æquo <lb/>in punctis <var>.f.</var> et <var>.g.</var> ita diſtent, vt globus circa eoſdem, in ſitu longitudinis mundi vol-<lb/>ui poſſit. </s> + <s xml:space="preserve">Huiuſmodi 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 ſit <var>.h.e.K.</var> cum primo ad angulos rectos in puncto <var>.e.</var> & <lb/>in ſuo oppoſito connexus, & is appellabitur æquator, & poli <var>.f.g.</var> primi poli dicentur.</s> + </p> + <p> + <s xml:space="preserve">Circa huiuſmodi duos circulos, alios <reg norm="etiam" type="context">etiã</reg> duos exiſtere <reg norm="vellem" type="context">vellẽ</reg> ſimul <reg norm="coniunctos" type="context context">cõiũctos</reg> medio <lb/>ad angulos rectos. </s> + <s xml:space="preserve">In quibus <reg norm="quidem" type="context">quidẽ</reg> interſectionis punctis ſint duo poli, qui hos duos <lb/>circulos cum ſecundo priorum ideſt cum æquatore in duobus punctis inuicem op-<lb/>poſitis connectant; </s> + <s xml:space="preserve">quæ æquatoris puncta à punctis interſectionis eiuſdem cum ſuo <lb/>deferente, ratione vna quarta diſtent, quorum duorum circulorum primus ſit <var>.n.i.<lb/>m.</var> quem deferentem azimut appellabimus; </s> + <s xml:space="preserve">ſecundus <var>.r.n.s.m.</var> azimut locorum no<lb/>minabimus. </s> + <s xml:space="preserve">eorundem interſectionis rectæ, puncta ſint <var>.n.</var> et <var>.m.</var> à quibus duo poli ex <lb/>aurichalco confecti ſimiles primis <var>.n.h.</var> et <var>.m.K.</var> vſque ad puncta <var>.h.</var> et <var>.K.</var> æquatoris <lb/>perueniant, qui ſpisſitudinem æquatoris diſtantem 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 ſitu latitudinis mundi verti queat. </s> + <s xml:space="preserve">Et <lb/>hos, ſecundos polos nominabimus.</s> + </p> + <p> + <s xml:space="preserve">Alius deinde circulus <var>.q.i.p.</var> duos poſteriores circulos ambiat, cum deferente ta-<lb/>men azimut mediantibus duobus polis in puncto <var>.i.</var> & in ſuo oppoſito exęquo diſtan <lb/>tibus à ſecundis polis vnius quartæ ſpatio 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 vniuerſa <lb/>lem vocabimus. </s> + <s xml:space="preserve">Hic vero orizon ſuper quatuor quartas circuli, aut ſuper quatuor <lb/>paruis columnis, ut fieri ſolet innixis ſuæ baſi, 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 conſtet, incipiendo ab interſectionibus <var>.e.</var> & eius op <lb/>poſito æquatoris, & numeri in polis <var>.f.</var> et <var>.g.</var> globi finem ſortiantur. </s> + <s xml:space="preserve">Diuidatur etiam <lb/>æquator <var>.h.e.K.</var> in .360. partes incipientes à puncto <var>.e.</var> verſus <var>.K.</var> deferens autem azi <lb/>mut <var>.n.i.m.</var> ab omni diuiſione liber maneat, ſed azimut <var>.n.s.m.r.</var> in .360. gradus inci <lb/>piendo à puncto <var>.n.</var> verſus <var>.r.</var> diuidatur.</s> + </p> + <p> + <s xml:space="preserve">Orizon autem <var>.q.i.p.</var> diuidatur in quartas, quarum quælibet ſit nonaginta <reg norm="graduum" type="context">graduũ</reg> <lb/>incipiendo à puncto <var>.i.</var> & eius oppoſito ideſt à polis poſtremis & terminando in pun-<lb/>ctis <var>.q.</var> et <var>.p.</var> in medio ipſorum polorum, & quarta <var>.i.p.</var> orientalis ſeptentrionalis, et <var>.i.<lb/>q.</var> orientalis meridiana appellentur. </s> + <s xml:space="preserve">& ſic ordine ſeruato occidentales.</s> + </p> + <p> + <s xml:space="preserve">Præterea pręparata ſit quædam quarta, ex aurichalco, circuli æqualis ipſi orizon-<lb/>ti, & in .90. gradus diſtincta quæ cum quauis ſuarum extremitatum ipſi zenit, in azi-<lb/>mut applicari poſſit, quemadmodum circa globos cęleſtes fieri ſolet; </s> + <s xml:space="preserve">quę quidem ad <lb/>cognoſcendam altitudinem poli ipſius globi ab orizonte nobis inſeruiet.</s> + </p> + <p> + <s xml:space="preserve">Atque hac ratione hanc noſtram machinam perfectè abſoluemus <reg norm="quam" type="context">quã</reg> appellan-<lb/>dam eſſe Armillam nauticam ſentio. </s> + <s xml:space="preserve">Hic autem illud non omittam, concauum <lb/>duorum priorum circulorum à ſuperficie globi non nimis diſtare debere & con-<lb/>cauum aliorum à ſuperficie conuexa priorum longe poſitos eſſe <reg norm="non" type="context">nõ</reg> debere, & con <lb/>cauam orizontis à conuexa ſecundorum procul abeſſe non debere.</s> + </p> + <p> + <s xml:space="preserve">Neque illud etiam prætermittendum eſt, opere pretium fore ſi in interſectione <lb/>e. priorum, erit foramen elicum, vt clauo elico ex aurichalco confecto, poſſimus <lb/>ſiſtere globum, quando oportuerit, ne amplius circa primos ſuos polos <var>.f.g.</var> circun-<lb/>uoluatur, cum <reg norm="ſitus" type="simple">ſitꝰ</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, ideſt <var>.f.e.g.</var> aliud quoddam foramen huiuſmodi ſit poſitum, 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 vſque ad circulum <var>.f.e.g.</var> perueniens, æquatorem ſiſtere poſſit, ne circa <lb/>ſecundos polos <var>.h.</var> et <var>.K.</var> amplius moueatur quum noluerimus eum mutare ſitum.</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">Deuſu Armillæ nauticæ.</head> + <p> + <s xml:space="preserve">V Tautem noſtra Armilla nautica vti poſſimus pyxidem nos prius oportebit <lb/>habere, diuerſam tamen ab ijs, quibus nautæ hactenus vſi 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, ſed ratione graduum orizontis in .360. partes diſtincti, atque ob hanc cauſam <lb/>ſentio, vt ima pars pyxidis penitus detecta videatur, & in .360. partes dinidatur, <reg norm="nilque" type="simple">nilq́</reg> <lb/>aliud quam quandam lanceo lam ſupra eius acum eſſe volo, quæ dum mouebitur na <lb/>uis, per gradus quamlibet orizontis partem oſtendet; </s> + <s xml:space="preserve">hos autem .360. gradus, ita ſe <lb/>habere volo, vt quęlibet quarta .90. contineat, <reg norm="ſupputatioque" type="simple">ſupputatioq́;</reg> à linea meridiana inci-<lb/>piat, & in verticali deſinat, vt huiuſmodi diuiſio cum ea, quæ eſt orizontis Armillæ <lb/>eadem ſit.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Pręſupponantur" type="context">Pręſupponãtur</reg> nunc in globo duo loci extra æquatorem, & in diuerſis meridia-<lb/>nis quomodolibet a dinuicem diſtantes, à quorum vno ad alium ſit nauigandum iti-<lb/>nere quo ad fieri poterit breuiori, ideſt per gyrum circuli maioris, dixi autem extra <lb/>æquatorem, ideſt vt ambo, nec in æquatore, nec in uno <reg norm="eodemque" type="simple">eodemq́;</reg> meridiano <reg norm="exiſtant" type="context">exiſtãt</reg>, <lb/>quia vt aliàs dixi in huiuſmodi 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">circũuolutione</reg> globi circa primos polos <var>.f.g.</var> & æqua <lb/>toris circa ſecundos <var>.h.K.</var> hoc eſt per longitudinem, & latitudinem, hi duo loci in <lb/>globo propoſiti ſub azimut <var>.r.n.s.m.</var> ſecundorum circulorum <reg norm="ſitum" type="context">ſitũ</reg> ſortiantur, qui azi-<lb/>mut orizontem in punctis <var>.q.</var> et <var>.p.</var> ſemper ad angulos rectos diſpeſcit <reg norm="ibique" type="simple">ibiq;</reg> globum <lb/>ita quieſcere vt circa polos <var>.f.g.</var> non voluatur, & æquatorem etiam ſic ſirmare, vt cir-<lb/>ca ſecundos polos <var>.h.K.</var> non vertatur faciamus.</s> + </p> + <p> + <s xml:space="preserve">Quod cum factum fuerit, ſecundorum circulorum primus, qui eſt <var>.n.i.m.</var> deferens <lb/>azimut, circa tertios polos <var>.i.</var> & eius oppoſitum, eo uſque voluatur quouſque prior <lb/>globi locus, ideſt is a quo iter eſt incohandum per .90. gradus azimut diſtet ab ori-<lb/>zonte, ideſt ſub zenit orizontis <var>.q.i.p.</var> ſit poſitus, quemadmodum, exempli gratia, ſi <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 diſtans ab <var>.q.</var> et <var>.p.</var> <lb/>ideſt per .90. gradus poneretur ſub <var>.r</var>.</s> + </p> + <p> + <s xml:space="preserve">Conſideretur deinde vbi æquator <var>.h.e.K.</var> ſecundus circulus duorum primorum, <lb/>ab orizonte <var>.q.i.p.</var> ſecabitur, exempli gratia, in puncto <var>.c.</var> quartę orientalis ſepten-<lb/>trionalis eiuſdem orizontis. </s> + <s xml:space="preserve">Videatur deinde quot nam gradibus conſtabit ar-<lb/>cus <var>.i.c.</var> & per totidem gradus conſtituatur extremitas ſeptentrionalis lineæ meridia-<lb/>nę pyxidis nauticę, diſtantis à cuſpide ſeptentrionali ipſius lanceolæ orientem ver-<lb/>ſus, mediante nauis circunuolutione. </s> + <s xml:space="preserve">vnde ipſamet nauis in huiuſmodi ſitu azimut, <lb/>qui per duos hos loc<unsure/>os tranſit, dirigetur, eſſiciendo vt eius prora verſus <reg norm="locum" type="context">locũ</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/>ſeu leucarum iter conficiemus, quot quarta pars vnius gradus requirit. </s> + <s xml:space="preserve">& dum <lb/>hociter abſoluitur, ille qui pręeſt naui, defferentem azimut <var>.n.i.m.</var> circa ſuos polos <var>.i.</var> <lb/>& eius oppoſitum, ſic circunuoluat, vt interſe ctio azimut <var>.r.n.s.m.</var> cum orizonte <var>.q.i.<lb/>p.</var> diſtet à prima ratione dictæ quartę partis vnius gradus, conſtituendo ſecundum lo <lb/>cum, proximiorem zenit, ratione dictæ quartæ partis gradus azimut. </s> + <s xml:space="preserve">Hiſce ita pera <lb/>ctis, obſeruetur deinde vbiæquator <var>.h.e.K.</var> hac ſecunda vice interſecabit orizontem <lb/><var>q.i.p.</var> quod quidem interſectionis punctum ſemper appelletur <var>.c.</var> quod dico non am <lb/>plius in eadem diſtantia manſurum, ut prius à puncto <var>.i.</var> ſed aut longius diſtabit, aut <lb/>propius accedet, vt in præſenti exemplo. </s> + <s xml:space="preserve">quemadmodum ex ſe manifeſtum eſt, cú <lb/>poli globi, ideſt ęquatoris ſint extra azimut, vt præſupponitur, quia loci ſunt in diuer <lb/>ſis meridianis.</s> + </p> + <p> + <s xml:space="preserve">Pro huiuſmodi autem diſtantiæ ratione denuo dirigatur nauis prout æquator <var>.h.<lb/>e.K.</var> in orizonte <var>.q.i.p.</var> nobis oſtendet, atque hoc modo omnium iter quaſi breuiſſi-<lb/>mum fiet. </s> + <s xml:space="preserve">dico autem, quaſi, quia omnibus modis neceſſariò conficitur iter contor-<lb/>tum & in formam ſerpigineæ lineæ. </s> + <s xml:space="preserve">Applicantes deinde per vices extremitatem <lb/>quartæ appoſitæ (de qua ſuperius mentionem fecimus) ipſi zenit <var>.r.</var> efficientes ut per <lb/>ſitum poli globi pertranſeat, deueniemus in cognitionem altitudinis eiuſdem ab <lb/>orizonte, & per conſequens quantum itineris per latitudinem eiuſdem globi pere-<lb/>gerit. </s> + <s xml:space="preserve">mediante deinde interſectione orizontis <var>.q.i.p.</var> cum æquatore, cognoſcemus <lb/><reg norm="quantum" type="context">quãtum</reg> itineris per longitudinem eiuſdem globi, in ipſo ęquatore 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, & occaſum Lunacognoſcendum <lb/>qualibet anni die.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">ECce tibi vir Illuſtriſs. <reg norm="modum" type="context">modũ</reg> <reg norm="conficiendi" type="context">conficiẽdi</reg> inſtrumenti nuper à me <reg norm="inuenti" type="context">inuẽti</reg>, vt tibi ſi <lb/>gnificaui, quo ſcire poſſis fermè in dies, qua hora (de aſtronomicis loquor) ad <lb/>determinatum parallelum & abſque multa ſupputatione, etiam abſque Aſtrolabio <lb/>Luna oriatur <reg norm="occidatque" type="simple">occidatq́;</reg>. </s> + <s xml:space="preserve">In quo inſtrumento poteris etiam videre quo in ſigno Sol, <lb/>& ſæpius itidem Luna permeat, & huiuſce aſpectus cum Sole, atque longitudinem <lb/>diei <reg norm="noctisque" type="simple">noctisq́;</reg> toto anni tempore exactè diſcernere.</s> + </p> + <p> + <s xml:space="preserve">Circularis lamina ex argento, aut ære, aliaúe materia paranda eſt, in cu-<lb/>ius ſuperficie ambarum facierum Zodiacus delineabitur, modo inferius depicto, <lb/>deinde pro anno quinque circuli ſibi inuicem <reg norm="concentrici" type="context">cõcentrici</reg>, at reſpectu Zodiaci excen <lb/>trici cęlabuntur in ea, adeo vt vtriuſque centri diſtantia ſit pro .32. parte ſemidia-<lb/>metri concauitatis Zodiaci è regione locis augis, temporis qui noſtra ætate circa ſi-<lb/>nem ſecundi gradus cancri inuenitur, eandem viam, in hoc, ſequuti, quam Stofle-<lb/>rus in dorſo Aftrolabij docet. </s> + <s xml:space="preserve">At nomina menſium media ponantur inter duos <lb/>maiores circulos, poſtea inter ſecundum, & tertium ab vna facierum laminæ, ar-<lb/>cus ſemidiurni, ab altera vero arcus ſeminocturni, per quinos quoſque dies collo-<lb/>centur, ita exactè, vt hic ſubtus videbis. </s> + <s xml:space="preserve">adeo vt numeri dierum & ipſorum dierum <lb/>ſigna ſint in interuallis vicinioribus centro communi dictorum quinque circulorum.</s> + </p> + <p> + <s xml:space="preserve">Poſteaquam ab vna & altera <reg norm="facierum" type="context">facierũ</reg> laminæ hæc inſculpra fuerint, aliæ duæ circu <lb/>lares laminæ, magnitudinis ſemidiametri minimi quinque circulorum accipiantur: <lb/></s> + <s xml:space="preserve">quarum vna pro ortu, & altera pro occaſu Lunæ deſeruiet. </s> + <s xml:space="preserve">In qualibet ipſarum <lb/><reg norm="conſtituentur" type="context">conſtituẽtur</reg> circuli quatuor, eo modo qui paulo inferius cernitur, quos omnes diui-<lb/>demus in triginta ſpacia æqualia: </s> + <s xml:space="preserve">& in interuallo <reg norm="quod" type="wordlist">qđ</reg> inter duos primos circulos poſi <lb/>tum eſt, triginta dies <reg norm="annotabimus" type="simple">annotabimꝰ</reg> qui ipſos Lunætriginta dies <reg norm="pręſcribent" type="context">pręſcribẽt</reg>, vt in figara.</s> + </p> + <p> + <s xml:space="preserve">Poſtmodum in lamina quæ ortus Lunæ indicabit, ac duorum maiorum <reg norm="circulo- rum" type="context">circulo-<lb/>rũ</reg> interuallo è regione numeri .1. videlicet primi diei, ponemus horas .12. & minuta <num value="48">.<lb/>48.</num> ex aduerſo diei ſecundi ho .13. et min .36. ex oppoſito tertij ho 14. min .24. & ſic <lb/>ſucceſſiuè augendo per min .48. & indicem è diuerſo diei .30. <reg norm="ſtatuendo" type="context">ſtatuẽdo</reg>, qui coitus <lb/>Lunæ cum Sole ſigniſicabit: </s> + <s xml:space="preserve">atque lineas aſpectuum, vt inferius videre eſt facilè in <lb/>ueniemus.</s> + </p> + <p> + <s xml:space="preserve">Altera in lamina quæ occaſum Lunæ indicabit, poſtquam diſtincta fuerit, vt alte-<lb/>ra .30. dies ac cęteræ lineæ, eo modo quo in ſuperiori collocabuntur, at numeri in-<lb/>terualli maioris, aliter diſponentur, vt potè ex aduerſo diei primi ſolum .48. minu-<lb/>ta deſcribi debent, è directo ſecundi diei ponenda erit hora vna cum minutis .36. & <lb/>è regione tertij inſcribentur .2. horæ, & min .24. & ſic ex ordine per .48. minuta au-<lb/>gendo.</s> + </p> + <p> + <s xml:space="preserve">Nunc lamina ortus Lunę, cum anno arcuum ſeminocturnorum, & illa occaſus cum <lb/>anno arcuum ſemidiurnorum <reg norm="concentrari" type="context">concẽtrari</reg> debet, & ita noſtrum inſtrumentum perfe-<lb/>ctum erit & abſolutum.</s> + </p> + <p> + <s xml:space="preserve">Quoties igitur voluerimus medio inſtrumento dignoſcere fermè in tali orizonte <lb/>qua hora Luna oriatur, ita neceſſe erit volubilem rotam ortus flectere, ut index ve <lb/>niat è regione diei menſis in quo talis operatio fit & talirota ſirma manente perſpi- + <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 aduerſo diei Lunæ, numerum horarum & minutorum in maiori interuallo <lb/>ipſius rotæ notatorum, qui cum arcu ſeminocturno anni, quo cum in ipſa rectitudi-<lb/>ne centri conueniet colligetur, & ſumma quæ ex tali ſupputatione proueniet aper-<lb/>tas faciet horas aſtronomicas, quibus ferè etſi non exactè in die propoſito Luna <lb/>orietur. </s> + <s xml:space="preserve">Idipſum fiet pro occaſu Lunæ.</s> + </p> + <p> + <s xml:space="preserve"><emph style="sc">DIem</emph> ætatis Lunæ iam totus orbis ſcit inuenire, media ſupputatione nu-<lb/>meri Epactę currentis cum numero menſium, ſumpto principio à Martio, <lb/>adiunctis diebus menſis currentis, & detracto numero .30. à ſumma prędicta, ſi ab ip <lb/>ſa dictus numerns .30. ſuperatur.</s> + </p> + <p> + <s xml:space="preserve">Sed ne aliquis putet ſufficere tantummodo additionem quatuor quintarum ho-<lb/>rę qualibet die. à nouilunio inchoando, ſciendum eſt huiuſmodi receſſum Lunæ <lb/>(quamuis non ita exactæ fiat) non <reg norm="computandum" type="context">computandũ</reg> eſſe ab orizonte aliquo, ſed à recto, <lb/>ſeu à meridiano quod idem eſt, quemadmodum <reg norm="vnicuique" type="simple">vnicuiq;</reg> mediocriter erudito pa-<lb/>tere poteſt. </s> + <s xml:space="preserve">At propoſitum nobis non eſt ſcire qua hora Luna in meridiano repe-<lb/>riatur, ſed in noſtro obliquo orizonte, in parte orientali ſeu occidentali, </s> + <s xml:space="preserve">propterea <lb/>igitur addendus eſt, ei ſummæ temporis, qua Luna diſtat à meridiano, arcus ſemi-<lb/>diurnus, vel ſeminocturnus illius loci Zodiaci, in quo Luna reperitur illa die in pro <lb/>poſito parallelo, vt ſciatur proximę, qua hora (ex aſtronomicis) Luna erit in ori-<lb/>zonte <reg norm="orientali" type="context">oriẽtali</reg>, vel occidentali dicti paralleli. </s> + <s xml:space="preserve">ſupra dicta enim additio quatuor quin <lb/>tarum horæ tantummodo, ſufficiens erit temporibus æquinoctij, ſed 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 occaſu. <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 collendiſs. anno .1570. conſtruxi.</head> + <head style="it" xml:space="preserve">CLARISS. FRANCISCO BARBARO VENETORVM <lb/>apud Sereniſſimum Sabaudiæ Ducem Oratori Illuſtriſſimo.</head> + <p> + <s xml:space="preserve"><emph style="sc">HEron</emph> varias ac diuerſas hydraulicas, & ſpiritales machinas propoſuit, in <lb/>ter quastamen nullam ſimilem, ei <reg norm="quam" type="context">quã</reg> ego Sereniſſimo Sabaudiæ Duci <lb/>conſtruxi, deſcribit, quæ quidem fuit Lucerna, & erat huiuſmodi, vt à <lb/>magno aliquo vaſe oleo pleno ſupra alicuius triclinij tabulatum poſito, <lb/>ſubtilis 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 vſque ad <lb/>medium deſcenderet, ita tamcn <lb/>vt hic ſolus 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, ſimile co <lb/>operculo alicuius pyxidis, è cuius <lb/>ambitu prope baſim multi diuer-<lb/>ſi quæ tubi æquales & orizonta-<lb/>les, cuiuſuis longitudinis proſili-<lb/>rent, quorum quilibet in extremi <lb/>tate ſua, 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">chnũ</reg> eſſet <reg norm="cum" type="context">cũ</reg> mixo. </s> + <s xml:space="preserve"><reg norm="oleum" type="context">oleũ</reg> deinde <lb/>medio <reg norm="perpendicularis" type="simple context">ꝑpẽdicularis</reg> tubi ad rece <lb/><reg norm="ptaculum" type="context">ptaculũ</reg> extrinſece deſcendebat, <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">cũ</reg> eædem erant extinctæ <lb/>ne minima quidem olei gutta de <lb/>ſcendebat: </s> + <s xml:space="preserve">id quod eos qui aſta-<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 ſic erat <reg norm="con" type="context">cõ</reg> <lb/>ſtructa. </s> + <s xml:space="preserve">Vas <reg norm="olearium" type="context">oleariũ</reg> cylindricum <lb/>vt in ſubſcripta figura patet, cuiuſ <lb/>uis magnitudinis, omni ex parte <lb/>clauſum faciendum curaui, ita ta <lb/>men vt eius coopereulum <reg norm="aliquam" type="context">aliquã</reg> <lb/><reg norm="tulum" type="context">tulũ</reg> concauum eſſet, in cuius me <lb/>dio erat foramen <var>.e.</var> quod erat os <lb/>tubi <var>.e.g.</var> qui ſub <reg norm="eiuſdem" type="context">eiuſdẽ</reg> vaſis fun-<lb/>do vſque ad <var>.g.</var> tranſibat, ſed po-<lb/>ſtea ſurſum, quaſi <reg norm="vſque" type="simple">vſq;</reg> ad cooper <lb/>culum in ſitu <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 poſtea 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">contignationemq́;</reg> vſque <lb/>ad centrum ſupradictireceptaculi (circa quod <reg norm="tuborum" type="context">tuborũ</reg> ope appenfæ erant ellychnio-<lb/>rum pyramides) tranſibat, atque huiuiuſmodi tubi <var>.n.u.</var> extremitates tam ſuperius <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 vaſis fundo feceram, cui <lb/>paruum <reg norm="tubum" type="context">tubũ</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">ꝑ</reg> hunc reflexum tu <lb/>bum <var>.a.o.t.</var> oleum vaſis exibat, <reg norm="quod" type="simple">ꝙ</reg> <lb/>per oſculum <var>.t.</var> in quendam cana-<lb/>lem tubo <var>.n.u.</var> inſertum, ab extra <lb/>oleum effundebat, & ab exteriori <lb/>parte arundinis <var>.n.u.</var> ingredieba-<lb/>tur receptaculum appenſum ex-<lb/>tremo dicti tubi <var>.n.u.</var> quod extre-<lb/>mum <reg norm="apertum" type="context">apertũ</reg> erat, ut dixi, & à fun-<lb/>do receptaculi tantum <reg norm="diſtans" type="context">diſtãs</reg>, <reg norm="quam" type="context">quã</reg> <lb/>tum volebamus oleum ab ipſius <lb/>receptaculi fundo altum exiſtere <lb/>quod quidem oleum ſtatim vt ad <lb/>oſculum 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 conſequens, neque <lb/>amplius <reg norm="oleum" type="context">oleũ</reg> <reg norm="per" type="simple">ꝑ</reg> <reg norm="tubum" type="context">tubũ</reg> <var>.a.o.t.</var> eflue-<lb/>bat, nec etiam per <reg norm="tubum" type="context">tubũ</reg> <var>.e.g.c.</var> aer <lb/>ingredi poterat, cum <var>.c.g.</var> ſemper <lb/>oleo exiſteret <reg norm="plenum" type="context">plenũ</reg> <reg norm="quum" type="context">quũ</reg> <var>.g.</var> magis <lb/><reg norm="quam" type="context">quã</reg> <var>.t.</var> <reg norm="adimum" type="context">adimũ</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">cũ</reg> fumitate digiti claudere oſcu-<lb/>lum <var>.t.</var> exiguitubi <var>.a.o.t.</var> vnde aer <lb/><reg norm="impulsus" type="simple">impulsꝰ</reg> ab oleo tubi <var>.e.g.c.</var> extra <lb/>per tubum <var>n.u.</var> quouſque oleum <lb/>vaſis ad ęquilibrium ipſius <var>.n.</var> per-<lb/>ueniebat, <reg norm="pertubum" type="context">pertubũ</reg> <var>.n.u.</var> ingredie-<lb/>batu. </s> + <s xml:space="preserve">& quando dictum oleum <reg norm="qui" type="simple">ꝗ</reg> <lb/>dictum tubum <var>.n.u.</var> extrinſecè 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">ꝑ</reg> ipſum <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> & peripſum <var>.t.</var> denuo tantum olei exire permitte-<lb/>bamus, quantum in receptaculo ad claudendum foramen <var>.u.</var> idoneum exiſteret. </s> + <s xml:space="preserve">Ra-<lb/>tio vero, quę me mouit, ut punctum <var>.g.</var> inferius ipſius <var>.t.</var> conſtituam, eſt, quia <reg norm="cum" type="context">cũ</reg> clau-<lb/>ſum 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 vſque 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> ſurſum verſusreflexum conſtitui, vt aerem ab ingreſſu per foramen <var>.t.</var> <reg norm="arcerem" type="context">arcerẽ</reg>, <lb/>quia huiuſmodi aer nunquam deſcendit ſi corpus magis denſum non deſcendat.</s> + </p> + <p> + <s xml:space="preserve">Verum eſt, <reg norm="quod" type="simple">ꝙ</reg> melius erit, vt maiores difficultates euitemus, ſtatuere dictum <reg norm="tubum" type="context">tubũ</reg> <lb/><var>a.o.t.</var> ita curuum vt eſt .ω qui cum ſuo extremo inferiori ipſi <var>.n.u.</var> ſit contiguus ita ta-<lb/>men ut dictum extremum inferius <reg norm="non" type="context">nõ</reg> ſit inferius quam <var>.o.</var> quia totum oleum exiret.</s> + </p> + <p> + <s xml:space="preserve">Volui etiam vt ſuperior extremitas <var>.n.</var> tubi <var>.n.u.</var> ſit in aere vaſis & non in oleo, ne <lb/>per eam oleum exeat, quia cum extremitas <var>.u.</var> inferior ſit <var>.g.</var> totum oleum quod ſu-<lb/>peraret oſculum <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 philoſophicis rebus verſato innoteſcet.</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 ſcripta quædam, quorum titulus <emph style="sc">animadver-<lb/>siones in ephemeridas</emph>. </s> + <s xml:space="preserve">& breuis alia diſputatio de er <lb/>roribus calculorum Aſtronomicorum. ac <reg norm="demum" type="context">demũ</reg> Theſes quæ-<lb/>dam typis datæ .11. Auguſti .1581. quę omnia cum ad manus <lb/>meas perueniſſent, non potui, non eis animum admouere, <reg norm="cum" type="context">cũ</reg> <lb/>ibi de his ſtudijs <reg norm="ageretur" type="simple">ageret̃</reg>, in quibus partem non exiguam an-<lb/>norum meorum conſumpſi. </s> + <s xml:space="preserve">nec tamen ſcribere aliquid ſta-<lb/>tueram; </s> + <s xml:space="preserve">tum quod exiſtimarem viros Aſtronomiæ 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 ſi ingenue meam ſententiam profer <lb/>re voluiſſem<unsure/>, non poteram, ſine maxima authoris moleſtia ferè omnia reprobare. <lb/></s> + <s xml:space="preserve">Quandoquidem ſolet vnuſquiſque indignationem concipere ex his, quæ ſuæ opi-<lb/>nioni repugnant, id omne maleuolentiæ potius quam veritatis ſtudio tribuens. </s> + <s xml:space="preserve">Qui <lb/>nimo cum nec deeſſent qui dicerent in meipſum directa ea tela fuiſſe, nullam fidem <lb/>eis adhibendam duxi, nec enim qui in ephemeri das inuehitur, me arguere poteſt, <lb/>qui nullas ephemeridas ſcripſi, nec tabulas compoſui. </s> + <s xml:space="preserve">Nec ſi author quidpiam ex <lb/>his quæ à nobis edita fuiſſent impugnare voluiſſet, ægrè ferre debuiſſem, modo à ve <lb/>ritate nuſquam deuiaſſet. </s> + <s xml:space="preserve">Liberum enim eſt cuique ſcribere quodlibet. </s> + <s xml:space="preserve">nec Ari-<lb/>ſtotelem afficit iniuria, quicunque illi fidem ſuam non accommodat, & ſi valdè ini-<lb/>quus ſit, quiſquis maiorum opiniones veras, & ab omnibus merito comprobatas <reg norm="non" type="context">nõ</reg> <lb/>admittit. </s> + <s xml:space="preserve">Hinc mihi ſatis <reg norm="oimbus" type="context">oĩbus</reg> feciſſe videbar, cum his qui de ſcriptis illis me <reg norm="inter- rogauerant" type="context">inter-<lb/>rogauerãt</reg> reſpondiſſem, ea non ſatis 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 deſideres tibi clarius, quæ nam de his mea ſit ſententia explicari, non tam tuis <lb/>precibus deuictus quam mea ipſius cupiditate de te benemerendi impulſus, non ſu-<lb/>ſtineo diutius animum tuum hęſitantem relinquere. </s> + <s xml:space="preserve">Atque vt tibi adeo honeſta cu <lb/>pienti <reg norm="morem" type="context">morẽ</reg> geram paucis hiſce ſcriptis incultis quidem, vt ab homine omni pror-<lb/>ſus facundia deſtituto exaratis, ſed ex quibus nihilominus facile, atque perſpicue, vt <lb/>ſpero, conceptum animi noſtri percipere poſſis, ſi tamen eam præſtantis ingenij tui <lb/>aciem adhibueris, qua ſoles intima quæque ſcientiarum penetrare, noſtræ opinio-<lb/>nis ſummam perſtrinxi, quę ad te mittere decreui. </s> + <s xml:space="preserve">& quamuis ipſa res de qua agitur, <lb/>quæ exactiorem deſiderat expoſitionem, prolixiorem me eſſe cogerit quam voluiſ-<lb/>ſem. </s> + <s xml:space="preserve">multa tamen me obmiſiſſe intelliges, non admodum neceſſaria his quibus <lb/>Aſtrologiæ noti ſunt termini, vt tuarum occupationum rationem me etiam habere <lb/>intelligeres, at que vt ſummam oblectationem <reg norm="concipiam" type="context">cõcipiam</reg> animo ſi me tibi aliqua ex <lb/>parte ſatisfeciſſe intellexero ita humanitati tuæ gratiam habebo, quę mihi occaſio <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 ſimilibus, <lb/>ideſt <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">eorũ</reg> animis falſam 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, ſi quam fortaſſe ex <reg norm="illorum" type="context">illorũ</reg> ſcriptorum lectione <reg norm="conceperunt" type="context">cõceperunt</reg> circa ea, de qui-<lb/>bus nunc ſum acturus.</s> + </p> + <p> + <s xml:space="preserve"><reg norm="Quemadmodum" type="context">Quemadmodũ</reg> igitur ab hoc authore ter ſcriptum fuit de <reg norm="contradictionibus" type="context">cõtradictionibus</reg>, ſiue <lb/>erroribus Ephemeridum, & earum calculos ſequentium, & de ratione qua cognoſci <lb/>poteſt ſitus & locus alicuius ſuperioris planetæ, diuerſus ab eo, qui ab ipſis Epheme <lb/>ridibus aſſignatus eſt, ita diſputationem hanc meam diuidam in tres partes, quo ſci <lb/>licet minus confusè, & magis diſtinctè à me ſcribatur, <reg norm="praeſupponendo" type="simple">p̃ſupponendo</reg>, vt animaduer-<lb/>tere potes, huius ſcriptoris intentionem, aliam non fuiſſe, <reg norm="quam" type="context">quã</reg> <reg norm="oſtendere" type="context">oſtẽdere</reg>, quod ſcripto <lb/>res Ephemeridum diuerſimode eiuſdem temporis locum planetæ aſſignauere, & <lb/>quod eum faciant modo nimium velociter currere, modo nimium in vno ſigno mo-<lb/>rari, vt (exempli gratia) Martem interdum faciunt morari ſex, aut ſeptem menſibus <lb/>in vno ſigno. </s> + <s xml:space="preserve"><reg norm="Idque" type="simple">Idq;</reg> poſtea in cauſa eſſe ait, vt Aſtrologi indiciarij <reg norm="fallantur" type="context">fallãtur</reg>, & ſimul <lb/>careant certis fundamentis rationum quibus futura indicent, & prædicant. </s> + <s xml:space="preserve">Primum <lb/>ergo videndum eſt, quam rectè hic vſus ſit arte, & ſcientia, vt aliorum opiniones, & <lb/>ſcripta redarguere poſſet. </s> + <s xml:space="preserve">Deinde videbimus quomodo verum ſit, & poſſibile id <lb/>quod ab Aſtrologis hactenus creditum, atque traditum eſt, & qua ratione poſſint ſie <lb/>ri veri calculi à peritis regularum ſcientiæ.</s> + </p> + <p> + <s xml:space="preserve">In primo igitur tractatu inſcripto Animadnerſiones, præſupponit Author pro-<lb/>feſſores huius ſcientiæ neſcire inuenire vera loca planetarum, quia vtuntur Ephe-<lb/>meridibus, in quibus eorum loca non rectè ſunt notata. </s> + <s xml:space="preserve">Quod ſecundum ipſum ori <lb/>tur, ex errore calculatorum, ſeu computiſtarum, potius quam ex varietate tabula-<lb/>rum, à quibus Ephemerides ſumptæ ſunt, hoc tamen verum non eſt, Ephemeridas, <lb/>ſcilicet, ita inter ſe differre, ratione errorum computiſtarum tantummodo, ſed po-<lb/>tius ratione ipſarum tabularum, & ſi interdum contingere poſſit error aliquorum mi <lb/>nutorum, nec non graduum, non propterea Ephemerides ita ſpernendæ ſunt. </s> + <s xml:space="preserve">In <lb/>multis enim calculis, tales errores excuſabiles ſunt, cum ab innumerabilibus propè <lb/>accidentibus oriri poſſint, præſertim in calculis prutenicis.</s> + </p> + <p> + <s xml:space="preserve">Videatur deinde vbi is profert quinquageſimum enuntiatum centiloquij Ptole-<lb/>męi, ſatis mendoſe. </s> + <s xml:space="preserve">Ptolemęus enim ibi ſic ait.</s> + </p> + <quote> + <s xml:space="preserve">Non obliuiſcaris eſſe centum viginti coniunctiones, quæ ſunt in ſtellis erraticis, <lb/>in illis enim eſt maior ſcientia eorum quæfiunt in hunc <reg norm="mundum" type="context">mũdum</reg> ſuſcipiendi incre-<lb/>mentum, & decrementum.</s> + </quote> + <p> + <s xml:space="preserve">Nam, neque eo in loco, neque alibi, Ptolemęus quidquam eius dicere voluit <lb/>quod ab hoc profertur.</s> + </p> + <p> + <s xml:space="preserve">Pergatur poſtea in pag .2. & videbitur hunc exiſtimare abſurdum quod Saturni, <lb/>& Iouis coniunctio vera anni .1563. potuerit eſſe in Leone ſigno igneæ triplicitatis <lb/>cum eorum coniunctio vera anni .1544. fuerit in Scorpione, ſigno triplicitatis <lb/>aqueæ, & cum coitus eorum anni .1583. futurus ſit in Piſcibus, ſigno pariter tripli-<lb/>citatis aqueæ. </s> + <s xml:space="preserve">Ita enim ait.</s> + </p> + <quote> + <s xml:space="preserve">Nam poſtquam duæ ſtellæ coiuerint, non prius ſub alio alterius triplicitatis ſigno <lb/>inter ſe ſunt conuenturæ, quam per omnia ſigna quæ eiuſdem ternarij cum primo ex <lb/>titerint prius <reg norm="coniungantur" type="context">coniungãtur</reg>. </s> + <s xml:space="preserve">Ita ſentit Ptolemęus, <reg norm="cæterique" type="simple">cæteriq́;</reg> non aſpernendi nominis <lb/>Aftronomi.</s> + </quote> + <p> + <s xml:space="preserve">Et <reg norm="tamen" type="wordlist">tamẽ</reg> Ptolemęus nunquam <reg norm="quidquam" type="context">quidquã</reg> huius rei attigit, & quamuis Albumaſar <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, ſed de mediocribus aut æqualibus, & quidem re-<lb/>ctè dicunt, quia lineæ eorum mediorum motuum non coeunt in aliquo ſigno alte-<lb/>rius triplicitatis, prius quam pertranſiuerint omnia ſigna illius, in qua incęperunt. <lb/></s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> nullum inconueniens ſequitur, ſi in veris <reg norm="coniunctionibus" type="context">cõiunctionibus</reg> non reperitur hæc re-<lb/>gula. </s> + <s xml:space="preserve">Fieri enim poteſt, vt lineæ mediorum motuum coniungantur in vno ſigno, cor <lb/>pora verò eorum planetarum coeant in alio, cum rarò eueniat, vt linea medij mo-<lb/>tus, eadem ſit 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">vanũ</reg> eſſe cre <lb/>dere id eueniſſe propter ſimilem naturam, aut qualitatem ſignorum. </s> + <s xml:space="preserve">Hunc enim <lb/>reſpectum non habent illi planetæ in verisſuis coniunctionibus. </s> + <s xml:space="preserve">Exempli autem <lb/>cauſa ponamus, quod rectè ſupputatæ fuerint coniunctiones annorum .1484. 1504. <lb/>& 1524. quod attinet ad differentiam duodecatemorij, ſcilicet prima in .24. gradu <lb/>Scorpij, ſecunda in .20. Cancri, tertia in .10. Piſcium. </s> + <s xml:space="preserve">Cum ſecunda anticipauerit <lb/>trigonum perfectum cum prima, gradibus .4. & tertia anticipauerit trigonum perfe-<lb/>ctum <reg norm="cum" type="context">cũ</reg> ſecunda gradibus .10. ſi forte prima vt facta fuit in .24. gradu Scorpij facta <lb/>fuiſſet in .2. gradu eiuſdem, <reg norm="planum" type="context">planũ</reg> eſt <reg norm="quod" type="simple">ꝙ</reg> <reg norm="ſecunda" type="context">ſecũda</reg> facta fuiſſet in .28. gradu geminorum <lb/>& tertia in .18. Aquarij, quæ ſigna ſunt diuerſæ triplicitatis ab illa Cancri. </s> + <s xml:space="preserve">Inſuper <lb/>ſi coniunctio anni .1544. quæ fuit in .28. gradu Scorpij fuerit recta <reg norm="correſpondens" type="context">correſpondẽs</reg> <reg norm="prae cedenti" type="simple">prę<lb/>cedenti</reg>, anni .1524. per gra .18. ſine dubio ſi coniunctio anni .1524. facta fuiſſet in <lb/>18. gradu Aquarij, illa anni .1544. fuiſſet in .6. Scorpij ſigni alterius triplicitatis <reg norm="quam" type="context">quã</reg> <lb/>ſint Gemini. </s> + <s xml:space="preserve">Præterea, vt anno .1544. <reg norm="coniunctio" type="context">cõiunctio</reg> facta eſt in .28. gra. Scorpij, & 1563 <lb/>in .29. Cancri, ponendo eas eſſe rectas, quod attinet ad ſuperandum trigonum vno <lb/>gradu, ſi anno .1544. facta fuiſſet in .30. Scorpij, anno .1563. proculdubio facta fuiſ-<lb/>ſet in primo gradu Leonis. </s> + <s xml:space="preserve">Et ſuppoſitis ijs interuallis, quæ ſuperſunt, aut deſunt per <lb/>fectis trigonis, ſi coniunctio anni .1524. fuiſſet in .20. gradu Piſcium, anno .1544. <lb/>fuiſſet in .8. Sagittarij. </s> + <s xml:space="preserve">Quæ quidem omnia aduerſantur opinioni huius ſcriptoris.</s> + </p> + <p> + <s xml:space="preserve">Quodautem opinatur coniunctionem anni .1583. fore in Ariete, ſic dicens pagi <lb/>na ſecunda.</s> + </p> + <quote> + <s xml:space="preserve">Non erit ab re ſi & eandem Saturni, & Iouis coniunctionem in primo igneæ tri-<lb/>plicitatis ſigno, quod eſt Aries futuram afferamus anno .1583. ſi ab accidentibus no <lb/>bis licet, vt ab omnibus paſſim conceditur, planetarum loca diſcernere.</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">ꝑ</reg> <lb/>4. <reg norm="quidem" type="context">quidẽ</reg> in <reg norm="qui" type="simple">ꝗ</reg> buſuis Ephemeridibus aut tabulis. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> videbit eiuſmodi <reg norm="coniunctionem" type="context context">cõiunctionẽ</reg> <lb/>contra <reg norm="ſententiam" type="context">ſententiã</reg> <reg norm="ſuam" type="context">ſuã</reg> fieri in Piſcibus, <reg norm="non" type="context">nõ</reg> <reg norm="autem" type="wordlist">aũt</reg> in Ariete. (vt poſtea res ipſa nos docuit <lb/>ſub <reg norm="menſe" type="context">mẽſe</reg> Aprili. poſt <reg norm="quidem" type="simple context">ꝗdẽ</reg> <reg norm="ſcriptam" type="context">ſcriptã</reg> <reg norm="hanc" type="context">hãc</reg> <reg norm="epiſtolam" type="context">epiſtolã</reg>, <reg norm="vulgarique" type="simple">vulgariq́;</reg> ſermone <reg norm="tranſmiſsam" type="context context">trãſmiſsã</reg>, ſed <reg norm="an- tequam" type="context">an-<lb/>tequã</reg> in <reg norm="latinum" type="context">latinũ</reg> <reg norm="tranſlata" type="context">trãſlata</reg>, & huic volumini inſerta <reg norm="cum" type="context">cũ</reg> alijs Typographo <reg norm="committeretur" type="context">cõmitteretur</reg>.)</s> + </p> + <p> + <s xml:space="preserve">Vbi poſtea 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">antiquorũ</reg>, vt videri <reg norm="pont" type="context">põt</reg> apud <reg norm="Albumaſarem" type="context">Albumaſarẽ</reg> & <reg norm="Alchibicium" type="context">Alchibiciũ</reg>, <lb/>ſed etiam perpendendum an eſſet vera, <reg norm="priuſquam" type="context">priuſquã</reg> ei adhæreret. </s> + <s xml:space="preserve">Hic enim fuit vnus <lb/>ex erroribus illius ætatis, quæ nondum penetrauerat intima huius ſcientiæ. </s> + <s xml:space="preserve">Sunt <lb/>tamen illi antiqui excuſatione aliqua digni. </s> + <s xml:space="preserve">Ponebant enim vigeſimo quoque an-<lb/>no præcisè fieri mediam coniunctionem Saturni <reg norm="cum" type="context">cũ</reg> Ioue, & in quolibet ſigno eiuſ-<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 ſecundum primum ſuppoſitum finiebatur ſpacio annorum .240. qui nume <lb/>rus fit. ex .20. duodecies multiplicatis. </s> + <s xml:space="preserve">Et quia triplicitates ſunt .4. ideo credebant <lb/>in ſpacio annorum .960. qui numerus fit ex .240. quater multiplicato, perfici .48. <reg norm="con- iunctiones" type="context">cõ-<lb/>iunctiones</reg>, priuſquam redirent ad ſe coniungendos in eodem loco, ubi prius iun-<lb/>cti fuiſſent. </s> + <s xml:space="preserve">Primum autem ſuppoſitum, quod vigeſimo quoque anno <reg norm="iungerentur" type="context">iũgerentur</reg>, <lb/>colligebant ſic ratiocinantes. </s> + <s xml:space="preserve">Si Saturnus annis .30. peragit ſuum curſum per om-<lb/>nia ſigna Zodiaci, Iupiter autem peragit eum annis .12. Saturnus ambulauerit .4. ſi <lb/>gna, et .4. quintas partes ſigni, ſiue gra .24. dum Iupiter peragit integrum ambitum <lb/>ideſt annis .12. </s> + <s xml:space="preserve">Itaque deſunt ei anni .8. ad perueniendum ad .20. quibus .8. annis Sa-<lb/>turnus <reg norm="perambulat" type="simple">ꝑambulat</reg> ſigna tria & <reg norm="quintam" type="context">quintã</reg> <reg norm="partem" type="context">partẽ</reg> unius ſigni <var>.i.</var> gradus .6. qui iuncti dictis ſi <lb/>gnis .4. & gra .24. faciunt ſigna .8. quæ Iupiter item percurrit in annis .8. atque ita in <lb/>annis .20. </s> + <s xml:space="preserve">Iupiter percurrit .20. ſigna <reg norm="antequam" type="context">antequã</reg> perueniat ad <reg norm="Saturnum" type="context">Saturnũ</reg>, cum Saturnus eo <lb/>dem tempore perfecerit curſum ſignorum .8. </s> + <s xml:space="preserve">Eandem concluſionem etiam fortaſ <lb/>ſe collegerant ex dictis ſuppoſitis, dicentes, ſi Saturnus annis .30. ambulat .12. ſigna <lb/>proculdubio annis .20. ambulat .8. ſigna, quo tempore Iupiter perambulat .20. ad ra <lb/>tionem .12. ſignorum in annis .12.</s> + </p> + <p> + <s xml:space="preserve">Verum hoc ſuppoſitum non eſt bonum, quoniam, ſi ita eſſet, coniunctiones <reg norm="horum" type="context">horũ</reg> <lb/>duorum planetarum nunquam exirent ex vna triplicitate, & non modo .960. <reg norm="quoque" type="simple">quoq;</reg> <lb/>anno, ſed etiam ſexageſimo rurſus coniungerentur in eodem puncto. </s> + <s xml:space="preserve">nec coniunctio <lb/>nes eorum (ſemper autem intelligo de medijs) unquam egrederentur ex illis tribus <lb/>ſignis Zodiaci.</s> + </p> + <p> + <s xml:space="preserve">Sed periodus æqualis Saturni, eſt dierum circiter .10740. atque ita minor an .30. <lb/>atque etiam .29. cum dimidio periodus autem æqualis Iouis, eſt circiter .4328. vt <lb/>ego eam comperio, quidquid alij <reg norm="dicant" type="context">dicãt</reg>, <reg norm="vtque" type="simple">vtq́;</reg> planius alias oſtendam. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> hæc per <lb/>iodus Iouis, etiam minor eſt ann .12. prætermittendo in ſupputatione <reg norm="tam" type="context">tã</reg> Saturni <reg norm="quam" type="context">quã</reg> <lb/>Iouis quaſdam 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 cognoſcere 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 ſunt periodus <lb/>Iouis, conficiet gradus .145. & min .4. ideſt min .8704. & eadem regula inueniemus <lb/><reg norm="quod" type="simple">ꝙ</reg> Saturnus .30. <reg norm="quibuſque" type="simple">quibuſq́;</reg> diebus, conſiciet min .60. & ſecunda .20. </s> + <s xml:space="preserve">Iupiter autem ſin-<lb/>gulis .30. diebus, conficiet min .149. & ſecunda .43. vnde ſubtrahendo minuta Satur-<lb/>ni à minutis Iouis, ſupererunt min .89. cum ſecun .23. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> Iupiter .30. <reg norm="quibuſque" type="simple">quibuſq́;</reg> die-<lb/>bus velocitate curſus, ſuperabit Saturnum minutis .89. cum ſecundis .23. </s> + <s xml:space="preserve"><reg norm="Atque" type="simple">Atq;</reg> dicen-<lb/>do, ſi minuta .89. cum ſecundis .23 dant nobis dies .30. ſupradicta, minuta .8704. da-<lb/>bunt nobis dies .2921. quibus iunctis cum diebus .4328. periodi Iouis, efficientur dies <lb/>7249. ideſt 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 oſtendatur hanc operationem rectam eſſe (nam demonſtrationem <reg norm="ſpeculatiuam" type="context">ſpeculatiuã</reg> <lb/>huius operationis in .113. Theoremate noſtræ Arithmeticę <reg norm="cuique" type="simple">cuiq;</reg> videre licet) fieri <lb/>poteſt his alijs calculis.</s> + </p> + <p> + <s xml:space="preserve">Si Saturnus diebus .10740. tranſit per gra .360. in ſpacio dierum .2921. tranſibit <lb/>per gradus .97. min .54. quibus iunctis cum gra .145. min .4. ſupra notatis, efticientur <lb/>gra .242. min .58. </s> + <s xml:space="preserve">Deinde, ſi Iupiter ſpatio dierum .4328. tranſit per gra .360. igitur <lb/>ſpatio .2921. per eandem regulam inueniemus eum tranſire gradus .242. mi .58. qui <lb/>numerus par eſt illi Saturni. </s> + <s xml:space="preserve">Cum ergo Iupiter confecerit vnum ambitum poſt 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 rurſus aſſequatur Saturnum, tranſeundum ei erit <reg norm="per" type="simple">ꝑ</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/>termiſimus quaſdam exiguas particulas periodorum perfectorum cuiuſque planetę <lb/>in ſuperioribus ſupputationibus.) </s> + <s xml:space="preserve">Illos verò gradus .43. Iupiter conſiciet 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 Alfonſi conueniant cum hoc <reg norm="numero" type="wordlist">nr̃o</reg> calculo, <reg norm="con- ſiderabimus" type="context">cõ-<lb/>ſiderabimus</reg>, <reg norm="quod" type="simple">ꝙ</reg> Era (vt vocant) dicti temporis annorum .19. cum diebus .314. eſt dua-<lb/>rum tertiarum ſexagenarum, ſecundæ nullius, & 53. <reg norm="primarum" type="context">primarũ</reg> ſiue 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">ꝙ</reg> <reg norm="vtriuſque" type="simple">vtriuſq;</reg> planetæ lineæ eiuſmodi motus tranſi <lb/>bunt per min .56. tertij gradus Sagittarij, ideſt coniunctæ erunt.</s> + </p> + <p> + <s xml:space="preserve">In fine poſtea ſecundæ periodi, cuius era erit .4. tertiarum, ſecundæ .1. et .47. pri-<lb/>marum ſexagenarum, locus mediocris <reg norm="vtriuſque" type="simple">vtriuſq;</reg> erit in min .56. gra. ſexti Leonis. </s> + <s xml:space="preserve">In <lb/>fine verò tertiæ periodi, cuius era erit .6. tertiarum .2. ſecundarum, 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">cuiusq́;</reg> periodi, locus <reg norm="eorum" type="context">eorũ</reg> mediocris coniunctim ſemper diſtabit à loco me <lb/>diocri præcedentis coniunctionis gradibus .117. ideſt in trigono antecedenti, minus <lb/>gra .3. </s> + <s xml:space="preserve">Vnde apparet has coniunctiones procedere in contrariam partem reſpectu or <lb/>dinis <reg norm="ſignorum" type="context">ſignorũ</reg> Zodiaci, ſed reſpectu ordinis graduum <reg norm="ſignorum" type="context">ſignorũ</reg>, ſemper <reg norm="progrediuntur" type="simple">progrediunt̃</reg> or <lb/>dine per ternos gradus nunquam retrogradientes. </s> + <s xml:space="preserve">Hinc ſe quitur, vt non duodecies <lb/>in omni triplicitate coniungantur hi duo planetę, vt antiqui putauerunt, ſed decies <lb/>tantum. </s> + <s xml:space="preserve">& ad ſummum ter in ſingulo ſigno, ſpatio 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 ſigno ſequenti alterius triplicitatis. </s> + <s xml:space="preserve">Exempli gratia, po-<lb/>namus <reg norm="quod" type="simple">ꝙ</reg> prima vice <reg norm="coniungantur" type="context simple">cõiungant̃</reg> in gra .2. Arietis, ſecunda 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/>ſeptima in .20. Arietis, octaua in .23. Sagittarij, nona in .26. Leonis, decima in .29. <lb/>Arietis, et vndecima erit in gra .2. Capricorni ſigni <reg norm="ſequentis" type="context">ſequẽtis</reg> triplicitatis. </s> + <s xml:space="preserve">Decem igi<lb/>tur interualla ſingula annorum .19. & dierum .314. faciunt annos .198. & dies .220. <lb/>Immo pertabulas Alfonſi, eiuſmodi periodus non modo non reperitur <reg norm="annorum" type="context">annorũ</reg> .242 <lb/>nec .240. vt antiqui credidere, ſed tribus diebus minor annis .198. & diebus .220. id-<lb/>eſt 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 ſe 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">aggregatũ</reg> <lb/>annorum .794. & dierum .138. cum annis .198. & diebus .217. quorum ſumma eſt <num value="992">.<lb/>992.</num> & dies .355. inuenietur locus mediocris ipſorum <reg norm="planetarum" type="context">planetarũ</reg> in dicto vltimo gra <lb/>du Sagittarij. </s> + <s xml:space="preserve">Sed ſi quęſiuerimus eorum locum mediocrem per aggregatum anno <lb/>rum .198. & dierum .217. cum annis .960. quod erit ſumma annorum .1158. & <reg norm="dierum" type="context">dierũ</reg> <lb/>217. reperiemus Iouem in gradu .18. </s> + <s xml:space="preserve">Sagittarij & Saturnum in .16. Leonis diſtanti-<lb/>bus inter ſe 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">ꝙ</reg> coniunctio eorum mediocris fuerit multis annis ante <lb/>omittendo (vt dixi) radices, quia ſatis eſt 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 animaduerſionum non quaſi cæcus eæcos ſequi, ſed 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 eſt, <reg norm="quod" type="simple">ꝙ</reg> idem author paulo inferius citat coniunctiones horum duorum <lb/>planetarum anni .1493. et .1512. quas neſcio vnde ſumpſerit.</s> + </p> + <p> + <s xml:space="preserve">Nam, etſi inter hos annos eſt interuallum annorum .19. tamen tantum abeſt, vt <lb/>coiuerint dictis annis, vt Saturnus anno .1493. ante finem Auguſti fuerit in .28. gra-<lb/>du Aquarij, Iupiter verò in .28. Leonis ex diametro oppoſiti. </s> + <s xml:space="preserve">Et anno .1512. per to <lb/>tum menſem Iunium & Auguſtum, Saturnus fuerit in Libra, Iupiter verò in Ariete, <lb/>itaque inter ſe ſimiliter oppoſiti, & ſi perfecta oppoſitio non fuit poſtea niſi 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, eſt mendoſus. </s> + <s xml:space="preserve">Nam ipſe Montere gius po <lb/>nit dictam coniunctionem in mi .42. gra .24. non autem in min .4. ipſius gradus. </s> + <s xml:space="preserve">Sed <lb/>hic error nullius eſt momenti, fortaſſe qui impræſſorum incuria irrepſit.</s> + </p> + <p> + <s xml:space="preserve">Pergatur poſtea obſecro ad paginam .3. ipſarum Animaduerſionum, vbi hic co-<lb/>natur oſtendere calculatores non obſeruaſſe verum modum, ſic dicens.</s> + </p> + <quote> + <s xml:space="preserve">Anno .1484. Nouembris .25. Saturno locum conſtituit Monteregius in grad .23. <lb/>min .4. Scorpij. </s> + <s xml:space="preserve">Anno poſtmodum ſubſequenti qui eſt .1485. eundem in min .7. Sa-<lb/>gitarij collocat .21. Februarij die. </s> + <s xml:space="preserve"><reg norm="Interque" type="simple">Interq́;</reg> tempora duo interſunt menſes dies .26. <lb/></s> + <s xml:space="preserve">At cum ex motus ſui natura Saturnus hoctemporis ſpacio gradus .4. non debeat <reg norm="tran- ſcendere" type="context">trã-<lb/>ſcendere</reg>, ſit tamen inter <reg norm="vtrunque" type="simple">vtrunq;</reg> tempus differentia graduum .7. minutorum .3. quæ <lb/>ratione ſui motus requirunt menſes .6. vt eos perficiat, conſtat pluſquam tribus men <lb/>ſibus fallere nos Saturnum.</s> + </quote> + <p> + <s xml:space="preserve">Hic videre licet quam veram viam hic ſecutus ſit ad aperiendos errores Epheme <lb/>ridum, & miſeri Monteregij, qui Saturnum claudum facit tantum itineris conficere <lb/>tribus <reg norm="menſibus" type="context">mẽſibus</reg>, <reg norm="quantum" type="context context">quãtũ</reg> vix confeciſſet <reg norm="menſib" type="context">mẽſib</reg>. ſex. </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 cęlum annis .30. igitur menſi-<lb/>bus .30. conficiet duodecimam partem circuitus, cum menſes .30. ſint duodecima <lb/>pars annorum .30. & quia duodecima pars circuitus cęli intelligitur conſtare ex .30. <lb/>gradibus, igitur quilibet menſis poſtulabit gradum vnum. </s> + <s xml:space="preserve">Ideo illi .6. aut .7. gradus <lb/>poſtulant tempus, amplius menſium ſex.</s> + </p> + <p> + <s xml:space="preserve">Atque eiuſmodi mira ratiocinatio poteſt in .2. exemplo eius, inſcripto.</s> + </p> + <quote> + <s xml:space="preserve">Deeodem ex eodem</s> + </quote> + <p> + <s xml:space="preserve">Vbi miratur, <reg norm="quod" type="simple">ꝙ</reg> Monteregius faciat Saturnum ambulare gra .9. min .10. in menſi-<lb/>bus .7. & diebus .6. </s> + <s xml:space="preserve">Ad quod iter Saturnc ſeni opus eſſet ſaltem menſibus .9. eius <lb/>iudicio.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi hoc miratur, quid dicturus fuiſſet, ſi animaduertiſſet, quod idem calculator <lb/>Monteregius facit Saturnum ambulare immo volare gra .9. min .48. non in 7. ſed in <lb/>2. menſibus cum dimidio, videlicet à .10. die Iunij vſque ad .26. Auguſti <reg norm="eiuſdem" type="context">eiuſdẽ</reg> an-<lb/>ni .1504.</s> + </p> + <p> + <s xml:space="preserve">Quid ſi etiam animaduertiſſet <reg norm="quod" type="simple">ꝙ</reg> à .10. die Iunij ſupradicti vſque ad .16. Ianuarij <lb/>anni ſequentis, faciunt Saturnum, ſurſum, deorſum curſitare amplius gra .17. mi .54. <lb/>Imino ſi animaduertiſſet, quod anno .1524. </s> + <s xml:space="preserve">Stoflerinus ab initio anni, <reg norm="vſquem" type="context">vſquẽ</reg> ad <lb/>medium Maium, ideſt menſib .4. cum dimidio, facit Saturnum ambulare gra .15. </s> + <s xml:space="preserve">Pro <lb/>fectò ob has velocitates, eius iudicio, tam abſurdas, obſtupuiſſet.</s> + </p> + <p> + <s xml:space="preserve">Vbi autem in tergo eiuſdem paginę ait, quod gradibus .13. min .42. reſpondent <lb/>menſes .19. errauit in calculo, nam ex eiuſmodi tempore ſecundum 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 ſinguli 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">ꝙ</reg> in primo exemplo circa <reg norm="annum" type="context">annũ</reg>. <lb/>1484. repręhendit Monteregium, quia facit Iouem ambulare gradus .14. cum min <num value="6">.<lb/>6.</num> in menſibus .2. diebus .4. ad quod iter, vt ipſe ait, opus eſſet ſaltem <reg norm="menſibus" type="context">mẽſibus</reg> .11. <reg norm="atque" type="simple">atq;</reg> <lb/>ita ſecundum ipſum, Ioui opus eſſet anno vno pro ſingulo medio ſigno. </s> + <s xml:space="preserve">Vbi bonus <lb/>hic vir pariter cæcutit.</s> + </p> + <p> + <s xml:space="preserve">Idem in ſecundo exemplo ſumpto à Stoflerino ait, <reg norm="quod" type="simple">ꝙ</reg> Ioui ad curſum vnius gra-<lb/>dus, & min .5. opus eſt diebus .30. non autem menſibus .7. & diebus .28. vbi oſtendit, <lb/>ſe paruum diſcrimen facere inter Iouem, & Saturnum.</s> + </p> + <p> + <s xml:space="preserve">Miratur poſtea <reg norm="quod" type="simple">ꝙ</reg> Stoflerinus faciat laborare generoſum Iouem ferè menſibus <lb/>ſex in vno gradu. </s> + <s xml:space="preserve">Sed multo magis, vt puto, miratus eſſet, ſi vidiſſet, quod idem <lb/>Stoflerus in eodem anno facit, quod Iupiter die .4. </s> + <s xml:space="preserve">Ianuarij ſit in eodem puncto, in <lb/>quo poſtea reperitur die vltima Auguſti. </s> + <s xml:space="preserve">At fortaſſe dici poſſet, quod Iupiter pro-<lb/>pter prudentiam, & bonitatem ſuam factus eſt R ex omnium Deorum, vt ait Home-<lb/>rus, & ideo expulit è ſede Saturnum, & aſcendit in altiori cœlo. </s> + <s xml:space="preserve">Vnde euenit vt fa-<lb/>ctus fuerit lentior in curſu, 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 ſenem, & pariter tardio-<lb/>rem in ſe <reg norm="mouendo" type="context">mouẽdo</reg>. </s> + <s xml:space="preserve">aut <reg norm="tunc" type="context">tũc</reg> temporis illum detentum fuiſſe in ſibi dilecta Arcadia <reg norm="cum" type="context">cũ</reg> <lb/>Caliſto. </s> + <s xml:space="preserve">Aut fortaſſe erat in alta ſpecula intentus audiendo ingenti certami-<lb/>ni Timoclis & Damidis, vnde pendebat exitium aut gloria familiæ ſuæ, nam alio-<lb/>quin Stoflerus non depræhendiſſet eum tam otioſum & 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 ſemper bella & ignes ſpirare ſolet, etiam, & ipſe ab <lb/>Aſtrologis factus eſt piger, & languidus, vt velint eum nonnunquam commorari <lb/>in vno ſigno ſex aut ſeptem menſibus; </s> + <s xml:space="preserve">quod nullo pacto placet authori Animaduer-<lb/>ſionum, cum pag .4. ita ſcribat.</s> + </p> + <quote> + <s xml:space="preserve">Quod citra notam, ab omnibus creditur poſſe obſeruari, 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 ſuum 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 cœleſtium obſeruatores, ipſis oculis certi fa <lb/>cti ſunt tam de his effectibus Martis, quam aliorum, vnde coacti ſunt fingere tantam <lb/>magnitudinem eius epicycli, cum ipſe nunquam obſeruauerit motum, nec huius nec <lb/>alterius planetæ, ſed tantum viderit eius moram in Ephemeride ſcriptam. </s> + <s xml:space="preserve">Si enim <lb/>ſaltem diceret, ſe aliquo tempore obſeruaſſe iter Martis, & comperuiſſe aliorum opi <lb/>nionem falſam, attuliſſet aliquem colorem ſententiæ ſuæ. </s> + <s xml:space="preserve">Sed ſi obſeruaſſet, non <lb/>ſcripſiſſet poſtea contra, vt puto. </s> + <s xml:space="preserve">Res enim ita ſe habet, quod Mars in omni circui <lb/>tu ſui epicycli tranſiens per inferiorem partem ipſius epicycli, ſemper commoratur <lb/>multis menſibus in vno duodecatemorio Zodiaci, ſcilicet .6. et .7. menſibus, atque <lb/>etiam amplius, quod quidem ego ſæpe obſeruaui, præſertim anno .1565. et .1566. <lb/>hoc ordine. </s> + <s xml:space="preserve">Primum inſpiciens Ephemeridas ſtadij, reperi <reg norm="quod" type="simple">ꝙ</reg> Mars ſecundum eum <lb/>egrediebatur retrogradationem circa diem .12. Ianuarij anni 1566. in .16. grad. <lb/>Geminorum. </s> + <s xml:space="preserve">Et ſimiliter quod anno .1565. die vltima Auguſti Mars futurus erat in <lb/>eodem ſupradicto loco, priuſquam retrogradi inciperet. </s> + <s xml:space="preserve">Poſtea inueni, quod poſt <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 conſumebatur ſpatium <lb/>menſium .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 ſupputato, ſumpſi inſtrumenta, & ad experimentum me paraui, & vltima <lb/>nocte menſis Auguſti anni .1565. reperi Martem eſſe in dicto gradu geminorum vt <lb/>ſcribebat Stadius. </s> + <s xml:space="preserve">Deinde ſingulis ebdomadibus obſeruans retrogradationem; </s> + <s xml:space="preserve">vidi <lb/>circa finem Octobris quod retrogradi incipiebat, & ea retrogradatio perſeuerauit <lb/>vſque ad medium menſem Ianuarium, aut circiter, anni .1566. obſeruaui poſtea <lb/>etiam ſitum eiuſdem planetæ die .11. </s> + <s xml:space="preserve">Aprilis ſequentis <reg norm="eumque" type="simple">eumq́;</reg><unsure/> inueni in gradu .16. <lb/>Cancri, vti eum poſuerat Stadius. </s> + <s xml:space="preserve">Atque ita experimentum meum conuenit cum <lb/>calculo Stadij, <reg norm="comperique" type="simple">comperiq́;</reg> eum non erraſſe: </s> + <s xml:space="preserve">Et ſic quiſque binis <reg norm="quibusque" type="simple">quibusq́;</reg> annis pote-<lb/>rit certior fieri de veritate. </s> + <s xml:space="preserve">Si autem delectationis cauſa id experiri volueris, expe-<lb/>ctato primam retrogradationem Martis, cuius initium ſecundum Stadium futurum <lb/>eſt 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 aſpice quum erit in grad .9. Leonis, quod erit circa diem .7. Mai .1583. & vide-<lb/>bis <reg norm="quod" type="simple">ꝙ</reg> ipſe Mars in his gra .30. <reg norm="morabitur" type="simple">morabit̃</reg> <reg norm="per" type="simple">ꝑ</reg> menſes .7. & dies .18. <reg norm="atque" type="simple">atq;</reg> vt eius rei pericu-<lb/>lum facias, obſerua noctem præcedentem diei .19. </s> + <s xml:space="preserve">Septem .1582. locum <reg norm="longitudinis" type="context">lõgitudinis</reg> <lb/>eius ſtellæ, & idem poſtea obſerua nocte præcedente diei .7. Mai, aut nocte ſequen-<lb/>ti .1583. & inter duos hoſce terminos obſerua aliqua alia nocte ſtatum eius. </s> + <s xml:space="preserve">Mani-<lb/><reg norm="feſtoque" type="simple">feſtoq́;</reg> videbis Martem conſumere totum dictum tempus in hoc duodecatemorio. <lb/></s> + <s xml:space="preserve">Et quicunque aliquid intelligit in hac facultate quamuis non viderit Ptolomęi <lb/>Almageſtum, minori labore poſſet per calculos ſcientificos colligere verita-<lb/>tem, ſuppoſitis tamen terminis ſcriptis in theoricis planetarum. </s> + <s xml:space="preserve">Qui enim vidit <lb/>Almageſtum vel reuolutiones orbium cœleſtium Nicolai Copernici, non poteſt de <lb/>hoc vllo pacto dubitare. </s> + <s xml:space="preserve">Sed qui nondum tantopere progreſſus eſt, <reg norm="ſaltem" type="context">ſaltẽ</reg> capiat <reg norm="huius" type="simple">huiꝰ</reg> <lb/>rei notitiam vniuerſalem, hoc modo. </s> + <s xml:space="preserve">Supponat primum eccentricitatem deferen-<lb/>tis epicycli Martis, eſſe .6. partium <reg norm="talium" type="context">taliũ</reg>, quales ſunt ſexageſimæ ſemidiametri ipſius <lb/>deferentis, & ſemidiametrum epicycli eſſe, partium <reg norm="ſupradictarum" type="context">ſupradictarũ</reg> .39. cum dimidia, <lb/>& quod argumenta vera, in temporibus primarum ſtationum ( cum epicyclus eſt in <lb/>auge, aut in eius oppoſito, aut in <reg norm="lungitudinib" type="context">lũgitudinib</reg>. me dio cribus ) <reg norm="iam" type="context">iã</reg> ab antiquis rectè ſuppu <lb/>tata ſint, ſicuti ſunt. </s> + <s xml:space="preserve">Et præſupponat motum diurnum centri epicycli. min .31. cum di <lb/>midio, quamuis reuera ſit min .31. & ſecundorum .27. aut circiter, nunc <reg norm="quidem" type="context">quidẽ</reg> præter <lb/>mittens, quod vnus habeat reſpectum ad augem mediam epicycli, & alter ad cen-<lb/>trum æquantis. </s> + <s xml:space="preserve">Atque his præſuppoſitis fingat ( exempli gratia ) quod centrum <lb/>epicycli ſit in quauis longitudinum mediarum, & Mars in prima maxima æqua-<lb/>tione argumenti, ſcilicet 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="proficiſcens" type="context">proficiſcẽs</reg> <lb/>Mars perget ad lineam primæ ſtationis, vt poſtea retrogradiatur, veluti ſi in infrapo <lb/>ſita figura maiori, <reg norm="centrum" type="context context">cẽtrũ</reg> <reg norm="mundi" type="context">mũdi</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> ſint notatæ <var>.o.i.</var> et <var>.o.t.</var> & li-<lb/>nea primæ ſtationis <var>.o.n.b.</var> & linea ſecundæ <var>.o.u.d.</var> ſi 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 Alfonſi ponatur eſſe gra .41. <lb/>minut .10. quod euenit quia calculatores ipſarum tabularum interuallum <var>.o.c.</var> quod <lb/>in eo ſitu epicycli interponitur inter centrum mundi, & centrum dicti epicycli, ac-<lb/>ceperunt partium ſexaginta præcisè, nihili facientes minuta illa .18. aut circiter, quę <lb/>verè ſunt 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 ſitu epicycli ſit baſis vnius trianguli orthogonij, cuius vnum ex illis duobus late-<lb/>ribus eſt ſemidiameter eccentrici partium .60. pręcisè, aliud eſt interuallum eccen-<lb/>tricitatis partium .6. eiuſmodi. </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 ſecundum ſitum epicycli. </s> + <s xml:space="preserve">& cum centrum <lb/>eius eſt in auge eccentrici. eſt minimus <reg norm="quam" type="context">quã</reg> eſſe poſſit. </s> + <s xml:space="preserve"><reg norm="eſtque" type="simple">eſtq́;</reg> tantum grad .36. min <num value="46">.<lb/>46.</num> & in oppoſito ipſius augis eſt grad .47. min .1. maximus quam alibi vnquam ſit, <lb/>& ſic continuò variatur, ſecundum ſitum, 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> eſt ſemper rectangulus in puncto <var>.i.</var> & latus <var>.c.i.</var> reſpectu <lb/>ſemidiametri eſt 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> ſit linea veri motus epicycli, & latus ſimiliter <lb/>vnius trian guli, cuius duo latera ſunt ſupradicta, ſcilicet ſemidiameter eccentrici, & <lb/>eccentricitas, inter ſe compræhendentes angulum datum. </s> + <s xml:space="preserve">Nam ſemper præſuppo <lb/>nitur datus locus centri ipſius epicycli, cum ipſe eſt extra augem aut oppoſitum eius <lb/>quia in auge linea <var>.o.c.</var> conſtat ex ſemidiametro eccentrici & interualli eccentricita-<lb/>tis. </s> + <s xml:space="preserve">& in eius oppoſito, ipſa linea <var>.o.c.</var> eſt minor dicto ſemidiametro eccentrici per in <lb/>teruallum dictæ eccentricitatis. </s> + <s xml:space="preserve">Vnde etiam poſſumus extra augem, vel oppoſitum <lb/>eius cognoſcere <var>.o.c.</var> tanquam latus dicti trianguli duorum laterum <reg norm="cum" type="context">cũ</reg> angulo cogni <lb/>torum. </s> + <s xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</reg> per .49. propoſitionem libri primi <reg norm="eiuſdem" type="context">eiuſdẽ</reg> Monteregij cum ſcilicet dictus <lb/>angulus <reg norm="non" type="context">nõ</reg> fuerit rectus. </s> + <s xml:space="preserve">Nam ſi fuerit rectus videbitur per .27. et .28. ſupra 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 reſpondet arcus <var>.i.g.</var> epicycli confectus à Marte <lb/>in diebus circiter .105. ad rationem min .28. aut circiter in ſingulos dies, prætermiſ-<lb/>ſis nunc quidem minutijs cum exigui momenti ſit error .15. aut .20. dierum ad verifi <lb/>cationem longæ morę 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">cũ</reg> <lb/>dimidio in ſingulos dies. qui numerus graduum .55. min .7: differt à numero <reg norm="graduum" type="context">graduũ</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 reſpectum ad centrum æquantis, magis quam ad centrum <reg norm="mundi" type="context">mũdi</reg>, <lb/>quia differentia non eſt tanta, vt poſſit inducere errorem menſium. </s> + <s xml:space="preserve">Hinc ſequitur <lb/>quod in fine dictorum dierum .105. </s> + <s xml:space="preserve">Mars erit in linea <var>.o.c.</var> veri motus epicycli, ſed <lb/>gradibus .14. min .12. vlterius quam in primo loco, in quo erat in Zodiaco, & erit in <lb/>medio ſuæ retrogradationis. </s> + <s xml:space="preserve">Sed quoniam Mars manifeſtè retrogradi non incipit <lb/>in puncto <var>.i.</var> conting entiæ, imo ab illo puncto vſque ad terminum primæ ſtationis li <lb/>neæ <var>.o.n.</var> interponitur arcus <var>.i.n.</var> epicycli, qui eſt graduum .32. minu .14. </s> + <s xml:space="preserve"><reg norm="Idque" type="simple">Idq́;</reg> cogno-<lb/>ſcitur ſubtrahendo arcum <var>.f.i.n.</var> graduum .163. mi .9. qui eſt inter augem, & primam <lb/>ſtationem, à gradibus .180. ( qui arcus <var>.f.i.n.</var> erit verum argumentum, quod ſi-<lb/>militer variatur ſecundum ſitum epicycli, etſi eiuſmodi varietas, nobis <reg norm="non" type="context">nõ</reg> eſt magni <lb/>momenti, vnde poſſumus præſupponere, quod <var>.c.</var> centrum epicycli non alteret <reg norm="in- teruallum" type="context">in-<lb/>teruallũ</reg> <var>.c.o.</var> à centro <reg norm="mundi" type="context">mũdi</reg>, <reg norm="cum" type="context">cũ</reg> non posſit intercedere, error <reg norm="menſium" type="context context">mẽſiũ</reg> reliquum verò <var>.g.<lb/>n.</var> graduum .16. min .51. ſubtrahendo 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 eiuſmodi tamen ſitu 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 ſupradictam, omittendo quod ipſa ſtella habeat reſpectum ad <lb/>augem mediocrem epicycli, & quod dicta aux mediocris mutet diſtantiam 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 ſupradictam. </s> + <s xml:space="preserve">Reſtat nunc no-<lb/>bis inuenire angulum <var>.b.o.c.</var> in centro <reg norm="mundi" type="context">mũdi</reg> inter duas lineas, <var>b.o.</var> et <var>.c.o.</var> <reg norm="quarum" type="context">quarũ</reg> prior <lb/>eſt primæ ſtationis, altera eſt 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> ſint 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 ſi-<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 ſinus anguli <var>.c.n.h.</var> reſtantis ex vno recto, erit partium .95706. dicendo <lb/>poſtea ſi <var>.n.e.</var> tanquam ſinus 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 ſubtractis extota <var>.c.o.</var> quę eſt <reg norm="partium" type="context">partiũ</reg> .60. mi .18. reliqua <lb/>erit nobis <var>.h.o.</var> partium .22. min .30. capiendo poſtea radicem <reg norm="quadratam" type="context">quadratã</reg> ſummæ 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> eſt partium .11. min .27. ſi igitur ad <var>.o.n.</var> tanquam partium .25. min .12. reſpondet <var>.<lb/>n.h.</var> partium .11. minuti .27. linea <var>.n.h.</var> ad <var>.o.n.</var> tanquam partium 100000. reſpon <lb/>debit part .45436. tanquam ſinus anguli <var>.n.o.h.</var> qui angulus erit gra .27. minut .1. ſub <lb/>tracto poſtea 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/>ſtationis in eiuſmodi ſitu. </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. ( præſuppoſita ſemper eadem di-<lb/>ſtantia <var>.o.c.</var> quamuis nonnulla ſit 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. reſtabunt gra .22. min .19. itaque in diebus .69. </s> + <s xml:space="preserve">Mars promo-<lb/>tus fuerit a primo ſitu 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/>ſtatione vſque ad <lb/><reg norm="lineam" type="context">lineã</reg> veri motus <lb/>epicycli ſunt gra <num value="16">.<lb/>16.</num> min .51. ipſius <lb/>epicycli, vt ſupra <lb/><reg norm="vidimus" type="simple">vidimꝰ</reg> quos Mars <lb/>tranſit in diebus <lb/>36. aut circiter ad <lb/>rationem min .28. <lb/>in ſingulos dies, <lb/>quo tempore cen <lb/>trum epicycli, in <lb/>tali diſtantia 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 ſingulos dies, <lb/><reg norm="quibus" type="simple">quibꝰ</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. ſubtractis à <lb/>grad .22. minu .19. per quos Mars progreſſus erat, ſupererunt grad .14. minut .12. <lb/>quibus ipſe in media retrogradatione exiſtens in linea <var>.o.c.</var> veri motus epicycli pro-<lb/>motus erit à principio primi ſitus. </s> + <s xml:space="preserve">Quod cum eo concordat quod ſupra diximus cir <lb/>ca hos gradus .14. min .12. vltra primum ſitum in ſpatio dierum .105. vt ſupra, ad <reg norm="tam" type="context">tã</reg> <lb/>tum enim aſcendunt .69. et .36. nunc fingendo <reg norm="quod" type="simple">ꝙ</reg> Mars pergat in ſuo motu compo-<lb/>ſito qui conſtat ex his duobus circulis, vi eccentrici, & epicycli ( quanquam <lb/>vt dixi omittimus illam ſummam ſubtilitatem ſeu ſcrupuloſi<unsure/> atem <reg norm="continuæ" type="context">cõtinuæ</reg> inæqua-<lb/>litatis diſtantiæ 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 ſuo pergat, vſque ad punctum ſecundæ ſta-<lb/>tionis, præteribunt alij dies .36. vt prius, quibus iunctis cum .105. fiet ſum-<lb/>ma dierum .141. & Mars retrogreſſus erit per alios grad .8. minut .7. quibus ſub-<lb/>tractis a gradibus .14. m in .12. per quos progreſſus erat ſupererunt. gra .6. min .5. qui <lb/>bus ipſe Mars in fine ſuæ retrogradationis promotus erit à primo loco vnde moueri <lb/>cœpit. </s> + <s xml:space="preserve">Inter hanc igitur ſecundam ſtationem lineæ <var>.o.u.d.</var> & lineam <var>.o.t.</var> ſecundæ con <lb/>tingentiæ, Mars diebus .69. vt prius, confecerit gradus .32. min .14. ſui epicycli, & eo-<lb/>dem tempore linea contingens <var>.o.t.</var> ambulauerit gra .36. min .13. vt prius, à quo itine <lb/>re ſubtracto angulo ſecundo <var>.d.o.t.</var> graduum .13. min .54. ſupererunt 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. ideſt in .141. et .36. qui ſunt ferè m. n <lb/>ſes .6. </s> + <s xml:space="preserve">Itaque Mars partim ſurſum partim, deorſum ambulans detentus erit <reg norm="menſibus" type="context">mẽſibus</reg> <lb/>6. in grad .28. </s> + <s xml:space="preserve">Zo-<lb/>diaci, <reg norm="atque" type="simple">atq;</reg> ſi 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/>ſus oppoſitum au-<lb/>gis, <reg norm="longior" type="context">lõgior</reg> erit mo <lb/>ra planetæ in eiuſ <lb/>modi duodecate-<lb/>morio, propter au <lb/>gumentum æqua-<lb/>tionis argumenti. <lb/></s> + <s xml:space="preserve">Itaque probata à <lb/>nobis eſt poſſibi-<lb/>litas huius moræ <lb/>Mattis. </s> + <s xml:space="preserve">Quod qui <lb/>dem mihi ſuffice-<lb/>re videtur <reg norm="non" type="context">nõ</reg> mo-<lb/>do cibi, ſed etiam <lb/>cuiuis, qui harum <lb/>ſcientiarum prin-<lb/>cipia teneat. </s> + <s xml:space="preserve">Ne-<lb/>que enim <reg norm="nunc" type="context">nũc</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 ſunt conſumati, nec curam mihi ſuſcipere erudiendi imperitos. </s> + <s xml:space="preserve">Satis igitur ſit <lb/>oſtendiſſe, quod qui ſcripſit Martem commorari poſſe tam multos menſes in vno ſi <lb/>gno, non impoſſibilem rem tradidit. </s> + <s xml:space="preserve">Immo per obſeruationes huius veritatis mil-<lb/>lies factas, Aſtrologi fecere ſupradictas ſuppoſitiones neceſſarias ad <reg norm="reducendum" type="context">reducẽdum</reg> in <lb/>ſuas cauſas, & ad regulam, eiuſmodi 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 ſcriptorem harum animaduerſionum tantopere eiuſmodi <lb/>mora commoueri, ſed cogitare <reg norm="quod" type="simple">ꝙ</reg> fortaſſe calculi facti fuerunt eo tempore quo mi-<lb/>ſer Mars à Vulcano rete vinctus erat. </s> + <s xml:space="preserve">Vnde cum nonita celeriter ſe expedire poſſet <lb/>iter eius ſegnius peractum fuit. </s> + <s xml:space="preserve">Aut <reg norm="quod" type="simple">ꝙ</reg> <reg norm="quum" type="context">quũ</reg> vulneratus fuit in bello Troiano, vis eius <lb/>& agilitas per aliquantulum temporis imminuta fuit. </s> + <s xml:space="preserve">Atque ſi hic etiam intellexiſ <lb/>ſet eum <reg norm="aliquando" type="context">aliquãdo</reg> fuiſſe in poteſtate Othi, & Ephialtis <reg norm="vinctum" type="context">vinctũ</reg> & carceri incluſum <reg norm="men- ſes" type="context">mẽ-<lb/>ſes</reg> tredecim, dum ab Eribea ſolutus fuit, vt tu, antiquos ſequens, eleganter ſcribis <lb/>in illis tuis pulcherimis dialogis. non exiſtimaſſet, credo, tam abſurdum quod alius <lb/><reg norm="eum" type="context">eũ</reg> detinuiſſet ſex aut ſeptem menſibus, ſed operam dediſſet vt a te intelligeret quid <lb/>ſibi 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 diſtat à me-<lb/>dio circiter dies .8. ſupponens medium motum eſſe dierum .683. <reg norm="quod" type="simple">ꝙ</reg> etiam falſum eſt. <lb/></s> + <s xml:space="preserve">Sed vtcunque ſit, 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 poteſt etiam <lb/>eſſe multo breuior, ſicuti erit à die 3. </s> + <s xml:space="preserve">Decembris anni .1593. <reg norm="vſque" type="simple">vſq;</reg> ad initium Iunij <num value="1595">.<lb/>1595.</num> </s> + <s xml:space="preserve">Tunc enim erit tantum dierum .545. & non quidem ſine ratione, nam dicto <lb/>initio Decembris Mars paulo ante cæperiteſſe directus, cum centrum epicycli erit <lb/>circa medium Tauri, & eius ſtella in principio Arietis & initio Iunij .1595. </s> + <s xml:space="preserve">Mars pa <lb/>rum diſtabit ab initio retrogradationis, regreſſus tamen ad initium ipſius Arietis, & <lb/>centrum epicycli erit circa medium Aquarij, in cuius ſigni medio, hac ætate repe-<lb/>ritur oppoſitum augis, & in quo ſitu, æquationes <reg norm="argumenti" type="context">argumẽti</reg> ſunt, quam maximę eſſe <lb/>poſſint, quum centrum epicycli circuiuerit ſolum circiter tres quartas totius ambi-<lb/>tus, & Mars circuiuerit per partem ſuperiorem epicycli circiter gradus .252. </s> + <s xml:space="preserve">Hoc au <lb/>tem dico, vt oſtendam 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 poſtea idem author miratur interualla, quæ <reg norm="ponuntur" type="context">ponũtur</reg> inter coniunctio <lb/>nes Iouis, & Martis in eodem ſigno, <reg norm="eaque" type="simple">eaq́;</reg> vocat errores maximos, oſtendit ſe non re <lb/>ctè conſideraſſe motus eorum. </s> + <s xml:space="preserve">Et præcipuè primum miratur <reg norm="quod" type="simple">ꝙ</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 ſe diſtent ann .25. afferens pro ratione, quod hæc duo ſydera, altero quoque anno <lb/>coniunguntur, ſic dicens.</s> + </p> + <quote> + <s xml:space="preserve">Qui ſciet has duas ſtellas ſecundo quoque anno inter ſe coniungendas, mirabitur <lb/>quomodo non poterunt numeratores, huiuſmodi 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> conſiderauit, <reg norm="quod" type="simple">ꝙ</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">aũt</reg> in eo ſæpe fuit. </s> + <s xml:space="preserve">Vnde im <lb/>poſſibile eilvidetur eos non conueniſſe in dicto ſigno. </s> + <s xml:space="preserve"><reg norm="Idemque" type="simple">Idemq́;</reg> dici poteſt de alijs <lb/>coniunctionibus eorundem planetarum, atque has differentias temporum inter di-<lb/>ctas coniunctiones ipſe tribuit erroribus calculorum Ephemeridum, non autem ta-<lb/>bularum, vt ſupra dixit. </s> + <s xml:space="preserve">ſed neſcio quare vellet dictos planetas coire in Leone, ſi <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 ſignis <reg norm="vſque" type="simple">vſq;</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">Idq́;</reg> fortaſſe, Iupiter data opera fecit, vt huiuſ-<lb/>modi Aſtrologos 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 poſtea ait, eos ſecundo quoque anno coniungi, animaduertendum eſt, <reg norm="quia" type="simple">ꝗa</reg> <lb/>(vt <reg norm="iam" type="context">iã</reg> dixi) duæ ſunt ſpecies coniunctionum, quarum vna eſt linearum eorum me-<lb/>diorum motuum, altera corporum eorum, ſaltem in longitudine, cum ambo inue-<lb/>niuntur in eodem circulo, qui tranſit per polos ecclipticæ, nam eos inueniri in <reg norm="eadem" type="context">eadẽ</reg> <lb/>linea recta <reg norm="tranſeunte" type="context context">trãſeũte</reg> per centrum mundi, rarisſimum eſt. </s> + <s xml:space="preserve">Atque coniunctio ſupra di-<lb/>ctarum linearum vocatur media, & inter Iouem & Martem fieri ſolet ſpatio <reg norm="dierum" type="context">dierũ</reg>. <lb/>816. cum dimidio, aut circiter. </s> + <s xml:space="preserve">Altera <reg norm="dicitur" type="simple">dicit̃</reg> vera, ſiue apparens, & irregulatiſſima, <lb/>quæ quidem non ſeruat 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 tranſeat per totum Zodiacum, non ideo <lb/>neceſſe eſt, vt in ſpatio .24. annorum coniungantur in ſingulis ſignis, nunquam in eo <lb/>deſicientes, vtipſe credit loquens de veris coniunctionibus apparentibus, eo quod <lb/>ſint irregulatiſſimæ, vt dixi.</s> + </p> + <p> + <s xml:space="preserve">Atque ſi quis velit inuenire periodum coniunctionum mediocrium horum duo-<lb/>rum planetarum, ita faciendum erit. </s> + <s xml:space="preserve">Sumat periodum motus mediocris Iouis, quę <lb/>eſt dierum .4328. & Martis, quæ eſt dierum .687. in quo tempore Martis, Iupiter am <lb/>bulat gra .57. min .8. & diebus .30. conficit. grad .2. minut .29. & ſecun .23. ad ratio-<lb/>nem gra .360. in diebus .4328. </s> + <s xml:space="preserve">Mars verò ad rationem <reg norm="graduum" type="context">graduũ</reg> .360. in diebus .687. <lb/>ſingulis .30. diebus conficit. gra .15. mi .43. <reg norm="ſecum" type="context">ſecũ</reg> .14. vnde differentia inter eos eſt gra <lb/>duum .13. mi .15. <reg norm="ſecum" type="context">ſecũ</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 hęc eſt periodus infallibilis mediarum con <lb/>iunctionum Iouis cum Marte.</s> + </p> + <p> + <s xml:space="preserve">Nunc venientes ad tabulas Animaduerſionum, videbimus hæc mirabilia eius, in <lb/>quo conſiſtant & vbi ſint tam multi inſignes 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, & ſimiliter Iouem. </s> + <s xml:space="preserve">ſed 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">mirũ</reg> videtur, quod vnus ex his calculatoribus detineat Saturnum plu-<lb/>ribus menſibus in vno ſigno, & alter in alio, non animaduertens dictam <reg norm="differentiam" type="context">differentiã</reg> <lb/>eſſe eius rei cauſam. </s> + <s xml:space="preserve">Miratur item, quod vnus ex is faciat Saturnum morari paucis <lb/>menſibus in vno ſigno, alter vero eum ibi detineat integris annis. </s> + <s xml:space="preserve">Vt exempli gra-<lb/>tia, verſus finem ſuæ tabulæ Saturni, dicit quod Leouitius eum carceri includit in <lb/>geminis annis .2. menſe vno, & diebus .9. </s> + <s xml:space="preserve">Stadius vero clementior eum liberat intra <lb/>menſes .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 vſque ad diem .6. Septembris, <lb/>eiuſdem anni gra .6. min .34. <reg norm="eumque" type="simple">eumq́;</reg> poſtea retrogradum inde exire die .22. Decem. <lb/>eiuſdem anni, cum ingreditur in Taurum, vbi partim retrogradus, & partim dire-<lb/>ctus manet vſque ad diem .20. Februa .1560. rediens poſtea in geminos, in quibus <lb/>manet vſque ad diem Iunii .1561. & inde ingreditur in Cancrum, <reg norm="ambulatque" type="simple">ambulatq́;</reg> dire-<lb/>ctus. gra .4. min .59. vſque ad diem .4. Octob. </s> + <s xml:space="preserve">Vnde retrogradiens rurſum intratin Ge <lb/>minos die .28. Decemb. eiuſdem anni, at que ibi partim retrogradus partim directus <lb/>manet vſque ad diem .12. Apr .1562. itaque in pluribus vicibus facit eum morari in <lb/>Geminis dies circiter .816. ideſt circiter menſes .27. ſumpſit autem hic ſcriptor 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 cauſa 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">ꝙ</reg> planeta non perambulet totum ſignum, in quod eſt ingreſſus <lb/>vel directus vel totum retrogradus. </s> + <s xml:space="preserve">Atque hæc opinio ſimilis eſt ſuperiori de con <lb/>iunctionibus veris Saturni, & Iouis, vbi dicit quod nunquam coniunguntur in vno ſi <lb/>gno alterius triplicitatis, niſi perfecerit coniunctionem in omnibus ſignis primæ tri <lb/>plicitatis. </s> + <s xml:space="preserve">Verum vt ſuperſedeam vlterius diſputare, mihi videtur, quod hactenus <lb/>dixi, poſſe tibi ſatisfacere, quod attinet ad ſciendam ſententiam mean ſuper dictis <lb/>Animaduerſionibus latinè ſcriptis. </s> + <s xml:space="preserve">Hoc tamen non prætermittam, <reg norm="quod" type="simple">ꝙ</reg> hic non ani-<lb/>maduertit, <reg norm="nempe" type="context">nẽpe</reg> <reg norm="quod" type="simple">ꝙ</reg> <reg norm="differentiæ" type="context">differẽtiæ</reg> locorum <reg norm="planetarum" type="context">planetarũ</reg> quæ <reg norm="suntinter" type="context">sũtinter</reg> ephemeridas Leouitij <lb/>& Stadij, euenere, quia vnus ſupputat <reg norm="cum" type="context">cũ</reg> radicibus, & <reg norm="fundamentis" type="context">fundamẽtis</reg> Alfonſi alter verò <lb/>Reinoldi ex Copernico recentius obſeruatis, ita idem euenire poterit futuris tem-<lb/>poribus, ſi ſupputati fuerint dicti motus, & loci cum recentioribus obſeruationibus <lb/>cum impoſſibile ſit tam ſubtiliter, tanq́ue perfectè ſupputare loca & <reg norm="motus" type="simple">motꝰ</reg> eorum, <lb/>vtlungo interuallo temporis non comperiantur in eis aliquæ differentię, cuius rei re <lb/>medium eſt ſemper ſequi recentiores obſeruationes & tabulas.</s> + </p> + <p> + <s xml:space="preserve">Atque vt tibi ſatisfaciam etiam circa alia ſcripta vulgari lingua edita menſibus .4 <lb/>poſt latina, etſi intelligere potes, qualia poſſint eſſe alia eius ſcripta, ex ijs quæ ſupra <lb/>dicta ſunt, atque etiam ex eo, quod dicit ſemiſiſſe multa exempla ſuarum Animad <lb/>uerſionum in varias terras, illis qui profitentur has ſcientias, aut earum ſtudioſi ſunt, <lb/>nec <reg norm="quenquam" type="context">quenquã</reg> inueniſſe qui ad <reg norm="tam" type="context">tã</reg> laudabilem prouinciam motus ſit, nec vidiſſe, <reg norm="quod" type="simple">ꝙ</reg> ali <lb/>quis reſponderit eius rationibus; </s> + <s xml:space="preserve">laudabilem prouinciam, autem puto, <reg norm="quod" type="simple">ꝙ</reg> intelligat <lb/>correctionem ephemeridum, verens, ne culpa calculatorum, qui eas ſumpſere e ta-<lb/>bulis, tam differentes ſint, vt <reg norm="quibuſdam" type="context">quibuſdã</reg> locis cap .1. </s> + <s xml:space="preserve">Videtur, & præcipuè vbi ſic ait.</s> + </p> + <quote xml:lang="it"> + <s xml:space="preserve">Perche eſſendo impoſſibile alli ſtudioſi di dette ſcientiæ di non ſeruirſi delle <lb/>ephemeridi, maggiormente a quelli che non ſanno ſeruirſi delle tauole, e cono-<lb/>ſcendo d'incorrere in errori ſenza hauerui altro rimedio, ſarebbono forzati di ab <lb/>bandonare i ſtudij loro.</s> + </quote> + <p> + <s xml:space="preserve">Quanquam circa finem dicti capitis redeat in meliorem viam & aduerſetur ſi-<lb/>bijpſi vbi ſic ait.</s> + </p> + <quote xml:lang="it"> + <s xml:space="preserve">Che poi eſſi poſſeſſori della ſcienza, & c.</s> + </quote> + <p> + <s xml:space="preserve">Etiam aperiam tibi, quæ mea ſit de ijs ſententia.</s> + </p> + <p> + <s xml:space="preserve">Hicigitur in ſcriptis Italicis, vt morderet aliquem ex ijs, qui eius ſuperiora ſcripta <lb/>non laudauerant, occaſionem capit aperiendi aliquos illius errores, per editionem <lb/>collationis quorundam calculorum a ſe collectorum illius, atque etiam aliorum, cu <lb/>ius calculi ſunt in ſecunda, & ſeptima figura. </s> + <s xml:space="preserve">Sed prius quam veniamus ad defenſio <lb/>nem harum duarum figurarum vide obſecro quam alienum ei videatur, quod alij <lb/>dixerint differentiam ephemeridum non eſſe magni momenti, non afferens reſpe-<lb/>ctum vllum, qui enim dixerunt eiuſmodi differentiam non eſſe magni momen-<lb/>ti id dixerunt habito reſpectu ad ſignum in quo eſt 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 ſigno, tuncid nullius momenti eſt, <lb/>& ſic in coniunctionibus aut alijs aſpectibus duo, aut .3. gradus non faciunt alteratio <lb/>nem ſenſibilem, cum virtus coniunctionum, & aſpectuum inſit, & duret per mul-<lb/>tos gradus ante aut poſt ipſum punctum. </s> + <s xml:space="preserve">Nec quicquam <reg norm="tamen" type="wordlist">tamẽ</reg> eſt qui dubitet, quin <lb/>præſtaret ſcire ſubtiliter ipſum 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 correſpondeant gradibus æquatoris. </s> + <s xml:space="preserve">Et præterea in ephe <lb/>meridibus videntur certè motus & aſpectus luminarium, quamuis inſit <reg norm="differentia" type="context">differẽtia</reg> mi <lb/>nutorum. </s> + <s xml:space="preserve">Nam non differunt gradibus, præter ſitum parum diſtantem à vero om-<lb/>nium planetarum, quorum cognitio in cœlo, quamuis circa eorum locum error eſ-<lb/>ſet gra .10. tamen in hoc prodeſſet, & tempus aſpectus eorum, etſinon diei præcisè, <lb/>quia influentia eiuſmodi a ſpectuum, præterquam Lunæ durat multis diebus, & non <lb/>vno tantum. </s> + <s xml:space="preserve">præterquam quod ipſæ ephemerides <reg norm="oſtendunt" type="context">oſtendũt</reg> nobis tempus <reg norm="ecclipſium" type="context">ecclipſiũ</reg>, <lb/>in quo certènon differunt nec diebus nec multis horis, & itidem multa alia.</s> + </p> + <p> + <s xml:space="preserve">Non ſunt 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 ſignis alterius triplicitatis prius quam peręgerit <lb/>præcedentem, quam de faciendo currere <reg norm="Saturnum" type="context">Saturnũ</reg>, & de retinendo Ioue, de <reg norm="detinem" type="context">detinẽ</reg> <lb/>do Marte .6. aut .7. menſibus in vno ſigno, de Marte, & Ioue non coeuntibus ſingu-<lb/>lis .24. annis in quolibet ſigno, & cius generis alia, minime verum eſt quod ſint er-<lb/>rores, quamuis huic præbuerint occaſionem toties errandi.</s> + </p> + <p> + <s xml:space="preserve">Comparatio poſtea inter eius calculos ſumptos partim ex tabulis Iunctini, & par <lb/>tim ex ephemeridibus Stadij tan quam calculis Copernici, & calculos figurarum ſu-<lb/>per eis poſitarum ſupputatarum à diuerſis per ephemeridas Alfonſinas, etiam pro-<lb/>poſita ab eo eſt ad oſtendendum magnam & monſtruoſam differentiam, vt ait cap <num value="2">.<lb/>2.</num> vbi miratur, quod cum ex communi ſententia calculi Copernici meliores ſint, cal <lb/>culatores dictarum figurarum potius eos ſumpſerint à tabulis Alfonſi, quam Coper <lb/>nici. </s> + <s xml:space="preserve">Quæ admiratio quam aliena ſit, conſiderandum permittam cuiuis intelligen-<lb/>ti harum ſacultatum, cum ſæpe accidere poſſit. </s> + <s xml:space="preserve">vt cum aliquis velit ſcire ſolum vni <lb/>uerſalia alicuius geneſis, ſiue natiuitatis, cum non inueniantur ephemerides Coper-<lb/>nici, ſed tantum Alfonſi, calculator vtatur tantum ephemeridibus, quas inuenit, <reg norm="tum" type="context">tũ</reg> <lb/>cauſa vitandi tædij calculi tabularum, qui magni laboris eſt, pręcipuè in tabulis Pru <lb/>tenicis Reinoldi. </s> + <s xml:space="preserve">tum quia ſuperflua ei eſt ſumma ſubtilitas, cum non curet laborare <lb/>circa directiones vt <reg norm="factum" type="context">factũ</reg> eſt pro <reg norm="ſecunda" type="context">ſecũda</reg> figura ab hoc propoſita, quæ erat anni .1551 <lb/>quo non <reg norm="inueniebantur" type="context">inueniebãtur</reg> ephemerides Copernicæ, quæ non editæ ſunt ante annum <lb/>1554. præter quam quod ille nobilis vir pro quo ſupputata fuit dicta ſecunda natiui <lb/>tas dubitabat de anno, vt hic ſimiliter ſcit. </s> + <s xml:space="preserve">quare potuiſſet perdi tempus, & labor, ſi <lb/>ſupputata fuiſſet per tabulas Reinoldi, nam Iunctini tabulæ nondum editæ fuerant. <lb/></s> + <s xml:space="preserve">Calculus poſtea ſeptimæ figuræ, qui erat reuolutio dictæ ſecundæ natiuitatis, <reg norm="duabus" type="simple">duabꝰ</reg> <lb/>de cauſis non factus eſt 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="deſinunt" type="context">deſinũt</reg> ann .1576. & <reg norm="continuatæ" type="context">cõtinuatæ</reg> poſtea quæ perueniunt vſque <lb/>ad annum .1600. non peruenere ad manus calculatoris ante hunc annum .1581. </s> + <s xml:space="preserve">Al-<lb/>tera ratio eſt, 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 ſolum non curant eas ſupputare per <lb/>tabulas, ſed nec exquiſitè quidem per ephemeridas. </s> + <s xml:space="preserve">Calculi poſtea ab hoc ſumpti <lb/>ex tabulis Iunctini, & poſiti ſub dicta ſecunda figura, adeò rectè facti ſunt, vt cum ſe <lb/>cundum ip ſas tabulas oporteat Saturnum eſſe circa .32. minutum gradus .23. Aqua-<lb/>rij, ipſe eum ſcribat in gra .11. mi .3. dicti ſigni. Iupiter ſimiliter qui ſecundum dictas <lb/>tabulas inuenitur circa finem gradus .5. Cancri, ab eo ponitur in min .28. gra .19. eiuſ <lb/>dem. ex quibus planetis Saturnus in figura poſitus eſt in min .27. grad .23. Arietis, Iu <lb/>piter autem in min .3. gra .6. Cancri, </s> + <s xml:space="preserve">Vnde ſecun 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"/> + ſi & Iunctini in Saturno non erat differentia plus quam minu .5. & in Ioue min .4. ſed <lb/>ſecundum calculum huius in Saturno fuiſſet differentia gra .11. minu .54. & in Ioue <lb/>grad .13. min .35. </s> + <s xml:space="preserve">Atque hæ ſunt quidem differentiæ magnæ, & monſtruoſæ vtipſe <lb/>eas vocat, vt etiam eſtilla Veneris, & Mercurij inter tertiam figuram, & eius calcu-<lb/>lum ſumptum, non quidem à tabulis laborioſis, ſed à ſimplicibus ephemeridibus Sta <lb/>dij, quæ differentia <reg norm="non" type="context">nõ</reg> eſt quidem paucorum graduum, cum ſit tertiæ partis cœli in <lb/>quolibet dictotum planetarum. </s> + <s xml:space="preserve"><reg norm="Huiuſmodique" type="simple">Huiuſmodiq́;</reg> monſtra certè non ſunt orta à ta-<lb/>bulis ſiue ephemeridibus diuerſis, ſed ſunt partus huius authoris.</s> + </p> + <p> + <s xml:space="preserve">Pergens poſtea aſſiduè bonus hic vir hominibus dare ſpecimen doctrinæ ſuæ ape <lb/>riendo (vt conatur) aliorum errores, proponit duas differentias inter primam fi-<lb/>guram, & ſuum calculum ſuppoſitum Saturni, & Iouis. </s> + <s xml:space="preserve">Primum de Saturno ait, <reg norm="quod" type="simple">ꝙ</reg> <lb/>cum differentia ſit gra .1. min .30. oſtendit in directione, <reg norm="quod" type="simple">ꝙ</reg> accidens ſit euenturum <lb/>anno vno, & menſibus ſex ante, aut poſt, quaſi eiuſmodi differentia eſſet partium æ-<lb/>quatoris, ſicuti eſt partium Zodiaci. </s> + <s xml:space="preserve">Idem dico de differentia Iouis. </s> + <s xml:space="preserve">Quod quidem, <lb/>manifeſtum eſt inditium ſcientiæ ſuæ, & quantum ea intelligat de quibus loquitur.</s> + </p> + <p> + <s xml:space="preserve">Quod poſtea attinet ad differentiam inter Copernicum & Alfonſum, circa <reg norm="Solem" type="context">Solẽ</reg>, <lb/>nullus eſt harum ſcientiarum peritus, qui id neſciat, & ſimiliter de differentia ſitus <reg norm="cae li" type="simple">cę<lb/>li</reg> in reuolutionibus annuis.</s> + </p> + <p> + <s xml:space="preserve">Quod vero ait ſeptimam ſiguram malè <reg norm="ſupputatam" type="context">ſupputatã</reg> fuiſſe, ſi non eſt maximus cer-<lb/>tè non eſt minimus monſtruoſorum eius errorum. </s> + <s xml:space="preserve">Vbi itidem videri poteſt, quam <lb/>alienus hic ſit ab hac ſcientia. </s> + <s xml:space="preserve">Nam ſi ſaltem curaſſet ſibi ab aliquo ſupputandum <lb/>locum Solis per tabulas Alfonſi in inſtanti 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/>ideſt præterierat gra .10. cum min .54. vel ſi curaſſet ſibi inueniendum tempus, per <lb/>dictas tabulas cum grad .10. min .54. </s> + <s xml:space="preserve">Geminorum vt faciendum eſt, ſequendo <reg norm="tamen" type="wordlist">tamẽ</reg> <lb/>Alfonſum, & non per calculum Solis poſitum in ephemeridibus, vt parum periti fa <lb/>cere ſolent, vidiſſet <reg norm="quod" type="simple">ꝙ</reg> inuenta eſſent min .36. pomeridiana. </s> + <s xml:space="preserve">Leuis tamen occaſio hu <lb/>ic fuit ſuſpicandi eiuſmodi tempus eſſe falſum, quod viderit in illa figura <reg norm="Solem" type="context">Solẽ</reg> po <lb/>ſitum eſſe cum gra .11. & non cum gra .10. min .54. non animaduertens ita notatum <lb/>fuiſſe Solem vt omnes alios planetas, ſcilicet ſine minutis, quum, vt dixi, in reuolu <lb/>tionibus non adhibeatur tanta ſcruploſitas.</s> + </p> + <p> + <s xml:space="preserve">Quod deinde ait, in illa figura Solem poſitum eſſe in decima domo, & non in .9. <lb/>id relinquam iudicio eorum qui ſciunt numerare domos, ſaltem poſuiſſet authori-<lb/>tate ſua Solem in dicta decima diuersè ab exemplo ei dato ab amico, vt oſtenderet <lb/>ſe dicere verum, vt in ſecunda figura diſcrepat ab ipſo exemplo in collocando Leo-<lb/>ne, Virgine, & Libra, & Scorpio, quos malè locauit, & ſi alii bene ſe habent.</s> + </p> + <p> + <s xml:space="preserve">Atque quod hactenus à me dictum eſt, ſatis ſit ad <reg norm="intelligendum" type="context">intelligẽdum</reg> quale ſit <reg norm="reliquum" type="context">reliquũ</reg> <lb/>dictæ eius diſputationis. </s> + <s xml:space="preserve">Sienim velim pergere notare omnia eius errorum loca, eſ <lb/>ſet mihi inanis labor, & tibi nimia moleſtia. </s> + <s xml:space="preserve">Et quamuis non defuerint præſtantiſſi-<lb/>mi viri, qui viſis eius ſcriptis familiariter eum monuere, & tu ipſe, vt audiui, cum <reg norm="in- ſtrumento" type="context">in-<lb/>ſtrumẽto</reg> theoricę in <reg norm="manibus" type="simple">manibꝰ</reg> ei <reg norm="oſtenderis" type="context">oſtẽderis</reg> quo <reg norm="mon" type="context">mõ</reg> Mars poſſit morari amplius ſex <reg norm="mem" type="context">mẽ</reg> <lb/>ſibus in vno ſigno. </s> + <s xml:space="preserve">& præterea cum iam ab initio Taurinum aduenit, mecum com-<lb/>municauerit illa ſua prima ſcripta, <reg norm="egoque" type="simple">egoq́;</reg> eum monuerim, quod in varijs <reg norm="rebus" type="simple">rebꝰ</reg> falleba <lb/>tur, <reg norm="diſſuaſerimque" type="simple">diſſuaſerimq́;</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/>uerſiones differentiarum ephemeridum attinet, quod id iam <reg norm="oens" type="context">oẽs</reg> animaduerterant. <lb/></s> + <s xml:space="preserve">Mihi reſpondit ſe decreuiſſe illa edere, vt poſtea 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/>ieſcens, die .11. </s> + <s xml:space="preserve">Auguſti edidit chartam illam impreſſam inuitans ad diſputa-<lb/>tionem quotquot adhęrerent contrariæ ſententiæ, volens ſuſtinere <reg norm="Martem" type="context">Martẽ</reg> non poſ <lb/>ſe commorari in vno ſigno amplius duobus menſibus, ſupponens partem princi-<lb/>piorum ab omnibus admiſſorum, & in fine paginæ exponens modum, quo vtitur ad <lb/>probationem ſuæ intentionis. </s> + <s xml:space="preserve">Puto autem quodſecum ratiocinabatur de Marte, <lb/>vt fecit de Saturno in ſcripto latino, hoc modo. </s> + <s xml:space="preserve">Si Mars in duobus annis ambulat <lb/>per omnia 12. ſigna, neceſſe eſt igitur, vtin menſibus duobus ambulet per vnum ſi-<lb/>gum, cum menſes .2. ſint duo decima pars annorum duorum. </s> + <s xml:space="preserve">Sedibi ſtatim in ipſo <lb/>initio commitcit errorem graduum ferè .7. dicens, quod medius motus Martis inue-<lb/>niebatur ſignorum .4. & gra .17. cum eo tempore dictus medius motus non eſſet reue <lb/>ra plus quam ſign .4. grad .10. mi .36. verum hoc ad ea, quæ ſequuntur exigui eſt mo <lb/>menti. </s> + <s xml:space="preserve">Is poſtea particulatim colligit medium motum Martis ad diem .29. </s> + <s xml:space="preserve">Mai an-<lb/>ni .15 14. quem ait eſſe ſignorum .9. gra .27. min .53. & tamen reuera erat <reg norm="tantum" type="context">tantũ</reg> ſigno <lb/>rum .9. gra .21. mi .29. ſed 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, ſimiliter errat, nam <lb/>centrum epicycli nunquam poni debet vbi eſt linea medij motus, niſi ſit in auge, aut <lb/>in oppoſito aug is eccentrici, quia debebat collocare ipſum centrum <reg norm="tanto" type="context">tãto</reg> poſt linec<unsure/><reg norm="am" type="context">ã</reg> <lb/>medij motus, quanta erat æquatio centri, quia medium centrum Martistunc erat mi <lb/>nus ſignis ſex, & aux eccentrici eius erat in ſexto minuto grad .16. Leonis. </s> + <s xml:space="preserve">Tamen <lb/>hoc etiam leue eſt. </s> + <s xml:space="preserve">Præſupponamus igitur quod centrum epicycli cſſet in grad .28 <lb/>Capricorni, vt ipſe credidit, ideſt gradibus .7. vlterius quam erat reuera. </s> + <s xml:space="preserve">Ait poſtea <lb/>ſe comperiſſe Martem ambulaſſe ſigna .4. & grad .22. eius epicycli, ſed non explicat <lb/>an intelligat de argumento medio, an de vero, quod vocatur æquatum, nam ſi intel <lb/>ligatur de medio, hoc eſſe non poteſt, cum <reg norm="medium" type="context">mediũ</reg> eſſet ſignorum .4. gra .24. mi .35. <lb/>ſed ſi intelligatur de vero, vt iure credendum eſt (alioquin etiam erraſſet) cer-<lb/>tè falſum eſt. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg>, verum, erat ſignorum .4. grad .29. minu .39. </s> + <s xml:space="preserve">Itaque Mars non di-<lb/>ſtabat à linea veri motus epicycli amplius gradibus .30. & minu .21. ipſius epicycli, <lb/>& æquatio argumentiſecundo correcta erat gra .44. minu .2. à quo ſubtracta æquatio <lb/>ne centri, quæ erat gr .5. minu .4. (cum centrum epicycli deberet tanto ſpacio eſſe <lb/>poſt lineam medij motus quantum ſupra dixi) ſupererant 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. Piſcium. </s> + <s xml:space="preserve">Sed præſuppoſito <reg norm="ſecundum" type="context">ſecundũ</reg> <lb/>ipſum, quod medius motus eſſet grad .28. </s> + <s xml:space="preserve">Capricorni, & quod Mars eſſet non <reg norm="ſolum" type="context">ſolũ</reg> <lb/>vbi hic ait, ſed etiam in prima linea contingentiæ epicycli, ideſt in prima linea ma <lb/>ximæ æquationis argumenti, & præſuppoſito etiam quod dicta æquatio eſſet æqua-<lb/>lis illi, quam haberet ad medium Aquarij ſcilicet grad .47. quum centrum epicycli <lb/>eſt in oppoſito augis, manifeſtum eſt, quod eiuſmodi linea contingentiæ non tranſi <lb/>ret vltra grad .15. Piſcium, & 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 ſi verum eſſet, & <var>.o.c.</var> etiam eſſet partium .54. ſecundum diſtantiam pro-<lb/>ximiorem centro mundi, ſemidiameter epicycli eſſet eiuſmodi partium .52. minut <num value="49">.<lb/>49.</num> & quum Mars eſſet in <var>.g.</var> ideſt in oppoſito veræ augis epicycli, dum centrum epi <lb/>cycli eſſet in eiuſmodi diſtantia à terra, diſtantia <var>.o.g.</var> ideſt à terra ad Martem non <lb/>eſſet plus, quam vna ſola pars ex dictis, cum minut .11. cum partes .52. minu .49. <lb/>ad .54. ſint vt ſinus anguli gra .78. qui eſt partium .97814. ad ſinum totalem partium <lb/>100000. </s> + <s xml:space="preserve">Nam iam ſupra dixi, quod triangulus <var>.o.c.i.</var> eſt rectangulus. </s> + <s xml:space="preserve">Hinc ſeque- + <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">vniꝰ</reg> <lb/> + <anchor type="figure" xlink:label="fig-0257-01a" xlink:href="fig-0257-01"/> + partis, & min .11. <lb/>reſpectu <var>.o.c.</var> par-<lb/>tium .54. </s> + <s xml:space="preserve">Colloca <lb/>retur ſemidiame-<lb/>ter terrę cum ſpiſ-<lb/>ſitudine aeris, i-<lb/>gnis, cęlorum Lu-<lb/>næ, Mercurij, Ve-<lb/>neris, & Solis, <reg norm="prae terquam" type="simple">prę<lb/>terquam</reg> quod vt <lb/>inter Solem, & <lb/>terram ſunt circa <lb/>605. diametri ip <lb/>ſius terræ, inter <lb/>terram, & Mar-<lb/>tem cum eſſet in <lb/>auge ſui epicycli, <lb/>& epicyclus in au <lb/>ge eccentrici, in-<lb/>uenirentur cir--<lb/>ca .60000. dia--<lb/>metri eiuſdem ter <lb/>ræ, & tamen ea diſtantia ſiue interuallum non poteſt continere .5000. diametri ter-<lb/>ræ. </s> + <s xml:space="preserve">Et quod plus eſt, hic tam vaſtum facit hunc ſuum 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. menſ. </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 ſint 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 falſis ſuppoſitis huius) Videamus ordine ſcientifico vbi poterat eſſe verus lo-<lb/>cus Martis, aut vero proximus, die .29. Mai anni .1514. quem hic exempli cauſa ſu-<lb/>mit. </s> + <s xml:space="preserve"><reg norm="Idque" type="simple">Idq́</reg> tam ad defenſionem tabularum Alfonſi, quam ephemeridum ex eis col <lb/>lectarum. </s> + <s xml:space="preserve">quæ quidem exactæ ſunt, vt quiſque peritus ſacile videre poterit, non au-<lb/>tem calculatæ à tam ſtupidis hominibus, vt à vero aberrent etiam gradibus .46. vt <lb/>hic ait ſe depræhendiſſe.</s> + </p> + <p> + <s xml:space="preserve">Primum igitur ſupponemus eoſdem illos terminos, quos ipſe nec d@bet, nec po <lb/>teſt negare, præter ea quæ ſupra ſuppoſita ſunt, nempe quod ſemidiameter epicycli <lb/>ſit partium .39. minu .30. & eccentricitas partium .6. talium qualium eſt ſemidiame-<lb/>ter deferentis diuiſus in .60. & quod dicto tempore aux eccentrici Martis eſſet cir-<lb/>ca minutum .5. grad .16. Leonis, ſcilicet 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">centrũ</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 oſtendam, neque tabulas, neque ephemerides errare, ne quidem vno gra-<lb/>du, ac ne quidem multis minutis, non modò tam monſtruoſa differentia, vt ipſe <lb/>ait.</s> + </p> + <p> + <s xml:space="preserve">Quare primum nobis ſcientificè inueniendum eſt, quanta eſſet diſtantia <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"/> + præciſe ideſt interuallum inter centrum mundi, & centrum epicycli Martis in huiuſ-<lb/>modi ſitu.</s> + </p> + <p> + <s xml:space="preserve">Fingemus igitur eccenticum Martis ſignificatum per <var>.p.c.m.</var> cuius centrum ſit <var>.r.</var> <lb/>& lineam augis <var>.p.r.o.m.</var> in qua <reg norm="centrum" type="context">centrũ</reg> mundi ſit <var>.o.</var> centrum autem verum epicycli, <lb/>comprehendatur ab angulo <var>.p.o.c.</var> qui ſit graduum .151. min .30. ſecundum ſuppoſi-<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> quouſque 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 cognoſcemus angulum <var>.r.o.k.</var> tanquam reliquum ex duobus rectis, qui <lb/>erit gra .28. min .30. & ſimiliter angu-<lb/> + <anchor type="figure" xlink:label="fig-0258-01a" xlink:href="fig-0258-01"/> + lum <var>.o.r.k.</var> tanquam reſiduum vnius <lb/>recti, qui erit gra .61. min .30. cuius ſi-<lb/>nus ideſt <var>.o.k.</var> erit partium .8788 1. et <var>.k.<lb/>r.</var> vt ſinus anguli <var>.r.o.k.</var> partium .47715 <lb/>talium qualium <var>.o.r.</var> eſſet 100000. ſed <lb/>vt <var>.o.r.</var> eſt .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/>cſt <reg norm="partium" type="context">partiũ</reg> 60. eiuſmodi, ſi ab eius qua-<lb/>drato ſubtractum fuerit quadratum ip <lb/>ſius <var>.r.k.</var> reliquum erit nobis <reg norm="quadratum" type="context">quadratũ</reg> <lb/>ipſius <var>.k.c.</var> cuius radix, ideſt <var>.k.</var> erit par-<lb/>tium .59. min .56. à qua <var>.c.k.</var> ſubtrahen-<lb/>do poſtea <var>.k.o.</var> partium .5. minu .16. re-<lb/>manebit <var>.o.c.</var> partium .54. min .40. pro <lb/>diſtantia 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 poſtea epicyclum <var>.f.n.g.</var> <lb/>in quo argumentum verum graduum <num value="149">.<lb/>149.</num> minu .39. ſit arcus <var>.f.n.</var> vbi Mars inueniatur in <var>.n.</var> per quem punctum tranſeat 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, ideſt ducendo ſinum <var>.n.h.</var> arcus <var>.n.g.</var> qui arcus tanquam reliquus <lb/>argumenti veri, iam præſuppoſiti, ex dimidio circulo, erit graduum 30. minu .21. & <lb/><var>n.h.</var> eius ſinus partium .50528. ſinus ſimiliter anguli <var>.n.c.h.</var> et <var>.c.h.</var> tanquam ſinus an-<lb/>guli <var>.c.n.h.</var> reſtantis ex uno recto grad .59. minu .39. erit partium .86295. <reg norm="talium" type="context">taliũ</reg> qua-<lb/>lium <var>.c.n.</var> ſinus totus eſſet partium .100000. ſed vt partium .39. & min .30. ſinus <var>.c.h.</var> <lb/>erit partium .34. min .5. et <var>.n.h.</var> partium .19. mi .57. reliquum poſtea <var>.h.o.</var> ex <var>.o.c.</var> par-<lb/>tium .20. min .35. quia iam ſupra inuenimus <var>.o.c.</var> eſſe partium eiuſmodi .54. minu .40. <lb/>vnde <var>.o.n.</var> vt radix quadrata ſummæ 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 poſtea partium, <lb/>69552. talium qualium <var>.n.o.</var> partium .100000. & ſumpta dicta <var>.n.h.</var> vt ſinus dictarum <lb/>partium, dabit nobis angulum <var>.n.o.h.</var> quæſitum gra .44. min .4. qui per tabulas Alfon <lb/>ſi inuentus eſt gra .44. min .2. par huic, vt dici poteſt. </s> + <s xml:space="preserve">Quiangulus gra .44. minu .4. <lb/>collectus cum angulo veri centri iam ſuppoſito graduum .151. minu .20. & cum an-<lb/>gulo augis eccentrici Martis, ſimiliter ſuppoſitæ grad .135. min .5. dabit nobis ſum-<lb/>mam veræ diſtantiæ Martis à principio Arietis grad .330. min .29. quod aliud non <lb/>ſignificat, niſi quod Mars inuenietur in minu .29. primi gradus Piſcium. </s> + <s xml:space="preserve">Et Stofle-<lb/>rus in ſuis 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/>eſt minut .5. tan <lb/> + <anchor type="figure" xlink:label="fig-0259-01a" xlink:href="fig-0259-01"/> + <reg norm="tum" type="context">tũ</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 ſumere ip-<lb/>ſum angulum æ-<lb/>quationis à tabu <lb/>lis propter duas <lb/>rationes, <reg norm="primum" type="context">primũ</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">ĩ</reg> ſua <lb/>theorica Martis. <lb/></s> + <s xml:space="preserve">Vbi <reg norm="inuenit" type="wordlist">ĩuenit</reg> quod <lb/>linea <var>.o.n.</var> tranſit <lb/>per gra .16. </s> + <s xml:space="preserve">Arie <lb/>tis. </s> + <s xml:space="preserve">Secunda ra-<lb/>tio eſt, vt videa-<lb/>tur quod dictæ tabulæ rectè ſupputatæ ſunt, ſuper dictis ſuppoſitis.</s> + </p> + <p> + <s xml:space="preserve">Sed vt videat quantus ſit medius motus Martis die .29. </s> + <s xml:space="preserve">Mai colligit fruſtatim, <reg norm="quod" type="simple">ꝙ</reg> <lb/>eleganter colligere poterat, vna opera in columellis ipſius medij motus eiuſmodi <lb/>ſtellæ per eram eiuſdem temporis, quæ erat .2. primarum ſexagenarum .33. ſecunda <lb/>rum .32. tertiarum, et .52. quartarum.</s> + </p> + <p> + <s xml:space="preserve">Primum deinde ſuppoſitum quod ſcribit, ſcilicet, quod diameter epicycli ſum-<lb/>ptus in longitudine media ſit ſignorum .2. & grad .19. vti ſuperfluum eſt, ita etiam fal <lb/>ſum, nam eiuſmodi diameter in dicto loco, non occupat ad centrum mundi plus <reg norm="quam" type="context">quã</reg> <lb/>gra .66. min .28. ideſt ſigna .2. gra .6. min .28. quia proportio <var>.o.c.</var> ad ſemidiametrum <lb/>epicycli in eiuſmodi loco eſt ut partium .60. minu .18. ad partes .39. min .30. quę duę <lb/>lineæ intellectę, vt latera vnius trianguli rectanguli, habebunt pro baſi aliam lineam <lb/>partium ſimilium .72. mi .5. </s> + <s xml:space="preserve">Quæ intellecta vt ſinus totus dabit ſemidiametrum epi-<lb/>cycli partium .54798. tanquam ſinum ſubiectum 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 ſpatium duorum ſignorum cum gradibus. <lb/>19 quæ quidem ratio alia eſſe non poteſt, niſi quod cum iſte inuenerit, in <reg norm="comentarijs" type="context">comẽtarijs</reg> <lb/>Theoricarum, ſemidiametr um huiuſmodi epicycli eſſe partium .39 min .30. talium <lb/>qualium ſunt .60. illæ quæ ſunt ſemidiametri huius eccentrici, dictas igitur partes .39 <lb/>mi .30. hic putauit eſſe gradus Zodiaci, & propterea dixit <reg norm="diametrum" type="context">diametrũ</reg> huiuſmodi epicy <lb/>cli eſſe <reg norm="ſignorum" type="context">ſignorũ</reg> <reg norm="duorum" type="context">duorũ</reg>, & gra .19. qui numerus .79. duplus eſt numero .39. <reg norm="cum" type="context">cũ</reg> dimidio <lb/>hoc autem dixit accidere in longitudinibus medijs, quia ſi hic intellexiſſet de pro-<lb/>portione horum duorum diametrorum, quæ eſt ut .120. ad .79. non ſpecificaſſet 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 ipſa proportio nullo modo alteratur exiſtente epicyclo vbi volue <lb/>ris ipſius circunferentiæ eccentrici, ſed angulus in centro mundi, cui ſubiacet dictus <lb/>diameter epicycli, bene alteratur, propter inæqualem diſtantiam centri epicycli <lb/>ab ipſo centro mundi. </s> + <s xml:space="preserve">At ſi de tali angulo inferre voluiſſet, iam probaui ipſum <reg norm="con- tinere" type="context">cõ-<lb/>tinere</reg> ſolum gra .66. minu .28. exiſtente centro epicycli in longitudinibus medijs & <lb/>non gra .79. vt ipſe dicit.</s> + </p> + <p> + <s xml:space="preserve">Omitto poſtea, quod vbi mentionem facit coniunctionum Solis cum Marte au-<lb/>gium & earum oppoſitorum, non explicat an intelligat de veris an de medijs. </s> + <s xml:space="preserve">Nam <lb/>ſi ex eius modo loquendi accipiatur eum loqui de veris multum erraret.</s> + </p> + <p> + <s xml:space="preserve">Sed quia iam tibi moleſtum eſſe inciperet ſi diutius te detinerem in his conten-<lb/>tionibus aſtronomicis, vlterius non diſputabo. </s> + <s xml:space="preserve">Satis enim hactenus explicaui <reg norm="ſenten- tiam" type="context">ſentẽ-<lb/>tiam</reg> meam, vt oſtendiſſe videor quam mihi iucundum ſit 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">quũ</reg> petitione tua occaſionem mihi de deris <lb/>efficiendi, vt tum amici tui (amant enim te omnia ſublimia ingenia) tum alij, ſi <reg norm="quam" type="context">quã</reg> <lb/>falſam opinionem ex huius Benedicti Altæuillæ ſcriptis ſumpſiſſent, <reg norm="eam" type="context">eã</reg> relinquant, <lb/>& per te hoc beneficium à me conſequantur, & huiuſmodi occaſionem, & iuuandi <lb/>hominum ſtudia & tibi gratum faciendi, honorificum, & per gratum mihi fuiſſe in-<lb/>telligant. </s> + <s xml:space="preserve">Vale & me vt ſoles 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 diuiſionis numerorum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">INter alia quæ à me ſcire cupis, vir doctiſſime, hoc vnum eſt, vt ex literis tuis ac-<lb/>cepi, vnde ſit vt priſci noſtri probatione numeri nouenarij potius quam ſepte-<lb/>narij vſi fuerint, & qua ratione non idem proueniat ex probatione numerorum <lb/>octonarij, ſenarij, vel quinarij, aut cuiuslibet alterius: </s> + <s xml:space="preserve">Vnde pariter oriatur quod in <lb/>partitionis probatione neceſſum ſit probationum euentus multiplicare cum proba-<lb/>tione diuiſoris, ac eam quæ eſt producti poſtea cum probatione fractionis in ſum-<lb/>mam colligere, & c. </s> + <s xml:space="preserve">Ad hæc in primis reſpondeo, cum aliquoties accidere poſſit ta <lb/>les probationes nos fallere poſſe, <reg norm="idque" type="simple">idq́;</reg> fi in tali ſumma ſimilis numerus, ut puta ſe-<lb/>ptem, aut nouem, plus vel minus æquo iuſtouè poſitus fuerit, attamen per raro eueni <lb/>re poteſt, vt quis per nouenarium potius quam per ſeptenarium decipiatur. </s> + <s xml:space="preserve">Exem-<lb/>pli gratia, ponamus ſummam eſſe .100. quam numerus nouenarius vndecies ſolum <lb/>ingreditur, at ſeptenarius quatuordecies, vnde quis <reg norm="ſępius" type="simple">ſępiꝰ</reg> ex ſeptenario, hacratione, <lb/>quam ex nouenario numero ſe poſſe errare facile depræhendet, efſi ex probatione <lb/>nouenarij magis quam ſeptenarij, vt practici ſcribunt, duabus de cauſis errare poſſi-<lb/>mus. </s> + <s xml:space="preserve">Alia tamen ratio mihi ſuppetit, ob quam credibile eſt ipſos potius nouena-<lb/>rio adiutos fuiſſe, quam ſeptenario, quæ eſt ob ſui cum velocitatem tum <reg norm="facilitatem" type="context">facilitatẽ</reg>, <lb/><reg norm="neque" type="simple">neq;</reg> enim in ſeptenario eſt adeo facilis. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> quamuis, tam vna quam altera aliud <lb/>non ſit, quam numerorum ordines diuidere (ſi de ſummis primo loquamur) aut è <lb/>ſumma ſuperſluum ordinum colligere, & videre an idemmet ſuperfluum ex eadem <lb/>ſumma emanet, attamen cum modus, qui in hoc adhiberi poteſt in nouenario <reg norm="quam" type="context">quã</reg> <lb/>in ſeptenario velocior ſit, & ob id probationem nouenarij ſeligunt potius quam <lb/>ſeptenarij.</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, quæfalſa eſt, opinione conſiſtere, nonidem, & cum octona-<lb/>rio, ſenario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra <lb/>tio, quæ in ſeptenario, aut nouenario, <reg norm="ent" type="context">ẽt</reg> in cæteris perhibeatur. </s> + <s xml:space="preserve">Ponamus <reg norm="exemplum" type="context">exemplũ</reg> <lb/>hos tres or dinum numeros velle ſupputare, quorum primus ſit .679. ſecundus .846. <lb/>& tertius .935. & illorum <reg norm="ſummam" type="context">ſummã</reg> .2460. nunc maiorem numerum primi ordinis ab <lb/>octonario menſi, proijciendo, remanebit .7. deinde maiorem numerum demendo à <lb/>ſecundo or dine, reſiduum erit .6. ac ſi idem in tertio ordine fecerimus, erit nobis re-<lb/>liquum .7. </s> + <s xml:space="preserve">Demum tria hæc reſidua in vnum collecta .20. efficient, à quibus ſi nume <lb/>rum maiorem ab octonario menſum dempſeris, ſupererunt .4. & totidem à ſumma <num value="2460">.<lb/>2460.</num> remanebunt, reiecto maiori numero ab octonario menſo. </s> + <s xml:space="preserve">Atque idem me-<lb/>dio quouis alio numero, euenire poteſt.</s> + </p> + <p> + <s xml:space="preserve">Cuius ratio tam perſe clara atque euidens eſt, quod ſi ſummam trium <reg norm="reliquorum" type="context">reliquorũ</reg>, <lb/>quæ eſt .20. à ſumma .2460. ſubduxeris, remanebunt .2440. pro ſumma trium nume <lb/>rorum dictorum trium ordinum ab octonario menſorum, cui numero addito .16. pro <lb/>maiori numero ſummę <reg norm="reliquorum" type="context">reliquorũ</reg>, qui ab octonario menſus ſit, ſupererunt .4. </s> + <s xml:space="preserve">At ſi per <lb/><reg norm="ſenarium" type="context">ſenariũ</reg> <reg norm="experimentum" type="context context">experimẽtũ</reg> feceris, remanebit <var>.o.</var> & ſic de reliquis per ordinem <reg norm="procedendo" type="context">procedẽdo</reg>.</s> + </p> + <p> + <s xml:space="preserve">Verum poſſes ſciſcitari, quare velocius, exceſſus ordinum, potius per <reg norm="noue- narium" type="context">noue-<lb/>nariũ</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 ſummæ, deinde alias duas. </s> + <s xml:space="preserve">Exem-<lb/>plum ſit primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc ſumma ſit dua <lb/>rum figurarum, ſupputantur & ipſæ, è quibus prodeunt .4. & conſimilis erit proba-<lb/>tio numeri .67. facta per .9. quod idem eſt, ac ſi quis diuidat .67. per .9. ex quo reli-<lb/>qui erunt ſemper .4.</s> + </p> + <p> + <s xml:space="preserve">At quo ratio huiuſce perſpicuè dignoſci poſſit, in primis ſciendum eſt, cuique <lb/>ex ſe cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium ſu <lb/>perare, & ex hoc ſequitur, ſex denarios continere in ſe ſex nouenarios, & ſex vni-<lb/>tates.</s> + </p> + <p> + <s xml:space="preserve">At ſex vnitates, vna cum .7. faciunt .13. & quia in .13. eſt denarius, igitur in illo erit <lb/>vnitas ſupra .9. </s> + <s xml:space="preserve">Quæ vnitas addita ternario, præbet nobis ſuperfluum, per quod .67. <lb/>ſuperat .54. iunctum cum .9. ſcilicet ſummam .63.</s> + </p> + <p> + <s xml:space="preserve">Idem dicinon poteſt de octonario, ſeptenario, vel ſenario, & de reliquis, quo-<lb/>niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-<lb/>merum exceſſus maioris numeri, qui à numero probationis menſus eſt. </s> + <s xml:space="preserve">Et quod di <lb/>co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta ſub-<lb/>tractionis, multiplicationis, & partitionis ſeu diuiſionis.</s> + </p> + <p> + <s xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus ſit probationem euentus <lb/>cum diuiſionis probatione multiplicare, & productum cum fractionis probatione <lb/>ſupputare, ſeu aggregare, tibi non erit ignotum, quoties animaduerteris, quod <lb/>productum ipſius euentus cum diuiſore, adiunctum fractioni, perpetuo ſe æquat nu <lb/>mero diuiſibili. </s> + <s xml:space="preserve">Et quoniam numeri probationum ſunt partes, quæ remanent ex <lb/>ipſis totis, detractis maioribus numeris ab eo dimenſis, quo pro communi men-<lb/>ſura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eſt vt ex ip-<lb/>ſarum remanentibus partibus, velut ex ipſis 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 ſphericorum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod diebus præteris tibi ſignificaui, idem nunc confirmo, ſcilicet ſphærico-<lb/>rum triangulorum operationem ſæpe nos fallere, vt exempli gratia, ſi pro <lb/>poſitus 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/>ſi reliquos angulos cum tertio latere etiam cognoſcere voluerimus, ex methodo .11 <lb/>primi Copernici propoſitum 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> falſus eſſet, eo quod operatio paruorum triangulorum in <lb/>cauſa eſt, quotieſcunque eorum latera tam breuia ſint, ut non eccedant vnum gra-<lb/>dum, quare ipſorum angulorum veram quantitatem non tribuunt. </s> + <s xml:space="preserve">propterea igitur <lb/>cum voluerimus veram <reg norm="quantitatem" type="context">quãtitatem</reg> ipſius anguli <var>.B.</var> oportet poſt quam inuenerimus <lb/>angulum <var>.C.</var> mediante arcu <var>.D.E.</var> ſupponere alium polum in <var>.B.</var> deinde producere. </s> + <s xml:space="preserve">B <lb/>A. vſque ad <var>.d.</var> et <var>.B.C.</var> vſque ad <var>.e.</var> imaginando <var>.B.d.</var> et <var>.B.e.</var> duas quartas eſſe magno-<lb/>rum circulorum, extendendo poſtea <var>.d.e.</var> vſque ad interſectionem cum <var>.A.C.</var> & <reg norm="eum" type="context">eũ</reg> <lb/>dem ordinem proſequendo, </s> + <s xml:space="preserve">tunc <var>.e.d.</var> nobis oſtendet 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, eſto, exempli gratia, quod nobis datus ſit 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 ſi ordi-<lb/>nem .11. dicti lib. ſe quemur, obtinebimus intentum, hoc modo ſcilicet ſupponendo <lb/>in <var>.A.</var> polum, & non in <var>.B.</var> ducendo etiam <var>.A.B.</var> et <var>.A.C.</var> ſed <var>.A.B.</var> <reg norm="vſque" type="simple">vſq;</reg> ad gra .90. du-<lb/>cendo poſtea <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 ſemicirculi magnorum circulorum. </s> + <s xml:space="preserve">quare <var>.C.<lb/>D.</var> nobis cognitus erit gra .44. min .50. & ſic etiam angulus <var>.D.C.f.</var> gra .57. min .14. ex <lb/>4. dicti lib. poſtea habebimus <var>.F.l.</var> gra .60. min .54. & angulum <var>.f.</var> gra .53. mi .24. aggre <lb/>gatum poſtea <var>.f.C.</var> cum <var>.C.B.</var> habebimus <var>.f.B.</var> gra .150. min .24. qui ſi a ſemicirculo <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">cũ</reg> ſit 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 oppoſito. </s> + <s xml:space="preserve">Vnde ex dicta .4. co-<lb/>gnoſcemus angulum <var>.B.</var> gra .40. min .31. <lb/>qui ferè æ qualis eſt ſuperiori iam inuen-<lb/>to, nec ab ipſo differt niſi per min .9. quæ <lb/>quidem differentia parua eſt reſpectu 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 ſupra 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> ſit 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. falſus eſſet, qui <reg norm="quidem" type="context">quidẽ</reg> poſtea nobis daret <var>.D.E.</var> gra .45 <lb/>minu .16. falſum ſimiliter.</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> eſt quin paſſiones circuli innumerabiles penè ſint, quę <lb/>quidem omnes ferè caſu inueniuntur, vt mihi nunc accidit, quam tibi mitto, <lb/>hæc autem eſt, quòd quadratum lineæ <var>.a.g.</var> in figura hic ſubſcripta ſemper æquale <lb/>eſt ei producto, quod fit ex <var>.a.e.</var> in diametro circuli <var>.g.c.b.</var> ſimul ſumpto cum quadra <lb/>to inſcriptibili in dicto circulo, & ſimul cum quadrato lineæ <var>.a.b.</var> <reg norm="contingentis" type="context">contingẽtis</reg> ipſum <lb/>circulum, ſupponendo <var>.a.g.</var> per centrum ipſius circuli tranſire.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius demonſtratione à centro <var>.e.</var> duco ſemidiametrum <var>.e.c.</var> <reg norm="perpendicularem" type="context">perpendicularẽ</reg> <lb/>ipſi <var>.g.a.</var> & à puncto <var>.c.</var> ad <var>.a.</var> duco <var>.c.a.</var> quæ ſecabit circunferentiam ipſius circuli in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.d.</var> eo, quod angulus <var>.c.</var> acutus eſt. </s> + <s xml:space="preserve">Nunc ex .35. tertij, productum <var>.c.a.</var> in <var>.a.d.</var> æqua <lb/>le eſt quadrato <var>.a.b.</var> productum autem <var>.a.c.</var> in <var>.d.c.</var> æquale eſt quadrato inſcriptibili in <lb/>circulo <var>.g.c.b.</var> ex .130. primi Vitellionis, <reg norm="in" type="wordlist">ĩ</reg> qua propoſitione ipſe Vitellio ſupplet pro <lb/>eo, quod in quinta propoſitione libri de lineis ſpirabilibus Archimedis deſideratur, <lb/>ſed quadratum <var>.a.c.</var> æquale eſt ijs duobus productis. per .2. ſecundi Eucli. ergo qua-<lb/>dratum <var>.a.c.</var> æquale erit quadrato inſcriptibili in circulo <var>.d.c.g.</var> & quadrato <var>.a.b.</var> ſed <lb/>quadratum lineæ <var>.a.c.</var> æquale eſt duobus quadratis, hoc eſt 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> & lineę <var>.e.c.</var> hoc eſt <lb/>lineæ <var>.e.g.</var> quod idem eſt, æqualia erunt duobus iam dictis, hoc eſt inſcriptibili, <lb/>& ei, quod fit ex <var>.a.b.</var> ſed quadratum lineæ <var>.a.g.</var> æquale eſt 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 eſt producto <var>.a.e.</var> in <lb/>diametrum. </s> + <s xml:space="preserve">Quare quadratum lineæ <var>.a.g.</var> æquale eſt quadrato circunſcriptibili, & <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 demonſtrare 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 eſ-<lb/>ſe quadrato circunſcriptibili, & quadrato lineæ <var>.<lb/>a.b.</var> ducendo lineam <var>.e.b.</var> quæ æqualis eſt lineæ <var>.<lb/>e.g.</var> tali methodo, hoc eſt, conſiderando, quod <lb/>quadratum inſcriptibile ſemper duplum eſt qua <lb/>drato ſemidiametri, vel medietati circumſcri-<lb/>ptibili, quod quidem nihil aliud eſt, niſi æquale <lb/>eſſe ijs duobus quadratis, hoc eſt lineæ <var>.e.b.</var> & li-<lb/>neæ <var>.e.g.</var> ſed quadratum lineæ <var>.a.e.</var> æquale eſt iis <lb/>duobus quadratis, hoc eſt 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 eſt quadrato circunſcriptibili, ſimul 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 propoſitionum de quibus agit <lb/>Cardanus capite primo libro .16. de <lb/>ſubtilitate.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">EA quæ Cardanus in primo cap. lib. 16. de ſubtilitate ita ſcribit, quod ſi diame-<lb/>tros producatur extra quantumlibet, alia verò diametro in centro ſecetur ad <lb/>rectos, ex huius fine <reg norm="&c." type="unresolved">&c.</reg> quæ quidem ſecundum illum eſt vndecima proprietas cir <lb/>culi, quoniam te id non intelligere ſcribis, <reg norm="idemque" type="simple">idemq́;</reg> dicis etiam de duodecima, & ſi-<lb/>militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, ſeu <lb/>ellipſi, & hyperboli, tibi breuiter reſpondebo.</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, diuiſum <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">ducanturq́;</reg> ab ijſdem punctis tot perpendiculares diametro <var>.d.g.</var> quæ ſint <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> vſque ad concurſum 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 vſque 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 ſint ipſi <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. eiuſdem <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> parallelę, & 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 eſt igitur propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Duodecima vero <reg norm="proprietas" type="simple">ꝓprietas</reg> eſt, ut ſi fuerit circulus <var>.a.b.e.q.</var> cuius duo diametriad <lb/>rectos coniuncti ſint <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 ſi ab extremo <var>.b.</var> diametri <var>.q.b.</var> ducta fuerit <var>.b.n.u.</var> extra circulum, ſeu <var>.b.u.n.</var> in <lb/>tra circulum, vt in ſubiecta figura patet, ita vt ſecta ſit à circunferentia circuli in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.n.</var> vel à diametro in puncto <var>.u.</var> ſemper id quod fit ex <var>.u.b.</var> in <var>.b.n.</var> æquale erit qua-<lb/>drato inſcriptibili in dicto circulo, hoc autem diuerſimodè cognoſci poteſt, tribus <lb/>enim modis ego inueni, quorum primus ita ſe habet. </s> + <s xml:space="preserve">Nam ſi 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> ſimi <lb/>les inuicem, eo, quod angulus <var>.b.</var> communis ambobus exiſtit, & angulus <var>.b.n.e.</var> æqua <lb/>lis eſt 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, ſed ex quinta primi angulus <var>.b.<lb/>e.a.</var> ęqualis eſt 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> ęquatur duobus <lb/>rectis, ſed ex .13. eiuſdem 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. eiuſdem reliquus angulus <var>.b.u.e.</var> æqualis erit reliquo angulo <lb/><var>b.e.n.</var> latera igitur erunt proportionalia ex .4. ſexti, vnde ita ſe habebit <var>.u.b.</var> ad <var>.b.<lb/>e.</var> vt <var>.b.e.</var> ad <var>.b.n.</var> ex .16. ſexti igitur <reg norm="verum" type="context">verũ</reg> erit propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi punctus <var>.u.</var> intra circulum fuerit, triangulus <var>.b.e.n.</var> ſimilis 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> ęqualis eſt 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. ſexti eadem proportio erit ipſius <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. eiuſdem patebit propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Secundus autem modus ita ſe habet, ducta <var>.q.n.</var> habebimus duo triangula ortho-<lb/>gonia ſimilia inuicem <var>.b.q.n.</var> et <var>.b.u.o.</var> eo quod angulus <var>.b.</var> communis ambobus exi-<lb/>ſtit, </s> + <s xml:space="preserve">quare ex .4. ſexti ita ſe habebit <var>.u.b.</var> ad <var>.b.o.</var> vt <var>.q.b.</var> ad <var>.b.n.</var> vnde ex .15. eiuſdem <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. eiuſdem, <reg norm="quod" type="simple">ꝙ</reg> <lb/>fit ex <var>.q.b.</var> in <var>.b.o.</var> ęquatur quadrato <var>.b.e.</var> quia <var>.b.e.</var> media proportionalis eſt inter dia <lb/>metrum & ſemidiametrum eiuſdem circuli. ex .4. eiuſdem, </s> + <s xml:space="preserve">quare quod fit ex <var>.u.b.</var> in <lb/><var>b.n.</var> æquale erit quadrato ipſius <var>.b.e</var>.</s> + </p> + <p> + <s xml:space="preserve">Tertius modus adiungitur, & eſt quod cum quadratum <var>.u.b.</var> exiſtente <var>.u.</var> extra cir-<lb/>culum æquale ſit ei, quod ſit ex <var>.u.b.</var> in <var>.b.n.</var> ſimul ſumpto cum eo, <reg norm="quod" type="simple">ꝙ</reg> fit ex <var>.u.b.</var> in <var>.u.n.</var> <lb/>ex ſecunda ſecundi, & 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. ſcilicet et <var>.o.b.</var> ſed 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 ſe <lb/>ipſam ex .6. ſecundi, </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> ſci <lb/>licet. et <var>.o.e.</var> & ei quod fit ex <var>.a.<lb/>u.</var> in <var>.u.e.</var> ſed quod fit ex <var>b.u.</var> in <var>.u <lb/>n.</var> æquale eſt 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">qđ</reg> fit ex <var>.u.b.</var> in <var>.b.n.</var> æqua-<lb/>le ſit <reg norm="duobus" type="simple">duobꝰ</reg> quadratis <var>.o.b.</var> et <var>.o.<lb/>e</var>. </s> + <s xml:space="preserve">quare & quadrato ipſius <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">pũctũ</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> ſed vice qua-<lb/>drati <var>.o.e.</var> dicemus <reg norm="quadratum" type="context">quadratũ</reg> <var>.o.<lb/>u.</var> cum eo quod fit ex <var>.a.u.</var> in <var>.u.e.</var> <lb/>ex .5. ſecundi, id eſt quadratum <var>.<lb/>o.u.</var> <reg norm="cum" type="context">cũ</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> ideſt quadrato <lb/><var>b.u.</var> ex Pitagorica ſimul <reg norm="cum" type="context">cũ</reg> pro-<lb/>ducto <var>.b.u.</var> in <var>.u.n.</var> ideſt producto <lb/><var>n.b.</var> in <var>.b.u.</var> quod æquale eſt qua <lb/>drat <var>o.b.u.</var> cum producto <var>.b.u.</var> in <lb/><var>u.n.</var> ex .3. ſecundi.</s> + </p> + <p> + <s xml:space="preserve">Circa tres paſſiones commu-<lb/>nes poſtea circulo hyperboli, & <lb/>defectioni notandum eſt <reg norm="primam" type="context">primã</reg> <lb/>patere ex .36: primi Pergei, ſe- + <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. eiuſdem, </s> + <s xml:space="preserve">propterea quod in .37. probat mediante maiori <lb/>diametro ipſius hyperbolis & defectionis, In .38. autem mediante minori diametro <lb/>ordinatè ad maiorem.</s> + </p> + <p> + <s xml:space="preserve">Tertia autem paſſio, non niſi circulo conuenit; </s> + <s xml:space="preserve">pace ipſius Cardani dictum ſit.</s> + </p> + <p> + <s xml:space="preserve">Quapropter ſit circulus <var>.q.o.b.</var> cuius diameter ſit <var>.q.b.</var> contingentes vero ab extre <lb/>mitate diametri ſint <var>.d.b.</var> et <var>.q.g.</var> per punctum autem <var>.o.</var> quoduis, ipſius <reg norm="circunferentiæ" type="context">circũferentiæ</reg>, <lb/>tranſeant <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> ęquale eſ-<lb/>ſe quadrato <var>.q.b.</var> quod ita probo.</s> + </p> + <p> + <s xml:space="preserve">Nam angulus <var>.q.b.d.</var> ſeu <var>.b.q.g.</var> rectus eſt ex .17. tertij Eucli. et <var>.b.o.q.</var> ſimiliter re-<lb/>ctus ex .30. ipſius lib. angulus verò <var>.b.q.d.</var> ſeu <var>.q.b.g.</var> communis eſt. </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 ſequetur <lb/>propoſitum ex .16.6. Eucli.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi circa diametrum <var>.q.b.</var> mente fingamus aliquam elipſim, quætangat ipſum <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">pũ</reg> <lb/>ctus <var>.o.</var> erit extra <reg norm="circunferentiam" type="context">circunferentiã</reg> <lb/>ipſius defectionis, </s> + <s xml:space="preserve">quare ipſa cir <lb/>cunferentia ſecabit <var>.b.g.</var> vel <var>.q.<lb/>d.</var> in alio puncto, vnde ipſi 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 ſum, ipſum <lb/>Cardanum dicere hyperbolem <lb/>ita vocari, eo quod angulus con <lb/>tentus ab axe ipſius figuræ, & à <lb/>latere trigoni in hyperbole ma-<lb/>ior ſit quam in parabole, quod <lb/>eriam confirmat paulo inferius, <lb/>nam hoc verum non eſt, imo fal <lb/>ſiſſimum. </s> + <s xml:space="preserve">Talis enim ſectio ita <lb/>nominata fuit, hoc eſt hyperbo <lb/>les, ſimili ratione, qua elipſis ſeu <lb/>defectio etiam vocata fuit, nam <lb/>ſicut in ipſa defectione quadra-<lb/>tum ordinatę <var>.l.m.</var> minor eſt pro <lb/>ducto lineæ <var>.e.m.</var> in <var>.e.t.</var> per figu <lb/>ram ſimilcm producto <var>.d.e.</var> in <var>.e.<lb/>t.</var> quæ eandem obtineat <reg norm="altitu- dinem" type="context">altitu-<lb/>dinẽ</reg> ipſius <var>.e.m.</var> vt ipſe Pergeus <lb/>monſtrat in .13. primi lib. ita in <lb/>hyperbole <reg norm="dictum" type="context">dictũ</reg> quadratum ex <lb/>cedit quantitatem illius figuræ, <lb/>per ſimilem dictæ vt in .12. <reg norm="ipſius" type="simple">ipſiꝰ</reg> <lb/>Pergei facilè videre eſt. </s> + <s xml:space="preserve">ſed <reg norm="prae ter" type="simple">prę<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 ſupradicto capite, multæ aliæ ſunt, cum corollarium primæ tertij Eu-<lb/>cli. ſit paſſio propria ipſius circuli, & idem dico de propoſitione .3. 4. 7. 8. 9. 11. 12. <lb/>13. 14. 15. 17. 18. 19. 20. 30. 31. ipſius tertij lib. nec non de .8. 9. ct .10. tertijdecimi, & <lb/>de prima .3. 4. 5. 6. et .7. quartidecimi eiuſdem. </s> + <s xml:space="preserve">Idem infero de ea quod ſcrip ſi Ma <lb/>rio Nizzolio, Franciſco Vimercato, Franciſco Contareno, Angelo Agrimenſori, & <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">Illuſtri vire, Philiberto Pingonio Sabaudo Cuſiacenſium <lb/>Baroni.</head> + <p> + <s xml:space="preserve">CVm antea meo nomine Sebaſtianus noſter omnia ferè tibi retuliſſet, inter alia, <lb/>quæ relin quebantur tibi <reg norm="dicenda" type="context">dicẽda</reg>, hoc vnum erat, quod ſi abſque lumine ſupe-<lb/>riori, in quem finem facta fuerint corpora cœleſtia ſcire deſideras, & humanam ra-<lb/>tionem ſequi volueris, putandum tibi non erit ea ſolum effecta eſſe, vt tam vile cor <lb/>pus, vt eſt terra aquis irrigata, animalia, & plantas regant, cum ea corpora ſint diuina, <lb/>in numero incompręhenſibilia, maximis ma gnitudinibus, & motibus velocisfimis, <lb/>prædita, id etiam minus putabunt hij, qui opinionem Ariſtarchi Samij, & Nicolai <lb/>Copernici ſequuntur, quorum ratione fieri non poteſt, vt credant, eius, quod ex vni <lb/>uerſo reliquum eſt, 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 ſi centra aliorum epicyclorum <lb/>planetarum tali regimine priuarentur, id quod nullo modo cum ratione conſentit, <lb/>ſi tam vera eſt ea opinio, quemadmodum <reg norm="rationabiliorem" type="context">rationabiliorẽ</reg> eam <reg norm="exiſtimant" type="context">exiſtimãt</reg>. </s> + <s xml:space="preserve"><reg norm="Neque" type="simple">Neq;</reg> quid <lb/>quam valet opinio Ariſtotelis, qui corpora cœleſtia, ab ortu, & interitu libera eſſe <lb/>fentit. </s> + <s xml:space="preserve">dicens ſuperioribus fęculis, à noſtris antiquis nullam vnquam animaduerſam <lb/>fuiſſe alterationem in cœlo, cum non videat ſi quis eſſet in cœlo, <reg norm="neque" type="simple">neq;</reg> etiam obſerua <lb/>re poſſet alterationes quæ in terra, & circa terram fiunt, quæ in partibus, & non in to <lb/>to ſpectantur: </s> + <s xml:space="preserve">vnde etiam fieri poteſt, vt in cœlo ſint particulares alterationes, <lb/>quæ à nobis tamen, qui ab illis longè diſtamus, non compręhendantur, terra, <reg norm="mareque" type="simple">mareq́;</reg> <lb/>(quamuis minimum reſpectu ipſius terræ) ratione totius ita ſe ſemper <reg norm="habuerunt" type="context">habuerũt</reg> <reg norm="quem" type="context">quẽ</reg> <lb/>admodum ſeſe habere corpora cœleſtia videmus, ſed alteratio, ratione tantum ali-<lb/>quarum minimarum partium quaſi inſenſibilium, ſi cum toto comparentur fit. </s> + <s xml:space="preserve">Quis <lb/>enim ſcit, vt iam tibi dixi, quin, quemadmodum Luna circa terram voluitur, <reg norm="ipſaque" type="simple">ipſaq́;</reg> <lb/>terra ſit veluti centrum epicycli maioris eiuſdem, vt Ariſtarchus Samius, & Nico-<lb/>laus Copernicus cenſuerunt; </s> + <s xml:space="preserve">ſic etiam Saturnus, Iupiter, Mars, Venus, <reg norm="atque" type="simple">atq;</reg> Mercu <lb/>rius circa alia huiuſmodi corpora, huic terræ ſimilia, in orbem agantur, quaſi ſpecu-<lb/>la, lumen Solis ſuo centro ex reflexione, deferentia (fuppoſita dico vera illorum opi <lb/>nione) <reg norm="Nollem" type="context">Nollẽ</reg> tamen tibi è mente excidere, vt aliàs te monui, <reg norm="quod" type="simple">ꝙ</reg> ſi communis opinio <lb/>vera eſt, neceſſario <reg norm="fatendum" type="context">fatendũ</reg> ſit corpus ſolare, <reg norm="dum" type="context">dũ</reg> in æquatore reperitur moru diurno <lb/>quolibet horę minuto, magis <reg norm="quam" type="context">quã</reg> <reg norm="decem" type="context">decẽ</reg> & <reg norm="ſeptem" type="context">ſeptẽ</reg> mille milliaria <reg norm="peragere" type="simple">ꝑagere</reg>, ideſt paulo mi <lb/>nus quam .18000. milliaria, Saturnum verò cum ſimiliter eſt in æquatore, eodem <reg norm="tem- poris" type="context">tẽ-<lb/>poris</reg> ſpatio, quaſi tercentamille milliaria Italica conficere, & ſic per gradus alia cor <lb/>pora velociora alijs moueri; </s> + <s xml:space="preserve">quæ quidem omnia, <reg norm="cum" type="context">cũ</reg> ſimplici gyro terræ circa ſuum + <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 ſufficit ad recipiendum lumen, & influentias illo-<lb/>rum corporum. </s> + <s xml:space="preserve">Et ita, veluti princeps corporum vniuerſi, intra vnum an-<lb/>num circa eam vertitur. </s> + <s xml:space="preserve">Ita etiam ſuſſiceret, vt ipſa terra circa dictum diuinum <lb/>corpus ſolare, interſecando axem diurnum cum axe annuali (cum ab eo lumen, ca-<lb/>lorem, & influentiam ſuſcipere debeat) <reg norm="circunuolueretur" type="simple">circunuolueret̃</reg>. </s> + <s xml:space="preserve">Rationes autem a Ptolo-<lb/>meo in contrarium adductæ apud ipſos, nullę ſunt, quia quęlibet pars (vt inquiunt) <lb/>retinet naturam totius, præterquam <reg norm="quod" type="simple">ꝙ</reg> aer, & aqua, quæ ipſam terram <reg norm="circundant" type="context">circundãt</reg>, pla <lb/>nè eundem naturalem impetum motus obtineant, quitanto lentior eſt, quanto lon <lb/>gius diſtat aer ab ipſa terra, ſecundum etiam talem opinionem, nulla neceſſitas fo-<lb/>ret, vt locus fixarum terminaretur aliquibus ſuperficiebus, conuexa ſcilicet, & 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 ſit lux aliqua, quæ à corpore lucido non proue-<lb/>niat, mihi facilè ad conſiderandum videtur. </s> + <s xml:space="preserve">hic enim oportet, vt nos ad id <lb/>quod perpetuò videmus referamus, exiſtimo autem te velle dicere lumen, non lu-<lb/>cem, quia propriè lux, qualitas ea viſibilis appellatur, quæ eſt 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 ipſum corpus reperitur, à luce, quæ in dicto corpore manet emanans. </s> + <s xml:space="preserve">vnde pa-<lb/>tet, nullam lucem abſque corpore ſubiecto eſſe poſſe, id quod cum fieri quîret, <reg norm="idem" type="context">idẽ</reg> <lb/>de quolibet alio accidente dici poſſet, id eſt quod ex ſe, & abſque aliquo ſubiecto <lb/>ſubſiſteret.</s> + </p> + <p> + <s xml:space="preserve">Lumen deinde à luce proficiſci patet, <reg norm="quod" type="simple">ꝙ</reg> penetrat diaphanum, neque aliquo mo-<lb/>do ſuum actum oſtendit, niſi, aut per incidentiam, aut ratione opaci, ex reflexione, <lb/>cuius ſuperficiei colorem induit. </s> + <s xml:space="preserve">Atque hæc eſt cauſa, vt inter crepuſculum matu-<lb/>tinum, aut veſpertinum, nox etiam ſi ſit ſerena, adeo obſcura nobis appareat, quam-<lb/>uis totum vniuerſum diaphanum, extra conum vmbræ, quæ ex terra prouenit ſit vn <lb/><reg norm="dique" type="simple">diq;</reg> radijs luminoſis Solis colluſtratum; </s> + <s xml:space="preserve">qui quidem radij, non niſi à ſuamet reflexio <lb/>ne à Luna, & ab alijs ſtellis (vt corporibus opacis, quæ reſiſtunt lumini, ne vlterius <lb/>penetrare poſſit, vnde retrò redit) comprehenduntur.</s> + </p> + <p> + <s xml:space="preserve">Ais etiam propria viſus obiecta plura eſſe, nominans pro vno, colorem, & lucem <lb/>pro alio. </s> + <s xml:space="preserve">Ego autem reſpondeo, obiectum oculi eſſe vnicum tantum, ideſt lumen. <lb/></s> + <s xml:space="preserve">Quod ad lucem ſpectat, iam tibi dixi, eam eſſe quandam qualitatem in corpore luci <lb/>do, & non extra ipſum poſitam, à quo quidem corpore, cum non exeat, oculi obie-<lb/>ctum eſſe nequit, ſed lumen quidem ab ipſa luce productum. </s> + <s xml:space="preserve">Color etiam, qui eſt in <lb/>corpore colorato, obiectum oculi eſſe non poteſt, cum dictum corpus non deſerat, <lb/>ſed lumen quidem ab eodem corpore reflexum, & huiuſmodi corporis colore tin-<lb/>ctum: </s> + <s xml:space="preserve">vnde tam lumen incidens, quam reflexum colore eſt ſemper imbutum.</s> + </p> + <p> + <s xml:space="preserve">Illud quidem coloratum eſt qualitate lucis corporis lucidi, a ut medij, per quod <lb/>tranſit, ſed 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">priisq́;</reg> ſuperficiebus corporum coloratorum, eo <reg norm="quod" type="wordlist">qđ</reg> pauca corpora tam opaca repe-<lb/>riuntur, ut immediatè lumen à ſuperficie propriè <reg norm="reflectant" type="context">reflectãt</reg>, ſed 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 propoſitum redeamus, dico lum en tantum eſſe viſus obiectum, quod ſi <lb/>colore eſt imbutum, aut tale eſt ratione color ris lucis, quæ eum mittit, aut ratione me <lb/>dij per quem tranſit, aut ratione corporis, vnde reflectitur, etſi ſuperficies corporis <lb/>vnde lumen reflectitur eſſet omnino priuata colore, ſub aſpectum non caderet, vt <lb/>etiam cum huiuſmodi ſuperficies læuigata, & polita eſt ſecundum <reg norm="continuitatem" type="context">continuitatẽ</reg> ſua-<lb/>rum partium, videlicet, vt ſpeculi radio tamen non profundante, & ideo perfectiffi <lb/>morum quorundam ſpeculorum ſuperficies non cernuntur, ſed lumen tantum re-<lb/>flexum, colore alicuius alterius ſuperficiei, aut à luce, corporis lucidi, aut à me-<lb/>dio per quem tranſit, conſpicitur. </s> + <s xml:space="preserve">Ego verò non aſſero colorem non eſſe quid di-<lb/>uerſum à lumine, ſed imagineris lumen eſſe veluti animam, aut ſubſtantiam & colo <lb/>rem corporis formam accidentalem, cum nullum lumen à ſenſu viſus percipi poſ-<lb/>ſit, quod aliquo modo colore non ſit imbutum: </s> + <s xml:space="preserve">& eundem reſpectum quem ſonus <lb/>ad auditum, lumen ad oculum habet, quia vt ſonus ſecundum eam velocitatem, quæ <lb/>à motione aeris, aut aquæ, ex colliſione <reg norm="aliorum" type="context">aliorũ</reg> corporum producitur ad euitan dum <lb/>vacuum, a cutus, vel grauis ſentitur, ita lumen originem ducens à corpore lucido per <lb/>medium diaphanum aeris, aut aquæ, aut alterius huiuſmodi corporis ad oculum tran <lb/>ſit colorem lucis, aut medij per quod tranſit, aut vnde reflectitur induit.</s> + </p> + <p> + <s xml:space="preserve">Quod verò Luna nullum ex ſe habeat lumen, ſufficiens inditium eſt nos ipſam <lb/>tantò magis obſcuram videre, quantò magis in cono vmbræ terræ immergitur, & <lb/>ſi eo tempore ipſam videmus rubeo colore affectam, hoc enim accidit, quia radij ſo <lb/>lares vndequaque refranguntur à vaporibus ipſam terram circundantibus, quæ qui-<lb/>dem refractio fit verſus axem coni vmbræ terræ, </s> + <s xml:space="preserve">& propterea vmbra dicti coni non <lb/>eſt æqualiter obſcura, ſeu tenebroſa, circa vero <reg norm="axem" type="context">axẽ</reg> ipſius coni, magis quam circa eius <lb/><reg norm="circunferentiam" type="context context">circũferentiã</reg>, obſcura <reg norm="verum" type="wordlist">vr̃</reg>, & quia corpus lunare tale eſt, vt facillimè recipiat <reg norm="qualecum" type="context">qualecũ</reg> <lb/>que lumen, quod etiam manifeſtè videtur dum ipſa Luna reperitur ſecundum lon-<lb/>gitudinem inter Solem, & Venerem, quod pars Lunæ lumine Solis deſtituta, à lumi <lb/>ne Veneris aliquantulum illuſtratur, quod ego ſæpè vidi, & multis oſtendi. </s> + <s xml:space="preserve">Propte-<lb/>rea dum ipſa Luna in cono vmbræ terræ reperitur adhuc videtur. </s> + <s xml:space="preserve">Rubedo etiam il-<lb/>la nubium poſt Solis occaſum, vel ante ortum, aliunde non prouenit, niſi à qualitate <lb/><reg norm="vaporum" type="context">vaporũ</reg>, per quos ſolares radij tranſeunt, à quibus vaporibus, tali colore ipſi radij <lb/>afficiuntur, eomet modo quo radius, cuiuſuis corporis lucidit, <reg norm="tranſiens" type="context">trãſiens</reg> per vitrum, ſeu <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/>diuerſas eleuationes. Et de quibuſdam erroribus Nico-<lb/>lai Tartaleæ, circa idem.</head> + <head style="it" xml:space="preserve">Fllustri D. Ioſepho Cambiano ex Ruffia Dominis, aquiti <lb/>ſtrenuo, & tormentis bellicis Serenißimi Ducis <lb/>Sabaudia Prafecto.</head> + <p> + <s xml:space="preserve">EXcogitaui quędam dum ocio frui licuit per abſentiam Ducis Sereniſſimi, <lb/>quæ ad te ſcribere placuit, vt ſi probaueris in lucem quandoque profer-<lb/>re non dubitem, ſi deſpexeris, ocius ſupprimam, ſunt autem huiuſmodi.</s> + </p> + <p> + <s xml:space="preserve">Vnde fiat vt tormentum bellicum vehementi<unsure/> feriat ictu ſuperius delato <lb/>quam orizontali, vt Tartalea ſcribit, quæſito ſecundo libr. primi quæſitorum, à ne-<lb/>mine adhuc (quod ſciam) traditum eſt.</s> + </p> + <p> + <s xml:space="preserve">Rationes verò Tartaleæ nullius ſunt momenti, quia ſi validæ eſſent, ſequeretur <lb/>vt inclinata bombarda, adeo vt angulus ſub orizonte factus æqualis eſſet ei, qui ſu <lb/>pra orizontem eſt, ictum bombardę in vtroque huiuſmodi ſitu eundem eſſe <reg norm="futurum" type="context">futurũ</reg>. <lb/></s> + <s xml:space="preserve">& ſi aliqua differentia oriretur ratione granitatis pilæ ab ipſa bombarda emiſſæ, hoc <lb/>fieret, vt ſcilicet 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 cauſa vnde fiat, vt bom <lb/>barda eleuata vehementius feriat, quàm ea quæ eſt minus alta, eadem eſt ferè, in ge-<lb/>nere, cum ea, qua aliquod corpus materia magis denſa, ſed ſimile & ęquale alteri cor <lb/>pori materiæ minus denſæ velocius mouetur ab vna eademq́ue, aut æquali vi <lb/>compulſum. </s> + <s xml:space="preserve">Eſt eadem etiam in ſpecie ei, qua maiorem effectum producit <lb/>puluis, qui in locis ſubterraneis ponitur quum vaſis optimè colligatis ferro in-<lb/>cluditur. </s> + <s xml:space="preserve">Eſt etiam ſimilis ei, qua longius impellitur pila, qua ludimus, ab ali-<lb/>quo inſtrumento ligneo, quando percutitur contra, quam cum ſecundum ſuum mo-<lb/>tum proijcitur. </s> + <s xml:space="preserve">Id quod inde fit, quia virtus mouens maiori vi, & intenſiori huiuſ-<lb/>modi corpus percutit, quia corpus quod moueri debet, quanto magis reſiſtit virtu-<lb/>ti mouenti (certum tamen terminum præſcribendo) in exiguo eo temporis ſpatio, <lb/>tanto maiorem virtutem colligit, quæ ipſum deinde tanto cum impetu mo-<lb/>uet, & tanto magis impellens concomitatur, vt maiorem effectum efficiat, quam ſi <lb/>ad mouendum ſeſe facilè reddidiſſet. </s> + <s xml:space="preserve">Atque hoc ſupradictis ictibus eleuatis acci-<lb/>dit, quia grauitas pilæ, ea eſt quæ reſiſtens 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">ĩmpelleret</reg>. </s> + <s xml:space="preserve">Et quia huiuſmodi 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 ſufficeret, 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, ſed vt dixi, tanta cum veloci-<lb/>tate <reg norm="adaugetur" type="simple">adauget̃</reg>, vt huiuſmodi virtus longè ſuperet <reg norm="reſiſtentiam" type="context context">reſiſtẽtiã</reg> <reg norm="ponderis" type="context">põderis</reg> pilæ, & ſic eſt cau <lb/>fa, ut effectus, quod <reg norm="experientia" type="context">experiẽtia</reg> innot eſcit <reg norm="producatur" type="simple">producat̃</reg>. </s> + <s xml:space="preserve">Sed ea ratio, qua ſeſe <reg norm="idem" type="context">idẽ</reg> author <lb/>in tertio quæſito ad aliquod impoſſibile, circa iter ipſius pilæ Legatum Hiſpanum + <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 eſt ſemper dicendum, quod <reg norm="quam" type="context">quã</reg> <lb/>to velocior ſit quædam pila, tanto rectius moueatur, quia ei dici poſſet, vſque ad cer <lb/>tum quendam terminum velocitatis, per tantum ſpatij eam aptam eſſe, vt recta per <lb/>fectè moueatur, ſed ſi velocius iret, non tamen futurum, vt per idem ſpatium re-<lb/>ctius moueretur, ſed quod per longius ſpatium recta motum perageret, & ſic nihil <lb/>haberet quod replicaret, præter quam quod ipſe ſupponit id quod in 18. quæſito <lb/>negat, in quo ait pilam uicinam orificio, non adeo uelocem eſſe, quam cum aliquan-<lb/>tulum ab eodem eſt rem ota, ratione reſiſtentiæ ſui 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 diſtat à recto huiuſmodi <reg norm="angulus" type="simple">angulꝰ</reg>, ſiue ſit acutus ſiue obtuſus, tanto minorem uim <lb/>habet, eodem planè ferè modo quem tertio capite mei tractatus de rebus mechani <lb/>cis deſcripſi. </s> + <s xml:space="preserve">Quia in ictibus eleuatis, iter inclinationis violentæ ipſius pilæ verſus <lb/>terminum ad quem, incipiendo à loco ipſius pilæ cum itinere inclinationis natura-<lb/>lis, angulum obtuſum, & in ictibus inclinatis acutum conſtituit. </s> + <s xml:space="preserve">Neque etiam hic <lb/>prætermittam notatu dignum errorem, quem Tartalea eodem loco committit, <reg norm="cum" type="context">cũ</reg> <lb/>putet indifferenter aliquod corpus impellere, aut percutere maiori <reg norm="cum" type="context">cũ</reg> impetu cum <lb/>eſt in itinere recto. </s> + <s xml:space="preserve">Quia ſequeretur quod aliquod corpus graue perpendiculari-<lb/>ter ſurſum verſus proiectum, in qualibet parte ſui itineris, ſemper fortius percute-<lb/>ret, quam in qualibet parte itineris alterius cuiuſuis eleuationis obliquæ, quod <reg norm="quam" type="context">quã</reg> <lb/>ſit falſum, tibi conſiderandum relinquo.</s> + </p> + <p> + <s xml:space="preserve">Eſt etiam falſa ea ratio, quam in quarto quæſito idem adducit, quia aer in motu <lb/>non tantum durat, quantum ipſe putat, imò huiuſmodi violenta agitatio, citò ceſſat <lb/>& citius etiam, quam ſi extra aliquam bombardam cum tanta violentia impulliſſet <lb/>ſaceum plumis plenum.</s> + </p> + <p> + <s xml:space="preserve">Ratio etiam quam in .18. quæſito de eo, quod pila pertran ſeat illud corpus cyllin <lb/>dricum aereum adducit, eſt planè vana, quia ſtatim aer, qui prius in <reg norm="bombarda" type="context">bõbarda</reg> erat <lb/>incluſus, extra ipſam <reg norm="erumpit" type="context">erũpit</reg>, cedit, à <reg norm="pilaque" type="simple">pilaq́;</reg> diuiditur, vt ſi nunquam eam figuram in-<lb/>duiſſet, neque aer ambiens ei reſiſtit. </s> + <s xml:space="preserve">Sed quod velocior ſit in certa quadam diſtan <lb/>tia, quam in principio erat, ſi hoc <reg norm="verum" type="context">verũ</reg> eſſet, ab alia cauſa dependeret, quæ partim ſi <lb/>milis eſſet ei, quæ efficit, vt corpora in motibus naturalibus, cum longius diſtant à ter <lb/>mino vnde naturaliter ſeſe mouerunt, ſint velociora, quia per aliquod ſpatium hu-<lb/>iuſmodi corpus moueretur quemadmodum motu naturali cietur.</s> + </p> + <p> + <s xml:space="preserve">Ratio autem eius quare pila, aut globus bombardæ ſibiletab eodem in ſeptimo <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 ſignum, ictus altius tenderet, quod <lb/>poteſt etiam eſſe falſum, cum hocnon ſit neceſſarium, quia pila dum deſcendit, for-<lb/>taſſe tangeret ſcopum.</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. ſuarum tabularum Prute-<lb/>nicarum, à Nicolao Copernico ſumpſit pag .64. à tergo in libr. reuolutionum <lb/>cœleſtium, ſed ipſe Stadius eam non intellexit, omitto, quod mutauerit characte-<lb/>res ipſius figurę, vt illa ſua videatur, quod nihil refert, alterat etiam <reg norm="demonſtrationem" type="context">demonſtrationẽ</reg>, <lb/>ſed ipſum putare <var>.i.K.</var> perpendicularem à centro circuli ſemper dependere, eſt intol <lb/>lerabilis error; </s> + <s xml:space="preserve">nec vnquam verificatur hoc, niſi quando punctum <var>.K.</var> interſectionis <lb/>diametrorum parallelorum, forte reperitur in axe mundi. </s> + <s xml:space="preserve">Reliqua verò ſuæ demon <lb/>ſtrationis, ſi non intelligis, minimè miror, eo quod ipſemet Stadius ſeipſum confun <lb/>dit. </s> + <s xml:space="preserve">Veram autem demonſtratio nem huiuſmodi 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 ſit maxima ecclipticæ declinatio, vt puta <var>.a.c.</var> ſi latitudo <reg norm="etiam" type="context">etiã</reg> <lb/>ſtellæ nobis data fuerit, vt puta <var>.c.e.</var> cognitus nobis erit totalis arcus <var>.a.e.</var> & eius ſinus <var>.<lb/>e.m.</var> & quia notus etiam nobis eſt ſinus arcus <var>.a.c.</var> hoc eſt <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 ſinus arcus <var>.e.p.</var> cognitus etiam nobis eſt ſinus <var>.<lb/>q.g.</var> declinationis <var>.a.g.</var> datæ, cui æqualis eſt <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> ſint, 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 ſcientia anguli <var>.c.</var> et <var>.e.</var> ſunt æquales, cum ex .29. dicti lib. <lb/><reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> æqualis ſit 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">quorũ</reg> <lb/>vnuſquiſque æqualis eſt angulo <var>.ω.x.t.</var> & ſic de alijs dico, co quod vnuſquiſque <reg norm="eorum" type="context">eorũ</reg> <lb/>æqualis eſt angulo <var>.m.</var> vnde cum cognitum nobis ſit latus <var>.n.c.</var> et <var>.c.i.</var> et <var>.t.e.</var> <reg norm="notum" type="context">notũ</reg> etiam <lb/>nobis erit <var>.e.K.</var> ex .19. ſeptimi, eo <reg norm="quod" type="simple">ꝙ</reg> <lb/>ex .4. ſexti ſunt inuicem proportio-<lb/> + <anchor type="figure" xlink:label="fig-0272-01a" xlink:href="fig-0272-01"/> + nalia, detrahendo poſtea <var>.e.K.</var> ab <var>.<lb/>e.l.</var> cognito, vel ècontra, hoc ab il-<lb/>lo, nobis innoteſcet <var>.K.l.</var> ſinus longi <lb/>tudinis ſtellæ.</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. ſcri-<lb/>bit, hoc eſt, Solem maiorem <lb/>eſſe Luna, ſolum .1644. vici-<lb/>bus, </s> + <s xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</reg> cum affirmet So-<lb/>lem maiorem eſſe terra (vt etiam <lb/>in Almageſto videre eſt) 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 cognoſcendam latitudinem ſtellæ, <reg norm="eiusque" type="simple">eiusq́</reg> declinationem, Monteregius in <lb/>10. propoſitione .8. li. </s> + <s xml:space="preserve">Almageſti <reg norm="methodum" type="context">methodũ</reg> ſatis docuit, ſed ſi alia aliqua metho <lb/>do hoc idem cognoſcere voluerimus, oportebit nos prius altitudinem poli cogno-<lb/>ſcere, deinde altitudinem meridianam ipſius ſtellæ, nec non horam, <reg norm="quando" type="context">quãdo</reg> ipſa ſtel <lb/>la in meridiano ſupra terram reperitur, qua hora mediante, illicò cognoſcemus pun <lb/>ctum ecclipticæà meridiano interſecto, eo tempore, quo ſtella cœlum mediat ſu-<lb/>pra terram. </s> + <s xml:space="preserve">Et quia ex cognita altitudine poli, illico cognoſcitur altitudo æqua-<lb/>toris, cuius altitudinis differentia ab altitudine ſtellæ eſt declinatio ipſius ſtellæ, ha-<lb/>bebimus ideo eius declinationem cognitam; </s> + <s xml:space="preserve">qua mediante ad <reg norm="cognoſcendum" type="context">cognoſcendũ</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 aſtri <var>.u.o.</var> verò eius declinatio ab æquatore, et <var>.e.a.</var> arcus ęcclipticæ inter <lb/>æquatorem, & meridianum, hoc eſt minor quarta, et <var>.a.u.</var> aſcenſio recta ipſius arcus, <lb/>et <var>.u.e.</var> ſit declinatio puncti <var>.e.</var> ęcclipticę ab æquatore, <reg norm="reſiduum" type="context">reſiduũ</reg> vero declinationis ſtel-<lb/>lę ſit <var>.o.e.</var> quæ <reg norm="omnia" type="wordlist">oĩa</reg> nobis cognita erunt, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.t.</var> polus ęcclipticus, à quo per <var>.o.</var> vſque ad <lb/>ęcclipticam tranſeat quarta <var>.t.i.</var> in qua quęrendus 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 ſunt, cum angulo <var>.a.</var> declinatio <lb/>nis ęclipticę, & cum angulo <var>.u.</var> recto, vnde ex .4. primi Copernici, cognoſcemus 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> cognoſcemus <reg norm="angulum" type="context">angulũ</reg> <lb/>e. et deinde <var>.i.</var> vt <reg norm="rectum" type="context">rectũ</reg>, & latus <var>.o.e.</var> ergò ex <reg norm="eadem" type="context">eadẽ</reg> .4. cognoſcemus <reg norm="arcum" type="context">arcũ</reg> <var>.o.i.</var> quæſitum, <lb/>& ſimiliter arcum <var>.e.i.</var> qui coniunctus vel <reg norm="demptus" type="context">dẽptus</reg> ab <var>.a.e.</var> tribuet nobis longitudinem <lb/>ſtellę, ſed quia huiuſmodi operatio in paruis triangulis valde fallit. </s> + <s xml:space="preserve">Ideo tibi ſua-<lb/>deo alia methodo, hoc facere, hoc eſt 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 ſunt, cum angulo <var>.p</var>. </s> + <s xml:space="preserve">Nam <var>.o.p.</var> eſt complementum de <lb/>clinationis ſtellæ, et <var>.p.t.</var> eſt arcus coluri ſolſtitiorum inter duos polos, & angulus <var>.p.</var> <lb/>reſiduum ex recto <var>.t.p.a.</var> duorum colurum dempto angulo. a, <var>p.u.</var> cognito aſcenſionis <lb/>recte, vnde angulus <var>.u.o.s.</var> vt contrapoſitus cognitus remanet. </s> + <s xml:space="preserve">angulus verò <var>.u.</var> rectus <lb/>eſt, & 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">cũ</reg> an-<lb/>gulo <var>.a.s.i.</var> reſiduo ex duobus rectis. </s> + <s xml:space="preserve">Et quia <lb/>etiam angulus <var>.s.a.i.</var> cognitus eſt, cum ſit an <lb/>gulus maximę 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">cũ</reg> arcu <var>.a.i.</var> ſed <lb/><var>a.i.</var> erit longitudinis ſtellæ dempto poſtea <var>.<lb/>s.i.</var> ex <var>.s.o.</var> iam inuento habebimus arcum <var>.i.<lb/>o.</var> latitudinis ipſius ſtellæ.</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 ſcribo non vt ipſis vta-<lb/>ris, ſed potius vt tibi morem <reg norm="geram" type="context">gerã</reg>, cum bre <lb/>uiſſima methodus ſit illa, <reg norm="quam" type="context">quã</reg> Monteregius <lb/>ſcripſit <reg norm="in" type="wordlist">ĩ</reg> .10. <reg norm="propoſitione" type="simple">ꝓpoſitione</reg> .8. li. in Almageſt.</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 deſignari poßit alios duos circulos <lb/>propoſitos includens.</head> + <head xml:space="preserve">CLARISS. PETRO PIZZAMANO.</head> + <p> + <s xml:space="preserve">SVperioribus diebus per tuas literas à me quæſiuiſti, vt modum tibi ſcribere vel-<lb/>lem, quo circulus deſignari poſſit circunſcribens alios duos propoſitos circulos. <lb/></s> + <s xml:space="preserve">Qua in re vt tibi ſatisfaciam quod maximè cupio ita rem accipe.</s> + </p> + <p> + <s xml:space="preserve">Propoſiti circuli ſint, aut inter ſe contigui, aut interſecantes vel ſeparati. </s> + <s xml:space="preserve">Eſto <reg norm="pri- mum" type="context">pri-<lb/>mũ</reg> contiguos eſſe, qui ſint <var>.d.b.</var> et <var>.f.q.</var> <reg norm="quorum" type="context">quorũ</reg> <var>.d.b.</var> maior ſit et <var>.f.q.</var> minor, <reg norm="eorum" type="context">eorũ</reg> vero <lb/>centra ſint .a et <var>.o.</var> <reg norm="punctum" type="context">punctũ</reg> autem <reg norm="contingentię" type="context">cõtingentię</reg> ſit <var>.i</var>. </s> + <s xml:space="preserve"><reg norm="Nunc" type="context">Nũc</reg> <reg norm="protrahatur" type="simple simple">ꝓtrahat̃</reg>. <var>b.a.o.q.</var> per <reg norm="centra" type="context">cẽtra</reg> eo <lb/>rum ab vna circunferentia ad aliam, quę quidem linea tranſibit per punctum <var>.i.</var> ex <lb/>11, tertij Eucli. </s> + <s xml:space="preserve">deinde à diametro maiori abſcindatur <var>.i.e.</var> ad æqualitatem minoris <lb/>ſemidiametri, quo facto ſumatur diſtantia inter <var>.e.</var> et <var>.b.</var> circino mediante <reg norm="factoque" type="simple">factoq́</reg> cen <lb/>tro <var>.o.</var> ſcindatur, alio circini pede, circunferentia maioris circuli in puncto <var>.u.</var> à quo ſi <lb/>mente concipiemus duas lineas <var>.u.a.d.</var> et <var>.u.o.f.</var> tranſeuntes per eorum centra <var>.a.</var> et <var>.o.</var> <lb/>vſque ad circunferentias in punctis <var>.d.</var> et <var>.f.</var> ipſę <reg norm="erunt" type="context">erũt</reg> inuicem ęquales, eo quod <var>.e.i.</var> <reg norm="sum- pta" type="context">sũ-<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> ſed <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 ſemidiameter, con-<lb/>tiguus erit ipſis propoſitis circulis ex conuerſo .11. iam dictæ. </s> + <s xml:space="preserve">Idem dico pro circu-<lb/>lis ſe inuicem ſecantibus.</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 ſi circuli propoſiti ſeiuncti fuerint, ſumatur <var>.b.i.</var> diameter maioris, qui fiat ſe-<lb/>midiameter vnius circuli circa centrum <var>.o.</var> & hic circulus vocetur <var>.h.x.</var> coniunga-<lb/>tur deinde ſemidiameter <var>.o.i.</var> minoris circuli cum ſemidiametro <var>.a.i.</var> circuli maio-<lb/>ris, & ex huiuſmodi compoſita linea, fiat vnus ſemidiameter <var>.a.x.</var> circuli <var>.x.n.</var> concen <lb/>trici cum maiori, & à puncto <var>.x.</var> interſectionis horum circulorum (poſito quod ſe in-<lb/>uicem interſecent) ducantur per eorum centra <var>.x.a.</var> et <var>.x.o.</var> vſque ad ipſorum circun-<lb/>ferentias in punctis <var>.d.</var> et <var>.f.</var> duę <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, & ſemidiametri am-<lb/>borum circulorum, facto deni <lb/>que centro <var>.x.</var> vnius circuli, cu <lb/>ius ſemidiameter ęqualis ſit <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 diſtantia duorum <lb/>propoſitorum circulorum tanta fuerit, quod ſecundi circuli nequeant ſe inuicem <lb/>tangere, vel ſecare, tunc alia via incedendum erit, quę talis eſt & 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="ſignentur" type="context">ſignẽtur</reg> duo puncta ab ipſo ęquidi <lb/>ſtantia <var>.K.</var> et <var>.p.</var> diſtantia vero <var>.a.K.</var> facta ſit ſemidiameter eſſe vnius circuli <var>.K.x.</var> circa <lb/>centrum <var>.a.</var> diſtantia autem <var>.o.p.</var> ſemidiameter alterius circuli <var>.p.x.</var> circa cen-<lb/>trum <var>.o.</var> qui quidem circuli ſe inuicem ſecent in puncto <var>.x.</var> à quo cum ductę fue-<lb/>rinc <var>.x.a.d.</var> et <var>.x.o.f.</var> per centra dictorum circulorum, ipſe erunt <reg norm="inuicem" type="context">inuicẽ</reg> ęquales, eo <reg norm="quod" type="wordlist">qđ</reg> <lb/>cum <var>.b.K.</var> æqualis ſit <var>.q.p.</var> igitur <var>.x.d.</var> et <var>.q.p.</var> erunt inuicem ęquales, ſed <var>.f.x.</var> æqualis eſt <lb/><var>q.p</var>. </s> + <s xml:space="preserve">quare <var>.x.f.</var> æqualis erit <var>.x.d.</var> tunc ſi <var>.x.</var> centrum fuerit vnius circuli, cuius ſemidia-<lb/>mer ſit vna dictarum, problema ſolutum erit.</s> + </p> + <p> + <s xml:space="preserve">Talis etiam ſoiutio commo-<lb/>da erit ad inueniendum dictum <lb/> + <anchor type="figure" xlink:label="fig-0275-02a" xlink:href="fig-0275-02"/> + circulum cuiuſuis magnitudinis, <lb/>dato tamen <reg norm="quod" type="simple">ꝙ</reg> eius diameter, ma <lb/>ior ſit <var>.b.z.</var> cum in noſtra poteſta <lb/>te ſit accipere puncta <var>.K.</var> et <var>.p.</var> pro <lb/>xima vel remota ab ipſo <var>.z.</var> ad li-<lb/>bitum. </s> + <s xml:space="preserve">Vnde abſque vlla diuiſio <lb/>neipſius <var>.q.b.</var> per medium, ſatis <lb/>erit ſignare puncta <var>.K.</var> et <var>.p.</var> dua-<lb/>bus diſtantijs mediantibus <var>.b.K.</var> <lb/>et <var>.q.p.</var> inuicem æqualibus, & <lb/>etiam propoſitis.</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 ſuperficialem ellipſi ſimilem, ex datis axibus cir-<lb/>cino mediante delineari poſſe.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">FIguram ſuperficialem ellipſi ſimilem, ex datis axibus, circino mediante delinea <lb/>re cum volueris, ita facito.</s> + </p> + <p> + <s xml:space="preserve">Sit <var>.e.c.</var> ſemiaxis maior <var>.a.e.</var> verò minor, ad angulum rectum inuicem coniuncti, <lb/>tunc <var>.a.e.</var> producatur vſque ad <var>.o</var>. </s> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> <var>.a.o.</var> maior ſit quam diſtantia inter <var>.o.</var> et <var>.c.</var> quę <lb/>quidem <var>.a.o.</var> poſſet etiam dari, deſcribatur poſtea circulus <var>.a.d.b.</var> circa centrum <var>.o.</var> à <lb/>quo puncto protrahatur ſemidiameter <var>.o.b.</var> quæ cum <var>.a.o.</var> angulum rectum conſti-<lb/>tuat, quę <var>.o.b.</var> erit æquidiſtans <var>.e.c.</var> ex .28. primi, ducatur poſtea <var>.b.c.d.</var> et <var>.o.t.d.</var> vnde <lb/>angulus <var>.t.c.d.</var> ęqualis erit angulo <var>.o.b.d.</var> ex .29. eiuſdem. </s> + <s xml:space="preserve">ex quinta autem anguli <var>.b.</var> <lb/>et <var>.d.</var> ſunt inuicem æquales, </s> + <s xml:space="preserve">quare etiam <lb/>& anguli <var>.d.</var> et <var>.c.</var> inuicem ęquales erunt, <lb/> + <anchor type="figure" xlink:label="fig-0276-01a" xlink:href="fig-0276-01"/> + & ex .6. eiuſdem <var>.t.c.</var> ęqualis erit <var>.t.d.</var> duca <lb/>tur poſtea <var>.d.x.h.</var> perpendicularis lineæ <var>.c.<lb/>e.</var> ita diſtans ſub ipſa <var>.c.e.</var> vt arcus circula-<lb/>ris circa <var>.t.</var> delineatus ex ſemidiametro <var>.t.<lb/>d.</var> aptus ſit eam ſecare, ſumpto poſtea <var>.r.</var> <lb/>tam diſtante ab <var>.e.</var> vt <var>.t.</var> reperitur ab ipſo <lb/>e. et <var>.z.</var> ab <var>.e.</var> vt <var>.o.</var> ab eodem, ducendo po-<lb/>ſtea duos alios arcus magnitudinis <reg norm="priorum" type="context">priorũ</reg> <lb/>circa centra <var>.r.</var> et <var>.z.</var> habebimus propoſi-<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 cuiuſuis magnitudinis, vt <lb/>in ſecunda figura videre eſt, poſito tamen quod eorum diameter, minor ſit minore <lb/>axe ipſius figurę, quorum circulorum vnus ſit <var>.c.d.</var> circa <var>.t.</var> eius centrum, deinde in axe <lb/>minori ſumatur <var>.a.x.</var> æqualis <var>.c.t.</var> & protrahatur <var>.t.x.</var> quę per ęqualia diuidatur in pun-<lb/>cto <var>.n.</var> à quo poſtea 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> vſque ad interſectionem 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 ſecunda figura, ſumpto <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, ipſum, centrum erit alterius arcus oppoſi-<lb/>ti, poſſemus etiam <reg norm="abſque" type="simple">abſq;</reg> diuiſione ipſius, <var>t.x.</var> conſti <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 propoſite portionis datæ ſphæræ.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">VTaxem propoſitæ alicuius datæ ſphæræ inuenire poſſis ita tibi operandum eſt <lb/>vt gratia exempli. </s> + <s xml:space="preserve">Propoſita nobis eſt ſphæra <var>.c.i.e.t.</var> diametri cognitæ. </s> + <s xml:space="preserve">pro <lb/>poſita etiam eſt nobis eius portio <var>.n.e.u.</var> axis <var>.e.a.</var> cognitæ minoris ſemidiametro, da-<lb/>ta etiam nobis eſt proportio alterius portionis minoris hemiſphærio <var>.i.e.t.</var> ad por-<lb/>tionem <var>.n.e.u.</var> quæritur nunc quantus ſit axis <var>.e.x.</var> ſecundæ portionis hoc eſt deſidera-<lb/>mus cognoſcere proportionem <var>.e.x.</var> ad <var>.e.a.</var> vel ad diametrum ipſius ſpheræ.</s> + </p> + <p> + <s xml:space="preserve">Cuius gratia reperiatur primò proportio <reg norm="circunferentiæ" type="context">circũferentiæ</reg> maioris circuli ipſius <reg norm="ſphae ræ" type="simple">ſphę<lb/>ræ</reg> adeius diametrum, quæ ferè eſt vt .22. ad .7. ex Archimede.</s> + </p> + <p> + <s xml:space="preserve">Quo facto, inueniatur quantitas ſuperficialis huiuſmodi maioris circuli, quæ ſem-<lb/>per æqualis eſt producto quod fit ex ſemidiametro in dimidium circunferentiæ ip-<lb/>fius circuli, ex eodem Archimede. </s> + <s xml:space="preserve">Et ſic cognoſcemus quartam partem ſuperficiei <lb/>ſphæricæ ſphærę propoſite ex .31. primi lib. de ſphæra, & cyllindro Archimedis.</s> + </p> + <p> + <s xml:space="preserve">Deinde ſumatur tertia pars producti, quod fit ex ſemidiametro in ſuperficiem <lb/>maioris circuli, & habebimus conum, cuius baſis erit circulus maior, altitudo verò <lb/>ſemidiameter propoſitæ ſphæræ ex .9. duodecimi Eucli.</s> + </p> + <p> + <s xml:space="preserve">Quadruplum poſtea huiuſmodi coni, erit quantitas ſoliditatis, ſeu corporeitas to <lb/>tius ſphærę ex .32. dicti lib. Archimedis.</s> + </p> + <p> + <s xml:space="preserve">Imaginemur poſtea <reg norm="in" type="wordlist">ĩ</reg> ſphærica portione <var>.n.e.u.</var> <reg norm="lineam" type="context">lineã</reg> <var>.e.u.</var> à <reg norm="summitate" type="context">sũmitate</reg> ad <reg norm="extremitatem" type="context">extremitatẽ</reg> <lb/>baſis, cuius <var>.e.u.</var> quantitatem cognoſcemus, hoc modo ſcilicet, 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 eſt 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 eſt producto quod <lb/>fit ex <var>.c.a.</var> in <var>.a.e.</var> ex .34. tertij <reg norm="eiuſdem" type="context">eiuſdẽ</reg>, <lb/>& quadrato <var>.a.e.</var> hoc eſt producto, <lb/>quod fit ex <var>.c.e.</var> in <var>.e.a.</var> ex .3. ſecundi <lb/>eiuſdem.</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 poſtea <var>.e.u.</var> ponamus eam <lb/>vnius circuli ſemidiametrum eſſe, cu <lb/>ius ſuperficialis quantitas etiam inue <lb/>niatur, vt ſupra dictum eſt, quæ qui <lb/><reg norm="dem" type="context">dẽ</reg> æqualis erit ſuperficiei portionis <lb/><var>n.e.u.</var> ex .40. primi li. </s> + <s xml:space="preserve">Archimedis de <lb/>ſphæra, & 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 ſe-<lb/>midiametri datæ ſphæræ, & habebi-<lb/>mus ſoliditatem vnius coni æqualis <lb/>aggregato ſoliditatis portionis <var>.n.e.<lb/>u.</var> ſimul ſumptę<unsure/>, <reg norm="cum" type="context">cũ</reg> ſoliditate vnius co <lb/>ni, cuius axis ſit <var>.a.o.</var> <reg norm="reſiduum" type="context">reſiduũ</reg> ſemidia-<lb/>metri noſtræ ſphæræ 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"/> + ſis verò eadem quæ eſt portionis, cuius diameter eſt <var>.n.u.</var> ex .9. 12. Eucli. & ex .42. id-<lb/>eſt vltima primi Archimedis de ſphæra, & cyllindro.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius <var>.o.a.</var> eſt <lb/>axis et <var>.n.u.</var> diameter baſis, qui quidem conus nobis cognitus eſt, cum <var>.a.n.</var> ſemidia-<lb/>meter eius baſis, nobis cognita ſit ex .34. 3. Eucli. </s> + <s xml:space="preserve">& ſic quantitas eius baſis, & ita ter-<lb/>tia pars <var>.a.o.</var> eius axis, quę multiplicata cum dicta baſi, cuius <var>.n.u.</var> eſt diameter, produ <lb/>cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-<lb/>gre gato, relinquet nobis ſoliditatem portionis <var>.n.e.u.</var> vnde cognoſcemus proportio <lb/>nem iſtius portionis ad totam ſphæram propoſitam.</s> + </p> + <p> + <s xml:space="preserve">Sed cum nobis propoſita ſit proportio portionis <var>.n.e.u.</var> ad portionem <var>.i.e.t.</var> cogno <lb/>ſcemus etiam ſoliditatem huius ſecundę portionis <var>.i.e.t.</var> & ſimiliter <reg norm="proportionem" type="context">proportionẽ</reg> hu-<lb/>ius ad totam ſphęram, & ad <reg norm="reſiduum" type="context">reſiduũ</reg> <reg norm="etiam" type="context">etiã</reg> ipſius ſphęrę hoc eſt 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="vſque" type="simple">vſq;</reg> quo <var>.e.f.</var> æqualis ſit <var>.e.o.</var> ſemidiame <lb/>tro ſphęrę, 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 ſit proportioni portionis <var>.i.c.t.</var> ad portionem <var>.i.e.t.</var> quod quidem hoc modo efficie<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">pũ-<lb/>cto</reg> <var>.f.</var> in qua accipiemus duas lineas <var>.f.p.</var> et <var>p.q.</var> inuicem ita relatas, vt ſe habent in pro <lb/>portione duæ iam dictæ portiones, hoc eſt, vt <var>.i.c.t.</var> portio ad portionem <var>.i.e.t.</var> ducen <lb/>do poſtea <var>.q.e.</var> et <var>.p.h.</var> parallelam ad ipſam <var>.q.e.</var> diuiſam habebimus <var>.f.e.</var> in eadem pro <lb/>portione vt dictum eſt ex .2. ſexti, & .11 quinti Euclidis, vnde <var>.c.e</var>: <var>e.f.</var> et <var>.f.h.</var> nobis co <lb/>gnitę erunt.</s> + </p> + <p> + <s xml:space="preserve">Oportebit nos nunc cognoſcere quantitatem <var>.c.x.</var> hoc modo, videlicet, quęramus <lb/>quadratum, cuius <var>.c.x.</var> eius ſit radix, cui quadratum lineę <var>.c.e.</var> cognitum, ita ſit propor-<lb/>tionatum, vt eſt linea <var>.x.f.</var> ad lineam <var>.f.h.</var> quę nobis cognita eſt, quod rectè factum erit <lb/>ex eo, quod ſcripſit Archimedes in .4. ſecundi de ſphęra, & cyllindro.</s> + </p> + <p> + <s xml:space="preserve">Sed quia Archimedes eo in loco ſupponit id, quod necipſe, nec alius adhuc inue <lb/>nit, niſi via naturali, hoc eſt tres partes ęquales 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 ſemidiametro, diuidemus<lb/>q́ue <var>.f.e.</var> in puncto <var>.h.</var> vt ſupra factum fuit, applicabimus poſtea <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> quęremus 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">ctaq́;</reg> <var>.d.c.</var> quæ à diametro <var>.e.c.</var> deducta ab <var>.c.</var> incohando vſque ad <var>.x.</var> relinquat nobis <var>.<lb/>x.f.</var> ęqualem <var>.c.m</var>.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ratio eſt, quia quadratum <var>.c.e.</var> ſe habet ad quadratum <var>.c.d.</var> vt <var>.c.e.</var> ad <var>.c.<lb/>b.</var> ex .4. et .18. ſexti Eucl. </s> + <s xml:space="preserve">ſed ex .4. ita ſe habet <var>.m.c.</var> ad <var>.c.g.</var> vt <var>.e.c.</var> ad <var>.b.c.</var> & cum ſit <var>.c.<lb/>g.</var> ęq alis <var>.f.h.</var> ſi <var>.c.m.</var> ęqualis fuerit <var>.f.x.</var> habebimus propoſitum. </s> + <s xml:space="preserve">Quod ſi quis per di-<lb/>ſcretum vel et hoc facere, ita ei agendum erit.</s> + </p> + <p> + <s xml:space="preserve">Ponamus exempli gratia totum diametrum <var>.c.e.</var> propoſitæ ſphæræ eſſe ut decem, <lb/><reg norm="proportionemque" type="simple">proportionemq́;</reg> reſiduę portionis <var>.i.c.t.</var> ad ſecundam <var>.i.e.t.</var> hoc eſt <var>.f.h.</var> ad <var>.h.e.</var> ſeſqui-<lb/>alteram eſſe, vnde <var>.e.h.</var> bis tertia erit ìpſius <var>.f.h.</var> <reg norm="totaque" type="simple">totaq́;</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 poſtea quadratum lineæ <var>.c.x.</var> cui quadratum <var>.c.e.</var> hoc eſt .100. ita pro-<lb/>portionatum ſit vt <var>.f.x.</var> ad <var>.f.h.</var> hoc eſt ad .3. ſi autem cogitauerimus <var>.c.x.</var> eſſe nouem <lb/>partium talium qualium <var>.c.e.</var> eſt decem, eius quadratum erit .81. et <var>.x.f.</var> erit .6. par-<lb/>tium talium qualium <var>.c.f.</var> eſt .15. dicendo poſtea ſi .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 eſt .6. dabit .4. integra cum <num value="86">.<lb/> + <anchor type="figure" xlink:label="fig-0279-01a" xlink:href="fig-0279-01"/> + 86.</num> centeſimis, ſed nos vellemus no-<lb/>bis prouenire tria, eo <reg norm="quod" type="simple">ꝙ</reg> ita eſt <var>.f.h.</var> <lb/>qua propter deſcendere 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> poſtea ex regula de <lb/>tribus, ſi 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">cũ</reg> 296. de .300. hoc eſt .2. <lb/>cum circa .49. quinquageſimis, quæ <lb/>quidem quantitas, cum propinquiſſi <lb/>ma ſit 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> reſiduum, hoc eſt 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 inſularijs.</head> + <head style="it" xml:space="preserve">Excellentißimo D. lo. Baptiſtæ Fæmello Ciui <reg norm="Decurionique" type="simple">Decurioniq́ꝫ</reg> Tau-<lb/>rinenſi Philoſopho, Medico, & in Accademia eius <lb/>Ciuitatis Medicinæ Practicæ Ordinario, <reg norm="Pri- marioque" type="simple">Pri-<lb/>marioq́ꝫ</reg> profeßori celeberrimo.</head> + <p> + <s xml:space="preserve">DIj perdant tuas adeo moleſtas, & aſſiduas curas, quæ te nimis à ſuauiori-<lb/>bus ſtudijs diſtrahunt, & 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 promiſi, nunc mitto.</s> + </p> + <p> + <s xml:space="preserve">Negari quidem non poteſt, quin fuerit laborioſum opus Porcachij, & Benedicti <lb/>Bordonij, hoc eſt inſularium, qui rectè etiam feciſſent, cum loqui eos oportebat de <lb/>terminis ſphæræ ratione ſitus locorum, ſi ſeipſos alicuius excellentis Coſmographi <lb/>conſilio ſubmiſiſſent. </s> + <s xml:space="preserve">Conſidera quæſo, quomodo admitti poſſit, id quod ait Por-<lb/>cachius initio ſui operis, ideſt Iſlandiam ſub Polo arctico iacere, inter auſtrum, & <lb/>boream: </s> + <s xml:space="preserve">omittamus etiam quod idem in Proęmio lib. ſecundi, vbi ait Biarmiam, <lb/>(& non Iſlandiam) eſſe ſub dicto polo arctico: </s> + <s xml:space="preserve">in <reg norm="eodemque" type="simple">eodemq́;</reg> principio repetit ipſam <lb/>Iſlandiam inter auſtrum, & boream per centum leucas Germanicas extendi, dein-<lb/>de verſus occidentem, ea duo ſtupenda miracula conſpici. </s> + <s xml:space="preserve">Vide quæſo, quomodo <lb/>incolę ſub aliquo ex polis, habere poſſint occidentem, orientem, magiſtrum, <reg norm="auſtrum" type="context">auſtrũ</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 poteſt, vt in-<lb/>ſula Iſlandiæ ſit ſub polo, eius tamen dies, & nox maior non ſit longior ſpatio <reg norm="trium" type="context">triũ</reg> <lb/>menſium? </s> + <s xml:space="preserve">vt ipſe pagina .62. in proęmio ſecundi lib. affirmat, quamuis hoc à Bordo <lb/>ne deſumat. </s> + <s xml:space="preserve">In quo <reg norm="vterque" type="simple">vterq;</reg> fallitur, <reg norm="ſentientes" type="context">ſentiẽtes</reg> huiuſmodi diem ab ingreſſu Solis, in <lb/>principium geminorum incipere, & in egreſſu à Leone terminari, ideſt à .12. Maij ad <lb/>14. Auguſti, quaſi ſi ab æquatore finis Leonis ita declinaret, vt principium gemino-<lb/>rum, & finis Aquarij, vt initium Sagittarij, nam ratio poſtulat, tantum de-<lb/>clinari ab æquatore finem quantum initium diei, vbi maximus dies .24. horas ex <lb/>cedit, & ſic dico de noctibus: </s> + <s xml:space="preserve">vnde in huiuſmodi regione, vbi per tres menſes conti <lb/>nuos Sol radios emittit, huiuſmodi dies à medietate Tauri incipit, & in medietate <lb/>Leonis terminatur, quæ quidem loca æqualem declinationem habent, & ſic nox <lb/>trium menſium 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 ſolſtitiales eſſe circa .24. Iunij. </s> + <s xml:space="preserve"><reg norm="quod" type="wordlist">Qđ</reg>, an <reg norm="tunc" type="context">tũc</reg> eſſet <reg norm="verum" type="context">verũ</reg>, <lb/>tu ipſe videto. </s> + <s xml:space="preserve">Is præterca modus <reg norm="quem" type="context">quẽ</reg> ad <reg norm="inueniendum" type="context context">inueniẽdũ</reg> <reg norm="orientem" type="context">orientẽ</reg>, & <reg norm="occidentem" type="context">occidẽtem</reg> præſcribit <lb/>in eodem proęmio pag .63. eſt tædioſus, cum ſemper expectare nos cogat æquino-<lb/>ctij tempus, cum alij modi reperiantur breuiores, qui in qualibet reuolutione primi <lb/>mobilis obſeruari poſſunt, quorum vnus erit mediante inuentione lineæ meridiane <lb/>orizontalis, eo modo, quo ſcriptum eſt ab antiquis mediante Sole, aut Luna, quæ <lb/>luminaria in quolibet alio loco, <reg norm="præterquam" type="context">præterquã</reg> ſub polo efficiunt, vt extremitas vmbræ <lb/>rectæ <reg norm="gnomonum" type="context">gnomonũ</reg> <reg norm="gyrum" type="context">gyrũ</reg> <reg norm="oxigonium" type="context">oxigoniũ</reg>, ſeu <reg norm="eclipticum" type="context">eclipticũ</reg> ducat, ideſt in ijs locis, <reg norm="quorum" type="context">quorũ</reg> zenit eſt <lb/>inter polum, & circulum arcticum, quemadmodum facit, vt alijs, exiſtentibus ipſis <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 ſub polo vi-<lb/>detur, ideſt vt multis rationibus, vt ipſe dicit, fieri non poſſit, ut fiat immediata quę <lb/>dam, & ſubita mutatio à continuo die ad continuam noctem abſque eo quod ijs, <lb/>ſaltem ſemel conceſſa ſint dies, & nox terminata duodecim horarum, eſt magis ad <lb/>mirandum impoſſibile, quod imaginari poſſimus, nam neceſſarium eſſet, ut orizon-<lb/>habitatorum ſub polo ſecaret æquatorem contra id, quod ſuperius admiſerat, id-<lb/>eſt <reg norm="orizontem" type="context context">orizõtẽ</reg> Biarmiæ, eſſe <reg norm="eundem" type="context context">eũdẽ</reg> <reg norm="cum" type="context">cũ</reg> circulo æquinoctiali. </s> + <s xml:space="preserve">Vide etiam quid is ab anti-<lb/>quis colligat, loquens de iis, quæ in inſula Taprobana ad finem pag .186. admirabi <lb/>lia ſunt, ſcribens eiuſdem inſulę habitatoribus, Lunam ſuper terram non apparere <lb/>ab octauo uſque ad decimumſextum diem: </s> + <s xml:space="preserve">pręter quam, quod etiam ſcribit, in <lb/>eadem inſula, tramuntanam non uideri, quod falſum eſt, quia hæc à polo arctico <lb/>circiter quatuor gradibus diſtat noſtris temporibus. </s> + <s xml:space="preserve">unde ab ijs qui ſunt ſub æqua-<lb/>tore, cum ea ſupra orizontem eſt, conſpici poteſt, cum ijſdem ſingulis diebus oria-<lb/>tur, & occidat. </s> + <s xml:space="preserve">Idem etiam pro re admirabili ſcribit, uideri Canopum, qui à po-<lb/>lo antarctico plus quam quadraginta gradibus diſtat.</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 Philalthęus tam eximius Mathematicus ſit, ut ipſum Anto-<lb/>nius Berga facit, ego quidem non uideo. </s> + <s xml:space="preserve">In ſuis enim commentariis de <lb/>Cœlo, dicit primum, Pyramidem, quę 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"/> + ſex baſibus conſtare, 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 ſpectet: </s> + <s xml:space="preserve">Omitto etiam quod idem neget aſtrologiam pag .74. <lb/>79. & quod etiam dicat pag .89. </s> + <s xml:space="preserve">Deum eſſe ad orientem, non conſiderans aliqui-<lb/>bus populis noſtrum orientem eſſe occidentem.</s> + </p> + <p> + <s xml:space="preserve">Quod idem ait pag .241. </s> + <s xml:space="preserve">Aſtrologiam eſſe antiquiorem Aſtronomia eſt falſiſſi-<lb/>mum, quia iudiciaria ſemper præſupponit cognitionem ſitus ſtellarum, quæ ab A-<lb/>ſtronomia petitur. </s> + <s xml:space="preserve">Mouebit tibi riſum quod ait pag .307. his verbis.</s> + </p> + <quote> + <s xml:space="preserve">Verum propriè media dicitur illa, quæ rectam ſphæram omninò habet, quæ eun <lb/>dem polum orizontis & mundi obtinet, quæ orizontem habet diuidentem <reg norm="ſphæram" type="context">ſphærã</reg> <lb/>æquè ſecundum angulos rectè.</s> + </quote> + <p> + <s xml:space="preserve">Paulo inferius continuans ſermonem de ſphæra recta, ait.</s> + </p> + <quote> + <s xml:space="preserve">Et niſi tumor terræ, & gibum eſſet, ijs perpetuus eſſet dies ſine nocte.</s> + </quote> + <p> + <s xml:space="preserve">Linea verò .56. ait habitatores ſphęrę rectè habere .4. ſolſtitia, ſeſe ipſum huius <lb/>rei planè ignarum prodens .310. autem pag. ſic ſcribit.</s> + </p> + <quote> + <s xml:space="preserve">Quoniam repercutiuntur radij, & peridem centrum tranſeunt, ob id ſtupam ap <lb/>poſitam centro radius accendit.</s> + </quote> + <p> + <s xml:space="preserve">Quem quidem errorem ab Euclide deſumit, 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 ſubſtantiam corporis lucidi & corpoream, ſubijciam 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 ſi ſolum inci-<lb/>dat ſine repercuſſu, neorecta incidere iuuerit.</s> + </quote> + <p> + <s xml:space="preserve">Quod falſum eſt cum radius incidens longè magis quam reflexus calefaciat. </s> + <s xml:space="preserve">In fi <lb/>ne autem .405. ſic ſcribit.</s> + </p> + <quote> + <s xml:space="preserve">Sol in ortu & in occaſu longius apparet, iccircò reuolui creditur. </s> + <s xml:space="preserve">Hinc etiam in <lb/>abſide ſtare putatur, & in oppoſito abſidis, vnde ſolſtitia vocant, ſed nobis in Can-<lb/>cro, antipodibus verò in Capricorno tum Sol abeſſe longius apparet vtriſque.</s> + </quote> + <p> + <s xml:space="preserve">An hoc quid peius dici poteſt? </s> + <s xml:space="preserve">Circa vero .40. lineam pag .459. ſic ſcribit.</s> + </p> + <quote> + <s xml:space="preserve">Si enim alij planetæ, & ſtellæ fixę reciperent à Sole lumen, dum accederent ad So <lb/>lem, vel recederent, aut contra, Sol ad eas appropinquaret, & abſcederet, eaſdem-<lb/>lucis viciſſitudinis ſubiret, quas Luna.</s> + </quote> + <quote> + <s xml:space="preserve">Hoc autem nondum depręhenſum eſt, quin etiam Mercurius, Venus, ſuo interpo <lb/>ſitu, Solem occultarent nobis, vt Luna.</s> + </quote> + <quote> + <s xml:space="preserve">Paulo inferius ſic ait. </s> + <s xml:space="preserve">Rurſus æquè Saturnus, Jupiter, Mars, ſubire deliquium, <lb/>more Lunæ, aut ſaltem obiectu terræ inter Solem & ipſos, quia tum ob interpoſitam <lb/>terram non poſſent haurire lumen à Sole.</s> + </quote> + <p> + <s xml:space="preserve">Hæc verò omnia, talia ſunt, qualia ab ijs qui incipiunt intelligere ſphæram non <lb/>proferrentur. </s> + <s xml:space="preserve">Omittamus, quod ait deinde.</s> + </p> + <quote> + <s xml:space="preserve">Accedit quod ſi aſtra lumen à Sole acciperent eiuſdem caloris eſſent. </s> + <s xml:space="preserve">Itaque om <lb/>nia ſiccarent, & nulla eſſent frigidæ conſtitutionis contra Aſtrologos.</s> + </quote> + <p> + <s xml:space="preserve">Quia hac ratione, Luna, quæ negari non poteſt, quin ab ipſo Sole <reg norm="lumem" type="context">lumẽ</reg> accipiat, <lb/>eiuſdem 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. quaſi 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 ſit effecta quam ad illuminandam ſuperficiem huius excrementi ip-<lb/>ſius vniuerſi ad vtilitatem hominum, imò, vt rectius dicam, <reg norm="animalium" type="context">animaliũ</reg>. </s> + <s xml:space="preserve">vide etiam <lb/>pag .632. et .633. vbi Ariſtotelem de implendo loco non intellexit, cum citet ſphæ-<lb/>ram, loco pyramidis, & inter .46. et .47. lineas dicat <reg norm="quadratum" type="context">quadratũ</reg> eſſe quid multiplex, <lb/>cum ſit vnicum tantum in ſpecie, quia ſpecies eſt quadrilateri, & quadranguli, ſed <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. ſcribens.</s> + </p> + <p> + <s xml:space="preserve">Pyramis, ſiue planum, ſiue ſolidum, habet acutiſſimum, & in .2. libr. de anima <lb/>pag .215. dicat de die poſſe videri ſtellas in ſpeculo poſito in vaſe aqua pleno, quod <lb/>reuera eſt valde abſurdum. </s> + <s xml:space="preserve">Alios eiuſdem errores tibi non patefacio, quia iam ni-<lb/>hil amplius otij mihi eſt, ſed eos tu ipſe perſpicere, & cognoſcere 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 ſine ijs exploratum mihi erat. </s> + <s xml:space="preserve">Sed conce <lb/>do tantum eſſe dicere vbi eſt maius lumen, minus non diſcerni, quantum inter <lb/>diu ſtellas non videri: </s> + <s xml:space="preserve">immo eſt etiam magis vniuerſale, quia idem multis aliis lumi-<lb/>nibus, præter ea quæ ſunt ſtellarum, ea ratione contingit, quia ingrediente per pupil-<lb/>lam, tam lumine maiori, quam minori, reflexum ipſius maioris in oculo, in ſitu mino <lb/>ris, efficit, vt ipſum minus confundatur, & diſtingui nequeat, quemadmodum aper-<lb/>te cognoſci poteſt in aliquo cubiculo, cuius parietes dealbati ſint, in quo, vnicum <lb/>tantum ſit exiguum foramen, per quod aliqua lumina reflexa ab obiectis extrinſecis <lb/>intra ipſum cubiculum ingredi poſſint, vnde imagines <reg norm="obiectorum" type="context">obiectorũ</reg> in parietibus con-<lb/>ſpiciuntur, ſed ſi per idem foramen ingrederetur etiam primarius radius Solis, re-<lb/>flexus huiuſmodi radij efficeret, vt dictæ imagines, magis aut minus euaneſcerent, <lb/>prout dictus reflexus radij ſolaris, maiori, minoríue vi polleret.</s> + </p> + <p> + <s xml:space="preserve">Ad hoc tamen propoſitum, nolo tibi ſilentio inuolui mirabilem quendam effe-<lb/>ctum eiuſmodi rei. </s> + <s xml:space="preserve">Hoc eſt vt fiat foramen illud rotundum, magnitudinis tamen <lb/>vnius ſpecilli, quod foramen obturetur mediante vno illorum ſpecillorum, quæ pro <lb/>ſenibus (non breuis viſionis) conficiuntur, hoc eſt quorum ambæ ſuperficies con <lb/>uexæ ſunt, non autem concauæ. </s> + <s xml:space="preserve">Deinde opponatur folium album papiri, adeo di <lb/>ſtans à foramine, vt extrinſeca obiecta in eo appareant. </s> + <s xml:space="preserve">Quæ quidem obiecta ſi à <lb/>Sole illuſtrata fuerint, tam clara, & diſtincta videbuntur, vt nihil pulchrius dele-<lb/><reg norm="ctabiliusque" type="simple">ctabiliusq́;</reg> videri poterit, inuerſa tamen. </s> + <s xml:space="preserve">Sed ſi ea directa videre voluerimus. </s> + <s xml:space="preserve">hoc <lb/>optimè faciemus, mediante reflexione alicuius ſpeculi 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 ſequatur ac<unsure/>tatem in qua <lb/>calor viguerit.</head> + <head xml:space="preserve">NOBILISSIMO, NECNON INGENIOSISSIMO <lb/>Gabrieli Buſchæ, Mediolanenſi.</head> + <p> + <s xml:space="preserve">QVod dixi hyemem valde frigidam ſequi <reg norm="æſtatem" type="context">æſtatẽ</reg>, in qua calor viguerit, inde na <lb/>ſcitur, quia calor terrę, aquæ, & aeris, non eſt naturalis horum corporum, vt <lb/>eſt frigus, cum calor à Sole procedat, qui ea calefacit ſuo 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 ſortiatur, & cum eandem poſtea deſerit, ad aliam partem æquatoris <reg norm="tranſmigrans" type="context">tranſmigrãs</reg> <lb/>terra ad ſuam qualitatem reddit, maiori cum impetu, eo modo, quo res in mo-<lb/>tibus localibus naturalibus, qui etiam terminos ſibi pręfixos, & conſtitutos exce-<lb/>dunt, hinc etiam hyeme fit glacies, ex calefacta prius aqua, quæ durior poſtea eſt <lb/>atque frigidior alia. </s> + <s xml:space="preserve">Aeſtas etiàm quæ ſequitur hyemem valde frigidam, non <lb/>erit admodum calida, quia Sol inueniens contrarium naturale valde potens, non <lb/>tam facile illud pellere poteſt, vnde etiam ſi in Geminis, Cancro, & Leone, moram <lb/>trahat, non ſufficit tamen ut magnum calorem imprimere poſſit. </s> + <s xml:space="preserve">Vnde ſequitur duas <lb/>æſtates quarum una ſequatur aliam, in eodem loco, uehementi calore præditas eſ-<lb/>ſe 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 ſcribit Tattalea quęſito quinto primi lib. necnon quæſito <lb/>21. et .24. maxima cum ratione eſſe uidetur, non tamen ea quam ipſe in <lb/>quinto profert, quia uerum non eſt, vt quanto aliquid fit calidius, <reg norm="tanto" type="context">tãto</reg> ue-<lb/>hementius attrahat, eo quod ſi etiam huiuſmodi res, in eodem calore, in <lb/>quo ſemel 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 conſequens circumcirca <reg norm="undique" type="simple">undiq;</reg> <lb/>trudit, & partes uaſis debiliores cedunt. </s> + <s xml:space="preserve">dum uerò dictum corpus re frigeratur, re-<lb/>ſtringitur, & dum in unum cogitur, ſi reperiatur in uaſe, quod aer, aqua, aut aliud <lb/>aliquod corpus ingredi nequeat, dictum uas à quo circundatur frangit, ne aliqua <lb/>pars loci uacua remaneat, ſed ſi aliquod corpus ingredi poteſt, illud ipſum ad ſe at-<lb/>trahit, quemadmodum uidere licet in cucurbitulis. </s> + <s xml:space="preserve">Vnde ſequitur eam propoſi-<lb/>tionem, qua dicitur, calidi eſt attrahere, ueram non eſſe, quia ſi 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 ſequitur, ut <lb/>quanto calidior facta fuerit materia aliqua, aliquo loco determinata, redeundo po-<lb/>ſtea ad ſuam priorem frigiditatem, tanto minori loco indigeat, ſimiliter etiam <lb/>è conuerſo accidit, ut quanto frigidior <reg norm="reperitur" type="simple">reꝑitur</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"/> + ſtea egeat ipſa 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">tuasq́;</reg> dubitationes conſideraui, quas quidem non inutiles <lb/>inueni, quo uerò ad primam, dico te oportere illud Theorema ſpeculari or <lb/>dine huiuſmodi methodi, uidelicet quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> habuerimus <reg norm="angulum" type="context">angulũ</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, ſi ab aliquo puncto inter ipſas poſito, puta <var>.u.</var> quod etiam uocetur <var>.i.</var> du <lb/>ctę fuerint .4. lineę ipſis dictis lateribus, hac ſcilicet <reg norm="conditione" type="context">cõditione</reg>, <reg norm="quod" type="wordlist">qđ</reg> duę ex dictis .4. ſint <lb/>parallelę ipfis <reg norm="lateribus" type="simple">lateribꝰ</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> reliquę uero duę <lb/>ſeccent ipſa 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, quę <var>.E.V.</var> ad <var>.E.I.</var> <lb/>Nam ſcimus proportionem <lb/><var>E.i.</var> ad <var>.E.i.</var> eandem eſſe quę <lb/><var>e.i.</var> ad <var>.e.A.</var> ex fimilitudine <lb/><reg norm="triangulorum" type="context">triangulorũ</reg>, ſimiliter <reg norm="propor" type="simple">ꝓpor</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> quę <lb/><var>e.a.</var> ad <var>.e.u.</var> </s> + <s xml:space="preserve">quare aggregata <lb/>ex iſtis erunt inuicem <reg norm="aequa- lia" type="simple">ęqua-<lb/>lia</reg>, uel ſi mauis ex ęqua pro <lb/>portionalitate, quod idem <lb/>eſt, ita ſe 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">Suppoſito nunc plano orizontali <var>.V.E</var>. </s> + <s xml:space="preserve"><reg norm="Altitudineque" type="simple">Altitudineq́;</reg> inacceſſibili <var>.A.E</var>. </s> + <s xml:space="preserve">Duę ue-<lb/>rò ſtationes oculorum ſint <var>.V.</var> et <var>.I.</var> lineę autem uiſuales ſint <var>.V.A.</var> et <var>.I.A</var>. </s> + <s xml:space="preserve">Et quadra-<lb/>tum geometricum ſit <var>.b.e</var>. </s> + <s xml:space="preserve">Supponatur nunc pro prima dubitatione, quod in am-<lb/>babus ſtationibus filum perpendiculare ſeccet latus <var>.e.c.</var> non autem <var>.b.c.</var> (nam quan-<lb/>do in ambabus ſtationibus filum ſecat latus <var>.b.c.</var> nullum tibi dubium oritur, imo ma <lb/>nifeſtè 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> ſumpto <var>.E.</var> pro <var>.b.</var> et <var>.I.V.</var> pro punctis ſecatis à filo, ex <reg norm="euidenti" type="context">euidẽti</reg> ſimilitudine trian-<lb/>gulorum quadrati cum triangulis <var>.A.E.V.</var> et <var>.A.E.I.</var>) Sed cum in pręſenti caſu repe-<lb/>riatur triangulum <var>.u.e.a.</var> minus, in ſtatione remotiori, ſimile triangulo maiori <var>.V.E.<lb/>A.</var> & triangulum maius <var>.i.e.a.</var> proximioris ſtationis, ſimile triangulo minori <var>.I.E.A.</var> <lb/>(quod in alio iam dicto, caſu non accidit, ut unum triangulorum, minus ſcilicet, ſi-<lb/>mile ſit uno triangulorum, maiori ſcilicet & è conuerſo) Non omnino abſque ratio <lb/>ne dubitas quo pacto fieri poſſit ut <var>.a.e.</var> remotioris ſtationis ad <var>.a.e.</var> propinquioris ita <lb/>ſe habeat quema dmodum <var>.I.E.</var> ad <var>.E.V</var>. </s> + <s xml:space="preserve">Quapropter ſi pręcedentem 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 inſpexeris, omnis tua dubitatio euaneſcet. in qua figura apertè vi debis cor-<lb/>reſpondentiam talium triangulorum inter ſe, nec magis, nec minus quam in infra-<lb/>ſcripta hic figura cernere licet, quamuis in hac, triangula quadrati, ſeparata ſint ab <lb/>imaginarijs <var>.A.E.V.</var> et <var>.A.E.I.</var> in ſupradicta 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 ſe 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 proſequi, vt in cognitionem <lb/>peruenias ipſius <var>.I.E.</var> ipſius <var>.A.E.</var> & ipſius <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 ſtatione latus <var>.b.c.</var> in remotiori vero latus <var>.c.e.</var> ſecatur à fi <lb/>lo (pro ſecunda dubitatione) </s> + <s xml:space="preserve">Tunc oportet imaginatione conſiderare latus <var>.b.c.</var> in <lb/>re motiori ſtatione diſtentum eſſe vſque ad filum in puncto <var>.n.</var> vbi videbis <reg norm="triangulum" type="context">triangulũ</reg> <var>.<lb/>u.b.n.</var> ſimile triangulo <var>.A.E.V.</var> ita vt <var>.i.b.a.</var> ſimile ſuo <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> correſpondebit ipſi <var>.A.E.</var> et <var>.b.n.</var> ipſi <var>.E.V.</var> et <var>.b.a.</var> ipſi <var>.E.I.</var> <lb/>quapropter iubeo, vt quæras quantum ſit latus <var>.b.n.</var> ex regula de tribus, dicens ſi <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> ſimile eſt <var>.u.b.n.</var> reperto autem <lb/>latere <var>.b.n.</var> ex quo dempto <var>.b.a.</var> breuioris diſtantię, reſiduum reſpondebit ipſi <var>.I.V.</var> vt <lb/>ſcis, vnde proſequendo operationem tibi cognitam, obtinebis intentum, hoc eſt co <lb/>gnoſces reliqua interualla. </s> + <s xml:space="preserve">Nihil enim miror demonſtrationem Tartaleæ circa hu <lb/>iuſmodi operationem te minime ſatisfeciſſe.</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 propoſitio illius ſcriptoris, de quo nuper <reg norm="mecum" type="context">mecũ</reg> locutus es, vt <lb/>mihi dixiſti, tua ſit, hoc enimego, nec affirmare, nec negare audeo, quamuis in mul <lb/>tis cum tua manuſcripta concordet. </s> + <s xml:space="preserve">Nam ſępæ cogitationes hominum in idem co-<lb/>incidunt, vt pluries cenſuit Ariſto.</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 ſubiecta.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">AMor erga te meus ſanè ſingularis, nullo modo <reg norm="permittit" type="simple">ꝑmittit</reg>, vt ea quæ Tartaleę ſcri <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 ſunt circa quæ <lb/>dam illius Authoris problemata, quorum primum ab ipſo Tartalea <lb/>ſcriptum in .3. quæſito libr .4. tale eſt, is vult locare .3500. homines, eodem modo, <lb/>quo præſupponit locatos eſſe .1000. ita vt quilibet hominum ordo ſiue vt vulgo di-<lb/>citur filtia ſit .49. quapropter multiplicat quadratum ipſius .49. quod eſt .2401. <reg norm="cum" type="context">cũ</reg> <lb/>numero .3500. propoſito, productum verò .8403500. diuidit per .1000. vt proue-<lb/>niat .8403. cuius radix quadrata eſt .91. pro numero hominum vniuſcuiuſque ordinis <lb/>propoſiti 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> ſit vna filtia ſiue ordo .49. <reg norm="hominum" type="wordlist">hoĩum</reg>, cuius <reg norm="quadratum" type="context">quadratũ</reg> ſit <var>.b.c.</var> 2401. imaginemur <reg norm="etiam" type="context">etiã</reg> re <lb/>ctangulum <var>.A.B.</var> 3500. hominum, quod ſupponemus ſimile 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"/> + ſit quadratum ipſius <var>.D.B</var>. </s> + <s xml:space="preserve">Nunc ſupponendo <var>.A.B.</var> ſimile <var>.a.b.</var> clarum erit ex diffini-<lb/>tione ſimilium figurarum, quod eadem proportio erit <var>.A.D.</var> ad <var>.D.B.</var> quę <var>.a.d.</var> ad <var>.d.<lb/>b.</var> hoc eſt <var>.A.D.</var> ad <var>.D.C.</var> vt <var>.a.d.</var> ad <var>.d.c.</var> hoc eſt <var>.A.B.</var> ad <var>.B.c.</var> vt <var>.a.b.</var> ad <var>.b.c.</var> ex prima <lb/>ſexti, vel .18. ſeu .19. ſeptimi, </s> + <s xml:space="preserve">tunc cum dixerimus ſi <var>.a.b.</var> ita reſpondet ad <var>.b.c.</var> ergo <var>.A.<lb/>B.</var> correſpondet 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. ſexti vel .20. ſeptimi, cuius <lb/>prouentus radix quadrata erit quod quærebatur.</s> + </p> + <p> + <s xml:space="preserve">Sed aliter idem poſſe fieri ſpeculatus ſum, hoc eſt multiplicando numerum .49. <lb/>ordinis .1000. hominum <reg norm="cum" type="context">cũ</reg> radice quadrata numeri .3500. propoſiti, productum ve-<lb/>rò diuidere per radicem quadratam ipſius .1000. vnde prouentus .91. erit numerus <lb/>vnius ordinis .3500. numeri <reg norm="propoſiti" type="simple">ꝓpoſiti</reg>.</s> + </p> + <p> + <s xml:space="preserve">Cuius <reg norm="operationis" type="simple">oꝑationis</reg> ſpeculatio eſt iſta. <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> ſua <lb/>radix et <var>.a.d.</var> rectangulum propoſi-<lb/>tum ipſius .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">rectangulũ</reg> <lb/>ipſius numeri .3500. propoſiti, ſimile <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">Cũ</reg> enim <var>.a.b.</var> æquale ſit <lb/><var>a.d.</var> et <var>.A.B</var>: <var>A.D.</var> <reg norm="tunc" type="context">tũc</reg> <var>.a.c.</var> erit media <lb/>proportionalis inter <var>.a.e.</var> et <var>.e.d.</var> & ſic <lb/><var>A.C.</var> erit etiam media proportiona <lb/>lis inter <var>.A.E.</var> et <var>.E.D.</var> per .16. ſexti, <lb/>ſeu .20. ſeptimi, & quia proporrio. A <lb/>E. ad <var>.E.D.</var> æqualis eſt proportioni <var>.<lb/>a.e.</var> ad <var>.e.d.</var> cum <var>.A.D.</var> ſupponatur ſi-<lb/>mile <var>.a.d.</var> ergo proportio <var>.A.E.</var> ad <var>.A<lb/>C.</var> ęqualis erit proportioni <var>.a.e.</var> ad <var>.a.<lb/>c.</var> quę medietates ſunt <reg norm="totorum" type="context">totorũ</reg> æqua-<lb/>lium, rectè igitur fiet ſi procedamus <lb/>ex regula de tribus, dicendo ſi <var>.a.c.</var> <lb/><reg norm="correſpondet" type="context">correſpõdet</reg> <var>.a.e.</var> tùc <var>.A.C.</var> <reg norm="correſpon" type="context">correſpõ</reg> <lb/>det <var>.A.E.</var> ex ſupradictis .15. ſexti. vel <lb/>20. ſeptimi.</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 ſe <lb/>patet, quod eſt inuenire <reg norm="pauimentum" type="context">pauimentũ</reg> <lb/>ſeu aream quadratam, in qua poſſint <lb/>locari quot homines volueris, ita in <lb/>ter ſe ſiti, ut vnuſquiſque occupet <num value="7">.<lb/>7.</num> pedes ipſius areę in longitudinem <lb/>et .3. per latitudinem à lateribus.</s> + </p> + <p> + <s xml:space="preserve">Seu ex propoſito hominum nume <lb/>ro inuenire numerum ipſorum loca-<lb/>bilem in aliqua area quadrata, ita, <lb/>vt vnuſquiſque occupet .21. pedes <lb/>quadratos ipſius 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 eſt <reg norm="multiplicando" type="context">multiplicãdo</reg> radicem quadratam pro-<lb/>poſiti numeri hominum per .21. & productum item multiplicando per eandem radi <lb/>cem, & huiuſmodi 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 huiuſmodi numeri radicem <lb/>quadratam hoc eſt .60. per .21. hoc <lb/> + <anchor type="figure" xlink:label="fig-0288-01a" xlink:href="fig-0288-01"/> + eſt per productum quod fit ex .7. <reg norm="cum" type="context">cũ</reg> <lb/>3. & reſultabit nobis .1260. quod ſi <lb/>multiplicabitur, per .60. hoc eſt per <lb/>eandem radicem, reſultabit nobis <num value="75600">.<lb/>75600.</num> cuius producti radix qua-<lb/>drata eſt ferè .275. qua diuiſa 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 eſt iſta, cogitemus nu <lb/>merum .3600. propoſitum eſſe qua <lb/>dratum <var>.a.b.</var> (ſed non areæ) cuius ra <lb/>dix .60. ſit <var>.a.c.</var> & quia hic numerus <num value="60">.<lb/>60.</num> intelligitur eſſe hominum, quo-<lb/>rum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> occupat .21. pedes <lb/>quadratos ſuperficiales ex ſuppoſi-<lb/>to, </s> + <s xml:space="preserve">& propterea multiplicatur, 60. <lb/>cum .21. vnde nobis veniat .1260. <lb/>quadrati ſuperficiales pro vnoquo-<lb/>que ordine, & <reg norm="quia" type="simple">ꝗa</reg>.b.c. vt. latus qua-<lb/>drati <var>.a.b.</var> habet tot ordines homi-<lb/>num ſimiliter, hoc eſt .60. igi-<lb/>tur multiplicando .60. cum .1260. <lb/>habebimus totalem ſuperficiem <var>.a.<lb/>b.</var> ex .75600. quadratis ſuperficiali-<lb/>bus, quæ quadrata imaginemur lo-<lb/>cata eſſe in quodam totali quadra-<lb/>to, quod ſit <var>.e.f.</var> cuius radix ſit <var>.e.</var> g <num value="275">.<lb/>275.</num> pedum qui diuidantur per .3. <lb/>hoc eſt per numerum pedum latitu-<lb/>dinis & prouenient nobis .91. pro <lb/>numero <reg norm="hominum" type="context">hominũ</reg> <reg norm="vniuſcuiuſque" type="simple">vniuſcuiuſq;</reg> ordi-<lb/>nis, diuidendo poſtea latus <var>.f.g.</var> per <lb/>numerum ſpatij inter vnum, & <reg norm="alium" type="context">aliũ</reg> <lb/>ordinem, quod eſt .7. proueniet <lb/>nobis .39. pro numero ordinum.</s> + </p> + <p> + <s xml:space="preserve">Aliter, & breuius etiam poſſumus idem inuenire, hoc eſt multiplicando <reg norm="nume- rum" type="context">nume-<lb/>rũ</reg> <reg norm="propoſitum" type="context">propoſitũ</reg> <reg norm="hominum" type="context">hominũ</reg> <reg norm="cum" type="context">cũ</reg> rectangulo .21. vnde venietnobis <reg norm="productum" type="simple context">ꝓductũ</reg> .75600 quod pro <lb/><reg norm="ductum" type="context">ductũ</reg> ſi accipiemus vt <reg norm="quadratum" type="context">quadratũ</reg>, cuius radix erit .275. quæ diuidatur <reg norm="per" type="simple">ꝑ</reg> .3. habebi-<lb/>mus <reg norm="propofitum" type="simple context">ꝓpofitũ</reg>. </s> + <s xml:space="preserve">Cuius ratio <reg norm="pendet" type="context">pẽdet</reg> à ſupradicta, eo <reg norm="quod" type="wordlist">qđ</reg> loco <reg norm="multiplicandi" type="context">multiplicãdi</reg> <var>.a.c.</var> (hoc eſt <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">productũ</reg> etiam multiplicare per <var>.b.c.</var> (hoc eſt .60.) breuius erit <lb/>multiplicare totum numerum .3600. per .21. cętera verò facere, vt diximus.</s> + </p> + <p> + <s xml:space="preserve">Sed <reg norm="vnaquæque" type="simple">vnaquæq;</reg> iſtarum operationum, aliquid imperfectionis patitur, eo quod <reg norm="cum" type="context">cũ</reg> <lb/>aliquis cuperet quadratum perfectum ſuperficiale habere, <reg norm="abſque" type="simple">abſq;</reg> aliquo defectu, vel <lb/>exceſfu, aliquid aliud adhuc facere oporteret, hoc eſt, inuentum cum fuerit quadra <lb/>tum <var>.e.f.</var> cum ſuis radicibus <var>.e.g.</var> et <var>.g.f.</var> pedum .275. vnaquaque, vt in dicto exemplo <lb/>factum eſt, oportebit <reg norm="numerum" type="context">numerũ</reg> quærere minorem ipſo .275. ſed proximiorem men-<lb/>furabilem ab .3. & ab .7. quod facilè fiet ſi diuiſerimus .275. per .21. detrahendo fra-<lb/>cta diuiſionis ab ipſo .275. quæ quidem fracta in hoc exemplo ſunt .2. vnde remane-<lb/>bit .273. pro numero laterum quadrati ſuperficialis, in quo poſſent locari .3549. ho-<lb/>mines, eo ordine quo ſupra dictum eſt, quorum ſcilicet vnuſquiſque obtineat .21. <lb/>pedes ſuperſiciales.</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 Muſico celeberrimo.</head> + <p> + <s xml:space="preserve"><emph style="sc">OPinio</emph> Hectoris Euſonij Cypriane mi dilectiſſime, vera non eſt, quod ali <lb/>quisrectè poſſit intelligere rationes conſonantiarum muſicæ, ablque co <lb/>gnitione illarum mediante ipſo ſenſu, imo nemo <reg norm="pont" type="context">põt</reg> calere <reg norm="theoriam" type="context">theoriã</reg> mu <lb/>ſices, niſi aliquo <reg norm="mon" type="context">mõ</reg> verſatus ſit in praxi. </s> + <s xml:space="preserve"><reg norm="Quando" type="wordlist">Qũo</reg> enim cognoſci <reg norm="poterunt" type="context">poterũt</reg> <lb/>quid nam ſint diapaſon, diapente, diateſſeron, ditonus, ſemiditonus, hexacordum <lb/>maius, aut minus, & conſonantiæ ex ijs cum diapaſon compoſitæ, abſque earum <lb/>praxi? </s> + <s xml:space="preserve">vnde ſequetur <reg norm="neque" type="simple">neq;</reg> etiam cognoſci poſſe interualla diſſonantia. </s> + <s xml:space="preserve">Et purus <lb/>practicus non intelliget quid ſit octaua, quinta, quarta, tertia maior, tertia minor, <lb/>ſexta maior, ſexta minor, decima maior, decima minor, vndecima, duode-<lb/>cima, decimatertia maior, aut minor, aut decimaquinta, & aliæ, ita vt ad <lb/>comparandam perfectionem muſicæ neceſſarium ſit, & thęoriam & praxim ad-<lb/>diſcere. </s> + <s xml:space="preserve">Cum pręterea Ludouicus Folianus apertè monſtrarit (etiam ſi id à diato-<lb/>nico ſintono Ptolomei deſumpſerit) reperiri duos tonos, maiorem, & minorem, id-<lb/>eſt ſeſquioctauum, & ſeſquinonum, & tria ſemitonia, maius, minus, & mini-<lb/>mum, ideſt ſeſquiquintumdecimum, qui eſt maius, ſeſquiuigeſimum quartum id-<lb/>eſt minimum, & mediocre, vt .27. ad .25. quæ proportio ſuperbipartiens vigeſi-<lb/>maſquintas appellatur, & cum cognouerit ſemiditonum conſonantem eſſe ſeſqui-<lb/>quintum, ditonum ſeſquiquartum, & hexachordum minus, vt .8. ad .5. quæ propor-<lb/>tio dicitur ſupertripartiens quintas, & hexachordum maius, vt .5. ad .3. hęc autem vo <lb/>catur ſuperbipartienstertias; </s> + <s xml:space="preserve">omnium ſimplicium conſonantiarum cognitioni, ex-<lb/>tremam impoſuit manum. </s> + <s xml:space="preserve">Et quia tibi etiam oſtendere promiſi 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 ſeruari, ideo ad te mitto ſeptem hic ſubſcripta exempla, in <lb/>quorum primo, & ſecundo, inter dieſim, et <var>.b.</var> in ſuperiori, agnoſces interuallum mi <lb/>nimi ſemitonij, & ſi ibi ſit dieſis, tanquam terminus ad quem, et <var>.b.</var> tanquam termi-<lb/>nus à quo: </s> + <s xml:space="preserve">quod autem inter dieſim et <var>.b.</var> ſit ſemitonium minimum, facilè agnoſces <lb/>ſi ſubtraxeris <reg norm="decimam" type="context">decimã</reg> <reg norm="minorem" type="context">minorẽ</reg> à maiori, <reg norm="quam" type="context">quã</reg> facit ſuperius <reg norm="cum" type="context">cũ</reg> inferiori, ideſt <reg norm="cum" type="context">cũ</reg> baſſu.</s> + </p> + <p> + <s xml:space="preserve">Qua quidem modulatione tu etiam vſus 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 ſermo-<lb/>ne compoſitis, quæ Moreta vocantur aliquando vſus ſum.</s> + </p> + <p> + <s xml:space="preserve">Sed in tertio exemplo inuenies ſemitonium maius, neceſſariò genitum in ſupe-<lb/>riori, ſi ſextam maiorem cum baſſu eſſicere volueris, quia tenor, à ditono cum <lb/>ſuperiori ad diapentem, & ad vniſonum cum baſſu procedit, vbi quieſcit, progre-<lb/>diendo poſtea baſſus ad ſemiditonum cum tenore, </s> + <s xml:space="preserve">tunc ſi à proportione huius ſep-<lb/>timæ, quæ eſt vt .9. ad .5. hoc eſt ſuperquadripartiensquintas demptum fuerit hexa-<lb/>chordum maius, ſeu ſexta maior, quæ eſt vt .5. ad .3. remanebit proportio .27. ad .25. <lb/>quæ maior eſt quam .32. ad .30.</s> + </p> + <p> + <s xml:space="preserve">In quarto <reg norm="exemplo" type="context">exẽplo</reg> habebis ſemitonium minus in ſuperiori, quod quidem remanet <lb/>ex ſubtractione ditoni <reg norm="conſonantis" type="context context">cõſonãtis</reg> ab diateſſaron <reg norm="compręhenſa" type="context">cõpręhenſa</reg> à ſuperiori cum tenore.</s> + </p> + <p> + <s xml:space="preserve">In quinto exemplo videbis tonum minorem, & tonum maiorem ſucceſſiuè vnum <lb/>poſt alium in tenore, detrahendo primo <reg norm="ſemiditonum" type="context">ſemiditonũ</reg> à diateſſaron, quod ſuperius fa-<lb/>cit cum tenore, vel detrahendo diapente ab hexachordo maiori, quod facit tenor <lb/>cum baſſu, vnde remanet tonus minor ſeſquinonus, detrahendo poſtea diateſſaron <lb/>à diapente, quod ſuperius facit cum tenore, remanebit tonus maior ſeſquioctauus.</s> + </p> + <p> + <s xml:space="preserve">In ſexto exemplo deinde videbis tenorem aſcendere per duos tonos minores ſuc <lb/>ceſſiuè vnum poſt alium in tenore, ſi <reg norm="demp" type="context">dẽp</reg> ſeris <reg norm="ſemiditonum" type="context">ſemiditonũ</reg> à diateſſaron <reg norm="cum" type="context">cũ</reg> ſuperiori.</s> + </p> + <p> + <s xml:space="preserve">In .7. <reg norm="exemplo" type="context">exẽplo</reg> demum videbis <reg norm="ſuperiorem" type="context">ſuperiorẽ</reg> aſcendere per duos tonos maiores ſucceſ-<lb/>ſiuè <reg norm="vnum" type="context">vnũ</reg> poſt <reg norm="alium" type="context">aliũ</reg>, ſi dempſeris diateſſaron à diapente, quod facittenor <reg norm="cum" type="context">cũ</reg> ſuperiori.</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 ſubiecto.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod aliàs tibi dixi, verum eſt, quod neceſſarium nullo modo ſit, vt modulan-<lb/>do, deſinat cantilena in eodem tono (quod Græci phthongum appel-<lb/>lant) à quo incępit. </s> + <s xml:space="preserve">immo neceſſariò ſemper ferè, altius, aut depræſ-<lb/>ſius terminatur, per differentiam alicuius interualli æqualis, vel multiplicis ipſi com <lb/>mati ſeſquioctuageſimæ, quod quidem comma, quamuis cantabile non ſit, inſenſi-<lb/>biliter tamen generatur, & toties ab aliqua parte ipſius cantilenæ poſſet <reg norm="dictum" type="context">dictũ</reg> com-<lb/>ma gcnerari, verſus acutum, vel graue, quod in fine ipſius cantilenę, vocis phtongus <lb/>reperiatur diſtans à primo per interuallum alicuius toni ſeſquinoni, ſeu ſeſquioctaui <lb/>plus, minúsue, vt in ſubſcripto exemplo clarè videre potes in prima figura, vbi ſu-<lb/>perius à <var>.g.</var> primę cellulæ ad <var>.g.</var> ſecundæ, intereſt vnum <reg norm="comma" type="context">cõma</reg>, eo quod progrediens <lb/>ſuperius in prima cellula ipſius cantilenæ à quarta ad quintam cum tenore, aſcendit <lb/>per tonum ſeſquioctauum, à prima cellula deinde ad ſecundam, tenor aſcendit ſimi-<lb/>liter per tonum ſeſquioctauum cum tranſeat à quinta ad quartam, quod facit cum <lb/>ſuperiori, in ſecunda cellula poſtea, cum ſuperius deſcendat à maiori ſexta ad quin <lb/>tam, quod facit cum baſſu, ſeu à quarta ad tertiam minorem, quod facit cum teno-<lb/>re, </s> + <s xml:space="preserve">tunc deſcendit per tonum ſeſquinonum, 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, ſed reperitur per <reg norm="vnum" type="context">vnũ</reg> <reg norm="coomma" type="context">coõma</reg> altius, <reg norm="quod" type="wordlist">qđ</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/><reg norm="comma" type="context">cõma</reg> eſt differentia inter <reg norm="tonum" type="context">tonũ</reg> <reg norm="ſeſquioctauum" type="context">ſeſquioctauũ</reg> & <reg norm="ſeſquinonum" type="context">ſeſquinonũ</reg>, vt alias tibi <reg norm="demonſtraui" 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 tranſeat à tertia minori ad quartam, quod facit cum ſuperiori, deſcendit per <lb/>tonum ſeſquinonum, vnde in tertia cellula altius remanet quam in prima per <reg norm="vnum" type="context">vnũ</reg> <lb/>comma, in qua tertia cellula, cum iterum tranſeat ſuperius à quarta ad quintam, <reg norm="quod" type="simple">ꝙ</reg> <lb/>facit cum tenore, eleuatur per tonum ſeſquioctauum, proſequendo deinde tali ordi <lb/>ne, vidc<unsure/>bis in quarta cellula cantilenam auctam per duo commata, in ſexta, <reg norm="autem" type="wordlist">aũt</reg> cel-<lb/>lula per tria commata, in octaua verò per .4. commata, vnde hac merhodo, ſi can-<lb/>tilena prolixior debito eſſet, vel ſi 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 ſuperant tonum ſeſquinonum, & ſi eſſent .10. commata<unsure/> ſuperarent tonum <lb/>ſeſquioctauum, eo quod aggregatum ex .9. commatibus continetur ſub iſtis duobus <lb/>terminis hoc eſt .150094635296999121. et .134217728000000000. quæ qui-<lb/>dem proportio maior eſt proportione ſeſquinona, ſumma verò .10. commatum con <lb/>tinetur ſub .12157665459056928801. et .10737418240000000000. quæ pro <lb/>portio maior eſt tono ſeſquioctauo, quod autem dico de aſcenſu cantilenæ, idem aſ-<lb/>ſero de eiuſdem deſcenſu, & hoc non tantum per interuallum illius commatis, quod <lb/>eſt differentia toni maioris à minori, ſed etiam per illud quod eſt differentia ſemito <lb/>nij maioris à minori, vt in ſecundo exemplo hic ſubſcripto videre eſt in deſcenſu <lb/>cantilenæ per comma & comma, vt differentia inter ſemitonia maiora<unsure/> & minora, <lb/>vbi in prima cellula diſcedens baſſus à quinta cum ſuperiori, & ab vniſono cum te-<lb/>nore deſcendens ad tertiam minorem cum ipſo tenore, facit cum ſuperiori <reg norm="ſeptimam" type="context">ſeptimã</reg> <lb/>maiorem, quæ eſt vt .9. ad .5. ſuperquadripartiensquintas ſcilicet, à qua diſcedens <lb/>poſtea ſuperius, vt faciat cum baſſu ſextam maiorem, deſcendit per ſemitonium ma <lb/>ius, à qua ſexta maiori deſcendens baſſus, & aſcendens per quartam, efficit cum di-<lb/>cto ſuperiori <reg norm="tertiam" type="context">tertiã</reg> maiorem, à qua diſcedens ſuperius, vt efficiat quartam cum ipſo <lb/>baſſu (qui quidem baſſus tranſit in tenorem) aſcendit per ſemitonium 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 deſcendit, & ſic toties facere poſſet, vt poſtremo 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 ſupradictum eſt, clrca inſtrumenta artificialia non accidit, qua <lb/>propter organa, & clauicimbula concordantur certo quodam ordine, ita vt omnes <lb/>conſonantiæ, excepta diapaſon, ſeu octaua, ſint imperfectæ, hoc eſt, aut diminutę, <lb/>aut ſuperantes à inſto, vt exempli gratia, omnes quintæ ſunt diminutæ, quartæ verò <lb/>ſunt exceſſiuę, quod quidem fit, vt tertiæ, & ſextæ, non multum auribus diſſonent, <lb/>eo quod ſi quintæ omnes, & quartæ, perfectæ eſſent, </s> + <s xml:space="preserve">tunc omnes ſextę, & tertiæ in-<lb/>tollerabiles eſſent, & à perfectis differrent per vnum comma, quod manifeſtum no-<lb/>bis erit hoc modo, accipiamus tres diapentes, ſeu quintas, conſequenter ſucceſſiuas <lb/>vnam poſt aliam, hoc eſt tres proportiones ſeſquialteras, quarum aggregatum erit <lb/>vt .27. ad .8. quæ proportio, dicitur tripla ſupertripartiensoctauas, & 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 ſecundo ela <lb/>mi, </s> + <s xml:space="preserve">tunc talis tertiadecima valde odioſa eſſet ſenſui auditus, à qua, ſi dempta ſuerit <lb/>diapaſon, ſeu octaua, remaneret quoddam hexachordum maius, ſeu ſexta maior, au-<lb/>ribus valde inimica, ſub proportione .13. ad 8. ſed hæc proportio differret à propor <lb/>tione ſuperbipartientetertias perfecti hexachordi maioris, hoc eſt ſextæ maioris <lb/>conſonantis, per proportionem ſeſquioctuageſimam, hoc eſt per vnum comma, <lb/>quod quidem eſt etiam differentia aggregati trium ſeſquialterarum, à tertiadeci-<lb/>ma maiori conſonanti, hoc eſt exceſſus proportionis triplæ ſupertripartientis octa-<lb/>uas, ſupra triplam ſeſquitertiam, quæ eſt ſumma ipſius duplæ cum ſuperbi partien-<lb/>tetertias.</s> + </p> + <p> + <s xml:space="preserve">A tali ſumma igitur trium ſeſquialterarum efficitur tertiadecima maior diſſonans <lb/>excedens conſonantem per vnum comma (cuius proportio eſt .81. ad .80.) quæ con-<lb/>fonans continetur in proportione .10. ad .3. vt ſupra dixi.</s> + </p> + <p> + <s xml:space="preserve">Hæcigitur eſt vera ratio, propter quam debemus comma diſtribuere in organis <lb/>& clauicymbalis, cum ab aggregato trium quintarum producatur talis exceſſus ſu-<lb/>pra perfectam, ſeu conſonantem tertiamdecimam maiorem, quod quidem aggre-<lb/>gatum, cum demptum fuerit à quintadecima, relinquet nobis tertiam minorem <lb/>diſſonantem, & mancam, per eundem exceſſum à conſonanti. </s> + <s xml:space="preserve">quæ quidem tertia <lb/>minor diſſonans ſubtracta à diapente ſeu quinta perfecta, relinquet nobis tertiam <lb/>maiorem diſſonantem, qu@ conſonantem excedit per eundem exceſſum comma-<lb/>tis, & hæc demum tertia maior diſſonans, dempta ex diapaſon, ſeu octaua, relin-<lb/>quet nobis hexachordum minus, hoc eſt ſextam minorem diſſonantem, & muti-<lb/>lam à conſonanti per eundem exceſſum commatis. </s> + <s xml:space="preserve">De huiuſmodi verò commatis <lb/>diſtributione doctiſſimè ſcripſit Excellentiſſimus Zarlinus in ſecunda parte Inſtitu-<lb/>tionum Harmonicarum.</s> + </p> + <p> + <s xml:space="preserve">Sed quia ſenſus auditus non poteſt exactè cognoſcere debitam quantitatem ex-<lb/>ceſſus, vel defectus, intendendo vel remittendo chordas inſtrumentorum, ideo hanc <lb/>viam ſequutus ſum.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia, hic ſubſcriptus ordo lignorum tangentium ſeu pinarum inci-<lb/>piens ab <var>.G.</var> deſinens ad <var>.g.</var> ita quod inter ipſos terminos ſit ea conſonantia quæ vo-<lb/>catur vigeſimaſecunda, quæro primum <var>.b.</var> inter <var>.D.E.</var> quod eſt nigrum ipſius Ela-<lb/>mi grauiſſimum, quod groſſo modo facio conſonans cum <var>.G.</var> grauiſſimo per ſex-<lb/>tam <reg norm="minorem" type="context">minorẽ</reg>, deinde <reg norm="cum" type="context">cũ</reg> ipſo pt<unsure/>imo <var>.b.</var> ipſius elami concordo ſuum octauum & quin-<lb/>tumdecimum, quo perfectius poſſum, deinde accipio <var>.b.</var> molle ſecundum ipſius. b <lb/>fabmi quod concordo cum <var>.b.</var> primo ipſius Elami per quintam imperfectam, dein-<lb/>de cum hoc <var>.b.</var> ſecundo ipſius bſabmi concordo ſecundum <var>.f.</var> per quintam ſimiliter <lb/>imperfectam, cum quo <var>.f.</var> poſtea concordo tertium <var>.c.</var> per ſimilem quintam, quem <lb/>tertium <var>.c.</var> poſtea confero <reg norm="cum" type="context">cũ</reg> ſecundo <var>.b.</var> ipſius elami, ita quod inter ſe conſonent per <lb/>ſextam maiorem tolerabilem, & ſi ſic 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"/> + eſt <var>.b.</var> ſecundum ipſius bfabmi, f. ſecundum, et <var>.c.</var> tertium, ſed ſi dictum tertium <var>.c.</var> <lb/>valde diſſonans eſſet cum <var>.b.</var> ſecundo ipſius elami, </s> + <s xml:space="preserve">tunc ipſum <var>.c.</var> intendo, aut re-<lb/>mitto, quouſque aliquo modo ſit conſonans per ſextam maiorem aliquantulum ex <lb/>ceſſiuam cum <var>.b.</var> ſecundo ipſius elami, cum quo poſtea <var>.c.</var> conſonare aliquantulum fa <lb/>cio <var>.f.</var> ſecundum per quintam defectiuam, & cum hoc<unsure/> demum <var>.b.</var> ſecundum ipſius <lb/>bfabmi, quo facto concordo ſecundum <var>.c.</var> cum tertio per octauam, cum quo ſecun-<lb/>do <var>.c.</var> poſtea concordo tertium <var>.g.</var> per talem quintam, quod ipſum tertium <var>.g.</var> cum ſe-<lb/>cundo <var>.b.</var> ipſius bfabmi conſonet tolerabiliter per ſextam maiorem <reg norm="aliquantulum" type="context">aliquãtulum</reg> ex-<lb/>ceſſiuam, </s> + <s xml:space="preserve">deinde cum iſto tertio <var>.g.</var> concordo tertium <var>.d.</var> per talem quintam, ita quod <lb/>ipſum <var>.3.d.</var> concordet tolerabiliter cum <var>.2.f.</var> per ſextam maiorem exceſſiuam, poſtea <lb/>cum hoc <var>.3.d.</var> concordo <var>.2.d.</var> per octauam perfecte, cum quo <var>.2.d.</var> poſtea concordo <var>.<lb/>3.a.</var> per quintam, vt in alijs <reg norm="factum" type="context">factũ</reg> eſt, ita vt <reg norm="cum" type="context">cũ</reg> <var>.2.c.</var> conſonet talis ſexta maior, vt ſupra <lb/>dictum eſt, cum quo <var>.3.a.</var> poſtea concordo <var>.3.e.</var> per quintam, vt dictum eſt, ita quod <lb/>cum <var>.3.g.</var> faciat ſextam maiorem vt ſupra, poſtea 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 eſt, ita quod <reg norm="cum" type="context">cũ</reg> <lb/><var>2.d.</var> faciat ſextam maiorem ſimilem alijs ſuperius dictis, cum quo <var>.b.</var> quadrato tertio <lb/>concordo tertium nigrum ipſius <var>.f.</var> per quintam, ita quod cum <var>.3 a.</var> faciat ſextam ma-<lb/>iorem, vt ſupra, 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 ſextam dictam, demum <reg norm="cum" type="context">cũ</reg> <lb/>hoc concordo <var>.4.g.</var> nigrum per quintam, ita quod faciat cum <var>.3.b.</var> quadrato ſextam <lb/>dictam, & ſic ad vltimam quintam peruenio, ſupra quod <var>.g.</var> nigrum nulla quinta am-<lb/>plius reperitur, poſtea cum iſtis 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 eſt, quod perſectiores quæque conſonan <lb/>tiæ, ita in harmonica diuiſione ſibi inuicem conueniant, vt dia paſon cum diapente, <lb/>cum diapaſondiapente, cum ditono, cum hexachordo maiori cum bisdiapaſon, <reg norm="cum" type="context">cũ</reg> <lb/>decimaleptima maiori. </s> + <s xml:space="preserve">Nam in ipſa diapaſon, harmonicè <reg norm="locantur" type="context">locãtur</reg> diapente in par <lb/>te grauiori, & diateſlaron in acutiori. </s> + <s xml:space="preserve">In diapente verò harmonicè locantur ditonus <lb/>in parte grauiori, & ſemiditonus 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 diapaſondia-<lb/>pente, harmonic<unsure/>è locantur diapaſon in parte grauiori, & diapente in acutiori. <lb/></s> + <s xml:space="preserve">In bisdiapaſon, ha@monicè locantur decima maior in parte grauiori & hexachor-<lb/>dum minus in acutiori. </s> + <s xml:space="preserve">In decimaſeptima maiori, harmonicè locantur diapaſondia-<lb/>pente in parte grauiori, & hexachordum maius in parte acutiori. </s> + <s xml:space="preserve">Ita quod ronus <lb/>ſeſquioctauus in ditono, proportionalis eſt ipſi ditono in diapente. </s> + <s xml:space="preserve">Tonus verò ſeſ-<lb/>quinonus in ipſo ditono, proportionalis eſt triemitonio, vel ſeſquitonio ſeu ſemidi-<lb/>tono (quod idem eſt) in diapente. </s> + <s xml:space="preserve">Ditonus autem in diapente, proportionalis eſt <lb/>ipſi diapente in diapaſon. </s> + <s xml:space="preserve">Seſquitonus verò in diapente, proportionalis eſt diateſ-<lb/>ſaron in diapaſon. </s> + <s xml:space="preserve">Et ſic de ſingulis. </s> + <s xml:space="preserve">Ita quod tonus ſeſquioctauus in ditono, dito-<lb/>mus in diapente, diateſſaron in hexachordo maiori, diapente in diapaſon, diapaſon <lb/>in diapaſondiapente, decimamaior in bisdiapaſon, diapaſondiapente 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"/> + ſeptima maiori, omnia ſibi inuicem ſunt proportionalia, idem etiam dico de reli-<lb/>quis partibus, cum relatæ fuerint ad ſua tota.</s> + </p> + <p> + <s xml:space="preserve">Nec alienum mihi videtur à propoſito inſtituto, ſpeculari modum generationis <lb/>ipſarum ſimplicium <reg norm="conſonantiarum" type="context">conſonantiarũ</reg>; </s> + <s xml:space="preserve">qui quidem modus fit ex quadam æquatione per <lb/>@uſſionum, ſeu æquali concurſu vndarum aeris, vel conterminatione earum.</s> + </p> + <p> + <s xml:space="preserve">Nam, nulli dubium eſt, quin vniſonus ſit prima principalis <reg norm="audituque" type="simple">audituq́</reg> amiciſſima, <lb/>nec non magis propria conſonantia; </s> + <s xml:space="preserve">& ſi intelligatur, vt punctus in linea, vel vnitas <lb/>in numero, quam immediate ſequitur diapaſon, ei ſimillima, poſt hanc verò diapen <lb/>te, <reg norm="cæteræque" type="simple">cæteræq́;</reg>. </s> + <s xml:space="preserve">Videamus igitur ordinem concurſus percuſſionum terminorum, ſeu <lb/>vndarum aeris, vnde ſonus generatur.</s> + </p> + <p> + <s xml:space="preserve">Concipiatur igitur mente monochordus, hoc eſt chorda diſtenta, quæ cum diuiſa <lb/>fuerit in duas æquales partes à ponticulo, </s> + <s xml:space="preserve">tunc <reg norm="vnaquæque" type="simple">vnaquæq;</reg> pars eundem ſonum pro-<lb/>feret, & ambæ formabunt vniſonum, quia eodem tempore, tot percuſſiones in aere <lb/>faciet vna partium illius chordæ, quot & altera: </s> + <s xml:space="preserve">ita vt vndæ aeris ſimul eant, & æqua <lb/>liter concurrant, abſque ulla interſectione, vel fractione illarum inuicem.</s> + </p> + <p> + <s xml:space="preserve">Sed cum ponticulus ita diuiſerit chordam, vt relicta ſit eius tertia pars ab vno la-<lb/>tere, ab alio vero, duę tertię, </s> + <s xml:space="preserve">tunc maior pars, dupla erit minori, & <reg norm="ſonabunt" type="context">ſonabũt</reg> ipſam dia <lb/>paſon conſonantiam, percuſſiones vero terminorum ipſius, tali proportione ſe inui-<lb/>cem habebunt, ut in qualibet ſecunda percuſſione minoris portionis ipſius chordæ, <lb/>maior percutiet, ſeu concurret cum minori, eodem temporis inſtanti, cum ne-<lb/>mo ſit qui neſciat, quod quo longior eſt chorda, etiam tardius moueatur, </s> + <s xml:space="preserve">quare <lb/>cum longior dupla ſit breuiori, & eiuſdem intenſionis 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 diuiſerit chordam, ut ab uno latere relinquantur duæ <lb/>quintæ partes, ab alio verò tres quintæ, ex quibus partibus generatur conſonantia <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 eſt tempus maioris interualli ad tempus minoris <lb/>erit <reg norm="ſeſquialterum" type="context">ſeſquialterũ</reg> quare non <reg norm="conuenient" type="context">cõuenient</reg> ſimul, niſi 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 ipſius chordæ <lb/>in numerum interuallorum motus ipſius portionis, æquale erit producto numeri <lb/>portionis maioris in numerum interuallorum ipſius maioris portionis; </s> + <s xml:space="preserve">quæ quidem <lb/>producta ita ſe habebunt, vt in diapaſon, ſit binarius numerus; </s> + <s xml:space="preserve">in diapente verò <lb/>ſenarius; </s> + <s xml:space="preserve">in diateſſaron duodenarius, in hexachordo maiori quindenarius; </s> + <s xml:space="preserve">in di-<lb/>tono vicenarius, in ſemiditono tricenarius, demum in hexachordo minori quadra <lb/>genarius: </s> + <s xml:space="preserve">qui quidem numeri non abſque mirabili analogia conueniunt inuicem.</s> + </p> + <p> + <s xml:space="preserve">Voluptas autem, quam auditui afferunt conſonantiæ fit, quia leniuntur ſenſus, <lb/>quemadmodum <reg norm="contra" type="context">cõtra</reg>, dolor qui à diſſonantijs oritur, ab aſperitate naſcitur, id quod <lb/>facilè vide<unsure/>re poteris cum conchordantur organorum fiſtulæ.</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">Franciſco Ferrario Anciſa Iuriſconſulto <reg norm="ſenatorique" type="simple">ſenatoriq́ꝫ</reg> apud <lb/>ſubalpinos 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 comparaſti loquuti ſu <lb/>mus. </s> + <s xml:space="preserve">Cum vero nonnunquam de mirabili iuſtitiæ commutatíuæ inſtitu <lb/>to non ingratus incidiſſet ſermo, dixi modum, quo formam ſuam à proportionali-<lb/>tate arithmetica diſiuncta, & non a coniuncta deſumat, à nemine literis proditum <lb/>eſſe, libet autem nunc per otium latius explicare. </s> + <s xml:space="preserve">dixi enim à diſiuncta, & non con-<lb/>iuncta proportionalitate, quia in coniuncta, ſeu continua nullo pacto fieri poteſt <lb/>talis commutatio, cum ſemper quatuor terminos ad minus tranſeat, vt nunc vide-<lb/>bimus.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, Petrus ex ſuis bonis tribuat Ioanni aliquid valoris quinquagin <lb/>ta aureorum.</s> + </p> + <p> + <s xml:space="preserve">Vnde priuſquam Ioannes aliquid ex ſuis bonis retribuat Petro, bona ipſius Pe-<lb/>tri diminuta erunt per quinquaginta aureos, bona verò ipſius Ioannis, aucta toti-<lb/>dem aureis.</s> + </p> + <p> + <s xml:space="preserve">Ecce nunc quo pacto conftituti ſunt .4. termini in proportionalitate aritmetica, <lb/>per quos ſit talis permutatio, ſed nondum æquata, niſi 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 ſignificentur bona Ioannis <var>.B.</var> vero Petri, prius quam Petrus aliquid ex bo <lb/>nis ſuis tribuat Ioanni. </s> + <s xml:space="preserve">Tunc Petrus ſecat partem vnam ex <var>.B.</var> <reg norm="eamque" type="simple">eamq́;</reg> dat ipſi Ioan-<lb/>ni, vnde ipſi Petro remanet <var>.D</var>. </s> + <s xml:space="preserve">Ioanni autem <var>.C.</var> quatuor igitur termini conſtituti <lb/>ſunt <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">aũt</reg> ſecundus, ſed <lb/>B. et <var>.A.</var> ſunt in ſua naturali mediocritate abſque defectu vel exceſſu ſui ipſius. </s> + <s xml:space="preserve">Non <lb/>ita tamen ſe 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 iſti .4. termini conſtituti ſunt in ipſa aritmetica proportionalitate, nam <lb/>eadem quantitate qua <var>.D.</var> diminuta eſt à <var>.B.</var> eadem <var>.C.</var> aucta eſt ſupra <var>.A</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed quia <var>.B.</var> et <var>.A.</var> tantummodo iuſti ſunt termini <var>.C.</var> uerò et <var>.D.</var> iniuſti, vt ad ſuam <lb/>priorem æqualitatem reuertantur, oportebit ex <var>.C.</var> ſecare aliquam partem æqualis <lb/>valoris ei, qua <var>.C.</var> ſuperat <var>.A.</var> vel qua <var>.D.</var> minor eſt <var>.B.</var> & ipſam partem addere ipſi <var>.D.</var> <lb/>vt bona Petri reuertantur ad priorem ſuam quantitatem ipſius <var>.B.</var> & bona Ioannis <lb/>remaneant æqualia <var>.A.</var> vt prius.</s> + </p> + <p> + <s xml:space="preserve">Quare neceſſarium non eſt, vt talis proportionalitas ſit coniuncta (vt inquit Eu <lb/>f<gap/>atius ſeu Michael Epheſius, <lb/>ſuper 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 ipſa <lb/>diſiuncta ſit, ut diximus, vbi <lb/>non eſt neceſſe quod <var>.A.</var> æqualis ſit <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 ſpacio extra <lb/>Cœlum, Cœ<reg norm="lique" type="simple">liq́;</reg> figura.</head> + <head style="it" xml:space="preserve">Illust. Ioanni Paulo Capra Nouarienſi Sabaudia Ducis boſpicij <lb/>Magistro, viro ingeny praſtantia, & morum cando-<lb/>re, non minus quam familia nobili-<lb/>tate conſpicuo.</head> + <p> + <s xml:space="preserve">SI vera eſſet animorum illa tranſmigratio quam ſibi Italicæ ſapientiæ Pa-<lb/>ter Pythagoras effinxerat, tuam, <reg norm="meamque" type="simple">meamq́;</reg> exiſtimarem 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, ſi ſuper aliquod punctum, quaſi <reg norm="mathematicum" type="context">mathematicũ</reg>, quieſceret, poſſet eſſe perpetuus, <lb/>cum aliquando eſſet mota, ſupponendo etiam eandem eſſe perfectè rotundam, & <lb/>lęuigatam. </s> + <s xml:space="preserve">Reſpondeo huiuſmodi 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/>reſiſtentiam facit ſtringitur, eſt etiam reſiſtentia partium illius corporis moti, quæ <lb/>cum motæ ſunt, natura, impetum habent efficiendi iter directum, vnde cum ſimul <lb/>iunctæ ſint, & earum vna continuata cum alia. </s> + <s xml:space="preserve">dum circulariter mouentur patiuntur <lb/>violentiam, & in huiuſmodi motu per vim vnitæ manent, quia quanto magis mo-<lb/>uentur, tanto magis in ijs creſcit naturalis inclinatio recta eundi, vnde tanto magis <lb/>contra ſuammet naturam voluuntur, ita vt ſecundum naturam quieſcant, quia cum <lb/>eis proprium ſit, quando ſunt motæ, eundi recta, quanto violentius voluuntur, tan-<lb/>to magis vna reſiſtit alteri, & quaſi retrò reuocat eam, quam antea reperitur habere.</s> + </p> + <p> + <s xml:space="preserve">Ab eiuſmodi inclinatione rectitudinis motus partium alicuius corporis rotundi <lb/>fit, vt per aliquod temporis ſpacium, trochus cum magna violentia ſeipſum circun-<lb/>agens, omninò rectus quieſcat ſuper illam cuſpidem ferri quam habet, non incli-<lb/>nans ſe verſus mundi centrum, magis ad vnam <reg norm="partem" type="context">partẽ</reg>, quam ad aliam, cum quælibet <lb/>ſuarum partium in huiuſmodi motu non inclinet omnino verſus <reg norm="mundi" type="context">mũdi</reg> centrum, ſed <lb/>multo magis per tranſuerſum ad angulos rectos cum linea directionis, aut verticali, <lb/>aut orizontis axe, ita vt neceſſariò huiuſmodi corpus rectum ſtare debeat. </s> + <s xml:space="preserve">Et quod <lb/>dico ipſas partes non omninò inclinare verſus mundi centrum, id ea ratione dico, <lb/>quia non abſolutè ſunt unquam priuatæ huiuſmodi inclinatione, quę efficit vt ipſum <lb/>corpus eo puncto nitatur. </s> + <s xml:space="preserve">Verum tamen eſt, quod quanto magis eſt velox, tan-<lb/>to minus premit ipſum punctum, imò ipſum corpus <reg norm="tanto" type="context">tãto</reg> magis leue remanet. </s> + <s xml:space="preserve">Id <reg norm="quod" type="wordlist">qđ</reg> <lb/>apertè patet <reg norm="ſumendo" type="context">ſumẽdo</reg> <reg norm="exemplum" type="context context">exẽplũ</reg> pilę alicuius arcus, aut <reg norm="alicuius" type="simple">alicuiꝰ</reg> alterius <reg norm="inſtrumenti" type="context">inſtrumẽti</reg>, ſeu ma <lb/>chinæ miſſilis, quæ pila quanto eſt velocior, in motu violento, tanto maiorem pro-<lb/>penſionem habet rectius eundi, vnde verſus mundi centrum tanto minus inclinat, <lb/>& hanc ob cauſam leuior redditur. </s> + <s xml:space="preserve">Sed ſi clarius, hanc veritatem videre cupis, <lb/>cogita illud corpus, Trochum ſcilicet, dum velociſſime circunducitur ſecari, ſeu <lb/>diuidi in multas partes, vnde uidebis illas omnes, non illico uerſus 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"/> + deſcendere, ſed recta orizontaliter, vt ita dicam, moueri. </s> + <s xml:space="preserve">Id quod à nemine ad-<lb/>huc (quod ſciam) in trocho eſt obſeruatum. </s> + <s xml:space="preserve">Ab huiuſmodi motu trochi, aut hu-<lb/>ius generis corporis, clarè perſpicitur, quàm errent peripatetici circa motum uio-<lb/>lentum alicuius corporis, qui exiſtimant aerem qui ſubintrat ad occupandum locum <lb/>à corpore relictum, ipſum corpus impellere, cum ab hoc, magis effectus contrarius <lb/>naſcatur.</s> + </p> + <p> + <s xml:space="preserve">Quod deinde ampullæ iungantur in aqua, non fit ratione ſimpathiæ, de qua lo-<lb/>quitur Fracaſtorus, nam per accidens iunguntur, quia cum alia ad aliam accedit, quę <lb/>libet earum tentat aſcendere ab ea parte, à qua inuicem hærent, quemadmodum <lb/>efficiunt iuxta labrum vaſis, ea enim ſuperficies a quæ vicina circunferentiæ vaſis ali <lb/>quantulum aſcendit in vaſe, qui non eſt omnino plenus.</s> + </p> + <p> + <s xml:space="preserve">Ad id deinde quod de claritate noctium ſcribis, miror cur non videas, quod <reg norm="quam" type="context">quã</reg> <lb/>to magis obſcura nox apparet, non dico ratione nubium, ſed diſtantiæ Solis ſub <lb/>orizonte ab eodem orizonte, tanto magis claram, & luminoſam ſeſe nobis oſten-<lb/>dit Luna in quintadecima, quia cum Sol eſt in Sagittario, & Capricorno, Luna eſt <lb/>in Geminis, & in Cancro, vnde in media nocte, eius radius per valde exiguam <reg norm="quam" type="context">quã</reg> <lb/>titatem vaporum tranſit, quia tunc ipſa eſt valde propinqua axi orizontis, & præ-<lb/>terea in huiuſmodi tempore anni & noctis, aer eſt magis purgatus, quàm in qualibet <lb/>alia temporis parte, quia hieme Sol non poteſt excitare multos vapores, & ij, qui <lb/>at tolluntur, nocte à frigore ſtatim congellati ratione grauitatis <reg norm="decidunt" type="context">decidũt</reg>, </s> + <s xml:space="preserve">unde rema-<lb/>net aer multo clarior, qua ratione apparent ſtellæ minutæ, & Cœlum ijſdem ma-<lb/>gis ornatum, quàm in quolibet alio anni tempore.</s> + </p> + <p> + <s xml:space="preserve">Dicere deinde, quemadmodum hic mundus eſt ætatis ſeptem, aut octomillium <lb/>annorum, ita nunc potuiſſet eſſe (ſi Deus voluiſſet) ætatis quinquagintamillium; </s> + <s xml:space="preserve">er <lb/>go erat tempus; </s> + <s xml:space="preserve">ita ſe habet, ac ſi diceremus, quemadmodum hic mundus eſt tan-<lb/>tæ magnitudinis, ita etiam quinquagies maior eſſe potuiſſet, ergo eſt ſpatium, aut <lb/>interuallum corporeum, quod eum capere potuiſſet.</s> + </p> + <p> + <s xml:space="preserve">Illud, nihil, Ariſtotelis extra Cęlum, nullo modo nobis inſeruit pro eiuſdem Cœ <lb/>li ſphęrica rotunditate, cum cuiuſque alterius ex infinitis figuris Cęlum ipſum eſſe <lb/>poſſit, ſecundum ſuam ſuperficiem conuexam. </s> + <s xml:space="preserve">Nam Cœlum ea ratione ſphęricum <lb/>non eſt, quod magis ſit capax, quia ei innumerahiles alias figuras adeo magnas po <lb/>terat concedere cauſa diuina: </s> + <s xml:space="preserve">ſed ſphæricum eſt effectum, ne partem aliquam habe <lb/>ret ſui termini ſuperfluam, quia nullum corpus à breuiori termino quam à ſphærico <lb/>terminari poteſt.</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 appenſa eſt ſitula, longè facilius axi inuoluitur, ſi ipſi axi affixa ſit 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, ſi ipſa rotæ extrema circunferentia, ex materia minori, <lb/>& denſiori, ac proinde grauiori conſtabit. </s> + <s xml:space="preserve">Cuius rei ratio multiplex eſt. </s> + <s xml:space="preserve">Nem-<lb/>pe quia omne corpus graue, aut ſui natura, aut vi motum, in ſe 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 ſeparatum à virtute mouente per aliquod temporis <lb/>ſpatium ex ſeipſo moueatur. </s> + <s xml:space="preserve">nam ſi ſecundum naturam motu cieatur, ſuam veloci-<lb/>tatem ſemper augebit, cum in eo, impetus & impreſſio ſemper 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">eaq́;</reg> eam remouendo rota ſtatim non quieſcet, ſed per aliquod temporis ſpatium <lb/>circunuertetur.</s> + </p> + <p> + <s xml:space="preserve">Secunda cauſa eſt, quia quoduis gr aue corpus, aut per naturam, aut per vim mo-<lb/>tum, rectitudinem itineris naturaliter appetat, quod clarè cognoſcere 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 inſito 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 ipſius rotæ, tanto facilius cir-<lb/>cunuoluatur, & ex ſeipſo tanto longiori tempore moueatur, quanto longius diſtat à <lb/>centro, cum eius iter tanto minus ſit curuum. </s> + <s xml:space="preserve">Hanc igitur ob cauſam, rota, quanto <lb/>maior erit, <reg norm="eiuſque" type="simple">eiuſq́;</reg> pondus tanto magis vicinum circunferentiæ, tanto magis durabit <lb/>impetus motus aſſumptus.</s> + </p> + <p> + <s xml:space="preserve">Tertia cauſa eſt, quod funis dum circunuoluitur, vicinius axi mathematico reuo-<lb/>lutionis, quam corpus graue circunferentiæ rotæ, ratione vectis, cum rota eſt in mo <lb/>tu, eius impetus non obtinet reſiſtentiam æqualem à contrario pondere aquæ in ſitu <lb/>la poſitæ.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head style="it" xml:space="preserve">De machina, qua aquam impellit & ſubleuat.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">VNde ſit vt in fonte mandauerim, <lb/> + <anchor type="figure" xlink:label="fig-0299-01a" xlink:href="fig-0299-01"/> + vas ſeu mortarium in quod in-<lb/>greditur inſtrumentum, quod aquam <lb/>impellit, diametrum ſuæ concauitatis, <lb/>habere non oportere maiorem dia-<lb/>metro fiſtulæ, per quam debet aſcende <lb/>re aqua, ratio eſt, quia ſi maius eſſet, <lb/>neceſſarium eſſet aliquod inſtrumen-<lb/>tum quo aqua impelleretur multo gra <lb/>uius toto corpore aqueo, quod aptum <lb/>eſſet implere aliquam fiſtulam adeo <lb/>altam, vt eſt fons, quæ tamen eſſet <lb/>adeo lata vt eſt 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 fiſtula, ſeu <lb/>hirundo, per quam aſcendit aqua <var>.f.</var> <lb/>mortarium verò ſit <var>.a.u.</var> quod tam <reg norm="altum" type="context">altũ</reg> <lb/>ſit vt <var>.f.</var> ſed <var>.f.</var> anguſtior ipſo <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 vaſa, ma-<lb/>nifeſtum erit, quod aqua ipſius <var>.f.</var> ſuffi-<lb/>ciens erit ad <reg norm="reſiſtendum" type="context">reſiſtẽdum</reg> toti aquæ <reg norm="ipſius" type="simple">ipſiꝰ</reg> <lb/><var>a.u.</var> & aqua <var>.a.u.</var> reſiſtet aquæ <var>.f.</var> quam-<lb/>uis aqua <var>.a.u.</var> maioris quantitatis ſit, & <lb/>ponderis ipſa <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 ſuo pondere, </s> + <s xml:space="preserve">propterea quod pondus diuiditur proportionaliter ſupra ba-<lb/>ſim vaſis.</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, ſeu <reg norm="truncus" type="context">trũcus</reg> coni concaui aqua <lb/>plenus, cuius orificij diameter ſit <var>.b.d.</var> & multiplex diametro <var>.m.n.</var> infimæ baſis. </s> + <s xml:space="preserve">co-<lb/>gitemus etiam <var>.b.d.</var> diuiſum in tot partes, quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis ſit <var>.m.n.</var> imagi-<lb/><reg norm="nemurque" type="simple">nemurq́;</reg> tot lineas perpendiculares deſcendere verſus mundi centrum ad puncta <var>r.<lb/>c.m.</var> et <var>.t.x.m.</var> vt in ſubſcripta hic figura videre eſt, per quas cogitemus tot ſuperfi-<lb/>cies curuas <reg norm="conicasque" type="simple">conicasq́;</reg>, inter quas, mente concipienda eſt aqua, quę pondere ſuo quie <lb/>ſcet ſupra maiorem ſuperficiem illa, quæ æque diſtans eſſet mundi centro, ſeu quam <lb/>ſupra baſim <var>.m.n.</var> vt exempli gratia conſideretur aqua inter <var>.g.m.</var> et <var>.s.r.</var> cuius pondus <lb/>diſtribuitur fecundum latitudinem <var>.m.r.</var> quæ maior eſt <var>.g.s.</var> cogitemus igitur <var>.m.c.</var> æ-<lb/>qualem eſſe <var>.g.s.</var> manifeſtum erit, quod <var>.m.c.</var> non ſuſtinebit 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/>ſus mundi centrum, quapropter fundus ſeu baſis <var>.m.n.</var> non ſuſtinet aliud pondus <reg norm="quam" type="context">quã</reg> <lb/>aquæ <var>.f.m.</var> ſed ſi quis hoc in dubium reuocaret dicens, quod aqua circunſcribens ſi-<lb/>tum corporis aquei <var>.f.m.</var> impellit lateraliter dictum corpus aqueum, reſpondendum <lb/>eſt, quod ex æquo huius corporis <var>.f.m.</var> aqua impellit etiam aquam circunſtantem, <lb/>eo, quod ſunt corpora homogenea, cum in corporibus homogeneis æquales partes <lb/>habeant æquales vires.</s> + </p> + <p> + <s xml:space="preserve">Sed redeundo ad vaſa <var>.a.u.</var> et <var>.f.</var> dico quod ſicut aqua <var>.f.</var> ſufficit ad <reg norm="reſiſtendum" type="context">reſiſtendũ</reg> aquæ <lb/><var>a.u.</var> ita quodlibet aliud pondus ęquale <var>.f.</var> cuiuſuis materiæ, in fiſtula <var>.f.</var> poſitum, ſuffi-<lb/>ciens erit, dummodo illud corpus ita ſit adæquatum concauitati fiſtulæ <var>.f.</var> quod non <lb/>permittat tranſitum aliquem aquæ vel <lb/> + <anchor type="figure" xlink:label="fig-0300-01a" xlink:href="fig-0300-01"/> + aeris inter conuexum ipſius corporis, <lb/>& deuexum fiſtulæ <var>.f.</var> & hoc ex ſe ſatis <lb/>patet, ſed in vaſe <var>.a.u.</var> cum ex hypothe <lb/>ſi latius ſit ipſo <var>.f.</var> nullum aliud corpus <lb/>ſufficiens erit ad reſiſtendum aquæ ip-<lb/>ſius <var>.f.</var> quin tam graue ſit, quam tota <lb/>aqua <var>.a.u.</var> exiſtente <var>.a.u.</var> tam alto quam <lb/>f. </s> + <s xml:space="preserve">Vnde ſi aqua ipſius <var>.f.</var> nil plus eſſet <lb/>quam vna tantummodo libra, & vas <var>.a.<lb/>u.</var> exiſteret latius ipſo <var>.f.</var> in decupla pro <lb/>portione, </s> + <s xml:space="preserve">tunc in ipſo <var>.a.u.</var> oporteret <lb/>corpus adæquatum ipſi concauitati po <lb/>nere, cuius pondus eſſet decem libra-<lb/>rum, vt ſufficeret ad ſuſtinendum <reg norm="aquam" type="context">aquã</reg> <lb/>ipſius <var>.f.</var> & ad im<unsure/><reg norm="pellendum" type="context">pellendũ</reg> ipſam <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"/> + denſius eſſe aqua, vt maius <reg norm="interuallum" type="context">interuallũ</reg> <lb/>non occupet, quam <var>.o.e.</var> corpus igitur <lb/><var>o.e.</var> ſufficiens 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 poſtulas eſt problema ab alijs iam ſcriptum, ſed illud tibialio <lb/>medio ſoluam.</s> + </p> + <p> + <s xml:space="preserve">Proponitur vas <reg norm="plenum" type="context">plenũ</reg> liquore aliquo, puta aqua, <reg norm="quod" type="simple">ꝙ</reg> tres habeat fiſtulas <lb/>ad baſim, quarum vnaquæque poſſit euacuare ipſum vas, inæquales ta-<lb/>men, ita quod prima tam lata ſit, vt ſpatio vnius horæ poſſit ipſum euacuare to-<lb/>tum; </s> + <s xml:space="preserve">ſecunda vero ſpatio duarum horarum, tertia autem ſpatio trium hora-<unsure/>-<lb/>rum. </s> + <s xml:space="preserve">Tunc quæritur quanto tempore omnes tres fiſtulæ ſimul apertæ euacua-<lb/>bunt ipſum vas. </s> + <s xml:space="preserve">Ad hoc volo vt quæratur primo quanta pars aquæ vnaquęquę fi-<lb/>ſtula euacuabit in aliquo dato tempore, quod facilè eſt, vt puta, prima fiſtu-<lb/>la, ſpatio dimidiæ horæ euacuabit dimidium vas, eo quod ſpatio integræ horæ po-<lb/>teſt totum euacuare, ſecunda fiſtula, eodem temporis ſpatio, euacuabit quartam <lb/>partem ipſius vaſis, tertia verò fiſtula, eodemmet ſpatio temporis dimidiæ horæ, <lb/>euacuabit ſextam partem ipſius vaſis, quæ omnia fracta ſimul collecta faciunt vnde-<lb/>cim duodecimas partes totius vaſis, vnde manifeſtum erit, quod omnes fiſtulæ pari-<lb/>ter apertæ, ſpatio dimidię horæ euacuabunt vndecim duodecimas partes totius a-<lb/>quæ, ſed nos cupimus ſcire, quanto tempore, totum vas euacuabitur, apertis omni <lb/>bus fiſtulis, quapropter dicemus ita; </s> + <s xml:space="preserve">Si vndecim duodecimæ partes conſumunt mi-<lb/>nuta .30. temporis, quantum conſument omnes partes aquæ? </s> + <s xml:space="preserve">quæ ſunt .12. quare ex <lb/>regula de tribus prouenient nobis minuta .32. cum .8. vndecimis vnius minuti, hoc <lb/>eſt cum .43. ſecundis horæ ferè, vel ſiaccipiemus tres quartas vnius horæ, </s> + <s xml:space="preserve">tunc pri-<lb/>ma fiſtula emittet tres quartas partes totius aquæ, ſecunda, tres octauas <reg norm="eiuſdem" type="context">eiuſdẽ</reg> aquę, <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 abſu-<lb/>mit .45. minuta temporis, ergo illud ſolum integrum abſumet idem vt ſupra hoc eſt <lb/>min .32. cum .8. vndecimis vnius minuti vel .43. ſecundis. </s> + <s xml:space="preserve">Cuius rei ſpeculatio <reg norm="tam" type="context">tã</reg> con <lb/>iuncta eſt operationi, quòd vna cognita, reliqua ſtatim cognoſcitur.</s> + </p> + <p> + <s xml:space="preserve">Idem eueniet de implendo vaſe tribus ſimilibus fiſtulis mediantibus.</s> + </p> + <p> + <s xml:space="preserve">Secundum quæſitum ab alijs traditum, tuum etiam, aliter quoque poteſt ſolui, <lb/>propterea non prętermittam tibi ſatisfacere.</s> + </p> + <p> + <s xml:space="preserve">Problema itaque tale eſt, vt ſit vas aliquod in <reg norm="quod" type="wordlist">qđ</reg> infunditur aqua per tres fiſtu-<lb/>las, ſed dum infunditur aqua, eadem egreditur per duas alias fiſtulas in fundo <lb/>vaſis poſitas, ſed tres ſuperiores ſint inuicem proportionatæ, vt ſupradictum <lb/>eſt, primaq́ue inferiorum talis ſit, vt ſpatio .4. horarum poſſit totum vas euacua-<lb/>re, ſecunda autem poſſit ſpatio .6. horarum idem facere, vnde ex ſupradictis, vas im <lb/>plebitur à tribus fiſtulis ſuperioribus, clauſis exiſtentibus inferioribus, ſpatio tempo <lb/>ris <reg norm="minutorum" type="context">minutorũ</reg> .32. <reg norm="cum" type="context">cũ</reg> .8. vndecimis hoc eſt min .32. cum .43. ſecundis, deinde per duas <lb/>fiſtulas inferiores poſſet euacuari ſpatio <reg norm="temporis" type="context">tẽporis</reg> horarum .2. et. mi .24. exſupradictis.</s> + </p> + <p> + <s xml:space="preserve">Supponamus igitur omnes fiſtulas operari ſpatio temporis minutorum .32. cum <lb/>ſecundis .43. </s> + <s xml:space="preserve">tunc manifeſtum eſt quod vas non implebitur, eo ſpatio min .32. cum <lb/>ſecundis .43. ſed tanta aqua deficiet, quanta ab inferioribus fiſtulis eo ſpatio tempo <lb/>ris min .32. ſecun .43. poteſt euacuari, </s> + <s xml:space="preserve">quare proportio partis vaſis vacuæ, ad totum <lb/>vas, erit vt min .33. ferè ad horas .2. min .24. quod per ſe patet, </s> + <s xml:space="preserve">tunc ſi 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 vaſe 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 ſi .111. indigent mi-<lb/>nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-<lb/>tum vas omnibus fiſtulis 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 aſcendendum ignis, & ad <reg norm="deſcendendum" type="context">deſcendendũ</reg> quicquid graue natum eſt, ita ad <lb/>ſpeculandum humanus intellectus. </s> + <s xml:space="preserve">nec quieſcit, dum poteſt, eſt enim ver-<lb/>ſatile, <reg norm="agitandoque" type="simple">agitandoq́;</reg> ſeſe cauſis rerum immiſcere, & abditum aliquid rimari, <lb/>conatur, & eſt in nobis, quaſi Diogenes quidam in Dolio.</s> + </p> + <p> + <s xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias ſcilicet nouas circuli paſſiones, <lb/>quæ ita ſe <reg norm="habent" type="context">habẽt</reg>. </s> + <s xml:space="preserve">Sit circulus <var>.a.b.c.</var> in quo ſit <var>.a.d.</var> latus quadrati inſcriptibilis in ipſo <lb/>circulo, ct <var>.b.c.</var> ſit 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. ſit ſimiliter <var>.a.f.</var> contingens ipſum circulum in puncto <var>.a.</var> quæ <lb/>protracta ſit vſque ad punctum <var>.f.</var> interſectionis cum diametro protracto, quod ita <lb/>eueniet cum anguli <var>.a.e.f.</var> et <var>.f.a.e.</var> minores ſint duobus rectis, eo quod angulus <var>.f.a.e.</var> <lb/>acutus ſit, cum <var>.a.d.</var> tranſeat 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> ipſius, æqualis erit produ-<lb/>cto ipſius <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">ſed ex .31. eiuſdem angulus <var>.f.a.<lb/>c.</var> æqualis eſt angulo <var>.b</var>. </s> + <s xml:space="preserve">quare æqualis erit angulo <var>.d.a.c.</var> & ita habebimus per .3. ſexti <lb/>eandem proportionem <var>.f.c.</var> ad <var>.c.e.</var> quæ <var>.f.a.</var> ad <var>.a.e.</var> ſed <var>.a.f.</var> eſt æqualis ſemidiametro <lb/>circuli propoſiti, </s> + <s xml:space="preserve">propterea quod ſi producta fuerit à puncto <var>.a.</var> ad centrum <var>.o.</var> ſemi <lb/>diameter <var>.a.o.</var> hæc cum <var>.o.e.</var> faciet dimidium angulirecti, cum ex ſuppoſito <var>.a.d.</var> la-<lb/>tus ſit quadrati inſcriptibilis in ipſo circulo. </s> + <s xml:space="preserve">& cum <var>.a.f.</var> rectum ex .17. tertij, vnde an <lb/>gulus <var>.f.</var> erit ſimiliter medietas recti ex .32. primi, </s> + <s xml:space="preserve">quare ex .6. eiuſdem <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> ſit. vt <var>.f.a.</var> ad <var>.a.e.</var> erit ſimiliter vt <var>.b.c.</var> ad <var>.a.d.</var> <lb/>hoc eſt ut dupli ad duplum, vnde ex .15. ſexti <lb/>manifeſtum erit propoſitum, ex quo alia paſ-<lb/> + <anchor type="figure" xlink:label="fig-0302-01a" xlink:href="fig-0302-01"/> + ſio oritur, hoc eſt, quod productum <var>.f.c.</var> in <var>.a.<lb/>d.</var> æ quale ſit qua drato ipſius <var>.a.c.</var> ratio eſt, quia <lb/>quadratum <var>.a.c.</var> æ quale eſt producto <var>.b.c.</var> in <var>.c.<lb/>e.</var> eo quod <var>.a.c.</var> media proportionalis eſt inter <var>.<lb/>b.c.</var> et <var>.c.e.</var> ex ſimilitudine 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 ſunt <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, ſequitur etiam, quod <var>.a.c.</var> ſit 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> eſt latus octago-<lb/>ni igitur tale latus <reg norm="medium" type="context">mediũ</reg> proportionale erit <lb/>inter latus quadrati. et <var>.f.c.</var> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> circuli, quę <lb/>quidem <var>.f.c.</var> eſt una portio diametri quadrati circunſcriptibilis ipſum circulum inter <lb/>circulum & angulum ipſius 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 ſolarium, non fiat in cen <lb/>tro ſpeculi ſpharici, & aliquid contra Cardanum, & <lb/>de moturadiorum ſolarium.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">ITerum tibi dico, quod radij illi ſolares, quià diuerſis punctis ipſius ſolaris corpo-<lb/>ris veniunt, tranſeuntes per centrum ſpeculi ſphærici concaui, quamuis à ſuper-<lb/>ficie ſpeculi ad centrum ipſum reflectantur, vt alíâs tibi dixi, nihilominus nullo mo <lb/>do poſſunt aliquod obiectum incendere duabus ex cauſis, quarum vna eſt, quia cum <lb/>Sol valde remotus ſit à nobis, valde etiam acutus generatur angulus coni radiorum <lb/>in centro ſpeculi, vnde à parua ſuperficie ipſius ſpeculi reflectuntur, </s> + <s xml:space="preserve">quare pauciſſi-<lb/>mi radij ſunt qui reflectantur in ipſo centro, & propterea non ſufficiunt ad combu <lb/>ſtionem alicuius obiecti. </s> + <s xml:space="preserve">Alia verò cauſa eſt, quod quamuis multi, & ſufficientes <lb/>radij fuiſſentad <reg norm="comburendum" type="context context">cõburendũ</reg> velociter quoduis obiectum. </s> + <s xml:space="preserve">impoſſibile tamen omnino <lb/>eſset, vt aliquod obiectum comburerent, propterea quod cum radij incidentes de-<lb/>beant per centrum tranſire, obiectum combuſtibile, vt opacum, obſtaret ipſis radijs, <lb/>ne vlterius tranſirent, vnde nulla fieret reflexio, ſed etiam ſi dicti radij in centro re <lb/>flexi, ſufficerent ad combuſtionem, incidentes hoc magis eſſicerent. </s> + <s xml:space="preserve">& ita abſque <lb/>vllo ſpeculo, omnia & in quolibet loco comburerentur, quod manifeſtè falſum eſt. <lb/></s> + <s xml:space="preserve">Deſine igitur mihi citate Lucillum Philalteum, qui in philoſophia mathematica <lb/>fuit omnium imperitiſſimus. </s> + <s xml:space="preserve">Verum ſpeculum vſtorium illud eſt quod ab Alhazem <lb/></s> + <s xml:space="preserve">Deinde à Vitellìone deſcribitur.</s> + </p> + <p> + <s xml:space="preserve">Quod deinde verum ſit, <reg norm="vmbram" type="context">vmbrã</reg> vniuſcuiuſque corporis opaci à Sole productam <lb/>ſemper eſſe centum <reg norm="nouemque" type="simple">nouemq́;</reg> vicibus maiorem diametro eiuſdem corporis, nego.</s> + </p> + <p> + <s xml:space="preserve">Imaginemur <var>.s.</var> 1. diametrum eſſe illius circuli, quo vltimi radij ſolares veniunt tan <lb/>gentes corpus cuius diameter ſit <var>.c.e.</var> et <var>.a.i.</var> ſit diameter alterius circuli eiuſdem cor-<lb/>poris ſolaris à quo vltimi radij veniunt tangentes corpus, cuius diameter ſit <var>.f.g.</var> in <lb/>eadem diſtantia, & eodem ſitu prioris corporis. </s> + <s xml:space="preserve">Tunc conus vmbræ ipſius <var>.f.g.</var> ſit <var>.f.<lb/>g.q.</var> & ipſius <var>.c.e.</var> ſit <var>.c.n.e.</var> centrum autem ſolare ſit <var>.o.</var> conorum verò axes ſint <var>.t.n.q</var>. <lb/></s> + <s xml:space="preserve">tunc ex ſuppoſito <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 ſolari, 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 ſi vmbra <var>.t.q.</var> tanto maior eſt <var>.f.g.</var> quanto .109. eſt <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> quę <var>.n.t.</var> ad <var>.t.<lb/>c.</var> ſed cum angulus <var>.t.</var> communis ſit ambobus triangulis <var>.q.t.f.</var> et <var>.n.t.c.</var> ſequitur ex .6. <lb/>ſexti dictos triangulos æ quiangulos eſſe. </s> + <s xml:space="preserve">Vnde ſi anguli <var>.t.n.c.</var> et <var>.t.q.f.</var> æ quales inui <lb/>cem ſunt, ergo <var>.q.f.</var> æquidiſtans erit <var>.n.c.</var> quod eſt impoſſibile, quia nunc demonſtra-<lb/>uimus ipſas concurrere in puncto <var>.u</var>. </s> + <s xml:space="preserve">Quare non eſt 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 ſubtilitate.</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 eſt, quæ ſit cauſa, quod nos videmus <lb/>radium ſolarem tardiſſimè moueri, cum alias tibi dixerim ipſum qualibet hora cir-<lb/>ca terram quindecim gradus perficere, reſpondeo, quod radius ille quem videmus, <lb/>exempli gratia, in aliquo cubiculo, nunquam eſt idem numero, ſed quia ipſi radij <lb/>nullo modo differunt inter ſe, niſi in numero, proptera putamus eundem ſemper eſſe, <lb/>cum ſemper alius, atque alius ſit, quorum vnuſquiſque (de illis loquor, qui ad hunc <lb/>terræ globum perueniunt) circa terram reuoluitur ſpatio .24. horarum, & cum quili <lb/>bet circulus diuidatur in .360. gradus, quorum vigeſimaquarta pars eſt .15. verum <lb/>eſt 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 ſcribit in .3. lib. pag .282. non nego <lb/>quin pulchrum ſit, ſed alijs pulchrioribus modis poſſumus illud idem de-<lb/>monſtrare; </s> + <s xml:space="preserve">cogita igitur ſuperficiem rectangulam, cuius medietas ſit <reg norm="triam" type="context">triã</reg> <lb/>gulus rectangulus <var>.a.b.g.</var> vnde ex ſuppoſito nobis cognita erit ſuperficies <lb/>ipſius trianguli, tanquam dimidium totius parallelogrammi rectanguli cogniti. <lb/></s> + <s xml:space="preserve">Quare ex .25. ſecundi triangulorum <reg norm="Monteregij" type="context">Mõteregij</reg>, cognita nobis <reg norm="erunt" type="context">erũt</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 ſit ſuperficies trianguli ſimul <reg norm="cum" type="context">cũ</reg> baſi <var>.a.g.</var> cognita etiam <lb/>nobis fit perpendicularis <var>.b.d.</var> à puncto <var>.b.</var> ad baſim, & conſequenter cognitum no-<lb/>bis erit productum ipſius <var>.a.d.</var> in <var>.d.g.</var> & quia nobis cognita eſt <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> ſimiliter nobis cognita erit ex quinta ſecundi <lb/>Eucl. </s> + <s xml:space="preserve">vnde ex penultima primi habebimus propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Poſſumus item circulum mente concipere cuius <var>.a.g.</var> ſit diameter, & ab eius cen-<lb/>tro <var>.e.</var> protracta cum fuerit <var>.e.b.</var> quæ nobis cognita erit, vt medietas ipſius <var>.a.g.</var> de cu <lb/>ius potentia, dempta <reg norm="cum" type="context">cũ</reg> fuerit potentia <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>b.o.</var> remanebit nobis potentia ipſius <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">huiuſmodi 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 ſubſcriptus <var>.a.b.c.u.</var> ſuperficiei cognitę ſimul cum diametro <var>.a.<lb/>c.</var> extendatur imaginatione <var>.b.c.</var> vſque ad, f. ita quod <var>.c.f.</var> æqualis ſit <var>.c.u.</var> intelligan-<lb/><reg norm="turque" type="simple">turq́;</reg> quadrata <var>.g.f</var>: <var>g.u.</var> ct <var>.u.f.</var> vnde <reg norm="summa" type="context">sũma</reg> <reg norm="quadratorum" type="context">quadratorũ</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="ſummam" type="context">ſummã</reg> <var>.g.u</var>:u.b: et <var>.u.f.</var> cognoſce-<lb/>mus, cui <reg norm="summæ" type="context">sũmæ</reg> addito ſuplemento <var>.d.e.</var> æ quali <var>.u.<lb/>b.</var> dabit nobis <reg norm="cognitum" type="context">cognitũ</reg> quadrarum <var>.g.f.</var> totale, qua <lb/> + <anchor type="figure" xlink:label="fig-0305-01a" xlink:href="fig-0305-01"/> + re cognoſcetur eius radix <var>.b.f.</var> cognita igitur <var>.b.f.</var> <lb/>cum pro ducto <var>.b.u.</var> illico ex .5. ſecundi cognoſce-<lb/>tur <var>.b.c.</var> et <var>.c.f.</var> forte cognita <var>.b.f.</var> diuiſa <reg norm="per" type="simple">ꝑ</reg> æqualia <lb/>in puncto <var>.t.</var> & per inæqualiz in <reg norm="puncto" type="context">pũcto</reg> <var>.c</var>. </s> + <s xml:space="preserve">Nam qua <lb/><reg norm="dratum" type="context">dratũ</reg> ipſius <var>.t.f.</var> cognitum, ęquatur <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.b.u.</var> <lb/><reg norm="cum" type="context">cũ</reg> quadrato ipſius <var>.t.c.</var> <reg norm="dempto" type="context">dẽpto</reg> igitur rectangulo, <var>b.<lb/>u.</var> ex quadrato ipſius <var>.t.f.</var> relinquetur quadratum <lb/><reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.t.c.</var> cognitum & eius radix <var>.t.c.</var> qua addita ipſi <lb/>medietati <var>.b.t.</var> & <reg norm="dempta" type="context">dẽpta</reg> ex medietate <var>.f.t.</var> relinque-<lb/>tur propoſitum.</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 eiuſdem 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, ſimul cum proportione late <lb/>rum <var>.b.c.</var> et <var>.b.a.</var> nobis data ſit. </s> + <s xml:space="preserve">cum autem ſcire voluerimus eius ſuperficiem <var>.b.u.</var> cla-<lb/>rum eſt, quod cum nobis data ſit proportio <var>.b.c.</var> ad <var>.b.a.</var> illico cognoſcemus <reg norm="etiam" type="context">etiã</reg> pro-<lb/>portionem quadrati ipſius <var>.b.c.</var> ad quadratum ip-<lb/> + <anchor type="figure" xlink:label="fig-0305-02a" xlink:href="fig-0305-02"/> + ſius <var>.b.a.</var> cum dupla ſit ei quæ <var>.b.c.</var> ad <var>.b.a.</var> ita etiam <lb/>& aggregati dictorum quadratorum ad quadra-<lb/>tum ipſius <var>.b.a.</var> hoc eſt nota erit nobis proportio <lb/>quadrati ipſius <var>.a.c.</var> diagonalis ad quadratum ip-<lb/>ſius <var>.a.b.</var> idem dico de quadrato <var>.b.c.</var> ideſt quod <lb/>proportio quadrati ipſius <var>.a.c.</var> ad quadratum <var>.b.c.</var> <lb/>cognita nobis erit, ſed <var>.a.c.</var> data nobis fuit, qua-<lb/>re cognoſcemus etiam omnia dicta quadrata eo-<lb/><reg norm="rumque" type="simple">rumq́;</reg> radices <var>.a.b.</var> et <var>.b.c.</var> </s> + <s xml:space="preserve">quare & ſuperficiem 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 poteſt <lb/>ſolui, propterea, quod cum nobis cognita ſit ba-<lb/>ſis trianguli cum ſumma reliquorum laterum, & <lb/><reg norm="cum" type="context">cũ</reg> angulo oppoſito baſi ipſius reliqua cognita no <lb/>bis emergunt ex .15. problemate ſecundi lib. de Triangulis ipſius 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 ſi tibi placet, accipe hanc aliam methodum à me excogitatum.</s> + </p> + <p> + <s xml:space="preserve">Duplicetur <reg norm="triangulum" type="context">triangulũ</reg> <var>.a.b.c.</var> <reg norm="orthogonium" type="context">orthogoniũ</reg>, & fiat <reg norm="rectangulum" type="context">rectangulũ</reg> <var>.b.u.</var> vt in mea figura <lb/>ſecundi exempli hic vides. </s> + <s xml:space="preserve"><reg norm="producaturque" type="simple">producaturq́;</reg> <var>.b.c.</var> quouſque <var>.c.f.</var> æqualis ſit <var>.c.u.</var> vnde <var>.b.f.</var> <lb/>cognita nobis erit ex hypotheſi, </s> + <s xml:space="preserve">quare cognoſcemus etiam quadratum <var>.g.f.</var> à quo <lb/><reg norm="demptum" type="context">demptũ</reg> cum fuerit <reg norm="aggregatum" type="context">aggregatũ</reg> quadratorum <var>.g.u.</var> et <var>.u.f.</var> nobis <reg norm="cognitum" type="context">cognitũ</reg> (nam quadra <lb/>ta <var>.g.u.</var> et <var>.u.f.</var> æqualia ſunt quadrato ipſius <var>.a.c.</var> diagonalis datę) remanebit aggrega-<lb/>tum <reg norm="ſupplementorum" type="context context">ſupplemẽtorũ</reg> cognitum, </s> + <s xml:space="preserve">quare eius medietas cognoſcetur ideſt <var>.b.u.</var> vndæ ex <num value="5">.<lb/>5.</num> ſecundi Eucli. vt ſuperius diximus cognoſcetur etiam <var>.b.c.</var> et <var>.c.f.</var> diſtinctæ.</s> + </p> + <p> + <s xml:space="preserve">Idem aſſero de <reg norm="exemplo" type="context">exẽplo</reg> Gemmæ Friſij à Stifelio citato in Appendice regulæ falſi.</s> + </p> + <p> + <s xml:space="preserve">Sit gratia exempli rectangulum hicſubſcriptum <var>.a.b.</var> datæ ſuperficiei data etiam <lb/>nobis ſit proportio <var>.a.e.</var> ad <var>.e.b.</var> laterum producentium, <reg norm="cogitemusque" type="simple">cogitemusq́;</reg> <var>.a.e.</var> producta <lb/>vſque ad <var>.o.</var> ita vt <var>.e.o.</var> æqualis ſit ipſi <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 ſexti ſeu .18. vel .19. ſeptimi vel .15. quinti <lb/>eadem proportio erit ipſius <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/>ſcemus quadratum <var>.b.o.</var> & eius <reg norm="radicem" type="context">radicẽ</reg> <var>.e.o.</var> & ex ea <lb/>demregula cognoſcemus <var>.a.e.</var> cum cognita nobis ſit <var>.e.o.</var> ſimul 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 ſit figura infinitorum angulorum hoc eſt <lb/>ultima poligoniarum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">SEd quod idem Stifelius in Appendice ſecundi libri dicat circulum eſſe figuram <lb/>poligoniam, non eſt ita mirandum, nam & alij multi doctiſſimi viri hanc <lb/>veritatem cognouerunt, de Leone Baptiſta Alberto nihil dicam, cum ipſe fateatur <lb/>hoc accepiſſe à philoſophis, vt etiam refert Ariſt. de ſphæratertio de cœlo. </s> + <s xml:space="preserve">conſi-<lb/>dera quæſo in circulo, quod cum angulus contingentiæ ſit angulus, quamuis <reg norm="omnium" type="context">omniũ</reg> <lb/>acutorum rectilineorum anguſtiſſimus, vnde ex communi ratione ſequitur reliquum <lb/>ex duobus rectis rectilineis eſſe angulum, & ſi omnium obtuſorum rectilineorum ſit <lb/>ampliſſimum, tanto magis igitur erit angulus, id quod remanet ex duobus rectis re <lb/>ctilineis, detractis <reg norm="cum" type="context">cũ</reg> fuerint duobus angulis contingentiæ, qui quidem angulus erit <lb/>in quouis puncto circunferentiæ ipſius circuli, idem intelligendum eſt de ſphæra, <lb/>cuius angulus eſt reſiduum ex quatuor rectis ſolidis, detractis cum fuerint quatuor <lb/>angulis contingentiæ <reg norm="ſolidisque" type="simple">ſolidisq́;</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 eſt, eo quod quandoque bonus dormitat Homerus. </s> + <s xml:space="preserve">Puto enim il-<lb/>lud problema ab ipſo Monteregio non fuiſſe viſitatum. </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">methodũ</reg> a me aliter <reg norm="etiam" type="context">etiã</reg> <reg norm="excogitatam" type="context">excogitatã</reg> in eadem ipſius 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="Propoſitum" type="context">Propoſitũ</reg> ſit nobis triangulum <var>.a.b.g.</var> cuius baſis data ſit cum area, ſeu perpendi-<lb/>culari <var>.a.d.</var> cum angulo etiam <var>.a.</var> ad cognoſcendum autem <var>.a.b.</var> et <var>.b.g.</var> cogitemus circu <lb/>lum <var>.a.b.q.g.</var> circunſcribere ipſum triangulum cuius diameter <var>.p.q.</var> ad rectos ſe-<lb/>cet baſim <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 ipſo cognoſces, & <lb/><reg norm="angulus" type="simple">angulꝰ</reg> <var>.p.b.m.</var> ſimiliter cognoſcetur, <lb/></s> + <s xml:space="preserve">quare ex .29. primi eiuſdem Montere <lb/>gij cognoſcemus <var>.p.m.</var> et <var>.p.b.</var> (nam <var>.b.<lb/>m.</var> datum fuit, vt dimidium totius ba-<lb/>ſis <var>.b.g.</var>) ducta poſtea <var>.b.q.</var> ex <reg norm="eadem" type="context">eadẽ</reg> .29. <lb/>cognoſcemus <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">cũ</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 cognoſcemus <var>.a.t.</var> quæ ſinus <lb/>eſt arcus <var>.a.p.</var> vnde cognitus erit arcus <lb/><var>a.p.</var> ſed arcus <var>.p.g.</var> cognitus eſt median <lb/>te angulo <var>.p.b.g.</var> cognito, qui quidem <lb/>arcus <var>.p.g.</var> ſi coniunctus fuerit cum arcu <var>.p.a.</var> cognoſcemus compoſitum <var>.a.g.</var> & eius <lb/>chorda ſimiliter (hoc eſt <reg norm="ſecundum" type="context">ſecundũ</reg> latus) qua cognita, illico cognoſcemus chordam <lb/><var>a.b.</var> hoc eſt tertium latus trianguli propoſiti.</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/>henſibilem eſſe, vide igitur, quod ſcribit in quinto lemmate in decimam <lb/>propoſitionem libr .2. de inſidentibus aquæ Archimedis, volens demonſtra-<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 eſt aliquo modo pro-<lb/>lixum, mediante linea <var>.c.p.</var> cum ſuis partibus, citans etiam antecedens lemma extra <lb/>propoſitum, eo quod nec in antecedente lemmate, nec in alio, ipſe vnquam proba <lb/>uerit proportionem <var>.c.d.</var> ad <var>.d.q.</var> eſſe, vt <var>.l.b.</var> @d.b.m. ſed ne putes me falli, tibi demon <lb/>ſtrabo non eſſe neceſſarium ducere lineam <var>.c.m.p.</var> vel <var>.q.p.</var> eo quod <reg norm="cum" type="context">cũ</reg> per quintam <lb/>lib. de quadratura parabolę Archimedis, ita ſit <var>.c.d.</var> ad <var>.d.e.</var> vt <var>.l.b.</var> ad <var>.b.m.</var> exiſtente <lb/><var>a.c.</var> dupla ipſi <var>.d.c.</var> et <var>.e.c.</var> dupla ipſi <var>.g.c.</var> et <var>.l.d.</var> dupla ipſi <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> ideſt <var>.g.c.</var> ad <var>.g.d.</var> vt <var>.b.d.</var> ideſt <var>.l.b.</var> <lb/>ad <var>.b.m.</var> ſed <var>.c.g.</var> ad <var>.g.d.</var> eft vt <var>.c.e.</var> ad <var>.e.a.</var> ratio eſt, quia componendo ita eſt <var>.c.d.</var> ad <var>.d.<lb/>g.</var> vt <var>.c.a.</var> ad <var>.a.e.</var> & hoc eſt, quia permutando, ita eſt <var>.a.c.</var> ad <var>.d.c.</var> vt <var>.a.e.</var> ad <var>.d.g.</var> & hoc <lb/>verum eſt ex .19. quinti eo quod totius <var>.a.c.</var> ad totum <var>.d.c.</var> eft vt abſciſſi <var>.e.c.</var> ad abſciſ <lb/>ſum <var>.g.c.</var> vt ſupradixi.</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 vniuerſaliori modo potes probare, quod ita ſit <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 ſimiliter <var>.y.</var> in <lb/>tali ſitu 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> diuiſa eodem modo etiam ſit à puncto <var>.s.</var> ex ſupradicta 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">totiꝰ</reg> <var>.a.c.</var> ad totum <var>.y.c.</var> vt abſcisſi <var>.s.c.</var> ad abſciſsum <var>.s.c.</var> </s> + <s xml:space="preserve">quare reſidui <lb/><var>a.e.</var> ad reſiduum <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 ſupradictam <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 ſequitur, <lb/>primum, ſecundum, tertium, & quartum lemma ſuperflua 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/>poteſt demonſtrari, hoc mode. </s> + <s xml:space="preserve">Nam ſuperius demonſtrauimus 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 æquidiſtantium ad <var>.h.e.</var> quibus rationibus mediantibus codem modo ſcies, <reg norm="quod" type="simple">ꝙ</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="æquidiſtantibus" type="context">æquidiſtãtibus</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, ſed cum ſit <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> & euerſim <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 eſt propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Non video etiam, quare ipſe ducat lineam <var>.s.r.</var> cum in ipſo contextu nihil ſaciac <lb/>de dicta <var>.s.r</var>.</s> + </p> + <p> + <s xml:space="preserve">Comentum poſtea contextus <var>.P.</var> pulchrius eſſet, ſi diceret, quod cum ita ſit totius, <lb/><var>l.a.</var> ad totum <var>.a.d.</var> ſic ſe habebit abſciſſum <var>.a.i.</var> ad abſciſſum <var>.a.z.</var> eo quod ita eſt, vt ſcis, <lb/>hoc eſt in proportione dupla, ergo reſidui <var>.i.l.</var> ad reſiduum <var>.d.z.</var> erit vt totius <var>.a.l.</var> ad <lb/>totum <var>.a.d.</var> hoc eſt in proportione dupla.</s> + </p> + </div> + <div type="letter" level="4" n="5"> + <head style="it" xml:space="preserve">De Viſu.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">RAtio vnde ſiat, vt videamus diſtinctè omnes eolores, cum in qualibet aeris par <lb/>te, quo lumina reſlexa poſſunt peruenire mixta ſint, & non diſtincta, oritur à <lb/>paruitate ipſius pupillæ oculorum, & à magna expanſione virtutis viſiuæ in ſuperſi-<lb/>cie concaua orbis continentis humores diaphanos oculorum per ramuſculos nerui <lb/>optici remotè ab ipſa pupilla. </s> + <s xml:space="preserve">& quamuis radii luminoſi frangantur ab vnoquoque <lb/>humore diuerſimodè, hoc nihilominus maximè iuuat ad diſtinctionem radiorum, <lb/>ſed & ſi directè procederent, idem ferè eueniret, non tamen ſuis locis, cogita exem-<lb/>pli gratia lineam <var>.a.u.e.</var> vt communis ſectio cuiuſdam plani ſecantis ſphæram oculi, <lb/>per centrum ipſius, & pupillæ, et <var>.o.</var> punctum ſit proximum centro ipſius pupillæ, <lb/>ſed interius aliquantulum, extra <reg norm="autem" type="context">autẽ</reg> <reg norm="oculum" type="context">oculũ</reg>, ſint varij colores, vt <var>.c.n.t.</var> in dicto plano.</s> + </p> + <p> + <s xml:space="preserve">Iam nulli dubium eſt quod lumina quæ producuntur ab <var>.c.n.t.</var> ad <var>.o.</var> in ipſo <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 diſtincta, procedendo igitur vlteriusipſi radij citra <var>.o</var>. </s> + <s xml:space="preserve">tunc <reg norm="diſgregantur" type="context">diſgregãtur</reg>, <lb/>& ſeparantur abinuicem, & <reg norm="cum" type="context">cũ</reg> perueniunt ad lineam <var>.a.u.e.</var> ſentiuntur diſtincti alij <lb/>ab alijs. </s> + <s xml:space="preserve">Cuius quidem rei, exemplum manifeſtum accipere poſſumus à quouis <lb/>cubiculo ex omni parte clauſo, quod tranſitum <reg norm="nullum" type="context">nullũ</reg> permittat radijs luminoſis, ni <lb/>ſi per aliquod paruum foramen, in quo foramine, & extra ipſum cubiculum, omnes <lb/>radij mixti erunt, ſed in obiecto pariete ipſius cu-<lb/> + <anchor type="figure" xlink:label="fig-0309-01a" xlink:href="fig-0309-01"/> + biculi videb untur diſtincti, vnde ſequitur, 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> & ſuus contrapoſitus ſimili-<lb/>ter, & per conſequens linea <var>.e.u.a.</var> breuior erit, & <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.o.</var> propinquius etiam erit ipſi lineæ <var>.a.u.e.</var> <lb/>quæ omnia efficiunt, vt nobis obiectum <var>.c.t.</var> <reg norm="paruum" type="context">paruũ</reg>, <lb/>& minus diſtinctum, ſeu magis confuſum 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/>hæmiſphærij.</head> + <head style="it" xml:space="preserve">Anſelmo Fucaro.</head> + <p> + <s xml:space="preserve">GRatæ mihi tuæ literæ <reg norm="fuerunt" type="context">fuerũt</reg>, quibus <reg norm="oſtendis" type="context">oſtẽdis</reg> non <reg norm="paruum" type="context">paruũ</reg> <reg norm="deſiderium" type="context">deſideriũ</reg> <reg norm="ſciendi" type="context">ſciẽdi</reg> vnde <lb/>fiat, quod cum dies illuceſcit, & eſt ſerena pars Cœli, circa axem orizontis <lb/>demiſſior appareat, quam aliæ partes, <reg norm="cum" type="context">cũ</reg> ab alijs (quod <reg norm="ſciam" type="context">ſciã</reg>) ſatis expreſſum <reg norm="non" type="context">nõ</reg> fue <lb/>rit, ſed quia de eo à me aliquid ſcire deſideras dicam quod mihi <reg norm="verum" type="wordlist">vr̃</reg>. </s> + <s xml:space="preserve">Scias non <reg norm="ſolum" type="context">ſolũ</reg> <lb/><reg norm="multitudinem" type="context">multitudinẽ</reg> <reg norm="obiectorum" type="context">obiectorũ</reg> <reg norm="oppoſitorum" type="context">oppoſitorũ</reg> efficere, vt aliqua res alia longius diſtare videat, <lb/>vt alij <reg norm="putarunt" type="context">putarũt</reg>, ſed etiam diuerſitates colorum, quamobrem cum decipiamur, cre-<lb/>dentes Cœlum eſſe præditum colore cęruleo, cum is color, aeri, non Cœlo <lb/>conueniat, & videntes huiuſmodi colorem circa axem <reg norm="orizontis" type="context">orizõtis</reg> magis denſum, <reg norm="quam" type="context">quã</reg> <lb/>verſus ipſum orizontem, ratione exiguæ reſlexionis, à pauca quantitate vaporum <lb/>inter noſtrum ſitum, & reſlexionis locum, iudicamus Cœlum proximiorem eſſe cir-<lb/>ca dictum axem, quam ſint aliæ partes; </s> + <s xml:space="preserve">præterquam, quod is color, qui videtur <lb/>terminare, aut impedire radium viſualem (aduertas tamen me hac in re platonicum <lb/>non eſſe) eo ſemper propinquior eſſe videtur, qui ei locum dat, & hanc ob cauſam <lb/>videntes nos <reg norm="denſitatem" type="context context">dẽſitatẽ</reg> cęrulei 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">contrariũ</reg> <lb/>euenire poteſt, ideſt vt eis magis profundum videa <lb/>tur <reg norm="Cœlum" type="context">Cœlũ</reg>, circa axem orizontis, quam vicinum gy <lb/>ro <reg norm="eiuſdem" type="context">eiuſdẽ</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 ſeſe magis <reg norm="obſcuram" type="context">obſcurã</reg> oculo <reg norm="demonſtrat" type="context">demõſtrat</reg>, & <lb/><reg norm="eam" type="context">eã</reg> <reg norm="propinquiorem" type="context">propinquiorẽ</reg> quę ſeſe <reg norm="clariorem" type="context">clariorẽ</reg> oſtendit, vt ei <reg norm="ent" type="context">ẽt</reg> <lb/>cont<unsure/>ingere poteſt, qui <reg norm="ſubſcriptam" type="context">ſubſcriptã</reg> <reg norm="ſiguram" type="context">ſigurã</reg> <reg norm="cubicam" type="context">cubicã</reg> non <lb/><reg norm="quidem" type="context">quidẽ</reg> <reg norm="ductam" type="context">ductã</reg> ſecundú <reg norm="ordinem" type="context">ordinẽ</reg> opticè, ſed ita, vt om-<lb/>nia latera oppoſita <reg norm="inuicem" type="context">inuicẽ</reg> ſint parallela, proſpiciet, <lb/>ideſt <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 ſequitur, vt aliquando quadratum <var>.a.o.</var> <lb/>videbitur citra, et <var>.i.n.</var> vltra <reg norm="dictum" type="context">dictũ</reg> cubum aliquando <lb/>verò èconuerſo.</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 Piſano.</head> + <p> + <s xml:space="preserve">SI omnia vno colore conſtarent, & corporum vmbræ à luminibus non di-<lb/>ſtinguerentur, neque diuerſitas ſitus, lumina, quæ veniunt ad oculum non al-<lb/>teraret; </s> + <s xml:space="preserve">perinde eſſet, ac ſi eſſemus cœci. </s> + <s xml:space="preserve">Miror quod cum in Ariſtotele ſis <lb/>verſatus, in tuis tamen ſcriptis philoſophum à Mathematico ſepares, quaſi mathe-<lb/>maticus non ſit adeò philoſophus, vt eſt naturalis, & metaphyſicus, cum multo ma <lb/>gis quam ij philoſophus ſit appellandus, ſi ad veritatem ſuarum concluſionum reſpi <lb/>ciamus. </s> + <s xml:space="preserve">Verum <reg norm="quidem" type="context">quidẽ</reg> eſt, te in huiuſmodi errore <reg norm="ſolum" type="context">ſolũ</reg> non verſari; </s> + <s xml:space="preserve">ſed grauius eſt, <lb/>quod cum vos videatis etiam res morales ſub philoſophię <reg norm="appellationem" type="context">appellationẽ</reg> cadere, non <lb/>animaduertatis diuinas ſcientias mathematicas etiam philoſophiæ nomine ornan-<lb/>das eſſe. </s> + <s xml:space="preserve">Quod ſi eiuſdem nomen penitius conſiderare velimus, inueniemus aper-<lb/>tè, mathematico magisillud ipſum quàm cuilibet alio conuenire, cum nullus ex alijs <lb/>tam certo ſciat id quod affirmat quam mathematicus, neque aliquis ſit, qui in co-<lb/>gnitionis, & ſcientiæ cupiditatem magis ducatur, vt apertè patet, cum nec etiam ipſi <lb/>ſenſui det locum, neque aliquid præſupponat, quod non ſit ita verum & intellectui <lb/>notum, vt nulla quæuis porentia, illud eſſe falſum oſtendere queat. </s> + <s xml:space="preserve">Sed quia <lb/>Græci, qui ad placitum nomina rebus impoſuerunt, voluerunt etiam, non ſolum <lb/>mathematica, ſed etiam naturalia, metaphyſica, & moralia, ſub communi philoſo-<lb/>phiæ nomine contineri. </s> + <s xml:space="preserve"><reg norm="Vtaunt" type="context">Vtaũt</reg> tibi ſatisfaciam authoritate Ariſtotelis, quem tanto-<lb/>pere colis, primum conſidera, nunquam eum de philoſopho <reg norm="mentionem" type="context">mẽtionem</reg> facere quin <lb/>prius aperiat de quo philoſopho loquatur, atque hoc ſemper præſtat, exceptis qui-<lb/>buſdam locis, vt cap .2. lib. 4. <reg norm="Metaphyſicorum" type="context">Metaphyſicorũ</reg>, vbi de philoſopho in genere <reg norm="loquens" type="context">loquẽs</reg>, <lb/>ait, proprium eſſe philoſophi. vt res omnes ſpeculetur atque hoc in principio quin <lb/>ti textus aſſerit, cum in quarto iam oſtenderit mathematicum eſſe philoſophum: <lb/></s> + <s xml:space="preserve">omitto quod in .2. textu ſecundi phyſicorum idem affirmet, æquum eſſe appellare <lb/>philoſophiam ſcientiam veritatis, & finem ſpeculatiuæ exiſtere veritatem. </s> + <s xml:space="preserve">An non <lb/>idem in primo cap .6. metaphiſicæ philoſophiam ſpeculatiuam, mathematicis phy <lb/>ſicis & ſupernaturalibus rebus contineri? </s> + <s xml:space="preserve">An non idem paulo inferius ſcribit phyſi-<lb/>cam primam futuram, ſi aliæ ſubſtantiæ quam naturales non reperirentur? </s> + <s xml:space="preserve">conſidera <lb/>deinde quid dicat in fine tertij cap. lib. 11. quo loco nil clarius eſſe poteſt, lege etiam <lb/>quæ .6. cap. eiuſdem libri ab eodem adducuntur, & quæ in .8. cap .12. libri textu .44. <lb/>apertè ponuntur. </s> + <s xml:space="preserve">Quod ſi hæc tibi non ſufficiunt, vereor ne tuus morbus deſpe-<lb/>ratus euadat.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">De imaginatione ſpecierum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod dixi domino Tadeo eſt, quod aliquas particularium ſpecies, perfectè & <lb/>integrè imaginari poſſumus, alias non item, id tibi melius exemplo innote-<lb/>ſc<unsure/>et. </s> + <s xml:space="preserve">Proponatur tibi triangulus æquilaterus datæ magnitudinis, <reg norm="datique" type="simple">datiq́;</reg> coloris, hu <lb/>iuſmodi enim particularis, potes imaginatione tibi fingere integram ſpeciem, tota <lb/><reg norm="lemque" type="simple">lemq́;</reg> ei adæquatam, ſed ſi aliquam ſpeciem aliquando vniuerſaliorem 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 ęquilateri, tali magnitudine, <lb/>ſed non præfinito colore conſtantis, hoc minime præſtare poſſes. </s> + <s xml:space="preserve">quia nullam rem <lb/>viſibilem priuatam colore imaginari poſſumus. </s> + <s xml:space="preserve">nec etiam potes imaginari <reg norm="ſpeciem" type="context">ſpeciẽ</reg> ali <lb/>cuius trianguli æquilateri, indeterminatæ magnitudinis, & indefiniti coloris, quæ <lb/>cuilibet particulari cuiuſuis magnitudinis, & coloris poſtea applicari queat. </s> + <s xml:space="preserve">Species <lb/>deinde alicuius trianguli ęquicruri, aut vnius trianguli laterum inęqualium, aut <reg norm="triam" type="context">triã</reg> <lb/>guli in genere, aut tandem figuræ, conſiderato tu ipſe, an poſſit ſub imaginationem <lb/>cadere. </s> + <s xml:space="preserve">Poſſumus quidem huiuſmodi ſpeciem (ratione mediante) intelligere, vn <lb/>de quamlibet ſpeciem rei particularis viſibilis, compoſitæ, ex figura, magnitudine, <lb/>& colore, perfectè imaginari poſſimus, & huiuſmodi conceptus erit ſpecialiſſima <lb/>ſpecies, quia in infinito ſuorum 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 ſe <lb/>partem aliquam habeat, quod aliquid aliud non obtineat, quin dicta differentia ſit <lb/>ſpecifica, eius tamen ſolum partis quæ differt ab alia duorum indiuiduorum, vnius, <lb/><reg norm="eiuſdemque" type="simple">eiuſdemq́;</reg> ſpeciei. </s> + <s xml:space="preserve">quia ſi eſt in magnitudine nulla planè magnitudo reperitur, quę <lb/>ſua ſpecie non ſit dotata, quod ſi non eſſet, inter res omnes nulla æqualitas <lb/>eluceret: </s> + <s xml:space="preserve">& ſi in figura, & colore, idem affirmo, aliter nulla res fimilis eſſet <lb/>alteri, neque aliqua ſimilitu lo reperiretur. </s> + <s xml:space="preserve">Idem de quolibet alio obiecto <lb/>ſenſibili dico. </s> + <s xml:space="preserve">Ratio autem eorum omnium quæ dixi eſt, quia imaginatiua nihil <lb/>aliud intellectui oſtendere poteſt, quam id quod recipit à ſenſu, & cum ſenſus, <lb/>alio modo moueri non poſſit quam ſupradicto, hanc ob cau ſam verum eſt, <reg norm="quicquid" type="simple">ꝗcquid</reg> <lb/>ſcripſi. </s> + <s xml:space="preserve">Vnde triangulum ęquilaterum datę magnitudinis, erit genus triangulorum <lb/><reg norm="ęquilaterum" type="context">ęquilaterũ</reg> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> datæ magnitudinis, ſed <reg norm="diuerſorum" type="context">diuerſorũ</reg> <reg norm="colorum" type="context">colorũ</reg>, erit <reg norm="etiam" type="context">etiã</reg> ſpecies trianguli <lb/>æquilateri indeterminatæ magnitudinis, & hic deinde erit ſpecies trianguli, & hic <lb/>poſtea ſpecies figuræ. </s> + <s xml:space="preserve">Idem de alijs omnibus rebus per gradus dico, quę ſicut à ſen <lb/>ſu, ita etiam ab imaginatione longè recedunt, adeo vt has ſpecies ſpecialiſſimas tan-<lb/>tum, ideſt eas ſolùm, quas hic ſuperius deſcripſi, integrè capere poſſit: </s> + <s xml:space="preserve">at verò gene <lb/>ra, quanto vniuerſaliora ſunt, ab eadem imaginatione, tanto longius diſtant.</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 ſunt, quàm partes ipſius Lunę magis perſpicuæ, à qui <lb/>bus, lumen non refleſſum, ſed penetrans, nobis occultatur; </s> + <s xml:space="preserve"><reg norm="quemadmodum" type="context">quemadmodũ</reg> <lb/>via lactea, nihil aliud eſt, quam pars octaui orbis magis opaca, à qua lumen Solis re-<lb/>fleſſum, ſeſe nobis oſtendit. </s> + <s xml:space="preserve">Quod autem Maurolicus ſcribit folio .64. cap. de aſtro <lb/>rum fulſionibus, circa <reg norm="Lunam" type="context">Lunã</reg>, eſt falſum; </s> + <s xml:space="preserve">primo, quia non conſiderat <reg norm="differentiam" type="context">differentiã</reg> inten-<lb/>ſionis luminum inter Venerem, & Lunam, cum <reg norm="lumem" type="context">lumẽ</reg> illius ſit magis intenſum, quam <lb/>Lunæ, quia quilibet qui ſano ſit <reg norm="onculo" type="context">õculo</reg>, facile poteſt compræhendere, ſi Lu-<lb/>na eſſet, vbi eſt Venus, aut Venus vbireperitur Luna (quibus in locis eiuſdem ma <lb/>gnitudinis nobis apparerent) ipſa Luna à Venere longè ſuperaretur, & excedere-<lb/>tur ſplendore, & lumine, ita vt ſi etiam verum eſſet, quod per tres gradus inter-<lb/>ualli ſeſe nobis proderet ſexageſima pars luminis (quod in quadraturis nec in vllo <lb/>alio ſitu verum e<unsure/>uadit, reſpectu ad Solem, ideſt vt tres gradus differentiæ ſitus, con <lb/>ſtituant ſexag eſimam partem differentiæ luminis reſpectu noſtri) 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 conſpiceretur, quia non ſufficit extenſio luminis, cum eiuſdem inten <lb/>ſio ſit etiam neceſſaria. </s> + <s xml:space="preserve">Sed id quoque tibi dico, quod etiam ſi dicta ſexageſima <lb/>pars totius luminis lunaris, eadem intenſione ſplendoris, & luminis Veneris, in tali <lb/>diſtantia trium graduum à Sole prædita eſſet, non eam <reg norm="tamen" type="wordlist">tamẽ</reg> videremus, ratione ob <lb/>liquitatis curuę, & ſphæricę ſuperficiei Lunæ, reſpectu noſtri, in huiuſmodi ſitu: </s> + <s xml:space="preserve">id <reg norm="qui" type="simple">ꝗ</reg> <lb/>tibi ita demonſtratum volo.</s> + </p> + <p> + <s xml:space="preserve">Pars ſuperficialis lunaris globi, quæ nos reſpicit ſit <var>.a.p.u.</var> quam accipere poſſu-<lb/>mus pro medietate ipſius ſuperficiei totalis, eo quod reſpectu noſtri viſus, inſenſibi <lb/>liter, ab ipſa medietate differat, pars autem à Sole viſa ſit <var>.u.q.a.</var> cogitemus etiam cir <lb/>culum <var>.a.p.u.q.</var> vnum eſſe ex maioribus ipſius globi, cuius ſuperficies <reg norm="tranſeat" 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 ipſius Lunæ à Sole receptum, ita quod pars illumi <lb/>nata, erit medietas <var>.u.q.a.</var> exceſſus verò, cum noſtro viſui incompræhenſibilis ſit, pro <lb/>nihilo reputetur, cuius cauſa eſt, maxima illa diſtantia, quæ inter Solem, & Lunam <lb/>reperitur, quamuis Sol maior ſit 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 ipſius <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 ſexageſima pars ipſius <var>.a.p.u.</var> <reg norm="quod" type="wordlist">qđ</reg> <lb/>à vero ſenſibiliter non diſcedit. <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 eſt quod linea <var>.d.u.</var> ſe-<lb/>cabit <reg norm="arcum" type="context">arcũ</reg> <var>.t.u.</var> ſed ita propinqua <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.u.</var> quod erit ei ferè contingens, <lb/>vnde abſque ſenſibili 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, ſub an-<lb/>gulo <var>.t.d.u.</var> quem quidem angulum <lb/>oportet nos videre, cuius magnitu-<lb/>dinis exiſtat, reſpectu 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> vſque 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> ſed quia, cum dia-<lb/>meter <var>.a.u.</var> tam breuis ſit reſpectu di <lb/>ſtantiæ à terra, tempore interlunij, <lb/>vnde minor <reg norm="centeſima" type="context">cẽteſima</reg> parte ipſius di-<lb/>ſtantiæ exiſtit, <reg norm="ſequitur" type="simple">ſequit̃</reg> nos poſſe <reg norm="abſque" type="simple">abſq;</reg> <lb/>ſenſibili errore cogitare, à puncto <var>.d.</var> <lb/>ad quoduis punctum ipſius diametri <lb/>omnes lineas ad angulos rectos cum <lb/>ipſo diametro, & inſenſibilis 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 ſinu arcus <var>.t.u.</var> qui eſt graduum .3. hoc <lb/>eſt ſexageſima pars ſemicirculi graduum .180. quapropter <var>.t.i.</var> erit partium .5233. ta-<lb/>lium qualium <var>.o.u.</var> eſt .100000. cuius <var>.t.i.</var> quadratum demptum cum fuerit à quadra-<lb/>to ſemidiametri <var>.o.t.</var> relinquet nobis quadratum ipſius <var>.o.i.</var> quæ quidem <var>.o.i.</var> vt radix <lb/>quadrata, erit partium .99862. talium qualium ſemidiameter eſt .100000. vnde <var>.i.u.</var> <lb/>reſiduum diametri, remanebit partium .138. </s> + <s xml:space="preserve">Vel ſic, cum cognitus ſit nobis arcus <var>.<lb/>t.u.</var> illicò cognoſcemus ſinum arcus <var>.p.t.</var> <reg norm="complementum" type="context">complementũ</reg> vnius quartæ, qui ſinus æ qua-<lb/>lis erit ferè arcui <var>.o.i.</var> partium .99862. vnde <var>.i.u.</var> erit, vt dictum eſt, partium .138. quę <lb/>quidem <var>.i.u.</var> æqualis eſt ferè ſinui arcus <var>.u.e.</var> & ita etiam <var>.u.e.</var> </s> + <s xml:space="preserve">quare ſi diuiſa fuerit to-<lb/>ta <var>.a.u.</var> <reg norm="partium" type="context">partiũ</reg> .200000. per .138. proueniet nobis .1449. & ſic 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 poteſt, 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">Poſt eas literas quas proximè ad te dedi, Franciſcus Monardus mihi retulit tuas <lb/>nonnullas dubitationes circa noſtrum Theorema Arithmeticum .116. quarum <lb/>prima eſt, quod ſi numerus <var>.a.</var> <reg norm="cogitatus" type="simple">cogitatꝰ</reg>, eſſet æqualis .4. </s> + <s xml:space="preserve">tunc ipſe non eſſet multiplex <lb/>ipſi .4. de quo tamen nullam mentionem feci. </s> + <s xml:space="preserve">Idem etiam inquis, ſi <var>.a.</var> fuiſſet .5. 6. 7. <lb/>nec non .1. 2. et .3. </s> + <s xml:space="preserve">Cui reſpondi, quod <reg norm="quanuis" type="context">quãuis</reg> nullam fecerim mentionem de æqua <lb/>litate ipſius <var>.a.</var> cum .4. nihiltamen refert, </s> + <s xml:space="preserve">propterea quod quando ita fuiſſet, nihi-<lb/>lominus eaſdem conditiones ſubiret, <reg norm="quemadodum" type="context">quemadodũ</reg> ſi fuiſſet duplus, triplus, aut qua <lb/>druplus. eo quod à genere multiplici, æqualitas, formam diuerſam non induat. </s> + <s xml:space="preserve">Qua <lb/>re idem eueniet ſi <var>.a.</var> fuerit .4. 5. 6. 7. vt ſi eſſet .8. 9. 10. et .11. & ſic de cæteris, excepto <lb/>quod in proprijs multiplicibus, vel in ſuperantibus ipſis multiplicibus <var>.a.</var> menſurare <lb/>tur ab ipſo .4. plus quam ſemel. </s> + <s xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eſt, quia, <lb/>vt in ſecunda ſumma, hoc eſt in tertio termino maximo, reliquus tertius terminus, <lb/>ideſt .9. non compræhendetur, ita nobis indicabit primum numerum ſumptum mi <lb/>norem eſſe quaternario. </s> + <s xml:space="preserve">Quæ omnia, exipſa noſtra thęoria ibidem expreſſa ma-<lb/>nifeſtantur. </s> + <s xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neſcio, quia ipſius opus <lb/>ad manus meas nunquam peruenit, ſatis 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 ſcripta ego nunquam vidi, vt Leonardi Piſani, Proſdocimi, Petri Borghi, <lb/>Fratris Lucæ, Ioannis Sfortunati, <reg norm="cæterorumque" type="simple">cæterorumq́;</reg> ſimilium.</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">Illuſtri Uiro Franciſco Mendo Zzæ</head> + <p> + <s xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noſter ſufficie-<lb/>bat. </s> + <s xml:space="preserve">Sed quia, quæ nunc à me petis, talia ſunt, vt ſine tripartita <reg norm="aequa- liter" type="simple">ęqua-<lb/>liter</reg> aliqua data proportione non poſſit aliquis exactè intentum perfice-<lb/>re, nihilominus, ſuppoſita di <lb/> + <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/> + cta diuiſione, reliqua facilia <reg norm="erunt" type="context">erũt</reg>. </s> + <s xml:space="preserve"><reg norm="Primum" type="context">Primũ</reg> <lb/>enim eſt. </s> + <s xml:space="preserve">Propoſitam Ellipſim 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> Ellipſis propoſita <var>.a.b.d.c.</var> cu-<lb/>ius axes ſint <var>.a.b.</var> et <var>.d.c.</var> dati, ſeu <reg norm="reperti" type="simple">reꝑti</reg> ex <lb/>47. <reg norm="ſecundi" type="context">ſecũdi</reg> Pergei, <reg norm="ſintque" type="simple">ſintq́;</reg> duo circuli <var>.a.e.<lb/>b.f.</var> et <var>.g.d.h.c.</var> circa eaſdem diametros, <lb/><reg norm="tunc" type="context">tũc</reg> proportio <var>.a.b.</var> ad <var>.d.c.</var> <reg norm="dimidium" type="context">dimidiũ</reg> erit <lb/>proportionis circulorum ex .2. 12. Eu-<lb/>clid. </s> + <s xml:space="preserve">ſed proportio <var>.a.b.</var> ad <var>.d.c.</var> æqualis <lb/>eſt proportioni maioris circuli ad Elli <lb/>pſim .ex .5. Archimedis in lib. de cono<lb/>idalibus, quapropter proportio Elli-<lb/>pſis ad minorem circulum altera me-<lb/>dietas erit totius proportionis circulo-<lb/>rum, hoc eſt maioris ad minorem, qua <lb/>re Ellipſis media proportionalis erit <lb/>inter eos circulos. </s> + <s xml:space="preserve">Nunc verò cum <lb/>ex Archimede repertę fuerint duæ fi-<lb/>guræ rectilineæ æquales duobus circu <lb/>lis iam dictis, & inter has, reperta fue <lb/>rit alia media proportionalis propoſi-<lb/>tum obtinebimus.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Spheroidem propoſitam cubare.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">PRopoſita ſphæroides erit, aut prolata, aut oblonga, ſit prius prolata, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.a.b.</var> <lb/>diameter circuli, qui eam per æqualia ſecat, circa quam <var>.a.b.</var> vt circa axem in-<lb/>telligatur ſphæroides oblonga, cuius ſpiſſitudo ſit <var>.d.c.</var> axis prolatæ, cogitemus <reg norm="nunc" type="context">nũc</reg> <lb/>duas ſphæras <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 eſt duas ſphæras, & duas ſphæroides, quas probabo continuas proportio-<lb/>nales inuicem eſſe.</s> + </p> + <p> + <s xml:space="preserve">Conſideremus igitur duos conos rectos, quorum <var>.a.b.</var> diameter ſit eorum baſium, <lb/>altitudo autem maioris, æqualis ſit ſemidiametro majori, hoc eſt 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 ſit ſemidiametro minori, hoc eſt medietati <var>.d.c.</var> vnde <lb/>habebimus proportionem coni maioris ad conum minorem, <reg norm="eandem" type="context">eãdem</reg> quæ eſt diame <lb/>tri maioris ad diametrum minorem, quod ex .2. parte .11. duodecimi Eucli. </s> + <s xml:space="preserve">nec non <lb/>ex .9. eiuſdem manifeſtum eſt, ſed conus minor, eſt quarta pars ſphæroidis prolatæ <lb/>ex .29. </s> + <s xml:space="preserve">Archimedis in lib. de conoidalibus, & conus maior, eſt etiam quarta pars <lb/>ſphæræ, ex .32. primi lib. de ſphæra, & cyllindro, </s> + <s xml:space="preserve">quare ex communi ſcientia, <reg norm="eadem" type="context">eadẽ</reg> <lb/>proportio erit ſphæræ maioris ad ſphæroidem prolatam, quæ <var>.a.b.</var> ad <var>.d.c.</var> ſed pro-<lb/>portio <var>.a.b.</var> ad <var>.d.c.</var> eſt tertia pars proportionis maioris ſphæræ ad <reg norm="minorem" type="context">minorẽ</reg>. </s> + <s xml:space="preserve">Conſidere <lb/>mus <reg norm="nunc" type="context">nũc</reg> alios duos conos rectos, vnius & <reg norm="eiuſdem" type="context">eiuſdẽ</reg> baſis, <reg norm="cuius" type="simple">cuiꝰ</reg> diameter ſit <var>.d.c.</var> ſed altitu <lb/>do maioris, æqualis ſit ſemidiametroſphęrę maioris, altitudo verò minoris, ſit æqua <lb/>lis ſemidiametro minoris ſphæræ, 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 eſt <var>.o.b.</var> ad <var>.o.d.</var> hoc eſt vt <var>.a.b.</var> ad <var>.d.c.</var> & ex dictis <reg norm="pro poſitionibus" type="simple">ꝓ<lb/>poſitionibus</reg> ita ſe habebit ſphæroides oblonga ad ſphęram minorem vt <var>.a.b.</var> ad <var>.d.<lb/>c.</var> hoc eſt tertia pars proportionis ſphæræ maioris ad minorem. </s> + <s xml:space="preserve">Quare proportio <lb/>ſphæroidis prolatæ ad oblongam, erit reliqua tertia pars proportionis maioris <reg norm="ſphae ræ" type="simple">ſphę<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 eſt inter <var>.a.b.</var> & ſuas duas tertias partes vna media pro por-<lb/>tionalis, quæ ſit <var>.K.</var> & ex Archimede, inuentum ſit quadratum ęquale circulo, cuius <lb/>ſit <var>.K.</var> diameter. </s> + <s xml:space="preserve">Vnde proportio circuli (cuius <var>.a.b.</var> eſt diameter) ad circulum cu-<lb/>ius <var>.K.</var> eſt diameter, ſeſquialtera 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 ſphærę maiori, ex corellario .32. primi de ſphęra & <lb/>cyllindro, cuius corporis, latus cubus ſit <var>.m</var>.</s> + </p> + <p> + <s xml:space="preserve">Idem facere oportebit mediante <var>.d.c.</var> minoris ſphærę, cuius corporis cubica ra-<lb/>dix ſit <var>.n</var>.</s> + </p> + <p> + <s xml:space="preserve">Nunc verò inter <var>.m.</var> et <var>.n.</var> inueniantur duę medię proportionales <var>.s.t.</var> & ex <var>.s.</var> pro-<lb/>ducatur cubus, qui ęqualis erit ſphęroidi prolatæ propoſiti, cubus vero <var>.t.</var> æqualis <lb/>erit ſphęroidi oblongę, cuius axis eſſet <var>.a.b</var>.</s> + </p> + <p> + <s xml:space="preserve">Si autem ſphęroides oblonga nobis propoſita fuiſſet, eodem methodo ſoluere-<lb/>tur problema.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head style="it" xml:space="preserve">Quadratum circulis mediantibus deſignare.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">MOdus autem conficiendi quadratum ex circulis ſupra datam lineam, vt Do-<lb/>minum Gaſparem docui, facillimus eſt.</s> + </p> + <p> + <s xml:space="preserve">Sit enim linea <var>.b.a.</var> 46. propoſitionis primi Euclidis, <reg norm="poſitoque" type="simple">poſitoq́;</reg> pede immobli circi-<lb/>ni in puncto <var>.a.</var> ſecundum quantitatem lineæ <var>.a.b.</var> propoſitę fiat circulus, ſimiliter cir-<lb/>ca punctum <var>.b.</var> alius circulus eiuſdem magnitudinis, </s> + <s xml:space="preserve">erecta deinde ſola <var>.a.c.</var> perpendi <lb/>culari ipſi <var>.a.b.</var> ex puncto <var>.a.</var> ipſa ſecabitur à circunferentia circuli. cuius centrum eſt <var>.<lb/>a.</var> in puncto <var>.c.</var> vnde <var>.a.c.</var> æqualis erit <var>.a.b.</var> poſito demum pede immobili ipſius circi <lb/>ni in puncto <var>.c.</var> ſecundum longitudinem ipſius <var>.c.a.</var> fiat alius circulus, qui æqualis erit <lb/>reliquis duobus circulis cum eorum ſemidiametri æquales ſint, & hic vltimo factus <lb/>ſecabit circulum, cuius <reg norm="centrum" type="context">centrũ</reg> eſt <var>.b.</var> in <reg norm="puncto" type="context">pũcto</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 ſunt inuicem ęqualia ex condi-<lb/>ti onibus circuli, angulus autem <var>.a.</var> rectus effectus fuit, </s> + <s xml:space="preserve">tunc ſi 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. eiuſdem concludemus etiam reliquos angulos rectos eſſe.</s> + </p> + <p> + <s xml:space="preserve">Circa verò id quod mihi ſcripſiſti de igne perpetuo putans nugas eſſe, quod Ro-<lb/>mæ inuentæ fuerint lucernę ardentes in ſepulchris 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, ſed multi homines fide digniſſimi. </s> + <s xml:space="preserve">Prętera cum aisid nulla ratione poſſe <lb/>fieri. </s> + <s xml:space="preserve">Reſpondeo quod maxima ratione poſſibile eſſe puto, quam quidem ra-<lb/>tionem ita eſſe oportet, quod primum lucerna ſit perfectè circuncluſa, vt <lb/>materia in ea conſtituta nullo modo exire poſſit, </s> + <s xml:space="preserve">deinde quod materia inflamabilis <lb/>talis ſit, vt excrementum fuliginoſum ex flamma tranſmiſſum, tangendo ſuperfi-<lb/>ciem deuexam ipſius lucernæ, aptum ſit in <reg norm="priſtinum" type="context">priſtinũ</reg> <reg norm="humorem" type="context">humorẽ</reg> conge<unsure/>lari, ſiue transfor-<lb/>mari, vnde materia prima per tres formas perpetuò tranſibit, hoc eſt per humorem, <lb/>ſiue oleum tale, vt diximus, per ignem, ſeu flammam, & per vaporem, ſeu exhala-<lb/>tionem fuliginoſam aptam condenſari, 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/>propoſitam proportionem.</head> + <head style="it" xml:space="preserve">Michaeli Angelo Muciaſco.</head> + <p> + <s xml:space="preserve"><emph style="sc">QVod</emph> mihi proponis, tale eſt, vt ſcilicet tibi modum ſcribam diuidendi <lb/>triangulum propoſitum ſecundum datam proportionem à linea tranſeun <lb/>te per punctum notatum extra triangulum.</s> + </p> + <p> + <s xml:space="preserve">Tr<unsure/><reg norm="iangulum" type="context">iangulũ</reg> <reg norm="igitur" type="simple">igit̃</reg> à te mihi propoſitum ſit <var>.n.o.u.</var> conſidero <reg norm="primum" type="context">primũ</reg> quod ſi <lb/>quis ipſum diuiſerit 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 tranſiens <lb/>alia linea à puncto <var>.p.</var> propoſito, ita quod efficiat duo triangula <var>.m.r.e.</var> et <var>.r.s.x.</var> inui-<lb/>cem æqualia, problema ſolutum 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 reſiduum <var>.m.n.u.x.</var> etiam ęquale <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 propoſitum expe-<lb/>ditum fuiſſer, hoc eſt, <reg norm="quod" type="simple">ꝙ</reg> ſi à puncto p. <lb/>protracta eſſet linea <var>.p.m.</var> quę trian-<lb/>gulum <var>.n.o.u.</var> in duas partes inuicem <lb/>ita proportionatas diuiſiſſet, vt ſe ha <lb/>bet <var>.A.</var> et <var>.B.</var> ita ſe haberet <reg norm="productum" type="context">productũ</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 eſt diffi-<lb/>cile ſpeculari, ex methodo .24. ſexti, + <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 ſit proportio <var>.A.</var> ad <var>.B.</var> vt data, cognita etiam <lb/>nobis erit coniuncta, hoceſt <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> & ſimiliter productorum. </s> + <s xml:space="preserve">Quæſiui poſtea modum inueniendi duas <lb/>dictas lineas <var>.m.o.</var> et <var>.o.x.</var> & cognoui quod ſi producta fuerit <var>.p.i.</var> æquidiſtans li-<lb/>neæ <var>.o.x.</var> <reg norm="producendoque" type="simple">producendoq́</reg> <var>.o.n.</var> quouſque cum <var>.p.i.</var> ſe interſecarent in puncto <var>.i.</var> inuenien <lb/>do poſtea 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> poteſt fieri, quod cognitum dico, eo quod nobis <lb/>cognita eſt proportio data, & rectangulum etiam <var>.n.o.</var> in <var>.o.u.</var> deinde ſecando ab <var>.o.<lb/>n.</var> partem æqualem lineæ iam inuentæ, quæ ſit <var>.o.t</var>. </s> + <s xml:space="preserve">Inueniendo poſtea, ex .28. ſexti <lb/>lineam <var>.o.m.</var> cuius productum in <var>.m.t.</var> æquale ſit producto <var>.t.o.</var> in <var>.o.i.</var> vnde ex .15. eiuſ <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 ſe ha-<lb/>beret <var>.m.i.</var> ad <var>.m.o.</var> vt <var>.m.o.</var> ad <var>.o.t.</var> ſed ex .4. ſexti, 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. ſexti productum <var>.<lb/>o.x.</var> in <var>.m.o.</var> æquale eſſet producto. p, i. in <var>.o.t.</var> & ſic 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 ſi punctum <var>.m.</var> caderet in punctum <var>.n.</var> idem eſſet, ſi vorò punctum <var>.m.</var> tranſiret <lb/>n. oporteret nos facere hoc in latere <var>.n.u.</var> ipſum quærendo in linea <var>.n.u.</var> ducendo pri <lb/>mum lineam <var>.p.i.</var> <reg norm="æquidiſtantem" type="context">æquidiſtantẽ</reg> <var>.u.x.</var> & producendo <var>.u.n.</var> ad partem <var>.u.</var> proſequendo, <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſuperius iam dictum eſt.</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, ſecundum aliquam datam <lb/>proportionem à linea tranſeunte per punctum propoſitum.</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 ſit, quæ <var>.A.</var> ad <var>.B.</var> vt ſupra. </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 ſit <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 ſecabit 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 ſi intra dictos terminos tranſibit, vt in prima figura videre potes, <lb/>problema ſolutum erit, eo quod <lb/> + <anchor type="figure" xlink:label="fig-0317-01a" xlink:href="fig-0317-01"/> + ſi à puncto <var>.i.</var> protracta fuerit <var>.p.<lb/>d.</var> pa rallela ad <var>.u.x.</var> habebimus <lb/>ex prima ſexti 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 eſt vt <var>.A.</var> ad <var>.B.</var> ſed <reg norm="triangulus" type="context">triãgulus</reg> <lb/><var>i.e.d.</var> æqualis eſt 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"/> + ſcientia. </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 eſt vt <var>.A.</var> ad <var>.B.</var> ex .11. eiuſdem.</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 ſi punctum <var>.q.</var> fuerit extra ut in .2. figura videre eſt. </s> + <s xml:space="preserve">tunc manifeſtum erit, <reg norm="quod" type="simple">ꝙ</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">dictũ</reg> <var>.q.t.u.</var> quapropter <lb/>cum diuiſus fuerit triangulus <var>.e.<lb/>x.t.</var> mediante linea <var>.o.n.K.</var> ita <reg norm="quod" type="simple">ꝙ</reg> <lb/><reg norm="quadrilaterum" type="context">quadrilaterũ</reg> <var>.e.n.K.t.</var> ſit æquale <lb/>triangulo <var>.q.t.u.</var> ex doctrina præ <lb/>cedenti, habebimus propoſitum.</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 ſi quadrilaterum dictum eſſet fruſtum alicuius <reg norm="trianguli" type="context">triãguli</reg> ut in figura <var>.A.</var> hic ſub <lb/>ſcripta videre eſt, ſuppoſita, <var>b.d.</var> parallela ad <var>.u.p.</var> ita faciendum eſſet, ducendo <lb/>ſcilicet parallelam <var>.u.x.</var> ad <var>.b.p.</var> quæ producatur vſque ad concurſum cum <var>.b.d.</var> <lb/>in puncto <var>.x.</var> <reg norm="ſitque" type="simple">ſitq́;</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> ſit 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> ſe ha-<lb/>beat vt <var>.t.a.</var> ad <var>.a.e.</var> quo facto ex doctrina <reg norm="prae cedenti" type="simple">prę<lb/>cedenti</reg> diuidatur totum parallelogram--<lb/>mum <var>.p.x.</var> mediante linea <var>.o.q.</var> ſecundum <lb/>quod ſe habet <var>.t.r.</var> ad <var>.r.e</var>. </s> + <s xml:space="preserve">Atque ita ſolu-<lb/>tum erit problema, vt exte ipſo 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 ſi nullum latus parallelum reliquo erit, ita faciendum erit. </s> + <s xml:space="preserve">ſi ſit tale quadrila <lb/>terum <var>.b.d.u.p.</var> oportet vt ipſum conuertamus in triangulum, producendo duo <lb/>quęuis eius latera oppoſita uſque ad interſectionem ut pote <var>.u.p.</var> et <var>.d.b.</var> in puncto <var>.x.</var> <lb/>quo facto, ſupponemus <var>.o.</var> eſſe punctum datum, proportio verò data ſit <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> ipſa <var>.t.i.</var> ſe 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"/> + ſe habet ad triangulum <var>b.p.x.</var> ducatur poſtea <var>.o.q.</var> quæ diuidat totale triangulum <var>.d.<lb/>u.x.</var> in duas partes inuicem ita proportionatas, ut ſe habent <var>t.r.</var> et <var>.r.e.</var> quæ quidem <lb/>partes ſint <var>.c.d.u.q.</var> et <var>.c.q.x.</var> ut in primo problemate tibi monſtraui, & habebis pro-<lb/>poſitum, dato quod punctum <var>.c.</var> ſit inter <lb/>b. et <var>.d</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi forte linea <var>.o.q.</var> ſecabit <var>.b.x.</var> hoc <lb/> + <anchor type="figure" xlink:label="fig-0319-01a" xlink:href="fig-0319-01"/> + eſt ſi punctum <var>.c.</var> eſſet inter <var>.b.</var> et <var>.x.</var> mani-<lb/>feſtum eſt, quod <var>.c.q.</var> ſecaret <var>.b.p.</var> in pun-<lb/>cto <var>.y.</var> vnde in tali caſu, alio modo ope-<lb/>randum eſſet, hoc eſt ducendo <var>.b.u.</var> quæ <lb/>diuideret quadrilaterum in duo triangu-<lb/>la, & ut ſe haberet triangulum <var>.b.d.u.</var> ad <lb/>triangulum <var>.b.p.u.</var> vellem vt ita ſecaretur <lb/><var>t.i.</var> in puncto <var>.n.</var> vt ita ſe haberet <var>.t.n.</var> ad <var>.n.<lb/>i.</var> ut dictum eſt de iſtis duobus triangulis, <lb/></s> + <s xml:space="preserve">deinde prout ſe habet <var>.n.r.</var> ad <var>.r.i.</var> ita ſeca-<lb/>res triangulum <var>.b.p.u.</var> mediante linea <var>.o.<lb/>K.</var> ex doctrina primi problematis, & ita haberes propoſitum.</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, ſeu hexagonum, vel alias quaſuis multilateras figuras propoſitas its <lb/>diuidere, vt dictum eſt de trilateris, & quadrilateris.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia pentagonus <var>.a.d.u.p.b.</var> quem ſecare volumus <reg norm="mediante" type="context">mediãte</reg> linea <var>.o.<lb/>q.</var> in duas partes inuicem ſe habentes, vt ſe habent <var>.t.r.</var> et <var>.r.i.</var> oportet igitur ut ipſum <lb/>pentagonum reducas ad quadrilaterum <var>.x.a.d.u.</var> quod diuidatur ſecundum præce-<lb/>dentem doctrinam, vt ſe habet <var>.t.r.</var> ad <var>.r.e.</var> <lb/>vnde ſi punctum <var>.q.</var> incidit inter <var>.p.</var> et <var>.u</var>. </s> + <s xml:space="preserve">tunc <lb/>habebis propoſitum, ſi 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> ſecabit <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">diuidaturq́;</reg> <var>.t.i.</var> in puncto <var>.n.</var> ita <lb/>vt <var>.t.n.</var> ad <var>.n.i.</var> ſe 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">hũc</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 propoſitum. </s> + <s xml:space="preserve">Idem dico de hexa <lb/>gono, reducendo ipſum ad pentagonum, & <lb/>item de eptagono, ipſum reducendo ad exa <lb/>gonum, & idem infero de infinito ipſarum <lb/>ſuperficialium 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 ſuperficie pla-<lb/>na, non tamen inuicem æqu idiſtantes, proponis etiam <var>.n.t.</var> in eadem ſuperfi-<lb/>cie, quæ vnamquamque priorum ſecat, proponis etiam lineam <var>.h.</var> tali conditione, <lb/>quod nulli dictarum ſit parallela, </s> + <s xml:space="preserve">deinde ſcire cupis qua arte aliquis poſſet ducere <var>.<lb/>c.u.</var> parallelam ad <var>.h.</var> ita quod ſecando <var>.n.t.</var> conſtituat 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/>vſque ad concurſum 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> poſtea <lb/>ex .25. ſexti Eucli. conſtitues <reg norm="triangulum" type="context">triãgulum</reg> <var>.i.u.e.</var> ſimile triangulo <var>.i.c.n.</var> æquale tamen <lb/>triangulo <var>.i.t.n.</var> & ſolutum erit problema. <lb/></s> + <s xml:space="preserve">Velſic, 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 poſtea <var>.e.u.</var> paralle-<lb/>lam lineę <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">& ſo-<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. ſexti 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">triangulũ</reg> <var>.i.t.n.</var> ex pri-<lb/>ma ſexti, et .11. quinti. </s> + <s xml:space="preserve">Vnde <lb/>ex .9. eiuſdem <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">Thaodoſio à Raifestaim.</head> + <p> + <s xml:space="preserve"><emph style="sc">DVritandvm</emph> profecto non eſt, quin quotidie hominibus ſtudioſis ali-<lb/>quid noui deſit, quemadmodum, quod tibi nunc occurrit, mihi non-<lb/>nunquam accidit, hoc eſt inuenire orizontem, cui aliqua propoſita ſtel <lb/>la oriatur cum gradu ipſius longitudinis. </s> + <s xml:space="preserve">pro <reg norm="cuius" type="simple">cuiꝰ</reg> rei operatione te prius <lb/>ſcire oportebit vtrum ſtella in ſignis aſcendentibus, vel deſcendentibus reperiatur, <lb/>hoc eſt in ſignis, quę à Capricorno ad Cancrum procedunt, vel in illis, quę à Can-<lb/>cro ad Capricornum numerantur, </s> + <s xml:space="preserve">propterea quod ſi in ſignis aſcendentibus inue-<lb/>nitur, ſciendum eſt, quod ſupra talem orizontem polus mundi auſtralis attollitur, <lb/>ſed ſi in ſignis deſcendentibus reperitur, </s> + <s xml:space="preserve">tunc polus borealis eleuatur ſupra 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 vigeſimi gradus Cancri, quapropter polus borealis eleuatur ſupra ori-<lb/>zontem, cui ipſa oritur cum eodem gradu, & minuto eclipticę illius ſigni. </s> + <s xml:space="preserve">ſed <lb/>quia volumus etiam ſcire veram quantitatem arcus eleuationis huiuſmodi poli, pro <lb/>pterea accipiemus in tabula generali Monteregij numerum qui vocatur radix aſcen <lb/>ſionum, èregione numeri longitudinis ipſius ſtellæ, qui quidem numerus in præſen <lb/>ti exemplo erit gra .107. cum minutis .53. qui eſt <reg norm="cuiuſdam" type="context">cuiuſdã</reg> arcus æquatoris, qui inci-<lb/>pit in principio Arietis, & in circulo latitudinis deſinit, hoc eſt ab orizonte quæſi-<lb/>to, ita quod talis numerus erit aſcenſio obliqua huiuſmodi puncti eclipticæ illi ori-<lb/>zonti, qua aſcenſione mediante, ſimul cum gradu, & minuto longitudinis in tabulis <lb/>aſcenſionum obliquarum, inueniemus gradum, & minutum altitudinis pollaris, <reg norm="quod" type="wordlist">qđ</reg> <lb/>quærebatur, eodem ordine ac methodo, quo vtimur ad inueniendum in tabulis po-<lb/>ſitionum, polum circuli poſitionis alicuius aſtri, mediante declinatione & diſtantia <lb/>à meridiano ciuſdem aſtri, vt ſcis. </s> + <s xml:space="preserve">Vnde in præſenti exemplo eleuatio poli borea <lb/>lis ſupra talem orizontem erit gra .7. cum minutis .45.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi ſtella fuerit in medietate aſcendente, tunc certi erimus polum auſtralem ſu <lb/>per dictum orizontem attolli, nam idem eſt quærere altitudinem vnius <reg norm="polorum" type="context">polorũ</reg> mun <lb/>di à tali orizonte, quod diſtantiam dicti poli à circulo ſecundum quem longitudo <lb/>terminatur, qui etiam latitudinis dicitur, eo quod tunc temporis talis circulus vnus <lb/>& idem eſt cum orizonte. </s> + <s xml:space="preserve">Sumatur ergo exempli gratia ſtella, quæ in ore piſcis au <lb/>ſtralis eſt, quę, pro nunc, ſit 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 ſtella oritur cum eiuſmodi puncto eclipticæ, depreſſum eſſe à parte <lb/>auſtrali ſub <reg norm="illoque" type="simple">illoq́</reg> polo, ſed quia propoſitum eſt ſcire etiam quantitatem huiuſmo-<lb/>di depræſſionis, reperiemus in tabula generali gradum, & minutum æquatoris, cor-<lb/>reſpondentem 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 eſt aſcen. obli. ad dictum orizontem, vbi polus auſtralis attollitur, & deſcenſio <lb/>obliqua, vbi polus borealis eleuatur. </s> + <s xml:space="preserve">Quapropter ſi à .317. gradibus cum minutis <lb/>46. demptus fuerit dimidius circulus gra .180. remanebunt gra .137. cum minutis .46 <lb/>& punctus oppoſitus gradibus .20. cum .14. minutis Aquarij eſt in eodem numero <lb/>Leonis, & mediantibus iſtis gradibus .137. min .46. aſcenſionis, cum grad .20. min <num value="14">.<lb/>14.</num> Leonis inueniemus eleuationem poli borealis ab orizonte in tabulis aſcenſio-<lb/>num obliquarum Monteregij, hoc eſt gra .17. min .53. & eadem altitudo erit poli <lb/>auſtralis ſupra orizontem à quo Fomahant cum dicto puncto eclipticæ oritur, in qua <lb/>longitudine dicta ſtella reperitur.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi propoſitus nobis fuerit punctus eclipticæ, cum quo aliqua ſtella oritura ſit, <lb/>& oporteat inuenire vbi, hoc eſt orizontem huiuſmodi ortus, eleuatione poli arti <lb/>ci, ſeu antarctici ſupra 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 ſtella <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 ſtella oritura ſit <var>.e.</var> orizon de <lb/><reg norm="mum" type="context">mũ</reg> <var>.o.e.</var> vbi ſtella oriri poſſit <reg norm="cum" type="context">cũ</reg> <lb/>puncto <var>.e</var>. </s> + <s xml:space="preserve">Nam cum ſtella 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 ſupponatur arcus <var>.a.<lb/>o.</var> eſſe circuli latitudinis, et <var>.a.o.</var> <lb/>Iatitudo ipſius ſtellæ, & angulus <lb/>a. rectus erit, & quia punctum <var>.e.</var> <lb/>datur, ergo arcus <var>.a.o.</var> & arcus <var>.a.<lb/>e.</var> ſimul <reg norm="cum" type="context">cũ</reg> angulo <var>.a.</var> recto cogni-<lb/>ti ſunt, vnde ex .11. primi lib. co<lb/>pernici, angulus <var>.a.e.o.</var> cognoſce-<lb/>tur, & angulus <var>.q.e.o.</var> ſimiliter, vt <lb/>reſiduum ex duobus rectis quo. e <lb/>mediante cum angulo <var>.q.</var> declina <lb/>tionis ab æquatore, <reg norm="medianteque" type="simple">medianteq́;</reg> <lb/>latere <var>.q.e.</var> cognito, cognitus quo <lb/>que nobis erit angnlus <var>.e.t.q.</var> ex <lb/>12. eiuſdem. </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 poſtea gradum eclipticę, cum quo ſtella data oriatur ad orizontem pro <lb/>poſitum, nullius eſt difficul@atis.</s> + </p> + <p> + <s xml:space="preserve">Ponamus exempli gratia, aliquem ſcire velle gradum eclipticæ, cum quo canicu-<lb/>la oritur ad orizontem, cui polus boreus eleuatur per gradus .44. quæ canicula ſup-<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/>ſeptentrionalis, </s> + <s xml:space="preserve">deinde inuenire eius aſcenſionem rectam ex doctrina .4. problema <lb/>tis eiuſdem 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 aſcenſionalium ſub polo .44. accipiemus differen-<lb/>tiam aſcenſionum, qua differt recta ab obliqua, quæ in præſenti exemplo erit gra <num value="5">.<lb/>5.</num> min .55. quæ dempta ab aſcenſione recta ſtellæ, vt præſens exemplum exigit, relin <lb/>quet nobis aſcenſionem obliquam ſtellæ propoſitæ ad polum. gra .44. quæ erit gra <num value="102">.<lb/>102.</num> minu .47. qua mediante, in tabulis aſcenſionum obliquarum poli .44. habebi-<lb/>mus gradum & minutum eclipticę cum quo ſtella oritur. </s> + <s xml:space="preserve">quod in caſu noſtro erit <lb/>gra .1. min .8. Leonis, ſed ſi tecum non fuerint tabulæ dictæ, potes eleganter omnia <lb/>hæc perficere via triangulorum ſphæricorum.</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> propoſitus <reg norm="autem" type="wordlist">aũt</reg> orizon ſit <var>.o.c.d.</var> <lb/>& ſtella data ſit <var>.o.</var> in orientali parte orizontis, circulus verò <var>.a.o.</var> ille ſit, qui <reg norm="tranſiens" type="context">tranſiẽs</reg> <lb/>per polos eclipti<unsure/>cæ & per centrum ſtellæ terminat longitudinem ipſius ſtellæ, & in <lb/>ipſo ſit eius latitudo. </s> + <s xml:space="preserve">Nunc propoſitum ſit inuenire arcum <var>.d.q.</var> eo quod illicò ſcie <lb/>mus punctum <var>.d.</var> qua propter oportet nos prius cognoſcere arcum <var>.d.a.</var> qui demptus, <lb/>vel additus arcui <var>.a.q.</var> prius cognito ex ſuppoſito (nam data nobis eſt longitudo, & <lb/>latitudo ſtellæ) dabit nobis <var>.d.q</var>.</s> + </p> + <p> + <s xml:space="preserve">Cum igitur voluerimus arcum <var>.d.a.</var> cognoſcere, ita faciemus. </s> + <s xml:space="preserve">nam <var>.q.a.</var> cognitus <lb/>nobis eſt ex ſuppoſito vt dictum eſt. </s> + <s xml:space="preserve">angulus quoque <var>.a.q.b.</var> qui declin tionis eclipti <lb/>cæ ab æquatore eſt, angulus deinde <var>.a.</var> (trianguli <var>.a.b.q.</var>) rectus eſt, 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> reſiduus ex duobus re-<lb/>ctis in duobus primis hic ſubſcriptis <lb/>figuris nobis itidem cognitus erit, <lb/>etiam & arcus <var>.b.o.</var> reſiduus ſiue com <lb/>poſitus ex ar cu <var>.a.o.</var> cognito ex ſup-<lb/>poſito <reg norm="cum" type="context">cũ</reg> ſit arcus latitud nis ab ecli-<lb/>ptica. </s> + <s xml:space="preserve">Tunc<unsure/> in triangulo <var>.o.b.e.</var> co-<lb/>gnoſcimus latus <var>.o.b.</var> & angulum <var>.o.<lb/>b.e.</var> nec non angulum <var>.b.e.o.</var> qui eſt <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">Conſideremus <lb/>deinde triangulum <var>.a.o.d.</var> cuius angu <lb/>lus <var>.a.</var> rectus eſt, & angulus <var>.a.o.d.</var> <reg norm="cum" type="context">cũ</reg> <lb/>latere <var>.a.o.</var> etiam cognitus, vnde ex <lb/>ſupradicta .4. nobis cognitus erit ar-<lb/>cus <var>.a.d.</var> & conſequenter cognoſce-<lb/>mus at cum <var>.d.q.</var> eius reſiduum, ſeu <lb/>compoſitum, 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 ſi hac via inuenire deſideras, <lb/>cui orizonti propoſita ſtella oriatur <lb/>cum eodem eclipticę puncto <var>.a.</var> lon-<lb/>gitudinis, hoc aliud nihil eſſet, quam <lb/>cognoſcere 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 ſci-<lb/>ret vnum illorum angulorum quem <lb/>æquator efficit cum orizonte, reli-<lb/>qua illicò ei innoteſcent, ſed dictus <lb/>angulus <var>.b.</var> iam diximus quomodo <lb/>cognoſcatur.</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 ſcire velle <reg norm="punctum" type="context">pũctum</reg> <lb/> + <anchor type="figure" xlink:label="fig-0324-01a" xlink:href="fig-0324-01"/> + eclipticę, cum quo Procyon oritur <lb/>polo .44. o. dato, quod ſtella in gra <num value="19">.<lb/>19.</num> cum min .24. </s> + <s xml:space="preserve">Cancri, reperiatur <lb/>diſtans ab ecliptica per gra .16. min <num value="10">.<lb/>10.</num> meridiem verſus. </s> + <s xml:space="preserve">vnde <reg norm="arcus" type="simple">arcꝰ</reg> <var>.a.q.</var> <lb/>erit gra .70. min .36. <reg norm="eiusque" type="simple">eiusq́;</reg> ſinus par-<lb/>tium 94321. talium qualium totalis <lb/>eſt .100000. arcus verò <var>.a.o.</var> gra .16. <lb/>minut .10. ſinus erit 27845. angu-<lb/>lus autem <var>.a.q.e.</var> declinationis zodia <lb/>ci ab ęquatore grad .23. min .30. cu-<lb/>ius ſinus eſt .39875. </s> + <s xml:space="preserve">Quare ex ſupra-<lb/>dictis rationibus angulus <var>.a.b.q.</var> erit <lb/>gra .82. mi .24. cuius <reg norm="ſinus" type="simple">ſinꝰ</reg> erit .99122. <lb/>arcus vero <var>.a.b.</var> gra .22. minu .17. cu-<lb/>ius ſinus erit .37945. </s> + <s xml:space="preserve">angulus deinde <lb/><var>o.e.b.</var> trianguli <var>.o.e.b.</var> eſt gra .46. mi <var>.<lb/>o.</var> altitudinis æquatoris ab orizonte, <lb/>cuius ſinus eſt .71934. angulus ſimili <lb/>ter <var>.o.b.e.</var> medio coniuncti, quibus <lb/>rectus perſicitur, arcus etiam <var>.o.b.</var> no <lb/>tus eſt grad .6. min .7. cuius ſinus eſt <num value="10655">.<lb/>10655.</num> cum ſit 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 ſupradicta an-<lb/>gulus <var>.e.o.b.</var> hoc eſt <var>.a.o.d.</var> erit. grad <num value="36">.<lb/>36.</num> min .39. cuius ſinus 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="ſinus" type="simple">ſinꝰ</reg> <lb/>erit .81998. arcus verò <var>.d.o.</var> gra .19. <lb/>min .51. cuius ſinus erit .33957 ar-<lb/>cus autem gra .11. min .42. cuius <reg norm="ſinus" type="simple">ſinꝰ</reg> <lb/>erit .20270. vnde <reg norm="arcus" type="simple">arcꝰ</reg> <var>.d.q.</var> reſiduus <lb/>ex <var>.a.q.</var> erit gra .58. min .54. complementum <reg norm="autem" type="wordlist">aũt</reg> quartæ erit gra .31. mi .6. hoc eſt gra <num value="1">.<lb/>1.</num> ſigni Leonis. cum min .6.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">De ſphæroide duplæ ſpbær æ propoſit æ.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">MOdus autem inueniendi ſphæroidem ex dato axe, quod duplum ſit ſphæ-<lb/>ra propoſita, talis eſt.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia <var>.a.b.c.</var> ſphæra propoſita. cuius ſemidia meter ſit <var>.o.c.</var> ſemiaxis <lb/>vero ſphæroidis ſit <var>.d.x.</var> cuius dimidium ſit <var>.u.x.</var> </s> + <s xml:space="preserve">tunc ex doctrina .9. ſexti 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 ſicut ſe 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 ſphæræ ita ſe habcat ad <var>.e.f.</var> ex .10. ſexti, quæ <lb/><var>e.f.</var> erit reliqua axis quæſita. </s> + <s xml:space="preserve">Vnde conſtituta cum fuerit ellipſis <var>.d.f.t.e.</var> ex dictis axi-<lb/>bus, </s> + <s xml:space="preserve">deinde circumuertendo ellipſim circa maiorem axem, conſtituemus ſphæroi-<lb/>dem oblongam, ſi autem circumuertemus ipſam circa minorem axim conſtituemus <lb/>ſphæroidem prolatam.</s> + </p> + <p> + <s xml:space="preserve">Quod autem talis operatio rationalis ſit, nulli dubium erit, que<unsure/>tieſcunque co-<lb/>gnoſcet 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 ſit cono <var>.e.u.f.</var> ex lemmate collecto ab <lb/>11. duodecimi, conus <var>.e.d.f.</var> duplus exiſtit etiam cono <var>.a.c.b.</var> ex .7. quinti. </s> + <s xml:space="preserve">Cum de-<lb/>inde ex .32. primi lib. de ſphæra, & cyllindro ſphæra <var>.a.c.b.q.</var> quadrupla ſit cono <var>.a.<lb/>c.b.</var> ipſa conſequenter dupla erit cono <var>.e.d.f.</var> ſed ex .29. primi de conoidalibus, dimi <lb/>dium ſphæroidis <var>.e.d.f.t.</var> hoc eſt <var>.e.d.f.</var> dupla eſt cono <var>.e.d.f</var>. </s> + <s xml:space="preserve">Quare talis medietas <lb/>æqualis eſt ſphæræ propoſitæ, totaq́ue ſphæroides dupla erit ſphærę datæ. </s> + <s xml:space="preserve">Quod <lb/>autem dico de proportione dupla, idem infero de qualibet alia, ſumendo <var>.u.x.</var> ita pro <lb/>portionatam ad <var>.d.x.</var> vt proponitur.</s> + </p> + <p> + <s xml:space="preserve">Sphęram autem inuenire quæ dimidia ſit ſphæroidis propoſitæ nullius erit nego-<lb/>tij, quotieſcunque inuentus fuerit modus diuidendi vnam datam proportionem in <lb/>tres æquales partes.</s> + </p> + <p> + <s xml:space="preserve">Sit propoſita ſphæroides <var>.e.f.d.t.</var> cuius axes ex conſequentia dantur <var>.e.f.</var> et <var>.d.t.</var> quę <lb/>quidem ſphæroides ſit primo oblonga, et <var>.u.x.</var> ſit dimidium axis maioris. </s> + <s xml:space="preserve">imagine-<lb/>tur etiam conus <var>.e.u.f.</var> vt ſupra. </s> + <s xml:space="preserve">Imaginetur etiam factum eſſe, quod proponitur, hoc <lb/>eſt, vt ſphæra <var>.a.b.c.q.</var> ſit dimidium ipſius ſphæroidis, vnde conus <var>.a.c.b.</var> æqualis erit <lb/>cono <var>.e.u.x.</var> vt ſupra demonſtratum eſt, & ſit <var>.g.h.</var> media proportionalis inter <var>.u.x.</var> et <lb/><var>o.c</var>. </s> + <s xml:space="preserve">Iam viſum ſuperius fuit, quod eadem proportio erat ipſius <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> ſed <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 eſt <var>.o.b.</var>) medię proportionales ſunt, eo quod cum <var>.g.h.</var> media <lb/>proportionalis ſit inter <var>.u.x.</var> et <var>.o.c.</var> & proportio <var>.o.b.</var> ad <var>.e.x.</var> æqualis ſit ei, quæ <var>.u.x.</var> <lb/>ad <var>.g.h.</var> hoc eſt ei quæ <var>.g.h.</var> ad <var>.o.c.</var> vel. ad <var>.o.b.</var> </s> + <s xml:space="preserve">quare quotieſcunque 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> ipſa <var>.o.c.</var> vel <var>.o.b.</var> erit ſemi <lb/>diameter ſphæræ quæſitę. </s> + <s xml:space="preserve">eodem modo faciendum erit ſi ſphęroides 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 ſe habet, duo ſcilicet triangula inuenire, æqualia dua-<lb/>bus ſuperficiebus rectilineis propoſitis, quę quidem triangula ſint eiuſdem <lb/>alritudinis, & quod <reg norm="vnunquodque" type="context">vnũquodque</reg> habeat angulum æqualem angulo pro<unsure/>poſito, & <reg norm="quod" type="simple">ꝙ</reg> <lb/>alius angulus vnius, cum alio alterius, æquetur duobus rectis.</s> + </p> + <p> + <s xml:space="preserve">Sint exempli gratia duæ propoſitæ ſuperflcies <var>.c.y.</var> duo verò anguli dati ſint <var>.r.s.</var> <lb/>cum voluerimus inuenire duo triangula (quæ ſint <var>.a.i.u.</var> et <var>.n.t.x.</var> ) tali conditio-<lb/>ne prædita, quod angulus, a. æqualis ſit angulo <var>.s.</var> & angulus <var>.t.</var> angulo <var>.r.</var> & quod <lb/>angulus <var>.x.</var> ſimul 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">gulũ</reg> <var>.a.i.u.</var> æquale ſit ſuperficiei <var>.<lb/>c.</var> reliquum verò ſuperficiei <var>.y.</var> <lb/>Ex duabus ſuperficiebus <var>.c.</var> et <var>.y.</var> <lb/>conſtituemus duo quadrata, per vl <lb/>timam ſecundi Eucli. accipiemus, <lb/></s> + <s xml:space="preserve">deinde duo latera tetragonica ip-<lb/>ſorum quadratorum, & inuenie-<lb/>mus tertiam lineam in continua <lb/>proportionalitate cum illis lateri-<lb/>bus ex .10. ſexti, ſeruabimus po-<lb/>ſtea extremas illarum, quæ ſint <var>.z.</var> <lb/>et <var>.l.</var> quarum proportio, <reg norm="eadem" type="context">eadẽ</reg> erit, <lb/>quæ inter duas propoſitas ſuperfi-<lb/>cies reperitur ex .18. ſexti, accipie <lb/>mus, deinde lineam aliquam cu-<lb/>inſuis longitudinis, quæ ſit <var>.q.g.</var> ſu-<lb/>pra quam conſtituemus in puncto <lb/>q. angulum <var>.m.q.g.</var> ęqualem angu-<lb/>lo <var>.s.</var> & angulum <var>.m.q.K.</var> æqualem <lb/>angulo <var>.r.</var> ex .23. primi, poſtea ve-<lb/>rò à quouis puncto ipſius lineæ <var>.q.<lb/>m.</var> puta <var>.o.</var> ducetur <var>.o.f.</var> vſque ad <var>.q.<lb/>g.</var> quorſum volueris, producendo <lb/>ipſam <reg norm="vſque" type="simple">vſq;</reg>. ad <var>.d.</var> ita quod propor-<lb/>tio <var>f.o.</var> ad <var>.o.d.</var> ſit vt <var>.z.</var> ad <var>.l.</var> ex .10. <lb/>ſexti, ducendo poſtea 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> ſecatur ab <var>.q.K.</var> in puncto <var>.b.</var> <lb/>protrahemus <var>.b.o.p.</var> vnde ex ſimi-<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 eſt vt <var>.z.</var> ad <var>.l.</var> hoc eſt vt <var>.c.</var> ad <var>.y</var>. </s> + <s xml:space="preserve">quare <reg norm="triangulum" type="context">triangulũ</reg> <var>.p.q.o.</var> ita erit proportio <lb/><reg norm="natum" type="context">natũ</reg> triangulo <var>.o.q.b.</var> vt <var>.c.</var> ad <var>.y.</var> conſtituo deinde ex .25. fexti duo triangula ſimi-<lb/>lia duobus <var>.p.q.o.</var> et <var>.o.q.b.</var> <reg norm="æqualiaque" type="simple">æqualiaq́</reg> <var>.c.</var> et <var>.y.</var> quę ſint <var>.a.i.u.</var> et <var>.n.t.x.</var> ſecetur poſtea <var>.q.<lb/>g.</var> in puncto <var>.æ.</var> ita, quod <var>.q.æ.</var> æqualis ſit <var>.i.a.</var> duco poſtea <var>.æ.e.</var> æquidiſtantem. ad <var>.p.b.</var> <lb/>& ſic 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> eaſdem 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/>uerſione in Ptolomeum. Deincendio carbo-<lb/>num à vento.</head> + <head style="it" xml:space="preserve">Clariβimo Dominico Moreſino.</head> + <p> + <s xml:space="preserve"><emph style="sc">SCio</emph> propoſitam tibi quæſtionem te diu agitauiſſe, nectamen ſolutio-<lb/>nem aſſequi potuiſſe, aduerte igitur ipſam falſam, ideſt impoſſibilem <lb/>eſſe, quemadmodum etiam decimumoctauum quæſitum propoſitum à <lb/>Cardano Tartaleæ, ab ipſo Tarralea ſolutum minimè fuit. </s> + <s xml:space="preserve">Quiquidem <lb/>Tartalea vult circulum deſcribi circa triangulum per quintam libri quarti Euclidis, <lb/>vt in fine ferè quintæ partis ſuarum menſurarum affirmat, neque videt in quinta <lb/>quarti Euclidem vti vndecima primi, & in vndecima primi, quarta aut octaua eiuſ-<lb/>dem, quas, ipſe Euclides oſtenſiuè non demonſtrauit. </s> + <s xml:space="preserve">Quapropter oportebat Tar-<lb/>taleam demonſtraſſe omnes propoſitiones ad hoc neceſſarias oſtenſiuè <reg norm="vſque" type="simple">vſq;</reg> ad pri-<lb/>mas indemonſtrabiles, quia ad demonſtrandam ſcientificè <reg norm="aliquam" type="context">aliquã</reg> propoſitionem, <lb/>aut à propoſitione in propoſitionem vſque ad prima principia vniuerſalia ( vt ali-<lb/>quando ego feci) eſt retrogradandum, aut ab ipſis principijs incipiendum ſucceſſi-<lb/>uè eouſque progrediendo donec ad propoſitionem quam demonſtrare volumus <lb/>perueniamus.</s> + </p> + <p> + <s xml:space="preserve">Quod ad Ptolomeum in geographia attinet, dico eum mihi non ſatisfacere, cum <lb/>ſumit portionem arcus circuli maioris inter vnam ciuitatem, & aliam, ea ratione <lb/>quam deſcribit. </s> + <s xml:space="preserve">Quod ſi vſus fuiſſet modo Menelai, ab ipſomet deinde in <reg norm="ſuum" type="context">ſuũ</reg> Al-<lb/>mageſtum vſurpato, aut Monteregij triangulorum ſphęricorum, 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 ſuum illud inſtrumentum geometricum attinet, eſt <reg norm="imperfectum" type="context">imperfectũ</reg>, <lb/>vt oſtendi domino Pandulfo.</s> + </p> + <p> + <s xml:space="preserve">Motum autem aeris, aut mauis ventum, accendere ignem, non ſolum ratione an <lb/>tiperiſtaſis, quam affers euenit, ſed etiam quia à carbonibus accenſis 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 propoſitionis .25. lib. 2. Monteregij.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">SCribiste non intelligere .25. propoſitionem lib. 2. Monteregij. cum necſcias <lb/>reperire diametrum circuli circunſcriptibilis circa propoſitum 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 ſit b aſis tantummodo ſimul cum angulo, qui ipſi baſi opponitur.</s> + </p> + <p> + <s xml:space="preserve">Imagineris igitur triangulum datum eſſe obtuſiangulum <var>.a.b.g.</var> cuius baſi <var>.b.<lb/>g.</var> ſit nobis data ſimul cum angulo <var>.a.</var> ei oppoſito, obtuſoq́ue; </s> + <s xml:space="preserve">Conſidera etiam cir-<lb/>culum <var>.a.b.g.q.</var> ipſum trian gulum circunſcribentem, cuius diameter <var>.q.e.p.</var> tranſeat <lb/>per <var>.m.</var> punctum medium ipſius <var>.b.g.</var> <reg norm="tunc" type="context">tũc</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">angulumq́</reg> <var>.q.e.g.</var> duplum eſſe an <lb/>gulo <var>.q.p.g.</var> ex .19. eiuſdem, vnde æqualem angulo <var>.a.</var> qui etiam duplus eſt 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 ſinum <var>.m.g.</var> arcus <var>.g.p.</var> & quia <var>.m.g.</var> vt <lb/>dimidium ipſius <var>.b.g.</var> tibi data eſt, cognoſces <lb/>etiam <var>.p.g.</var> vt <var>.m.g.</var> & ſic 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> eſt æquale ei quod fit ex <var>.b.m.</var> <lb/>in <var>.m.g.</var> ideo cum diuiſum 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, ſi <reg norm="cum" type="context">cũ</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 ſignificati, talis eſt. <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/>ſitus <var>.a.c.</var> minor autem <var>.b.d.</var> cum ergo <lb/>volueris inuenire punctum circunfe-<lb/>rentiæ correſpondentem 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 ſit <var>.h.</var> <reg norm="latusque" type="simple">latusq́</reg> <reg norm="tetragonicum" type="context">tetragonicũ</reg> pro-<lb/>ducti <var>.a.e.</var> in <var>.e.c.</var> quod ſit <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 ſecundi axis ita ſe habeat, quæ po <lb/>ſtea iuncta axi maiori, ad angulosrectos in puncto <var>.e.</var> dabit ſitum puncti <var>.f.</var> quæſiti ex <lb/>dicta .21. primi lib. Pergei, ſed talis modus prolixus eſt.</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 propoſitus maior axis <var>.q.p.</var> minor verò <var>.e.c.</var> ad angulos rectos ſe inuicem <lb/>ſecantes in puncto <var>.o.</var> deſcribatur circulus <var>.q.n.p.a.</var> cuius diameter ſit 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> deſignetur poſtea ſeparatim circulus <var>.u.b.n.</var> cuius diameter æqualis ſit ſe <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 ſit <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/>ſecabitur à minimo circulo in puncto <var>.i.</var> cum itaque longitudo <var>.u.i.</var> menſurata fue-<lb/>rit in <var>.u.b.</var> maximi circuli à puncto <var>.u.</var> habebimus propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Cuius reiratio eſt, quia <var>.u.b.</var> mediocris circuli diuiditur à gyro minimi in puncto <lb/>i. eadem proportione, qua diuiſa eſt <var>.u.n.</var> in puncto <var>.c.</var> quod manifeſtum eſt exſimi-<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> ſed ita <lb/>eſſe oportet parallelas maximi circuli, quotieſcunque circunferentia ipſius ellipſis <lb/>tranſitura ſit per <var>.c.</var> vt in .51. cap. meæ gnomonicæ oſtenſum 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 deſignandi angulum, certo modo conditionatum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">NVllius reuera difficultatis mihi videtur eſſe, quotieſcunque nobis propoſita <lb/>fuerint duo puncta <var>.a.</var> et <var>.b.</var> ſimul 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="conſtituant" type="context">conſtituãt</reg> <lb/>angulum æqualem dato, & ipſæ directè <reg norm="con" type="context">cõ</reg> <lb/>iunctæ conſtituant lineam æqualem da-<lb/>tæ. </s> + <s xml:space="preserve">Nam ducatur linea indefinita per <lb/>puncta propoſita, cuius lineæ, pars illa, quę <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 ſit <var>.o.c.</var> reliqua verò ſit <var>.o.e.</var> </s> + <s xml:space="preserve">deinde aperiatur circinus quantum <var>.<lb/>o.c.</var> <reg norm="poſitoque" type="simple">poſitoq́;</reg> vno pede in <var>.b.</var> <reg norm="deſignentur" type="context">deſignẽtur</reg> cum altero duo arcus <var>.n.K.</var> poſito iterum vno <lb/>pede in <var>.a.</var> deſignentur alij duo arcus inter-<lb/>ſecantes 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> tranſibit, quam <var>.n.K.</var> mente <reg norm="con- cipio" type="context">cõ-<lb/>cipio</reg>, vt axis minor <reg norm="vnius" type="simple">vniꝰ</reg> ellipſis, cuius <var>.e.c.</var> <lb/>ſit axis maior, quibus axibus mediantibus <lb/>deſignetur ellipſis <var>.n.c.K.e.</var> conſidero dein <lb/>de <var>.a.b.</var> vt <reg norm="chordanvnius" type="context">chordãvnius</reg> circuli, ſeu portio-<lb/>nis circularis, quæ capax ſit vnius anguli <lb/>æqualis angulo <var>.d.</var> propoſito, ex .32. tertij <lb/>Euclid. cuius circunferentia, circunferen-<lb/>tiam ipſius ellipſis ſecabit in <reg norm="duobus" type="simple">duobꝰ</reg> punctis <lb/>quorum <reg norm="vnum" type="context">vnũ</reg> ſit <var>.i.</var> à quo protractæ cum fue <lb/>rint duæ lineæ <var>.a.i.</var> et <var>.i.b.</var> habebis propoſitum, 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 monſtrauit <lb/>neſcio quis, ni fallor, talis eſt. </s> + <s xml:space="preserve">Deſignato meridiano <var>.l.b.m.q.</var> <reg norm="ductisque" type="simple">ductisq́;</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> ſit <lb/>verticalis <var>.b.q.</var> vero <reg norm="orizontalis" type="context">orizõtalis</reg>, <reg norm="ductoque" type="simple">ductoq́;</reg> diametro <var>.f.h.</var> tropici Cancri ſe-<lb/>cundum altitudinem poli datam, <reg norm="deſcriptoque" type="simple">deſcriptoq́;</reg> dimidio circulo <var>.f.z.h.</var> ipſius paralleli, <lb/><reg norm="inuentoque" type="simple">inuentoq́;</reg> puncto <var>.z.</var> horæ propoſitę, & 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 poſtea <var>.f.ω.</var> et <var>.<lb/>r.t.</var> vſque ad orizontalem <var>.q.b.</var> parallelas ad diametrum <var>.l.m.</var> verticalem. </s> + <s xml:space="preserve">Determi-<lb/>nato poſtea gnomone <var>.g.s.</var> in orizontis axe, <reg norm="ductaque" type="simple">ductaq́;</reg> vmbrarum linea <var>.s.K.</var> parallela <lb/>orizontali, <reg norm="ductaque" type="simple">ductaq́;</reg> <var>.y.g.K.</var> ad terminandam <var>.s.K.</var> delineas deinde ſeparatim circulum <lb/><var>q.x.b.n.</var> magnitudinis prioris, qui quidem circulus ſignificet orizontem ipſum, 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">cẽ<lb/>tro</reg> <var>.g.</var> diſtantia, vt ſunt in diametro orizontali prioris circuli, ducis poſtea 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/>tranſire facis circunferentiam circuli per <reg norm="quintam" type="context">quintã</reg> quarti Euclidis, poſtea in dicto dia-<lb/>metro accipis punctum <var>.t.</var> ita diſtans à centro, & ex eadem parte, vt in priori circulo, <lb/>à quo puncto ducis <var>.t.u.</var> parallelam <var>.x.n.</var> vſque 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> ipſius circularis circunferentiæ <var>.g.u.</var> inde-<lb/>terminatam, quam poſtea terminas in puncto <var>.K.</var> ita quod <var>.g.K.</var> æqualis ſit <var>.s.K</var>. </s> + <s xml:space="preserve">Dicis <lb/>poſtea punctum <var>.K.</var> in eodem ſitu reperiri, reſpectu duorum diametrorum <var>.q.b.</var> me-<lb/>ridiani. et <var>.l.m.</var> verticalis, vt decet, & oportet punctum horæ propoſitę exiſtere.</s> + </p> + <p> + <s xml:space="preserve">Quod quidem dico eſſe falſum, propterea quod perpendiculares quas cogita-<lb/>mus cadere à punctis circunferentiæ cuiuſuis paralleli ſupra 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 ſecantem æquatorem, omnes caduntin gyro elliptico, oxygonio, ſeu de-<lb/>fectionali, & non circulari. </s> + <s xml:space="preserve">Vnde per ſupradicta tria puncta <var>.n.ω.x.</var> oporteret tranſi <lb/>re talem circunferentiam, & non <reg norm="circularem" type="context">circularẽ</reg>, quæ circunferentia eſſet vnius ellipſis, <lb/>cuius minor axis in diametro <var>.b.q.</var> eſſet .ab <var>.ω.</var> vſque ad <var>.i.</var> terminum ſini <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="tranſiſ- ſet" type="context">trãſiſ-<lb/>ſet</reg> per punctum <var>.c.</var> medium inter <var>.ω.</var> et <var>.i.</var> quę 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">tiguaq́;</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 ſinus altitudinis horę, et <var>.t.u.</var> æqualis eſſet <var>.r.z.</var> communi ſectioni paralle <lb/>li cum almicantarat ex .34. primi Euclid. et <var>.u.g.</var> æqualis eſſet <var>.o.y.</var> communi ſectioni <lb/>almicantarat cum meridiano, vel cum azimut illius horæ ex .4. primi, cum <var>.g.t.</var> æqua <lb/>lis ſit ipſi <var>.o.r.</var> et <var>.t.u.</var> ipſi <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 ſe haberent, diſtan-<lb/>tia verò inter <var>.K.</var> et <var>.g.</var> <reg norm="iam" type="context">iã</reg> rectè ſum-<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">plurimũ</reg> <lb/>(in gyro circulari ſumptum) extra <lb/>puncta interſectionum ipſius 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> falſos, & non æquales il-<lb/>lis, qui fiunt ab azimut horæ cum <lb/>verticali, & cum meridiano, quæ <lb/>omnia ex cap .52. meæ gnomonicę <lb/>facilè videre potes.</s> + </p> + <p> + <s xml:space="preserve">Nectacere volo quod punctum <lb/>u. verum, hoc eſt ellipticum, inue-<lb/>niri poſſet ea via quam ſcripſi in <lb/>eodem .52. cap. qua mediante do-<lb/>cui demum inuenire punctum <var>.π.</var> <lb/>orizontis, quamuis in præſenti ca<lb/>ſu <var>.ω.λ.</var> perpendicularis eſſet ſupra <lb/>minorem axem ipſius ellipſis, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> ſupra maiorem axem, quod ta-<lb/>men minimè mutat ordinem, imò <lb/>rationes eędem ſunt, tam in vna, <lb/>quam in alia operatione, ſed vt il-<lb/>licò <reg norm="idipsum" type="context">idipsũ</reg> habeas, fac vt <var>.t.u.</var> æqua <lb/>lis. ſit <var>.r.z</var>. </s> + <s xml:space="preserve">& tunc punctum <var>.K.</var> erit <lb/><reg norm="quæſitum" type="context">quæſitũ</reg>, quod ego in .52. cap. meę <lb/>gnomonicę, ijs verbis ſignificaui.</s> + </p> + <quote> + <s xml:space="preserve"><reg norm="Itaque" type="simple">Itaq;</reg> mediis binis triangulis ijs, <lb/><reg norm="medioque" type="simple">medioq́;</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 ſcribere promiſi delineandi lineas horarias communes in <lb/>pariete perpendiculariter ad orizontem rectum, declinantem à meridiano, <lb/>ſumendus eſt ex .46. cap. meæ gnomonicæ, hocſcilicet ordine.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia, orizon hic ſubſcriptus. or. oc <var>.M.S.</var> diuiſus à meridiana <var>.M.<lb/>S.</var> et verticali ſeu æquinoctiali. or. oc. </s> + <s xml:space="preserve">Sitq́ue <var>.e.t.</var> communis ſectio muri cum ori-<lb/>zonte, et <var>.g.n.</var> ſit gnomon perpendicularis ipſi muro, vnde ex dictis in mea gnomo-<lb/>nica, cognoſcemus in ipſa murali orizontali totam <var>.e.t.</var> inter meridianam orizonta-<lb/>lem, & æquinoctialem orizontalem, cognoſcemus etiam partem <var>.g.t.</var> ipſius æ-<lb/>quinoctialis orizontalis, quam quidem accipiamus in rectitudine ipſius muralis ori <lb/>zontalis, quæ quidem ſit <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> deſignetur vna medietas circuli verſus <var>.e.</var> cuiuſuis magnitudinis, quæ di-<lb/>uiſa in .12. partes ęquales, ſignificabit <reg norm="medietatem" type="context">medietatẽ</reg> æquatoris, <reg norm="protrahanturque" type="simple">protrahanturq́;</reg> lineæ oc <lb/>cultæ à centro <var>.G.</var> per ſectiones circunferentię dimidij circuli, quæ fignificabunt <reg norm="con" type="context">cõ</reg> <lb/>munes ſectiones 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> ſignifi-<lb/>cabit communem ſectionem æquatoris cum muro propoſito, & erit ęquędiſtans me <lb/>ridianæ murali ex .6. vndecimi Eucli. eo quod ex .19. eiuſdem <reg norm="vnaquæque" type="simple">vnaquæq;</reg> illarum, per <lb/>pendicularis eſt tali orizonti. </s> + <s xml:space="preserve">Videantur nunc puncta communia iſti <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> ſignificat punctum axis mundi, & meridianæ in mu-<lb/>ro propoſito, eo quod in tali ſitu ſphæræ rectæ, dictum punctum reperitur in orizon <lb/>re, cum <var>.M.s.</var> non ſolum ſit meridiana orizontalis, ſed etiam axis mundi, </s> + <s xml:space="preserve">deinde nul <lb/>li dubium eſt quin meridiana muralis ſit 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">linearũ</reg> horariarum erit <lb/>ſub orizontali <var>.e.t.G.</var> alterum <lb/>vero <reg norm="dimidium" type="context">dimidiũ</reg> ſupra ipſam, opor <lb/>tet quod quę ſupra ſunt produ-<lb/>cantur à parte .oe. ſub orizonta-<lb/>li, ab alia parte meridianæ, & <lb/>talis erit effigies horologij mura <lb/>lis in hoc ſphęrę ſitu, hoc eſt ver <lb/>ſus quartam orientalem auſtra-<lb/><reg norm="lemque" type="simple">lemq́;</reg>, vnde orizontalis <var>.e.t.</var> <lb/>erit ſemper horæ .6. matutinæ, <lb/>ſecunda verò ab ipſa erit horæ. <lb/>7. tertia <reg norm="autem" type="context">autẽ</reg> horæ .8. & ſic 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">Quotieſcunque verò angu-<lb/>lus <var>.n.g.e.</var> minor erit maxima <lb/>Solis declinatione, & Sol fuerit <lb/>in parte auſtrali ab æquatore <reg norm="cum" type="context">cũ</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 ipſo Sole à mane vſque ad veſperam.</s> + </p> + <p> + <s xml:space="preserve">Huius quidem rei ſpeculatio, vnicuique manifeſta erit, qui rationes .46. cap. no-<lb/>ſtræ gnomonicæ prius intellexerit, vbi manifeſtè apparet proportionem ſemidiame <lb/>tri horologij (ſi ita eam appellare licet) ad ſemidiametrum æquatoris horarij ſem-<lb/>per eſſe, vt <var>.e.t.</var> ad <var>.t.g.</var> hoc eſt 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 ſemper iuſto breuiores ſint.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head style="it" xml:space="preserve">Declar atio quorundam verborum noſtræ<unsure/> Gnomonicæ. <lb/><reg norm="Defenſioque" type="simple">Defenſioq́ꝫ</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 ſignificas te totum <num value="52">.<lb/>52.</num> caput meæ gnomonicæ intelle-<lb/>xiſſe, præter illa verba, quæ etiam <lb/>ſuperioribus diebus ad te ſcripſi, <lb/>hoc eſt.</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">medioq́;</reg> azimut Solis, pariter ho-<lb/>rologia fabricari poterunt.</s> + </quote> + <p> + <s xml:space="preserve">Quapropter nealiquid tibi de-<lb/>ſit, ſcire debes, menihil aliud, eo <lb/>in loco inferre voluiſſe, quàm <reg norm="quod" type="wordlist">qđ</reg> <lb/>punctum horæ propoſitæ in plano <lb/>horologij orizontali reperiri po-<lb/>teſt, ope longitudinis vmbræ gno-<lb/>monis, & eius declinationis à ver-<lb/>ticali linea, ſeu à meridiana orizon <lb/>tali, iam in ipſo horologij plano <lb/>ductis.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſit analemma <var>.<lb/>l.q.m.b.</var> in quo <var>.l.m.</var> ſit verticalis <var>.q.<lb/>b.</var> verò orizontalis <var>.f.n.h.</var> autem ſit <lb/>ſemicirculus, cuiuſuis paralleli æqui <lb/>noctiali, cuius diameter fit <var>.f.h.</var> et <var>.<lb/>n.</var> ſit Solis locus in ipſo parallelo: </s> + <s xml:space="preserve">n.<lb/>r. autem ſit rectus ſinus arcus <var>.f.n</var>: et <var>.<lb/>r.o.z.</var> ſectio communis ipſius almi-<lb/>cantarat cum meridiano, et <var>.s.a.</var> <reg norm="con" type="context">cõ</reg> <lb/>munis ſectio 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 ſinus alti-<lb/>tudinis ipſius 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 cognoſcitur longitudo vmbræ gnomonis hoc eſt <var>.s.a</var>.</s> + </p> + <p> + <s xml:space="preserve">Cum autem dico, <reg norm="medioque" type="simple">medioq́;</reg> azimut Solis, nihil aliud ſigniſicare volo, niſi angu-<lb/>lum, quem terminat linea azimutalis horologij, hoc eſt vmbra gnomonis cum li-<lb/>nea meridiana, ſeu cum verticali in ipſo plano horologij. </s> + <s xml:space="preserve">qui quidem anguli, æqua <lb/>les ſunt ijs, qui in triangulo conſtituto 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 eſt, angulus verò terminatus ab <var>.n.r.</var> et <var>.o.z.</var> il <lb/>le eſt quem conſtituit azimut cum verticali, vel ipſi æqualis, vt coalternus, reliquus <lb/>verò in <reg norm="puncto" type="context">pũcto</reg> <var>.o.</var> ille eſt <reg norm="quem" type="context">quẽ</reg> azimut facit <reg norm="cum" type="context">cũ</reg> meridiano, vel ipſi ęqualis vt coalternus.</s> + </p> + <p> + <s xml:space="preserve">Vnde quotieſcunque volueris in aliquo plano, orizonti parallelo, lineas hora-<lb/>rias ducere, iudico optimum fore ſi ſeparatim deſignatæ fuerint hæ tres figuræ, hoc <lb/>eſt analemma meridianum, vel azimutale, vt ita dicam, </s> + <s xml:space="preserve">deinde parallelus <reg norm="inſeruiens" type="context">inſeruiẽs</reg> <lb/>pro tropicis, vt ego feci cap .51. meæ gnomonicæ, quæ duæ figuræ, <reg norm="ſufficientes" type="context">ſufficiẽtes</reg> <reg norm="erunt" type="context">erũt</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 ſeparatam <lb/>deſignaui, quę erit circulus <var>.H.I.<lb/>K.</var> eiuſdem magnitudinis cum ana-<lb/>lemmate, in quo, ductis duobus <lb/>dia metris inuicom ad rectos, quo-<lb/>rum vnus <var>.H.I.</var> ſignificet orizon-<lb/>talem lineam, reliqua verò <reg norm="ver- ticalem" type="context">ver-<lb/>ticalẽ</reg>, ducatur poſtea <var>.s.a.</var> tam di-<lb/>ſtans ab <var>.H.I.</var> quanta eſt 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/>ſius zenit, accipiantur arcus æqua-<lb/>les ijs ipſorum azimut, quos termi-<lb/>nat zenit, & ipſi almicantarat, vt <lb/>exempli gratia, accipiemus arcum <lb/><var>k.L.</var> æqualem arcui <var>.L.z.</var> ipſius ana-<lb/>lemmatis, ducta poſtea linea occul <lb/>ta <var>.o.L.</var> ſignabimus azimutalem <var>.s.a.</var> <lb/>in puncto <var>.a.</var> vbi hæ duæ lineæ ſein <lb/>uicem ſecant, & ſic habebimus iu-<lb/>ſtam <reg norm="quantitatem" type="context">quãtitatem</reg> ipſius vmbræ gno <lb/>monis <var>.s.o.</var> tali hora, </s> + <s xml:space="preserve">deinde in ori-<lb/>zontali <var>.H.I.</var> ſumatur <var>.o.r.</var> à centro <var>.<lb/>o.</var> ęqualis ei quę in analemate repe <lb/>ritur, quæ vna portio eſt commu-<lb/>nis ſectionis 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> vſque ad circunferentiam, in qua accipiatur <var>.r.n.</var> æqua <lb/>lis ei quæ eſt in parallelo, ducatur poſtea <var>.o.n.M.</var> & habe bimus triangulum <var>.o.r.n.</var> ſi-<lb/>milem <reg norm="æqualemque" type="simple">æqualemq́;</reg> triangulo iam ſupradicto. </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 ęqualis, quem azimut con-<lb/>ſtituit cum verticali, ita quod ſi talis circulus <var>.H.k.I.</var> eſſet planum horologij orizon-<lb/>talis, ſuppoſito <var>.o.</var> pro pede gnomonis, ſecando poſtea <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 ſit ad partem ſibi conuenientem, reſpectu <var>.o.<lb/>k.</var> ipſa pro verticali ſuppoſita, quod tibirelinquo, cum hoc facillimum ſit, 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 ſignificas Chriſtophorum Clauium, me duobus in locis meæ gno <lb/>monicæ redarguere, iam vidi. </s> + <s xml:space="preserve">Circa primum locum igitur, qui eſt in pagin .161. <lb/>ita inquit.</s> + </p> + <quote> + <s xml:space="preserve">Non enim deſunt, qui vel omninò negent, inter quos eſt Ioannes Baptiſta Be-<lb/>nedictus in ſua gnomonica cap .70. et .71. vbialia, & multo longiore ratione cona-<lb/>tur arcus ſignorum deſcribere, vel certe dubitent, hoc modo rectè poſſe deſcribi ar-<lb/>cus ſignorum, cum rationem non videant, qua hæc noſtra deſcriptio quam quidem <lb/>omnes ſcriptores ſine vlla demonſtratione tradunt nitatur.</s> + </quote> + <p> + <s xml:space="preserve">Abſque dubio raptim tranſcurrit illa capita .70. 71. </s> + <s xml:space="preserve">Reuerendus Clauius alio quin <lb/>non ſcripſiſſet, quòd ego alia & multo longiore ratione conatus ſim arcus ſignorum <lb/>deſcribere & c. præſertim cum eadem prorſus ratio, quæ ibi à me tradita eſt, illa ſit, <lb/>quam ip ſe ſuis ſcriptis inſeruit.</s> + </p> + <p> + <s xml:space="preserve">Meus igitur modus in dictis capitibus traditus, minime diſcrepat ab eo, ſed ab il-<lb/>lorum modo, quorum opinio eſt 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. poſitæ, æqualia eſſe interuallis <var>.e.h</var>:h.u:u.n:n.m. et <var>.m.d.</var> præcedentis figuræ, <lb/>qui etiam ſupponunt <var>.t.e.</var> meæ figuræ .75. eſſe directè coniuncta cum linea <var>.e.h.u.n.<lb/>m.d.</var> & propterea verſus finem .73. pag. dixi.</s> + </p> + <quote> + <s xml:space="preserve">Aduertat autem quam diligentiſſime quiſque ne ſe decipi patiatur à ſubſcripta fi <lb/>gura ſemicirculi <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 vetuſtioribus (quorum ſcripta ad meas manus <lb/>peruenerunt) qui ſumentes interualla <var>e.h</var>:h.u. & c. figuræ. pag .75. æqualia illis figu-<lb/>ræ pag .74. <reg norm="putauerunt" type="context">putauerũt</reg> lineam <var>.t.e.</var> directè coniunctam eſſe cum <var>.e.h.</var> & c. quod quidem <lb/>maximi erroris cauſa erat, </s> + <s xml:space="preserve">& propterea cap .71. verum modum oſtendi, ſeruando il <lb/>lam eandem ſuppoſitionem, hoc eſt quod interſtitia <var>.e.h</var>: <var>h.u</var>: & c. figurę pag .75. æqua <lb/>lia ſint interſtitijs <var>.e.h</var>: <var>h.u.</var> & c. præcedentis figuræ, & ideò in dicto cap .71. dixi.</s> + </p> + <quote> + <s xml:space="preserve">Suppoſito deinde <var>.f.e.b.</var> lineam meridianam eſſe in plano orizontali, cęterę lineę <lb/>horarię erunt prędictę.</s> + </quote> + <p> + <s xml:space="preserve">Stantibus igitur his ſuppoſitis, vt habeantur omnia ſcientificè, 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) poſuit in figura horologij, vt ipſe docuit <lb/>propoſitione ſecunda, lib. ſecundi, cuius baſis eſt triangulum <var>.H.I.F.</var> ſuæ figurę (exem <lb/>pli gratia pro quinta hora poſt 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, eſt in ſua <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 ſunt, quamuis in plano æquatoris <lb/>tres ſint, & c.</s> + </quote> + <p> + <s xml:space="preserve">Angulus verò <var>.e.</var> quem dico rectum eſſe, in ſua figura eſt 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, eſt ſua <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 ſua eſt triangulum <var>.D.F.H.</var> vnde mea <var>.f.t.</var> reſpon-<lb/>det ſuę <var>.H.D.</var> & mea <var>.f.d.</var> ſuæ <var>.H.F.</var> & mea <var>.t.d.</var> ſuę <var>.D.F</var>. </s> + <s xml:space="preserve">Quartum autem triangulum <lb/><var>f.t.e.</var> in mea figura, reſpondet ſuo <var>.H.D.I.</var> & meum punctum <var>.t.</var> ſuo. </s> + <s xml:space="preserve">D, Nunc triangu <lb/>lum rectangulum, quod dico ſeparatim conſtituere, eſt illud tertium dictum corre-<lb/>ſpondens ſuo <var>.D.F.H.</var> vt ipſe facit in <reg norm="ſequenti" type="context">ſequẽti</reg> figura, quod ipſe vocat <var>.D.C.H.</var> & <reg norm="meus" type="simple">meꝰ</reg><unsure/> <lb/>radius <var>.t.x.</var> in ſua figura, ille eſt qui terminatur ab <var>.D.</var> & ab initio Tauri, & Virginis.</s> + </p> + <p> + <s xml:space="preserve">Et quamuis ego non ſcripſerim talem ſiguram, vt ipſe fecit, nihilominus ipſam <lb/>verbis deſcripſi eomet modo, & propterea dixi.</s> + </p> + <quote> + <s xml:space="preserve">Quam <reg norm="diuiſionem" type="context">diuiſionẽ</reg>, ſi in triangulo ſeorſum deſcripto inuenire voluerimus, res erit <lb/>inuentu facillima, cum rectum angulum <var>.f.t.d.</var> (reſpondentem ſuo <var>.H.D.C.</var>) prędicti <lb/>trianguli tertij ea ratione diuiſerimus, & c.</s> + </quote> + <p> + <s xml:space="preserve">Quapropter Reuerendus Clauius non animaduertit meam rationem aliam non <lb/>eſſe, nec puncto longiorem ſua, cum eademmet ipſa ſit.</s> + </p> + <p> + <s xml:space="preserve">Citaui etiam Munſterum cap .30. eo quod in ea impreſſione, quam tunc prę mani <lb/>bus habui, vidi in ea figura, quam ipſe vocat fundamentum horologiorum, literam <lb/>c. poſitam eſſe loco <var>.f.</var> et <var>.f.</var> loco <var>.c.</var> quod cauſæ fuit, vt omnia mendoſa viderentur, re <lb/>centiores autem impreſſiones correctæ ſunt.</s> + </p> + <p> + <s xml:space="preserve">Rurſus alio in loco mihi accidit vt repręhenderim Alexandrum Piccolomineum <lb/>in libris de ſphęra, qui quidem dicebat eas figuras ſuperficiales, quæ paucioribus an <lb/>gulis circunſcriberentur, capaciores eſſe alijs, dummodo earum periphæriæ eſſent <lb/>æquales.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem correctę ſunt eo in loco impreſſiones, & qui non viderit primas, pu-<lb/>tabit me immeritò ipſum repræhendere.</s> + </p> + <p> + <s xml:space="preserve">Idem etiam dico de eo capite ipſius Piccolominei, in ijſdem libris, vbi tractat de <lb/>modo, quo vſi ſunt antiqui ad diuidendum zodiacum in .12. ſigna, quod erat circa <lb/>finem quarti libri.</s> + </p> + <p> + <s xml:space="preserve">Nunc verò, in recentioribus impreſſionibus, illud caput poſitum non eſt. </s> + <s xml:space="preserve">Impreſ <lb/>ſiones autem illæ, vbi talia dixit, duæ fuerunt, quarum prima erat anni .1540. ſecun <lb/>da verò .1552 Venetijs apud Andream Puteum.</s> + </p> + <p> + <s xml:space="preserve">Alius verò locus ipſius Reuerendi Clauij, contra meas repręhenſiones, eſt circa <lb/>finem pag .298. & circa .299. vbi ita ſcribit.</s> + </p> + <quote> + <s xml:space="preserve">Ex his liquido conſtat, non rectè à Ioan. Baptiſta Benedicto in ſua gnomonica ca <lb/>pit .49. repręhendi hancrationem deſcribendi horologij declinantis, qua omnes fe-<lb/>rè alij ſcriptores vtuntur, quoniam, vt ex demonſtratione à nobis allata conſtat, re-<lb/>ctè per eam lineæ ho rarię 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/>eſt, ſed nulla ratione noſtro contrarius, quia nos conſtituimus <var>.D.E.F.</var> angulum de-<lb/>clinationis plani à verticali circulo propriè dicto, ipſe autem loco huius anguli aſſu-<lb/>mit angulum declinationis eiuſdem plani à Meridiano circulo, vnde mirum non eſt <lb/>modum ipſius à noſtro diſcrepare. </s> + <s xml:space="preserve">Quod ſi <reg norm="conſtitueremus" type="context">cõſtitueremus</reg> <var>.D.E.F.</var> angulum decli-<lb/>natio nis plani à Meridiano, ut ipſe (quemadmodum forſitan ab alijs putauit fieri) <lb/>& in reliqua deſcriptione progrederemur, vt tradidimus, proculdubio <reg norm="horologium" type="context">horologiũ</reg> <lb/>declinans perperam deſcriberetur, vt rectè docet.</s> + </quote> + <p> + <s xml:space="preserve">Optimè ſcripſiſſet Reuerendus Clauius, ſi verum fuiſſet, quod antiqui ſumerent <lb/>declinationem ſuperius <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. meę gnomonicę ſumere 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 ſidera primum in Munſtero cap .16. ſuæ horologiographiæ, vbi clarè docet <lb/>accipere angulum compræhenſum inter meridianum, & planum propoſitum, vbi <lb/>etiam ponit quandam figuram ædificij cum pariete ſuper quo deſignatum eſt quod <lb/>dam horologium, & vbi ſe manifeſtè declarar, ita dicens.</s> + </p> + <quote> + <s xml:space="preserve">Nam ipſarum partium complementum. propoſitum indicabit angulum, quan-<lb/>tus videlicet fuerit arcus eiuſdem circuli <var>.d.e.f.g.</var> à puncto <var>.g.</var> vſque ad productam li-<lb/>neam meridianam interceptus, qui vnà cum ipſo <var>.f.g.</var> quadrantem integrare videtur, <lb/>vt in ſequenti figura: </s> + <s xml:space="preserve">quoniam arcus <var>.f.g.</var> eſt ſexaginta partium, qualium <var>.e.</var> f<unsure/>. <reg norm="quadrans" type="context">quadrãs</reg> <lb/>nonaginta, vnde concluditur reliquam partem hoc eſt, datum inclinationis <reg norm="angulum" type="context">angulũ</reg>, <lb/>fore partium triginta ſimilium.</s> + </quote> + <p> + <s xml:space="preserve">Orontius verò cap .13. <reg norm="ijſdem" type="context">ijſdẽ</reg> vtitur verbis, cum figura ſimili ad reliqua autem ipſius <lb/>R. Clauij, videnda nondum mihi otium fuit. </s> + <s xml:space="preserve">quod ſi dabitur, tibi libenter<unsure/> dicam <lb/>quid ſentiam.</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/>ſuper 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 ſuper cyllindro immobili, ad orizontemq́ue <lb/>perpendiculariter erecto difficile tibi non erit, (quod à me poſtulaſti) ſi <lb/>modum .53. cap. meæ gnomonicæ obſeruaueris, accipiendo tamen pro <lb/>linea orizontali in tabula non aliquam rectam lineam, ſed circularem, <lb/>ſimilemque circunferentiæ ipſius cyllindri, dico autem ſimilem, eo quod ſi gn o-<lb/>mon <var>.o.x.</var> ſupra tabulam ſignatus, & perpendicularis ipſi orizontali circulari <var>.b.i.x.</var> <lb/>eſſet dimidia, vel tertia vel quarta pars gnomonis cyllindro infixi, oporteret, vt <lb/>ſemidiameter 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 ſemidiametri cyllindri, vt <lb/>omnes arcus huiuſmodi circuli in <lb/>ter ipſos azimut intercepti ſimiles <lb/>ſint arcubus cyllindri, quod à <lb/>te ipſo facilè videre ſcientificè po <lb/>teris. </s> + <s xml:space="preserve">reliqua nihil mutanda erunt <lb/>ab eo, quod ſcripſi circa figuram <num value="53">.<lb/>53.</num> cap. vt dixi. </s> + <s xml:space="preserve">Vnde inuenta <lb/>cum fuerit diſtantia 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 ſemper ab orizontali per <lb/>pendiculariter deſcendit, illicò <lb/>punctum <var>.t.</var> horæ propoſitæ 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 horarię lineæ inuenta fuerint, quæ à Solis ſi-<lb/>tu in diuerſis parallelis efficiuntur, ſi voluerimus ipſam lineam <reg norm="horariam" type="context">horariã</reg> ducere, ſcien <lb/>dum primò eſt ipſam lineam horariam eſſe communem ſectionem circuli horarij, <lb/>illius horæ cum ſuperficie cyllindrica, </s> + <s xml:space="preserve">& propterea ellipticam, vt oſtendit Serenus <lb/>in .19. primi lib. quod etiam ellicere poſſumus ab eo, quod Archimedes in .10. pro-<lb/>poſitione libr. de conoidalibus, ſcribit. </s> + <s xml:space="preserve">Quapropter oporter nos inſtrumen-<lb/>tum prius componere, modo circini, ſed trium crurum, quæ omnia in eadem <lb/>plana ſuperficie ſint, ea tamen arte factum, vt quodlibet illorum poſſimus pro-<lb/>longare, necnon contrahere, ut cum duo extrema firmata fuerint, media poſ-<lb/>ſit circunduci circa centrum, ſeu punctum commune illarum interſectionum <reg norm="ſimulque" type="simple">ſimulq́;</reg> <lb/>poſſit produci, necnon abbreuiari vel augeri, & diminui, vt mediante ſua extremi-<lb/>tate inſeriori poſſimus delineare gyrum ellipticum horarium, dum <reg norm="centrum" type="context">cẽtrum</reg> ipſorum <lb/>crurum adhæreat extremitati gnomonis, reliquæ vero extremitates ipſorum <reg norm="crurum" type="context">crurũ</reg> <lb/>ſint ſupra puncta inuenta ipſius horæ. </s> + <s xml:space="preserve">oportet etiam vt hoc inſtrumentum à tergo <lb/>ipſorum crurum habeat in ſuperiori parte ſuperficiem quandam <reg norm="ſemicircularem" type="context">ſemicircularẽ</reg>, quę <lb/>ſit vice vnius partis illius ſuperficiei, in qua ſupponuntur omnia crura inſtrumenti, <lb/>& hoc quantum fieri poteſt, quod quidem fieri debet, ne crus medium, hoc eſt mo <lb/>bile, exeat à tali ſuperficie, ſeu declinet ab ea, quæ ſemper ſupponitur in ſitu circuli <lb/>horarij talis horæ. </s> + <s xml:space="preserve">oportet etiam, vt iuxta circunferentiam dimidij circuli ſint duo <lb/>gyri eiuſdem materiæ inter ſe parum diſtantes, ita ut crura poſſint moueri, intra hos <lb/>gyros, & dimidium circulum, & quod inter hos gyros locatæ ſint duæ cochleæ, ſeu <lb/>duo helices, vt quando voluerimus, poſſimus fir-<lb/>mare ipſa crura extrema, dum eorum extremitates <lb/> + <anchor type="figure" xlink:label="fig-0338-01a" xlink:href="fig-0338-01"/> + fuerint ſupra puncta inuenta illius horæ, </s> + <s xml:space="preserve">deinde in <lb/>dorſo iſtius inſtrumenti, circa centrum coniunctio <lb/>nis, rectè factum erit ſi 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 inſtrumentum excogitaui ad fugiendum <lb/>tædium inueniendi dictam ellipticam ex punctis.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem ſciendum eſt, quod vnus tantum-<lb/>modo gnomon ſufficiens non erit pro tota die æſti-<lb/>ua, neque duo, niſi valde breues fuerint reſpectu <lb/>ſemidiametri cyllindri, & in ſitu medio quartarum <lb/>meridionalium noſtro 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="diuiſusque" type="simple">diuiſusq́;</reg> à duobus diametris <var>.d.e.</var> et <var>.c.<lb/>f.</var> quarum <var>.c.f.</var> ſit pro meridiana: </s> + <s xml:space="preserve">d.e. autem pro verticali, <reg norm="ſitque" type="simple">ſitq́;</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> ſeptentrionalis, <reg norm="computeturque" type="simple">computeturq́;</reg> maxima. <lb/></s> + <s xml:space="preserve">Solis amplitudo ab <var>.f.</var> verſus <var>.e.</var> quæ terminetur ab <var>.q.</var> ita <reg norm="quodarcus" type="simple">quodarcꝰ</reg> <var>.f.q.</var> minor ſit <reg norm="quam" type="context">quã</reg> <lb/>graduum .45. aliter impoſſibile eſſet duobus <reg norm="tantummodo" type="context">tantũmodo</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> vſque ad contingentem <var>.q.p.</var> vnde <var>.u.i.</var> longitu <lb/>do erit vniuſcuiuſque 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">Huiuſmodi rei ratio per ſe nota erit quotieſcunque cogitauerimus verum arcum <lb/><var>e.b.</var> amplitudinis æſtiuæ, <reg norm="protractaque" type="simple">protractaq́;</reg> <var>.o.b.</var> quę parallela erit <var>.q.p.</var> vnde cum Sol tem-<lb/>pore æſtiuo orietur, tunc radios ſuos emittet via iſtarum æquidiſtantium linearum.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi 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="vnuſquiſque" type="simple">vnuſquiſq;</reg> poteſt eſſe maior <lb/>tertia parte ſemidiametri cyllin-<lb/>dri, ſed ſi voluerimus ſcire <reg norm="quan- tum" type="context">quan-<lb/>tũ</reg> ad plus poſſit eſſe longus vnuſ-<lb/>quiſque 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">quadratũ</reg> <var>.o.a.h.u.</var> ex <lb/>ſemidiametro dicti circuli, a dia-<lb/>metro poſtea <var>.o.h.</var> huiuſmodi qua <lb/>drati ſubtrahatur ſemidiameter <var>.<lb/>o.e.</var> circuli, reſiduum verò <var>.e.h.</var> ip-<lb/>ſius diametri <var>.o.h.</var> quadrati, erit <reg norm="lon" type="context">lõ</reg> <lb/>gitudo gnomonis, vbi ſimul appa <lb/>ret huiuſmodi rei ratio, eo quod <lb/>cum gnomon <var>.e.h.</var> orientalis deſi-<lb/>net operari, illico meridionalis <var>.<lb/>f.g.</var> ſubintrabit, poſt hunc verò occidentalis <var>.d.K.</var> monſtrabit reliquum diei.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Earundem line arum deſcriptio ſaper conum rectum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">CVm ſuper datum conum re-<lb/> + <anchor type="figure" xlink:label="fig-0339-02a" xlink:href="fig-0339-02"/> + ctum idem facere volueris <lb/>eſto conus <var>.A.</var> & R. qui diuiſus ima <lb/>ginatione ſit à quodam plano per <lb/>axem, & communis ſectio ſit trian <lb/>gulus <var>.A.</var> & R. in quo plano cogite <lb/>mus gnomonem infixum ad rectos <lb/>vbi volueris, qui ſit <var>.p.t.o.</var> cogitem <lb/>etiam <var>.l.t.m.</var> aliud eſſe planum (in <lb/>quo ſit gnomon) quod conum ſe <lb/>cet, quæ quidem ſectio, circularis <lb/>erit, ex .4. primi Pergei. </s> + <s xml:space="preserve">imagine-<lb/>mur etiam ſuperficiem <var>.p.s.</var> eſſe azi <lb/>mut in quo gnomonreperirur, ſu-<lb/><reg norm="perficiemque" type="simple">perficiemq́;</reg> <var>.e.s.</var> azimut propoſitæ <lb/>horę, <reg norm="angulumque" type="simple">angulumq́;</reg> <var>.e.o.a.</var> contrapoſi <lb/><reg norm="tum" type="context">tũ</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 ver<lb/>ſus baſim tranſit per medium latitudinis ipſius gnomonis, concipiamus etiam mente <lb/><var>e.a.</var> communem ſectionem eſſe trianguli ſupra dicti cum azimut horæ, necnon pun-<lb/>ctum <var>.K.</var> eſſe commune radio Solis <var>.o.a.</var> & ſuperficiei conicæ, quod quidem eſt illud <lb/>quod quæritur, hoc ſcilicet modo. </s> + <s xml:space="preserve">Primum cognoſcimus angulum <var>.p.A.t.</var> vt medie <lb/>tas anguli totius coni, & angulum <var>.p.</var> rectum, vnde <var>.t.</var> tam intrinſecus, quam extrinſe-<lb/>custrianguli <var>.A.p.t.</var> nobis cognitus erit. </s> + <s xml:space="preserve">Nunc cum angulus <var>.A.t.o.</var> cognoſcatur, ſi <lb/>gnomon <var>t.o.</var> fixus fuerit in ſuperficie conica, ita qd cum latere <var>.A.t.</var> eſſiciat <reg norm="angulum" type="context">angulũ</reg> <lb/><var>A.t.o.</var> & lateraliter faciat angulosrectos cum ſuperficie conica, ad quod efficiendum <lb/>nulla eſt difficultas, cognoſcendo deinde <var>.A.t.</var> ſimul cum angulis <var>.A.</var> et <var>.t.</var> intrinſecis <lb/>trianguli ortogonij <var>.A.p.t.</var> cognoſcemus <var>.p.t.</var> et <var>.A.p.</var> vnde etiam tota <var>.o.p.</var> ſed cogno <lb/>ſcendo <var>.o.p.</var> cum angulo <var>.p.o.e.</var> (angulus enim <var>.p.o.e.</var> cognoſcitur ex hypotheſi cum <lb/>ſit inter azimut Solis & azimut gnomonis) cum angulo <var>.o.p.e.</var> recto cognoſcemus <var>.p.<lb/>e.</var> et <var>.o.e.</var> </s> + <s xml:space="preserve">deinde cum nobis nota ſit <var>.o.e.</var> cum angulo altitudinis Solis <var>.e.o.a.</var> & angu-<lb/>lo <var>.o.e.a.</var> recto cognoſc emus longitudinem azimutalis <var>.e.a.</var> necnon quantitatem <var>.a.o.</var> <lb/>Imaginata poſtea <var>.a.q.</var> æquidiſtante <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="cognitiſque" type="simple">cognitiſq́;</reg> cum angulo recto <var>.p.</var> cogno <lb/>ſcemus <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"/> + gnoſcemus <var>.A.i.</var> et <var>.q.i.</var> quę <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> ſemper <lb/>ſunt in eadem ſuperficie ſecante co <lb/>num, quæ etiam ſecat ſuperficiem <lb/>trianguli <var>.A.q.x.</var> ad rectos ex .18. vn <lb/>decimi, cum linea <var>.u.n.</var> perpendicu <lb/>laris ſit ſuperficiei trianguli <var>.A.q.i.</var> <lb/>ex .8. dicti, quia parallela eſt <var>.l.p.</var> quę <lb/>perpendicularis eſt ſuperficiei <reg norm="trian- guli" type="context">triã-<lb/>guli</reg> <var>.o.p.q.</var> ex .4. eiuſdem, ſequitur, <lb/>quod talis ſectio ( quæ intelligatur <lb/>per <var>.u.K.i.n.</var>) ſemper erit elliptica, <lb/>vel parabole, ſeu hyperbole, <reg norm="prout" type="simple">ꝓut</reg> <lb/>linea <var>.o.i.q.</var> ſecabit latus coni, oppo <lb/>ſitum lateri <var>.A.i.</var> diſtento in ipſa ſuperficie conica, ſeu ad ſuperiorem partem produ <lb/>ctum, velipſi 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> ſecare dictum oppoſitum latus lateri <var>.A.i.</var> <lb/>verſus baſim, vnde ſectio <var>.u.K.i.n.</var> erit elliptica. </s> + <s xml:space="preserve">quod facile cognitu eſt <reg norm="mediante" type="context">mediãte</reg> com <lb/>paratione angulorum <var>.A.q.i.</var> et <var>.q.A.i.</var> interſe, eo quod ſi eſſent ęquales, dicta ſect o <lb/>barabola eſſet ex .27. primi Eucli. et .11. primi Pergei, ſed ſi angulus <var>.A.q.i.</var> maior eſ-<lb/>ſet angulo <var>.q.A.i.</var> ſectio eſſet ellipſis, ex ultimo poſtulato primi Euclid. </s> + <s xml:space="preserve">& ex .13. pri-<lb/>mi Pergei, ſed ſi dictus angulus <var>.A.q.i.</var> minor eſſet angulo <var>.A.</var> tunc ſectio eſſet hyper-<lb/>bole ex dicto poſtulato & ex .12. primi Pergei. </s> + <s xml:space="preserve">Sit ergo primum vt <reg norm="dictum" type="context">dictũ</reg> eſt, hoc <lb/>eſt, quod ſectio eſſet oxygonia, ideſt elliptica, ſeu defectio (quod idem eſt,) ſepa-<lb/>ratim oportebit nos ellipſim deſignare <reg norm="ſimilem" type="context">ſimilẽ</reg> <reg norm="ęqualemque" type="context simple">ęqualẽq́;</reg> ei, quæ eſt <var>.u.K.i.n.</var> <reg norm="quod" type="wordlist">qđ</reg> <reg norm="quidem" type="context">quidẽ</reg> <lb/>difficile non erit, quotieſcunque ſuos axes inuenerimus, maiorem ſcilicet, & 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 ſit <var>.i.A.b.</var> quem diui-<lb/>demus per æqualia mediante <var>.A.q.</var> conſtituendo <var>.A.i.</var> huius anguli æqualem <var>.A.i.</var> ſu-<lb/>perficiei conicæ et <var>.A.q.</var> diuidentem, æqualem parti <var>.A.q.</var> axis coni, ducendo poſtea <lb/>ab <var>.i.</var> per <var>.q.</var> lineam vnam quouſque concurrat <var>.A.b.</var> in puncto <var>.b.</var> habebimus <var>.i.b.</var> pro <lb/>maiori axi ipſi ellipſis, quod per ſe clarum eſt, cuius medietas ſit <var>.i.c.</var> ſed <var>.i.q.</var> ipſius <var>.i.<lb/>b.</var> æqualis eſt ipſi <var>.q.i.</var> ipſius coni, ex quarta primi Eucli. et <var>.q.b.</var> ipſius <var>.i.b.</var> æqualis alte <lb/>ri parti inuiſibili. </s> + <s xml:space="preserve">Reliquum eſt, 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">æqualisq́;</reg> ei quæ eſt coni, & diui <lb/>ſa ſimiliter in <var>.a.</var> quæ <var>.u.n.</var> ipſius coni nobis cognita eſt ex lateribus <var>.A.u.</var> et <var>.A.n.</var> & ex <lb/>angulo coni, et <var>.a.q.</var> æqualis eſt <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 ipſius <var>.f.c.</var> quæ pro-<lb/>ducti ipſius <var>.i.q.</var> in <var>.q.b.</var> ad productum ipſius <var>.i.c.</var> in <var>.c.b.</var> & cum cognita nobis ſint <lb/>hæc tria producta hoc eſt <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 ſeipſa, cognoſcemus <reg norm="etiam" type="context">etiã</reg> <lb/>quartum ipſius <var>.f.c.</var> & fic <var>.f.c.</var> <reg norm="eiuſque" type="simple">eiuſq́;</reg> duplum <var>.f.r.</var> cogniti nobis itaque cum ſint hi duo <lb/>axes <var>.i.b.</var> et <var>.f.r.</var> formabimus ellipſim. </s> + <s xml:space="preserve">Deinde producemus axim <var>.b.i.</var> à part <var>e.i.</var> quo-<lb/>uſque <var>.i.o.</var> æqualis ſit ei quæ extra conum eſt, 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/>ſecabit in puncto <var>.K.</var> vnde habebimus quantita-<lb/>tem ipſius <var>.o.K.</var> et <var>.K.i.</var> rectam. </s> + <s xml:space="preserve">inde mediante cir-<lb/>cino ſi acceperimus rectam diſtantiam ab <var>.i.</var> ad <var>.K.</var> <lb/>in ellipſi, </s> + <s xml:space="preserve">deinde firmando pedem circini in pun-<lb/>cto <var>.i.</var> in ſuperficie conica, & cum alio ſignando <lb/>lineam vnam curuam ad partem <var>.K.</var> in ſuperficie <lb/>conica, ſumendo poſtea interuallum <var>.o.K.</var> extra el <lb/>lipſim, </s> + <s xml:space="preserve">deinde firmando vnum pedem circini in <lb/>extre mitate gnomonis, cum alio poſtea ſignan-<lb/>do aliam lineam curuam in ſuperficie ipſius coni, <lb/>quæ primam ſe cet in puncto <var>.K.</var> hoc erit punctum <lb/>quæſitum horę propoſitæ in ſuperficie conica <lb/>propoſita.</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 ſi talis ſectio fuerit parabole, vel hyperbo <lb/>le, tunc mediante ſuo diametro <var>.i.q.</var> cum baſi <var>.u.<lb/>q.n.</var> cognita, deſignabimus ipſam ſectionem <var>.u.i.</var> n <lb/>ope mei <reg norm="inſtrumenti" type="context">inſtrumẽti</reg> in calce meę gnomonicæ de <lb/>ſcripti, </s> + <s xml:space="preserve">deinde diuiſa <var>.u.q.</var> in <var>.a.</var> <reg norm="pro" type="simple">ꝓ</reg><reg norm="ductaque" type="simple">ductaq́;</reg> <var>q.i.</var> <reg norm="vſque" type="simple">vſq;</reg> <lb/> + <anchor type="figure" xlink:label="fig-0341-02a" xlink:href="fig-0341-02"/> + ad <var>.o.</var> <reg norm="ductaque" type="simple">ductaq́;</reg> <var>.o.a.</var> habebimus punctum <var>.K</var>. </s> + <s xml:space="preserve">Reli-<lb/>qua facienda ſunt, vt dictum eſt de ellipſi.</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 eiuſ-<lb/>dem horæ propoſitę, ducemus ab vno ad a-<lb/>liud, lineam horariam mediante circino trium <lb/>crurum, quem tibi ſcripſi nudius tertius pro cyl <lb/>lindro, quæ <reg norm="quidem" type="context">quidẽ</reg> linea crit portio gyri ellipſis, <lb/>ſeu hyperbolę, vel parabolę, vt à te ipſo 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 Sabaudiœ Ducis apparitore.</head> + <p> + <s xml:space="preserve">EX tuis literis cognoui quo erga me animo eſſes, <reg norm="qualique" type="simple">qualiq́;</reg> voluntate, ſed ne <lb/>tua pulcherrima ſtudia aliquo modo imperfecta <reg norm="relinquantur" type="simple">relinquant̃</reg>, vel ego tibi <lb/>deeſſe videar, dum Problemata geographica Magni Ptolomei conſi-<lb/>deras, aduerte, quod ſi putares in figura .6. cap. libr .7. geographię eiuſ-<lb/>dem (vt multi credunt) lineam <var>.V.*.</var> ſecare circunferentiam <var>.A.D.</var> in puncto <var>.G.</var> ita <lb/>vt punctus <var>.G.</var> ſit tropici æſtiui, ideſt arcum <var>.D.G.</var> eſſe graduum .24. cum illis inci-<lb/>deres in maximum errorem. </s> + <s xml:space="preserve">Quapropter conſidera quæ nunc tibi ſcribo.</s> + </p> + <p> + <s xml:space="preserve">Sit circulus <var>.A.B.C.D.</var> huius centrum <var>.E.</var> <reg norm="ſupponaturque" type="simple">ſupponaturq́;</reg> ſemidiameter <var>.E.D.</var> eſſe <lb/>partium 120. quarum <var>.E.Z.</var> in alio ſemidiametro <var>.C.E.</var> ei orthogonaliter coniuncto, <lb/>talium ſit .17. in ſemidiametro 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. ſimilium.</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/>ſectione lineę ductæ per puncta <var>.Z.D.</var> in puncto <var>.*.</var> ducatur demum <var>.V.*.</var> quæ ſeca-<lb/>bit circunf rentiam <var>.A.D.</var> in puncto <var>.G</var>. </s> + <s xml:space="preserve">Quæratur nunc quantitas ipſius <var>.G.D.</var> <lb/>Ad quod efficiendum quærenda primum eſt quantitas ipſius <var>.S.*.</var> quam illico co <lb/>gnoſcemus ex regula de tribus, cum dixerimus, ſi@ 17. dat nobis .120. quid dabit .41. <lb/>(nam duo triangula <var>.Z.E.D.</var> et <var>.Z.S.*.</var> ſunt inuicem ſimilia, cum <var>.S.*.</var> parallela ſit <lb/>ipſi <var>.E.D.</var>) vnde <var>.S.*.</var> proueniet nobis ex ſimilibus partibus .289. cum fracto, quod <lb/>r<unsure/>eijciamus ob minorem laborem.</s> + </p> + <p> + <s xml:space="preserve">Producantur poſtea <var>.V.*.</var> et <var>.E.D.</var> vſque ad eorum concurſum in puncto <var>.ω.</var> quæ-<lb/><reg norm="remusque" type="simple">remusq́;</reg> quanta ſit <var>.E.ω.</var> ex eadem regula, cum dixerimus, ſi .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> ſunt inuicem ſimilia 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 ipſius <var>.E.V.</var> quod eſt .4096. cum quadrato ipſius <var>.<lb/>E.ω.</var> quod eſt .213444. & habebimus quadratum ipfius <var>.V.ω.</var> talium <reg norm="partium" type="context">partiũ</reg> .217540.</s> + </p> + <p> + <s xml:space="preserve">Dicemus poſtea ſi .217540. dat nobis .4096. quid dabit quadratum ipſius <var>.V.ω.</var> vt <lb/>ſinus totus quod eſt .10000000000. vnde veniet pro quadrato ipſius <var>.V.E.</var> talium <lb/>partium, ſuperficialium ſcilicet .18827211. cuius radix erit .13721. & erit ſinus 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 ſinus erit partium .99054.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem quia angulus <var>.E.V.ω.</var> eſt acutus, imaginemur <var>.E.Ŕ.</var> ductam eſſe ad re<lb/>ctos ipſi <var>.V.ω.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> etiam ducta ipſa <var>.E.G</var>. </s> + <s xml:space="preserve">Vnde habebimus angulum <var>.Ŕ.E.V.</var> gra-<lb/>duum .7. min .53. eius vero ſinus <var>.Ŕ.V.</var> partium .13721. (propter ſimilitudinem trian<lb/>gulorum <var>.E.Ŕ.V.</var> et <var>.ω.E.V.</var>) talium ſcilicet, qualium <var>.E.V.</var> fuerit .100000. </s> + <s xml:space="preserve">Sed qua-<lb/>lium <var>.E.V.</var> eſt .64. talium erit .8. cum tribus quartis, cuius <var>.Ŕ.V.</var> quadratum erit par<lb/>tium .76. cum dimidio ſimilium ſed ſuperſicialium, quo quidem quadrato dempto <lb/>ex quadrato ipſius .64. quod eſt .4096. remanebit quadratum ipſius <var>.E.Ŕ.</var> partium <num value="2871">.<lb/>2871.</num> quo etiam quadrato <var>.E.Ŕ.</var> dempto ex quadrato <var>.E.G.</var> partium .14400. remane<lb/>bit quadratum ipſius <var>.Ŕ.G.</var> partium .11529. cuius radix <var>.Ŕ.G.</var> erit partium .107. <reg norm="talium" type="context">taliũ</reg> <lb/>qualium <var>.E.G.</var> eſt .120. ſed qualium <var>.E.G.</var> erit .100000. talium <var>.Ŕ.G.</var> erit partium <num value="89166">.<lb/>89166.</num> quæ vt ſinus anguli <var>.Ŕ.E.G.</var> habebit pro ipſo angulo, gra .63. min .5. qui colle<lb/>cti cum gra .7. min .53. anguli <var>.V.E.Ŕ.</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 Philoſopbo Franciſco Vimercato.</head> + <p> + <s xml:space="preserve"><emph style="sc">QVoniam</emph> non videbatur quieſcere animus tuus, cum paucis ab hinc die-<lb/>bus tibi ſiſcitanti reſpondiſſem, nec tamen rationem omnium, quæ dixe-<lb/>ram exactè explicare per tem poris anguſtiam potuiſſem, cogitaui ad te <lb/>per hanc occaſionem ſcribens, & iam dicta repetere, & omnium tibi ra-<lb/>tionem ſubiungere, & vt mihi plenius ſatisfaciam, & tibi commodè perlegenti faci <lb/>lius ſit veritatem intueri. </s> + <s xml:space="preserve">Scripſiſti enim in tuis diſputationibus, vir doctiſſime, quod <lb/>omnis res viſa per <reg norm="ſpeculum" type="context">ſpeculũ</reg> <reg norm="quodcunque" type="context">quodcũque</reg>, ſub breuiſſimis lineis <reg norm="compræhendatur" type="context">cõpræhendatur</reg> à vifu.</s> + </p> + <p> + <s xml:space="preserve">Propoſitio hæc non eſt vniuerſaliter vera (quamuis etiam ab alijs omnibus pro ta <lb/>li poſita ſit) cum in ſpeculis concauis non ſemper verificetur, vt nunc tibi demon-<lb/>ſtrabo.</s> + </p> + <p> + <s xml:space="preserve">Eſto 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 ſit communis ſectionis ſup crficiei re <lb/>flexionis, & ſphæricę alicuius ſpeculi ſphærici <lb/>concaui, & punctum contingentiæ ſit <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 viſa primò ſit in ipſa circunferen-<lb/>tia huiuſmodi circuli in puncto <var>.n.</var> oculus vero in <lb/>puncto <var>.q.</var> ipſius circunferentię. </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> ſimul ſumptas longiores eſſe omnibus alijs lineis exeuntibus ab ip <lb/>ſis punctis <var>.q.n.</var> quæ in aliquo puncto dictæ circunferentiæ ſimul 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 ſimul ſumptas, eſſe minores dua <lb/>bus ſimul ſumptis <var>.q.b.</var> et <var>.n.b</var>. </s> + <s xml:space="preserve">Nam ex .20. tertij Eucli. cognoſcimus angulos <var>.q.b.n.</var> <lb/>et <var>.q.o.n.</var> inuicem æquales eſſe, & ſimiliter angulos <var>.b.n.o.</var> et <var>.b.q.o</var>. </s> + <s xml:space="preserve">deinde ex .15. pri <lb/>mi eiuſdem habemus angulos contra ſe poſitos, <lb/>circa <var>.a.</var> eſſe etiam inuicem ęquales. </s> + <s xml:space="preserve">Vnde ex .4 <lb/> + <anchor type="figure" xlink:label="fig-0344-01a" xlink:href="fig-0344-01"/> + ſexti, habebimus proportionem <var>.a.b.</var> ad .a<lb/>o. eandem eſſe, quæ <var>.a.n.</var> ad <var>.a.q.</var> & ſic .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> ſed cum <var>.a.n.</var> maior ſit <var>.q.a.</var> ex .18. primi, <lb/>eo quod angulus <var>.b.q.n.</var> (qui æqualis eſt angulo <var>.<lb/>b.n.q.</var> ex .5. eiuſdem) maior eſt angulo <var>.a.n.q.</var> <lb/>qui pars eſt ipſius <var>.b.n.q.</var> ergo latera ſimul ſum-<lb/>pta <var>.a.b.n.</var> maiora erunt lateribus <var>.a.o.q.</var> ſed 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 ſequi-<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 ſi oculus eſſet in <var>.u.</var> quemadmodum in ſubſcripta hic <reg norm="ſecunda" type="context">ſecũda</reg> figura videre eſt, <lb/>res autem viſibilis in <var>.n.</var> ambo extra dictum circulum, eſto etiam primum <var>.b.u.</var> æqua-<lb/>lis <var>.b.n.</var> probabo ſimiliter <var>.u.b.n.</var> maiores eſſe <var>.u.o.n</var>. </s> + <s xml:space="preserve">Nam angulus <var>.o.</var> maior eſt angu-<lb/>lo <var>.b.</var> eo quod ſi circulum <var>.u.b.n.</var> cogitemus circunſcribere triangulum <var>.u.b.n.</var> ducen-<lb/>do vſque ad ſuam 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> ſed <reg norm="cum" type="context">cũ</reg> angulus <var>.u.o.n.</var> exterior trian <lb/>guli <var>.u.o.s.</var> exiſtat, ipſe maior erit angulo <var>.s.</var> ex .16. primi. </s> + <s xml:space="preserve">duco poſtea <var>.o.q.</var> parallelam <lb/>ad <var>.u.s.</var> quæ ſecabit <var>.a.u.</var> in puncto <var>.q.</var> & habebimus angulum <var>.a.o.q.</var> ęqualem angulo <var>.<lb/>n.s.u.</var> ex .29. eiuſdem, hoc eſt angulo <var>.n.b.u.</var> fed ex ſu-<lb/> + <anchor type="figure" xlink:label="fig-0344-02a" xlink:href="fig-0344-02"/> + pradictis rationibus, lineæ <var>.q.b.n.</var> ſimul ſumptæ maio-<lb/>rem efficient longitudinem, quam <var>.q.o.n</var>. </s> + <s xml:space="preserve">Nunc cum <lb/>ipſi <var>.q.b.</var> addita fuerit <var>.u.q.</var> & vice <var>.q.o.</var> ſumpta fuerit ali-<lb/>qua linea minor ipſa <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 eſt ambabus <var>.o.<lb/>q.</var> et <var>.q.u.</var> ex .20. primi, ita enim habebimus <reg norm="propoſitum" type="context">propoſitũ</reg>. <lb/></s> + <s xml:space="preserve">ſed breuiori modo hoc ipſum videbis ex pręcedenti, <lb/>& ex .21. primi Euclid. </s> + <s xml:space="preserve">Nam ex præcedenti <var>.u.b.n.</var> lon-<lb/>gior eſt ipſa <var>.u.s.n.</var> ex .21. autem primi <var>.u.s.n.</var> longior eſt <lb/>ipſa <var>.u.o.n.</var> ergo verum eſt propoſitum.</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, ſit vt in hac ſubſcripta tertia figura vide <lb/>re eſt <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/>manifeſtabitur verum eſſe propoſitum, 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 viſibilis <reg norm="oculusque" type="simple">oculusq́;</reg> ambo fuerint intra circulum, <reg norm="tunc" type="context">tũc</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 ipſa <var>.u.o.n.</var> <reg norm="nunc" type="context">nũc</reg>. <lb/></s> + <s xml:space="preserve">Quod etiam affirmo de <var>.u.b.p.</var> ſimiliter etiam eueniet ſi vnus terminorum <var>.u.</var> vel <var>.n.</var> <lb/>fuerit intra circunferentiam, reliquus verò extra ipſam.</s> + </p> + <p> + <s xml:space="preserve">Conſideremus nunc hic inſraſcriptam .4. figuram vbi <var>.d.b.p.</var> ſit circunferentia oxy <lb/>gonia ſeu elliptica (quod idem eſt) cuius maior axis ſit <var>.d.p.</var> in quo, duo termini <var>.u.n.</var> <lb/>ſint centra eius generationis: </s> + <s xml:space="preserve">b.x. verò ſit minor axis. </s> + <s xml:space="preserve">Imaginemur etiam circulum <var>.<lb/>b.o.x.</var> cuius ſemidiameter ſit <var>.c.b.</var> non maior medietate minoris axis, ne circunferen-<lb/>tia huiuſmodi circuli ſecet circunferentiam oxygoniam. </s> + <s xml:space="preserve">Cogitemus etiam circu-<lb/>lum <var>.b.e.</var> cuius ſemidiameter, minor non ſit minori axe <var>.b.x.</var> ipſius oxygoniæ, ne ſe <lb/>inuicem ſecent huiuſmodi circunferentiæ, ſint etiam ambo eorum centra in linea <var>.b.<lb/>x.</var> minoris axis, & punctum <var>.b.</var> ſit commune vnicuique earum periphæriarum, vnde <lb/>minor circulus, totus intra, maior autem, totus extra ipſam <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 tranſire alium circulum, cuius cen-<lb/>trum in axe <var>.b.x.</var> ſit <var>.t.</var> <reg norm="omnesque" type="simple">omnesq́;</reg> iſti circuli imaginentur trium diuerſorum ſphærico-<lb/>rum ſpeculorum, vnde pro genera <lb/>tione <reg norm="ipſius" type="simple">ipſiꝰ</reg> oxygonię, ſeu 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> ęqualem eſſe longitudini <var>.u.b.<lb/>n.</var> & ei, quæ eſt <var>.u.o.n.</var> (vt minor ip <lb/>ſa <var>.u.r.n.</var> ex .21. primi Euclidis) mi-<lb/>nor ipſa <var>.u.b.n.</var> & longitudinem <var>.u.<lb/>e.n.</var> (vt maior ipſa <var>.u.r.n.</var> ex eadem <num value="21">.<lb/>21.</num> primi Eucli.) maior ipſa <var>.u.b.n</var>. <lb/></s> + <s xml:space="preserve">Sed ſi quis vellet hoc demonſtrare <lb/>ope circuli, <reg norm="vnius" type="simple">vniꝰ</reg> <reg norm="tantummodo" type="context context">tãtũmodo</reg> ſpeculi, <lb/><reg norm="multiplicando" type="context">multiplicãdo</reg> ipſas oxygonias <reg norm="quem- admodum" type="wordlist">quẽ-<lb/>admodum</reg> de ipſis circulis fecimus, obtineret ſimiliter propoſitum.</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 eſt dubitatio tua, <lb/> + <anchor type="figure" xlink:label="fig-0345-02a" xlink:href="fig-0345-02"/> + vtrum (<reg norm="cum" type="context">cũ</reg> circulus minor hoc <lb/>eſt <var>.b.o.</var> habeat ſuum centrum in mi <lb/>nori axe inter centrum oxygoniæ, <lb/>et .b: exiſtente <var>.b.</var> extremo axis mi-<lb/>noris, <reg norm="communeque" type="simple">communeq́;</reg> ambobus circun-<lb/>ferentijs circuli ſcilicet & oxigonię) <lb/>dictus circulus minor, plura puncta <lb/>communia habeat cum ipſis 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="reſpondeo" type="context">reſpõdeo</reg> quod <lb/>quotieſcunque 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 ſcilicet minoris axis, exiſtente <var>.b.</var> <lb/>ſua 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, ipſas circunferentias inuicem contiguas eſſe oportebit in puncto <var>.b.</var> <lb/>tantummodo.</s> + </p> + <p> + <s xml:space="preserve">Eſto primum quod centrum <var>.c.</var> commune exiſtat, vt dictum eſt. </s> + <s xml:space="preserve">ſit etiam centrum <lb/>vnius circuli, cuius diameter ſit <reg norm="idem" type="context">idẽ</reg> <reg norm="cum" type="context">cũ</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/>ipſi <var>.g.c.</var> vſque 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="ſecabitque" type="simple">ſecabitq́;</reg> gyrum circuli <var>.b.o.</var> minoris in puncto <var>.t.</var> quod di <lb/>co eſſe intra oxygoniam, <reg norm="ſeparatumque" type="simple">ſeparatumq́;</reg> ab <var>.f</var>. </s> + <s xml:space="preserve">Quapropter duco <var>.c.e.</var> quæ ſecabit cir-<lb/>cunferentiam circuli minoris in <reg norm="puncto" type="context">pũcto</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 conſidero, quod ex ra-<lb/>tionibus ab Archimede adductis in <lb/> + <anchor type="figure" xlink:label="fig-0346-01a" xlink:href="fig-0346-01"/> + quinta propoſitione libri de conoi-<lb/>dalibus, & ſphæroidibus, eadem <lb/>proportio erit <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.g.c.</var> ad <var>.b.c.</var> quę <lb/>ipſius <var>.e.a.</var> ad <var>.f.a.</var> vnde permutando <lb/>ita erit ipſius <var>.g.c.</var> ad <var>.e.a.</var> vel <var>.b.c.</var> ad <lb/><var>f.a.</var> hoc eſt ipſius <var>.e.c.</var> ad <var>.e.a.</var> vt <var>.o.c.</var> <lb/>ad <var>.f.a.</var> ſed ex ſimilitudine triangu-<lb/>lorum, & ex .11. quinti, ita <reg norm="etiam" type="context">etiã</reg> erit <lb/>ipſius <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 ſequitur <var>.o.i.</var> æqualem eſſe <var>.f.a.</var> <lb/>ſed ex .14. tertij Eucli <var>.t.a.</var> minor eſt <var>.<lb/>o.i</var>. </s> + <s xml:space="preserve">Quare minor etiam erit ipſa <var>.f.<lb/>a</var>. </s> + <s xml:space="preserve">Vnde punctum <var>.t.</var> intra oxygo-<lb/>niam erit, & conſequenter ſepara-<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 ſi centrum circuli minoris <lb/>fuerit inter <var>.c.</var> et <var>.b.</var> hoc eſt eccentri-<lb/>cum ipſius oxygoniæ, ipſe tanget concentricum in puncto <var>.b.</var> tantummodò, vt in .3. <lb/>Euclidis libro probatur. </s> + <s xml:space="preserve">Vnde tanto magis diſtans erit punctum <var>.t.</var> à puncto <var>.f.</var> quod <lb/>erit propoſitum.</s> + </p> + </div> + <div type="letter" level="4" n="3"> + <head style="it" xml:space="preserve">Alterius dubitationis ſolutio.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">VNde autem fiat, quod à ſpeculis planis, obiectorum imagines, ita diſtantes <lb/>vltra ſuperficiem ipſius ſpeculi videantur, vt obiecta citra ipſam ſuperficiem <lb/>reperiuntur.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ſcientia, tres cognitiones nos primum habere oportet, quarum pri-<lb/>ma eſt. </s> + <s xml:space="preserve">Vnde fiat, quod obiecti imago in catheto incidentiæ videatur. </s> + <s xml:space="preserve"><reg norm="Secunda" type="context">Secũda</reg>. </s> + <s xml:space="preserve">vn-<lb/>de efficiatur, quod angulus reflexionis, ſemper æqualis ſit angulo incidentiæ.</s> + </p> + <p> + <s xml:space="preserve">Terria demum. </s> + <s xml:space="preserve">Vnde naſcatur quod radius incidentiæ ſimul cum radio reflexio-<lb/>nis ſit in quodam plano ſecante ſuperficiem ſpeculi ſemper ad rectos, quod qui-<lb/>dem planum vocatur ſuperficies reflexionis. </s> + <s xml:space="preserve">Huiuſmodi tres paſſiones, ab omnibus <lb/>ſpecularijs conſideratæ ſunt, ſed rationes ab illis traditæ, mihi non ſatisfaciunt.</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 vniuerſaliter euenire à breuitate aggregati radiorum incidentiæ reflexio-<lb/><reg norm="nisque" type="simple">nisq́;</reg>. </s> + <s xml:space="preserve">Sed hoc naſcitur potius ab eo, quod cum radius incidentiæ non poſſit ſuper <lb/>ficiem corporis opaci penetrare, reflectit, vt citra ipſam <reg norm="cum" type="context">cũ</reg> angulo æquali ei, quem <lb/>faceret cum eadem ſuperficie vltra ipſam ſi tranſiuiſſet.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia ſit <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> ſuperficies ipſius <lb/>ſpeculi <var>.d.</var> verò ſit punctum ipſius ſuperficiei, à quo ad oculum reflectitur imago ip-<lb/> + <anchor type="figure" xlink:label="fig-0347-01a" xlink:href="fig-0347-01"/> + ſius <var>.a</var>. </s> + <s xml:space="preserve">Nunc ſi radius <var>.a.d.</var> incidentiæ, recta <lb/>incederet ſub <var>.c.e.</var> efficeret angulum <var>.e.d.h.</var> <lb/>æqualem angulo <var>.c.d.a.</var> eius contrapoſito, <lb/>ſed quia impeditur ipſæ radius ab opacitate <lb/>ipſius ſpeculi <var>.c.e.</var> ne vlterius incedat, propte <lb/>rea reflectitur ab ipſa ſuperficie ſpeculi, con-<lb/>ſtituens cum ipſa angulum <var>.e.d.b.</var> æqualem <lb/>angulo <var>.e.d.h.</var> ſed quia angulus <var>.c.d.a.</var> eſt <reg norm="etiam" type="context">etiã</reg> <lb/>ęqualis ipſi angulo <var>.e.d.h.</var> propterea angulus <var>.e.d.b.</var> ęqualis exiſtit angulo <var>.c.d.</var> a; </s> + <s xml:space="preserve">per <lb/>accidens igitur ſequitur <var>.a.d.</var> et <var>.d.b.</var> ſimul ſumptas, breuiorem facere longiludinem <lb/>omni alia, quæ ab ipſa ſuperficie <var>.c.e.</var> ad eadem puncta <var>.a.b.</var> ducta eſſet, </s> + <s xml:space="preserve">quare natu-<lb/>ræintentio eſt efficere angulum <var>.e.d.b.</var> æqualem angulo <var>.e.d.h.</var> vnde ex accidenti po <lb/>ſtea ſequitur, ipſum æqualem eſſe angulo <var>.c.d.a.</var> & deinde <reg norm="quod" type="wordlist">qđ</reg> lineæ <var>.a.d.</var> et <var>.d.b.</var> con-<lb/>ſtituant longitudinem breuiorem. </s> + <s xml:space="preserve">Quare illud quod omnes putabant eſſe primum <lb/>& perſe, vltimum eſt, & 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 ſuperficies, quæ vocatur reflexionis, in qua ſunt duæ lineę, hoc eſt <lb/>incidentię, <reg norm="reflexionisque" type="simple">reflexionisq́;</reg>, ſemper ſit perpendicularis ſuperficiei ipſius ſpeculi: </s> + <s xml:space="preserve">Hæc <lb/>eſt ratio, quia cum quilibet radius incidentiæ, perpendicularis ipſi ſuperficiei ſpe-<lb/>culi, in ſeipſo reflectit, ex ijſdem dictis rationibus, hoc eſt, quia cum tali angulo vult <lb/>reflecti, cum quali tranſiret, ita etiam purandum eſt, quodradius incidens obliquus, <lb/>cum in ſeipſum non poſſit redire, quia non eſt perpendicularis ſuperficiei ſpeculi, <lb/>reflectitur tamen per planum erectum ipſi ſuperficiei ſpeculi, vt in eo, cui magis re-<lb/>ſiſtit ſuperficies corporis opaci, quàm alicui alij plano ipſius infiniti inclinatorum <lb/>planorum, ab vtraque parte ipſius plani perpendicularis, quod vnum etiam tan-<lb/>tummodo eſt, & in quo, radius maiorem vim obtinet reflectendi, ſeu in eo, in quo <lb/>radius ipſe cum maiori reſiſtentia repercutitur à ſuperficie corporis opaci.</s> + </p> + <p> + <s xml:space="preserve">Poſtremo <reg norm="ſciendum" type="context context">ſciẽdũ</reg> vnde oriatur, <reg norm="quod" type="simple">ꝙ</reg> rei viſibilis imago, à ſpeculo plano reflexa, ſem <lb/>per in catheto incidentiæ videatur.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ratione cognoſcendum primò eſt, quo modo fit perfecta <reg norm="ſimplexque" type="simple">ſimplexq́;</reg> <lb/>viſio, & non reflexa, deinde proſequemur ad reliqua huius tertiæ propoſitionis.</s> + </p> + <p> + <s xml:space="preserve">Animaduertendum igitur eſt, quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> obiectum aliquod viſibile aſpi <lb/>cimus, nos nunquam perfectè illud comprehendere poſſumus, niſi in puncto con-<lb/>curſus, ſeu interſectionis axium viſualium, ſeu radialium ( vt ita loquar ) <reg norm="quam" type="context">quã</reg> <reg norm="inter- ſectionem" type="context">inter-<lb/>ſectionẽ</reg>, nos efficimus ope reuolutionis oculorum <reg norm="adinuicem" type="context">adinuicẽ</reg>, hoc eſt voluendo vnum <lb/>verſus alium, ita vt in ſitu ipſius obiecti, ſeinuicem ſecent axes iam dicti, </s> + <s xml:space="preserve">tunc enim <lb/>vtroque oculo mediante, exacte rem perſpicimus, cęteris .8. circunſtantijs non ob-<lb/>ſtantibus.</s> + </p> + <p> + <s xml:space="preserve">Vnde ſtantibus oculis in tali ſitu, altero reſpectu alterius, ſi eorum alter tectus; <lb/></s> + <s xml:space="preserve">ſeu velatus fuerit, tune alio tantummodo oculo mediante, videbimus obiectum, <lb/>in ea diſtantia, 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, ſecundum diſtantiam obiecti, oculum accommodat ad recipien-<lb/>dum quam exactiſſimè ſpeciem ipſius obiecti, & hoc voluendo ambos oculos, vnum <lb/>verſus alium, ita quod interſectio axium ſit in ſitu ſeu loco dicti obiecti, nam tunc vi <lb/>dent ambo vel aliquis eorum ſolus, in tali diſtantia exactè obiectum videbit.</s> + </p> + <p> + <s xml:space="preserve">Vnde ſequitur obiectum viſibile, compræhenſibile non eſſe ab vno tantummodo <lb/>oculo in quolibet ſitu axis ipſius oculi, ſed in eo, vbi alius axis interſecatur à dicto. <lb/></s> + <s xml:space="preserve">Quæ quidem interſectio poteſt fieri propinqua, vel remota à viſu, ad certos tamen <lb/>terminos vſque.</s> + </p> + <p> + <s xml:space="preserve">De huiuſmodi axium viſualium interſectione ſcribit Alhazem in .2. et .15. propo <lb/>ſitione tertij lib. Vitellio verò in .32. et .45. eiuſdem.</s> + </p> + <p> + <s xml:space="preserve">Quod igitur dico, verum eſt, ideſt, quod ſi vno tantummodo oculo aſpiciemus <lb/>obiectum aliquod, ipſum nunquam perfectè proſpicietur, niſi cum oculus ita ſitus <lb/>fuerit, vt eius axis cum axe alterius in loco obiecti ſe inuicem ſecent, quamuis alter <lb/>oculus nihil videat, <reg norm="cum" type="context">cũ</reg> <reg norm="autem" type="wordlist">aũt</reg> duobus oculis in tali ſitu <reg norm="conſtitutis" type="context">cõſtitutis</reg> <reg norm="obiectum" type="context">obiectũ</reg> videmus, vnum <lb/>tantummodo nobis cernere videbimur, & ſi extra talem punctum interſectionis ip-<lb/>ſum obiectum poſitum fuerit, tunc duo talia, obiecta nobis apparebunt, ſed huiuſ <lb/>modi rei cauſam alias tibi manifeſtabo.</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"/> + ſpeculi ſuperficiem eſſe <var>.g.h.</var> in figura <var>.B.</var> <lb/>obiectum autem viſibile <var>.b.</var> oculos vero <var>.a.</var> <lb/>et <var>.u.</var> punctum autem <var>.n.</var> in ſuperficie ſpecu <lb/>li, à quo imago ipſius <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/>ſit <reg norm="communis" type="context">cõmunis</reg> ſectio ſuperficiei reflexionis <lb/>radiorum <var>.b.n.a.</var> et <var>.c.f.</var> ſit communis ſectio <lb/>ſuperficiei reflexionis radiorum <var>.b.t.u.</var> qua <lb/>rum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> ſuperficies reflexionis, ere-<lb/>cta eſt ad ſuperficiem ſpeculi <var>.g.h.</var> vt ſupra <lb/>diximus. </s> + <s xml:space="preserve">Nunc ex .19. vndecimi Eucl. ſequitur communem ſectionem harum dua-<lb/>rum ſuperficierum. (b.c.d. ſcilicet) ad rectos etiam eſſe ſupra ſuperficiem ſpeculi <var>.g.<lb/>h.</var> cum qua <var>.b.c.</var> quælibet linearum <var>.a.n.</var> vel <var>.u.t.</var> reflexarum ( productę cum fuerint ) <lb/>ſeinuicem interſecabunt eo quod duo anguli <var>.d.c.n.</var> et <var>.d.n.c.</var> ſimul collecti minores <lb/>ſunt 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, ipſis con-<lb/>trapoſiti, æquales ſint angulis <var>.b.n.c.</var> et <var>.b.t.c.</var> incidentiæ, quorum <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg> ex .32. <lb/>primi, minor eſt 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 huiuſmodi catheti <var>.b.c.d.</var> in quo interſecabi-<lb/>tur à linea <var>.a.n.</var> in eodem ſecabitur à linea <var>.u.t.</var> & quod punctum dicti concurſus, tan-<lb/>tum depreſſum erit ſub ſuperficie ſpeculi <var>.g.h.</var> quantum <var>.b.</var> ſupra ipſam reperietur. <lb/></s> + <s xml:space="preserve">Nam anguli <var>.b.n.c.</var> et <var>.d.n.c.</var> ſunt inuicem æquales, <reg norm="angulique" type="simple">anguliq́;</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 ipſius 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 ſit: </s> + <s xml:space="preserve">d.c. igitur etiam erit, & ipſum <reg norm="vnum" type="context">vnũ</reg> <lb/>& idem, quod erit propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem cum hi duo radij ſeinuicem ſecent in puncto <var>.d.</var> ergo in ipſo puncto <var>.<lb/>d.</var> videbimur nobis videre <reg norm="imaginem" type="context">imaginẽ</reg> obiecti .b: <reg norm="cum" type="context">cũ</reg> ope <reg norm="duorum" type="context">duorũ</reg> <reg norm="iſtorum" type="context">iſtorũ</reg> <reg norm="radiorum" type="context">radiorũ</reg> <var>.n.a.</var> et <var>.t.<lb/>u.</var> ita inuicem <reg norm="ſitorum" type="context">ſitorũ</reg>, videamur nobis <reg norm="imaginem" type="context">imaginẽ</reg> proſpicere. </s> + <s xml:space="preserve">Vnde ſi in tali caſu, 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> ipſo <lb/>loco <var>.d.</var> & non in alio ex ſuperius dictis rationibus.</s> + </p> + <p> + <s xml:space="preserve">Et ſi ſtantibus ijs terminis volueremus pupillam oculi <var>.u.</var> verſus aliam <var>.a.</var> ad aſpi-<lb/>ciendum punctum <var>.n.</var> in ſuperficie <var>.g.h.</var> ipſius ſpeculi, hoc eſt ſi fecerimus quod axes <lb/>viſuales ſeinuicem ſecarent in ipſo puncto <var>.n</var>. </s> + <s xml:space="preserve">tunc videremur nobis videre duas <lb/>imagines ipſius obiecti <var>.b.</var> intra ſpeculum, eo quod obiectum, propter hoc non <lb/>ceſſaret reflectere ad oculos ab ipſis punctis <var>.n.</var> et <var>.t.</var> quapropter recipiendo ra-<lb/>dium <var>.t.u.</var> in ſitu axis oculi <var>.u.</var> & radium <var>.n.a.</var> in ſitu axis oculi <var>.a.</var> hi axes ex neceſſitate <lb/>(vt probauimus ) ſeinuicem ſecant in puncto <var>.d.</var> vnde vnam tantummodo imaginem <lb/>ipſius obiecti nobis apparebit.</s> + </p> + <p> + <s xml:space="preserve">Ex his igitur omnibus potes facilè videre omnem imaginem, cuiuſuis obiecti, re-<lb/>flexam à ſpeculo, reperiri in ipſo catheto incidentiæ, cum ipſe ſemper ſit communis <lb/>ſectio duarum ſuperficierum reflexionis, in quo catheto concurrunt ipſæ axes vi-<lb/>ſuales.</s> + </p> + <p> + <s xml:space="preserve">Exijſdem etiam dictis rationibus facile compræhendere poteris, vnde fiat, vt vi-<lb/>deamus imaginem reflexam à ſpeculis ſphęricis concauis citra <reg norm="ipſorum" type="context">ipſorũ</reg> ſuperficiem, & <lb/>non vltra. </s> + <s xml:space="preserve">Quod <reg norm="nunquam" type="context">nunquã</reg> euenit, niſi quando <reg norm="punctum" type="context">punctũ</reg> <var>.d.</var> interſectionis <reg norm="ipſorum" type="context">ipſorũ</reg> <reg norm="radiorum" type="context">radiorũ</reg> <lb/>viſualium (quod alio in loco non fit, niſi in catheto incidentiæ hoc eſt in communi <lb/>ſectione duarum ſuperficierum reflexionis. </s> + <s xml:space="preserve">Dato quod obiectum non ſit in vna ea-<lb/>demq́ue ſuperficie, in qua reperti fuerint axes viſuales, hoc eſt dato, <reg norm="quod" type="simple">ꝙ</reg> ambo axes <lb/>viſuales non ſint in vna <reg norm="eademque" type="simple">eademq́;</reg> ſuperficie reflexionis) reperitur citra & non vltra ſu <lb/>perficiem ipſius ſpeculi.</s> + </p> + <p> + <s xml:space="preserve">Ad cuius rei euidentiam non <reg norm="prætermittam" type="context">prætermittã</reg> dicere, quod cum debeant ſemper ſu-<lb/>perficies reflexionum perpendiculares eſſe, velad rectos ſecare ſuperficiem ipſius <lb/>ſpeculi, ipſarum communes ſectiones cum ſuperficie ſpeculi ſphęrici, ſemper erunt <lb/>circunferentiæ magnorum circulorum illius ſphæræ, cuius portio eſt ſpeculum <lb/>propoſitum, vt etiam Vitellio affirmat in prima ſexti libri. </s> + <s xml:space="preserve">Vnde vnuſquiſque ca-<lb/>thetus incidentiæ tranſibit per centrum ſpeculi, cum ipſe ſit communis ſectio dua-<lb/>rum ſuperficierum reflexionis, </s> + <s xml:space="preserve">quare in ipſo catheto erit punctum interſectionis ip <lb/>ſorum axium viſualium ex neceſſitate, vt videbimus, ſi vnam tantummodo <reg norm="imaginem" type="context">imaginẽ</reg> <lb/>obiecti nobis videremur videre.</s> + </p> + <p> + <s xml:space="preserve">Exempli gratia, ſint duæ ſuperficies reflexionis ſpeculi ſphærici concaui <var>.b.n.c.a.</var> <lb/>et <var>.b.t.c.u.</var> <reg norm="obiectumque" type="simple">obiectumq́;</reg> ſit <var>.b.</var> oculi autem ſint <var>.a.u.</var> punctum verò ſuperficiei ſpeculi, à <lb/>quo obiectum emittit reflexionem ſuę <lb/>imaginis ad oculum <var>.a.</var> ſit <var>.n.</var> <reg norm="punctum" type="context">pũctum</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> ſit <lb/>t. communis autem ſectio harum dua-<lb/>rum ſuperficierum ſit <var>.b.c.</var> ſed <var>.x.</var> <reg norm="centrum" type="context">centrũ</reg> <lb/>ſit ſpeculi, radius verò incidentię ſuper <lb/>ficiei <var>.b.n.c.</var> erit <var>.b.n.</var> cuius reflexus ſit <var>.n.<lb/>a.</var> radij autem alterius ſuperficiei erunt <lb/><var>b.t.</var> et <var>.t.u</var>. </s> + <s xml:space="preserve">Imaginemur nunc duos ſemi <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 ſup-<lb/>poſito.</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 ſuppoſitis, ſi 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 ſupradictis nos ipſam videre in <reg norm="communi" type="context">cõmuni</reg> concurſu ipſorum <lb/>axium viſualium, qui axes cum reperiantur vnà cum ipſis radijs reflexis <var>.n.a.</var> et <var>.t.u.</var> <lb/>ex neceſſitate ſeinuicem <reg norm="ſecabunt" type="context">ſecabũt</reg> in catheto <var>.b.c.</var> cum extendantur in ipſis ſuperficie-<lb/>bus reflexionum, quæ ſuperficies nihil aliud commune inuicem habent, quam cathe <lb/>tum dictum <var>.b.c.</var> ſit igitur in puncto <var>.d</var>.</s> + </p> + <p> + <s xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eſt, quod quotieſcunque vnam tantummo <lb/>do imaginem obiecti <var>.b.</var> videmus, dato quod duæ ſuperficies reflexionis ſint, & non <lb/>vna tantum, tunc angulos <var>.n.</var> et <var>.t.</var> ſemper 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. ſexti Euclid. quod eadem proportio erit ipſius <var>.b.n.</var> ad <var>.n.<lb/>d.</var> quę ipſius <var>.b.x.</var> ad <var>.x.d.</var> & ipſius <var>.b.t.</var> ad <var>.t.d.</var> ſimiliter, </s> + <s xml:space="preserve">quare ipſiusb <var>.n.</var> ad <var>.n.<lb/>d.</var> erit vt ipſius <var>.b.t.</var> ad <var>.t.d</var>. </s> + <s xml:space="preserve">Vnde ſequitur <var>.b.n.</var> æqualem eſſe ipſi <var>.b.t.</var> et <var>.n.d.</var> <lb/>ipſi <var>.t.d.</var> vt à medio circulo <var>.E.</var> potes videre, quamuis etiam <var>.b.</var> non eſſet extremum <lb/>diametri, ſed vbicunque volueris in ipſo diametro, vel <reg norm="etiam" type="context">etiã</reg> protracta, eo quod pun-<lb/>ctum <var>.n.</var> & punctum <var>.t.</var> in eodem ſemicirculo, vel in æqualibus ſemicirculis, non <reg norm="poſsent" type="context">poſsẽt</reg> <lb/>aliter in ipſa circunferentia locari, <reg norm="eandem" type="context">eãdem</reg> ſeruando 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 ſitu exiſtente puncto <var>.t.</var> ipſa <var>.b.t.</var> eſſet aut maior <lb/>aut minor ipſa <var>.b.n.</var> et <var>.t.d.</var> aut minor, aut maior ipſa <var>.t.d.</var> ex .7. & 14. tertij Eucli. </s> + <s xml:space="preserve">vnde <lb/>aut maior, aut minor proportio eſſet ipſius <var>.b.t.</var> ad <var>.t.d.</var> quam ipſius <var>.b.n.</var> ad <var>.n.d.</var> & non <lb/>eadem.</s> + </p> + <p> + <s xml:space="preserve">Nunc è conuerſo ſi <var>.b.n.</var> et <var>.b.t.</var> ſunt ſibi inuicem æquales, & ſic <var>.n.d.</var> cum <var>.t.d.</var> ſequi-<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 ijſdem ſpeculationibus potes etiam videre vnde accidat quod partes ſuperio <lb/>res alicuius obiecti reflexæ à tali ſpeculo concauo videntur nobis inferiores eſſe, & <lb/>inferiores appareant ſuperiores, & dextræ ſiniſtræ, & ſiniſtræ dextræ. </s> + <s xml:space="preserve">quod autem <lb/>hucuſque demonſtraui de ſpeculis planis, & ſphæricis concauis, ratiocinare tu ijſdem <lb/>medijs circa ſphærica conuexa, vbi clarè videbis puncta huiuſmodi ſpeculi conuexi, <lb/>à quibus reflectitur imago obiecti ad ambos oculos, ſemper oportere æquidiſtantia <lb/>eſſe à <reg norm="puncto" type="context">pũcto</reg> communi ipſius ſuperficiei ſpeculi, & catheto incidentiæ, dum unam tan <lb/>tummodo imaginem ipſius obiecti videmus, & à diuerſis ſuperficiebus reflexionum.</s> + </p> + <p> + <s xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi ſuccurrit, hoc eſt quod poſſet ali-<lb/>quis duos ſitus inuenire, vnum pro oculo, alterum verò pro obiecto, reſpectu alicu-<lb/>ius ſpeculi concaui, ſphęroidis prolatæ, vt reflexio ipſius obiecti videretur, vt linea <lb/>diuidens per æqualia ipſum ſpeculum. </s> + <s xml:space="preserve">Reſpectu verò alicuius ſpeculi concaui ſphæ-<lb/>roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota ſuperficie ipſius ſpecu-<lb/>li, vnde tota ſuperficies ipſius ſpeculi videretur colorata illo colore cuius eſſet <lb/>obiectum, quæ quidem paſſiones <reg norm="pendent" type="context">pendẽt</reg> à .48. tertij lib. ipſius Pergei, vt ex te ipſo fa <lb/>cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s> + </p> + <p> + <s xml:space="preserve">Opinio autem mea, quam ſcire cupis de imagine obiecti reflexa, quam putas eſ-<lb/>ſe in ſuperficie ſpeculi, hæc eſt, quod nec in ſuperficie, nec ultra, nec citra eam eſt ip <lb/>ſa imago, quod autem vltra non ſit, hoc puto nulli dubium eſſe. </s> + <s xml:space="preserve">eadem etiam ra-<lb/>tione non erit citra ſuperficiem ſpeculi concaui, quamuis ipſam nos compræhenda-<lb/>mus in concurſu radiorum viſualium, tam ab vno ſpeculo quam ab alio reflexione <lb/>facta. </s> + <s xml:space="preserve">Sed quòd ipſa neque ſit in ipſa ſpeculi ſuperficie, manifeſtum erit ex hoc, <reg norm="quod" type="simple">ꝙ</reg> <lb/>duo ſpectantes in eodem ſpeculo, duas diuerſas imagines vident, tres, <reg norm="autem" type="wordlist">aũt</reg> tres, qua-<lb/>tuor, quatuor, & ſic deinceps, vnde tot eſſent imagines ſupra ſuperficiem ſpeculi, <lb/>quot obiecta, <reg norm="quod" type="simple">ꝙ</reg> tamen ita non eſt, nec plus eſt in vno loco ipſa 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"/> + niſi in obiecto ipſo, lumen enim abipſo obiecto reflexum, ſeipſum diffundit vndi-<lb/>que, & radijipſius lu<unsure/>minis reflexi, vt plurimum ſeinuicem ſecant. </s> + <s xml:space="preserve">Vnde in ipſo ae-<lb/>re funt omnes miſti. </s> + <s xml:space="preserve">Quapropter natura ſagacifſima pupillam oculi animalibus tam <lb/>paruam conſtruxit ad ſuperficiem tam amplæ ſphæræ ipſius oculi, vt diſtinctæ vide-<lb/>rentur omnia obiecta.</s> + </p> + <p> + <s xml:space="preserve">Nolo etiam tibi tacere, quod <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> oculorum pupillæ poſitæ fuerint inter <lb/>cathetum incidentiæ, & ſuperficiem <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"/> + ſpeculi ſphærici 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, ſed duas nec non confu-<lb/>sè, </s> + <s xml:space="preserve">propterea <reg norm="quod" type="simple">ꝙ</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">cũ</reg> <lb/>ambobus axibus viſualibus, eo quod <lb/>axes vifuales nunquam poſſunt inui-<lb/>cem interſecari poſt viſum, ſed ſem-<lb/>per ante ipſum, 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 ſi obiectum inci <lb/>derit in eadem ſuperficie, in qua duo <lb/>axes viſuales, vel radij reflexi <reg norm="reperiuntur" type="context">reperiũtur</reg>, hoc eſt in vna <reg norm="eademque" type="simple">eademq́;</reg> ſuperficie 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 ipſo catheto ſed extra, in qua interſectione fit viſio vnius tan-<lb/>tummodo imaginis, quod antiqui non animaduerterunt. </s> + <s xml:space="preserve">Hoc autem dico deſpe-<lb/>culo ſphærico concauo.</s> + </p> + </div> + <div type="letter" level="4" n="4"> + <head style="it" xml:space="preserve">Speculatio cuiuſdam propoſitionis aritbmetica.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">SPeculatio vltimæ propoſitionis quam numerorum via inueni, hæc eſt. </s> + <s xml:space="preserve">Imagi-<lb/>nemur triangulum <var>.r.e.o.</var> abſciſum à circulo, in cuius circunferentia ſit punctum <lb/>r. ſuperioris anguli ipſius trianguli, vel etiam non ſit abſciſum dummodo protrahan <lb/>tur lineæ <reg norm="vſque" type="simple">vſq;</reg> ad circunferentiam, à quo ad oppoſitum 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">ꝙ</reg>. <var>K.o.</var> æqualis ſit <var>.f.e.</var> vnde hæc .4. lineæ ſecabuntur à circulo dicto in punctis <var>.n.<lb/>c.b.u</var>. </s> + <s xml:space="preserve">Dico nunc <reg norm="quod" type="simple">ꝙ</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 tranſire per centrum <var>.s.</var> ipſius circuli, ſi-<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 eſt <var>.e.a</var>: <var>i.d.</var> et <var>.o.q.</var> & vbi circulus ſecat <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> ſimi-<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">productumq́</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> ſunt in-<lb/>u<unsure/>icem ſimiles, ſed productum <var>.g.r.a.</var> ſimul cum producto <var>.g.r.q.</var> duplum eſt producto <var>.<lb/>g.r.d.</var> ex prima ſexti, eo quod <var>.a.r.q.</var> dupla eſt <var>.d.r.</var> & ideo productum <var>.e.r.u.</var> ſimul <reg norm="cum" type="context">cũ</reg> <lb/>producto <var>.o.r.n.</var> duplum erit producto <var>.i.r.x.</var> quod quidem æquale eſt producto <var>.g.r.<lb/>d.</var> ex ſimilibus rationibus iam ſupradictis. </s> + <s xml:space="preserve">Nunc ex ſimilibus 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 ſecun-<lb/>dis. </s> + <s xml:space="preserve">Quod eſt propoſitum.</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 huiuſmodi demonſtrat ione facilè videre poteris non eſſe generaliter verum, <lb/>id quod Nicolaus Tartalea inquit .43. quæſito vltimæ partis ſuorum tractatuum, hoc <lb/>eſt centrum circuli <var>.r.n.g.</var> ſemper eſſe in perpendiculari, quæ à puncto <var>.r.</var> ad lineam <var>.e.<lb/>o.</var> tranſit, protracta ipſa <var>.e.o.</var> quantum volueris, imò in quacunque alia linea ipſum eſ <lb/>ſe poteſt, nec non in aliqua parallela ipſi <var>.e.o.</var> quemadmodum ex te ipſo, medianti-<lb/>bus, hic ſupradictis rationibus videre poteris, vnde ex neceſſitate ſequitur illud pro <lb/>blema ſemper ferè falſum 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 ſpeculatio 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 demonſtrandum <lb/>breuitatem radiorum incidentium, & reflexorum in ſpeculo plano, nunc ad <lb/>te ſcribo, quamuis prolixior ali quantulum ſit eo, quod ab antiquis traditus eſt.</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> ſectione ſuperficiei reflexionis <reg norm="cum" type="context">cũ</reg> ſpe-<lb/>culo <var>.r.a.</var> verò et <var>.a.b.</var> pro radijs dictis, qui ſemper <reg norm="faciunt" type="context">faciũt</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 iiſdem punctis <var>.b.r.</var> ad aliud punctum, <lb/>quod volueris ipſius lineæ <var>.p.h.</var> quas probabo <reg norm="longiores" type="conjecture">longiores'</reg> (ſimul ſumptas) eſſe priori-<lb/>bus. </s> + <s xml:space="preserve">Imaginemur igitur duas perpendiculares, ſeu cathetos <var>.b.i.</var> et <var>.q.r.a.</var> punctis <var>.b.<lb/>r.</var> ad <var>.p.h.</var> <reg norm="abſciſſaque" type="simple">abſciſſaq́</reg> ſit linea <var>.o.b.</var> in puncto <var>.x.</var> ita quod <var>.b.x.</var> æqualis ſit ipſi <var>.b.a.</var> quod <lb/>nulli dubium erit poſſe effici, cum <var>.o.b.</var> <reg norm="longiot" type="context">lõgiot</reg> ſit <var>.b.a.</var> co quod opponatur angulo ob-<lb/>tuſo ipſius trianguli <var>.b.a.o.</var> quę <var>.o.b.</var> ſimiliter protrahatur vſque ad <var>.d.</var> ita quod <var>.b.d.</var> <lb/>æqualis ſit <var>.x.b.</var> </s> + <s xml:space="preserve">deinde protrahatur <var>.o.i.</var> quouſque <var>.i.h.</var> æqualis ſit <var>.a.i</var>. </s> + <s xml:space="preserve">In alia parte po-<lb/>ſtea idem faciendum eſt ſecando <var>.a.r.</var> in puncto <var>.u.</var> ita quod <var>.u.r.</var> æqualis ſit <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">productũ</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> exiſtis <lb/>rationibus. </s> + <s xml:space="preserve">Nam cum quadratum ipſius <var>.o.b.</var> æquale ſit duobus quadratis <var>.o.i.</var> et <var>.i.<lb/>b.</var> ex penultima primi Eucli. ipſa quadrata <var>.o.i.</var> et <var>.i.b.</var> æqualia erunt producto <var>.o.d.</var> in <lb/><var>o.x.</var> ſimul ſumpto cum quadrato <var>.b.x.</var> ex .6. ſecundi, hoc eſt ipſi producto ſimul ſum-<lb/>pto cum quadrato <var>.b.a.</var> hoc eſt ipſi producto ſimul ſumpto cum duobus quadratis <var>.a.<lb/>i.</var> et <var>.i.b.</var> ſed quia productum <var>.o.h.</var> in <var>.o.a.</var> ſimul ſumpto cum quadrato <var>.a.i.</var> ęquatur qua <lb/>drato <var>.o.i.</var> ideo productum <var>.o.h.</var> in <var>.o.a.</var> ſimul ſumptum 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> ſimul ſumpto <reg norm="cum" type="context">cũ</reg> duobus quadratis dictis <lb/>hoc eſt ipſius <var>.a.i.</var> et <var>.i.b.</var> quę 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 ipſi <var>.o.b.</var> & habebimus proportionem <lb/>ipſius <var>.a.b.</var> ad <var>.a.i.</var> maiorem eſſe ea quæ eſt ipſius <var>.a.e.</var> ad eandem <var>.a.i.</var> cum <var>.a.b.</var> maior <lb/>ſit ipſa <var>.a.e.</var> vt oppoſita angulo obtuſo, quapropter proportio <var>.x.b.</var> ad <var>.a.i.</var> maior erit <lb/>ea quæ eſt <var>.o.b.</var> ad <var>.o.i</var>. </s> + <s xml:space="preserve">Iam enim ſcis proportionem <var>.o.b.</var> ad <var>.o.i.</var> eſſe, vt <var>.a.e.</var> ad <var>.a.i.</var> ex <lb/>ſimilitudine 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">tũc</reg> ex .27. quinti <reg norm="permutando" type="simple context">ꝑmutãdo</reg> <reg norm="proportio" type="simple">ꝓportio</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="eiuſdem" type="context">eiuſdẽ</reg> <reg norm="componendo" type="context context">cõponẽdo</reg> maior <reg norm="proportio" type="simple">ꝓportio</reg> erit <var>.o.d.</var> ad <var>.o.b.</var> ea quę eſt <var>.o.h.</var> ad. o <lb/>i. & <reg norm="permutando" type="context">permutãdo</reg> maior ipſius <var>.o.d.</var> ad <var>.o.h.</var> ea quæ <var>.o.b.</var> ad <var>.o.i.</var> & ex .33. maior ipſius <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 eſt <var>.a.r.</var> ad <var>.a.q.</var> ex ſimilitudine <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 eſt <var>.o.b.</var> ad <var>.o.i.</var> & exijſdem <lb/>ſupradictis rationibus maior erit proportio ipſius <var>.s.a.</var> ad <var>.p.a.</var> ea quæ eſt <var>.a.r.</var> ad <var>.a.q.</var> <lb/>ſed 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 eſt <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 ipſius <var>.a.s.</var> ad <lb/><var>a.p.</var> ca quæ <var>.o.d.</var> ad <var>.o.h.</var> ſed cum ex .15 <lb/>ſexti, eadem ſit 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> quę <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 ipſius <var>.o.d.</var> ad <var>.o.h.</var> quæ <var>.o.a.</var> <lb/>ad <var>.o.x.</var> & ipſius <var>.a.o.</var> ad <var>.a.u.</var> quemad-<lb/>modum ipſius <var>.a.s.</var> ad <var>.a.p</var>. </s> + <s xml:space="preserve">Quare maior proportio erit ipſius <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 ſequitur <var>.o.x.</var> maiorem eſſe <var>.a.u.</var> ex .8. quinti, ergo <var>.b.x.o.r.</var> longior erit <lb/>ipſa <var>.b.a.u.r</var>. </s> + <s xml:space="preserve">Quod eſt propoſitum.</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 ellipſis tranſire per duo puncta <var>.r.</var> et <var>.b.</var> ſupponendo ipſa puncta, ea eſle, quæ ita <lb/>axem diuidunt, vt ſingula produ-<lb/> + <anchor type="figure" xlink:label="fig-0354-01a" xlink:href="fig-0354-01"/> + cta fectionum ſint, vt inquit Per-<lb/>geus. </s> + <s xml:space="preserve">imaginemur, etiam <var>.p.h.</var> con <lb/>tiguam eſſe ipſi ellipſi in <reg norm="puncto" type="context">pũcto</reg> <var>.a.</var> <lb/>vnde ſi protractæ fuerint duæ <var>.r.a.</var> <lb/>et <var>.b.a.</var> habebimus ex .48. tertijip-<lb/>ſius Pergei angulos <var>.b.a.h.</var> et <var>.r.a.<lb/>p.</var> inuicem æquales. </s> + <s xml:space="preserve">Ducendo <lb/>poſtea ad quoduis punctum ipſius <lb/><var>p.h.</var> duas <var>.b.o.</var> et <var>.r.o.</var> certi erimus, <lb/>quod ſecabuntur à gyro oxygo-<lb/>nio, quarum vna ſecta ſit in pun-<lb/>cto <var>.i.</var> ducta poſtea <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 ipſa <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 ſpeculum vstorium.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">VErum ſpeculum vſtorium, illud non eſt, quod ab Euclide traditum fuit, & <reg norm="quod" type="simple">ꝙ</reg> <lb/>tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie <lb/>ſphærica concaua ſeinuicem in centro ſpeculi interſecare. </s> + <s xml:space="preserve">Nam cum omnes lineę <lb/>recte à centro, & cir cunferentia alicuius ſphæræ terminatæ, ſint eidem circunferen-<lb/>tiæ perpendiculares, ſequeretur ex neceſſitate radios incidentiæ etiam perpendicu <lb/>lares eidem ſuperficiei eſſe, cum anguli incidentiæ ſemper æquales ſint angulis re-<lb/>flexionis, vnde etiam ex neceſſitate ſequeretur punctum corporis lucidi, à quo radij <lb/>luminoſi excunt, in centro ſpeculi reperiri. </s> + <s xml:space="preserve">quod quidem falſiſſimum eſt.</s> + </p> + <p> + <s xml:space="preserve">Alia etiam via poſſum hanc oſtendere impoſſibilitatem, & tibi probabo, quod <lb/>in nullo aliquo puncto poſſunt inuicem conuenire ipſi radijrefle xi omnes.</s> + </p> + <p> + <s xml:space="preserve">Sit igitur <var>.l.a.c.</var> <reg norm="conis" type="context">cõis</reg> ſectio ſuperficiei reflexionis cum ſpeculo, cuius centrum ſit <var>.o.</var> <lb/>punctum verò lucidum ſit <var>.g.</var> <reg norm="protrahaturque" type="simple">protrahaturq́</reg> <var>.g.o.a</var>. </s> + <s xml:space="preserve">Nunc autem primum dico, quod <lb/>radij reflexi à punctis diuerſarum <reg norm="diſtantiarum" type="context">diſtantiarũ</reg> ab <var>.a.</var> non <reg norm="coincident" type="context">coincidẽt</reg> inuicem in aliquo <lb/>puncto lineę <var>.g.o.a</var>: ſint ergo duo puncta <var>.u.</var> et <var>.r.</var> diuerſarum <reg norm="diſtantiarum" type="context">diſtantiarũ</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> ſit <var>.r.e.</var> protrahatur <lb/><var>u.e.</var> quam dico effe non poſſe radium reflexum ab <var>.u.</var> quotieſcunque eius incidens <lb/>deſcendat 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">reflexũ</reg> eſſe ipſius <var>.g.u.</var> igitur anguli <var>.g.u.o.</var> et <var>.o.u.e.</var> erunt inuicem æquales, & ſic <lb/>etiam erunt duo <var>.g.r.o.</var> et <var>.o.r.e.</var> vnde ex tertia ſexti & .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 demonſtra-<lb/>bo, eo quod cum <var>.g.u.</var> maior ſit <var>.g.r.</var> ex .8. tertij, erit ex .8. quinti proportio ipſius <var>.g.u.</var> <lb/>ad <var>.r.e.</var> maior proportione ipſius <var>.g.r.</var> ad <var>.r.e.</var> ſed ex .7. tertij <var>.u.e.</var> minor eſt <var>.r.e.</var> erit igi-<lb/>tur ex dicta .8. quinti maior proportio <reg norm="ipſius" type="simple">ipſiꝰ</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 ipſius <var>.g.u.</var> ad <var>.u.e.</var> <lb/> + <anchor type="figure" xlink:label="fig-0355-01a" xlink:href="fig-0355-01"/> + quam ipſius <var>.g.r.</var> ad <var>.r.e.</var> ergo non æqualis, <lb/>quapropter impoſſibile eſt <var>.u.e.</var> eſſe radium <lb/>reflexum incidentis radij <var>.g.u</var>. </s> + <s xml:space="preserve">Vnde ſequi <lb/>tur concurſum radiorum reflexorum à ſpe-<lb/>culo ſphærico concauo non eſſe in vno, & <lb/>e<unsure/>odem puncto ipſius catheti incidentiæ, <lb/>quando à ſitu non æquidiſtanti ab ipſo ca-<lb/>theto <reg norm="reflectuntur" type="context">reflectũtur</reg>, ex hac ſpeculatione <reg norm="etiam" type="context">etiã</reg> <lb/>videre licet, verum eſſe id quod in .3. Epiſto <lb/>la tibi ſcripſi nempe, quod quotieſcunque <lb/>axes viſuales, vel radij reflexi, in vna ea-<lb/><reg norm="demque" type="simple">demq́</reg> ſuperficie reflexionis fuerint, </s> + <s xml:space="preserve">tunc <lb/>imago obiecti nullo modo videbitur in ca-<lb/>theto incidentiæ, in ſpeculo ſphærico 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 ſolutio.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">NOn abſque ratione dubitas, vtrum etiam in ſphæricis ſpeculis conuexis idem <lb/>accidat, hoc eſt, an radij reflexi à punctis inęqualis diſtantiæ à catheto inciden <lb/>tiæ conueniant inuicem in eodem catheto.</s> + </p> + <p> + <s xml:space="preserve">Ad quod reſpondeo, non concurrere in dicto catheto, ſed extra ipſum, & ſimi-<lb/>liter extra ipſum vide bitur imago.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ratione, imaginemur ſuperficiem reflexionis alicuius ſpeculi ſphæ-<lb/>rici conuexi <var>.b.d.h.g.</var> cuius communis ſectio cum ſuperficie ſphærica ſit 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> ſit ſe <lb/>midiameter circuli <var>.d.g.h.</var> et <var>.o.c.</var> ſit plus medietate ipſius <var>.o.g.</var> <reg norm="accipiaturque" type="simple">accipiaturq́;</reg> linea <var>.e.c.</var> <lb/>minor ipſa <var>.o.c.</var> ſed maior ipſa <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 ipſum aliquantulum <lb/>plus quam <var>.c.g.</var> ſed minus quam <var>.c.<lb/>o.</var> ſignando circunferentiam <var>.d.e.h.</var> <lb/>in puncto <var>.e.</var> quod ex .7. tertij poſſi-<lb/>bile eſt, protrahatur poſtea <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 poſtea cum fuerit <var>.c.<lb/>e.K.</var> indeterminatè, <reg norm="habebimus" type="simple">habebimꝰ</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">qđ</reg> ſi <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 ſecabit cathetum <var>.b.o.</var> in puncto <var>.c.</var> intra ſpeculum, nec dubitandum eſt quin <lb/>linea <var>.e.b.</var> ſectura ſit <var>.b.o.</var> eo quod cum angulus <var>.o.e.c.</var> ſit maior angulo <var>.e.o.c.</var> ex .19. <lb/>primi, & ſimiliter angulus <var>.b.e.f.</var> ſequitur 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 eſt <lb/><reg norm="neceſſarium" type="context">neceſſariũ</reg> <reg norm="semper" type="context">sẽper</reg> exiſtat, vt linea <var>.c.e.</var> minor eſſe linea <var>.c.o</var>. </s> + <s xml:space="preserve">Sed vnde eueniat quod ip <lb/>ſa neceſſariò debeat ſemper maior eſſe ipſa <var>.c.g.</var> clarum eſt 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> ſit vt reflectatur. </s> + <s xml:space="preserve">Accipiamus <lb/>nunc <var>.p.c.</var> minorem medietate ipſius <var>.b.c.</var> & à puncto <var>.p.</var> imaginemur tangentem <var>.p.q.</var> <lb/>in puncto <var>.q.</var> prorractaq́ue ſit <var>.b.q.</var> vt radius incidentiæ, </s> + <s xml:space="preserve">tunc dico, radium reflexum <lb/>ipſius <var>.b.q.</var> <reg norm="non" type="context">nõ</reg> concurrere in eodem puncto <var>.c.</var> ipſius catheti, ſi vero dixeris <reg norm="quod" type="simple">ꝙ</reg> ſic. </s> + <s xml:space="preserve">Eſto <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. ſexti Vitellionis proportio <var>.b.i.</var> ad <var>.i.c.</var> erit, vt <var>.b.o.</var> ad <var>.o.c.</var> & ſimiliter erit <lb/>ipſius <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/>ſius <var>.b.p.</var> ad <var>.p.c.</var> erit vt ipſius <var>.b.i.</var> ad <var>.i.c.</var> ſed quia <var>.p.b.</var> vt pars ipſius <var>.b.i.</var> minor eſt ip-<lb/>ſa, ergo ex .14. dicti <var>.p.c.</var> minor erit ipſa <var>.c.i.</var> hoc eſt totum minus ſua parte, quod eſt <lb/>impoſſibile, </s> + <s xml:space="preserve">quare non in ipſo catheto videbitur imago ipſius 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 ipſis ſpeculis ſphæricis conuexis, hoc eſt <lb/>quod poſſibile ſit aliquoties, radium reflexum concurrere cum catheto incidentiæ <lb/>extra ſpeculum inter puncta <var>.g.</var> et <var>.p.</var> vt exempli gratia .ſi 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> ipſius concurſus cum catheto incidentiæ eſſet <lb/>inter <var>.g.</var> et <var>.p.</var> eo quod <reg norm="cum" type="context">cũ</reg> linea <var>.p.q.</var> debeat @iui lere <reg norm="angulum" type="context">angulũ</reg> <var>.b.</var> q, c. <reg norm="per" type="simple">ꝑ</reg> ęqualia, oportebit <lb/>c. poſitum eſſe inter <var>.g.</var> et <var>.p.</var> quia angulus <var>.g.q.p.</var> maior eſt 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 poſtea <var>.q.<lb/>p.</var> <reg norm="vſque" type="simple">vſq;</reg> ad <reg norm="alteram" type="context">alterã</reg> <reg norm="partem" type="context">partẽ</reg> circunferen-<lb/>tiæ ipſius circuli. </s> + <s xml:space="preserve"><reg norm="argumentando" type="context context">argumẽtãdo</reg> dein-<lb/>de mediante vltima ſexti, illud <reg norm="idem" type="context">idẽ</reg> <lb/>po@es etiam ſcire ex .22. quinti Alha <lb/>zeni. & ex .26. ſexti Vitellionis. </s> + <s xml:space="preserve">vn-<lb/>de ſi ad ambas pupillas venerint ra <lb/>dij reflexi ipſius obiecti <var>.b.a.</var> duobus <lb/>punctis huiuſmodi ſpeculi, ita di-<lb/>ſtantibus à puncto <var>.g.</var> vt <var>.q</var>. </s> + <s xml:space="preserve">tunc com <lb/>mune punctum concurſus axium vi <lb/>ſualium erit in catheto inter <var>.g.p.</var> <lb/>vbi apparebit imago ex ſuperius di <lb/>ctis rationibus, ita vt <reg norm="non" type="context">nõ</reg> ſolum con <lb/>cauis, ſed 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 poteſt euenire, vt tibi alias dixi, eo quod ſi <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> ſectione <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg> <var>.l.t.x.</var> reflexionis & <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg> ſpeculi, <reg norm="punctumque" type="context context simple">pũctũq́;</reg> <lb/>lucidum <var>.l.</var> <reg norm="protractoque" type="simple">protractoq́;</reg> catheto <var>.l.r.t.</var> <reg norm="lineisque" type="simple">lineisq́;</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 ſint, & ſic 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 ſemper in puncto <var>.t.</var> <reg norm="conuenient" type="context">conueniẽt</reg> omnes radij reflexi ipſius + <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 ſphærico concauo, ſeu conuexo, non <lb/>omnes radij reflexi conueniunt in vno, <reg norm="eodemque" type="simple">eodemq́;</reg> puncto catheti incidentiæ, quemad <lb/>modum in planis accidit, in quibus ſemper vnum, & idem punctum eſt ipſis commu <lb/>ne in ipſo incidentiæ catheto.</s> + </p> + <p> + <s xml:space="preserve">Non prætermittam etiam hunc alium breuiorem modum ſpeculandi <reg norm="æqualita- tem" type="context">æqualita-<lb/>tẽ</reg> depreſſionis imaginis ſub ſpeculo plano, ei quæ ſupra reperitur ipſius obiecti, in ca <lb/>theto incidentiæ, quemadmodum nu nc <lb/>vltimò diximus, hoc eſt 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 eiuſdem obiecti reflexa à pun-<lb/>cto <var>.m.</var> reperiatur in linea <var>.z.m.t.</var> & <lb/>iſtæ duæ lineę ſeinuicem ſecent in <lb/>puncto <var>.t.</var> ipſius catheti, exiſtente <var>.<lb/>r.t.</var> æquali <var>.r.l.</var> vt nunc vidimus, er-<lb/>go ſemper imago reflexa à ſpecu-<lb/>lo plano, nobis apparebit <reg norm="in" type="wordlist">ĩ</reg> ipſo ca <lb/>theto, tam vltra ſpeculum, quam ci <lb/>tra ipſum, <reg norm="repertum" type="simple context">reꝑtũ</reg> fuerit <reg norm="ipsum" type="context">ipsũ</reg> <reg norm="obiectum" type="context">obiectũ</reg> <lb/>quod nec Alhazem, nec Vitellio, <lb/>nec alius aliquis (quod ſciam) ad huc ſcientificè demonſtrauit. </s> + <s xml:space="preserve">exempla enim vel ex <lb/>perientia non faciunt ſcire. </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 ſint duobus rectis cum <lb/>æquales ſint 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 ecclipſibus Lunaribus.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">ROtunditas vmbræ in ecclipſi-<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, & ſi terra eſ-<lb/>ſet <reg norm="etiam" type="context">etiã</reg> cuiuſuis alterius figurę, <reg norm="quam" type="context">quã</reg> <lb/>ſphæricę, dummodo aqua impleret <lb/><reg norm="locum" type="context">locũ</reg> ſphęriceitatis à terra <reg norm="derelictum" type="context">derelictũ</reg>, <lb/>nihilominus vmbra eſſet rotunda, <lb/>quę 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 ſentiat in libro de magnitudi-<lb/>ne terrę, & 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">Sciẽdũ</reg> enim eſt, quod omne cor <lb/>pus in ſe habens <reg norm="aliquantulum" type="context">aliquantulũ</reg> opaci-<lb/>tatis, ſemper debilitat <reg norm="radium" type="context">radiũ</reg> lumino <lb/>ſum, & <reg norm="tanto" type="context">tãto</reg> magis, <reg norm="quanto" type="context">quãto</reg> magis in ip <lb/>ſo corpore radius penetrat, <reg norm="etiam" type="context">etiã</reg> & ſi <lb/>ad rectos incideret ipſe radius ſupra <lb/><reg norm="ſuperficiem" type="context">ſuperficiẽ</reg> <reg norm="ipſius" type="simple">ipſiꝰ</reg> corporis. </s> + <s xml:space="preserve"><reg norm="Exempli" type="context">Exẽpli</reg> gra <lb/>tia, eſto <var>.q.p.</var> corpus a <reg norm="queum" type="context">queũ</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"/> + deſcendat radius <var>.b.d.K.s.f.</var> ad <reg norm="libitum" type="context">libitũ</reg> hoc eſt rectè vel obliquè, cuius pars <var>.b.d.</var> in ipſo <lb/>aere exiſtat. </s> + <s xml:space="preserve">Nunc manifeſtum erit partem <var>.b.d.</var> ipſius radij clariorem ſeu minus im <lb/><reg norm="peditam" type="context">peditã</reg> eſſe quam <var>.d.K.</var> quod ex eo etiam cognoſcere poſſumus quia <var>.b.d.</var> reflectitur à <lb/>puncto <var>.d.</var> ſuperficiei corporis a quei, quapropter minus luminoſa remanebit pars <var>.d.<lb/>K.</var> cum non tota claritas <var>.b.d.</var> deſcendat in corpore aqueo, ſed vna eius pars reflecta-<lb/>tur, reliqua verò tantummodò deſcendat, </s> + <s xml:space="preserve">deinde pars <var>.K.s.</var> ex neceſſitate debilior <lb/>erit ipſa <var>.d.K.</var> eo quod ſuccedit poſt ipſam <var>.d.K.</var> propter hoc etiam, quia cum corpus <lb/>aqueum habeat aliquantulum opacitatis, radius <var>.d.K.</var> ab omni puncto ipſius ſpiſſitu-<lb/>dinis a quæ continuo reflectitur, quę quidem reflexio eſt illud lumen cęruleum, quod <lb/>in profunditate ipſius aquę nobis apparet. </s> + <s xml:space="preserve">Cum igitur reflexio ipſa ſemper detra-<lb/>hat ab ipſo radio luminoſo, reſiduum verò ſit id quod penetrat, ideo <var>.K.s.</var> erit vna <lb/>pars tantummodò luminis ipſius <var>.d.K</var>: in <var>.s.f.</var> verò aliqua pars luminis ipſius <var>.K.s.</var> & ſic <lb/>continuò debilitatur radius, ita quod ad nihilum vſque deuenit, & vltra tale cor-<lb/>pus remanebit vmbra, quaſi ſi ipſum corpus eſſet perfectè opacum, cuius rei cauſa, <lb/>eſt illa continua reflexio, vt diximus, quæ continuò adimit aliquid ex ipſo radio, <lb/>nec permittit eum totum tranſire.</s> + </p> + <p> + <s xml:space="preserve">Quapropter mirandum non eſt 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/>ſuperficie corporis lunaris, ab alia etiam ratione prouenire poſſet. </s> + <s xml:space="preserve">Imaginemur enim <lb/>aggregatum terrę, <reg norm="marisque" type="simple">marisq́;</reg> eſſe tantummodò aqueum, quod quidem eſſet perfectè <lb/>ſphæricum ratione centri grauitatis, <reg norm="ſupponamusque" type="simple">ſupponamusq́;</reg> <reg norm="ipsum" type="context">ipsũ</reg> eſſe valde diaphanum, ita <lb/>quod radij ſolares ipſum penetraſſent. </s> + <s xml:space="preserve">Tunc dico quod in ſuperficie corporis luna-<lb/>ris produceret vmbram. </s> + <s xml:space="preserve">Pro cuius intelligentia cogitemus ſubſcriptam hic figuram <lb/><var>b.h.q.a.e.</var> eſſe ſphęram aliquam cryſtallinam, & ad partem <var>.b.h.q.</var> ſit radius lumino-<lb/>ſus ſolaris qui ipſam illuminet, cuius radij extremitates ſint <var>.d.b.l.</var> et <var>.p.q.r.</var> ſupponen-<lb/>do <var>.d.l.</var> et <var>.p.r.</var> terminos eſſe vnius plani ſecantis ipſum radium per axem, </s> + <s xml:space="preserve">tunc vide-<lb/>bis ipſum radium <var>.b.p.q.d.</var> <reg norm="tranſeun- tem" type="context">tranſeũ-<lb/> + <anchor type="figure" xlink:label="fig-0358-01a" xlink:href="fig-0358-01"/> + tem</reg> ipſam ſphæram, congregari ſeu <lb/><reg norm="condenſari" type="context">condẽſari</reg>, ob vniformem refractio-<lb/>nem, vſque ad punctum <var>.o.</var> deinde; <lb/>propter rectitudinem ipſius diffu-<lb/>ſionis, vltra punctum <var>.o.</var> ipſum dila-<lb/>tari, diſgregari, ſeu rarefieri, <reg norm="quouſque" type="simple">quouſq;</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 vmbroſa remanerent. di-<lb/><reg norm="ſtantiaque" type="simple">ſtantiaq́;</reg> ab <var>.o.</var> ad ſuperficiem ſphęri <lb/>cam corporis <var>.b.e.d.q.</var> non ſolum <reg norm="non" type="context">nõ</reg> <lb/>maior eſt diametro ipſius ſphæræ; <lb/></s> + <s xml:space="preserve">imo minor, vt à te ipſo experiri po-<lb/>tes. </s> + <s xml:space="preserve">Poſito igitur aliquo obiecto <lb/>opaco in loco <var>.K.o.g.</var> eius ſuperficies <lb/>intercepta inter <var>.K.</var> et <var>.g.</var> adumbrata <lb/>erit, excepto puncto <var>.o</var>. </s> + <s xml:space="preserve">Poſito dein <lb/>de ipſo obiecto in loco <var>.n.y.x.m.</var> <reg norm="eius" type="simple">eiꝰ</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 deſtitutæ <reg norm="interuallumque" type="simple">interuallumq́;</reg> tantummodò inter <var>.y.x.</var> illuminatum erit, ſed ſi in <lb/>loco <var>.c.u.</var> poſitum fuerit, </s> + <s xml:space="preserve">tunc totum <var>.c.u.</var> illuminatum erit, ſed debili modo propter <lb/>detractionem factam à reflexione in ſuperficie corporis ſphærici, vt ſupra 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">Poſito 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, ſed <var>.z.H.</var> pauciſſimum habebit lumen, pro-<lb/>pter diſgregationem radiorum. </s> + <s xml:space="preserve">Poſito poſtea ipſo obiecto in loco <var>.t.l.r.s.</var> tanto <lb/>minus lumen habebit pars <var>.l.r.</var> propter dictam <reg norm="diſgregationem" type="context">diſgregationẽ</reg>, ſeu <reg norm="diſſipationem" type="context">diſſipationẽ</reg> radio <lb/>rum, & ſic ſucceſſiuè quanto remotius poſitum fuerit ipſum 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 ſcilicet, qui per ſphæram chryſtallinam tranſibunt, ſed videbi-<lb/>tur vmbra ipſius ſphęrę in obiecto propoſito, cum nullum actum illuminationis in <lb/>eo loco obiecti habeant radij tranſeuntes per dictam ſphęram. </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 deſtituta.</s> + </p> + <p> + <s xml:space="preserve">Quòd vero tolerabilior ſit oculis radius reflexus Solis à ſuperſicie aquæ, quàm <lb/>à ſuperficie alicuius ſpeculi, oritur ab eo, quod ſupra diximus, hoc eſt, quod ma-<lb/>gna parsipſius luminis penetrat in aquam, & non totum reflectit, quod quidem non <lb/>accidit ſpeculis 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/>cuiuſuis trianguli ſupra 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 cuiuſuis triangu-<lb/>li orthogonij, longiora ſint tertio latere, per diametrum circuli in eo in-<lb/>ſcripti, ab alijs iam demonſtratum fuit. </s> + <s xml:space="preserve">Sed quòd quælibet duo latera <lb/>cuiuſuis trianguli longiora ſint tertio per latus tetragonicum, quadrupli <lb/>producti cuiuſuis lineæ deſcendentis ab angulo contento à dictis duobus lateribus <lb/>ad oppoſitam partem circuli inſcripti, in partem extrinſecam ipſius lineæ, nullus <lb/>(quod ſciam) vnquam ſcripſit, vel animaduertit.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia triangulus <var>.a.b.c.</var> quem volueris, in quo deſcribatur circulus <var>.<lb/>u.s.n.</var> & puncta contingentiæ ſint eadem <var>.u.s.n.</var> à puncto vero <var>.a.</var> deſcendat linea <var>.a.<lb/>i.e.</var> quæ terminetur à circunferentia in puncto <var>.e.</var> ipſius 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">tetragonicũ</reg> qua-<lb/>drupli producti ipſius <var>.a.e.</var> in <var>.a.i</var>. </s> + <s xml:space="preserve">Nam certi ſamus ex vltima parte penultimæ ter-<lb/>tij Eucli <var>.n.c.</var> et <var>.s.c.</var> æquales inuicem eſſe, & ſimiliter <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"/> + ipſo <var>.b.c.</var> per <var>.a.u.</var> et <var>.a.n.</var> quæ duæ <lb/>partes ſunt inuicem æquales di-<lb/>cta ratione, & quadratum lineæ <lb/>æqualis aggregato earum, eſſet qua <lb/>druplum quadrato cuiuſuis earum <lb/>ex .4. ſecundi, ſed ex penultima ter <lb/>tij, productum <var>.a.e.</var> in <var>.a.i.</var> æquale eſt <lb/>quadrato ipſius <var>.a.u.</var> vel ipſius <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 eſt igitur quod <var>.a.b.</var> cum <var>.a.c.</var> longiores ſint ipſa <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 propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Illud etiam non eſt ſpernendum, quod quotieſcunque data fuerint omnia latera <lb/>alicuius trianguli, illicò poſſumus cognoſcere puncta <var>.u.n.s.</var> contingentiæ circuli in <lb/>ſcripti, ope vltimæ partis penultimæ tertij, eo quod ex illa iam ſcimus, 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 vnaquęque nota erit, cum illarum quælibet, medietas ſit reſidui cogniti, detra <lb/>hendo poſtea vnam <reg norm="illarum" type="context">illarũ</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> ęqualis <var>.b.s.</var> vel <var>.c.<lb/>s.</var> vnde ſimiliter nobis innoteſcet <lb/>punctum <var>.s.</var> cum duobus punctis <var>.u.</var> <lb/>ct <var>.n.</var> à quibus duobus punctis, ſi <lb/>duę perpendiculares ad talia latera <lb/>ductæ fuerint, vbi hæe perpendicu <lb/>lares ſeinuicem ſecabunt, ibi <reg norm="cen- trum" type="context">cen-<lb/>trũ</reg> circuli inſcriptibilis erit in trian <lb/>gulo propoſito.</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 eſt niſi <lb/>quod cum Iris videtur, non eodem loco ab omnibus videtur, quia reflexio eſt, & <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 Inſtrumento oxygonio, ſeu elliptico.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod aliquando à me audiuiſti falſum non eſt, ſcilicet poſſibile eſſe (vt <lb/>ſpeculatus ſum) particulare inſtrumentum fabricari ad deſignandum oxy-<lb/>goniam, ſeu ellipticam ſectionem, quæ à Pergeo defectio appellatur, quod quidem <lb/>inſtrumentum valde diuerſum eſt ab alijs, quę aliàs inueni, pro ipſis conicis ſectio <lb/>nibus delineandis. </s> + <s xml:space="preserve">Occaſionem <reg norm="autem" type="wordlist">aũt</reg> huiuſimodi inſtrumenti inueniendi mihi præ <lb/>buit <reg norm="ſecunda" type="context">ſecũda</reg> dubij ſolutio <reg norm="quam" type="context">quã</reg> feci ann .1568. grauiſſ. philoſopho Franciſco Vimer <lb/>cato, <reg norm="nam" type="context">nã</reg> <reg norm="cum" type="context">cũ</reg> viderim in ea figura <var>.f.a.</var> ſemper <reg norm="æqualem" type="context">æqualẽ</reg> eſſe <var>.o.i.</var> ſuæ parallelæ ſcilicet, <lb/>vnde cum recta linea fuerit protracta per <var>.o.</var> et <var>.f.</var> ipſa foret ſemper <reg norm="ęquidiſtans" type="context">ęquidiſtãs</reg> <var>.d.p.</var> ex <lb/>33. primi Eucli. </s> + <s xml:space="preserve">Venit mihi in mentem modus conſtruendi hoc ſubſcriptum inſtru-<lb/>mentum, tali ordine, videlicet, <reg norm="coniungendo" type="context">coniungẽdo</reg> ſeptem hic ſubnotatas 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> ſimul, hoc modo, ſcilicet <reg norm="ſabricando" type="context">ſabricãdo</reg> quadrila-<lb/>terum æquilaterum <var>.c.e.s.l.</var> hac conditione, quod immobili exiſtente puncto <var>.c.</var> in li <lb/>nea <var>.z.r.</var> reliqua omnia mobilia exiſtant, hoc eſt quod <reg norm="punctum" type="context">punctũ</reg> <var>.s.</var> moueatur per di-<lb/>ctam lineam <var>.z.r.</var> & immobili exiſtente puncto <var>.e.</var> vt extremum lineæ <var>.e.h.</var> hoc eſt <lb/>coniuncto extremo <var>.e.</var> lineæ <var>.e.h.</var> cum angulo <var>.c.e.s.</var> reliqua puncta lineæ ipſius <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> ſit compo-<lb/>ſita ex duplo vnius lateris ipſius quadrilateris. </s> + <s xml:space="preserve">Oportet deinde quod punctum <var>.f.</var> <lb/>ſemper vnum, & idem ſit ipſius 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 ſectonis + <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"/> + deſignabit, puncta verò <var>.o.</var> et <var>.K.</var> vt puncta laterum <var>.c.e.</var> et <var>.s.e.</var> æquædiſtantia à <lb/>punctis <var>.c.</var> et <var>.s.</var> <reg norm="eadem" type="context">eadẽ</reg> ſemper ſint, ita tamen vt puncta lineæ <var>.u.n.</var> ſemper diuerſa exi <lb/><reg norm="ſtant" type="context">ſtãt</reg>, & quodlibet ipſius quadrilateri latus, æquale ſit medietati maioris axis ipſius <lb/>oxygoniæ ſectionis delineandæ, et <var>.c.o.</var> ſeu <var>.s.K.</var> (quod idem eſt) ſit æqualis medie <lb/>tati axis minoris dictæ ſectionis, et <var>.z.r.</var> æqualis duplo <var>.e.h.</var> vnde, quando puncta <var>.e.</var> et <var>.<lb/>l.</var> coniuncta ſimul erunt, ſimiliter coniunctæ ſimul 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 eſt in aliquo extremorum maioris axis, & cum punct is <var>.s.</var> coniunct is fuerit cum <lb/>centro <var>.c.</var> punctus <var>.f.</var> parallelę <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 eſt, vt ſint duo <reg norm="puncta" type="context">pũcta</reg> fer <lb/>rea, ad firmandum ipſam <var>.z.r.</var> ſuper ſubiectam lineam ſignificantem maiorem axem <lb/>propoſitę ſectionis.</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 conſtituta ſit alia parallela ad <var>.z.<lb/>r.</var> & in æquali diſtantia ab ipſa quemadmodum <var>.u.n.</var> diſtat 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="vſque" type="simple">vſq;</reg> <lb/>ad <var>.c.</var> deſignabimus quartam partem ſectionis, conuerſo poſtea inſtrumento, hoc eſt <lb/>poſito puncto <var>.r.</var> vbi prius erat <var>.z.</var> et <var>.z.</var> vbi erat <var>.r.</var> aliam delineabimus quartam, & <lb/>ſic ad oppoſitam partem ipſius <var>.z.r.</var> faciendum erit. </s> + <s xml:space="preserve">Hoc inſtrumentum poſſumus <lb/>etiam ita conſtruere, 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">axiũ</reg> propoſita, oxygoniam <lb/>deſignare. </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 poſtulas, non eſt admodum difficile, cupis enim triangulum <lb/>orthogonium, exempli gratia <var>.o.i.e.</var> in figura <var>.A.</var> ita conſtituere, vt di-<lb/>uiſum ſit à perpendiculari <var>.a.i.</var> & quod proportio <var>.o.e.</var> ad <var>.o.i.</var> ſit vt <var>.o.i.</var> ad <lb/><var>i.e.</var> & quod quadrati <var>.o.i.</var> ad quadratum <var>.o.a.</var> ſit vt <var>.e.i.</var> ad <var>.e.a.</var> & quadra <lb/>tum <var>.o.i.</var> ad quadratum <var>.e.i.</var> ſit .ut <var>.o.a.</var> ad <var>.e.a</var>. </s> + <s xml:space="preserve">Quæ omnia in promptu veniunt, quo <lb/>tieſcunque <var>.o.e.</var> fuerit diameter alicuius circuli, <reg norm="diuiſaque" type="simple">diuiſaq́;</reg> in puncto <var>.a.</var> ſecundum pro <lb/>portionem habentem medium <reg norm="duoque" type="simple">duoq́;</reg> extrema, protracta deinde perpendiculari <var>.a.<lb/>i.</var> ad <var>o.e.</var> uſque ad circunferentiam, <reg norm="coniunctæque" type="simple">coniunctæq́;</reg> <var>.o.i.</var> et <var>.i.e</var>: tale triangulum, omnia <lb/>ſupradicta in ſe continebit.</s> + </p> + <p> + <s xml:space="preserve">Nam ex .30. tertij angulus <var>.i.</var> rectus erit, & ex .8. ſexti <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> ſed quia ex diuiſione facta in <reg norm="pum" type="context">pũ</reg> <lb/>cto <var>.a.</var> etiam <var>.o.a.</var> erit media proportio-<lb/>nalis inter totum & reſiduum, 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. eiuſdem <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> eſt, quę ipſius <var>.o.e.</var> ad <lb/><var>o.a</var>. </s> + <s xml:space="preserve">tunc videbis ex .18. ſexti, 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> ſint inuicem ſimiles <lb/>ex ſupradicta .8. ſexti, </s> + <s xml:space="preserve">tunc videbis ex <lb/>18. et .17. eiuſdem dictos <reg norm="triangulos" type="context">triãgulos</reg> ean <lb/>dem habere inter ſe proportionem, quę <lb/>eſt inrer quadrata ipſius <var>.o.i.</var> et <var>.i.e.</var> vnde <lb/>ex prima ſexti ita ſe 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 ſit 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> ipſius <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">tũc</reg> <lb/>habebimus ad partem <var>.o.u.i.</var> triangulum <var>.o.u.i.</var> diuiſum in duo triangula ſimilia ipſi <lb/>totali triangulo. </s> + <s xml:space="preserve">Idem etiam dico ad partem <var>.o.n.i.</var> vnde ex tali ſimilitudine habe-<lb/>bimus <var>.o.u.</var> mediam proportionalem inter <var>.o.i.</var> et <var>.o.a.</var> et ſic <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. ſexti, quadratum <var>.o.u.</var> æquale erit producto ipſius <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> ſed ex prima eiuſdem, ea <lb/>dem proportio eſt ipſius <var>.o.a.</var> ad <var>.o.e.</var> quæ producti ipſius <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 hęc <lb/>eſt alia circuli paſſio.</s> + </p> + <p> + <s xml:space="preserve">Reliqua verò difficultas quam te habere ſcribis, eſt, quare cum duæ lineę <lb/><var>a.u.</var> et <var>.b.s.i.</var> ſint inuicem ęquales, diuiſæ verò non æquali modo, ſed tali, quod <var>.a.</var> <lb/>maior ſit quam <var>.u.</var> et <var>.b.s.</var> maior quam <var>.i.</var> quomodo poteſt fieri, quod ſi <var>.u.</var> maior fue-<lb/>rit <var>.i.</var> proportio <var>.a.</var> ad <var>.i.</var> maior ſit quam ipſius <var>.b.s.</var> ad <var>.u</var>.</s> + </p> + <p> + <s xml:space="preserve">Hoc etiam ex neceſſitate cuenit, eo <lb/>quod ſi 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="ipſaque" type="simple">ipſaq́;</reg> abſciſa fuerit <var>.t.</var> æqualis <var>.i.</var> & ab <var>.<lb/>b.s.</var> abſciſa <var>.s.</var> æqualis <var>.n.</var> habebimus <var>.a.</var> et <lb/>b. inuicem æ quales, vnde habebis ma-<lb/>iorem propor tionem ipſius <var>.b.</var> ad <var>.t.</var> <reg norm="quam" type="context">quã</reg> <lb/>s. ad <var>.n.</var> quod cum clarum per ſe ſit, tibi <lb/>relinquo. </s> + <s xml:space="preserve">ſed ex .27. quinti, proportio <lb/>b. ad. s, maior erit quam <var>.t.</var> ad <var>.n.</var> & ex <lb/>28. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <reg norm="proportio" type="simple">ꝓportio</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 ipſius <var>.b.s.</var> ad <var>.n.t.</var> quam <var>.s.</var> ad <var>.n.</var> <lb/>ergo ex .33. maior erit ipſius <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 eſt maior ipſius <var>.a.</var> ad <lb/>i. quam <var>.b.s.</var> ad <var>.u.</var> quod eſt propo-<lb/>ſitum.</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/>diſtinctione orbium cęleſtium, ortum <lb/>habet à communi opinione motuum <lb/>fixarum. </s> + <s xml:space="preserve">Nam cum putauerint philo-<lb/>ſophi ipſas moueri, ſemper eandem <reg norm="ſeruando" type="context">ſeruãdo</reg> inuicem diſtantiam, non ſine ratione <lb/>crediderunt eas fixas eſſe eodem in orbe, idem etiam poſtea de planetis opinaue-<lb/>runt. </s> + <s xml:space="preserve">Hoc eſt, vnumquemque, aliquo in orbe, fixo exiſtere.</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/>propoſitam ſecundum 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, eſt quidem poſſibile, non tamen adhuc inuentum, quo <lb/>niam nemo ad <reg norm="hunc" type="context">hũc</reg> vſque diem diuiſit vnam datam proportionem in tres <lb/>æquales partes, ſed ſi 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="cupiſque" type="simple">cupiſq́;</reg> <lb/>ſcire modum diuidendi ipſam parabolem vna mediante linea parallela ipſi baſi, ita <lb/>vt eandem habeat proportionem tota parabola ad partem abſciſſam, quæ eſt inter <var>.<lb/>p.</var> et <var>.q</var>. </s> + <s xml:space="preserve">Ad quod faciendum, ſupponendum primò datam proportionem inter <var>.<lb/>p.</var> et <var>.q.</var> diuiſam 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 circunferentię <lb/>ipſius figuræ ducatur parallela baſi <var>.x.e.</var> poſtea verò per puncta media harum dua-<lb/>rum <reg norm="æquidiſtantium" type="context">æquidiſtantiũ</reg> protrahatur <var>.g.b.</var> quæ diameter erit ſectionis, ex 28. ſecundi Per-<lb/>gei, </s> + <s xml:space="preserve">diuidatur deinde hæc diameter in puncto <var>.a.</var> ita quod eadem proportio ſit ipſius <lb/><var>b.g.</var> ad <var>.b.a.</var> quæ ipſius <var>.p.</var> ad <var>.u.</var> quod tibi facile erit, ſecando à linea <var>.p.</var> partem <var>.i.</var> æqua <lb/>lem ipſi <var>.u.</var> tali modo poſtea diuidendo <var>.b.g.</var> ex .12. ſexti, ducatur a puncto <var>.a.</var> ipſa <var>.d.<lb/>h.</var> parallclam ipſi <var>.x.e.</var> & habebitur propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius reiratione, ſcies primum quod <var>.h.d.</var> diuiſa erit à diametro <var>.b.g.</var> per æqua <lb/>lia ex .7. primi Pergei, vel ſi cogitabimus aliquam lineam tangentem ipſam parabo <lb/>lam in puncto <var>.b</var>. </s> + <s xml:space="preserve">tunc ex quinta ſecundi ipſius Pergei habebimus ipſam eſſe paralle-<lb/>lam <var>.e.x.</var> & ex .30. primi Eucli. erit ſimiliter æquidiſtans <var>.d.h.</var> vnde ex .46. primi eiuſ-<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 ſuper <lb/>ficiei totalis parabolæ <var>.x.b.e.</var> ad trigonum <var>.x.b.e.</var> quæ portionis <var>.h.b.d.</var> ad ſuum <reg norm="tri- gonum" type="context">tri-<lb/>gonũ</reg>, eo quod <reg norm="tam" type="context">tã</reg> vna quàm alia erit ſeſquitertia, <reg norm="eius" type="simple">eiꝰ</reg> <reg norm="etiam" type="context">etiã</reg> medietates ſic ſe <reg norm="habebunt" type="context">habebũt</reg>.</s> + </p> + <p> + <s xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolę ad medietatem partia <lb/>lem ipſius, æqualis erit proportioni trianguli <lb/><var>g.b.e.</var> ad triangulum <var>.a.b.d.</var> ſed ex .20. primi <lb/>Pergei, eadem eſt proportio quadrati ipſius <var>.<lb/> + <anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/> + g.e.</var> ad quadratum ipſius <var>.a.d.</var> quæ <var>.b.g.</var> ad <var>.b.a.</var> <lb/>hoc eſt, vt <var>.g.e.</var> ad <var>.a.o.</var> ex ſimilitudine triangu-<lb/>lorum, & quia <var>.b.g.</var> ad <var>.b.a.</var> eſt ſicut <var>.p.</var> ad <var>.u.</var> ita <lb/>igitur erit quadrati ipſius <var>.g.e.</var> ad quadratum <lb/>ipſeus <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. ſexti Euclid. </s> + <s xml:space="preserve">ſed cum ex .24. eiuſdem <lb/>proportio trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> compoſita ſit ex proportione <var>.g.e.</var> ad <var>.a.<lb/>d.</var> er. ex <var>.g.b.</var> ad <var>.b.a.</var> hoc eſt <var>.g.e.</var> ad <var>.a.o.</var> & <lb/>quia <reg norm="proportio" type="simple">ꝓportio</reg> <var>.g.e.</var> ad <var>.a.o.</var> æqualis eſt 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 eſt ei quæ <var>.p.</var> ad <var>.n.</var> hoc eſt 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> compoſita erit ex ca quę <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 ſemper erit ſeſquialtera proportioni <lb/>ipſius <var>.b.g.</var> ad <var>.a.b.</var> eo quod cum ſit proportio totalis portionis ad partialem vt trian-<lb/>guli <var>.b.g.e.</var> ad <var>.b.a.d.</var> & hæc ſeſquialtera proportioni ipſius <var>.g.e.</var> ad <var>.a.o.</var> hoc eſt vt ip-<lb/>ſius <var>.b.g.</var> ad <var>.b.a.</var> ideo proportio ipſarum portionum erit ſimiliter ſeſquialtera pro-<lb/>portioni diametrorum.</s> + </p> + <p> + <s xml:space="preserve">Deinde ſi protractæ fuerint <var>.b.d.</var> et <var>.g.e.</var> quouſque 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 ſupra diximus.). <var>a.d.</var> media proportionalis ſit inter <var>.<lb/>g.e.</var> et <var>.a.o.</var> & proportio <var>.g.z.</var> ad <var>.g.e.</var> vt ipſius <var>.a.d.</var> ad <var>.a.o.</var> eo quodipſius <var>.g.z.</var> ad <var>.a.d.</var> <lb/>& ipſius <var>.g.e.</var> ad <var>.a.o.</var> eſt vt ipſius <var>.b.g.</var> ad <var>.b.a.</var> ex ſimilitudine triangulorum, ideo di-<lb/>ctæ <reg norm="proportiones" type="simple">ꝓportiones</reg> erunt <reg norm="inuicem" type="context">inuicẽ</reg> æquales. </s> + <s xml:space="preserve">Vnde permutatim ita erit ipſius <var>.g.z.</var> ad <var>.g.e.</var> <lb/>vt ipſius <var>.a.d.</var> ad <var>.a.o.</var> & ut ipſius <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 eſt, quę <lb/>cubi ipſius <var>.g.e.</var> ad cubum ipſius <var>.a.d.</var> & ex <reg norm="con" type="context">cõ</reg> <lb/>ſequenti, vt cuborum earundem baſium, eo <lb/>quod cum ſit, ex .36. vndecimi Euclid. pro-<lb/>portio cubi ipſius <var>.g.e.</var> ad cubum ipſius <var>.a.d.</var> <lb/>tripla ei quæ ipſius <var>.g.e.</var> ad <var>.a.d.</var> ideo æqualis <lb/>erit ei quę trianguli <var>.b.g.e.</var> ad triangulum <var>.b.<lb/>a.d.</var> cum proportio horum duorum triangu <lb/>lorum compoſita ſit (vt ſupra 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, ſed trianguli ita ſe in <lb/>uicem habenr, vt parabolę, </s> + <s xml:space="preserve">quare ipſæ para-<lb/>bolæ ſeinuicem habebunt, vt cubi ipſarum <lb/>baſium.</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 propoſitæ.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">CVbum fabricare æqualem propoſitæ pyramidi quadrilateræ, nullius erit diffi-<lb/>cultatis, ſuppoſita tamen pro reperta diuiſione cuiuſuis datæ proportionis in <lb/>tres partes æquales. </s> + <s xml:space="preserve">Nam ex .6. duodecimi Eucli. patet omne corpus ſerratile d-ui <lb/>ſibile eſſe in tres pyramides quadrilateras æquales, ſcimus etiam quod cuilibet py-<lb/>ramidi quadrilateræ poteſt reperiri ſuum ſerratile. </s> + <s xml:space="preserve">Sit igitur propoſita pyramis qua <lb/>drilatera <var>.m.g.f.h.</var> cuius ſerratile ita inueniemus, ducendo primum <var>.h.i.</var> parallelam <lb/>ipſi <var>.g.f.</var> et <var>.f.i.</var> ipſi <var>.g.h.</var> in ſuperficie trianguli <var>.f.g.h.</var> et <var>.m.K.</var> ipſi <var>.g.h.</var> in ſuperficie <lb/>trianguli <var>.m.g.h.</var> & æqualem dictæ <var>.g.h.</var> ducetur poſtea <var>.K.h.</var> et <var>.K.i.</var> & habebimus cor <lb/>pus <var>.f.K.g.</var> ſerratile, & triplum pyramidi propoſitæ. </s> + <s xml:space="preserve">Nunc duplicemus ipſum, du-<lb/>cendo <var>.K.x.</var> in ſuperficie trianguli <var>.i.k.h.</var> parallelam, <reg norm="æqualemque" type="simple">æqualemq́;</reg> ipſi <var>.i.h.</var> et <var>.m.y.</var> <lb/>in ſuperficie trianguli <var>.f.m.g.</var> parallelam, ę<reg norm="qualemque" type="simple">qualemq́;</reg> ipſi <var>.f.g.</var> ducatur poſtea <var>.g.y.</var> et <var>.h.<lb/>x.</var> quarum <reg norm="vnaquæque" type="simple">vnaquæq;</reg> æqualis erit ipſi <var>.f.m.</var> vnde habebimus corpus <var>.f.x.</var> parallelepe-<lb/>pidum, & ſexcuplum ipſi pyramidi propoſitæ.</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 ſextæ parti ſuperficiei <var>.f.i.g.h.</var> quod per <lb/>ſe 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 baſim <var>.f.g.h.</var> quę ſit <var>.n.e.</var> qua mediante, cum quadrato <var>.u.n.</var> fabri <lb/>cetur ſolidum 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 ſint duæ mediæ proportionales <var>.r.s.</var> inter <var>.n.e.</var> et <var>.n.p.</var> quarum <var>.s.</var> ſit <lb/>proximior ipſi <var>.u.p.</var> ex qua <var>.s.</var> ſi conſtitutus fuerit cubus, habebimus propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei ratione, cogitemus corpus <var>.u.e.</var> productum eſſe vſque ad <var>.a.o.</var> per lon-<lb/>gitudem <var>.s.</var> latus dicti cubi, qui quidem cubus ſit <var>.d.b.</var> vnde proportio corporis <var>.u.e.</var> <lb/>ad corpus <var>.e.o.</var> erit, vt ſuperficiei <var>.p.e.</var> ad ſuperficiem <var>.t.e.</var> ex .33. undecimi, ipſæ verò <lb/>ſuperficies ſibi inuicem erunt vt <var>.n.e.</var> ad <var>.e.a.</var> ex prima ſexti, </s> + <s xml:space="preserve">quare proportio corpo <lb/>ris <var>.u.e.</var> ad corpus <var>.e.o.</var> dupla erit proportioni ipſius <var>.s.</var> ad <var>.n.p.</var> ſed cum ex .33 vndeci-<lb/>mi, proportio cubi <var>.d.b.</var> ad corpus <var>.e.o.</var> ſit vt <reg norm="quadratum" type="context">quadratũ</reg> <var>.q.b.</var> ad quadratum <var>.o.a.</var> & cum <lb/>proportio <var>.q.b.</var> ad <var>.o.a.</var> dupla ſit ei quæ <var>.q.o.</var> ad <var>.o.t.</var> ex .18. ſexti, 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 eſt ei quæ <var>.s.</var> ad <var>.n.p.</var> ſed 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 eſt pyramidi propoſitæ.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi oportebit cubum maiorem vel minorem ipſa pyramide reperire, in qua <lb/>proportione tibi placuerit, </s> + <s xml:space="preserve">tunc opus erit aliud quadratum inuenire, quod in ea <lb/>proportione ſe habeat ad quadratum <var>.u.n.</var> quam volueris, quo mediante ſimul cum <lb/>altitudine pyramidis conſequemur propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Aduertendum tamen quod fabri-<lb/>care ipſum corpus ſerratile <var>.k.f.h.</var> & ſo <lb/> + <anchor type="figure" xlink:label="fig-0367-01a" xlink:href="fig-0367-01"/> + lidum <var>.f.x.</var> neceſſarium non eſt, niſi pro <lb/>demonſtratione. </s> + <s xml:space="preserve"><reg norm="idemque" type="simple">idemq́;</reg> dico de alijs <lb/>ſolidis, nam pro ſimplici operatione <lb/>huiuſmodi problematis, abſque ali-<lb/>qua re neceſſaria ad ſpeculandum, 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">eiꝰ</reg> <lb/>alitudinem à <reg norm="puncto" type="context">pũcto</reg> <var>.m.</var> vſque ad ſuper <lb/>ficiem baſis <var>.f.g.h.</var> quæ ſit <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 ipſius baſis <var>.f.<lb/>g.h.</var> quod latus ſit <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">quarũ</reg> <var>.<lb/>s.</var> proximior ſit <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 propoſito <lb/>una tantummodo statione ducendi.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">DVcere parallelam orizontalem alicui muro recto propoſito vna tantummodò <lb/>ſtatione, non ſolum poſſibile eſt ſed etiam facile.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia murus rectus <var>.a.d.</var> ſitus verò <var>.o.n</var>. </s> + <s xml:space="preserve">Si cupimus ducere <var>.n.u.</var> <lb/>parallelam dicto muro, accipiatur quadratum geometricum, ſeu ſcala altimetra <lb/>vel aliquod ſimile inſtrumentum, quo mediante à ſitu <var>.o.</var> videbimus punctum <var>.q.</var> <lb/>quod volueris ipſius muri, <reg norm="dexteram" type="context">dexterã</reg> <lb/>verſus, inferius tamen. ipſo <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="ſiniſtram" type="context">ſiniſtrã</reg> <lb/>cum eodem angulo <var>.n.o.q.</var> oporte-<lb/>bit nos inuenire punctum aliquod <var>.<lb/>p.</var> in dicta ſuperficie 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">duoq́;</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> ſint 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> extrinſe <lb/>cus trianguli <var>.p.q.n.</var> <reg norm="reſiduusque" type="simple">reſiduusq́;</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> ſunt inuicem æquales, & ex .32. eiuſdem, æquales ſunt extrinſeco <var>.q.n.<lb/>g.</var> & ex 27. <var>n.u.</var> erit parallela ipſi <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, ſumendo duo illa puncta in ſuprem a linea <lb/>orizontali ipſius muri ad ſuperiorem partem aſpiciendo, quemadmodum ad infe-<lb/>riorem, quod vnum & idem erit, dummodò non aſpiciamus orizontaliter, eo quod <lb/>nos oportet ſuperficiem conicam producere, linea viſuali mediante. </s> + <s xml:space="preserve">cognoſcere au<lb/>tem angulum <var>.q.n.p.</var> facile erit, conſtituendo primò inſtrumentum in ſitu trianguli <var>.<lb/>o.n.q.</var> <reg norm="aſpiciendoque" type="simple">aſpiciendoq́;</reg> punctum <var>.c.</var> in ſuperficie <var>.n.q.o.</var> & ſic in alia parte, exiſtente in-<lb/>ſtrumento in ſitu trianguli <var>.o.p.n.</var> aſpicere 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 ſi ſitus puncti <var>.n.</var> talis eſſet, vt ab eo non poſſet aliquis murum videre ad re-<lb/>ctos angulos, aſpiceremus punctum <var>.q.</var> ſub orizontali ab oculis noſtris, in orizontali <lb/>tamen puncti <var>.n.</var> ita quod angulus <var>.o.n.q.</var> rectus exiſtat, quo facto obſeruando angu-<lb/>lum <var>.n.o.q.</var> eo mediante, medianteq́ue <var>.n.o.</var> cum angulo <var>.o.n.q.</var> cognoſcemus <lb/>quantitatem diſtantiæ <var>.n.q.</var> idem etiam faciendum eſt cum alio puncto <var>.p.</var> quod <lb/>volueris, & mediantibus duobus punctis inuicem proximis <var>.c.e.</var> cognoſcatur 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/>cognoſcemus reliqua trianguli <var>.<lb/>q.p.n</var>. </s> + <s xml:space="preserve">Conſtituendo poſtea angu-<lb/>lum <var>.q.n.u.</var> æqualem angulo <var>.n.q.p.</var> <lb/>propoſitum 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/>ſterent 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">pũcta</reg> <var>.c.e.</var> proxima <var>.n.</var> in ijſdem <lb/>ſuperficiebus 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> ſectiones dictarum ſuperficierum cum ſuperficie pauimenti ſupra quam fit <lb/>ſtatio.</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 baſi ſecundum 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/>ſi ſecundum vnam datam proportionem, nullius tibi erit difficultatis, con <lb/>ceſſa <reg norm="tamen" type="wordlist">tamẽ</reg> pro inuenta diuiſione cuiuſuis propoſitę proportionis per tres <lb/>æquales partes.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia conus rectus <var>.a.b.c.</var> ſecandus vt dictum eſt, accipiatur latus <lb/>ipſius, quod ſit <var>.a.c.</var> <reg norm="ipſumque" type="simple">ipſumq́;</reg> diuidatur in puncto <var>.d.</var> ſecundum illam proportionem <lb/>quam deſideras, hoc eſt ipſius <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 ſit <var>.a.i.</var> </s> + <s xml:space="preserve">tunc ſi conus <var>.a.b.c.</var> ſectus fue-<lb/>rit à plano per punctum <var>.i.</var> parallelo baſi, habebimus quod quærebamus.</s> + </p> + <p> + <s xml:space="preserve">Cuius rei ratio, primò eſt, quia quotieſcunque conus aliquis ſectus fuerit ab ali-<lb/>quo plano parallelo baſi ipſius, pars ſuperior ſimilis ſemper erit totali cono, quod <lb/>ita probo, cogitemus conum ſectum 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 ſectio triangularis erit, quæ <lb/>ſit <var>.a.b.c.</var> et <var>.b.c.</var> diameter erit baſis.</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 ſectionem huiuſmodi trianguli cum <lb/>plano parallelo ipſi baſi, </s> + <s xml:space="preserve">tunc tale <reg norm="planum" type="context">planũ</reg>, <lb/>circulare erit ex .4. primi ipſius Pergei <var>.K.<lb/>i.</var> verò, eius diameter erit, et <var>.a.m.</var> <reg norm="ſuus" type="simple">ſuꝰ</reg> axis.</s> + </p> + <p> + <s xml:space="preserve">Cum verò <var>.a.l.</var> ſit perpendicularis ipſi <lb/>baſi conitotalis, eo quod rectus ſupponi-<lb/>tur, ideo eadem <var>.a.m.l.</var> erit perpendicula <lb/>ris eriam ipſi ſecundo plano circulari, ex <lb/>conuerſa .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"/> + ſecunda definitione eiuſdem 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 ipſi <var>.b.c.</var> ex .28. primi, quod etiam poteſt con <lb/>cludi mediante .16. vndecimi, cum <var>.k.i.</var> et <var>.b.c.</var> ſint communes ſectiones 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 poſtea ad <var>.a.</var> communes <lb/>ſunt 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. ſexti, eadem <lb/>proportio erit ipſius <var>.m.i.</var> ad <var>.l.c.</var> & ipſius <var>.m.k.</var> ad <var>.l.b.</var> vt ipſius <var>.a.m.</var> ad <var>.a.l</var>. </s> + <s xml:space="preserve">Quare ex <lb/>vndecima quinti, ita erit ipſius <var>.m.k.</var> ad <var>.l.b.</var> vt ipſius <var>.m.i.</var> ad <var>.l.c.</var> & ex .13. eiuſdem, ita <lb/>erit ipſius <var>.k.i.</var> ad <var>.b.c.</var> vt <var>.m.i.</var> ad <var>.l.c.</var> ſed ipſius <var>.m.i.</var> ad <var>.l.c.</var> eſt vt ipſius <var>.a.m.</var> ad <var>.a.l.</var> quod <lb/>iam dictum eſt, vnde ex .11. dicta, ita erit ipſius <var>.k.i.</var> ad <var>.b.c.</var> vt ipſius <var>.a.m.</var> ad <var>.a.l.</var> & ex <lb/>16. dicti ita erit ipſius <var>.a.m.</var> ad <var>.k.i.</var> vt ipſius <var>.a.l.</var> ad <var>.b.c</var>. </s> + <s xml:space="preserve">Quare ex definitione ab Eu-<lb/>cli. poſita in .11, lib. pars coni ſuperior ſimilis erit cono totali.</s> + </p> + <p> + <s xml:space="preserve">Deinde ſciendum eſt illud quod Euclid. ſcribit in .10. duodecimi lib. hoc eſt, <reg norm="quod" type="simple">ꝙ</reg> <lb/>proportio duarum pyramidum inuicem <lb/>ſimilium, triplicata eſt ei diametrorum <lb/> + <anchor type="figure" xlink:label="fig-0370-01a" xlink:href="fig-0370-01"/> + ſuarum baſium, hoc eſt, 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> ſed ita eſt ipſius <var>.a.c.</var> ad <var>.a.i.</var> vt ipſius <var>.b.<lb/>c.</var> ad <var>.k.i.</var> ex .4. ſexti cum trianguli <var>.a.b.c.</var> <lb/>et <var>.a.k.i.</var> ſint æquianguli, quod ex ijs, quę <lb/>ſuperius diximus facile compręhenditur. <lb/></s> + <s xml:space="preserve">Quare <reg norm="proportio" type="simple">ꝓportio</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 abſciſſam <var>.a.k.i.</var> ſed eadem proportio <lb/>ipſius <var>.a.c.</var> ad <var>.a.i.</var> erat etiam tertia pars pro <lb/>portionis ipſius <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 abſciſſam, æqualis erit pro-<lb/>portioni ipſius <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, ſed ex rationibus, quas tibi ſcribo conſidera, quod quo <lb/><reg norm="tieſcunque" type="simple">tieſcunq;</reg> craſſities vel <reg norm="denſitas" type="context">dẽſitas</reg> <reg norm="vaporum" type="context">vaporũ</reg>, ſeu altitudo, maior eſſet ea, quę nunc re-<lb/>peritur, </s> + <s xml:space="preserve">tunc minor differentia eſſet inter maiorem <reg norm="minoremque" type="simple">minoremq́;</reg> calorem Solis, quam <lb/>nunc ſentiamus. </s> + <s xml:space="preserve">Pro cuius rei euidentia, imaginemur in hac ſubſcripta figura, li-<lb/>neam <var>.o.a.</var> pro ſemidiametro terræ, et <var>.a.c.</var> pro craſſitie vaporum, vt nunc ſe <lb/>habet, et <var>.a.d.</var> pro maiori craſſitie, imaginemurq́ue lineam <var>.a.b.</var> quaſi perpen-<lb/>dicularem ad <var>.o.a.</var> quæ abſciſſa ſit 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 ipſius <var>.a.b.</var> ad <var>.a.d.</var> quam ipſius <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> ſecabit <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 ipſi <var>.d.b.</var> ex. ſecunda par-<lb/>te ſecundæ ſexti, vnde ex prima parte <lb/>ciuſdem, ita eritipſius <var>.b.i.</var> ad <var>.i.a.</var> vt <var>.d.<lb/>c.</var> ad <var>.c.a.</var> & coniunctim ita erit ipſius <var>.b.<lb/>a.</var> ad <var>.a.i.</var> vt ipſius <var>.d.a.</var> ad <var>.a.c.</var> & permu <lb/>tatim ipſius <var>.a.b.</var> ad <var>.a.d.</var> erit, vt <var>.a.i.</var> <lb/>ad <var>.a.c.</var> ſed cum <var>.a.u.</var> maior ſit ipſa <var>.a.i.</var> <lb/>vt omne totum maius eſt ſua parte. <lb/></s> + <s xml:space="preserve">maior proportio erit ipſius <var>.a.u.</var> ad <var>.a.<lb/>c.</var> quam ipſius <var>.a.i.</var> ad <var>.a.c.</var> hoc eſt quam <lb/>ipſius <var>.a.b.</var> ad <var>.a.d</var>. </s> + <s xml:space="preserve">Verum igitur eſt <lb/>propoſitum.</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 reſpectu altitudinis ipſius.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVodà me poſtulas deinde, ita ſe habet. </s> + <s xml:space="preserve">Inquis enim, quod cum differentia <lb/>inter maiorem, <reg norm="minoremque" type="simple">minoremq́;</reg> calorem, oriatur etiam ex differentia maioris <lb/>quantitatis vaporum ad minorem, per quam quantitatem vaporum rranſit lumen <lb/>Solis (vt alias etiam tibi dixi) velles nunc ſcire quantitatem ipſius differentię, quæ <lb/>inter duas Solis datas altitudines ſupra 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, ſupponatur etiam quod angulus <var>.z.o.d.</var> vel <var>.z.a.b.</var> qui ſunt inuicem fe-<lb/>rè æquales, ſit angulus diſtantiæ Solis à zenit, <var>z.a.</var> verò ſit ſpiſſitudo vaporum, et <var>.a.<lb/>b.</var> radius tranſiens 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 eſt 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, velipſo minore, vnde cogno <lb/>ſcemus differentiam duarum <var>.a.b.</var> quæ qui-<lb/>dem inæquales inuicem erunt, eo quod ſup <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 eſt quam <lb/>Alhazem docetinuenire, cognoſcimus <reg norm="etiam" type="context">etiã</reg> <lb/><var>o.a.</var> vt ſemidiametrum 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">erũt</reg> <lb/>ſimul cum angulo <var>.o.a.b.</var> reſiduo duorum re <lb/>ctorum, eo quod reliquus <var>.z.a.b.</var> datus eſt. <lb/></s> + <s xml:space="preserve">Quare <var>.a.b.</var> cognita erit reſpectu <var>.o.a.</var> et <var>.o.<lb/>b.</var> et <var>.a.z.</var> quæ eſt eorum differentia. </s> + <s xml:space="preserve">Nunc <lb/>ſi idem faciemus cum alia <var>.a.b.</var> ſub diuerſo <lb/>angulo, habebimus propoſitum.</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, baſibus oppoſiti, ijſdem baſibus propor <lb/>tionales eſſent, cuius opinionis cauſa fuit quod nunquam viderit vel me <lb/>minerit eius quod Ptolomeus ſcripſit lib. primo Almageſti, vbi de diſpro <lb/>portionalitate chordarum <reg norm="arcuumque" type="simple">arcuumq́;</reg> tractat, vel quod ſcribit Vitellio lib. primo pro <lb/>poſitione .35. ſeu lib. quarto, propoſitione .21. quod idem eſt. </s> + <s xml:space="preserve">Sed nec ego tibi pro <lb/>ponam id quod ſcribit Nicolaus Tartalea diuiſioni .28. quinti capitis quartæ partis <lb/>ſuorum tractatuum, eo quod non exactè ſcientificè ſcripſerit, nec vniuerſaliter, <reg norm="quan- uis" type="context">quã-<lb/>uis</reg> talis propoſitio poſſit ſcientificè ſcribi, 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 poſtea <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 vniuerſaliori modo <lb/>poſſumus hoc ſpeculari. </s> + <s xml:space="preserve">Nam manifeſtè ſcimus, 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> cuiuſuis amplitudinis, quorum latera <var>.e.a</var>: <var>e.b</var>: <lb/>et <var>.e.c.</var> ſint 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">ꝙ</reg> <lb/>faciat angulum <var>.g.b.c.</var> æqualem angulo <var>.e.b.a.</var> protracta poſtea <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 eſt angulo <var>e.b.a.</var> ex quinta eiuſdem, ideo ex .32. dicti, et .4. ſexti, horum <lb/>duorum triangulorum latera, erunt inuicem proportionalia. </s> + <s xml:space="preserve">Conſtituto deinde <var>.g.</var> <lb/>centro, & ſecundum ſemidiametrum <var>.g.b.</var> vel <var>.g.c.</var> quod idem eſt, deſcripto circu-<lb/>lo <var>.b.i.c.</var> necnon circulo <var>.b.c.a.</var> circa centrum <var>.e.</var> ope ſemidiametri <var>.e.b.</var> et <var>.e.a.</var> vn <lb/>de iſte circulus eritillo maior, cum <var>.e.b.</var> maior ſit <var>.g.b.</var> ex .14. quinti. cum ex .14. tertij <lb/><var>a.b.</var> longior ſit <var>.c.b.</var> ſed ex vltima definitione tertij, arcus <var>.b.i.c.</var> et <var>.b.c.a.</var> erunt in-<lb/>uicem ſimiles, hoc eſt 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> ſed proportio diame-<lb/>tri ad circunferentiam eſt vt diametri ad cir <lb/>cunferentiam, vt ſupra diximus; </s> + <s xml:space="preserve">Quare ex <lb/>proportionum æqualitate, vt ſemidiametri <lb/>ad circunferentiam erit, vt ſemidiametri <lb/>ad circunferentiam, & per eandem propor <lb/>tionum ęqualitatem, proportio <var>.e.b.</var> ad <reg norm="arcum" type="context">arcũ</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> ſed <lb/>arcus <var>.b.i.c.</var> maior eſt 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 ſcicntia. </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 ſexti et .12. quinti, proportio anguli <var>.a.e.b.</var> ad an-<lb/>gulum <var>.c.e.b.</var> maior erit quam chordæ, ſiue baſis <var>.a.b.</var> ad chordam ſiue baſim <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 Franciſco Venerio.</head> + <p> + <s xml:space="preserve">EGo enim non tantum miror ea quæ mihi ſcripſiſti de opinione Ortenſij <lb/>quantum quod Antonius Berga putat nubes à Sole ſupenſas teneri, id pla <lb/>nè falſum eſt, vera cauſa huiuſmodi effectus, alia nulla eſt, niſi earundem <lb/>raritas hoc eſt, cum rariores ſint ipſo aere ſubiecto, </s> + <s xml:space="preserve">propterea ſupra <reg norm="ipsum" type="context">ipsũ</reg> <lb/>natant & ſtant ſub eo qui rarior ipſis eſt, eo quod corpora rariora poſita in medio <lb/>non tam raro, aſcendunt, & denſiora in medio minus denſo deſcendunt. </s> + <s xml:space="preserve">Nam ſi <lb/>Sol ipſas nubes ſuſpenſas in aere teneret, hoc interdiu tantummodo fieret, ſed no <lb/>ctu, cur non deſcendunt vſque ad terram, & in eodem loco ſemper manent? </s> + <s xml:space="preserve">Scien-<lb/>dum igitur eſt nubes aſcendere in altum quouſque inueniant aerem eiuſdem ra-<lb/>ritatis cuius ipſæ ſunt. </s> + <s xml:space="preserve">Raritas enim & denſitas non ſunt res viſibiles niſi per acci-<lb/>dens, quemadmodum etiam leuitas, & grauitas, opacitas verò & diaphaneitas ma <lb/>gis <reg norm="compræhenduntur" type="context">compræhendũtur</reg>, opacitas enim ex reflexione radiorum luminoſorum, diapha <lb/>neitas verò compræhenditur ex penetratione ipſorum radiorum, opacitas autem nu <lb/>bis non eſt denſitas, cum valde diuerſa ſit denſitas ab opacitate, ſicut raritas ab dia-<lb/>phaneitate, vt aliàs dixi. </s> + <s xml:space="preserve">Et quando dicit, quod Sol calefaciendo aerem ipſam nu <lb/>bem ambientem, rarefaciat eum magis quam ipſam nubem reſpondeo, hoc verum <lb/>non eſſe, proptere<unsure/>a quodradius Solis non multum calefacit ea corpora, quæ ip ſi per <lb/>mittunt liberum tranſitum. </s> + <s xml:space="preserve">vnde corpora quanto magis diaphana ſunt tanto minus <lb/>ab ipſo radio luminoſo calefiunt, ſed ea quæ magis opaca ſunt, magis etiam calefiunt <lb/>& per conſequens magis rarefiunt, cum calidi ſit per ſe rarefacere, & non attrahere, <lb/>vt ipſe & 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/>cuiuſdam libramenti, & de quadam ſimboleita-<lb/>te circuli cum ellipſi.</head> + <head style="it" xml:space="preserve">Angelo Ferrario Serenißimi Ducis Sabaudia <lb/>Agrimenſori expertißimo.</head> + <p> + <s xml:space="preserve"><emph style="sc">TIbi</emph> in mentem veniet, quod cum ſuperioribus diebus in villa lucenti, in <lb/>qua degebat Sereniſſimus Dux noſter, dum viridarium ad æquilibrium <lb/>reducebas, eſſemus, à te quæſiui an ſcires vnde fieret, vt ſtante libramen-<lb/>to ad angulos rectos ſupra ſuum pedem, funis quæ extrema eiuſdem li-<lb/>bramenti cum pede in formam trianguli æquicruris coniungit, magis diſtentus exi-<lb/>ſteret, quam cum dictum libramentum cum pede obliquum remanet, ita vt huiuſ- + <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 ſcalenum conſtitueret.</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 iſocellum, & aliquan <lb/>do ſcalenum, eſſe <var>.d.e.c.</var> eſto etiam quod in figura <var>.A.</var> dictus triangulus <var>.d.e.c.</var> ſit iſo-<lb/>cellus, & in figura <var>.B.</var> ſcalenus. </s> + <s xml:space="preserve">Tunc quæſiui à te an ſcires rationem, quare <lb/>funis <var>.d.e.c.</var> in figura <var>.A.</var> eſſet diſtenſus, & in figura <var>.B.</var> laxus quemadmodum vide-<lb/>bamus. </s> + <s xml:space="preserve">cum mihireſponderis, neſcio quid, quod nunc memoria <reg norm="non" type="context">nõ</reg> teneo, ſed quia <lb/>pollicitus ſum metibi eam afferre, propterea nunc ad te mitto. </s> + <s xml:space="preserve">Scias ergo huiuſ-<lb/>modirationem nihil aliud eſſe niſi quod in figura <var>.A.</var> duæ lineæ <var>.c.e.</var> et <var>.d.e.</var> ſimul è <lb/>directo iunctæ longiores ſint illis, quę reperiuntur in figura <var>.B.</var> ſed quia funis tam in <lb/>figura <var>.B.</var> quam in figura <var>.A.</var> vnus, & idem eſt, ideo in figura <var>.B.</var> laxatus eſt, & non in <lb/>tenſus, ut in figura <var>.A</var>. </s> + <s xml:space="preserve">Sed vt huiuſmodi veritatis certam notitiam habeas, infraſcri <lb/>ptum circulum mente concipe <var>.f.e.i.</var> cuius ſemidiameter, æqualis ſit <var>.b.e.</var> & diame-<lb/>ter ſit <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> figurę <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">alicuiꝰ</reg> ellipſis <lb/>cuius quidem figuræ ponamus <var>.d.</var> et <var>.c.</var> <lb/>centra ipſius circunſcriptionis eſſe, cu <lb/>ius <reg norm="circunferentia" type="context">circunferẽtia</reg>, nullidubium eſt, quin <lb/>extra propoſitum circulum tranſitura, <lb/>& in vno tantummodo puncto ipſum <lb/>circulum tactura ſit, qui exiſtat <var>.e.</var> <lb/> figuræ <var>.A.</var> ſeparatum tamen à puncto <lb/>e. figuræ <var>.B</var>. </s> + <s xml:space="preserve">Tunc ſi protracta fue-<lb/>rit linea <var>.d.e.</var> figuræ <var>.B.</var> vſque 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 ſit linea <lb/><var>g.c</var>. </s> + <s xml:space="preserve">tunc <reg norm="manifeſtum" type="context">manifeſtũ</reg> erit duas lineas <lb/><var>d.e.</var> et <var>.e.c.</var> figuræ <var>.A.</var> ſimul iunctas, <lb/>æquales eſſe duabus <var>.d.g.</var> et <var>.g.c.</var> ſi-<lb/>mul poſitis, vt etiam ex .52. tertij <lb/>Pergei facilè videre eſt, ſed ex .21. <lb/>primi Euclid. iam certò ſcimus <var>.d.g.c.</var> longiores eſſe <var>.d.e.c.</var> ſiguræ <var>.B.</var> ergo <var>.d.e.c.</var> figu-<lb/>ræ <var>.A.</var> longiores ſunt <var>.d.e.c.</var> figuræ <var>.B.</var> quod eſt propoſitum.</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 ſuccurrit, tibi libenter ſignifico, hoc eſt, quod <lb/>ſicut in ellipſi duæ lineæ <var>.d.e.e.c.</var> figuræ <var>.A.</var> ſimul iunctæ, ſunt ſemper 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 ſunt in <lb/>potentia duabus <var>.d.e.e.c.</var> figurę <var>.B</var>.</s> + </p> + <p> + <s xml:space="preserve">Manifeſtum enim primum eſt ex penultima primi in figura <var>.A.</var> quadratum <var>.e.c.</var> <lb/>æquale eſſe duobus quadratis ſcilicet <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 ſunt quadratis <var>.e.b.</var> figuræ <var>.A.</var> et <var>.e.<lb/>b.</var> figurę. B et <var>.b.c.</var> et <var>.b.d.</var> hoc eſt 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> figurę <var>.B.</var> æqualia ſunt 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 eſt <reg norm="vtrum" type="context">vtrũ</reg> <reg norm="duplum" type="context">duplũ</reg> quadrati <var>.a.e.</var> <reg norm="cum" type="context">cũ</reg> duplo qua <lb/>drati <var>.b.a.</var> <reg norm="cum" type="context">cũ</reg> duplo quadrati <var>.b.c.</var> ſit æquale duplo quadrati <var>.a.e.</var> <reg norm="cum" type="context">cũ</reg> quadrato <var>.a.d.</var> & <lb/>cum quadrato <var>.a.c</var>. </s> + <s xml:space="preserve">Sed quia tam ex vna parte quàm 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> ſimul cum duplo qua-<lb/>drati <var>.b.c.</var> ęquale ſit quadrato <var>.a.c.</var> cum quadrato <var>.a.d.</var> ſed hoc manifeſtum eſt .ex .10. <lb/>ſecundi Euclidis, dato quod <reg norm="punctum" type="context">punctũ</reg> <var>.a.</var> ſit inter <var>.f.</var> et <var>.d.</var> ſed ſi fuerit inter <var>.d.</var> et <var>.b.</var> hoc <lb/>manifeſtum erit ex .9. ſecundi dicti, nihilominus accipe hunc alium modum.</s> + </p> + <p> + <s xml:space="preserve">Sit hic ſubſcriptum quadratum <var>.D.</var> ex <var>.a.c.</var> in ſeipſa producta, cuius diameter ſit <lb/><var>a.n.</var> <reg norm="protrahanturque" type="simple">protrahanturq́</reg> parallelę <var>.d.h</var>: <var>b.K</var>: <var>l.m.o.</var> et <var>.r.q.s.</var> <reg norm="eique" type="simple">eiq́;</reg> addatur <var>.c.p.</var> ad <var>.a.c.</var> æqua-<lb/>lis tamen <var>.d.a.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> protracta <var>.p.u.</var> vſque 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 eſt, <reg norm="vtrum" type="context">vtrũ</reg> <lb/>hęc duo quadrata æqualia ſint 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> ſint <var>.K.o.</var> et <var>.h.l.</var> videndum nunc eſt utrum reſiduum ęquale <lb/>ſit duobus quadratis lineę <var>.a.b.</var> quorum vnum ſit <var>.m.b.</var> alterum verò <var>.l.p.</var> quod ſupe-<lb/>rat <var>.l.c.</var> et <var>.s.p.</var> figuræ <var>.D.</var> per ſupplementum <var>.o.t.</var> cui æquale eſt parallelogrammum <var>.h.<lb/>m.</var> figuræ <var>.D.</var> ſed ſi punctus <var>.a.</var> poſitus fuerit inter <var>.d.</var> et <var>.b.</var> conſtituto quadrato <var>.d.u.</var> <reg norm="cum" type="context">cũ</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> ęquari duplo quadratorum <var>.l.n.</var> et <var>.r.l.</var> nam in quadrato <var>.r.n.</var> ipſa duo quadra-<lb/>ta <var>.l.n.</var> et <var>.r.l.</var> capiuntur, reliquum eſt igitur vt videamus an duo ſupplementa <var>.l.t.</var> et <var>.l.<lb/>s.</var> cum quadrato <var>.d.r.</var> ſint æqualia dictis <reg norm="quadratis" type="typo">q́uadratis</reg> <var>.l.n.</var> et <var>.r.l.</var> ſed quadratum <var>.d.l.</var> æ qua-<lb/>tur quadrato <var>.l.n.</var> videndum igitur eſt, <lb/>an duo ſupplementa <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> ſint æqualia duobus quadra <lb/>tis <var>.d.l.</var> et <var>.r.l.</var> ſed quadratum <var>.d.l.</var> æqua-<lb/>tur quadrato <var>.d.r.</var> & ſupplemento <var>.l.t.</var> <lb/>mediante <var>.q.l.</var> & ſupplemento <var>.r.b.</var> ſup-<lb/>plementum verò <var>.l.s.</var> ſuperat <reg norm="ſupplemem" type="context">ſupplemẽ</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 ſupplementa <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">cũ</reg> quadrato <var>.l.r.</var> verum igitur eſt duas <var>.d.e.e.c.</var> figuræ <var>.A.</var> æquales eſſe in <lb/>potentia duabus <var>d.e.e.c.</var> figurę <var>.D.</var> quæ quidem affectio circuli, à nemine fuit adhuc <lb/>(quod ſciam) 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 ſtateram appenſi, & quadam alia demonſtratione, <lb/>& quibuſdam erroribus Tartaleæ.</head> + <head style="it" xml:space="preserve">Mutio Groto.</head> + <p> + <s xml:space="preserve">SI ea quæ à me audiuiſti non credis, conſidera quæſo libram ſeu ſtateram <lb/><var>o.a.</var> cuius centrum non longitudinis ſed ponderum ſit <var>.i.</var> quę ſtatera, vt ori <lb/>zontaliter conſiſtat, oportebit pondus extremitatis <var>.o.</var> ita ſe habere <lb/>ad pondus extremitatis <var>.a.</var> ut <var>.a.i.</var> ſe habet ad <var>.o.i.</var> quod te ſcire puto, ima <lb/>ginemur nunc d uas lineas <var>.a.e.</var> et <var>.o.n.</var> paralle las <reg norm="infinitasque" type="simple">infinitasq́;</reg> & à puncto <var>.n.</var> immobili, <lb/>& fixo extra ſtateram, tranſeat per <var>.i.</var> linea <var>.n.i.e</var>. </s> + <s xml:space="preserve">Cogitemus etiam punctum <var>.e.</var> inter <lb/>ſectionis ipſius <var>.n.i.e.</var> cum <var>.a.e.</var> progredi vniformiter <reg norm="continuòque" type="simple">continuòq́;</reg> ab <var>.a.</var> per lineam <var>.a.e.</var> <lb/>vnde punctum <var>.i.</var> interſectionis ipſius <var>.n.i.e.</var> cum <var>.a.i.o.</var> ſemper 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 ſtateram <var>.o.i.a.</var> oporteat ſemper orizontalem eſſe virtute ponderis, <lb/>o. oportebit pundus <var>.o.</var> in infinitum etiam augeri, <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg> pondus <var>.a.</var> nunquam <lb/>diminui voluerimus vel econtra hoc in infinitum diminui, ſi illud nunquam augeri <lb/>voluerimus.</s> + </p> + <p> + <s xml:space="preserve">Sedre vera non putabam te indigere aliqua demonſtratione, quod linea <var>.b.h.</var> di-<lb/>uiſa ſit per æqualia à<unsure/> linea <var>.c.a.</var> cum hæc perpendicularis ſit ab <var>.a.</var> ad baſim <var>.g.d.</var> in <reg norm="triam" type="context">triã</reg> <lb/>gulo orthogonio <var>.g.a.d.</var> & cum ſit <var>.b.h.</var> perpendicularis ad <var>.a.o.</var> ex ſuppoſito quæ <var>.a.<lb/>o.</var> in ſe habet punctum medium baſis <var>.g.d.</var> nec <reg norm="non" type="context">nõ</reg> illud anguli recti <var>.a.</var> quod per ſe cla <lb/>riſſimum eſt, cum iam ſcis <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 ipſi <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. eiuſdem, angulus <var>.h.</var> æqua <lb/>bitur angulo <var>.d.</var> eo quod an gulus <var>.e.</var> rectus eſt, quemadmodum et <var>.a.</var> ſed angulus <var>.d.</var> <lb/>æqualis eſt 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 ipſi <var>.u.<lb/>a.</var> ex .6. primi, cum poſtea angulus <var>.<lb/> + <anchor type="figure" xlink:label="fig-0376-01a" xlink:href="fig-0376-01"/> + o.a.d.</var> æqualis ſitangulo <var>.d.</var> ex quin<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 eſt, & ex eadem æqualis <lb/>erit angulo <var>.d.a.c.</var> vnde <var>.u.b.</var> erit <lb/>æqualis ipſi <var>.u.a.</var> ex .6. dicti, & ideo <lb/>æqualis eric ipſi <var>.u.h</var>. </s> + <s xml:space="preserve">Reliqua ve-<lb/>rò illius propoſitionis credo ex te <lb/>omnia poſſe <reg norm="intelligere" type="context">ĩtelligere</reg>, excepto, <reg norm="quod" type="simple">ꝙ</reg> <lb/>vt tibi ſignificaui ſi à <reg norm="puncto" type="context">pũcto</reg> <var>.i.</var> com-<lb/>muni ipſi <var>.a.c.u.</var> & circunferentiæ, <lb/>ducta fuerit <var>.i.x.</var> ad <reg norm="punctum" type="context">pũctum</reg> <var>.x.</var> com <lb/>mune vni parallelæ à <reg norm="puncto" type="context">pũcto</reg> <var>.g.</var> ipſi <lb/><var>h.b.</var> & circunferentiæ, quod di-<lb/>cta <var>.i.x.</var> ad rectos erit ipſi <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"/> + ſit angulo <var>.a.h.b.</var> propter æquidiſtantiam 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> ſed angulus <var>.i.a.d.</var> com-<lb/>munis eſt 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 eſt <var>.i.x.</var> per <lb/>pendicularis erit ipſi <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 ſcripſi circa finem illius epiſtolæ, Tartaleam erraſſe in quinta propo-<lb/>ſitione primi lib. ſuæ nouæ ſcientiæ, non ſine ratione illud ſcripſi. </s> + <s xml:space="preserve">Nam, inquit ipſe, <lb/>nullum corpus æquè graue poteſt in aliquo temporis ſpatio moueri motu naturali, <lb/><reg norm="violentoque" type="simple">violentoq́;</reg> ſimul miſtis. </s> + <s xml:space="preserve">Vbi decipitur, eo quod non animaduertit incrementum ve <lb/>locitatis vnius motus, ſimul eſſe cum decremento velocitatis alterius, <reg norm="eodemque" type="simple">eodemq́;</reg> tem <lb/>pore, vt manifeſtè patet in itinere corporis, ab ipſo pro exemplo aſſumpto, hoc eſt <lb/>quod velocitas motus in ſpatio <var>.c.d.</var> creſcit vt naturalis, & decreſcit vt violenta. </s> + <s xml:space="preserve"><reg norm="nam" type="context">nã</reg> <lb/>creſcit orizontem verſus & decreſcit in remotione à linea <var>.a.b.</var> ſed ſi à puncto <var>.c.</var> ad <lb/>punctum <var>.d.</var> motus eſſet purè violentus, vt putat Tartalea, corpus illud minimè de-<lb/>ſcenderet, eo quod uirtus mouens, in <var>.a.</var> poſita, nullo pacto poteſt talem effectum ef-<lb/>ficere, vnde ab ipſa natura prouenit deſcenſio 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 ſcilicet, & non ex violentia aliqua. </s> + <s xml:space="preserve">Sed ſi <lb/>dixiſſet ipſe, illum motum eſſe purum naturalem, hoc eſſet falſum, eo quod purus <lb/>naturalis motus alicuius corporis non impediti, extra locum ſuum, ſit per lineam re <lb/>ctam, & non per curuam, vt videre eſt inter <var>.c.</var> et <var>.d</var>.</s> + </p> + <p> + <s xml:space="preserve">In vltima propoſitione deinde eiuſdem lib. quæ .6. eſt decipitur ſimiliter, & hæc <lb/>deceptio oritur ab ignoratione quintæ, & à putando motum naturalem non eſſe cau <lb/>ſam ipſius deſcenſus per ſpatium <var>.c.d</var>. </s> + <s xml:space="preserve">Sed quia tibi ſignificaui expeditiorem viam <lb/>repeririad <reg norm="cognoſcendam" type="context">cognoſcendã</reg> proportionem inter <var>.a.h.</var> et <var>.a.e.</var> in vltima propoſitione ſe-<lb/>cundi lib. ipſius Tartaleæ, ipſam nunc tibi ſcribo. </s> + <s xml:space="preserve"><reg norm="Nam" type="context">Nã</reg> iam ſcis angulum <var>.h.l.i.</var> diui-<lb/>ſum eſſe per æqualia ab <var>.P.l.</var> & quod <var>.a.h.</var> et <var>.h.p.</var> ęquales inuicem ſunt ex .6. primi Eu-<lb/>cli. </s> + <s xml:space="preserve">vnde <var>.p.i.</var> et <var>.a.h.</var> æquales erunt inuicem ſimiliter, ſed ex .3. ſexti ita eſt ipſius <var>.a.l.</var> <lb/>ad <var>.l.i.</var> vt ipſius <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> ſed <var>.a.l.</var> cogni <lb/>ta eſt ex eius quadrato, et <var>.l.i.</var> etiam, cum æqualis ſit ipſi <var>.a.i.</var> vnde ex regula de tribus <lb/>notam habebimus <var>.p.i.</var> reſpectu <var>.a.i.</var> & ita reſpectu <var>.a.e.</var> ſi hypotheſes ipſius Tartaleæ <lb/>veræ ſunt.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Alia demonstratio impoßibilitatis diuidendi per æqualia <lb/>proportionem ſuperparticularem in <lb/>diſcretis.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod à me poſtulas, hoc eſt ſcientiam impoſſibilitatis diuidendi per æqualia <lb/>proportionem ſuperparticularem in numeris ſatis à Campano in .8. octaui <lb/>potes habere, Iacobus Faber Stapulenſis etiam idem tractat<unsure/> in libello ſuę muſicæ <lb/>demonſtratæ. </s> + <s xml:space="preserve">Sed ſi etiam alia via idem deſideras, quamuis longiori, nih<unsure/>ilomi-<lb/>nus vniuerſaliori, conſidera duos numeros <var>.g.</var> et <var>.h.</var> inuicem relatos ſecundum pro-<lb/>portionem ſuperparcicularem, quam volueris. </s> + <s xml:space="preserve">Tunc dico impoſſibile eſle, vt per <lb/>æqualia diuidatur, quod ſi dixeris poſſibile eſſe, ſit 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 diuiſibilis eſſet ſi <var>.g.h.</var> minimi fuiſſent, quod non conceditur, ſint igitur mini <lb/>mi in dicta proportione <var>.a.</var> et <var>.b.</var> quorum differentia erit vnitas, vt ſcis, <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.c.</var> quadra <lb/>tum ipſius <var>.g.</var> et <var>.d.</var> quadratum ipſius <var>.K</var>. </s> + <s xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-<lb/>tio ipſius c. ad <var>.d.</var> eadem erit quæ <var>.g.</var> ad <var>.h.</var> hoc eſt vt ipſius <var>.a.</var> ad <var>.b.</var> vnde ſi vnus termi. <lb/>norum <var>.a.</var> vel <var>.b.</var> eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex <lb/>16. eiuſdem, inter <var>.a.</var> et <var>.b.</var> reperiretur aliquis medius numerus proportionalis, quod <lb/>fieri non poteſt ex hypotheſi, cum inter <var>.a.</var> et <var>.b.</var> nullus ſit numerus, quia differunt in <lb/>ter ſe per vnitatem tantummodo. </s> + <s xml:space="preserve">Nunc autem cum nullus numerorum <var>.a.</var> vel <var>.b.</var> qua <lb/>dratus ſit, ponatur quod <var>.f.</var> quadratus ſit ipſius <var>.b.</var> et <var>.e.</var> ſit productum ipſius <var>.a.</var> in <var>.b.</var> vn <lb/>de ex .18. ſeptimi, proportio ipſius <var>.e.</var> ad <var>.f.</var> erit vt. ipſius <var>.a.</var> ad <var>.b.</var> hoc eſt vt ipſius <var>.c.</var> ad <lb/>d. quapropter <var>.e.</var> erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet <reg norm="medium" type="context">mediũ</reg> <lb/>proportionale inter <var>.a.</var> et <var>.b.</var> ex .20. ſeptimi, quod eſt impoſſibile, vt iam dixi, cum <var>.a.</var> <lb/>et <var>.b.</var> ſint inui cem conſequentes, vnus poſt alium immediatè.</s> + </p> + <p> + <s xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerſalem, <lb/>hoc eſt quod hac methodo poſſumus in cognitionem <lb/>vcnire, quod non ſolum in duas æquales partes diui-<lb/> + <anchor type="figure" xlink:label="fig-0378-01a" xlink:href="fig-0378-01"/> + di non poſſit, ſed 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/>ipſo cognoſces ope <reg norm="cuborum" type="context">cuborũ</reg> vice <reg norm="quadratorum" type="context">quadratorũ</reg>, opevero <lb/><reg norm="cenſuum" type="context">cenſuũ</reg> <reg norm="cenſuum" type="context context">cẽſuũ</reg>, ve<unsure/>l qui cognouerit eam <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe indiuiſibilem per æqualia, illicò etiam cognoſcet <lb/>indiuiſibilem eſſe per quatuor partes, ope verò pri-<lb/>morum relatorum, cognoſcet non eſſe diuiſibilem per <lb/><reg norm="quinque" type="simple">quinq;</reg> partes, & ſic de cęteris, ſed mediantibus ijs <lb/>quas ſcripſi de iſtis dignitatibus in libro <reg norm="Thęorematum" type="context">Thęorematũ</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 Illuſtriſſimus Daniel Barbarus ſcri <lb/>bit in quinta parte ſuæ perſpectiuæ, ſi ſupra aliquo im <lb/>mobili, atque magno pariete facere volueris, te opor <lb/>tebit hoc ex reflexione radij ſolaris 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 circunſcribentis triangulum.</head> + <head style="it" xml:space="preserve">Francbino Triuultio.</head> + <p> + <s xml:space="preserve"><emph style="sc">QVod</emph> mihi nunc proponis eſt triangulum, cuius baſis cum angulo ſibi op <lb/>poſito dantur. </s> + <s xml:space="preserve"><reg norm="Vellesque" type="simple">Vellesq́;</reg> diametrum circuli apti eum triangulum circnn-<lb/>ſcribere inuenire in diſcreto.</s> + </p> + <p> + <s xml:space="preserve">Sit igitur triangulum <var>.a.b.g.</var> cuius baſis <var>.b.g.</var> ſimul cum angulo <var>.a.</var> ei op-<lb/>poſito data ſit in numeris. </s> + <s xml:space="preserve">Imaginetur ergo circulas circunſeribens ipſum triangu-<lb/>lum <var>.b.p.g.q.</var> cuius diameter ſit <var>.q.p.</var> perpendicularis eius baſi <var>.b.g.</var> vnde <var>.b.g.</var> diuiſa <lb/>erit per æqualia ab ipſo 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 ſit angulo <lb/><var>q.p.g.</var> medietati anguli ipſius portionis ex .19. tertij, ita quod angulus <var>.q.e.g.</var> nobis <lb/>cognitus erit, & ſimiliter arcus <var>.g.q.</var> & conſequenter ar-<lb/>cus <var>.p.g.</var> reſiduum medij circuli, & ſic <var>.m.g.</var> eius ſinus re <lb/> + <anchor type="figure" xlink:label="fig-0379-01a" xlink:href="fig-0379-01"/> + ctus, & etiam chorda <var>.p.g.</var> vt dupla ſinus dimidij arcus <var>.<lb/>p.g.</var> & ſic <var>.p.m.</var> eius ſinus verſus, vel vt tertium latus trian <lb/>guli orthogonij <var>.p.g.m.</var> vnde nobis cognita erit propor <lb/>tio ipſius <var>.b.g.</var> (quæ dupla eſt ipſi <var>.m.g.</var>) ad <var>.m.p.</var> & quia <lb/>productum <var>.p.m.</var> in <var>.m.q.</var> æquale eſt 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 ipſius <var>.b.g.</var> ad <var>.q.p.</var> qua <lb/>mediante cognoſcemus diametrum ſecundum partes il <lb/>las quibus propoſita ſuerit <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 ſed 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">QVotieſcunque etiam inuenire voluerimus triangulum aliquem, puta <var>.n.q.o.</var> <lb/>æqualem triangulo <var>.t.</var> (exempli gratia) propoſito, qui habeat angulum <var>.n.</var> æ-<lb/>qualem angalo <var>.a.</var> dato, latera vero continentia ipſum angulum <var>.n.</var> ſint 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, ſupra quam conſtituemus triangulum <var>.m.n.p.</var> æqualem triangulo <var>.t.</var> hac metho-<lb/>do, hoc eſt prolungando latus <var>.r.z.</var> trianguli <var>.t.</var> quod ſit <var>.r.e.</var> ita vt duplum ſit ipſi <var>.r.z.</var> <lb/>ducendo poſtea <var>.c.e.</var> habebimus ex .38. primi triangulum <var>.t.</var> eſſe dimidium totius <lb/>trianguli <var>.r.c.e.</var> deſignabimus deinde ex .44. dicti ſuperficiem <var>.p.n.m.b.</var> parallelo <lb/>grammam <reg norm="æqualemque" type="simple">æqualemq́;</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/>poſtea <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 poſtea <var>.n.p.</var> ita vt <var>.n.K.</var> <lb/>ſe 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 ſi oportuerit, <lb/></s> + <s xml:space="preserve">deinde eas ad æqualitatem ſe-<lb/>can<unsure/>do ipſis <var>.x.</var> et <var>.y.</var> efficiendo <lb/>exempli gratia quod <var>.n.i.</var> ſit <lb/>æqualis ipſi <var>.x.</var> et <var>.n.u.</var> ipſi <var>.y.</var> du <lb/>cendo poſtea <var>.u.i.</var> deinde à puncto <var>.m.</var> ducendo <var>.m.K.</var> æquidiſtanter <var>.u.i.</var> ex .31. <lb/>primi. </s> + <s xml:space="preserve">& ſic habebimus ex .4. ſexti 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 poſtea ex .9. eiuſ-<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 ſit <var>.n.o.</var> duces poſtea <lb/><var>o.q.</var> parallelam ipſi <var>.m.K.</var> & ha <lb/>bebis propoſitum, eo quod <reg norm="cum" type="context">cũ</reg> <lb/>ſit 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 ſexti, 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. eiuſdem <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. ſexti ſit 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 ſit quadrato alterius.</s> + </p> + <p> + <s xml:space="preserve">Vt exempli gratia ſi fuerint duæ lineæ <var>.e.d.</var> et <var>.e.f.</var> opor-<lb/><reg norm="teretque" type="simple">teretq́;</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">iũ<lb/> + <anchor type="figure" xlink:label="fig-0380-02a" xlink:href="fig-0380-02"/> + gere</reg>, ita longam, vt productum totius compoſiti <var>.e.c.</var> vel <var>.<lb/>f.b.</var> in <var>.f.c.</var> vel <var>.e.b.</var> eſſet æquale quadrato ipſius <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="quotieſcun- que" type="context">quotieſcũ-<lb/>que</reg> <var>.e.d.</var> coniuncta erit cum <var>.e.f.</var> ad rectos, <reg norm="diuiſaque" type="simple">diuiſaq́;</reg> per me <lb/>dium à puncto <var>.a.</var> à quo ducta <var>.a.d.</var> deinde ſecundum ſemi-<lb/>diametrum <var>.a.d.</var> deſignato circulo <var>.b.d.c.</var> & protracta <var>.e.f.</var> <lb/>à qua volueris parte vſque ad circunferentiam in <reg norm="puncto" type="context">pũcto</reg> <var>.c.</var> <lb/>ſeu in puncto <var>.b.</var> habebimus intentum, eò quod ſi produ-<lb/>cta fuerit <var>.e.f.</var> etiam ab alia parte, vſque ad circunferentiam, habebimus <var>.b.e.</var> æqua-<lb/>lem ipſi <var>.f.c.</var> ex communi conceptu, & productum <var>.e.c.</var> in <var>.e.b.</var> æqualem quadra-<lb/>to ipſius <var>.e.d.</var> ex .34. tertij, cum ex .3. eiuſdem <var>.e.d.</var> medietas ſit chordæ arcus dupli <lb/><var>b.d</var>.</s> + </p> + <p> + <s xml:space="preserve">De lapſu verò lapidis verſus mundi centrum, dum ipſum attingere, ac præterire <lb/>poſſet, de quo me interrogas. </s> + <s xml:space="preserve">Dico Nicolaum Tartaleam, nec non Franciſcum <lb/>Maurolicum rectè ſenſiſſe, malè verò Alexandrum Piccolhomineum, & exemplum <lb/>Maurolici optimum eſſe, quod tamen ſi capere non potes, crede ſaltem authoritati <lb/>bus talium virorum, qui tantum in ijs ſcientijs ſuperant ipſum Alexandrum Piccol-<lb/>homineum, quantum à Sole cætera ſuperantur aſtra.</s> + </p> + <p> + <s xml:space="preserve">Lapis igitur ille tranſiret centrum, <reg norm="reddiretque" type="simple">reddiretq́;</reg>, cum diminutione tamen motus im <lb/>preſſi, eo fermè modo vt ſcribunt iudicioſiſſimi illi viri, donec poſt multas reddi-<lb/>tiones ſurſum, <reg norm="deorſumque" type="simple">deorſumq́;</reg> quieſceret 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 ſilum illum (exempli adducti ab illis doctiſſimis viris) cui pon <lb/>dus appenſum eſt, æqualem eſſe axi orizontis, hoc eſt eius extremitatem immobi-<lb/>lem eſſe in primo mobili, & in ipſo zenit tui orizontis, </s> + <s xml:space="preserve">tunc arcus motionis ipſius la <lb/>pidis per tantum interuallum, quantum eſt diameter terræ, inſenſibiliter differret à <lb/>linea recta, & cum lapis diſtans à centro mundi per ſemidiametrum terræ, iret re-<lb/><reg norm="diretque" type="simple">diretq́;</reg>, vt ſcis, ergo idem faceret ſi ſilum longius eſſet per dictum terræ ſemidia-<lb/>metrum, ita vt poſſetipſum centrum attingere, nam differentia illa ſemidiametri <lb/>terræ, ferè nulla eſt reſpectu ſemidiametri ipſius 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/>deſcriptus æquiangulus ſit.</head> + <head style="it" xml:space="preserve">Conrado Neubart.</head> + <p> + <s xml:space="preserve">SI non credis Pentagonum ab Alberto Durero ſuper datam lineam deſi-<lb/>gnatum, æquiangulum non eſſe. </s> + <s xml:space="preserve">Fingamus hic ſubiectam ſiguram ſimi-<lb/>lem ei quæ à Durero ponitur, in qua primò, ducta ſit 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 ſuppoſito, arcus <var>.a.b.</var> eſt <lb/>ſexta pars totius circunferentiæ, angulus vero <var>.b.o.d.</var> rectus eſt, eo quod <var>.b.o.q.</var> rectus <lb/>etiam ſit, </s> + <s xml:space="preserve">quare angulus <var>.d.o.a.</var> reſiduus ex recto erit graduum .60. talium, ut rectus <lb/>eſt .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 ſinus 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> eſt partium .51762. talium qualium <var>.a.d.</var> vel <var>.a.c.</var> ſemidiameter eſt .100000.</s> + </p> + <p> + <s xml:space="preserve">Nam ſinus dimidij arcus <var>.a.o.</var> (exi <lb/>ſtente <var>.a.o.</var> graduum .30.) eſt <reg norm="partium" type="context">partiũ</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 ſi <lb/>nus eſt <var>.a.e.</var> erit graduum .26. min .38 <lb/>qui quidem angulus, ſumptus cum an <lb/>gulo <var>.a.o.d.</var> erit gra .86. min .38. </s> + <s xml:space="preserve">Dem <lb/>pta denique hac ſumma ex duobus <lb/>rectis gra .180. reliquum erit gra .93. <lb/>min .22. ideſt 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 ſic, cum inuentus fuerit angu-<lb/>lus <var>.a.d.o.</var> gra .26. min .38. ſi 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> reſiduo ex re cto dempto angulo <var>.a.o.e.</var> grad .60. qui <var>.e.a.o.</var> eſt 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> cognoſcemus vt ſinus anguli <var>.d.o.n.</var> gra .45. nam <lb/>angulus ei contrapoſitus <var>.q.o.p.</var> eſt dimidium recti, </s> + <s xml:space="preserve">quare <var>.d.n.</var> erit partium .70710. <lb/>talium qualium <var>.d.o.</var> fuerit .100000. ſed <var>.d.o.</var> eſt partium .115270. qualium <var>.a.d.</var> eſt <num value="100000">.<lb/>100000.</num> nam <var>.e.d.</var> vt ſinus anguli <var>.e.a.d.</var> gra .63. min .22. eſt partium .89389. <var>o.e.</var> ve-<lb/>ro eſt partium .50000. talium qualium <var>.a.o.</var> eſt .100000. vt ſinus anguli <var>.e.a.o.</var> gra .30. <lb/>ſed vt <var>.a.o.</var> eſt <reg norm="partium" type="context">partiũ</reg> .51762. </s> + <s xml:space="preserve">hoc eſt vt <var>.a.d.</var> eſt .100000. ipſa <var>.o.e.</var> erit <reg norm="partium" type="context">partiũ</reg> .25881 <lb/>quæ iuncta cum fuerit cum <var>.e.d.</var> efficiet <var>.d.o.</var> partium .115270. vt dictum eſt, quapro-<lb/>pter cum <var>.d.n.</var> ſit partium .70710. talium qualium <var>.d.o.</var> fuerit .100000. ipſa <var>.d.n.</var> erit <lb/>partium .81507. talium qualium <var>.d.o.</var> erit .115270. ideſt qualium <var>.d.a.</var> vel <var>.d.u.</var> erit <lb/>100000. quæ quidem <var>.d.n.</var> eſt ſinus anguli <var>.d.u.n.</var> graduum ſcilicet .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 ſupra, cui applica angu-<lb/>lum <var>.o.d.n.</var> gra .45. min <var>.o.</var> ſimul cum <lb/>angulo <var>.u.d.n.</var> reſiduo ex recto gra <lb/>duum .35. minu .24. & conficies an-<lb/>gulum <var>.a.d.u.</var> grad .107. minu .2. & <lb/>habebis propoſitum, 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 ſuppu-<lb/>tata ſint, ex ſumma omnium angulo <lb/>rum patere poteſt. </s> + <s xml:space="preserve">nam collectis om <lb/>nibus quinque angulis <var>.a.c.d.f.u.</var> ſi-<lb/>mul, hoc eſt grad .108. minu .22. cum <lb/>gra .107. min .2. cum grad .12. effi-<lb/>cient grad .540. min <var>.o.</var> ſumma æqua <lb/>lis ſex 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/>ſecundi Euclidis.</head> + <head style="it" xml:space="preserve">Petro Catenæ.</head> + <p> + <s xml:space="preserve"><emph style="sc">QVamvis</emph> nona ac decima ſecundi Euclid. aliter à Comandino & Mau-<lb/>rolico demonſtratæ fuerint, nihilominus mihi etiam viſum eſt non nihil <lb/>meo moræ in eas tibi ſcribere, vt ſenſibiliter quoque cognoſcas il-<lb/>las veras eſſe.</s> + </p> + <p> + <s xml:space="preserve">Eſto linc<unsure/>a.a.b. pro nona propoſitione, diuiſa per æqualia in <var>.c.</var> per inæqualia verò <lb/>in <var>.d.</var> quadratum autem <var>.a.d.</var> ſit <var>.d.e.</var> quadratum verò <var>.d.b.</var> ſit <var>.d.i.</var> quadratum <var>.a.c.</var> <lb/>ſit <var>.c.f.</var> & quadratum <var>.c.d.</var> ſit <var>.c.K.</var> clarum enim erit <var>.K.h.</var> æqualem exiſtere ipſi <var>.a.c.</var> <lb/>ſeccetur igitur <var>.e.h.</var> in <var>.g.</var> ita vt <var>.h.g.</var> ęqualis exiſtat ipſi <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">duplũ</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 ſint etiam duo quadrata partis <var>.a.c</var>.</s> + </p> + <p> + <s xml:space="preserve">In decima <reg norm="autem" type="wordlist">aũt</reg> propoſitione, <reg norm="quadratum" type="context">quadratũ</reg> totalis lineæ <var>.a.d.</var> ſit <var>.d.e.</var> & lineæ <var>.b.d.</var> ſit <var>.b.i.</var> et <lb/><var>c.d.</var> ſit <var>.d.n.</var> et <var>.a.c.</var> ſit <var>.c.f.</var> et <var>.f.e.</var> ſit <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 Aſtrologo Peritißimo.</head> + <p> + <s xml:space="preserve"><emph style="sc">POstqvam</emph> tua doctiſſima ſcripta perlegi, conſideraui, quod ſi à mul-<lb/>titudine exhalationum in regione elementari acciderit anno .1572. & <lb/>1573. vt totos ſex menſes ab omnibus per vniuerſum terrarum orbem <lb/>viſa fuerit ſtella illa, quæ eſt in angulo ſeptentrionali 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 ipſo lucifero videretur rutilantior <reg norm="atque" type="simple">atq;</reg> cæterarum (abſque vlla <lb/>aſpectus diuerſitate) magis ſcintillans. </s> + <s xml:space="preserve">Quî fieri poterat, vt ſtellæ quæ ab illa pa-<lb/>rum diſtant, alioqui multo maiores, non etiam illa clariores apparuerint? </s> + <s xml:space="preserve">ſed ſi ali-<lb/>quis diceret eam exhalationem non ita fortaſſe dilatari, vt inter nos, & aliam ali-<lb/>quam ſtellam interponeretur. </s> + <s xml:space="preserve">Tunc ego reſponderem neceſſariò ſequi debere ta-<lb/>lem exhalationem, tantam latitudinem occupare, quod aliquibus populis aliam <lb/><reg norm="aliquam" type="context">aliquã</reg> <reg norm="ſtellam" type="context">ſtellã</reg> circunuicinãhac ipſa de qua mentionem ſecimus redderet lucidiorem. <lb/></s> + <s xml:space="preserve">Sed cum hoc perſpectum fuerit nulli, ſequebatur lucem illam ab ipſis exhalatio-<lb/>nibus elementaribus haud poſſe oriri: </s> + <s xml:space="preserve">quod nobis ſcintillatio illa maxima perma-<lb/>gno fuit inditio, ſi phas eſt credere, <reg norm="nam" type="context">nã</reg> quo magis aliquod coeleſte corpus ſcintillat, <lb/>eo longius à nobis diſtare.</s> + </p> + <p> + <s xml:space="preserve">Verum quoniam efflagitaſti à me vt aliquid circa huiuſce rei ſpeculationem tibi <lb/>ſcribam, idcirco tibi morem gerere volens paucis ſubiungam.</s> + </p> + <p> + <s xml:space="preserve">Conſidera primo hanc ſubſcriptam primam figuram, in qua <var>.c.a.e.</var> ſignatur pro <lb/>Globo terreſtri cuius <var>.i.</var> centrum ſit et <var>.u.o.n.</var> pro conuexo ignis, ſed <var>.K.x.s.</var> pro orbe <lb/>octauo <var>.x.</var> autem pro ſtella iam ſuperius dicta, quæ ſemper fuit, eſt, & erit, quamuis <lb/>cæteris tribus nunc obſcurior ſit. </s> + <s xml:space="preserve">Accipiantur deinde duo loca in ſuperſicie terræ, <lb/>quę ſint <var>.c.</var> et <var>.e.</var> diametraliter inuicem oppoſita, ita quod circa eorum orizontes poſ <lb/>ſibile ſit ſtellam <var>.x.</var> videre, radijs ipſius ſtellæ mediantibus <var>.x.n.e.</var> et <var>.x.u.c.</var> <reg norm="quorum" type="context">quorũ</reg> par-<lb/>tes <var>.n.c.</var> et <var>.u.e.</var> ita breues ſint, reſpectu eorum <reg norm="totorum" type="context">totorũ</reg>, vt vix ſexcenteſima pars ſit vna <lb/><reg norm="quæque" type="simple">quæq;</reg> illarum, nec non <var>.c.e.</var> ita breuis reſpectu ſemidiametri octauæ ſphæræ, quod <lb/>vix ſit vna ex partibus decemmillibus, vt ſcis, ſequitur quod recta terminata ab <var>.u.</var> <lb/>et <var>.n.</var> minor ſenſibiliter non ſit ipſo terræ diametro <var>.c.e.</var> cum duo hæc interualla ex <lb/>triangulorum ſimilitudine ſe habeant vt <var>.x.i.</var> ad <var>.x.o.</var> hoc eſt 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 ſit. </s> + <s xml:space="preserve">ductę poſtea <reg norm="cum" type="context">cũ</reg> fuerint duæ diagonales <var>.e.u.</var> et <var>.n.c.</var> <reg norm="termi- nabunt" type="context">termi-<lb/>nabũt</reg> angulos <var>.n.e.u.</var> et <var>.e.n.c.</var> <reg norm="inuicem" type="context">inuicẽ</reg> ferè ęquales, <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 ſublimatam eſſe ad ſupremas par-<lb/>tes elementaris regionis circum circa lineam <var>.o.i</var>: </s> + <s xml:space="preserve">tunc clarum eſſet quod ſi ratione hu <lb/>iuſmodi exhalationis ſtella <var>.x.</var> ita lucida viſa fuerit tam aſpicientibus ab <var>.e.</var> quam ab <lb/>c. exhalatio minoris latitudinis quam <var>.u.n.</var> eſſe non poterat, hoc eſt, quam terræ dia-<lb/>meter, cum idem in longitudine ferè ſit, ſed punctum <var>.u.</var> ſatis videri poteſt 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 diuerſa diafana tranſeuntium. </s> + <s xml:space="preserve">Nunc producti cum fuerint ij duo radij <var>.e.u.</var> et <var>.c.<lb/>n.</var> vſque 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> ſed <var>.s.x.</var> ſubiacet 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 ſenſibiliter ac ſi eſſent in centro <var>.i</var>. </s> + <s xml:space="preserve">Et cum ſuperius di-<lb/>xerimus angulos <var>.s.e.x.</var> et <var>.k.c.x.</var> ſenſibiliter mi nime differre ab angulis <var>.c.n.e.</var> et <var>.e.<lb/>u.c.</var> ſi cognouerimus quantitatem iſtorum, cognita etiam nobis erit quantitas <lb/>ill orum.</s> + </p> + <p> + <s xml:space="preserve">Cum igitur ſemidiameter elementaris regionis maior ſit ſemidiametro terræ, vt <lb/>33. ad vnum, & cogitata <var>.c.n.</var> vt dicta ſemidiameter, quia ſenſibiliter ab ea minime <lb/>diffe<unsure/>rt, nunc ſi ſupponatur dicta <var>.n.c.</var> vt <reg norm="baſistrianguli" type="context">baſistriãguli</reg> orthogonij eſſe <reg norm="partium" type="context">partiũ</reg> .100000 <lb/>& dixerimus ſi <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">partiũ</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 ſtellam om-<lb/>nes aliæ non diſtant huiuſmodi 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 ſequuti ſunt <lb/>opinionem exhalationum, ad ſer <lb/>uandam nullitatem diucrſitatis <lb/>aſpectus, affirmant poſſe tam lon <lb/>ge à terra <reg norm="aſcendere" type="context">aſcẽdere</reg> exhalationes, <lb/>imo nec attingere ſupremas tertię <lb/>regionis a eris partes, ita ut non <reg norm="in- grediantur" type="context">in-<lb/>grediãtur</reg> ſuum <reg norm="igneum" type="context">igneũ</reg> orbem, qui <lb/>quidem orbis ſecundum illorum <lb/>opinionem incipit non valde lon-<lb/>gè à ſuperſicie terræ, vt <reg norm="iam" type="context">iã</reg> in mea <lb/>conſideratione contra Antonium <lb/>Bergam probaui, ſed demus, <reg norm="quod" type="simple">ꝙ</reg> <lb/>dictæ exhalationes aſcenderint <lb/>per decem ſemidiametros terræ, <lb/>diſcurrendo poſtea ſic, cum <var>.c.n.</var> ut <lb/>decem, nobis dat <var>.c.e.</var> vt duo, <reg norm="quid" type="simple">ꝗd</reg> <lb/>dabit nobisipſa <var>.c.n.</var> vt .100000. <lb/>& proueniet nobis <var>.c.e.</var> vt .200000. <lb/>cuius ſinus 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> & ſic eorum arcus <var>.s.x.</var> et <var>.k.x.</var> <lb/>ſed quis vnquam dubitabit <reg norm="quod" type="simple">ꝙ</reg> in <lb/>tanto interuallo à dicta ſtella non <lb/>fint aliæ multæ ipſa 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 aſpectus <lb/>diuerſitatis 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, exiſtente 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 ipſo oculo non videretur ob terræ globoſitatem. </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 ſecunda figura videre eſt, in qua <lb/>ducantur <var>.c.b</var>: <var>i.b</var>: et <var>.i.u.</var> quæ <var>.i.u.</var> ſecabit 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> ſimiliter in pun-<lb/>cto <var>.y.</var> quod nulli dubium eſt, cum <var>.c.u.</var> æqualis ſit <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> ſed ex .4. primi <var>.c.y.</var> æqua <lb/>lis erit ipſi <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 ſuppoſita <var>.c.i.</var> decima parte ipſius <var>.c.u.</var> nemi <lb/>ni dubium erit quod cum <var>.u.i.</var> ſubtendatur angulo <lb/>recto <var>.u.c.i.</var> (iam ſupra diximus angulum <var>.c.</var> ſenſibi-<lb/>liter minime differre à recto) ipſa vt ſinus totus erit <lb/>partium .100000. cuius quadratum cum diuiſum <lb/>fuerit in partes æquales centum & <reg norm="vnam" type="context">vnã</reg>, illarum vna <lb/>æqualis erit quadrto <var>.c.i.</var> reliquę vero quadrato ip-<lb/>ſius <var>.u.c.</var> ex proportione duplicata quadratorum ad <lb/>eam quam continent eorum latera. </s> + <s xml:space="preserve">Sed quadra-<lb/>tum ipſius <var>.u.i.</var> eſt 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 ipſius <var>.c.u.</var> erit <lb/>partium .9900990100. cuius radix <var>.u.c.</var> erit <reg norm="partium" type="context">partiũ</reg> <lb/>99500. vnde angulus <var>.c.i.u.</var> erit graduum .84. & mi <lb/>nu .17. & angulus <var>.c.u.i.</var> qui reſpondet ſinui <var>.c.i.</var> erit <lb/>gra .5. & min .43. cuius duplum, hoc eſt angulus <var>.c.<lb/>u.b.</var> erit grad .11. min .26. æqualis ferè angulo iam <lb/>ſupradicto. </s> + <s xml:space="preserve">ſed <var>.c.y.</var> ſinus anguli <var>.c.i.y.</var> erit ſimiliter <lb/>partium .99500. talium vt <var>.c.i.</var> ſunt .100000. ſed vt <lb/><var>c.i.</var> eſt partium .9950. </s> + <s xml:space="preserve">tunc <var>.c.y.</var> erit partium .9900 <lb/>hoc eſt quaſi decima pars ipſius <var>.c.u.</var> </s> + <s xml:space="preserve">quare ſi 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> abſque ſenſibili 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/>iſoperimetrarum.</head> + <head style="it" xml:space="preserve">Domino Ioanni Mariæ Agatio.</head> + <p> + <s xml:space="preserve"><emph style="sc">QVamvis</emph> à Theone ſupra Ptolomei Almageſtum ſufficienter traditum <lb/>ſit de magnitudinibus figurarum iſopetimetrarum, nihilominus vt tibi <lb/>morem geram, ea nunc ſcribo, quæ mihi in mentem venerunt contra <lb/><reg norm="Alexandrum" type="context">Alexãdrum</reg> Piccolhomineum, <reg norm="antequam" type="context">antequã</reg> aliquid ipſius 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 ſphæra vbi tractat de <reg norm="caeliro- tunditate" type="simple">cęliro-<lb/>tunditate</reg>, ita inquit.</s> + </p> + <quote xml:lang="it"> + <s xml:space="preserve">Oltre di queſto, douendo il decimo cielo contenere & in ſe chiudere tutte le co-<lb/>ſe, è conueneuol coſa il penſare, che foſſe fatto di quella più capace figura che eſ-<lb/>ſer poſſa, la qual è la figura rotunda, però che ſi può trar da molti luoghi d'Euclide <lb/>che ſi come ſe noi ciimmagineremo più figure ſuperficiali talmente che tutte le li-<lb/>nee de l'vna congionte inſieme, ſieno vguali à tutte le linee pur inſiememente com <lb/>poſte di qual ſi voglia de l'altre figure, ne ſeguirà, che quella figura ſarà più capa-<lb/>ce la qual haurà manco angoli, & quella capaciſſima che ſarà 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 iſoperimetris, ſit enim triangulus æ-<lb/>quilaterus <var>.o.b.g.</var> quadratum verò <var>.b.l.</var> quorum pe-<lb/>riferiæ inuicem æquales ſint. </s> + <s xml:space="preserve">Dico quadratum ma-<lb/> + <anchor type="figure" xlink:label="fig-0387-01a" xlink:href="fig-0387-01"/> + ioris ſuperficiei eſſe ipſo triangulo. </s> + <s xml:space="preserve">Accipio pri-<lb/>mum lineam <var>.f.h.</var> eiuſdem longitudinis quæ vnius <lb/>periferiæ dictarum figurarum, quam punctis <var>.r.K.</var> <lb/>mediantibus diuido in tres ęquas 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> ideſt 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> ſed <var>.f.l.</var> eſt quarta pars ip-<lb/>ſius <var>.f.h.</var> ergo <var>.k.i.</var> erit quarta pars ipſius <var>.k.h</var>. </s> + <s xml:space="preserve"><reg norm="Conium" type="context">Coniũ</reg> <lb/>gantur enim ambo iſtæ figuræ vt hic inferius vides, <lb/>vnde <var>.a.g.</var> erit quarta pars ipſius <var>.b.g.</var> diuiſa poſtea <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> quę per .8. primi, nec <reg norm="non" type="context">nõ</reg> ex definitione, <lb/>perpendicularis erit ipſi <var>.b.g.</var> ergo etiam <reg norm="quadratum" type="context">quadratũ</reg> <lb/>b q. ſupra <var>.b.g.</var> <reg norm="producoque" type="simple">producoq́;</reg> <var>.o.c.</var> vſque ad <var>.m.</var> nam nul <lb/>li dubium eſt quin <var>.o.c.</var> breuior ſit <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">ducoq́;</reg> <var>t.e.p.</var> æquidiſtantem <var>.b.g.</var> <lb/>vnde habebimus duo quadrata <var>.e.g.</var> et <var>.e.b.</var> ſed <lb/>quadratum <var>.b.l.</var> æquatur quadrato ipſius <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. ſecundi, hoc eſt æquatur quadrato <var>.c.a.</var> & re-<lb/>ctangulo <var>.t.g</var>. </s> + <s xml:space="preserve">Deinde vt ſe habet <var>.p.g.</var> ad <var>.o.e.</var> ita ſe habet <var>.u.p.</var> ad <var>.u.e.</var> ex ſimilitudine <lb/>triangulorum. </s> + <s xml:space="preserve">Sed <var>.p.g.</var> maior eſt ipſa <var>.o.e.</var> cum <var>.p.g.</var> æqualis ſit <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. ſexti. </s> + <s xml:space="preserve">Similiter dico maiorem eſſe trian <lb/>gulum <var>.b.d.t.</var> triangulo <var>.e.o.d.</var> vnde ſequitur rectangulum <var>.t.g.</var> maiorem eſſe triangu-<lb/>lo <var>.b.o.g.</var> ſed quadratum <var>.b.l.</var> eſt etiam maior ipſo rectangulo <var>.t.g.</var> ex quadrato ipſius <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 ſit periferia ipſius trianguli æquilateri. </s> + <s xml:space="preserve">Cogita triangulum æquilaterum hic <lb/>ſubſcriptum <var>.d.l.q.</var> cuius baſis <var>.l.q.</var> diuiſa ſit per æqualia à perpendiculari <var>.d.o.</var> <reg norm="deſcri- ptumque" type="context simple">deſcri<lb/>ptũq́;</reg> ſit rectangulum <var>.o.g.</var> quod æquale erit triangulo <var>.d.l.q.</var> ſed periferia trianguli <lb/>maior eſt periferia rectanguli, nam <var>.l.q.</var> æqualis eſt <var>.o.q.</var> cum <var>.d.g.</var> ſed <var>.q.d.</var> maior eſt <var>.o.<lb/>d.</var> ex .18. primi, vnde <var>.l.d.</var> maior etiam <var>.q.g.</var> cum ex .34. dicti latera oppoſita ipſius re <lb/>ctanguli ſint inuicem æqualia, accipiamus poſtea <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 pun<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 eſt quadratum ipſius <var>.c.i.</var> æquale erit ei quod fit .ex <var>.e.c.</var> <lb/>in <var>.c.h.</var> hoc eſt rectangulo <var>.g.o.</var> hoc eſt triangulo <var>.d.l.q.</var> ſed <var>.e.h.</var> eſt dimidium perife-<lb/>rię ipſius rectanguli <var>.g.o.</var> quæ minor eſt di midio periferiæ trianguli <var>.d.l.q.</var> vt vidimus <lb/>et <var>.i.k.</var> eſt dimidium periferię quadrati ipſius <var>.i.c.</var> & minor etiam ipſa <var>.e.h.</var> ex .14. tertij <lb/></s> + <s xml:space="preserve">quare verum eſt propoſitum.</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 ſunt inuicem æquales, poſſumus etiam breuiter videre id <lb/>quod ſupradiximus, hoc eſt, quod quadratum, maius ſit triangulo æquilatero. </s> + <s xml:space="preserve">Nam <lb/>cum <var>.b.g.</var> ſeſquitertia ſit 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 ipſo <reg norm="rectangulo" type="context">rectãgulo</reg> <var>.c.q.</var> ſed <var>.c.q.</var> maius eſt <reg norm="triam" type="context">triã</reg> <lb/>gulo <var>.b.o.g.</var> cum <var>.q.g.</var> quæ æqualis eſt <var>.o.g.</var> maior ſit <var>.o.c.</var> ex .18. vel penultima primi, <lb/>nam ſi <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 noſtri vſi ſunt quæ generalis eſt vt in Theone ſupra Al-<lb/>mageſtum videre eſt, medijs perpendicularibus à centris ad latera figurarum, ſed <lb/>quia <reg norm="differentia" type="context">differẽtia</reg> longitudinum ipſarum perpendicularium alio medio inueniri poteſt, <lb/>eo quo ipſi vſi ſunt, prætermittere nolo quin tibi ſcribam.</s> + </p> + <p> + <s xml:space="preserve">Ego enim ita diſcurro.</s> + </p> + <p> + <s xml:space="preserve">Sint duæ figuræ iſoperimetrę æquilaterę & æquiangulæ, puta primò trian-<lb/>gulum & quadratum quorum centra ſint <var>.e.</var> et <var>.o.</var> à quibus centris ad latera ſint 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 ſcis, ducantur <lb/>poſtea <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/>midiũ</reg> recti, et <var>.e.t.n.</var> tertia pars vnius recti, vt ex te ipſo 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. ſeſquialter erit angulo <var>.t.</var> quod vt clarius videas cogita lineam <var>.b.d.</var> cuius medietas <lb/>ſit <var>.c.d.</var> tertia verò pars illius ſit <var>.g.d.</var> </s> + <s xml:space="preserve">tunc dico <var>.c.d.</var> ſeſquialteram eſſe ipſi <var>.g.d.</var> ſit enim <lb/><var>f.d.</var> duplum ipſius <var>.g.d.</var> </s> + <s xml:space="preserve">quare <var>.f.d.</var> erunt duæ tertiæ totius lineę <var>.b.d.</var> & quia eadem pro <lb/>portio eſt 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> ſeſquialtera eſt, verum igitur erit quod an-<lb/>gulus <var>.a.</var> ſeſquialter ſit ipſi <var>.t.</var> deinde <var>.t.n.</var> eſt ſeſquitertia ipſi <var>.a.u.</var> vt ſuperius vidimus .<lb/>in eorum duplis. </s> + <s xml:space="preserve">ſcimus etiam <var>.n.e.</var> eſſe dimidium ipſius <var>.t.e.</var> co quod cum <var>.e.t.n.</var> ſit <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 inſcriptibilis circulo cuius diameter ſit <var>.e.t.</var> </s> + <s xml:space="preserve">quare <var>.e.t.</var> dupla erit <lb/>ipſi <var>.e.n.</var> in longitudine, ſed quadrupla in potentia: </s> + <s xml:space="preserve">t.n. vero tripla in potentia ipſi <var>.n.<lb/>e.</var> ex penultima primi, quæ omnia etiam ex .8. tertijdecimi. Eucli. elicere potes, ſed <lb/><var>c.n.</var> erat ſexquitertia ipſi <var>.a.u.</var> in longitudine, hoc eſt ipſi <var>.o.u.</var> nam <var>.o.u.</var> æqualis eſt ipſi <lb/><var>a.u</var>. </s> + <s xml:space="preserve">quare <var>.n.t.</var> erit minus quam dupla in potentia ipſi <var>.o.u.</var> hoc eſt, vt .16. ad .9. ergo <lb/>maior proportio erit ipſius <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 ipſius <var>.t.n.</var> ad <var>.n.e.</var> quam ad <var>.o.u.</var> vnde <var>.o.u.</var> longior erit <lb/>ipſa <var>.n.e.</var> quod eſt propoſitum.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi <var>.o.a.u.</var> eſſet pentagonus æquilaterus & æquiangulus, ſimiliter probabo per-<lb/>pendicularem <var>.o.u.</var> longiorem eſſe <var>.n.e.</var> ipſius trianguli æquilateri, dummodo ſint iſo-<lb/>perimetrę. </s> + <s xml:space="preserve">Sit enim <var>.a.u.</var> dimidium lateris pentagoni ex ſuppoſito, cuius centrum ſit <lb/>o. </s> + <s xml:space="preserve">tunc proportio <var>.t.n.</var> ad <var>.a.u.</var> erit ſuperbipartienstertias, vt ex ordine iam hic ſupradi <lb/>cto à te facillimè elicere potes, hoc eſt, 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">duorũ</reg> rectorum, hoc <lb/>eſt duæ quintæ vnius recti, vnde angulus <var>.a.</var> reſiduum vnius recti erit tres quin-<lb/>tæ vnius recti, </s> + <s xml:space="preserve">quare angulus <var>.a.</var> maior ericangulo <var>.o.</var> & conſequenter latus <var>.o.u.</var> ma-<lb/>ius latere <var>.a.u.</var> ſed <var>.t.n.</var> minor eſt quam tripla in potentia ad <var>.a.u.</var> eo quod erit vt .25. <lb/>ad .9. cum in longitudine ſit vt .5. ad .3. ſed dicta <var>.t.n.</var> tripla eſt in potentia ad <var>.e.n.</var> qua-<lb/>re <var>.a.u.</var> maior erit ipſa <var>.e.n.</var> ſed <var>.o.u.</var> maior eſt ipſa <var>.a.u.</var> vt diximus, igitur multo magis <var>.<lb/>o.u.</var> maior eſt ipſa <var>.a.u.</var> vt <reg norm="diximus" type="simple">diximꝰ</reg> & <reg norm="conſequenter" type="context context">cõſequẽter</reg> multo magis <var>.o.u.</var> maior erit ipſa <var>.n.e</var>.</s> + </p> + <p> + <s xml:space="preserve">Quotieſcunque enim cognoſcimus proportionem anguli <var>.o.</var> ad angulum <var>.a.</var> quod <lb/>quidem facillimum eſt, nec non proportionem <var>.t.n.</var> ad <var>.a.u.</var> quod, etiam illico cogno-<lb/>ſcitur, </s> + <s xml:space="preserve">tunc exſcientia cordarum & arcuum omnia etiam facillimè innueniuntur. <lb/></s> + <s xml:space="preserve">Verum circa <reg norm="triangulum" type="context">triangulũ</reg> æquilaterum, & pentagonum, alium <reg norm="modum" type="context">modũ</reg> inueni, ſed 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 incommenſur abilitate, in longitudine perpendicu-<lb/>laris trianguli æquilateri cum eiuſdem latere.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">ID quod à me poſtulas eſt omnino impoſſibile, velles enim duos numeros inueni <lb/>re inter ſe ita ſe habentes, vt ſe habent perpendicularis in triangulo æquilatero <lb/>cum vno eius laterum, quod vero hoc fieri non poſſit, conſidera in figura præcedenti <lb/>triangulum æquilaterum <var>.d.l.q.</var> cuius perpendicularis ſit <var>.d.o.</var> quæ diuidit <var>.l.q.</var> per <lb/>æqualia in <var>.o.</var> vnde ex .4. ſecundi Euclidis, quadratum <var>.l.q.</var> (ideſt <var>.d.q.</var>) quadruplum <lb/>erit quadrato <var>.o.q.</var> & ex penultima primi ęquale quadratis <var>.d.o.</var> et <var>.o.q.</var> </s> + <s xml:space="preserve">quare erit ſeſ-<lb/>quitertium quadrato ipſius <var>.d.o.</var> & ita quadratum <var>.d.o.</var> erit triplum quadrato ipſius <var>.<lb/>o.q.</var> hæe autem proportiones non ſunt vt numeri quadrati ad numerum quadratum <lb/>quod ſi ita fuiſſent, ſequeretur ternarium numerum eſſe quadratum ex .22. octaui. <lb/></s> + <s xml:space="preserve">Cum igitur non ſint vt numeri quadrati ad numerum quadratum, ſequitur ex ſepti-<lb/>ma decimi <var>.d.o.</var> eſſe incommenſurabilem ipſi <var>.l.q.</var> ſeu <var>.d.q.</var> in longitudine.</s> + </p> + <p> + <s xml:space="preserve">Vel dicamus ita, proportio quadrati ipſius <var>.l.q.</var> ad quadratum ipſius <var>.o.d.</var> eſt in ge <lb/>nere ſuperparticulari, cum ſit ſeſquitertia, vnde quadratum ipſius <var>.d.o.</var> numeris da-<lb/>ri non poteſt, eo quod ſi dabilis fuiſſet, ſequeretur, quod inter quadratum ipſius. l<unsure/> <var>.<lb/>q.</var> & ipſius <var>.d.o.</var> eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex <lb/>octaua eiuſdem vnitas diuiſibilis eſſet, quod fieri non poteſt.</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 conſideraui circa triangulum æquilaterum & pentagonum, vt <lb/>tibi ſignificaui ita ſe habet.</s> + </p> + <p> + <s xml:space="preserve">Probandum primò eſt pentagonum altiorem eſſe triangulo ſibi iſoperimetro. <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 diſtenta ſit <var>.K.z.</var> baſis autem <var>.m.<lb/>g.</var> bifariam diuiſa ſit in puncto <var>.a.</var> <reg norm="ductaque" type="simple">ductaq́;</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> (baſes 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>) ſint inuicem æquales ex .4. eiuſdem.</s> + </p> + <p> + <s xml:space="preserve">Accipiatur deinde vel intelligatur <var>.g.p.</var> æqualis duabus te<unsure/>rtijs ipſius <var>.a.g.</var> ducatur<lb/>q́ue <var>.b.p.</var> quam probabo maiorem eſſe duplo ipſius <var>.a.p.</var> vnde maior erit latere ipſius <lb/>trigoni æquilateris, cuius dimidium eſt <var>.a.p.</var> ſcimus enim ipſum latus ſe 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> ſit 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 ipſius <var>.a.p.</var> erit .103006. latus igitur dicti trigoni, quod <lb/>ab <var>.p.</var> erigitur, ſecabit perpendicularem <var>.a.b.</var> ſub <var>.b.</var> hoc eſt 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 iſoperimetro pentagono propoſito, <reg norm="ducaturque" type="simple">ducaturq́;</reg> <var>.u.p.</var> ducatur etiam <var>.u.n.</var> pa-<lb/>rallela ipſi <var>.b.g.</var> quæ concludet triangulum <var>.g.u.n.</var> ſimilem triangulo <var>.m.b.g.</var> eo quod <lb/>cum angulus <var>.m.b.g.</var> æqualis ſit 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="æquidiſtantes" type="context">æquidiſtãtes</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. eiuſdem, </s> + <s xml:space="preserve">vnde ex .4. ſexti proportio <var>.g.n.</var> ad <var>.g.m.</var> erit .vt <var>.g.u.</var> <lb/>ad <var>.m.b.</var> ſed cum <var>.g.u.</var> maior ſit dimidio ipſius <var>.b.g.</var> ex .20. primi, hoc eſt maior dimi-<lb/>dio ipſius <var>.b.m.</var> ergo <var>.g.n.</var> etiam maior erit ipſa <var>.g.a.</var> quapropter maior erit ipſa <var>.g.p.</var> <lb/>cum <var>.g.p.</var> minor ſit ipſa <var>.g.a.</var> ex hypotheſi, 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 ſexti <lb/>vel quia totum maius eſt ſua parte. </s> + <s xml:space="preserve">Triangulus igitur <var>.b.u.g.</var> maior eſt 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 ſibi iſo-<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 circunſcriptorum ab eodem circulo.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVod autem periferia quadrati in eodem circulo inſcripti, in quo ſit triangu-<lb/>lus æquilaterus, longior ſit periferia ipſius trianguli æquilateri, abſque 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 demonſtrare quod valde de-<lb/>ſideras.</s> + </p> + <p> + <s xml:space="preserve">Quapropter ſit circulus <var>.b.a.e.q.</var> in quo ſit <reg norm="triangulum" type="context">triangulũ</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 ſcire potes. </s> + <s xml:space="preserve">Sit etiam <lb/><reg norm="punctum" type="context">punctũ</reg> <var>.b.</var> commune tam anguli quadrati quam trianguli. </s> + <s xml:space="preserve">vnde ſequitur quod dictus <lb/>diameter ſecabit 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 ſit arcui <var>.b.n.</var> ex .27. tertij, remanet vt arcus <var>.q.e.</var> equalis ſit arcui <var>.q.n.</var> vnde <lb/>angulus <var>.q.b.e.</var> æqualis erit angulo <var>.q.b.n.</var> ex .26. eiuſdem. </s> + <s xml:space="preserve">quare ex .4. primi anguli <lb/>ad <var>.t.</var> erunt recti, et <var>.n.t.</var> æqualis erit ipſi <var>.t.e.</var> vt diximus.</s> + </p> + <p> + <s xml:space="preserve">Deinde <var>.b.e.</var> et <var>.q.a.</var> ſeinuicem <reg norm="ſecant" type="context">ſecãt</reg> in puncto <var>.o.</var> vt ex ſe clarum patet, ducatur po <lb/>ſtea <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 ipſa <var>.q.e.</var> et <var>.q.e.</var> longior erit ipſa <var>.e.t</var>. </s> + <s xml:space="preserve">quare <var>.q.o.</var> longior erit ipſa <var>.t.e</var>.</s> + </p> + <p> + <s xml:space="preserve">Vt probemus poſtea <var>.b.a.o.</var> longiorem eſſe ipſa <var>.b.e.</var> producatur <var>.b.a.</var> ita quod <var>.a.<lb/>p.</var> æqualis ſit ipſi <var>.a.o.</var> <reg norm="ducaturque" type="simple">ducaturq́;</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">rectꝰ</reg> ſit, erit angulus <var>.o.a.p.</var> ſimiliter <reg norm="rectus" type="simple">rectꝰ</reg> ex .13. primi, vnde ex .5. et .32. <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <lb/>angulus <var>.a.p.o.</var> erit dimidium recti, & ſimiliter, exijſdem, angulus <var>.b.q.a.</var> eſt dimidium <lb/>recti </s> + <s xml:space="preserve">quare angulus <var>.a.p.o.</var> æqualis erit angulo <var>.a.q.b.</var> ſed angulus <var>.a.e.b.</var> æqualis eſt 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 eſt 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 ſexti, <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> ſed ex .18. primi <var>.b.o.</var> maior eſt <lb/>ipſa <var>.b.a</var>. </s> + <s xml:space="preserve">quare ex .14. quinti <var>.b.p.</var> maior erit <lb/>ipſa <var>.b.e.</var> ſed <var>.b.p.</var> æquatur ipſis <var>.b.a.</var> cum <var>.a.</var> o <lb/>ex hypoteſi, ergo <var>.b.a.</var> cum <var>.a.o.</var> maior erit <lb/>ipſa <var>.b.e.</var> ſed <var>.q.o.</var> maior erat ipſa <var>.t.e.</var> vt ſupe <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 eſt ipſa <var>.b.e.</var> cum <var>.e.t.</var> hoc eſt dimidium <lb/>periferię ipſius quadrati, <reg norm="maius" type="simple">maiꝰ</reg> erit dimidio <lb/>periferię <reg norm="ipſius" type="simple">ipſiꝰ</reg> <reg norm="trianguli" type="context">triãguli</reg> propoſiti, </s> + <s xml:space="preserve">quare ex 14. <lb/>dicta tota periferia dicti trianguli, ſimiliter <lb/>probarem de omnibus alijs figuris regulari <lb/>bus eodem circulo inſcriptis.</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 aliàs dixi verum eſt, intellectum ſcilicet non omninò quieſcere cir <lb/>ca illas duas Archimedis propoſitiones, quæ in translatione Tartaleæ ſunt <lb/>ſub numeris .4. et .5. & in impreſſione Baſileæ ſub 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, ſeu ſtateræ: </s> + <s xml:space="preserve">A ſpice igitur in .4. ſupradicta, quod cum appen-<lb/>ſæ fuerint omnes illæ partes ponderum, partibus longitudinis ipſius <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 ipſi <var>.l.K.</var> hoc eſt reſpicientes 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 ſit paralle la ipſi <var>.l.k.</var> quæ diuiſa ſit in puncto <var>.i.</var> ſupra <var>.g</var>. </s> + <s xml:space="preserve">Hinc nulli dubium erit, <lb/>cum <var>.g.</var> fuerit centrum totius ponderis appenſi ipſi <var>.l.K.</var> quod <var>.i.</var> ſimiliter erit centrum <lb/>cum directe locatum ſit ſupra <var>.g.</var> hoc eſt in eadem directionis linea, quod quidem <lb/>non indiget aliqua demonſtratione, cum per ſe ſatis pateat. </s> + <s xml:space="preserve">Vnde ex communi <lb/>conceptu <var>.o.</var> erit centrum ponderis appenſi ipſi <var>.l.h.</var> et <var>.u.</var> erit centrum ponderis ap-<lb/>penſi. ipſi <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 eſt ipſius <var>.l.h.</var> & ipſius <var>.h.k.</var> con <lb/>tinuatorum per totam <var>.l.k</var>. </s> + <s xml:space="preserve">Nunc ergo ſi conſideremus <var>.l.k.</var> diuiſam eſſe, hoc eſt di-<lb/>ſiunctam in puncto <var>.h.</var> inueniemus nihilominus <var>.i.</var> centrum eſſe dictorum ponderum, <lb/>& quod tantum eſt, ipſam eſſe <reg norm="continuam" type="context">continuã</reg>, quantum diuiſam 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> diuiſa vt dictum eſt, habebimus propoſitum.</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 propoſitionem tibi relinquo.</s> + </p> + <p> + <s xml:space="preserve">Illa verò propoſitio, quam tibi dixi Archimedem tacuiſſe in huiuſmodi materia <lb/>eſt, quod ſi duo pondera æquilibrant ab extremis alicuius ſtateræ, in certis præfixis <lb/>diſtantijs à centro. </s> + <s xml:space="preserve">Tunc dico ſi eorum vno manente alterum moueatur remotius <lb/>ab ipſo centro quod illud deſcendet, & ſi vicinius ipſi centro appenſum fuerit aſcen-<lb/>det. </s> + <s xml:space="preserve">Hæc enim propoſitio quotidie omnibus in locis videtur, ipſam verſo4; </s> + <s xml:space="preserve">puto Ar <lb/>chimedem prætermiſiſſe ob facilitatem, cum ab antedicta ferè dependeat.</s> + </p> + <p> + <s xml:space="preserve">Sit exempli gratia ſtatera <var>.a.u.</var> cuius centr um ſit <var>.i.</var> & pondera <var>.u.a.</var> appenſa, ſein-<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> ſe inuicem habent. </s> + <s xml:space="preserve">Nunc dico quod ſi pondus ipſius <var>.u.</var> <lb/>poſitum fuerit vicinius centro vt puta in <var>.o.</var> inmoto exiſtente pondere, a. quod bra-<lb/>chium <var>.i.o.u.</var> aſcendet, & è conuerſo, ſi remotius poſitum fuerit, deſcendet.</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">dictũ</reg> eſt in <var>.o.</var> vicinius <reg norm="centro" type="context">cẽtro</reg>, quapropter brachium <var>.i.o.</var> <reg norm="breuius" type="simple">breuiꝰ</reg> erit <lb/>brachio <var>.i.u.</var> vnde minor proportio erit ipſius <var>.i.o.</var> ad <var>.i.a.</var> quàm.i.u. ad eundem <var>.a.i.</var> & <lb/>conſequenter quam ponderis ipſius <var>.a.</var> (quod ſit <var>.n.e.</var>) ad pondus ipſius <var>.u</var>. </s> + <s xml:space="preserve">Quare ſi cx <lb/>pondere <var>.n.e.</var> dempta fuerit <var>.e.</var> pars eius, ita quod reliqua pars <var>.n.</var> ſe habeat ad pondus <lb/>o. vt ſe habet. i<unsure/> <var>.o.</var> ad <var>.i.a.</var> tunc ſtatera non mouebitur; </s> + <s xml:space="preserve">addita verò parte <var>.e.</var> ex com-<lb/>muni conceptu, a. deſcendet vnde <var>.o.</var> aſcenderet conuerſum verò ex ſimilibus 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 ſatisfacerem, cum Eu-<lb/>totius citet ſextum librum Pergei, quem nunquam vidimus, <reg norm="ſupponatque" type="simple">ſupponatq́;</reg> ea, quæ nec <lb/>ipſe nec alius vnquam quod ſcimus probauit.</s> + </p> + <p> + <s xml:space="preserve">Deſideras enim demonſtrationem illius quod Archimedes dicit inter primam, <lb/>& ſecundam propoſitionem ſecundi libri, vbi tractat de centris grauium, propte-<lb/>rea quod illud ſupponit pro manifeſto.</s> + </p> + <p> + <s xml:space="preserve">Sit enim figura hic ſubſcripta, ferè ſimilis parabolæ poſitæ in .2. propoſitione di <lb/>cti libri, vt in impreſſione Baſileenſi habetur, <reg norm="ſintque" type="simple">ſintq́;</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">protractisq́;</reg> <var>.f.x.</var> et <var>.u.i.</var> ad <var>.b.d.</var> quæ inuicem etiam erunt parallelę <lb/>ex .30. primi Eucli. </s> + <s xml:space="preserve">vnde ipſæ etiam, diametri erunt ipſarum portionum: </s> + <s xml:space="preserve">vt ex eo col <lb/>ligere eſt, quod in .49. primi lib. Pergei probatur. </s> + <s xml:space="preserve">Imaginando poſtea ad puncta <var>.b.<lb/>f.</var> er<unsure/> <var>.i.</var> tres contingentes, manifeſtum erit punctum <var>.b.</var> illud eſſe quod terminat alti-<lb/>tudinem huiuſmodi portionis, et <var>.f.</var> et <var>.i.</var> terminantia altitudines partialium, ex .5. ſe<lb/>cundi ipſius Pergei, eo quod dictæ contingentes paralellæ erunt ipſis baſibus, vnde <lb/>trianguli inſcripti, eaſdem habebunt altitudines, quas portiones ipſæ, quod erit ex <lb/>mente Archimedis. </s> + <s xml:space="preserve">Et ſic deinceps poteris multiplicare angulos ſiguræ rectilineæ <lb/>in parabola, quæ deſignata erit vt deſiderat Archimedes, qui quidem dicit, quod <lb/>protractæ cum fuerint aliæ deinceps poſt <var>.f.i.</var> ipſæ inuicem ęquidiſtantes <reg norm="erunt" type="context">erũt</reg>, diuiſę-<lb/>q́ue peræqualia ab <var>.d.b.</var> quod <reg norm="quanuis" type="context">quãuis</reg> <reg norm="verum" type="context">verũ</reg> ſit, <reg norm="tantum" type="wordlist/context">tñ</reg> ab Eutotio non ſatis <reg norm="demonſtratum" type="context context">demõſtratũ</reg> <lb/>eſt, cum ſupponat <var>.a.f.b.</var> æqualem eſſe ipſi <var>.b.i.c.</var> probare volens eius diametros æqua <lb/>les eſſe abſque aliqua citata ratione, quæ quidem ratio eſſet conuerſum .4. propoſi-<lb/>tionis libri de conoidalibus. </s> + <s xml:space="preserve">Sed oporteret nos <reg norm="etiam" type="context">etiã</reg> videre .6. librum ipſius Pergei, <lb/>& propterea tibi non ſatisfacerem.</s> + </p> + <p> + <s xml:space="preserve">Eſto igitur, ut inuenta ſit linea <var>.K.</var> cuius productum in <var>.u.i.</var> æquale ſit qua drato ip <lb/>ſius <var>.u.c.</var> inuenta etiam ſit linea <var>.h.</var> cuius productum cum <var>.f.x.</var> æquale ſit quadrato ip-<lb/>ſius <var>.a.x.</var> vnde ex conuerſo .49. primi ipſius Pergei, proportio ipſius <var>.K.</var> ad <var>.b.c.</var> erit ut <lb/>ipſius <var>.b.c.</var> ad <var>.b.d.</var> & ipſius <var>.h.</var> ad <var>.a.b.</var> vt ipſius <var>.a.b.</var> ad <var>.b.d</var>. </s> + <s xml:space="preserve">Erit igitur ex .16. ſexti Eucl. <lb/>quadratum <var>.b.c.</var> æquale producto ipſius <var>.K.</var> in <var>.b.d.</var> & quadratum <var>.a.b.</var> æquale produ-<lb/>cto ipſius <var>.h.</var> in <var>.b.d.</var> & ex prima ſexti, ita erit ipſius <var>.K.</var> ad <var>.h.</var> vt producti quod fit ex <var>.K.</var> <lb/>in <var>.b.d.</var> ad productum ipſius <var>.h.</var> in <var>.b.d.</var> hoc eſt vt quadrati ipſius <var>.b.c.</var> ad quadratum ip <lb/>ſius <var>.b.a.</var> ex .16. et .11. quinti, hoc eſt vt quadrati ipſius <var>.u.c.</var> ad quadratum ipſius <var>.a.x.</var> <lb/>hoc eſt ut productum ipſius <var>.k.</var> in <var>.u.i.</var> ad productnm ipſius <var>.h.</var> in <var>.x.f</var>. </s> + <s xml:space="preserve">Nunc ſi ipſius <var>.k.</var> <lb/>ad <var>.h.</var> c<unsure/>ſt vt producti ipſius <var>.K.</var> in <var>.u.i.</var> ad productum ipſius <var>.h.</var> in <var>.f.x.</var> ergo ex .24. ſexti, <lb/>& communi conceptu, proportio ipſius <var>.k.</var> ad <var>.h.</var> compoſita erit ex ea quæ ipſius <var>.u.i.</var> <lb/>ad <var>.f.x.</var> & ex ea quæ ipſius <var>.k.</var> ad <var>.h</var>. </s> + <s xml:space="preserve">Cum ergo dempta fuerit proportio ipſius <var>.k.</var> ad <var>.h.</var> <lb/>(vt ſimplex) à proportione ipſius <var>.k.</var> ad <var>.h.</var> (vt compoſita) reliquum nihil erit. </s> + <s xml:space="preserve">Qua-<lb/>re <var>.f.x.</var> æqualis erit ipſi <var>.u.i</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed quod <var>.f.m.</var> æqualis ſit ipſi <var>.m.i</var>. </s> + <s xml:space="preserve">Videto in Eutotio, quia hoc ſatis ſui natura <lb/>facile eſt.</s> + </p> + <p> + <s xml:space="preserve">Sed accipe alium modum breuiorem ad probandum <var>.f.x.</var> eſſe æqualem ipſi <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> prolungatisq́ue diametris <var>f.<lb/>x.</var> et <var>.u.i.</var> vſque ad contingentem ipſam, habebis <var>.f.e.</var> æqualem ipſi <var>.f.x.</var> et <var>.g.i.</var> ipſi <var>.u.i.</var> <lb/>Ex .35. primi Pergei, producta poſtea <var>.x.u.</var> habeb is ex .2. ſexti Eucli <var>.x.u.</var> parallelam <lb/>ipſi <var>.a.c.</var> ſed <var>.e.g.</var> parallela eſt ipſimet <var>.a.c.</var> ex quinta ſecundi ipſius Pergei, </s> + <s xml:space="preserve">quare ex .30 <lb/>primi Euclid <var>.e.g.</var> parallela erit ipſi <var>.u.x.</var> & ex .34. eiuſdem æqualis erit <var>.e.x.</var> ipſi <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 deſideres probabo <var>.f.m.</var> æqualem eſſe ipſi <var>.m.i</var>. </s> + <s xml:space="preserve">Iam igitur ſcis 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 ſit <var>.f.x.</var> æqualis ipſi <var>.u.i.</var> vt tibi probaui, & inuicem parallelæ ideo <var>.f.i.</var> parallela <lb/>erit ipſi <var>.x.u.</var> ex .33. primi Euclidis. </s> + <s xml:space="preserve">Vnde ex .30. eiuſdem, parallela erit etiam ipſi <var>.a.<lb/>c.</var> ſed cum <var>.x.u.</var> diuiſa ſit ab <var>.d.b.</var> per æqualia, eo quod diuidit <var>.a.c.</var> eodem modo, quę <lb/>ipſi parallela eſt ex .2. ſexti. </s> + <s xml:space="preserve">Reliqua tibi conſideranda relinquo. </s> + <s xml:space="preserve">cum verò ambæ <var>.f.<lb/>x.</var> et <var>.u.i.</var> parallelæ ſint ipſi <var>.b.d.</var> ſequitur quod cum ex .34. primi <reg norm="vnaquæque" type="simple">vnaquæq;</reg> <var>.f.m.</var> et <var>.m.<lb/>i.</var> æqualis ſit medietati ipſius <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 ſatisfacere Eutocium in .3. propoſitione ſecundi <lb/>lib. de centris Grauium Archimedis, cum citet .6. librum de elementis conicis, ad-<lb/>de quod ſi aliud in ipſo .6. libro ab eo citato non eſſet magis ad propoſitum, quàm <lb/>ca quæ ab ipſo citata ſunt, nihilominus adhuc irreſolutus maneres.</s> + </p> + <p> + <s xml:space="preserve">Conſidera igitur eandem ipſam figuram præcedentem; </s> + <s xml:space="preserve">pro alia verò parabola ſi <lb/>mili dictæ, accipe ſecundam figuram ipſius tertiæ dictæ propoſitionis. </s> + <s xml:space="preserve">Deinde ima <lb/>ginabis duo latera <var>.o.x.</var> et <var>.o.p.</var> diuiſa eſſe per æqualia in punct is <var>.g.</var> et <var>.K.</var> <reg norm="protractisque" type="simple">protractisq́;</reg> <lb/>diametris <var>.g.y.</var> et <var>.K.u.</var> quæ, vt in præcedenti probaui, ſunt inuicem æquales, ſcire <lb/>debes quod ſimiles parabolæ inuicem aliæ non poſſunt eſſe, niſi eæ quæ diametros <lb/>proportionales ſuis baſibus habeant, <reg norm="ſimiliterque" type="simple">ſimiliterq́;</reg> poſitæ, hoc eſt, ut proportio ipſius <lb/><var>b.d.</var> ad <var>.a.c.</var> ſit eadem quæ ipſius <var>.o.r.</var> ad <var>.x.p.</var> & quod anguli ad <var>.r.</var> ſint æquales angulis <lb/>circa <var>.d</var>. </s> + <s xml:space="preserve">Notentur ergo primum puncta communia ip ſius <var>.o.g.</var> cum <var>.y.t.</var> & ipſius <var>.b.</var> x <lb/>cum <var>.f.m.</var> characteribus. ω<unsure/>. et <var>.n</var>. </s> + <s xml:space="preserve">Nunc igitur ſcimus <var>.f.m.</var> æqualem eſſe <var>.m.i.</var> tota <reg norm="mque" type="simple">mq́;</reg> <var>.f.<lb/>i.</var> parallelam eſſe ipſi <var>.a.c</var>. </s> + <s xml:space="preserve">Idem dico de <var>.y.t.u.</var> <reg norm="triangulique" type="simple">trianguliq́;</reg> <var>.x.f.n.</var> et <var>.g.y.</var> ω<unsure/>. eſſe ſimiles <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/>ſunt inuicem æquales, ex .29. verò eiuſdem anguli <var>.f.x.n.</var> et <var>.n.b.m.</var> ſimiliter 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 ſecunda figura, vnde ex .4. ſexti Eucli. proportio <var>.n.f.</var> ad <var>.m.n.</var> erit ea <lb/>dem quę <var>.f.x.</var> ad <var>.b.m.</var> & ipſius <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"/> + eiuſdem erit vt <var>.a.d.</var> ad <var>.d.b</var>. </s> + <s xml:space="preserve">Idem etiam dico in ſecunda parabola, ſed ipſius <var>.x.o.</var> ad <lb/><var>o.r.</var> eſt vt <var>.a.b.</var> ad <var>.b.d.</var> ex .6. ſexti 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 ſecunda parabola, ita ut <var>.h.</var> eſt 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. ſexti <var>.a.x.</var> media proportionalis eſt inter <var>.h.</var> et <var>.f.<lb/>x.</var> cum ſupponatur 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 ipſius <var>.f.n.</var> ad <var>.a.b.</var> ut <var>.y.ω.</var> ad <var>.x.o.</var> & ſic <var>.f.n.</var> ad <var>.d.a.</var> <lb/>vt <var>.y.ω.</var> ad <var>.x.r.</var> ſed <var>.f.m.</var> ad <var>f.n.</var> eſt 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 ijſdem 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 conſideranda 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 propoſitionibus illius lib. nullo pacto poteris dubitare: </s> + <s xml:space="preserve">Verum ne <lb/>in .4. aliquid tibi noui exurgat, te ſcire volo <ref id="ref-0025">corollarium .20. in libr. de quadratu<lb/>ra parabolę</ref> docere poſſibile eſſe inſcriptionem rectilineæ, ea tamen conditione <reg norm="quam" type="context">quã</reg> <lb/>dicit Archimedes.</s> + </p> + <p> + <s xml:space="preserve">In quinta poſtea animaduertendum eſt, quod prima pars, probat tantummodo de <lb/>centro trianguli, et .2. pars probat de centro pentagoni, à te ipſo deinde potes pro-<lb/>bare de centro nonanguli: </s> + <s xml:space="preserve">& ſic de cæteris: </s> + <s xml:space="preserve">eo quod cum probatum fuerit de centro <lb/>figuræ in medio locatæ ſi conſtitutæ poſtea fuerint ſimiles figuræ in portionibus la-<lb/>teralibus habebitur propoſitum in infinitum.</s> + </p> + <p> + <s xml:space="preserve">Idem intelligendum eſt in .3. propoſitione quamuis exemplum vlterius non ex-<lb/>tendatur quam ad pentagonos.</s> + </p> + <p> + <s xml:space="preserve">Sexta verò <reg norm="propoſitio" type="simple">ꝓpoſitio</reg> tibi ſacilis erit, quæ nihilominus <reg norm="pont" type="context">põt</reg> <reg norm="demonſtrari" type="context">demõſtrari</reg> hoc <reg norm="mon" type="context">mõ</reg> ſcili<lb/>cet. </s> + <s xml:space="preserve">Sint .4. <reg norm="quantitates" type="context">quãtitates</reg> <var>.a.b.c.d.</var> ipſius Archimedis <reg norm="ſupponendo" type="context">ſupponẽdo</reg> <var>.a.</var> pro figura rectilinea <lb/>inſcripta in parabola, et <var>.b.</var> pro reſiduo ipſius parabolę et <var>.c.</var> pro triangulo <var>.a.b.c.</var> in me <lb/>dio ipſius parabolę et <var>.d.</var> pro triangulo <var>.r</var>. </s> + <s xml:space="preserve">Nunc cum <var>.a.</var> maior ſit <var>.c.</var> prout totum ma-<lb/>ius eſt ſua 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 ſit <var>.d.</var> ex ſuppoſito, 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 cuiuſuis figuræ plenæ neceſſariò ſit intra <lb/>ipſam figuram, idcirco centrum reſidui ipſius parabolę intra ipſam reperietur. </s> + <s xml:space="preserve">quod <lb/>ita <reg norm="clarum" type="context">clarũ</reg> <reg norm="per" type="simple">ꝑ</reg> ſe eſt, <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> quoduis aliud axioma, & quia <reg norm="dictum" type="context">dictũ</reg> <reg norm="centrum" type="context">centrũ</reg> ex .8. primi <lb/>de centris, neceſſariò eſt 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 eſt <lb/>quam <var>.b.h.</var> ad <var>.f.</var> ex .12. quinti. </s> + <s xml:space="preserve">Sed <reg norm="cum" type="context">cũ</reg> <var>.h.b.</var> maior ſit ipſa <var>.h.g.</var> prout omne totum ma-<lb/>ius eſt ſua 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 ipſa <var>.f.</var> ex .10. <reg norm="quinti" type="context">quĩti</reg>.</s> + </p> + <p> + <s xml:space="preserve">Septima verò et .8. propoſitio 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 ſcribat ſuper duas vltimas lib. ſecundi de centris <reg norm="grauium" type="context">grauiũ</reg>, nihil <lb/>miror ipſum tibi non ſatisfacere. </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">primũ</reg> ſupponit 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>: ſupponit etiam quod proportio quæ eſt ipſius <var>.<lb/>e.b.</var> ad <var>.e.a.</var> <reg norm="eadem" type="context">eadẽ</reg> ſit quæ ipſius <var>.f.g.</var> ad tres quintas ipſius <var>.a.d.</var> & quod proportio com <lb/>poſiti dupli ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.c.</var> cum ſexcuplo ipſius <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"/> + ipſius <var>.b.e.</var> ad compoſitum quintupli ipſius <var>.a.b.</var> cum decuplo ipſius <var>.c.b.</var> cum decuplo <lb/>ipſius <var>.b.d.</var> cum quintuplo ipſius <var>.b.e.</var> eadem ſit quæ ipſius <var>.g.h.</var> ad <var>.a.d.</var> & vult proba-<lb/>re <var>.f.h.</var> eſſe duas quintas ipſius <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 ipſius <var>.a.c.</var> ad <var>.c.d.</var> & ipſius <var>.c.d.</var> ad <var>.d.e.</var> eſſe vt ipſius <lb/><var>a.b.</var> a<unsure/>d.b.c. & cętera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheſi ſit <lb/>ipſius, <var>a.b.</var> totalis ad <var>.c.b.</var> totalom vt ipſius <var>.c.b.</var> partialis (ſumptæ vt pars abſciſa ab <var>.a.<lb/>b.</var> pro nunc) ad <var>.d.b.</var> partialem (abſciſam ab <var>.c.b.</var>) erit ex .19. dicta ipſius <var>.a.c.</var> (reſidui <lb/>ex <var>.a.b.</var>) ad <var>.c.d.</var> (reſiduum ex <var>.c.b.</var>) vt ipſius <var>.a.b.</var> ad <var>.c.b.</var> & ita probabitur de pro-<lb/>portione ipſiuas <var>.c.d.</var> ad <var>.d.e.</var> eadem ratione.</s> + </p> + <p> + <s xml:space="preserve">Cum verò ex .18. quinti ſit ipſius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.c.b.</var> vt ipſius <var>.a.d.</var> ad <var>.d.e.</var> ergo ex <lb/>22. eiuſdem, ita erit ipſius <var>.a.b.</var> cum <var>.c.b.</var> ad <var>.d.b.</var> vt <var>.a.d.</var> ad <var>.d.e.</var> & exijſdem rationibus <lb/>eadem proportio erit ipſius <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 ipſius <var>.a.d.</var> <lb/>ad <var>.d.e.</var> quę dupli primi antecedentis cum ſimplo ſecundi antecedentis ad duplum <lb/>primi conſequentis cum ſimplo ſecundi conſequentis, hoc eſt dupli ipſius <var>.a.b.c.</var> <reg norm="cum" type="context">cũ</reg> <lb/>ſimplo <var>.c.b.d.</var> ad duplum ipſius <var>.d.b.</var> cum ſimplo <var>.e.b.</var> hoc eft dupli <var>.a.b.</var> cum triplo ip-<lb/>ſius <var>.b.c.</var> cum ſimplo <var>.d.b.</var> ad duplum ipſius <var>.d.b.</var> cum ſimplo <var>.e.b</var>. </s> + <s xml:space="preserve">Nunc duplum <var>.a.b.</var> <lb/>cum triplo <var>.b.c.</var> cum ſimplo <var>.b.d.</var> ſignatum ſit charactere <var>.D.</var> ſuum verò conſequens, <lb/>hoc eſt duplum <var>.d.b.</var> <reg norm="cum" type="context">cũ</reg> ſimplo <var>.e.b.</var> ſignificetur à charactere <var>.B.</var> hinc proportio ipſius <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, ſi quis ſumeret aliquod maius antecedens æquale ſci-<lb/>licet duplo ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.e.</var> cum quadruplo ipſius <var>.b.d.</var> cum du-<lb/>plo ipſius <var>.b.e.</var> <reg norm="compararetque" type="simple">compararetq́;</reg> illud cum <reg norm="conſequente" 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 eſt maiorem <lb/>quàm ipſius <var>.a.d.</var> ad <var>.d.e.</var> ex .12. quinti.</s> + </p> + <p> + <s xml:space="preserve">Nunc ſi ſumpta 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 ſecunda parte decimę quinti quod <var>.d.o.</var> minor erit ipſa <var>.d.e.</var> <lb/>Corrige igitur impreſſionem Baſileę locando characterem <var>.o.</var> inter <var>.d.</var> et <var>.e.</var> eo quod <lb/>ibi poſitum non fuit.</s> + </p> + <p> + <s xml:space="preserve">Volo nunc quod dictum maius antecedens æquale ſcilicet duplo ipſius <var>.a.b.</var> cum <lb/>quadruplo ipſius <var>.b.c.</var> cum quadruplo ipſius <var>.b.d.</var> cum duplo ipſius <var>.b.c.</var> ſignificetur à <lb/>charactere <var>.A</var>. </s> + <s xml:space="preserve">Hinc habebimus proportionem ipſius <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 poſtea habebimus <var>.A.B.</var> ad <var>.B.</var> vt <var>.a.o.</var> ad <var>.d.o.</var> & proportionalitate <lb/>euerſa 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"/> + ſe continet id quod Archimedes ſcribit, hoc eſt duplum ipſius <var>.a.b.</var> <reg norm="quadruplum" type="context">quadruplũ</reg> ipſius <lb/><var>b.c.</var> ſexcuplum ipſius <var>.b.d.</var> & triplum ipſius <var>.b.e</var>. </s> + <s xml:space="preserve">Conſequens verò <var>.A.</var> continet du <lb/>plum ipſius <var>.a.b.</var> quadruplum ipſius <var>.b.c.</var> quadruplum ipſius <var>.b.d.</var> & duplum ipſius <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 ſuppoſito deinde ipſius Archimedis & ex conuerſa proportionalitate in .19. <lb/>dicta, verum eſt id quod dicit Archimedes, videlicet quod eadem proportio eſt <lb/>ipſius <var>.a.d.</var> ad <var>.g.h.</var> quod quintupli ipſius <var>.a.b.</var> cum quintuplo ipſius <var>.b.e.</var> cum decuplo <lb/>ipſius <var>.b.c.</var> cum decuplo ipſius <var>.b.d.</var> (quod quidem antecedens ſignificetur per <var>.V.</var>) <lb/>ad duplum ipſius <var>.a.b.</var> cum quadruplo ipſius <var>.b.c.</var> cum ſexcuplo ipſius <var>.b.d.</var> cum triplo <lb/>ipſius <var>.b.e.</var> hoc eſt ad <var>.A.B</var>.</s> + </p> + <p> + <s xml:space="preserve">Erit igitur <var>.V.</var> ad <var>.A.B.</var> vt ipſius <var>.a.d.</var> ad <var>.g.h.</var> ſed ſuperius vbi ſignatum eſt <var>.T.</var> iam <lb/>probatum fuit ita eſſe <var>.A.B.</var> ad <var>.A.</var> vt ipſius <var>.a.o.</var> ad <var>.a.d</var>. </s> + <s xml:space="preserve">Ergò ex .23. quinti Archime <lb/>des verum ſcribit, hoc eſt quod ita erit ipſius <var>.V.</var> ad <var>.A.</var> vt ipſius <var>.a.o.</var> ad <var>.g.h</var>.</s> + </p> + <p> + <s xml:space="preserve">Clarum per ſe etiam eſt, id quod Archimed. dicit hoc eſt quod <var>.V.</var> ad <var>.A.</var> eſt 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 compoſito <var>.V.</var> ad quodlibet in-<lb/>gredientium in compoſito <var>.A.</var> ſit 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="eiuſdem" type="context">eiuſdẽ</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 ſcriptum eſt, rurſus quoniam <var>.o.a.</var> quia oportet dicere <lb/>Rurſus quoniam <var>.o.d</var>.</s> + </p> + <p> + <s xml:space="preserve">Archimedes igitur verum dicit, quod ipſius <var>.o.d.</var> ad <var>.d.a.</var> eſt vt ipſius <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"/> + conuerſa 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 ſpeculationis probatum iam fuit ita eſſe ipſius <var>.d.a.</var> ad <var>.d.e.</var> <lb/>vt ipſius <var>.D.</var> ad <var>.B.</var> vbi notatum eſt <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> ſe habeat ut <var>.D.</var> ad <var>.A.</var> erit ex conuerſa proportionalitate iam <lb/>dicta <var>.d.e.</var> ad <var>.d.o.</var> vt <var>.A.</var> ad <var>.D.</var> per euerſam vero erit <var>.d.e.</var> ad <var>.a.o.</var> vt <var>.A.</var> ad ſuum reſi-<lb/> + <anchor type="note" xlink:label="note-0400-01a" xlink:href="note-0400-01"/> + duum. </s> + <s xml:space="preserve">quod reſiduum componitur ex ſimplo <var>.b.c.</var> cum triplo <var>.b.</var> cum duplo <var>.b.o.</var> quod <lb/>à te ipſo videre poteris detrahendo numeros ipſarum quantitatum quæ in <var>.D.</var> <lb/>reperiuntur, ex numeris earundem, quæ in <var>.A.</var> quod quidem reſiduum ſigniſicetur <lb/>à charactere <var>.E</var>. </s> + <s xml:space="preserve">Vnde ex conuerſa proportionalitate verum dicit Archime. hoc eſt <lb/>quod ita ſe 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 ſit <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 ſuppoſito, ideo ex <lb/>17. quinti verum dicit Archim. </s> + <s xml:space="preserve">hoc eſt quod ita erit ipſius <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. eiuſdem eadem proportio erit tripli ipſius <var>.c.d.</var> ad triplum <lb/>ipſius <var>.d.b.</var> quæ dupli ipſius <var>.d.e.</var> ad. duplum ipſius <var>.e.b.</var> vt inquit Archi.</s> + </p> + <p> + <s xml:space="preserve">Ex qua .13. compoſitum ex <var>.a.c.</var> cum triplo ipſius <var>.c.d.</var> cum duplo ipſius <var>.d.e.</var> ean-<lb/>dem proportionem habebit ad <reg norm="compoſitum" type="context">compoſitũ</reg> ipſius <var>.c.b.</var> cum triplo <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.d.b.</var> cum duplo <lb/>ipſius <var>.e.b.</var> quam ipſius <var>.d.e.</var> ad <var>.e.b</var>. </s> + <s xml:space="preserve">Sed horum compoſitorum primum ſignificetur <lb/>per <var>.H.</var> ſecundum verò ſignificatum fuit per <var>.E.</var> vnde <var>.H.</var> ad <var>.E.</var> ſe habebit vt <var>.d.e.</var> ad <lb/><var>e.b.</var> ſed <var>.E.</var> ad <var>.A.</var> iam dictum eſt eſſe vt <var>.o.e.</var> ad <var>.d.e.</var> vbi ſignatum eſt <var>.λ</var>. </s> + <s xml:space="preserve">quare ex .23. <lb/>quinti eadem proportio erit ipſius <var>.o.e.</var> ad <var>.e.b.</var> quæ <var>.H.</var> ad <var>.A.</var> vt ipſe inquit.</s> + </p> + <note position="left" xml:space="preserve">X</note> + <!-- note-0400-02 --> + <p> + <s xml:space="preserve">Ex .18. poſtea eiuſdem 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 eſt quod ſi collectæ fuerint omnes partes compoſiti <var>.H.A.</var> hoc <lb/>eſt duplum <var>.a.b.</var> cum duplo <var>.b.e.</var> cum quadruplo <var>.b.c.</var> cum quadruplo <var>.b.d.</var> cum ſimplo <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> & ſexcuplum <lb/><var>b.c.</var> vt ipſe dixit. </s> + <s xml:space="preserve">Quod autem hoc verum ſit, cum diſtinctæ fuerint omnes partes, <lb/>vt in ſubſcriptis his lineis videre eſt, videbis quod ſi ex <var>.H.</var> detracta fuerit ſimplex <var>.a.<lb/>c.</var> quæ quidem poſtea iuncta vni ex partibus quadrupli <var>.b.c.</var> ipſius <var>.A.</var> reſultabit nobis <lb/>vna inte gra <var>.a.b</var>. </s> + <s xml:space="preserve">Vnde habebimus triplum ipſius <var>.a.b.</var> & in <var>.A.</var> remanebit triplum ip <lb/>ſius <var>.c.b</var>. </s> + <s xml:space="preserve">Deinde ſi ex <var>.H.</var> auferatur triplum ipſius <var>.c.d.</var> & ipſum addatur tribus parti-<lb/>bus quadrupli <var>.b.d.</var> ipſius <var>.A.</var> habebimus tres vices <var>.b.c.</var> quæ ſi iungantur tribus, quæ <lb/>remanebant in <var>.A.</var> vt dixi, habebimus ſexcuplum ipſius <var>.b.c.</var> & in <var>.A.</var> remanebit ſim <lb/>plum <var>.b.d.</var> cum duplo ipſius <var>.b.e</var>. </s> + <s xml:space="preserve">Vnde ſi ex <var>.H.</var> demptum fuerit duplum ipſius <var>.d.<lb/>e.</var> quod quidem iungatur cum duplo ipſius <var>.b.e.</var> habebimus duplum ipſius <var>.b.d.</var> quod <lb/>coniunctum cum ſimplo <var>.b.d.</var> quod in <var>.A.</var> relictum fuerat, habebimus triplum ipſius <lb/><var>d.b</var>. </s> + <s xml:space="preserve">Verum igitur eſt quod inquit Archimedes, hoc eſt, quod <var>.H.A.</var> eſt triplum ip-<lb/>ſius <var>.a.b.</var> ſexcuplum ipſius <var>.b.c.</var> & triplum ipſius <var>.b.d</var>.</s> + </p> + <p> + <s xml:space="preserve">Verum etiam dicit ex eo (vt ſupra probatum eſt) quod <var>.a.c</var>: <var>c.d</var>: et <var>.d.e.</var> ſe <reg norm="habebant" type="context">habebãt</reg> <lb/>in continua proportionalitate, </s> + <s xml:space="preserve">quare ex conuerſa proportionalitate erunt ſibi 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> ſint continuæ proportionales in ea proportione <lb/>in qua ſunt <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> & ſic etiam <var>.c.b.</var> ad <var>.e.b</var>. </s> + <s xml:space="preserve">Vnde ex .24. eiuſdem <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/>ſumpto, & cum <var>.b.d.</var> ad <var>.e.b</var>. </s> + <s xml:space="preserve">Quare ex conuerſa proportionalitate, vt ſe habet <var>.e.d.</var> <lb/>ad <var>.d.a.</var> ita ſe habebit <var>.e.b.</var> <reg norm="cum" type="context">cũ</reg> <var>.d.b.</var> ad <var>d.b.</var> <reg norm="cum" type="context">cũ</reg> <var>.b.c.</var> duplicato & <reg norm="cum" type="context">cũ</reg> <var>.b.a.</var> vt inquit Archi <lb/>medes. </s> + <s xml:space="preserve">Nunc antecedens vocetur <var>.M.</var> hoc eſt <var>.b.e.</var> cum <var>.d.b.</var> conſequens 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"/> + eſt <var>.d.b.</var> cum duplo <var>.b.c.</var> cum ſimplo <var>.b.a.</var> vocetur <var>.N</var>.</s> + </p> + <p> + <s xml:space="preserve">Animaduertendum tamen eſt quod impreſſio mendoſa eſt 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 eſt 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 ſe habet <var>.a.e.</var> ad <var>.d.a.</var> ita ſe 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 ſcriptum eſt <lb/><q open="„" close="">ad vtrunque ſimul <var>.b.d</var>: <var>d.a.</var> cum dupla <var>.b.c.</var></q> <lb/>dicendum eſt ita, <lb/>ad vtranque ſimul <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 ſicut ſe haber <var>.e.a.</var> ad <var>.d.a.</var> ita ſe habebit duplum <var>.M.N.</var> <lb/>ad duplum <var>.N</var>. </s> + <s xml:space="preserve">Quod quidem verum eſt ex .13. quinti, huiuſmodi verò antecedons <lb/>& conſequens, Archi. manifeſtat ex ſuis partibus, ſumendo 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/>quę ſimul 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 quàm du-<lb/>plum ipſius <var>.M.N</var>.</s> + </p> + <p> + <s xml:space="preserve">Verum etiam ſcribit, vbi dicit, quod proportio <var>.e.a.</var> ad tres quintas ipſius <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 ſuppoſito ita ſe habeat <var>.f.<lb/>g.</var> ad tres quintas ipſius <var>.a.d.</var> quemadmodum <var>.b.e.</var> ad <var>.e.a.</var> erit ex .16. quinti verum <reg norm="quod" type="simple">ꝙ</reg> <lb/>dicit Archimed. </s> + <s xml:space="preserve">hoc eſt, ita ſe habere <var>.b.e.</var> ad <var>.f.g.</var> vt <var>.e.a.</var> ad tres quintas ipſius <var>.a.d</var>.</s> + </p> + <p> + <s xml:space="preserve">Et per .11. eiuſdem verum etiam erit quod ſicut ſe habet <var>.e.b.</var> ad <var>.f.g.</var> ita ſe habe-<lb/>bit <var>.A.</var> ad tres quintas dupli <var>.N.</var> quod quidem duplum <var>.N.</var> ſignificetur per <var>.Q</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed ſuperius iam demonſtratum fuit (vbi <var>.X.</var>) quod <var>.o.b.</var> ad <var>.b.e.</var> ita ſe habebat vt <lb/><var>H.A.</var> ad <var>.A.</var> & <reg norm="nunc" type="context">nũc</reg> demum probatum fuit ita eſſe <var>.A.</var> ad tres quintas ipſius <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 ipſius <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 ſuis partibus videre eſt) ita ſe habet vt tres ad duo ex .13. <lb/>quinti, vt inquit Archimedes.</s> + </p> + <p> + <s xml:space="preserve">Ipſe etiam dicit proportionem <var>.H.A.</var> ad tres quintas ipſius <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> diuiſa 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> ipſius <var>.Q.</var> <lb/><reg norm="æqualem" type="context">æqualẽ</reg> eſſe duabus tertijs vniuſcuiuſque quintæ partis <var>.H.A.</var> vnde tres quintæ ipſius <lb/>Q. erunt, ex communi conceptu, ſex tertiæ vnius quintæ ipſius <var>.H.A.</var> hoc eſt duæ <lb/>quintæ. ipſius <var>.H.A</var>. </s> + <s xml:space="preserve">Quare <var>.o.b.</var> ita ſe 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 ſe habere vt <var>.H.A.</var> ad tres quintas ipſius <lb/>Q. (vbi <var>.Y.</var>) ſed iam probatum fuit (vbi. ω) quod <var>.o.a.</var> ad <var>.h.g.</var> erat etiam vt <lb/>quinque ad duo, hoc eſt quod <var>.f.h.</var> erit duæ quintę ipſius <var>.a.b</var>. </s> + <s xml:space="preserve">Quod eſt propoſitum.</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 propoſitione ſecundi lib. de ponderibus Archi. hoc modo intelli<lb/>gendus eſt, vt ſi diceret, <lb/>Sit paraboles <var>.a.</var> cuius baſis ſit <var>.a.c.</var> <reg norm="ſitque" type="simple">ſitq́;</reg> <var>.d.e.</var> recta parallela dictæ baſi <var>.a.c.</var> <reg norm="diameterque" type="simple">diameterq́;</reg> <lb/><var>b.f</var>. <lb/></s> + <s xml:space="preserve">Inquit deinde quod linea contingens in <var>.b.</var> parallela erit ipſi <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 ſit et <var>.a.c.</var> baſis, 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> diuiſa erit per æqua <lb/>lia à diametro <var>.b.f</var>. </s> + <s xml:space="preserve">Quare verum dicit ex quinta ſecundi ipſius Pergei hoc eſt 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 poſtea quod diuiſa cum fuerit pars diametri quę inter <var>.d.e.</var> et <var>.a.c.</var> poſita eſt <lb/>(hoc eſt <var>.g.f.</var>) per quinque partes æquales, <reg norm="quarum" type="context">quarũ</reg> partium media ſit <var>.h.k.</var> diuiſa etiam <lb/>imaginatione ſit in puncto <var>.i.</var> ita quod proportio ipſius <var>.h.i.</var> ad <var>.i.K.</var> eadem ſit quæ in-<lb/>ter duo ſolida quorum vnum (illud ſcilicet à quo relatio incipit, hoc eſt antecedens) <lb/>pro ſua baſi teneat quadratum ipſius <var>.a.f.</var> cuius etiam ſolidi altitudo compoſita ſit ex <lb/> + <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/> + duplo ipſius <var>.d.g.</var> cum ſimplo <var>.a.f</var>. </s> + <s xml:space="preserve">Aliud verò ſolidum habeat pro ſua baſi quadra-<lb/>tum ipſius <var>.d.g.</var> eius verò altitudo compoſita ſit ex duplo ipſius <var>.a.f.</var> cum ſimplo <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, oſtendet punctum <var>.i.</var> centrum eſſe <lb/>portionis abſciſſę à tota ſectione, quod <reg norm="fruſtum" type="context">fruſtũ</reg> <reg norm="nominatur" type="simple">nominat̃</reg> <reg norm="ſignatum" type="context">ſignatũ</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="ſitque" type="simple">ſitq́;</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 poſt <var>.o.n.</var> <lb/>hoc eſt quod ea proportio quæ eſt ipſius <var>.o.n.</var> ad <var>.n.t.</var> eadem ſit ipſius <var>.x.n.</var> ad <var>.n.o</var>. </s> + <s xml:space="preserve">Hinc <lb/>habebimus .4. lineas in continua proportionalitate ſibi 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> verſus <var>.g.</var> alia linea abſciſſa ſit, cui li-<lb/> + <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/> + neæ, ita proportionata ſit <var>.f.h.</var> vt <var>.t.m.</var> eſt ad <var>.t.n.</var> quæ quidem linea ſignata ſit <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 poſtea quod diameter <var>.b.f.</var> erit fortaſſe a xis vel aliqua reliquarum diame-<lb/>trorum, quod quidem in .46. primi Pergei videre eſt, cum omnes diametri ſint in-<lb/>uicem paralleli ipſi axi.</s> + </p> + <p> + <s xml:space="preserve">Cum poſtea dicit, quod <var>.a.f.</var> et <var>.d.g.</var> ſunt intentæ ductæq́ue, ibi vult id em infer-<lb/>re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipſius Pergei videre li-<lb/>cet, vnde ex .20. eiuſdem proportio <var>.b.f.</var> ad <var>.b.g.</var> erit vt quadrati <var>.a.f.</var> ad quadratum <lb/>ipſius <var>.d.g.</var> vt ipſe dicit.</s> + </p> + <p> + <s xml:space="preserve">Sed ita erit quadrati <var>.m.n.</var> ad qua <reg norm="dratum" type="context">dratũ</reg> <var>.x.n.</var> ex .18. ſexti 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 ipſius <var>.m.n.</var> ad quadratum ipſius <var>.n.x.</var> eandem habebit proportionem, <lb/>quam quadratum ipſius <var>.a.f.</var> ad quadratum ipſius <var>.d.g</var>. </s> + <s xml:space="preserve">Vnde ex .18. & ex communi <lb/><reg norm="ſcientia" type="context">ſciẽtia</reg>, eadem proportio erit ipſius <var>.m.n.</var> ad <var>.n.x.</var> quę ipſius <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 ipſius <var>.m.n.</var> ad cubum ipſius <var>.n.x.</var> erit vt cubi ipſius <var>.a.<lb/>f.</var> ad cubum ipſius <var>.d.g.</var> vt etiam dicit ex communi ſcientia, nec non ex .36. vndecimi.</s> + </p> + <p> + <s xml:space="preserve">Inquit poſtea quod proportio totius ſectionis <var>.a.b.c.</var> ad portionem <var>.d.b.e.</var> eadem <lb/>eſt quæ cubi ipſius <var>.a.f.</var> ad cubum ipſius <var>.d.g.</var> quod verum eſt, vt aliàs tibi monſtraui in <lb/>diuiſione parabolæ ſecundum aliquam propoſitam proportionem.</s> + </p> + <p> + <s xml:space="preserve">Quando autem dicit quod proportio cubi ipſius <var>.m.n.</var> ad cubum ipſius <var>.n.x.</var> eadem <lb/> + <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/> + eſt quæ ipſius <var>.m.n.</var> ad <var>.n.t.</var> verum dicit ex .36. vndecimi. </s> + <s xml:space="preserve">Vnde ex .11. quinti ita ſe <lb/>habebit totalis ſectio <var>.a.b.c.</var> ad portionem <var>.d.b.c.</var> vt <var>.m.n.</var> ad <var>.n.t.</var> & ex .17. eiuſdem ita <lb/>erit ipſius <var>.m.t.</var> ad <var>.t.n.</var> vt fruſti <var>.a.d.e.c.</var> ad ſectionem <var>.d.b.e.</var> quemadmodum ipſe di-<lb/>cit. </s> + <s xml:space="preserve">Sed quia ſuperius, vbi <var>.A.</var> ipſa <var>.f.h.</var> (quæ eſt tres quintæ ipſius <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 ipſius fruſti <var>.a.e.</var> ad ſectionem <var>.d.b.<lb/>e.</var> vt tres quintę ipſius <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 ſupradicti, quod pro ſua baſi habeat <lb/>quadratum ipſius <var>.a.f.</var> altitudinem verò compoſitam ex duplo ipſius <var>.d.g.</var> cum ſimplo <lb/><var>a.f.</var> ad cubum ipſius <var>.a.f.</var> eadem erit quæ dupli ipſius <var>.d.g.</var> cum ſimplo <var>.a.f.</var> ad <var>.a.f</var>. </s> + <s xml:space="preserve">Quod <lb/>quidem verum eſt ex .33. vndecimi & ex prima ſexti.</s> + </p> + <p> + <s xml:space="preserve">Sed ſuperius (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 conuerſa pro portion alitate ita erit ipſius <var>.x.n.</var> ad <var>.n.<lb/>m.</var> vt ipſius <var>.d.g.</var> ad <var>.a.f.</var> ſed dupli <var>.x.n.</var> ad ſimplum <var>.x.n.</var> eſt 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. eiuſdem ita erit <lb/>dupli <var>.x.n.</var> cum ſimplo <var>.m.n.</var> ad <var>.m.n.</var> vt dupli <var>.d.g.</var> cum ſimplo <var>.a.f.</var> ad <var>.a.f</var>. </s> + <s xml:space="preserve">Quare ſolidi <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 inſius <var>.a.f.</var> ex .11. quinti erit vt dupli <var>.x.n.</var> <reg norm="cum" type="context">cũ</reg> ſimplo <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. demonſtratum fuit ita eſſe ipſius <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 poſtea quod eadem proportio erit inter cubum <var>.d.g.</var> & corpus illud quod <lb/>pro baſi habeat quadratum inſius <var>.d.g.</var> altitudinem verò vt dictum eſt, quæ eſt inter <lb/><var>d.g.</var> & compoſitum ex duplo <var>.a.f.</var> cum ſimplo <var>.d.g.</var> quod compoſitum eſt altitudo di <lb/>cta, & <reg norm="verum" type="context">verũ</reg> dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius <var>.a.f</var>. <lb/></s> + <s xml:space="preserve">Quare etiam quemadmodum <var>.t.n.</var> ſe habet ad duplum ipſius <var>.o.n.</var> cum ſimplo <var>.t.n.</var> <lb/>ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de <var>.x.n.</var> cum <var>.m.n</var>.</s> + </p> + <p> + <s xml:space="preserve">Diſponantur <reg norm="nunc" type="context">nũc</reg> omnia tali ordine, ita vt <var>.u.</var> primum ſit corpus quod pro ſua ba <lb/>ſi habeat quadratum ipſius <var>.a.f.</var> & c.</s> + </p> + <p> + <s xml:space="preserve">Et <var>.y.</var> ſit cubus ipſius <var>.a.f.</var> et <var>.s.</var> ſit cubus ipſius <var>.d.g.</var> et <var>.z.</var> ſit corpus quod baſim ha-<lb/>bet quadratum ipſius <var>.d.g.</var> altitudinem verò vt ſupradictum eſt, et <var>.p.</var> ſit compoſitum <lb/>dupli <var>.n.x.</var> cum ſimplo <var>.m.n.</var> et <var>.l.</var> ſit compoſitum dupli ipſius <var>.n.o.</var> cum ſimplo <var>.t.n.</var> <lb/>Sed <var>.u.</var> locata ſit 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 ipſius <var>.u.</var> ad <var>.y.</var> vt <var>.y.</var> ad <var>.m.n.</var> & ipſius <var>.y.</var> ad <var>.s.</var> vt <var>.m.n.</var> ad <var>.<lb/>n.t.</var> quod ſuperius iam demonſtratum fuit, vbi, δ. et <var>.s.</var> ad <var>.z.</var> ita ſe 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 ſe 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 ſe 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 ſe <lb/>habet <var>.h.i.</var> ad <var>.i.K.</var> ita ſe habebit <var>.p.</var> ad <var>.l.</var> vt ipſe 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> ſe habebit ad <var>.h.K.</var> vt quintuplum <lb/>ipſius <var>.p.l.</var> ad <var>.p.l.</var> & ex .22. eiuſdem ita ſe habebit <var>.f.g.</var> ad <var>.i.k.</var> vt quintuplum ipſius <var>.p.<lb/>l.</var> ad <var>.l.</var> quintuplum autem ipſius <var>.p.l.</var> compoſitum eſt ex quintuplo ipſius <var>.n.m.</var> cum <lb/>decuplo ipſius <var>.n.x.</var> cum quintuplo ipſius <var>.n.t.</var> cum decuplo ipſius <var>.n.o.</var> vt à te facilè <lb/>computare potes.</s> + </p> + <p> + <s xml:space="preserve">Verum etiam erit ex communi ſcientia quod <var>.g.f.</var> ad <var>.f.k.</var> eſt ut quintuplum ipſius <lb/><var>p.l.</var> ad duplum ipſius <var>.p.l.</var> eo quod ſuperius ſuppoſitum 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/>poſitum eſt ex duplo ipſius <var>.m.n.</var> cum duplo ipſius <var>.n.t.</var> cum quadruplo ipſius <var>.n.x.</var> & <lb/>cum quadruplo ipſius <var>.x.o</var>.</s> + </p> + <p> + <s xml:space="preserve">Ex conuerſa proportionalitate deinde ita ſe habet, <var>i.K.</var> ad <var>.i.k.</var> ad <var>.f.g.</var> vt <var>.l.</var> ad quin-<lb/>tuplum ipſius <var>.p.l.</var> et <var>.k.f.</var> ad <var>.f.g.</var> vt duplum ipſius <var>.p.l.</var> ad quintuplum ipſius <var>.p.l</var>. </s> + <s xml:space="preserve">Vnde <lb/>ex .24. quinti <var>.i.f.</var> ſe habebit ad <var>.f.g.</var> vt <reg norm="duplum" type="context">duplũ</reg> ipſius <var>.p.l.</var> cum ſimplo <var>.l.</var> ad quintuplum <lb/>ipſius <var>.p.l</var>. </s> + <s xml:space="preserve">Deinde ex conuerſa proportionalitate quintuplum ipſius <var>.p.l.</var> ſe habebit <lb/> + <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/> + ad duplum ipſius <var>.p.l.</var> cum ſimplo <var>.l.</var> vt <var>.f.g.</var> ad <var>.f.i</var>. </s> + <s xml:space="preserve">Sed compoſitum dupli ipſius <var>.p.l.</var> <lb/>cum ſimplo <var>.l.</var> æquale eſt duplo ipſius <var>.m.n.</var> cum quadruplo ipſius <var>.x.n.</var> cum ſexcuplo <lb/>ipſius <var>.o.n.</var> cum triplo ipſius <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 ſumpta fuit <var>.i.r.</var> ad quam ita ſe haberet <var>.f.h.</var> hoc eſt tres quintæ ip-<lb/>ſius <var>.f.g.</var> vt <var>.m.t.</var> ad <var>.t.n</var>. </s> + <s xml:space="preserve">Quare ex conuerſa proportionalitate ita ſe habebit <var>.i.r.</var> ad tres <lb/>quintas ipſius <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> ſumpta fuit æqualis ipſi <var>.b.g.</var> et <var>.m.n.</var> ipſi <lb/><var>b.f.</var> ideo <var>.m.o.</var> ex communi ſcientia æ qualis erit ipſi <var>.g.f</var>. </s> + <s xml:space="preserve">Vnde proportio <var>.r.i.</var> ad tres <lb/>quintas ipſius <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 ſe habere <var>.i.f.</var> ad <var>.f.g.</var> vt duplum <reg norm="ipſius" type="simple">ipſiꝰ</reg> <var>.p.l.</var> cum ſim-<lb/>plo <var>.l.</var> ſe habet ad quintuplum ipſius <var>.p.l.</var> hoc eſt <var>.i.f.</var> ad <var>.m.o.</var> vt duplum ipſius <var>.p.l.</var> cum <lb/>ſimplo <var>.l.</var> ad quintuplum ipſius <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/>propoſitione ſupponit. </s> + <s xml:space="preserve">Vnde ex rationibus ibi allegatis ſequitur <var>.f.r.</var> eſſe duas quin-<lb/>tas ipſius <var>.m.n.</var> hoc eſt ipſius <var>.f.b</var>. </s> + <s xml:space="preserve">Quapropter punctum <var>.r.</var> centrum erit ponderis to-<lb/>tius ſectionis parabolæ ex .8. ſecundi lib. de ponderibus eiuſdem Archimedis.</s> + </p> + <p> + <s xml:space="preserve">Inquit nunc Archimedes, quod exiſtente <var>.q.</var> centro ponderis ipſius parabolæ <var>.d.<lb/>b.e.</var> partialis, centrum fruſti 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 ipſius <var>.r.f.</var> quæ reperitur inter centrum <var>.r.</var> & centrum <lb/>huius fruſti æqualis eſt proportioni totius parabolæ ad partialem. </s> + <s xml:space="preserve">Quod quidem ve <lb/>rum eſt ex .8. primi libri eiuſdem.</s> + </p> + <p> + <s xml:space="preserve">Inquit etiam punctum <var>.i.</var> illud eſſe, eo quod cum probatum ſit <var>.f.r.</var> duas quintas eſ-<lb/>ſe ipſius <var>.f.b.</var> ideo <var>.b.r.</var> tres quintas erit ipſius <var>.b.f.</var> vt ipſe 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> ſimiliter tres quintæ eſt ipſius <var>.d.b.</var> ex .8. prædicta. </s> + <s xml:space="preserve">Quare <var>.q.r.</var> tres quintæ <lb/>erit ipſius <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 ſe habeat vt abſciſ-<lb/>ſum <var>.b.g.</var> ad abſciſſum <var>.q.b.</var> ex .7. et .8. dicti primi libri eiuſdem ideo reſiduum <var>.f.g.</var> ex <lb/><var>f.b.</var> ad reſiduum <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 ſub. β. probauimus ita ſe habere fruſtum <var>.a.d.e.c.</var> ad parabolam <var>.d.b.e.</var> vt <lb/><var>m.t.</var> ad <var>.t.n.</var> ſed vt <var>.m.t.</var> ad <var>.t.n.</var> ita aſſ umpta fuit (vbi <var>.A.</var>). <var>i.r.</var> ad quam ſic ſe haberet <var>.f.<lb/>h.</var> hoc eſt tres quintæ ipſius <var>.f.g.</var> hoc eſt <var>.q.r</var>. </s> + <s xml:space="preserve">quare ex .11. quinti prop ortio fruſti <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">Exiſtente igitur <var>.r.</var> centro totius pa <lb/>rabolæ et <var>.q.</var> centro partialis, ergo <var>.i.</var> centrum erit fruſti propoſiti.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi nullo ſolido intercedente, voluerimus centrum <var>.i.</var> fruſti <var>.a.e.</var> citius inuenire, <lb/>inueniemus primò centrum <var>.r.</var> totius figuræ ex .8. ſecundi eiuſdem conſtituendo <var>.b.r.</var> <lb/>tres quintas totius axis <var>.b.f.</var> & centrum <var>.q.</var> parabolæ <var>.d.b.e.</var> partialis ſimiliter.</s> + </p> + <p> + <s xml:space="preserve">Nunc igitur manifeſtum eſt nobis, eandem proportionem fore ipſius <var>.q.r.</var> <lb/>ad <var>.r.i.</var> quæ fruſti <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 ſe habebit <var>.q.i.</var> ad <var>.i.r.</var> vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ſed vt <var>.a.b.c.</var> ad <var>.d.b.e.</var> ita ſe <lb/>habet <var>.m.n.</var> ad <var>.n.t.</var> eo quod vnaquæque harum duarum proportionum ſeſquialtera <lb/>eſt proportioni <var>.f.b.</var> ad <var>.b.g.</var> eo. quod <var>.f.b.</var> ad <var>.b.g.</var> ita ſe 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 ſe habebit vt <var>.g.i.</var> ad <var>.r.i.</var> vnde diſiunctim <var>.m.t.</var> ad <var>.t.n.</var> ita ſe 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 ſe habeat ad <var>.r.q.</var> vt <var>.t.n.</var> ad <var>.t.m.</var> vt <lb/>habeatur centrum fruſti.</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">Illuſtri Domino Horatio Muto.</head> + <p> + <s xml:space="preserve"><emph style="sc">INter</emph> ea quæ olim contra Antonium Bergam, ſermone Italico ſcripſi, <lb/>hoc vnum erat, quod ip ſe Berga non viderat quendam notatu dignum <lb/>errorem ipſius Pi ccolhominei, vbi ipſe 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 ipſum putare eandem proportionem inter diametros, <lb/>quæ inter ſphæras ipſas eſſe, nec amplius recordari eius quod ſcripſerat pag .24.</s> + </p> + <p> + <s xml:space="preserve">Piccolhom. igitur ibi <reg norm="ſupponens" type="context">ſupponẽs</reg> centrum <var>.D.</var> eſſe magnitudinis aquæ, & intra ſphæ <lb/>ram terreſtrem, putat omnino cauſam eſſe vt terra ſuperet aquam magnitudine, qua-<lb/>ſi quod ſi punctum <var>.D.</var> vt centrum ſphæræ aquæ, vnum <reg norm="idemque" type="simple">idemq́;</reg> eſſet cum puncto <var>.E.</var> <lb/>extremo diametri ipſius terræ, ſphæra <var>.A.G.H.</var> ſphæræ <var>.A.B.E.</var> dupla eſſe deberet, <lb/>quod quidem nullo pacto fieri poteſt, quamuis etiam proportio <var>.A.H.</var> ad <reg norm="diametrum" type="context">diametrũ</reg> <lb/><var>A.E.</var> ſuperbipartiensſeptimas exiſteret, quæ minor eſſet quam ſeſquitertia, ita quod <lb/>quando etiam <var>.D.E.</var> maior medietate ipſius <var>.D.H.</var> fuiſſet, nihilominus tamen <lb/>terra minor eſſet aqua, eo quod proportio dupla minor eſt, quam tripla ad <reg norm="propor- tionem" type="context">propor-<lb/>tionẽ</reg> ſuperbipartientenſeptimas, & maior <reg norm="quam" type="context">quã</reg> tripla ad proportionem ſeſquiquar-<lb/>tam. </s> + <s xml:space="preserve">Vnde ſi Piccolhom. ſuppòſuiſſet proportionem ipſius <var>.D.H.</var> ad <var>.C.E.</var> eſſe <lb/>ſeſquiquartam, rectè profectò dixiſſet, ſed dicere quod ubicunque exiſtat <reg norm="punctum" type="context">punctũ</reg> <var>.<lb/>D.</var> intra ſphæram terreſtrem, ſequitur ipſam eſſe maiorem aquea, verum non eſt.</s> + </p> + <p> + <s xml:space="preserve">Scripſi 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 exiſtimare non poteſt, quod vmbra quæ facit ori-<lb/>ri e cclipſes Lunæ, producta ſit à terra, & ab aqua ſimul, vt ab vno corpore aggre-<lb/>gato exijs duobus elementis, & ad vnam communem ſphæreceitatem reductis, pro <lb/>pterea quod cum vmbra produci debeat à corporibus opacis, quorum opacitas effi-<lb/>cit illa corpora vmbroſa, aqua autem, ſit corpus diaphanum, & tranſparens, nullam <lb/>vmbram poterit à ſe eminus producere.</s> + </quote> + <p> + <s xml:space="preserve">Hic enim decipitur Piccolhom. duabus rationibus, quarum prima eſt, quod ra-<lb/>dius luminoſus non poteſt multum in profundum mergi, vt probaui in .8. epiſtola ad <lb/>Vimercatum, altera verò eſt, quod cum ſphærica ſit aqua maris, ſupponatur etiam <lb/>quod ſub ea nulla terræ portio eſſet, & quod radij ſolares ipſam, non ſecus ac pilam <lb/>ex criſtallo fabręfactam penetrarent, cum autem ipſi radij, tam ab una, quam ab alia <lb/>parte ſup erficiei huiuſmodi globi <reg norm="frangantur" type="context">frãgantur</reg>, ob diſſimilem diaphaneitatem inter ae <lb/>rem & aquam, ipſi ſeinuicem interſecarent, vt poſt pilam criſtallinam videre eſt, de-<lb/>inde procedentes, diſgregarentur, <reg norm="diſciparenturque" type="simple">diſciparenturq́;</reg> quouſque nullam vim illuminatio <lb/>nis haberent, quod quilibet experiri poterit mediante aliquo vaſe uitreo ſphærico, <lb/>aqua pleno, cuiuſuis magnitudinis, ſoli expoſito.</s> + </p> + <p> + <s xml:space="preserve">Rationes etiam quas eodem loco Piccolho. adducit ad probandum quod ſi quis <lb/>in fundo maris exiſteret, nullum uideret lumen, nihil ualent. </s> + <s xml:space="preserve">Quarum prima eſt, <lb/>ubi ita dicit.</s> + </p> + <quote> + <s xml:space="preserve">Ille qui ſe in aquam mergit, cum maiorem lucem, quæ ſupra aquam eſt, relin-<lb/>quat, iudicat pro magno temporis ſpatio locum illum obſcurum, quemadmodum <lb/>accidit quando per multum temporis ſpatium fixis oculis in corpore Solis intuiti ſu <lb/>mus, ab eodem poſtea eoſdem amouentes, omnia obſcura nobis videntur.</s> + </quote> + <p> + <s xml:space="preserve">Ipſe autem non conſiderat quod talis obſcuritas quæ ſequitur viſionem maioris <lb/>luminis, parum durat, immo cito euaneſcit, ſed in aqua nunquam reuertimur ad vi-<lb/>dendum, ne que veſtigium aliquod luminis ibi videtur, in fundo maris dico, quem-<lb/>admodum nobis nuntiauerunt hi qui margaritas expiſcantur in imis partibus ingen <lb/>tium æquorum indicorum.</s> + </p> + <p> + <s xml:space="preserve">Secunda uerò ratio ipſius Piccolhom. eſt ubi ita dicit.</s> + </p> + <quote> + <s xml:space="preserve">Altera cauſa quod nobis obſcurus appareat locus ſub aqua, eſſe poteſt obſtacu-<lb/>lum quod aquæ habent ab opacitate terræ ſub eorum fundo, etenim ſicut <reg norm="chriſtallum" type="context">chriſtallũ</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="perſpicuum" type="context">perſpicuũ</reg> ſiue <reg norm="tranſparens" type="context">tranſparẽs</reg> ſit, nihilominus propter obſtaculum plumbi ſub <lb/>ipſo poſiti, efficit vt radij viſuales repercuſſi reuertantur. ita etiam quamuis aqua ſit <lb/>corpus tranſparens, nihilominus propter obſtaculum terræ opacæ, quæ ſubſidet in <lb/>fundo maris efficere poteſt obſcuras partes illas ſub aqua, illis hominibus qui in <lb/>ipſa aqua mèrguntur.</s> + </quote> + <p> + <s xml:space="preserve">In hac ſecunda ratione decipitur Piccolhom. </s> + <s xml:space="preserve">Primum quia ſi vſque ad imam par <lb/>tem maris, Solis radius ferri poſſet, ille qui ibi eſſet, attollens oculos ſurſum Solem <lb/>cerneret, </s> + <s xml:space="preserve">deinde aſpiciendo ipſum fundum Maris, videret illum, ratione reflexio-<lb/>nis luminis ab ipſo fundo, & ex eadem ratione ſpeculi ab ipſo adducta, quæ contra <lb/>ipſum eſt.</s> + </p> + <p> + <s xml:space="preserve">Decipitur etiam cum dicatradios viſuales à ſpeculo ſeu plumbo repercuti, eo <reg norm="quod" type="simple">ꝙ</reg> <lb/>non radij viſuales ſunt hi qui reflectuntur, ſed ſunt radij luminoſi primarij, ſeu ſecun <lb/>darij qui non ab oculis exeunt ſed à corpore lucido.</s> + </p> + <p> + <s xml:space="preserve">Scripſi etiam quod ſi verum eſſet proportionalitatem continuam <reg norm="quantitatum" type="context">quãtitatum</reg> ele-<lb/>mentorum ex proportione decupla conſtare, ignem pro maximo, terram verò pro <lb/>minimo terminorum ſumentes, totum aggregatum ex terra, aqua, aere, & igne, ita <lb/>eſſet maius terra, quemadmodum mille centum & vndecim ad vnum, vnde ſemidia <lb/>meter regionis elementaris eſſet quaſi aut paulo maior decuplo ſolum ſemidiame-<lb/>tro terræ, vnde inter conuexum ignis, & concauum minimi, ſeu inferioris orbis lu-<lb/>naris, relinqueretur quidam orbis vacuus ſpiſſitudinis vnius interualli plus quam vi-<lb/>ginti terræ ſemidiametrorum, quod ſpatium vacuum orbiculariter, maius exi-<lb/>ſteret ipſa totali regione elementari plus quam trigeſies millies, immo ſi ſemidia <lb/>meter dicti primi orbislunaris maior eſſet terreſtri vt trigintanouem ad unum, <reg norm="dictus" type="simple">dictꝰ</reg> <lb/>orbis vacuus maior eſſet elementari regione plus quam .58208. ad vnum, proportio <lb/>nalitatem igitur continuam quæ ex decupla proportionalitate reſultat in elementis <lb/>eſſe putare eſt 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 ſolidis (quaſi Bergę ſint quædam corpora quæ ſolida non extent) poſſit <lb/>dari medium æquale in geometrica proportione.</s> + </p> + <p> + <s xml:space="preserve">Sed vbi Plato ad ſermonem de numero elementorum ſe confert, poſtquam ra-<lb/>tione creationis ignis, & terrę ſe propoſuiſſe 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 ſuperficialium primò, deinde in quatuor, ratione corporearum <lb/>vtitur, ita dicens.</s> + </p> + <quote> + <s xml:space="preserve">Vinculorum verò ideſt aptiſſimum atque pulcherrimum quod exſe, & ex ijs quę <lb/>aſtringunt, quam maximè vnum efficit, <reg norm="&c." type="unresolved">&c.</reg></s> + </quote> + <p> + <s xml:space="preserve">Quo in loco Plato inſerre vult de proportionalitate geometrica trium termino-<lb/>rum, in qua ijdem ita ſe habent, vt medius, primi, <reg norm="vltimique" type="simple">vltimiq́;</reg> vice fungatur, ita vt vtriuſ-<lb/>que ipſorum extremorum particeps fiat, cum productum quod à medio termino in <lb/>ſeipſo progignitur idem ſit ei quod ab extremis fuit, vnde medius, potentia idem eſt <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 ſe habet <lb/>ad poſtremum vt primum ad medium, <reg norm="viciſſimque" type="simple">viciſſimq́;</reg> vt poſtremum cum medio, ita me-<lb/>dium cum primo congruit, </s> + <s xml:space="preserve">tunc quod medium eſt, & primum fit & <reg norm="poſtremum" type="context">poſtremũ</reg>, po-<lb/>ſtremum 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">animaduertendũ</reg> eſt 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 cenſent, <lb/>quia de numeris, magnitudinibus, <reg norm="viribusque" type="simple">viribusq́;</reg>, aut ut dici ſolet, virtutibus mentionem <lb/>fecerit. </s> + <s xml:space="preserve">Plato enim nihil aliud inferre voluit, quam eandem paſſionem (ut ipſe reci-<lb/>tat) inter medium <reg norm="extremaque" type="simple">extremaq́;</reg> vnius proportionalitatis continuæ geometricæ, tam <lb/>in quantitate, quam in qualitate <reg norm="reſultaturam" type="context">reſultaturã</reg>, cum tres termini eiuſdem eſſent ſpe-<lb/>ciei, & quia quantitas in duas principes <reg norm="primariasque" type="simple">primariasq́;</reg> partes, ideſt in continuam, & <lb/>diſcretam diuiditur, hanc ob cauſam Plato hoc præcipuè ſignificat numerorum ma-<lb/><reg norm="gnitudinisque" type="simple">gnitudinisq́;</reg> vocabulis vtens, quibus vniuerſum quantitatis genus complectitur.</s> + </p> + <p> + <s xml:space="preserve">Cum verò ait vires, uniuerſum qualitatis genus inferre uult. </s> + <s xml:space="preserve">Quia proportio & <lb/>proportionalitas tam continua quam diſcreta, non ſolum interterminos quanti, ſed <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">cũ</reg> geometrica magis ſim <lb/>bola eſt quam cum Arithmetica Plato minime intelligat, ex eiuſdem uerbis cum ita <lb/>ſcribit manifeſtè patet.</s> + </p> + <quote> + <s xml:space="preserve">Quando enim medium ita ſe habet ad poſtremum ut primum ad medium, <reg norm="uiciſ- ſimque" type="simple">uiciſ-<lb/>ſimq́;</reg> ut poſtremum 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/>poſtremum, & non ad medium, ita ſe habet geometricè ut differentia inter primum <lb/>& medium ad differentiam inter medium & ultimum.</s> + </p> + <p> + <s xml:space="preserve">Quod ſi clarum eſt ipſum 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 eſt & primum fit & poſtremum, poſtremum quoque, & pri-<lb/>mum media fiunt, <reg norm="&c." type="unresolved">&c.</reg></s> + </quote> + <p> + <s xml:space="preserve">Nihil aliud oſtendere uult, quam ſimilitudinem quæ inter huiuſmodi medium & <lb/>extrema intercedit, cum ipſum medium ad poſtremum, quem primus ad ſeipſum, <lb/>eundem reſpectum habeat, in quo eſt ſimilis primo, & contra ad primum <reg norm="terminum" type="context">terminũ</reg>, <lb/>eundem reſpectum, quem poſtremum ad ſeipſum habet, </s> + <s xml:space="preserve">unde hac ratione ultimum <lb/>repręſentat, uolens Plato inferre de conuenientia quę inter media elementa, & ex-<lb/>trema intercedit, ut aquæ inter aerem, & terram, cum aqua, ratione ſuæ frigiditatis, <lb/>terrę, ratione uero ſuæ humiditatis aeri ſimilis euadat. </s> + <s xml:space="preserve">Aer uero qui inter ignem, <lb/><reg norm="aquamque" type="simple">aquamq́;</reg> ponitur quod ad caliditatem attinet cum igne, quod uero ad humidita-<lb/>tem ſpectat cum aqua communicet.</s> + </p> + <p> + <s xml:space="preserve">Sed quia Plato multis in rebus doctrinam Pythagoricam ſequutus eſt, Pythago-<lb/>rici aut em omnia numeris metiebantur, & de omnire ſecundum numerorum ratio <lb/>nem diſſerebant, <reg norm="uidensque" type="simple">uidensq́;</reg> Plato quod inter duos numeros ſuperficiales, <reg norm="inuicemque" type="simple">inuicemq́;</reg> <lb/>ſimiles exiſtentes, unum tantum numerum medium in proportionalitate continua <lb/>geometrica cadere poteſt, ideo ſubiungit.</s> + </p> + <quote> + <s xml:space="preserve">Quod ſi uniuerſi corpus latitudinem habere debuiſſet, nullam uerò profundita-<lb/>tem, unum ſanè, tum ad ſeipſum, tum ad extrema uincienda interiectum medium <lb/>ſuffeciſſet.</s> + </quote> + <p> + <s xml:space="preserve">Sequitur poſtea ſic.</s> + </p> + <quote> + <s xml:space="preserve">Sed cum ſoliditatem mundus requireret, ſolida uerò non uno, ſed duobus ſem-<lb/>per modis copulentur, inter ignem, & terram, Deus, Aerem, Aquamq́ue 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 ſolidos, & inuicem <lb/>ſimiles, <reg norm="vnus" type="simple">vnꝰ</reg> <reg norm="tantum" type="context context">tãtũ</reg> medius proportionalis intercedere <reg norm="non" type="context">nõ</reg> poteſt, ſed duo neceſſariò re <lb/>quiruntur (vt exijs quæ Euclid .8. lib. 16. 17. 18. et .19. propoſitione proponit viden<lb/>tur) ita <reg norm="dictante" type="context">dictãte</reg> ratione inter igneum, <reg norm="terreumque" type="simple">terreumq́;</reg> corpus duo corpora interiecta <reg norm="eſsent" type="context">eſsẽt</reg>, <lb/>non ratione proportionalitatis continuæ in quantitate <reg norm="eorundem" type="context">eorũdem</reg> corporum, ſed pro <lb/>pter ſimilitudinem connexionis, cum productum ex duobus medijs proportionali-<lb/>bus æquale ſit producto ab extremis, & idem reſpectus, quem primum ipſorum qua <lb/>tuor ad ſecundum habet, ſecundi ad tertium extet, vnde ſecundum primo ſimile <lb/>euadit, & contra, reſpectus qui eſt quarti ad tertium, ſit etiam tertij ad ſecundum, vn-<lb/>de ipſum tertium, ratione vltimi ſubit, & eius imaginem induit, & hanc ob cauſam <lb/>ſic ſcribit Plato.</s> + </p> + <quote> + <s xml:space="preserve">Propterea ex huiuſmodi rebus numero quaternario concluſis, mundi corpus con <lb/>flatum eſt, ea connexum comparatione qua dixi. </s> + <s xml:space="preserve">Ex quo ſeipſum 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 cuiuſuis generis eſſe noluiſſe, clarè perſpicitur, ſed huiuſmo <lb/>di ſimilitudine, in eo quod media elementa cum extremis conueniunt eſt vſus, quæ <lb/>quidem conuenientia, nullibi maior, quam in proportionalitate continua geome-<lb/>trica reperitur. </s> + <s xml:space="preserve">Sed etiam ſi Plato de huiuſmodi corporea elementorum magnitu-<lb/>dine ſeipſum intelligi voluiſſet, ſi ſemidiameter regionis elementaris ex ęquo vt .39 <lb/>ad vnum, reſpectu ſemidiametri terræ fuiſſet, aqua, ipſam terram, magis quam tri-<lb/>geſies, & octies, non ſolum decies, & aer quoque eandem magis quam .1500. & <lb/>ignis magis quam .55000. partibus magnitudine ſuperaret.</s> + </p> + <p> + <s xml:space="preserve">Subſtantia vero rerum quas ſcripſeram circa finem illius conſiderationis talis fuit.</s> + </p> + <p> + <s xml:space="preserve">Nunc autem tempus eſſe videtur, vt ego etiam, ne tantum deſtruxiſſe, ſed etiam <lb/>conſtruxiſſe videar aliquid pro veritate diſſeram.</s> + </p> + <p> + <s xml:space="preserve">Non eſt igitur dubium, ſolidæ doctrinæ viris, quin præſtantiſſimus Piccolo. ſe-<lb/>cutus ſit tutam viam ad explorandum, quod terra maior ſit quam aqua, metiendo <lb/>vtriuſque horum corporum ſuperficiem detectam. </s> + <s xml:space="preserve">Omittamus autem compenſa-<lb/>tionem illam curuitatis, & concauitatis vallium, & montium, <reg norm="&c." type="unresolved">&c.</reg> quam ipſe Piccolo. <lb/>propè finem ſexti cap. vellet dare fluminibus, ſtagnis, fontibus, & eiuſmodi aquis. <lb/></s> + <s xml:space="preserve">eo enim in loco labitur Piccolo. vbi non conſiderat, quod eiuſmodi obliquis ſuper-<lb/>ficiebus non reſpondent anguli ſolidi centri ſphæræ, qui reſpiciunt eorum baſim ad <lb/>rectos angulos. </s> + <s xml:space="preserve">Sed poſtquam Piccolo. comperit ſuperficiem terrę detectam, eſſe <lb/>maiorem apparente ſuperficie ſphærica aquæ, proculdubio poterat concludere ter-<lb/>ram eſſe maiorem aqua, ſicuti fecit, <reg norm="etiam" type="context">etiã</reg> ſi aqua profunda eſſet pyramidaliter <reg norm="vſque" type="simple">vſq;</reg> <lb/>ad mundi centrum, ideſt .3500. milliaria, ſupponendo tantum eſſe huius globi ſemi <lb/>diametrum.</s> + </p> + <p> + <s xml:space="preserve">Verum quia poſſet aliquis dubitare circa diligentiam Piccolo. in hiſcæ duabus ſu <lb/>perficiebus dimetiendis, viſum eſt mihi non alienum ſequi aliam viam pro hac veri <lb/>tate probanda, ſupponendo verum eſſe, quod non vnus ſolus metitus fuerit, ſed mul <lb/>ti, ideſt ſupponendo <reg norm="verum" type="context">verũ</reg> eſſe quod maris profunditas menſurari poſſit, & præterea, <lb/>quod non modo ipſius maris maxima profunditas non perueniat ad quingentos paſ <lb/>ſus, ſicuti refert Piccolo. in fine ſui tractatus, & mihi aſſeruerunt Hiſpani multi, & <lb/>Luſitani præſtantiſſimi nautæ, tum Venetijs, tum Parmæ, in Aula Sereniſſimæ quon <lb/>dam Principis, inter quos, Venetijs fuit Illuſtris Rodericus Guzmanus, Dominus <lb/>Franciſcus Lopes, Dominus Garzias de Seuilia, <reg norm="multique" type="simple">multiq́;</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="recenſerem" type="context">recenſerẽ</reg> moleſtum eſſet. </s> + <s xml:space="preserve">Sed etiam ſupponendo quod maxima <lb/>pelagi profunditas ſit, non modo .500. paſſuum, ſed etiam .500. millium paſſuum, vt <lb/>dixi, & quod mare ſit huius profunditatis, non vno in loco tantum, aut multis, ſed <lb/>quod ſupra totam etiam faciem terræ, mare tantę profunditatis ipſam terram vn-<lb/>dique operiret, ideſt, quod vbicunque nunc terra detecta eſt, eſſet aqua, ſpiſſitudi-<lb/>nis .500. millium paſſuum. </s> + <s xml:space="preserve">Atque vt planius intelligar ſupponendo quod ſicuti to-<lb/>tus huius globi ſemidiameter eſt <reg norm="milliarium" type="context">milliariũ</reg> .3500. </s> + <s xml:space="preserve">Terreſtris partis ſemidiameter <lb/>eſſet <reg norm="tantum" type="wordlist">tm̃</reg> .3000. & reliquum ſemidiametri, id eſt quingenta milliaria eſſet craſſitudo <lb/>ſiue profunditas orbis aquei, in quo nihil neceſſe eſſet laborare in dimetiendis fon-<lb/>tibus, fluminibus, lacubus, ſtagnis, paludibus, & huiuſmodi particulis nullius momen <lb/>ti apud peritos, nec curare ſubterraneas aquas cauernarum, aut aliorum terræ cauo-<lb/>rum, ſeu terræ porroſitatum, quæ omnia ſunt circa ipſius terræ ſuperficiem. </s> + <s xml:space="preserve">Quia ve <lb/>riſimile non eſt naturam eiuſmodi caua ſiue ſpong oſitates produxiſſe demiſſius li-<lb/>bramenti maris. </s> + <s xml:space="preserve">Supponendo igitur ea quæ nunc dicta ſunt, terra tamen eſſet ferè <lb/>duplo maior aqua, hoc eſt, vt .12. ad .7. </s> + <s xml:space="preserve">Quod quidem, cuiuis mathematicæ philoſo-<lb/>phiæ mediocriter perito, ſupputatu facillnnum eſt. </s> + <s xml:space="preserve">Cum proportio diametrorum, <lb/>ſeu ſemidiametrorum, tertia pars exiſtat proportionis eorundem ſphærarum. </s> + <s xml:space="preserve">Sed <lb/>vt parum periti minore labore ſupputare poſſint.</s> + </p> + <p> + <s xml:space="preserve">Primum ſciendum eſt, quod ſupponendo diametrum globi, ex terra, & aqua com <lb/>poſiti, eſſe .3500. milliarium, & ſemidiametrum puræ terreſtris partis eſſe .3000. tan <lb/>tum, eiuſmodi proportio erit ut .7. ad .6. quia communis maior numerator horum <lb/>duum ſemidiametrorum erit .500. qui in maiorem ingredietur ſepties, in minorem <lb/>a utem ſexies. </s> + <s xml:space="preserve">Et eiuſmodi proportio ſuperparticularis, vocatur ſeſquiſexta, cuius <lb/>triplum erit vt .57. cum ſexta parte ad .36. & idem erit inter dictum globum compo-<lb/>ſitum, & partem terreſtrem ſimplicem. </s> + <s xml:space="preserve">Quare ſubtrahendo puram, ſeu ſimplicem <lb/>partem terreſtrem, ex compoſito, reliqua pars erit, vt .21. cum ſexta, pro quantitate <lb/>aquei orbis, ad quam, terreſtris 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> ſolum de parte illa maximæ e ius profunditatis, quæ nuſquam <lb/>ad quingentos paſſus peruenit, ſed de ficto illo orbe aqueo, profunditatis .500. paſ-<lb/>ſuum, qui totum terreſtrem orbem circundaret, & tegeret, ſupponendo quod per <lb/>quingentos paſſus profunditatis, quidquid eſt terra, eſſet aqua, ideſt ſuppoſito quod <lb/>ex totius orbis compoſiti ſemidiametro exiſtente .3500. milliarium, purę terræ ſemi <lb/>diameter eſſet milliarium .3499. cum dimidio. </s> + <s xml:space="preserve">Supponendo igitur, vt ſupradixi. <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 eiuſmodi fictum orbem aqueum, quæ quidem partes, <lb/>ſunt ampliores ſuperficię aquæ, vt obſeruauit Piccolo. atque alij <reg norm="præſtantes" type="context">præſtãtes</reg> viri, ideo <lb/>ſequetur, vt terra ſit 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 compoſitum circundant: </s> + <s xml:space="preserve">reſpondeo quod <lb/>non modò ei concedo computari eiuſmodi aquam, ſed ſupponendo etiam quodto <lb/>tus locus à vaporibus occupatus, qui attolluntur .52. milliaria ſupra ſuperficiem <lb/>huius globi, vt iam ſupradictum eſt, totus eſſet aqueus, & amplius, ſupponendo quod <lb/>orbis hic aqueus eſſet ſpiſſitudinis, ſiue altitudinis quingentorum milliarium ſupra <lb/>totum ipſum globum compoſitum. </s> + <s xml:space="preserve">Tamen terra eſſet maior ipſa aqua ferè duplo; <lb/></s> + <s xml:space="preserve">qua dere, quiſque eiuſmodi ſupputationum 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 ſolum maiorem eſſe aqua, ſed aqua & præte-<lb/>rea aere, ſi aer non tam altè pertingit, quam multi alij præter Piccolo. ſentiunt, qui <lb/>dicuntinde euenire quod aerea humiditas non tam altè aſcendere poteſt, quoniam <lb/>humiditas ipſa grauitatem ſecum affert, præterquam quod nubium ſitus oſtendit ſu <lb/>pra eas materiam eſſerariorem quam ſint ipſę nubes, infra vero denſiorem. </s> + <s xml:space="preserve">Corpo-<lb/>ra enim eouſque aſcendunt donec inueniunt conſtitutionem mediam formæ æqua-<lb/>lis (vt ita dicam) ſuis. </s> + <s xml:space="preserve">Quare materia illa quæ impropriè ignis vocatur (non enim <lb/>eſt ignis) incipit carere humiditate (qua mediante aer definitur) circa quinquage-<lb/>ſimum ſecundum milliarium ſupra ſuperficiem terræ, vt iam ſupradixi à Vitellione <lb/>demonſtratum fuiſſe. </s> + <s xml:space="preserve">Ariſto. autem affert <reg norm="rationem" type="context">rationẽ</reg> quare nubes altius <reg norm="non" type="context">nõ</reg> <reg norm="tranſcendant" type="context context">tranſcẽdãt</reg>. <lb/></s> + <s xml:space="preserve">Vnde apparet tertiam aeris regionem impropriè aerem appellari, ſi humiditate ca-<lb/>ret, vt ait Ariſt. qua mediante aer definitur, immo potius retinet ignis naturam, vt <lb/>etiam aſſerunt interpretes Ariſtotelis in primum Meteororum. </s> + <s xml:space="preserve">Qui Ariſto. in locis <lb/>ſupra citatis itidem oſtendit ſe etiam huius modi eſſe opinionis.</s> + </p> + <p> + <s xml:space="preserve">Quod autem attinet ad probandum quod ſuperficies terrę detecta ſit altior quam <lb/>ſuperficies detecta aquæ, id tam clarum eſt ſua ſponte philoſophis, qui ſciunt quid ſit <lb/>altum, quidue demiſſum, quod ſuperfluum eſſet quidquid ſuper hoc dicerem præ <lb/>terquam, quòd conſtat ex demonſtratione ab Ariſto. ſacta textu 31. li .2. de cœlo, in <lb/>quo agit de corporibus in aqua poſitis, vnde eiuſmodi veritas planiſſimè aperitur. <lb/></s> + <s xml:space="preserve">Omittimus etiam quod præſtantes Moderni omnes, eam pro manifeſtiſſima <reg norm="ponunt" type="context">ponũt</reg>, <lb/>ſicutiapud omnes ſani iudicij homines reuera exiſtimatur.</s> + </p> + <p> + <s xml:space="preserve">Hæc enim ſunt quæ in fine illius conſiderationis ſcripſeram.</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 Auguſtino Michele, Patritio Veneto, ad cor <lb/>roborandam opinionem antiquorum, vbi tot authoritates, <reg norm="totque" type="simple">totq́;</reg> rationes adducit, <lb/>vt nil amplius dici poſſit. </s> + <s xml:space="preserve">Atego ſenſum, <reg norm="rationemque" type="simple">rationemq́;</reg>, & non authoritatem <reg norm="aliquam" type="context">aliquã</reg> <lb/>ſequutus ſum: </s> + <s xml:space="preserve">cum verò dico ſenſum, de ſenſu illorum intelligo, qui profunditatem <lb/>maris metiti ſunt, vt non mihi ſolum, ſed, & Piccolo. & alijs permultis retulerunt, <lb/>de ratione vero à me adducta, aliorum ſit iudicium.</s> + </p> + <quote> + <s xml:space="preserve">Sediſte mirabilis & Excellentiſſimus vir, verba mea non accepit in eo ſenſu, vt <lb/>ego ſcripſi, ita vt omnino alienas conſequentias ſibi confingat, <reg norm="quemadmodum" type="context">quemadmodũ</reg> pag. <lb/>3. ſui tractatus inquit, me non concedere naturam produxiſſe in magna quantitate, <lb/>atque immenſa, id totum, quod bonum, & neceſſarium eſt. </s> + <s xml:space="preserve">Hanc enim conſequen <lb/>tiam ipſe colligit ex eo, quod ego pag .19. meæ conſiderationis contra Antonium <lb/>Bergam ſcripſeram, quod videntur multa corpora alijs nobiliora, nihilominus mi-<lb/>nora, eo quod quantitas non ſequitur nobilitatem, neque ab ea pender, ita vt res <lb/>illa quæ nobilior eſt, neceſſarium ſit vt etiam maior exiſtat. </s> + <s xml:space="preserve">Sed Excellentiſſimus <lb/>iſtæ vir ſcribit ita me dixiſſe.</s> + </quote> + <p> + <s xml:space="preserve">Multa immo infinita corpora ſunt nobilia, & neceſſaria, nihilominus ſunt paruę <lb/>molis.</s> + </p> + <p> + <s xml:space="preserve">Vide igitur quantum hoc diſtat ab illo.</s> + </p> + <p> + <s xml:space="preserve">Præterea cap .12. aliam conſequentiam facit, quam ego non tam amplam facio. <lb/></s> + <s xml:space="preserve">„ Ipſe enim me inferre vult in alijs terrę partibus cauernas non reperiri, eo quod Mon <lb/>„ tes ſint cauernoſi. </s> + <s xml:space="preserve">Aſpice quæſo. pag .29. meæ conſiderationis, & clarè videbis me <lb/>nullo modo negare illas concauitates ſeu porroſitates terræ extra montana loca, <lb/>circa ſuperficiem terræ, vſque ad æquilibrium, orbiculariter, infimæ profunditatis <lb/>maris.</s> + </p> + <p> + <s xml:space="preserve">Sed putare inferius has porroſitates reperiri, cum nulla ratio nobis perſuaſibilis + <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 ſit, idoneum nullo pacto eſſet. </s> + <s xml:space="preserve">Rationes autem ab ipſo Ex <lb/>cellentiſſimo Auguſtino adductas circa huiuſmodi rem, alij dijudicent, de authori-<lb/>tatibus verò, nihil dicam, quia ab illis petendæ ſunt, qui profitentur tales facultates, <lb/>quorum vnius tantummodo authoritas præualere deberet, contra omnes alias <reg norm="eorum" type="context">eorũ</reg> <lb/>qui nunquam attigerunt ſummis labris orificia harum <reg norm="ſcientiarum" type="context">ſcientiarũ</reg>. </s> + <s xml:space="preserve">Vt ſi exempli gra-<lb/>tia non ſolum authoritas illorum virorum, quos ipſe recenſuit, ſufficiens eſſet vt pu <lb/>ta Pioccolo. </s> + <s xml:space="preserve">Naibodæ, Bordini, Clauij, <reg norm="reliquorumque" type="simple">reliquorumq́;</reg> fautorum verę opinionis, <lb/>ſed Franciſci Maurolici tantummodo, qui in primo Dialogo ſuæ coſmographiæ ita <lb/>inquit.</s> + </p> + <quote> + <s xml:space="preserve">Exiſtimo autem totum terræ corpus rigidum eſſe ſaxum, nam ſi arena eſſet, aue <lb/>gleba fragilis, ita humorem imbiberet, vt cum eo quaſi confunderetur; </s> + <s xml:space="preserve">huc ac-<lb/>cedit, quod ſi mineræ, ac rupes, quæ ſunt grauiſſimæ partes in ipſa plerunque ſuper-<lb/>ficie comperiuntur, multo magis apud centrum eſſe debent. </s> + <s xml:space="preserve">Videtur ita ratio exi-<lb/>gere, vt grauiora centro quoque ſint propinquiora.</s> + </quote> + <p> + <s xml:space="preserve">Hæcigitur ſola authoritas, inſtar reliquarum omnium ſufficere poſſet. </s> + <s xml:space="preserve">Verum <lb/>de authoritatibus minime curandum eſt, vbi ſenſus, <reg norm="ratioque" type="simple">ratioq́;</reg> vera illis opponuntur.</s> + </p> + <p> + <s xml:space="preserve">Quod autem numerus animalium aquatilium maior exiſtat numero terreſtrium, <lb/>ſatis reſpondimus pag .41. noſtræ conſiderationis.</s> + </p> + <p> + <s xml:space="preserve">Sed in cap .14. Excellentiſſimus Auguſtinus ita inquit (vt etiam ſuperius dixerat) <lb/></s> + <quote> + <s xml:space="preserve">quod certiorem cognitionem homo non habet illa, quæ à ſenſu prouenit. </s> + <s xml:space="preserve">Et quod <lb/>nemo eſt qui aſpiciat terram, & aquam, quod hanc maiorem illa non iudicet, & <reg norm="non" type="context">nõ</reg> <lb/>exiſtimet.</s> + </quote> + </p> + <p> + <s xml:space="preserve">Quod autem certiorem cognitionem homo non habeat illa, quæ à ſenſu proue-<lb/>nit, concedendum non cenſeo. </s> + <s xml:space="preserve">Nam omnis cognitio mathematica (cum primum <lb/>gradum certitudinis obtineat) ab ipſo ſenſu fieret, quod omnino alienum <lb/>eſt à veritate. </s> + <s xml:space="preserve">Senſus enim nunquam vidit incommenſurabilitates magnitudinum, <lb/>vel incoincidentias linearum non tangentium cum curuitate hyperbolica, aut angu <lb/>lum contingentiæ aliquem, nec (vt vno verbo dicam) aliquam concluſionem ma-<lb/>thematicam, quam volueris. </s> + <s xml:space="preserve">Neque per ſenſum eſt ſcire, inquit Ariſtoteles. </s> + <s xml:space="preserve">Co-<lb/>gnitio igitur ſenſitiua, certior non eſt illa, quæ per habitum ſcientiſicum acquiritur.</s> + </p> + <p> + <s xml:space="preserve">Ad reliqua verò, ſupponamus nos tunc fuiſſe in Arca Noe, <reg norm="cum" type="context">cũ</reg> aquæ cooperiebant <lb/>omnia cacumina montium, vbi nullum terræ veſtigium videbatur, </s> + <s xml:space="preserve">quare proculdu <lb/>bio aquam iudicaremus, atque exiſtimaremus maiorem terra, <reg norm="dum" type="context">dũ</reg> nulla aliare vtere-<lb/>mur niſi ſenſu abſque alio diſcurſu intellectuali, ut reliqua illa animalia irrationalia, <lb/>quæ nobiſcum erant in dicta arca. </s> + <s xml:space="preserve"><reg norm="Non" type="context">Nõ</reg> ſufficit igitur ſuperficiem aquæ tantummodo <lb/>aſpicere, quia neque tunc temporis, aqua erat maior terra, etiam ſi non ſolum tot <lb/>cubitis attolleretur ſupra cacumina montium, ſed quingenta milliaria, vt ſupradi-<lb/>ximus.</s> + </p> + <p> + <s xml:space="preserve">Ratio autem illa, ex infinitis, ab ipſo, eo in loco adducta, talis eſt.</s> + </p> + <quote> + <s xml:space="preserve">Aqua eſt eccentrica ad terram, & pro <reg norm="centro" type="context">cẽtro</reg> habet centrum grauitatis terræ, aqua <lb/>igitur maioris eſt amplitudinis ipſa terra.</s> + </quote> + <quote> + <s xml:space="preserve">Hanc etiam conſequentiam alijs relinquo Philoſophis dijudicandam.</s> + </quote> + <p> + <s xml:space="preserve">Subſequitur poſtea dicens.</s> + </p> + <quote> + <s xml:space="preserve">Præterea proprius locus terræ, eſt ſuperſicies 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 ſuperficies aeris, & non aquæ exiſtet.</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 ſi locus terræ eſt ſub aqua, ergo locus aquæ proprius eſt ſub aere, & <lb/>non ſub terra, vnde non erit rationabile putare maiorem copiam aquarum exiſtere <lb/>in cauernis ſubterraneis, quam ſupra ſuperficiem terræ. </s> + <s xml:space="preserve">Adde quod locus illarum <lb/>aquarum non eſſet ſuperficies aeris, ſed terræ, vnde non minus locus aquę 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 eſt. </s> + <s xml:space="preserve">Ergo aqua maior eſt terra.</s> + </p> + <quote> + <s xml:space="preserve">Hæc enim conſequentia veriſſima eſſet. </s> + <s xml:space="preserve">Sed nullus vnquam Philoſophus (vt Phi-<lb/>loſophus dico) concedet totam materiam elementarem, in quatuor æquales partes <lb/>eſſe diuiſam.</s> + </quote> + <p> + <s xml:space="preserve">Cap. verò .21. inquit me dixiſſe non ſuffecturam paucam ſpiſſitudinem. </s> + <s xml:space="preserve">Eo enim <lb/>in loco pag .26. mei tractatus contradicens ipſi Bergæ, dixi, quod ſecundum ipſum <lb/>Bergam non ſufficeret pauca ſpiſſitudo.</s> + </p> + <p> + <s xml:space="preserve">Similiter etiam dixi, quod ſecundum ipſum, quanto remotius diffunditur lumen <lb/>fortaſſe tantò magis illuminat. </s> + <s xml:space="preserve">Putans ipſe Berga quod in propinquo debilius exi-<lb/>ſteret dictum lumen. </s> + <s xml:space="preserve">Et propter ea dixi, quod apud ipſum fortaſſe nihil valet illa <lb/>propoſicio, 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 Auguſtinus inquit, vnum tantummodo ele <lb/>mentum non ſufficere ad generationem miſtorum. </s> + <s xml:space="preserve">Hoc enim concedo, ſed hoc ni-<lb/>hil ad me ſpectat, eo quod meum reſponſum ad Bergam, erat circa tranſitum lumi-<lb/>nis, & non circa generationem elementorum.</s> + </p> + <p> + <s xml:space="preserve">Cap. demum .23. pag .20. linea .10. vbi ſcribit me dixiſſe, iudicare, oportebat <lb/>ſcribere, 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 expertusſum, difficillimum omnes Thy <lb/>pographi errores corrigere, neque (vt fertur) Argi oculi ſufficerent.</s> + </p> + <p> + <s xml:space="preserve">Hactenus enim in mei defenſionem hæc ſubiungere volui.</s> + </p> + <p> + <s xml:space="preserve">Ad defenſionem autem Piccolo. <reg norm="aliorumque" type="simple">aliorumq́;</reg> virorum meæ opinionis, nec non de <lb/>proportione duplicata profunditatis maris ad ſuam amplitudinem, ex conſequentia <lb/>pyramidali: </s> + <s xml:space="preserve"><reg norm="alijsque" type="simple">alijsq́;</reg> ſimilibus rationibus, prodeant alij. </s> + <s xml:space="preserve">Huiuſmodi tamen Doctiſſi-<lb/>mi viriingenium, memoriam, nec non doctrinam valde admiror, atque obſeruo.</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 Agrimenſores.</head> + <head style="it" xml:space="preserve">Anſelmo Roſemburg Agrimenſori Caſareo.</head> + <p> + <s xml:space="preserve"><emph style="sc">MEthodvs</emph> quàm mihi ſcribis in Prouincia tua maximè in vſu eſſe, nimis <lb/>longa atque prolixa eſt, Pedemontani verò Agrimenſores in produ-<lb/>ctione fractorum, valde breui methodo vti ſolent, quam libenter tibi <lb/>ſcribo, eo maxime, vt videas quam rationabiliter operentur.</s> + </p> + <p> + <s xml:space="preserve">Scire igitur primum te oportet illos, maximam eorum communem menſuram <lb/>vocare Trabucum, cuius ſextam 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">punctũ</reg>, <reg norm="duodecimam" type="context">duodecimã</reg> demum puncti; </s> + <s xml:space="preserve">Attomum.</s> + </p> + <p> + <s xml:space="preserve">Quotieſcunque igitur multiplicant trabucum, per trabucum nulli dubium eſt <lb/>quin producant trabucum ſuperficialem ſcilicet.</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 ſuperficialia rectangula oblonga, quorum lon <lb/>gitudo eſt ipſius trabuchi, latitudo vero lineæ dictarum ſpecierum.</s> + </p> + <p> + <s xml:space="preserve">Dum vero multiplicant pedem per pedem, nulli dubium eſt quin producant pe-<lb/>dem quadratum, ſed apud ipſos non vocatur quadratum, quamuis reuera ita ſit, ſed <lb/>illud vocant duas vncias, quæ quidem ſunt rectangula oblonga iam hic ſupradicta, <lb/>quarum vniuſcuiuſque longitudo ſit vnius trabuchi, latitudo vero vnius duodecimæ <lb/>partis ipſius pedis linearis.</s> + </p> + <p> + <s xml:space="preserve">Productum autem pedis per vnciam, vocant duo puncta, quæ etiam ſunt duo re-<lb/>ctangula oblonga, vt ſupra.</s> + </p> + <p> + <s xml:space="preserve">Productum deinde vnciæ per vnciam, vocant duos attomos, qui <reg norm="etiam" type="context">etiã</reg> ſunt duo re-<lb/>ctangula oblonga, vt dictum eſt, quæ omnia ſcientificè videbimus.</s> + </p> + <p> + <s xml:space="preserve">Pro cuius rei cognitione, ſit, 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> poſtea vnum punctum, et <var>.u.t.</var> vnus At-<lb/>tomus.</s> + </p> + <p> + <s xml:space="preserve">Vnde <var>.e.i.</var> erit ſexta pars ipſius <var>.a.e.</var> et <var>.i.o.</var> duodecima ipſius <var>.e.i.</var> et <var>.o.u.</var> duodecima <lb/>ipſius <var>.i.o.</var> et <var>.u.t.</var> duodecima ipſius <var>.o.u</var>. </s> + <s xml:space="preserve">Sit etiam <var>.a.b.</var> æqualis <var>.a.e.</var> lineæ & ſic <var>.e.d</var>: <var>i.<lb/>f</var>: <var>o.g</var>: <var>o.n.</var> & c. <reg norm="terminenturque" type="simple">terminenturq́;</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 ſupra, 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">trabuchũ</reg>, nulli dubium eſt quin ſit quadratum <var>.<lb/>a.d.</var> vt ſuperius diximus.</s> + </p> + <p> + <s xml:space="preserve">Productum autem trabuchi cum pede erit <var>.d.i.</var> ſexta pars ipſius <var>.a.d.</var> cum <var>.e.i.</var> ſit ſex <lb/>ta ipſius <var>.a.e.</var> ex prima ſexti vel .18. aut .19. ſeptimi, ſiue 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 eſt <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> ſexta pars eſt ipſius <var>.f.i.</var> ex ſuppoſito, et <var>.i.o.</var> duo-<lb/>decima ipſius <var>.e.i.</var> proportio igitur <var>.e.i.</var> ad <var>.o.i.</var> dupla eſt proportioni ipſius <var>.f.i.</var> ad <var>.K.i</var>. <lb/></s> + <s xml:space="preserve">quare <var>.K.e.</var> duplo maius eſt ipſius <var>.f.o.</var> eo quod ſi <var>.i.o.</var> vel <var>.f.g.</var> (quod idem eſt) duplo <lb/>maius eſſet ipſo latere pręſenti <var>.o.i.</var> vel <var>.f.g.</var> </s> + <s xml:space="preserve">tunc <var>.f.o.</var> æquale eſſet ipſi <var>.K.e.</var> ex .15. ſexti <lb/>vel .20. ſeptimi quod quidem <var>.f.o.</var> duplo maius eſſet ipſo præſenti <var>.f.o</var>. </s> + <s xml:space="preserve">Rectè igitur <lb/>inquiunt dicentes productum pedis cum pede eſſe duas vncias, vel ſi mauis, ita dicas <lb/><var>e.K.</var> ſexta pars eſt ipſius <var>.d.i.</var> ex iam dictis propoſitionibus <var>.f.o.</var> autem eſt duodecima <lb/>ipſius <var>.d.i.</var> ex ijſdem, cum exſuppoſito <var>.i.o.</var> duodecima ſit ipſius <var>.e.i.</var> </s> + <s xml:space="preserve">quare <var>.e.K.</var> <reg norm="duplum" type="context">duplũ</reg> <lb/>erit ipſius <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. ſit <var>.K.o.</var> quod probabimus ex ijſdem rationibus <lb/>duplum eſſe ipſius <var>.g.u.</var> puncti rectanguli oblongi. </s> + <s xml:space="preserve">Nam <var>.l.o.</var> ſexta pars ſimiliter eſt <lb/>ipſius <var>.g.o.</var> et <var>.o.u.</var> duodecima ipſius <var>.o.i.</var> </s> + <s xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.o.n.</var> dupla eſt propor <lb/>tioni <var>.g.o.</var> ad <var>.o.l.</var> ſequitur ergo ex prædictis rationibus <var>.k.o.</var> duplum eſſe ipſius <var>.g.u.</var> <lb/>vel ſic, vtlin præcedenti, cum <var>.K.o.</var> ſit ſexta pars ipſius <var>.f.o.</var> ex dictis propoſitionibus <var>.<lb/>g.u.</var> verò duodecima eiuſdem <var>.f.o.</var> ex ijſdem, nam <var>.o.u.</var> duodecima eſt ipſius <var>.o.i.</var> ergo <lb/><var>K.o.</var> duplo maius eſt ipſo <var>.g.u</var>.</s> + </p> + <p> + <s xml:space="preserve">Ex ijſdemmet rationibus productum <var>.l.u.</var> pedis cum puncto duplum eſt ipſius <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 eſt quadra-<lb/>tum, duplum eſſe ipſius <var>.c.t.</var> attomi rectanguli oblongi. </s> + <s xml:space="preserve">Nam <var>.i.n.</var> eſt pars vna ex .72. <lb/>ipſius <var>.c.u.</var> et <var>.u.t.</var> pars vna ex .144. ipſius <var>.o.i.</var> ex ſuppoſito, </s> + <s xml:space="preserve">quare proportio <var>.i.o.</var> ad <var>.u.t.</var> <lb/>dupla eſt proportioni ipſius <var>.c.u.</var> ad <var>.n.i.</var> ex dictis igitur rationibus <var>.o.n.</var> duplo maius <lb/>eſt ipſo <var>.c.t</var>. </s> + <s xml:space="preserve">Vel ſi placet dicas <var>.n.o.</var> eſt vna pars ex .72. ipſius <var>.f.o.</var> exſupradictis, eo <lb/>quod <var>.n.i.</var> ita ſe habet ad <var>.f.i.</var> vt vnitas ad .72. ſed ex ijſdem rationibus <var>.c.t.</var> pars vna ex <lb/>144. eſt ipſius <var>.f.o.</var> eo quod ita ſe habet <var>.u.t.</var> ad <var>.o.i.</var> </s> + <s xml:space="preserve">quare <var>.o.n.</var> duplo maius erit ipſo <var>.<lb/>c.t</var>.</s> + </p> + <p> + <s xml:space="preserve">Propoſitum ſit nobis nunc, exercitij gratia, quærere ſuperficiem alicuius rectan <lb/>guli, cuius vnum latus ſit <reg norm="trabuchorum" type="context">trabuchorũ</reg> .3. pedum .2. & vnciarum .3. aliud vero latus ſit <lb/>trabuchorum .2. pedum .3. vnciarum vero .2.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi autem methodo mediante, multiplicando primum latus <reg norm="dictum" type="context">dictũ</reg> .3. 2. 3. <lb/>per numerum trabucorum ſecundi lateris .2. ſcilicet producentur nobis primò trabu <lb/>cha ſuperficialia .6. pedes .4. & vnciæ .6. omnia rectagula, vt dictum eſt. </s> + <s xml:space="preserve">Multiplican-<lb/>do deinde idem primum latus .3. 2. 3. per pedes .3. ſecundi lateris. </s> + <s xml:space="preserve">Ex trabuchis .3. <lb/>primi lateris cum .3. pedibus ſecundi, producentur .9. pedes rectanguli, hoc eſt <lb/>vnus trabuchus cum tribus pedibus rectangulis. </s> + <s xml:space="preserve">Ex pedibus autem huius .2. cum <lb/>ijſdem alterius lateris .3. producentur .12. vnciæ rectangulæ ideſt vnus pes rectangu-<lb/>lus. </s> + <s xml:space="preserve">Exijſdem pedibus .3. ſecundi 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 eſt vna vncia cum .6. punctis rectangulis. </s> + <s xml:space="preserve">Deinde ex <lb/>multiplicatione vnciarum .2. ſecundi lateris, cum .3. trabuchis primi lateris, produ-<lb/>centur .6. vnciæ. </s> + <s xml:space="preserve">Ex multiplicatione poſtea dictarum .2. vnciarum ſecundi 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 ijſdem .2. vncijs ſecundi lateris cum .3. primi, producentur .12. atto-<lb/>mi, ideſt 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 cauſa, vt dixi, quomodo conueniat calculus iſte cum <lb/>calculo ordinario communi?</s> + </p> + <p> + <s xml:space="preserve">Nam quotieſcunque dicta latera, fracta fuerint in vncias, primum latus erit <lb/>vnciarum .243. ſecundum autem .182. productum vero vnius in alterum erit vn-<lb/>ciarum quadratarum .44 226. quod quidem productum cum diuiſum fuerit per <num value="5184">.<lb/>5184.</num> vncias quadratas vnius trabuchi quadrati, prouentus erit .8. trabucho-<lb/>rum, reliquus verò numerus, ſiue fractus, erit vnciarum quadratarum .2754. qui <lb/>cum diuiſus fuerit per numerum .144. vnciarum vnius pedis quadrati, prouenient <lb/>pedes .19. quadrati cum vncijs .18. ſuperabundantibus, dicti autem pedes .19. ſignifi-<lb/>cant tres pedes rectangulos oblongos cum vno pede quadrato, hoc eſt cum duabus <lb/>vncijs rectangulis oblongis, vt ſupra.</s> + </p> + <p> + <s xml:space="preserve">Videndum nunc eſt, vtrum illæ .18. vnciæ æquipolleant tribus punctis rectangu-<lb/>lis oblongis: </s> + <s xml:space="preserve">ſed hoc manifeſtè videre eſt, ex hoc, quia quęlibet vncia rectangula <lb/>oblonga componitur ex .72. quadratis, punctum autem rectangulum oblongum, <reg norm="cum" type="context">cũ</reg> <lb/>ſit duodecima pars ipſius vnciæ rectangulæ oblongæ, ipſum componetur ex .6. vn-<lb/>cijs quadratis .18. igitur vncijs quadratis, triplum erit ipſius puncti rectanguli dicti. <lb/></s> + <s xml:space="preserve">Vnde clarè patet, quod, quotieſcunque voluerimus ſcire proportionem ipſarum vn <lb/>ciarum quadratarum ſuperabundantium, ad punctum rectangulum oblongum, ſi <lb/>dixerimus ex regula de tribus, ſi .72. (vncia rectangula oblonga) dat .18. quid <reg norm="dabunt" type="context">dabũt</reg> <lb/>12? </s> + <s xml:space="preserve">puncta rectangula oblonga, quarum vnaquæque eſt duodecima pars ipſius vn-<lb/>ciæ rectangulæ oblongæ, in præſenti autem caſu prouenient .3. pro quarto termino <lb/>quæſito, & habebimus propſitum.</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 ſolutio quam neſcio quis te docuit, valde diuerſa eſt à vera. <lb/></s> + <s xml:space="preserve">quæſitum enim tale fuit.</s> + </p> + <p> + <s xml:space="preserve">Reperiuntur quatuor ſocij, Ludouicus, Hieronymus, Franciſcus, & Lau <lb/>rentius quorum primus, Ludouicus ſcilicet, poſuit aureos .6000. Hierony <lb/>mus verò aureos .5000. Franciſcus autem .2000. & Laurentius .1000. quorum ſum-<lb/>ma faciebat aureos .14000. </s> + <s xml:space="preserve">interim tamen de tali ſumma Ludouicus recepit aureos <lb/>2000. Hieronymus verò .1000. Franciſcus autem .900. & Laurentius .800. quapro-<lb/>pter in ſumma reſidua Ludouicus non habebat niſi aureos .4000. Hieronymus <reg norm="etiam" type="context">etiã</reg> <lb/>4000. Franciſcus .1100. & Laurentius .200. quorum ſumma erat .9300. </s> + <s xml:space="preserve">Nunc au-<lb/>tem iſti ſocij cupiunt augere hanc ſummam per aureos .20000. tali tamen conditio-<lb/>ne quod quilibet tantum tribuat vt in totali ſumma, tantam partem unus habeat, <lb/>quantam alter.</s> + </p> + <p> + <s xml:space="preserve">Hoc autem problema tam ſacile eſt, & cum ſuo theoremate ita coniunctum, quod <lb/>miror amicum noſtrum illud illico non vidiſſe.</s> + </p> + <p> + <s xml:space="preserve">Accipe igitur illos aureos .20000. & eos collige cum ſumma .9300. vnde habebis <lb/>aureos .29300. pro <reg norm="summa" type="context">sũma</reg> totali, cuius quarta pars erit .7325. <reg norm="quam" type="context">quã</reg> <reg norm="vnuſquisque" type="simple">vnuſquisq;</reg> poſtea <lb/>habebit in dicta ſumma. </s> + <s xml:space="preserve">Sed ut reperias quantitatem aureorum quam quilibet <lb/>prius debet contribuere, vt poſtea habeat aureos .7325. in dicta ſocietate. </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 ipſius <lb/>Ludouici. </s> + <s xml:space="preserve">Demptis ſimiliter aureis .4000. ex dictis .7325. <reg norm="remanebunt" type="context">remanebũt</reg> .3325. pro con <lb/>tributione ipſius Hieronymi. </s> + <s xml:space="preserve">Deinde ſi ex .7325. extracti fuerint aurei .1100. relin-<lb/>quent .6225. pro contributione Franciſci. </s> + <s xml:space="preserve">Demptis demum .200. ex .7325. reſidui <lb/>erunt .7125. pro contributione Laurentij, & ſic quilibet habebit æqualem portio-<lb/>nem in totaliſumma.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Speculatio cuiuſdam Methodire ductionis numiſmatum <lb/>unius ſpeciei in aliam.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">MIrum tibi videtur quo pacto verum ſit, quod ſumma <reg norm="mendietatis" type="context">mẽdietatis</reg> cuiuſuis <lb/>numeri illorum numiſmatum, quæ hic vocantur Blanci, cum ſexta parte eiuſ <lb/>dem medietatis, ſemper ſit numerus florenorum huius prouinciæ. </s> + <s xml:space="preserve">Vt exempli gra <lb/>tia, quotieſcunque reducere voluerimus .48. Blancos in Florenos, ſi medietati ip-<lb/>ſius .48. hoc eſt .24. adiecta fuerit ſexta pars ipſius medietatis, quæ eſt .4. </s> + <s xml:space="preserve">tunc habebi <lb/>mus .28. & ita dicemus quod .48. Blanci conſtituunt Florenos .28. quod quidem <lb/>verum eſt.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi autem rei ſpeculatio ita ſe habet. </s> + <s xml:space="preserve">Nam vnuſquiſque Blancus diuidi-<lb/>tur in .7. æquales partes, quarum .12. conſtituunt vnum Florenum, horum verò nu-<lb/>miſmatum communis menſura, vocatur Groſſus, vt ſcis, ex quo ſequitur, quod ſi + <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, noſtram figuram .79. Theorematis Arithmetici <var>.x.u.o.<lb/>e.n.</var> ſupponendo ambo producta <var>.u.x.</var> et <var>.n.e.</var> inuicem ęqualia exiſtere, & vnumquod-<lb/>que eſſe groſſorum .336. ſit 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/>ſexti vel .20. ſeptimi Euclidis eandem fore proportionem <var>.o.u.</var> ad <var>.o.e.</var> quæ <var>.o.n.</var> ad <var>.o.<lb/>x.</var> ſed <var>.o.n.</var> eſt ſumma medietatis ipſius <var>.o.x.</var> cum ſexta parte dictæ medietatis, ita igi-<lb/>tur erit <var>.o.u.</var> ipſius <var>.o.e.</var> hoc eſt ſumma medietatis <var>.o.e.</var> <reg norm="cum" type="context">cũ</reg> ſexta parte medietatis eiuſ-<lb/>dem, quæ ſumma in præſenti exemplo erit .28.</s> + </p> + <p> + <s xml:space="preserve">Hac enim ſpeculatione mediante, poteris methodum inuenire conuertendi Flo-<lb/>renos in Blancos. </s> + <s xml:space="preserve">Vt ſi nobis propoſiti fuerint Floreni .28. </s> + <s xml:space="preserve">Voluerimusq́ue inuenire <lb/>quot Blancos faciant, ſuppoſita menſura communi, iam ſupradicta. </s> + <s xml:space="preserve">Nam duplica-<lb/>bimus numerum Florenorum, à quo duplo detrahemus ſeptimam partem, <reg norm="reliquum" type="context">reliquũ</reg> <lb/>verò erit numerus quæſitus.</s> + </p> + <p> + <s xml:space="preserve">Huiuſmodi autem rei ratio eſt, quia, cum in ſupradicta figura, proportio <var>.o.e.</var> ad <lb/><var>o.u.</var> ęqualis exiſtat ei, quæ <var>.o.x.</var> ad <var>.o.n.</var> atque etiam <var>.o.x.</var> ſit minor duplo ipſius <var>.o.n.</var> <lb/>per ſeptimam partem ipſius dupli <var>.o.n.</var> minor erit <var>.o.e.</var> duplo ipſius <var>.o.u.</var> per <reg norm="ſeptimam" type="context">ſeptimã</reg> <lb/>partem eiuſdem dupli ipſius <var>.o.u</var>.</s> + </p> + <p> + <s xml:space="preserve">Idem affirmo de quauis conuerſione aliorum numiſmatum, quorum ſemper <var>.o.x.</var> <lb/>maior ſit <var>.o.n.</var> verò minor. </s> + <s xml:space="preserve">Vt ſi <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> menſuraretur ab <var>.o.n</var>.</s> + </p> + <p> + <s xml:space="preserve">Si cuperemus ſcire quot <var>.o.x.</var> ſint in <var>.o.n</var>. </s> + <s xml:space="preserve">Primo dicemus in <var>.o.n.</var> reperiri ſummam <lb/>medietatis ſex ſeptimorum ipſius <var>.o.x.</var> collectæ cum vna ſeptima parte ipſius <var>.o.x.</var> ſeu <lb/>(vt ita dicam) cum tertia ipſius medietatis. </s> + <s xml:space="preserve">Vnde dempta ſeptima parte ipſius .42. <lb/>quæ eſt .6. <reg norm="collectaque" type="simple">collectaq́</reg> cum medietate reſidui, quæ eſt .18. habebimus .24. res, quarum <lb/>vnaquæque æqualis erit ipſi <var>.o.x</var>.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi 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> ipſius <var>.o.u.</var> & habebit propoſitum. </s> + <s xml:space="preserve">Nam ita ſe 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 ſcire à me deſideras, eſt, quod cum vendideris libram vnam <lb/>mercis pro .4. ſolidis, & lucratus fueris .2. cum quarta parte vnius pro ſingu-<lb/>lis decem libris, ſcire velles quantum lucri facturus eſſes in libris <reg norm="decem" type="context">decẽ</reg> dan-<lb/>do ſingulam libram pro .6. ſolidis.</s> + </p> + <p> + <s xml:space="preserve">Nulli dubium eſt quin decima pars de .2. cum quarta vnius ſit lucrum libræ vnius. <lb/></s> + <s xml:space="preserve">Quæ decima pars ſunt <reg norm="nouem" type="context">nouẽ</reg> quadrageſimæ partes, & hæc ſubducta à ſolidis .4. reli-<lb/>qui erunt ſolidi .3. cum .31. quadrageſimis partibus pro ſorte vnius libræ. </s> + <s xml:space="preserve">Quę ſors <lb/>ſubtracta à ſolidis .6. remanebunt ſol .2. cum .9. quadrageſimis lucri pro libra, quod <lb/>multiplicatum per .10. proueniunt ſol .22. cum quarta parte vnius, & tantum aſcen-<lb/>deret lucrum, quod fieri poſſet in libris decem ſi quamlibet, ſol .3. cum .31. quadra <lb/>geſimis nobis conſtaret.</s> + </p> + <p> + <s xml:space="preserve">Vel ſic multiplicemus ſortem 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. ſortem cum lucro vnius librę quod eſt .4. pro-<lb/>ductum erit .40. differens à primo ſol .2. cum quarta parte, multiplicemus pariter <lb/>per .10. precium .6. ſolidorum proueniens erit .60. à quo deducendo productum ſor-<lb/>tis librarum .10. quod erat ſol .37. cum tribus quartis ſupererunt ſol .22. cum quar-<lb/>ta parte, vt ſupra.</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 diſtinctionem orbium, quæiam inualuit, nonteneas, ſed putes <lb/>totum eſſe quoddam continuum excipiens corpora ſtellarum, nouum <reg norm="non" type="context">nõ</reg> <lb/>eſt, nam nonnulli ſolidæ doctrinæ Philoſophi idem cenſuerunt. </s> + <s xml:space="preserve">Sed <lb/>quod attinet ad dignitates planetarum in ſignis zodiaci, ſcias huiuſmo-<lb/>di ordinem me compręhendere eſſe deſumptum ab ordine antiquo orbium <reg norm="ipſorum" type="context">ipſorũ</reg> <lb/>planetarum, quiquidem ordo erat, vt ſtatim poſt Lunam ſuccederet Sol, poſt So-<lb/>lem Mercurius, tum Venus deinde Mars, poſtea Iupiter, & tandem Saturnus per <lb/><reg norm="eoſdemque" type="simple">eoſdemq́;</reg> orbes, retro redibant, atque hoc cognoſcitur conſtituendo 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">eundemq́;</reg> tribuentes Saturno, Piſces, Ioui, <lb/>Arietem, Marti, Taurum, Veneri, & Gemellos, Mercurio, ſeptem Planetas cum <lb/>duodecim ſignis zodiaci concordes reddebant.</s> + </p> + <p> + <s xml:space="preserve">Quod deinde Ariſtoteles in libris de ſenſu & ijs quæ ſenſibus percipiuntur, dicit <lb/>pupillam oculi eſſe nigram, non ita ſe habet, nam idem eſt, ac ſi quis diceret <reg norm="nigrum" type="context">nigrũ</reg> <lb/>eſſe illud medium, quod permitteret tranſitum lumini per ſuam diaphaneitatem, nul <lb/>lum lumen à ſeipſo reflectens, & etiam ac ſi quis diceret nigrum eſſe aerem alicuius <lb/>cubiculi vndequaque clauſi tenebroſi.</s> + </p> + <p> + <s xml:space="preserve">Quod etiam idem Ariſtoteles volens adducere cauſam, cur oculus magis mate-<lb/>riam aquæ, quam aeris participet, dicensidea ratione fieri, quod aqua magis quam <lb/>aer ſeruari poſſit, eodem libro ſcribit, eſt reuera admirandum. </s> + <s xml:space="preserve">Ibi enim clarè de-<lb/>monſtrat ſe planè ignorare, & conſtructionem oculi, & cauſam diuerſitatis eorum <lb/>humorum tam in ſubſtantia, quam in figura, quæ non aliunde dependet quam quod <lb/>diuerſam refractionem radiorum luminoſorum producat, qui per pupillam ingre-<lb/>diuntur, vt ad proprios <reg norm="ſibique" type="simple">ſibiq́;</reg> deſtinatos locos dirigantur radij, vt à virtute viſiua per <lb/>fectius ſen tiantur.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">De ratione Frigiditatis locorum umbroſorum.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">VEra ratio vnde fiat, vt quanto magis ſentitur calor in locis expoſitis Soli, tan-<lb/>to minus ſentiatur in vmbra, vbi Solis radius non reflectitur, eſt quia cum ra <lb/>refactus eſt aer à vehementi calore radij ſolaris, ſeipſum colligit, & condenſatur in <lb/>locis, à quibus à calore, ratione rarefactionis, non expellitur, & quia naturaliter ca-<lb/>lor ſequitur rarum, rarum calorem, & frigidum <reg norm="densum" type="context">densũ</reg>, & <reg norm="densum" type="context">densũ</reg> frigidum, vt vnicui <lb/>que ſanę mentis patet, hanc ob cauſam ſequitur rem ita ſe habere vt diximus. </s> + <s xml:space="preserve">Poſſu <lb/>mus etiam abſque dubio credere huiuſmodi ratione fieri, vt frigus matutini tempo <lb/>ris, in crepuſculo maius eſſe eo, quod noctu viguit. </s> + <s xml:space="preserve">Nam materia conſiſtens in co-<lb/>no vmbræ terræ, ſemper denſior eſt ea, quæ extra reperitur, imo noua materia con <lb/>tinuo condenſatur, propter motum vmbrę, quæ ſemper corpori ſolari opponitur. </s> + <s xml:space="preserve">hęc + <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 condenſatio dico ſemper fit in crepuſculo matutino, hoc eſt in parte co <lb/>ni à Sole pulſa, in parte vero contrari a ipſius coni hoc eſt in parte crepuſculi ve-<lb/>ſpertini, contrarium accidit, quia potius aliquantulum in hac parte materia coni ra <lb/>rificatur, quia extrinſeca condenſatur, in parte vero matutina extrinſeca rarificatur; <lb/></s> + <s xml:space="preserve">& propterea intrinſeca conde nſatur.</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/>caſu non eſſe productum.</head> + <head style="it" xml:space="preserve">Hieronymo Condrumerio.</head> + <p> + <s xml:space="preserve">FErunt <reg norm="Ariſtippum" type="context">Ariſtippũ</reg> tempeſtate maris ad incognita littora delatum, cum in are-<lb/>na vidiſſet <reg norm="quaſdam" type="context">quaſdã</reg> figuras geometricas delineatas <reg norm="exultantem" type="context">exultantẽ</reg> lętitia dixiſſe: </s> + <s xml:space="preserve">Hæc <lb/>ſunt hominum veſtigia. </s> + <s xml:space="preserve">Nam conſonum rationi non erat, vt huiuſmodi figuræ ca-<lb/>ſu eſſent impreſſæ: </s> + <s xml:space="preserve">neque etiam credendum eſt ingentem hanc ma chinam tanto or <lb/>dine conſtantem fortuitò eſſe productam, cum nulla quantumuis minima eiuſdem <lb/>particula, dummodo nitatur ordine, aliquo modo caſu effecta fuerit; </s> + <s xml:space="preserve">cum caſus ni-<lb/>hil producat, quod regulam & ordinem ſeruet. </s> + <s xml:space="preserve">Non eſt igitur producta caſu admi <lb/>randa correſpondentia, quæ eſt obiectorum cum potentijs, luminis cum oculo, ſo-<lb/>ni cum auditu, ſaporis cum guſtatu, odoris cum odoratu, qualitatum tangibilium <reg norm="cum" type="context">cũ</reg> <lb/>tactu. </s> + <s xml:space="preserve">Si diligenter deinde cuiuſlibet rei naturalis operationem conſiderabimus, <lb/>eas tanta arte conſtructas videbimus, vt cogamur fateri aliquam prudentiſſimam, <lb/>& ſagaciſſimam mentem eas formaſſe, ſi ergo quælibet <reg norm="mundi" type="context">mũdi</reg> pars tanta cum ratione <lb/>& ordine eſt conſtructa: </s> + <s xml:space="preserve">quomodo fieri poterit, vt de toto ipſo mundo id in dubium <lb/>vocemus, <reg norm="certiſſimeque" type="simple">certiſſimeq́;</reg> non credamus diuiniſſimam aliquam <reg norm="mentem" type="context">mẽtem</reg> eſſe à qua exqui-<lb/>ſitiſſima huius vniuerſi harmonia, quæ ex tot <reg norm="tantisque" type="simple">tantisq́;</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 ſcribis dico, quod excrementa quæ ex corpore ſano prodeunt <lb/>in ſua <reg norm="ipſorum" type="context">ipſorũ</reg> qualitate ſenſibili ita ſe <reg norm="habent" type="context">habẽt</reg> ad <reg norm="facultatem" type="context">facultatẽ</reg> illius partis eiuſdem <lb/>corporis, ut <reg norm="eam" type="context">eã</reg> non lędant, <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> efficeret <reg norm="ſputum" type="context">ſputũ</reg>, ſi eſſet <reg norm="amarum" type="context">amarũ</reg>, aut quod ex <lb/>cernitur naſo <reg norm="fętidum" type="context">fętidũ</reg> eſſet. </s> + <s xml:space="preserve">Imagineris igitur <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg> poſſit eſſe <reg norm="verum" type="context">verũ</reg> id quod <reg norm="idem" type="context">idẽ</reg> <lb/><reg norm="amicus" type="simple">amicꝰ</reg> noſter ait. </s> + <s xml:space="preserve">Pręterea ſi aliquid tibi in <reg norm="oculum" type="context">oculũ</reg> inciderit, an neſcis quomodo ſtatim <lb/>affatim affluat humor, vt id foras <reg norm="propellat" type="simple">ꝓpellat</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 abſque myſterio naturam in tot miſerijs ſenectutem poſuiſſe, cum <lb/>ſæpiſſimè ſenex mori deſideret, ut huius vitæ calamitatibus liberetur, vnde fit, vt <lb/>cum eius aduentum ſentiat, minus affligatur. </s> + <s xml:space="preserve">Dicito etiam eidem, naturam non <lb/>fuiſſe tam ſolicitam de quibuſdam partibus quemadmodum eſt de toto, vnde ma-<lb/>gis rotunda, & polita poterat eſſe ſuperficies terræ, quam nunc eſt, quia natura ma <lb/>gis reſpicit totum, quam partes, & magis maiores, quam minores.</s> + </p> + <p> + <s xml:space="preserve">Dum tuas legerem, me continere non potui quin riſerim, id quod ſcribis te inter-<lb/>rogaſſe eum Philoſophum naturalem, vnde fit, vt ventus ſit frigidus, <reg norm="eumque" type="simple">eumq́;</reg> tibi re <lb/>ſpondiſſe, quod à remotiſſimis partibus veniat, <reg norm="genereturque" type="simple">genereturq́;</reg> à vaporibus terræ frigi-<lb/>dis. </s> + <s xml:space="preserve">( cum ipſa ſit 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/>proſilit, 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, ſcito <reg norm="naturam" type="context">naturã</reg> coniunxiſſe <reg norm="frigiditatem" type="context">frigiditatẽ</reg> <reg norm="cum" type="context">cũ</reg> denſitate, & <reg norm="caliditatem" type="context">caliditatẽ</reg> <reg norm="cum" type="context">cũ</reg> raritate, <lb/>vt ſup. diximus, ita vt cum aliquod corpus <reg norm="denſatur" type="simple">denſat̃</reg>, <reg norm="frigidum" type="context">frigidũ</reg> <reg norm="reddatur" type="simple">reddat̃</reg>, & dum rarefit ma-<lb/>iorem caliditatem acquirat, & ſic econtra fit, vt quanto magis aliquod corpus refri <lb/>geratur, tanto denſius reddatur, & quanto calidius fit tanto rarius efficiatur. </s> + <s xml:space="preserve">Quo-<lb/>ties igitur agitabitur aer, aut aliud corpus, quod ratione ſuæ ſubtilitatis, velociter <lb/>condenſari, & rarefieri poſſit, eius partes denſiores ſemper erunt frigidæ, & hanc <lb/>obrem quilibet ventus, qui per calida loca non tranſeat, natura ſua frigidus, calidus <lb/>autem per accidens erit. </s> + <s xml:space="preserve">Hinc fit vt vaſa 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, ſed hoc non ſequeretur, ſi in materia, qua huiuſ <lb/>modi vas conſtat, 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 cęleſtium, de albedine, <lb/>de ſphæra.</head> + <head style="it" xml:space="preserve">Clariβimo Antonio Nauaiero.</head> + <p> + <s xml:space="preserve"><emph style="sc">LVmen</emph> Lunæ etiam ſi ſit lumen reflexum Solis ab ipſa Luna, ab ea tamen <lb/>non ita reflectitur, vt à ſuperficie polita ſpeculi, <reg norm="cum" type="context">cũ</reg> eius luminis <reg norm="tantam" type="context">tantã</reg> <reg norm="quan titatem" type="context">quã<lb/>titatem</reg> ſuper ipſum corpus lunare videamus, & eo modo <reg norm="terminatam" type="context">terminatã</reg> quo <lb/>conſpicimus. per ſe lumen, cauſa oculi eſt 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> ſolaris reflexio exiſtat) <lb/>omnino deſtituta eſt, dico etiam, neque in ipſis ecclipſibus ſolaribus vel lunaribus, <lb/>in ſolaribus enim cum Soltot millia vices maior ſit Luna, Luna verò minor terra, ſe <lb/>quitur, vt terra non omnino priuata remancat lumine Lunæ, in ecclipſibus ve-<lb/>rò lunaribus Luna ſemper videtur, gratia luminis ſolaris, quamuis refracti. </s> + <s xml:space="preserve">Mo-<lb/>tus corporum cœleſtium fit ratione ſitus, & varietatis virtutis ſtellæ in diuerſis locis, <lb/>hæc autem varietas abſque diuerſo ſitu eiuſdem ſtellæ, nec diuerſus hic ſitus abſque <lb/>motu fieri poſſet, ita vt motus ſtellarum ſit ratione diuerſitatis ſituum ipſarum, er-<lb/>go motus, & diuerſitas ſituum, fit, ob diuerſam 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> ſimilitudinem 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 ſenſum. </s> + <s xml:space="preserve">Tertiò quia abſque reſiſtentia magis reci-<lb/>pit qualitatem aliorum colorum, quam alij colores. </s> + <s xml:space="preserve">Quartò quia maximus <lb/>eſt omnium colorum. </s> + <s xml:space="preserve">Quintò quia ſimplicior eſt reliquis. </s> + <s xml:space="preserve">Sextò quia diſgregat vi-<lb/>ſum. </s> + <s xml:space="preserve">Septimò quia qualitas quæ in niue alba eſſe videtur, nihil aliud eſt quam mul-<lb/>titudo quædam luminum reflexorum, & non albedo, ſimilis ei, quæ eſt lactis, aut <lb/>panni, quæ quidem ſeptima cauſa effecit, vt ipſam albedinem, magis quam alium <lb/>quemuis colorem cum ipſo lumine compararem, cum nihil ſit, quod eſſe ſuum <reg norm="tranſ" type="context">trãſ</reg> <lb/>mutans, aut apparenter, aut eſſentialiter, illud ipſum prius non tranſmutet in for-<lb/>mam ſibi propin quiorem, vt manifeſtè patet. </s> + <s xml:space="preserve">Eſt etiam huius rei octaua ratio, <lb/>magni ponderis, quia ſcilicet nullus ſit color, qui magis reſiſtat lumini, aut in quem <lb/>lumen minorem impreſſionem faciat, quam albedo. </s> + <s xml:space="preserve">Vnde ſequitur, obiecta alba, <lb/>minus eſſe combuſtibilia quam alia, cum quælibetres in ſuum 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"/> + ſibi ſimile valentius agat, vtrectè vidit Ariſtoteles cum dixit, omne contrarium @ <lb/>ſuo contrario patinatum eſt.</s> + </p> + <p> + <s xml:space="preserve">Inter corpora, multum ſimplicitatisretinet ſphæra.</s> + </p> + <p> + <s xml:space="preserve">Circa quod, præter rationes adductas ab Ariſtotele in libris de Cœlo, poſſumus <lb/>etiam ratiocinarià facilitate motus <reg norm="vndique" type="simple">vndiq́;</reg> ab eo quod violentiæ non reſiſtar, ab eo, <lb/>quod apta <reg norm="nataque" type="simple">nataq́;</reg> ſit quieſcere ſupra quoduis punctum ſuę ſuperficiei, ab eo quod ab <lb/>aliqua ſuperficie alterius corporis ſeſe tangi non permittat, quæ curuitate concaua <lb/>non adæquetur, niſi medio vnius puncti. </s> + <s xml:space="preserve">Verum eſt, quod licet hæc vltima ratio <reg norm="non" type="context">nõ</reg> <lb/>ſit propria ſphæræ, eſt tamen cauſa ſimplicitatis in eo, in quo reperitur, ſed proprię <lb/>paſſiones ſphæræ ſunt ſupradictæ, præter quam quod alia eiuſdem ſphæræ eſt pro-<lb/>prijſſima, quæ eſt diſtantia eiustermini ab vno tantummodo puncto ideſt ab <reg norm="eiuſdem" type="context">eiuſdẽ</reg> <lb/>centro, & etiam poſſe diuidere corpus aliquod medium, cum æquali reſiſtentia circa <lb/>punctum, quod prius in motu reperitur.</s> + </p> + <p> + <s xml:space="preserve">Aequalitas autem rerum, eſt etiam valde ſimilis ſimplicitati, & vnitati.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Comparatio uiſus, & auditus.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">QVodad viſum & auditum attinet, magis neceſſarium eſſe viſum, & nobilio-<lb/>rem quam auditum exiſtimo, primò quia ſi quis viſu orbatus eſſet, contra <lb/>frigus, & calorem, contra famen, & ſitim nil prouidere poſſet, neque aliud quic-<lb/>quam hoc vocabulum prouidere ſignificat, neque abſque periculo vitæ ab vno loco <lb/>ad alium ferri poſſet, neque aliquid arte facere.</s> + </p> + <p> + <s xml:space="preserve">Sed ſi quis deſtitutus eſſet facultate audiendi, ſupradictas tamen operationes <reg norm="prae- ſtare" type="simple">prę-<lb/>ſtare</reg> poſſet, neque modo careret, quo animi ſui ſenſa abſque beneficio ſoni, ſed <lb/>ope figurarum & characterum alteri aperiret: </s> + <s xml:space="preserve">neque etiam munere ſpeculandi ſcien <lb/>tias (excepta muſica) deſtitueretur. </s> + <s xml:space="preserve">Ad ſcientiam comparandam, longè magis ne <lb/>ceſſarius eſt viſus, quam auditus præterquam, quod viſus maiorem numerum obie-<lb/>ctorum, & differentiarum rerum percipit, & inter reliquos ſenſus velociſſimè imò <lb/>in inſtanti operatur, magis remotè quam alij, & exactius ſentit, <reg norm="minusque" type="simple">minusq́;</reg> quam reli-<lb/>qui afficitur, præterquam quod ſemperagit, dummodò non dormiat animal. </s> + <s xml:space="preserve">Præ-<lb/>terea ſeſe magis patefacit, & prodit anima per oculos, quam per aliud, cuiuslibet <lb/>ſenſus, inſtrumentum. </s> + <s xml:space="preserve">Oculo magis quam alia corporis parte, hominis natura co-<lb/>gnoſcitur: </s> + <s xml:space="preserve">& ſi aliquid ſpeculari volumus, quod ſine imaginatiua fieri non poteſt, <lb/>ſtatim imaginamur nos videre huiuſmodirem, ac ſi oculo fuiſſet compræhenſa, & <lb/>ab imagine quæ eſt vnum ex obiectis oculi, imaginatiua nuncupatur. </s> + <s xml:space="preserve">Beneficio <lb/>oculorum omnes ferè ſcientiæ ſunt adinuentæ. </s> + <s xml:space="preserve">Auditus nil aliud quam ſonum ca-<lb/>pit, auditus nunquam detulit intellectui figuram, corpus ſuperficiem, aut lineam, <lb/>materiam, formam, locum, dimenſionem, plenum inane, nec innumera alia acci-<lb/>dentia, quæ ab oculo compræhenduntur. </s> + <s xml:space="preserve">Quæ verò viſui, & auditui ſunt commu-<lb/>nia, ſunt etiam tactui communia, vt numerus, motus, maius, & minus, ſunt tamen ali <lb/>qua oculo & tactui communia, quæ auditus non poteſt capere, vt durum, molle, acu <lb/>tum, obtuſum, aſperum, lene, planum, curuum, concauum, conuexum, magnum, <lb/>paruum, & ſupradicta, ideſt figura corpus & cętera, vt ctiam rectum, obliquum, & <lb/>ſimilia.</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">Ariſtoteles circa finem primi capitis libri de ſenſu ait mediante viſu, magis <reg norm="quam" type="context">quã</reg> <lb/>quolibet alio ſenſu, nos percipere ſenſibilia communia. </s> + <s xml:space="preserve">Vbi eundem per ſe, & <lb/>non per accidens magis neceſſarium eſſe quam auditum, tam in ijs quæ ad victum, <lb/>quam in ijs quæ ad ſcientiam pertinent eſſe aſſerit, quia auditus intellectui confert <lb/>per accidens. </s> + <s xml:space="preserve">Vide etiam quod idem ſcribit primo metaphyſicorum. </s> + <s xml:space="preserve">Et ſi ad ali-<lb/>quid perfectè cognoſcendum, oculo ſeſe nobis offerrent ea omnia obiecta, quorum <lb/>ſpecies in imaginatiua formamus, ipſa imaginatiua non egeremus. </s> + <s xml:space="preserve">Sed quia hoc <lb/>fieri non poteſt, hunc <reg norm="theſaurum" type="context">theſaurũ</reg> imaginatiuè, ſeu memoriæ ad conſeruandam imagi <lb/>nem omnium obiectorum ſenſibilium nobis dedit natura, vt ope diſcurſus <reg norm="intellectus" type="simple">intellectꝰ</reg> <lb/>circa dictas imagines, rerum veritatem venari poſſimus. </s> + <s xml:space="preserve">Sed vt ad propoſitum re-<lb/>deamus, beneficio oculi animal liberum eſt, cum ſine ipſo locum mutare nequeat, <lb/>vt ſit tutum. tenebræ, <reg norm="priuatioque" type="simple">priuatioq́;</reg> viſus ſunt ferè vnum, & idem. </s> + <s xml:space="preserve">Neque vllus eſt@ſen-<lb/>ſus, qui ſit magis ſimilis intellectui quam viſus: </s> + <s xml:space="preserve">neque alij ſenſus habent obiecta vi-<lb/>ciſſim communia, quæ non ſint etiam oculo communia, ſed inter oculum, & quem <lb/>libet alium ex ſenſibus, inuenientur quidem obiecta communia, quæ cum alijs non <lb/>communicabunt, vt inter oculum & tactum, figura, acutum, obtuſum, & ſimilia, <lb/>quæ alijs ſenſibus non percipiuntur. </s> + <s xml:space="preserve">Mediante viſu, & auditu etiam, <reg norm="compræhendum" type="context">compræhendũ</reg> <lb/>tur variæ diſtantiæ, <reg norm="ſitusque" type="simple">ſitusq́;</reg> obiectorum, nec non proportiones, & alia quę ab alijs ſen-<lb/>ſibus non compræhenduntur. </s> + <s xml:space="preserve">Multa obiecta deinde ſunt ſubiecta guſtatui, quę alijs <lb/>accidentibus prędita ſunt, vnde cum fuerint ſemel deguſtata, talia, qualia ſunt ab o-<lb/>culo percipiuntur, quod nullus ex alijs ſenſibus præſtabit. </s> + <s xml:space="preserve">Idem de obiectis odora-<lb/>tus dico. </s> + <s xml:space="preserve">Senſuum nullus eſt qui maiorem ſimilitudinem gerat cum vigilia & cum <lb/>vita, quam viſus, neque aliquid eſt, quod magis repræſentet imaginem ſomni, & <lb/>mortis, quàm cęcitas.</s> + </p> + <p> + <s xml:space="preserve">Qui ſibi oculos eruit vt melius ſpecularetur maxima ſtultitia prius obcęcatus fuit <lb/>quia ſoni magis impediunt ſpeculationem quàm lumina, imò qui commodè vult <lb/>contemplari, quantum plus poteſt nititur longius eſſe ab omni ſtrepitu, magis quàm <lb/>à locis luminoſis, & animal magis lætatur lumine quam ſono: </s> + <s xml:space="preserve">& ad ſpeculationem <lb/>nos magis inuitat harmonia luminum variorum colorum & figurarum, quàm har-<lb/>monia ſonorum, præterquam quod inſtrumentum viſus totius corporis partium eſt <lb/>pulcherrima, & in eminentiori loco locata, ſi de inſtrumentis ſenſuum loquamur, & <lb/>veluti fineſtræ animæ. </s> + <s xml:space="preserve">Et ſi Ariſtoteles dicat oculos & aures in vno <reg norm="eodemque" type="simple">eodemq́;</reg> orbe <lb/>exiſtere, volens inferre quod in eodem æquilibrio ſint æqualiter alta non ita ſe ha-<lb/>bet, quia (ſi de homine loquamur) oculus eſt altior aure. </s> + <s xml:space="preserve">Beneficio huius ſenſus, eo <lb/>rum quæ abſunt, & longo iam tempore ſunt mortui, animi ſenſa, & conceptus intel-<lb/>ligimus, neque alia ratione rerum omnium memoria ſeruari poteſt. </s> + <s xml:space="preserve">Si cabala un-<lb/>quam vera fuit, nulla alia ratione eſt deleta, quam quia alicuius ſigni viſibilis medio <lb/>conſeruata non fuerit, & quæcunque non ſcribuntur, ideſt 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 ſemper pretio fuit pictura <reg norm="quam" type="context">quã</reg> muſi-<lb/>ca: </s> + <s xml:space="preserve">obiectis viſibilibus magis quam ijs quæ ſub 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 ſignificare Deum, <lb/>oculi medio id præſtabant.</s> + </p> + <p> + <s xml:space="preserve">Oculus, reſpectu aliorum inſtrumentorum ſenſuum, eſt quaſi 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 naſcitur, ſtatim cæcus euadat, neque aliqua eſt 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">Ariſtoteles ad finem cap .15. lib. pri <lb/>mi poſteriorum ait, clarum eſſe quod ſi aliquis ſenſus deficiat, futurum vt aliqua <lb/>quoque ſcientia deſit. </s> + <s xml:space="preserve">Conſidera, quot ſcientijs careret homo, ſi viſu orbaretur.</s> + </p> + <p> + <s xml:space="preserve">Et in tertio de anima ait, eum qui non ſentit, nihil intelligere poſſe; </s> + <s xml:space="preserve">id quod in-<lb/>de confirmat, quia nihil ſit in intellectu, quod prius non fuerit in ſenſu. </s> + <s xml:space="preserve">Plato in ti <lb/>meo ait, oculos nobis attuliſſe rerum optimarum notitiam, & ſi oculus non fuiſſet ni <lb/>hil eorum, quæ ad cœlum ſpectant inueniri potuiſſe, & <reg norm="cognitionem" type="context">cognitionẽ</reg> diei ac noctis ab <lb/>oculis ortum duxiſſe, vt reuolutiones menſium, & annorum metiri, & tempus co-<lb/>gnoſcere, & inueſtigare ordinem naturæ vniuerſalis poſſemus; </s> + <s xml:space="preserve">quibus <reg norm="philoſophiam" type="context">philoſophiã</reg> <lb/>nobis comparauimus, ut alia multa omittam, quæ ibi à Platone dicuntur. </s> + <s xml:space="preserve">Addam <lb/>hic & aliam ſpecialem differentiam inter auditum & viſum, quæ eſt, vt obiectum vi <lb/>ſus ſit permanens, & obiectum auditus tranſitorium ſiue ſucceſſiuum aut, vt alio mo <lb/>do idem dicamus, obiectum viſus particpes ſit æternitatis, illud autem quod eſt au-<lb/>ditus non item, nam auditus tempori ſubiectus eſt, viſus autem minimè. </s> + <s xml:space="preserve">Vel ſi di-<lb/>camus operationem auditus abſque tempore fieri non poſſe cum ſit motio, operatio <lb/>verò viſus, nullo indiget tempore, cum ip ſa ſit momentanea, & propterea inſtan-<lb/>tanea. </s> + <s xml:space="preserve">Nam momentum non eſt motus, nec inſtans 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 noſtrum videamus, & non æſtate, ratio eſt ab <lb/>eiuſdem halitus congelatione, quæ ab extrinſeco frigore fit. </s> + <s xml:space="preserve">Prius enim <lb/>ſcire debes aerem <reg norm="attractum" type="context">attractũ</reg> in pulmone, foras deinde erumpere cum alio <lb/>vapore aliquantulum craſſiore humido, & excrementitio expulſo à natu-<lb/>ra, quæ continuò noſtrum corpus euaporare facit, vnde ſequitur dum aer foras à pul <lb/>mone pellitur, maiorem ſemper materiæ portionem, ea quæ intus attracta eſt exire: <lb/></s> + <s xml:space="preserve">vnde ſtatim vt dicta materia foras expulſa, frigidum aerem offendit, cum conſtet ex <lb/>partibus craſſis, & obnoxiis congelationi, condenſatur in formam vaporis, ad dif-<lb/>ferentiam aeris ambientis qui in ſe eas partes craſſas non habet, à quibus <reg norm="quidem" type="context">quidẽ</reg> par-<lb/>tibus condenſatis, & redditis opacis reflectitur lumen, atque hanc ob cauſam æſtate <lb/>hoc non fit, quia calor vim condenſandi non habet.</s> + </p> + <p> + <s xml:space="preserve">Ventus nihil aliud eſt quam quidam aeris motus, cum condenſatur, ob defectum <lb/>caloris, neque (pace Ariſtotelis dicam) eſt exhalatio ſicca. </s> + <s xml:space="preserve">Exemplum à Vitruuio <lb/>allatum nil planè valet, quantum ſpectatad 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 eſt, quia eius beneficio, non ſolum arcemus à nobis aerem ambientem <lb/>calidum, ſed alium etiam aerem circa nos condenſamus: </s> + <s xml:space="preserve">& quia ordo naturæ eſt hu <lb/>iuſmodi quod quemadmodum calor ſequitur raritatem <reg norm="corporum" type="context">corporũ</reg>, ſic etiam frigus <lb/>eorundem denſitatem ſequatur. </s> + <s xml:space="preserve">Quod ſi vis vt exemplo illuſtrem, diligenter ob-<lb/>ſeruato 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 ſolaris fert, qui per vim, <lb/>dictum aerem rarefactum conſeruabat, ſtatim dictum aerem condenſari cognoſces: <lb/></s> + <s xml:space="preserve">& quia ea condenſatio homogenea non eſt, ob diuerſas rationes, hanc ob cauſam <lb/>percipimus eam aeris impulſionem, & inæqualiter, dum verò eadem vmbra diſce-<lb/>dit, ventus, ferè, ſtatim ceſſat, & ſæpe ante quam dicta vmbra diſcedat; </s> + <s xml:space="preserve">cuius rei cau <lb/>ſa eſt longa mora quam trahi vmbra, ita vt prius abſoluatur reditus aeris ad <reg norm="formam" type="context">formã</reg>, <lb/>quæ ei conuenit in huiuſmodi vmbra, quam faciet nubes dum Sol deregitur.</s> + </p> + <p> + <s xml:space="preserve">Vera non ſunt ea, quæ tibi Arnoldus dixit, vt mihi tuis literis ſignificaſti. </s> + <s xml:space="preserve">Nam ego <lb/>ita dixi, videlicet, quod quoti eſcunque aliquis aſpexerit aliquod punctum in ſuper-<lb/>ficie ſpeculi, </s> + <s xml:space="preserve">tunc imaginem ipſius poſt dictam ſuperficie<unsure/>m videbit duplicatam, ſi <lb/>verò aſpexerit imaginem intra ſpeculum, </s> + <s xml:space="preserve">tunc illud punctum videbit duplicatum, <lb/>huiuſmodi autem rei ratio pendet ab hijs quę ad Franciſcum Vimercatum ſcri-<lb/>pſi, quæ ſi memoria tenes, nullum tibi dubium remanebit. </s> + <s xml:space="preserve">Nam ea tibi omnia <lb/>oſtendi.</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 ſuper primum cap. noni lib. eiuſdem Authoris, ita dicens.</s> + </p> + <quote> + <s xml:space="preserve">Benè eſſe poteſt vt diagonalis (quadrati ſcilicet) numerorum via reperiatur, ſed <lb/>fortaſſe intercedent fracta.</s> + </quote> + <p> + <s xml:space="preserve">Miror te non memoria tenere quid ſint numeri rationales quidúe ſurdi, <reg norm="neque" type="simple">neq;</reg> con <lb/>ſideras, non ſolum non eſſe diuiſibilem in tres æquas partes omnem proportionem <lb/>rationabilem, ſed neque in duas, vt ſunt ſuperparticulares proportiones, necnon <lb/>aliæ innumeræ, ſed cum talia ſcribis te nimis parum verſatum in iſtis rebus oſtendis.</s> + </p> + <p> + <s xml:space="preserve">Id verò quod tibi dicere volebam nudiustertius de Mercurio erat, quod nullo pa <lb/>cto confidendum eſt calculis qui fiunt de curſu Mercurij, eo quod eius ſitus nullo mo <lb/>do obſeruabilis eſt, nam ipſe nunquam nec vbiuis locorum orbis terrarum viſibilis <lb/>eſt altior .18. gradibus ſupra orizontem, ſed neque confidendum eſſet ſi <reg norm="etiam" type="context">etiã</reg> ipſum <lb/>videremus altum .20. gradibus, </s> + <s xml:space="preserve">propterea quod magna refractio <reg norm="radiorum" type="context">radiorũ</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 eſt, propter diformem ſeu inæqualem craſſiciem vaporum quæ continuò muta <lb/>tur. </s> + <s xml:space="preserve">Imo multoties eum. videre putabimus ſupra orizontem, exiſtente ipſo ſub <lb/>orizonte.</s> + </p> + </div> + <div type="letter" level="4" n="2"> + <head style="it" xml:space="preserve">Quod Ouidius tr anſcurrit 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 ceſpitauit, cum pro itinere vnius diei efficiat, vt Phaeton à <lb/>patre edoctus ſit etiam de itinere annuali.</s> + </p> + <p> + <s xml:space="preserve">Nam, quod Phaeton petat pro curſu vnius diei, clarè patet <reg norm="ex" type="conjecture">ẽx</reg> diuerſis locis, & pri-<lb/>mò vbi ita ſcribit 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 eſt, & 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 eſt altiſſima cęlo. <lb/></s> + <s xml:space="preserve">Vnde mare, & terras ipſi mihi ſæpe videre. <lb/></s> + <s xml:space="preserve">Fit timor & pauida trepidat formidine pectus. <lb/></s> + <s xml:space="preserve">Vltima prona via eſt & 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">Dumq́;</reg> ea magnanimus <reg norm="Phaenton" type="context">Phaẽton</reg> miratur, <reg norm="opusque" type="simple">opusq́;</reg> <lb/>Perſpicit, ecce vigil nitido patefecit ab ortu. <lb/></s> + <s xml:space="preserve">Purpureas aurora fores, & plena roſarum. <lb/></s> + <s xml:space="preserve">Atria, diffugiunt ſtellæ, quarum agmina cogit. <lb/></s> + <s xml:space="preserve">Lucifer, & coeli ſtatione nouiſſimus exit.</s> + </quote> + <p> + <s xml:space="preserve">Necnon vbi ita inquit.</s> + </p> + <quote> + <s xml:space="preserve">Et ſi (modo credimus) vnum <lb/>Iſſe diem ſine Sole ferunt, incendia lumen Præbebant.</s> + </quote> + <p> + <s xml:space="preserve">Quod autem à Patre inſtruatur etiam de curſu annuali, <lb/>videbitur vbi ita dicit.</s> + </p> + <quote> + <s xml:space="preserve">Nitor in aduerſum, 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">Forſitan & lucos illic, <reg norm="vrbesque" type="simple">vrbesq́;</reg> Deorum. <lb/></s> + <s xml:space="preserve">Concipias animo <reg norm="delubraque" type="simple">delubraq́;</reg> ditia donis <lb/>Eſſe per inſidias iter eſt, <reg norm="formasque" type="simple">formasq́;</reg> ferarum. <lb/></s> + <s xml:space="preserve"><reg norm="Vtque" type="simple">Vtq́;</reg> viam teneas, <reg norm="nulloque" type="simple">nulloq́;</reg> errore traharis. <lb/></s> + <s xml:space="preserve">Per tamen aduerſi, gradieris cornua Tauri. <lb/><reg norm="Aemoniosque" type="simple">Aemoniosq́;</reg> arcus, <reg norm="violentique" type="simple">violentiq́;</reg> ora Leonis. <lb/><reg norm="Sæuaque" type="simple">Sæuaq́;</reg> circuitu curuantem brachia longo. <lb/>Scorpion atque aliter curuantem brachia cancrum. <lb/></s> + <s xml:space="preserve">Nec tibi quadrupedes animoſos 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 eſt 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 eſt lato curuamine limes. <lb/></s> + <s xml:space="preserve"><reg norm="Zonarumque" type="simple">Zonarumq́;</reg> trium contentus fine, <reg norm="polumque" type="simple">polumq́;</reg> <lb/>Effugit auſtralem <reg norm="iunctamque" type="simple">iunctamq́;</reg> aquilonibus arcton. <lb/></s> + <s xml:space="preserve">Hac ſit iter, manifeſta rotæ veſtigia 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 ſiniſterior preſſam rota ducat ad aram.</s> + </quote> + </div> + <div type="letter" level="4" n="3"> + <head style="it" xml:space="preserve">De ſupputatione quinque corporum regularium. <lb/>De aliquibus etiam eorum ſympathijs.</head> + <head xml:space="preserve">AD EVNDEM.</head> + <p> + <s xml:space="preserve">ID quod à me deſideras, ab alijs etiam factum eſt, ſed ne me putes laborem euita <lb/>re, non præter mittam aliquid tibi ſcribere, 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 iſtisrebus me aliquo modo exercebam.</s> + </p> + <p> + <s xml:space="preserve">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ <reg norm="cuiuſque" type="simple">cuiuſq;</reg> <reg norm="quinque" type="simple">quinq;</reg> cor-<lb/>porum regularium ab vna <reg norm="eademque" type="simple">eademq́;</reg> ſphæra terminatorum ſeu <reg norm="circunſcriptibilium" type="context">circunſcriptibiliũ</reg> cu-<lb/>rabis primum, cognoſcere quantitatem lateris <reg norm="cuiusque" type="simple">cuiusq́;</reg> eorum, talium partium, qua-<lb/>lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici. <lb/></s> + <s xml:space="preserve">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13. <lb/>lib. Eucli. & inuenies <var>.c.d.</var> tertiam partem ſemidiametri <var>.d.b.</var> eſſe partium .33333. æ-<lb/>qualem ſinui arcus <var>.f.e.</var> graduum .19. mi .28. qui quidem arcus <reg norm="demptus" type="context">dẽptus</reg> <reg norm="cum" type="context">cũ</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 cognoſces, ſumendo ſcilicet ſinum medietatis <var>.b.e.</var> hoc eſt ſinum 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 poſtea quadrato lateris exaedri, & quadrato totius diametri <var>.a.b.</var> reſi-<lb/>dui radix quadrata, erit <var>.a.e.</var> latus Tetraedri. </s> + <s xml:space="preserve">Vel ſi duplicaueris ſinum 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, ſiue, <lb/>vt reſiduus totius dimidij circuli, dempto <var>.b.e.</var> iam ſupra 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/>ſius <var>.d.b.</var> & habebis <var>.f.b.</var> latus quæſitum. </s> + <s xml:space="preserve">Vel ſi malis accipe duplum ſinus 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. ſecundi <lb/>Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s> + </p> + <p> + <s xml:space="preserve">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli <var>g.d.<lb/>a.</var> hoc eſt ipſius arcus <var>.b.n.</var> qui tali angulo ſubiacet, quod cum pluribus modis inue-<lb/>niri poſſit, nihilominus, hunc ſeruabis, inuenies primò quantitatem <var>.d.g.</var> quæ eſt ra <lb/>dix quadrata ſummæ duorum quadratorum hoc eſt <var>.d.a.</var> et <var>.a.g.</var> quæ <var>.a.g.</var> æqualis eſt <lb/>diametro <var>.a.b.</var> vt ſcis, dices poſtea, ſi <var>.d.g.</var> correſpondet ipſi <var>.g.a.</var> cui correſpondet <var>.d.<lb/>h.</var> ſemidiametro ſphæræ? </s> + <s xml:space="preserve">tibi veniet <var>.h.k.</var> ſinus arcus <var>.a.h.</var> hoc eſt <var>.b.n.</var> graduum .63 -<lb/>min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du <lb/>plum eſt partium .105142. pro latere Icoſaedri.</s> + </p> + <p> + <s xml:space="preserve">Incipiendo nunc à Tetraedro, ſcire debes, quod pars <var>.a.c.</var> totius diametri <var>.a.b.</var> æ-<lb/>qualis eſt axi ipſius Tetraedri, quæ quidem <var>.a.c.</var> vt ſubſeſquialtera ipſius <var>.a.b.</var> erit par <lb/>tium .13333.</s> + </p> + <p> + <s xml:space="preserve">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-<lb/>thodo, inueniendo primum radicem quadratam trium quartarum quadrati <lb/>ipſius <var>.a.e.</var> lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per<lb/>pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-<lb/>to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit <reg norm="partium" type="context">partiũ</reg> .141416. <lb/>& hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit <lb/>ſuperficiem quæſitam, hoc eſt baſim Tetraedri <reg norm="partium" type="context">partiũ</reg> <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg> .11546192152. <lb/><reg norm="Hanc" type="context">Hãc</reg> demum baſim 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 animaduerſionem, quæ eſt, <lb/>quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-<lb/>qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi <var>.b.f.</var> <lb/>quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo <lb/><var>b.f.</var> habebimus intentum. </s> + <s xml:space="preserve">Et quod hoc verum ſit poſſumus ita demonſtrare.</s> + </p> + <p> + <s xml:space="preserve">Primum, notum nobis eſt, ipſam 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, ipſum triangulum circunſcribentis, terminatur, & à baſi, vt in tertio <lb/>propoſito decimæſeptimæ quartidecimi Eucli. probatur, ex quo ſequitur proportio<lb/>nem huiuſmodi perpendicularis ad axem Tetraedri, hoc eſt ad <var>.a.c.</var> ſeſquioctauam <lb/>eſſe in potentia, ex penultima primi Eucli. </s> + <s xml:space="preserve">Sed cum <var>.d.c.</var> tertia pars ſit ipſius <var>.d.a.</var> vt <lb/>etiam ex .2. propoſito, ſeu corollario decimæſeptimæ .14. lib. diſcurrere licet, cum ex <lb/>dicto corollario <var>.d.c.</var> ſit ſexta pars ipſius <var>.a.b</var>. </s> + <s xml:space="preserve">Quare <var>.d.c.</var> quarta pars erit ipſius <var>.a.c.</var> vn <lb/>de <var>.a.c.</var> ſeſquitertia erit ipſi <var>.a.d.</var> in longitudine, <reg norm="ideoque" type="simple">ideoq́;</reg> quadratum ipſius <var>.a.d.</var> ad qua-<lb/>dratum ipſius <var>.a.c.</var> erit vt .9. ad .16: </s> + <s xml:space="preserve">& ita duplum quadrati ipſius <var>.a.d.</var> hoc eſt quadra-<lb/>tum ipſius <var>.b.f.</var> ad quadratum ipſius <var>.a.c.</var> erit, vt .18. ad .16. hoc eſt ſeſquioctauum, er-<lb/>go <var>.b.f.</var> æqualis erit dictæ perpendiculari, ex .9. quinti.</s> + </p> + <p> + <s xml:space="preserve">Cubus poſtea ipſius <var>.b.e.</var> erit partium .1539838576570176.</s> + </p> + <p> + <s xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in ſemidiametrum, quod <lb/>productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-<lb/>ductum, multiplicando per .100000. ſemidiametrum ſphæræ, tibi dabit columnam <lb/>quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum <lb/>erit ipſum Octaedron partium .1333333333333.</s> + </p> + <p> + <s xml:space="preserve">Pro Icoſaedro autem, oportet prius quantitatem perpendicularis inuenire, quæ <lb/>perpendicularis, per æqualia diuidit baſim ipſius Icoſaedri, quæ vt radix quadrata <lb/>trium quartarum quadrati lateris ipſius baſis, erit partium .91055. talium, qualium <lb/>dictum latus erit partium .105142. cuius medietas eſt .52571. quæ medietas ſi mul-<lb/>tiplicata fuerit cum dicta perpendiculari, dabit totam baſim ſuperficialem, hoc eſt <lb/>ſuperficiem vnius trianguli æquilateris partium ſuperficialium .4786852405. quo <lb/>facto, accipe quadratum duarum tertiarum ipſius, hic ſupra dictæ perpendicularis, <lb/><reg norm="ipſumque" type="simple">ipſumq́;</reg> deme ex quadrato ſemidiametri ſphæræ, hoc eſt, ex quadrato <reg norm="ipſius" type="simple">ipſiꝰ</reg> .100000 <lb/>radix poſtea quadrata reſidui, erit partium .79468. & hæc erit perpendicularis à cen <lb/>tro ſphærę ad vnam baſim ipſius Icoſaedri, quam volueris, quam perpendicularem <lb/>ſi multiplicaueris cum quantitate ſuperficiali, hic ſuperius reperta, vnius baſis, con-<lb/>ſequeris columnam trilateram partium .380401586920540. cuius tertia pars, erit <lb/>partium .126800528973513. pro vna ex .20. </s> + <s xml:space="preserve">Pyramidibus ipſum corpus compo-<lb/>nentibus. </s> + <s xml:space="preserve">Breuius tamen hoc efficiens, ſi multiplicaueris baſim dictam, cum tertia <lb/>parte ipſius perpendicularis, hanc poſtea pyramidem multiplicando per .20. habebis <lb/>totam corpulentiam ipſius Icoſaedri partium .2536010579470260.</s> + </p> + <p> + <s xml:space="preserve">Pro Duodecaedro demum, accipe ſinum gra .36. qui <reg norm="gradus" type="simple">gradꝰ</reg> ſunt pro dimidio quin <lb/>tæ partis totius gyri circularis, <reg norm="qui" type="simple">ꝗ</reg> <reg norm="quidem" type="context">quidẽ</reg> ſinus, erit partium .58778. cuius quadratum <lb/>ſi <reg norm="dem" type="context">dẽ</reg> pſeris ex quadrato <reg norm="ipſius" type="simple">ipſiꝰ</reg> .100000. ſemidiametri circuli <reg norm="circunſcribentis" 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 reſidui, erit perpendicularis du-<lb/>cta à centro dicti circuli ad medium vnius lateris ipſius pentagoni, quæ perp endicu <lb/>laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778. <lb/>Nunc verò dicendo ſi .58778. dat .80902. quid nobis dabit .35684? </s> + <s xml:space="preserve">medietas lateris <lb/>ipſius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipſius penta-<lb/>goni, ad latus ipſius Duodecaedri, quæ multiplicata cum me dietate ſupradicta ip-<lb/>ſius lat eris, hoc eſt cum .35684. producet vnum ex quinque triangulis componenti-<lb/>bus vn um pentagonum, ſeu vnam baſim ipſius Duodecaedri, quod quidem triangu <lb/>lum, erit partium .1752655344. ſu perficialium, quas ſi per quinque multiplicaueris <lb/>habeb is vnam baſim pentagonam dicti corporis partium .8763276720. </s> + <s xml:space="preserve">Dicendum <lb/>poſtea eſt, ſi ad .80901. conuenit ſemidiameter circularis partium .100000. quid <reg norm="con" type="context">cõ</reg> <lb/>ueniet partibus .49116. dabit .60711. pro tali ſemidiametro 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, ſi dempſeris ex quadrato ipſius .100000. ſemidiametro ſphęræ, </s> + <s xml:space="preserve">tuncradix qua-<lb/>drata reſidui, erit perpendicularis à centro ſphæræ ad centrum pentagoni partium, <lb/>79461. cuius tertia pars, ſi multiplicata fuerit cum pentagono ſupra reperto dicti cor <lb/>poris producet vnam ex .12. pyramidibus componentibus dictum Duodecaedron, <lb/>quæ pyramis, demum, multiplicata per .12. dabit totam corpulentiam ipſius Duo <lb/>decaedri partium .2785354925791680.</s> + </p> + <p> + <s xml:space="preserve">Nunc verò ſi experiri voluerimus vtrum iſti calculi duorum corporum maiorum <lb/>ſint rectè ſupputati, <reg norm="dicemus" type="simple">dicemꝰ</reg> ſi ad <reg norm="corpus" type="simple">corpꝰ</reg> .12. <reg norm="baſium" type="context">baſiũ</reg>, <reg norm="quod" type="wordlist">qđ</reg> eſt <reg norm="partium" type="context">partiũ</reg> .2785354925791680 <lb/>conuenit numerus partium .2536010579470260. ipſius Icoſaedri, quid conueniet <lb/>lateri cubi partium .115476. & inueniemus conuenire latus ipſius Icoſaedri partium <lb/>105138. eo quod probatum ſit in <ref id="ref-0026">.10. propoſitione .14. li. Eucl.</ref> eandem <reg norm="proportionem" type="context">proportionẽ</reg> <lb/>eſſe corpulentiæ ipſius Duodecaedri ad corpulentiam ipſius Icoſaedri, quæ lateris <lb/>cubi ad latus Icoſaedri.</s> + </p> + <p> + <s xml:space="preserve">Hæc autem corpora, ita ſibi inuicem, & cum eorum ſphæra harmonicè <reg norm="conueniunt" type="context">conueniũt</reg> <lb/>quemadmodum antiqui philoſophi inuenerunt, vt <reg norm="mirandum" type="context">mirandũ</reg> non ſit, ipſos credidiſ-<lb/>ſe omnia quæ natura conſtant, aliquo pacto exiſtis corporibus fieri. </s> + <s xml:space="preserve">Conſidera quæ-<lb/>ſo quomodo conueniant inuicem Tetraedron, Octaedron, & Icoſaedron, cum uniuſ-<lb/>cuiuſque baſes ſint triangulares æquilateræ intelli gendo ſemper hæc corpora ab ea-<lb/>dem ſphæra circunſcriptibilia.</s> + </p> + <p> + <s xml:space="preserve">Octaedron, cum Tetraedro etiam in hoc conuenit, quod latus Octaedri æquale <lb/>ſit ei perpendiculari, quæ diuidit baſim Tetraedri per æqualia, vtſupra demonſtra-<lb/>uimus.</s> + </p> + <p> + <s xml:space="preserve">Harmonicis etiam interua llis hæc duo corpora inuicem concordantur, cum baſis <lb/>Tetraedri ad baſim Octaedri ſeruet proportionem ſeſquitertiam, conſonantiæ dia-<lb/>teſſaron. </s> + <s xml:space="preserve">Et proportio omnium ſuperficierum ſiue baſium Octaedri ſimul ſumpta-<lb/>rum, ad omnes baſes ipſius Tetraedri ſimul ſumptas ſit ſeſquialtera, conſonantiæ dia <lb/>pentis. </s> + <s xml:space="preserve">Neque omittendum eſt, quod proportio Octaedriad triplum Tetraedri ſit, <lb/>vt latus Octaedri ad latus Tetraedri.</s> + </p> + <p> + <s xml:space="preserve">Proportio verò lateris Octaedri, ad axem Tetraedri, potentia eſt ſeſquioctaua, <lb/>vt ſupra vidimus interuallum ſcilicet harmonicum toni maioris.</s> + </p> + <p> + <s xml:space="preserve">Harmonia verò Tetraedri, & Exaedri <reg norm="cum" type="context">cũ</reg> eorum ſphæra, talis eſt, vt proportio dia <lb/>metriſphæræ, potentia, tripla ſit lateri Exaedri, & ſeſquialtera lateri Tetraedri, ex <lb/>quo ſequitur latus Tetraedri potentia duplum exiſtere lateri Exaedri. </s> + <s xml:space="preserve">Interuallum <lb/>enim triplum in harmonicis, componitur ex diapaſon, & diapente, & ſonat ſpeciem <lb/>diapentis. </s> + <s xml:space="preserve">Duplum verò eſt diapaſon, ſeſquialterum autem eſt di apente, quę con-<lb/>ſonantiæ perfectiſſimæ ſunt.</s> + </p> + <p> + <s xml:space="preserve">Proportio verò diametri ſphæræ, potentia dupla eſt lat eri Octaedri, conſonantię <lb/>diapaſon. </s> + <s xml:space="preserve">Ex quo ſequitur proportionem lateris Tetraedri ad latus Octaedri, po-<lb/>tentia, ſeſquitertiam eſſe, hoc eſt conſonantiæ diateſſaron, & proportionem lateris <lb/>Octaedri ad latus Exaedri, potentia, ſeſquialteram eſſe, ita quod quatuor iſtæ poten <lb/>tiæ, ideſt diametri ſphæræ, lateris Tetraedri, lateris Octaedri, & lateris Exaedri con-<lb/>ſtituunt harmoniam ferè perfectiſſimam, ijs terminis comprehenſam .6. 4. 3. 2. (dixi <lb/>ferè, quia ditonus ſupra terminum .3. vel ſemiditonus ſub 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 ſphæræ triplus eſt longitudine ad <reg norm="perpendicularem" type="context">perpendicularẽ</reg> ductam <lb/>à centro ſphæræ ad baſim Octaedri, quæ proportio, vt ſupra dictum eſt, dicitur dia-<lb/>paſondiapente, 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 ſphæræ ſeſquialter eſt longitudine axi Tetraedri, conſonantiæ <lb/>diapentis. </s> + <s xml:space="preserve">Axis autem Tetraedri ſeſquitertius eſt longitudinis ſemidiametro ſphæ-<lb/>ræ conſonantiæ diateſſaron. </s> + <s xml:space="preserve">Ita quod iſti tres termini, qui ſunt, diameter ſphæræ, <lb/>axis Tetraedri, & ſemidiameter ſphæræ conſtituunt etiam valde perfectam harmo-<lb/>niam huiuſmodi numeris contentam .6. 4. 3. corpulentia verò Exaedri ad corpu-<lb/>lentiam Tetraedri tripla eſt, conſonantiæ iam ſupradictæ diapaſondiapente. </s> + <s xml:space="preserve">Si ve-<lb/>rò de vniſono aliquid videre deſideras, conſidera æqualitatem dupli quadrati dia-<lb/>metri ipſius ſphæræ, cum omnibus baſibus Exaedri, vel potentia diametri ſphæræ <lb/>cum duabus potentijs ſimul ſumptis, quarum vna eſt lateris Tetraedri, reliqua verò <lb/>lateris Exaedri, vel æqualitatem numerorum laterum Tetraedri, cum baſibus Exae <lb/>dri. </s> + <s xml:space="preserve">Nec mihi videtur ſilentio inuoluendum eſſe, antequam vlterius progrediar no<lb/>tabilem ſympatiam inter triangulum æquilaterum, & Tetraedron (<reg norm="quanuis" type="context">quãuis</reg> <reg norm="triangulum" type="context">triangulũ</reg> <lb/>corpus non ſit) non ſolum ob <reg norm="inalterabilitatem" type="context">inalterabilitatẽ</reg> harum duarum figurarum. </s> + <s xml:space="preserve">(nam omnes <lb/>aliæ alterabiles eſſe poſſunt, ijſdem lateribns exiſtentibus, cum ex quadrato rom-<lb/>bus, vel ex pentagono ęquiangulo, pentagonum non æquiangulum & c. efficiatur) <lb/></s> + <s xml:space="preserve">ſed quod quemadmodum latus trianguli æquilateri ſeſquitertium potentia eſt per-<lb/>pendiculari ipſum per æqualia diuidenti, ita latus Tetraedri, ſeſquialterum eſt po-<lb/>tentia axi ipſius Tetraedri, vnde cum dempta fuerit illa proportio ſeſquitertia, ex <lb/>hac ſeſquialtera relinquetur nobis proportio ſeſquioctaua, inter perpendicularem <lb/>trianguli, & axem Tetraedri (quod etiam ſupra demonſtrauimus.) </s> + <s xml:space="preserve">Tranſeamus nunc <lb/>hęc, nec omittamus tamen ſympatias quaſdam inter Exaedron, Octaedron, & Tetra <lb/>edron, hoc eſt quod eadem proportio ſit inter corpulentias Exaedri, & Octaedri, <lb/>quæinter eorum ſuperficies, nec non, vt latus Exaedri ad ſemidiametrum ſphæræ. <lb/></s> + <s xml:space="preserve">Proportio verò baſis Exaedri ad baſim Tetraedri, vtlatus Tetraedri ad perpendicu <lb/>larem diuidentem per æqualia eius baſim.</s> + </p> + <p> + <s xml:space="preserve">Hactenus ſatis dictum ſit de Tetraedro, Exaedro, & Octaedro cum ſphæra. </s> + <s xml:space="preserve"><reg norm="Dicem" type="context">Dicẽ</reg> <lb/>dum nunc cenſeo aliquid de reliquis duobus mirabilibus corporibus, quamuis ferè <lb/>omnia hæc ab antiquis philoſophis inuenta ſint, quorum primum eſt, quod tam ba-<lb/>ſis Duodecaedri, quam Icoſaedri, ab vno <reg norm="eodemque" type="simple">eodemq́;</reg> circulo circunſcriptibiles ſunt, ve <lb/>rùm, talis paſſio accidit etiam baſibus Exaedri & Octaedri. </s> + <s xml:space="preserve">Præterea quemadmo-<lb/>dum in Duodecaedro, quilibet angulus ſolidus terminatur tribus angulis pentago-<lb/>norum æquiangulorum ita in Icoſaedro, quilibet angulus ſolidus viceuerſa 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 ſolidos angulos trian-<lb/>gulares, habet Duodecaedron, quot baſes triangulares continet Icoſaedron.</s> + </p> + <p> + <s xml:space="preserve">Et Icoſaedron, tot ſolidos angulos <reg norm="pentagonos" type="context">pẽtagonos</reg>, quot baſes <reg norm="pentagonas" type="context">pẽtagonas</reg> habet Duo <lb/>decaedron. </s> + <s xml:space="preserve">Et tam vnum quam alterum habet .60. angulos ſuperficiales. </s> + <s xml:space="preserve"><reg norm="Eademque" type="context simple">Eadẽq́;</reg> <lb/>proportio eſt omnium baſium ſimul <reg norm="ſumptarum" type="context">ſumptarũ</reg> Duodecaedri ad omnes baſes ſimul <lb/>ſumptas ipſius Icoſaedri, quæ corpulentiæ ipſius Duodecaedri ad corpulentiam <lb/>Icoſaedri (quamuis hęc paſſio accidat Exaedro cum Octaedro, vt ſpra diximus) quę <lb/>quidem proportio, eadem etiam eſt, quę lateris Exaedri ad latus Icoſaedri, vt ſu-<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 Ingenioſißimo Horologiorum Serenißimi <lb/>Sabaudiæ Ducis Artifici.</head> + <p> + <s xml:space="preserve"><emph style="sc">NOnnvnqvam</emph> conſideraui mirabilem pulchritudinem, ſimul cum vtili-<lb/>tate coniunctam, illorum horologiorum, quæin Germania conſtruuntur <lb/><reg norm="cum" type="context">cũ</reg> mobili Rete, ſeu Aranea Aſtrolabij <reg norm="ſuper" type="simple">ſuꝑ</reg> <reg norm="Tabulam" type="context">Tabulã</reg> regionis, in <reg norm="quibus" type="simple">ꝗbus</reg> <reg norm="conti" type="context">cõti</reg> <lb/>nuo <reg norm="videntur" type="simple">vident̃</reg> oriri, <reg norm="occidereque" type="simple">occidereq́;</reg> cæleſtia ſigna, cælum mediare ſupra orizon <lb/><reg norm="tem" type="context">tẽ</reg><unsure/>, necnon ſub eo, & vt vno verbo dicam, continuo erecta videtur tota coelifigura. <lb/></s> + <s xml:space="preserve">Sed quia talia horologia omnia eorum limbum diſtinctum habent in .24. horas, qua <lb/>propter diametrum limbi, minorem duobus palmis, ſeu ſemipede eſſe non oportet <lb/>neinterſtitia horarum iuſtò breuiora ſeu anguſtiora efficiantur, etiam ne interualla <lb/>dentium rotæ indicis nimis anguſta ſint. </s> + <s xml:space="preserve">Sed quia talis magnitudo vt plurimum in-<lb/>commoda exiſtit. </s> + <s xml:space="preserve">Ideo non inutile fore cogitaui, ſi modus aliquis inuentus fuerit, <lb/>vt ea omnia efficiantur in limbo diuiſo tantummodo in .12. horas æquales, <reg norm="ipſumque" type="simple">ipſumq́;</reg> <lb/>inueni, qui quidem erit, efficiendo vt Tabula (in qua deſignantur cęleſtes domus, <lb/>cum almicantarat, atque azimut) Reti ſubiectæ, mobilis ſit, tardior tamen ipſo Re-<lb/>te cum indice, pro duplo temporis, hoc eſt, quod eo tempore, quo Aranea cum in <lb/>dice circunuoluetur ſpacio .12. horarum vno gyro perfecto, ipſa Tabula efficiat tan <lb/>tummodo ſexinterſtitia horarum. </s> + <s xml:space="preserve">Ideſt dum Tabula dicta eſſicit vnam integram re <lb/>uolutionem, Aranea, ſeu Zodiacus cum indice, duas efficiat reuolutiones. </s> + <s xml:space="preserve">Ita quod <lb/>Aranea cum indice perficiet vnam reuolutionem ſpaci o temporis .12. horarum, Ta-<lb/>bula verò perficiet eam ſpacio temporis .24. horarum. </s> + <s xml:space="preserve">Vnde ſequetur quod Ara-<lb/>nea ſeu Zodiacus cum indice, ſpacio .24. horarum perfectè circunuoluetur ſupra Ta-<lb/>bulam, & ita huiuſmodi horologia, in hoc nihil differrent ab illis ſupradictis. </s> + <s xml:space="preserve">Vt au <lb/>tem facias dictam tabulam tardiorem duplo temporis Araneæ cum indice, quamuis <lb/>diuerſis modis hoc fieri poſſit, pręſtantiorem tamen iudico, ſi 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="vnaquęque" type="simple">vnaquęq;</reg> liberè poſſit volui, ſimiliter <lb/>ſi cum ea horologii particula (quę <reg norm="circumagit" type="context">circũagit</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">aũt</reg> ipſum vocabo, <emph style="sc">colinvm</emph>, qui ſubro-<lb/>ta fuſi reperitur) coniunxeris alium colinum quem, ſecundum vocabo, <reg norm="concentricum" type="context">concentricũ</reg> <lb/>verò cum primo, cum <reg norm="eoque" type="simple">eoq́;</reg> conſolidato, numerum verò dentium, tam Rotę adiunctę <lb/>quam ſecundi colini, varijs modis poteris inuenire, quorum primus erit, vt numerus <lb/>dentium ſecundæ Rotę duplus exiſtat numero dentium primę, efficiendo ſecundum <lb/>colinum eiuſdem numeri dentium quo primum, ſed quia interualla dentium huiuſ-<lb/>modi Rotę, nimis anguſta fortaſſe reſultabunt, </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> ſecun <lb/>dam <reg norm="Rotam" type="context">Rotã</reg> <reg norm="eiuſdem" type="context">eiuſdẽ</reg> numeri <reg norm="dentium" type="context">dentiũ</reg> cuius eſt prima. <reg norm="ſecundum" type="context context">ſecũdũ</reg> vero colinum, medietatis <lb/>numeri dentium cuius erit primus. </s> + <s xml:space="preserve">Attamen ſi primus colinus eſſet .4. dentium, ſecun <lb/>dum oporteret eſſe duorum dentium, vnde motus ſecundę Rotę non eſſet ita conti-<lb/>nuus. </s> + <s xml:space="preserve">Quapropter alium etiam mòdum excogitaui, hoc eſt, cupiendo vt ſecundus <lb/>colinus, extribus dentibus exiſtat, ſi primus ex .4. repertus fuerit, oportebit prius ex <lb/>regula de tribus, numerum quendam inuenire quo inuento ipſum duplicare, & hunc <lb/>duplicatum numerum conueniet ſecundam Rotam habere, vt ipſa poſſit ab illo co-<lb/>lino <reg norm="trium" type="context">triũ</reg> <reg norm="dentium" type="context">dentiũ</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, ſi prima Rota conſtaret ex .36. dentibus, dicendum eſſet, ſi <lb/>4. conuenit cum .36. cum quibus conuenient .3. & inueniemus .27. cum quo numero <lb/>dicta ſecunda 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 dictę ſecundæ Rotæ, vt duplo temporis circunuoluatur quo prima. </s> + <s xml:space="preserve">Sed <lb/>ſi primus colinus conſtaret ex .6. dentibus, exiſtente ſua Rota ex .36. <reg norm="vellemusque" type="simple">vellemusq́;</reg> <reg norm="quod" type="simple">ꝙ</reg> <lb/>ſecundus exiſteret ex .4. </s> + <s xml:space="preserve">tunc ſuam Rotam oporteret habere dentes .48. ex dicta re-<lb/>gula. </s> + <s xml:space="preserve">Si autem primus colinus conſtaret ex numero impari, nihil referret, dummo-<lb/>do huiuſmodi numerus impar, ſeu par, exiſteret pars propria numeri dentium, vel <lb/>ipſius dupli primæ Rotę, hoc eſt, eſſet pars aliquota numeri dentium ipſius primæ <lb/>Rotæ vel ipſius dupli. </s> + <s xml:space="preserve">In ijs verò horologiis in quibus duplum numeri dentium di-<lb/>ctę primę Rotę non erit multiplex numero dentium primi colini, hoc fieri non pote <lb/>rit. </s> + <s xml:space="preserve">Ratio enim tam clarè, tibi conſideranti, patebit, vt nullis verbis indigeat cum <lb/>ſemper numerus dentium ſecundę Rotę multiplex eſſe debeat numero dentium ſe-<lb/>cundi colini. </s> + <s xml:space="preserve">Idem autem non dico de prima Rota cum ſuo colino, hoc eſt, vt nu-<lb/>merus primę multiplex ſit numero ſui colini, nam hoc neceſſarium non eſt. </s> + <s xml:space="preserve">Pona-<lb/>mus exempli gratia primum colinum conſtare ſex dentibus, ſuam vero Rotam den-<lb/>tibus .21. cuius quidem numeri, 6. non eſt pars aliquota, ſed dupli ipſius .21. ipſe .6. <lb/>eſt pars aliquota. </s> + <s xml:space="preserve">Nunc verò ſi voluerimus numerum dentium ſecundæ Rotę inue-<lb/>nire, cuius colinus ex quinque dentibus exiſtat (ſuppoſito primo ex .6. conſtare) </s> + <s xml:space="preserve">tunc <lb/>ex regula de tribus, diuiſo producto, quod fit ex .21. in .5. per .6. exibit .17. cum di-<lb/>midio, cuius duplum eſſet .35. qui multiplex eſt ipſi quinque. </s> + <s xml:space="preserve">Reperto igitur nume <lb/>ro ſecundę Rotę, cum numero ipſius colini, oportet nunc ſcire modum compoſitio-<lb/>nis, ſeu coniunctionis harum rerum, hoc eſt duorum colinorum <reg norm="concentricorum" type="context">concentricorũ</reg> (ſed <lb/>de ijs ſatis iam ſuperius dictum fuit) duarum Rotarum concentricarum cum Tabula, <lb/>cum Zodiaco, & cum indice, ſeu Oſtenſore, cuius quidem Oſtenſoris medietas tan <lb/>tummodo nobis ſufficiet. </s> + <s xml:space="preserve">Sciendum igitur nunc eſt quod cum primus colinus re-<lb/>uoluat totam primam Rotam, ſpacio temporis .12. horarum, oportet vt eius axis, ſeu <lb/>arbor voluat oſtenſorem, <reg norm="Zodiacumque" type="simple">Zodiacumq́;</reg>, eodem temporis ſpacio, & quia Rota hęc <lb/>inalterabis eſt, propter eius coniunctionem cum ſuo colino, & nos oporteat indicem <lb/><reg norm="Zodiacumque" type="simple">Zodiacumq́;</reg>, quotidie ferè, dirigere, <reg norm="ſuisque" type="simple">ſuisq́;</reg> locis collocare, ideo nos oportet, indi <lb/>cem, Zodiacum, & primam Rotam, ita cum axe, ſeu arbore coniungere, vt poſſimus <lb/>dicta omnia efficere. </s> + <s xml:space="preserve">Pars igitur Arboris, ſeu axis dicti, quæ ingredi debet in prima <lb/>Rota, ſit rotunda, & contigua ipſi Rotæ, non autem continua, vel cum Rota conſoli-<lb/>data. </s> + <s xml:space="preserve">Pars verò quę per foramen Zodiaci, ſeu Araneę tranſibit, ſit quadrata vſque <lb/>ad <reg norm="ſummitatem" type="context">ſummitatẽ</reg> ipſius axis (tali ſpiſſitudine, vt in claui ipſius horologij ingredi poſ-<lb/>ſit) & ita foramen ipſius Araneę, quadratum ſit, Oſtenſor autem circa axem, com <lb/>poſitus ſit tali ordine, vt circa paruum circulum volui poſſit, qui paruus circulus ha-<lb/>beat quadratum foramen, per quod tranſeat axis, qui axis aliquantulum emineat <lb/>ſupra <reg norm="oſtenſorem" type="context">oſtenſorẽ</reg>. </s> + <s xml:space="preserve">Sub Aranea vero vel Zodiaco, locata erit Tabula, vt <reg norm="nunc" type="context">nũc</reg> dicemus, <lb/>ſed ſciendum eſt prius, quod inter Tabulam, & ſuam ſecundam Rotam, aliam lami-<lb/>nam immobilem interpoſitam eſſe oportet, quę circulare foramen habeat, per <reg norm="quod" type="wordlist">qđ</reg> <lb/>quędam breuis fiſtula tranſeat circundans axem & coniungens <reg norm="Tabulam" type="context">Tabulã</reg> cum ſua Ro-<lb/>ta, cuius quidem fiſtulæ ſuperficies concaua, rotunda ſit, ſuperficies verò extrinſe-<lb/>ca, nontota, niſi ea pars, quę ſecundam Rotam ingreditur, vt in rotundo foramine <lb/>ipſius Rotę, dicta fiſtula volui poſſit, pars vero extrinſeca quę Tabulam ingredi de-<lb/>bet, ſit 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, meridianę, & verticalis, vt acu mediante volui poſſit, prout oportebit.</s> + </p> + <p> + <s xml:space="preserve">Perfectum igitur cum fuerit op us hoc, te oportet ſcire modum ipſo vtendi. </s> + <s xml:space="preserve">Qua-<lb/>propter quotieſcunque volueris, aſpice Solis locum in Zodiaco, Ephemeridibus me <lb/>diantibus, idem dico de vnoquoque reliquorum planetarum. </s> + <s xml:space="preserve">Inuento poſtea So-<lb/>lis loco in noſtro Zodiaco horologij, manu mediante, volue oſtenſorem, ita, vt li-<lb/>nea fiduciæ tranſeat per gradum Solis, deinde, claui ipſius horologij mediante, vol-<lb/>ue indicem, ita cum Zodiaco coniunctum, vt linea fiducię, punctum, ſeu partem ho-<lb/>rę oſtendat in limbo horologii, quę quidem hora notanda eſt ſi fuerit ex ijs quę in-<lb/>cipiunt à meridie vſque ad mediam noctem, vel à media nocte vſque ad meri-<lb/>diem, </s> + <s xml:space="preserve">tunc acu ſupradicta mediante, poſita in aliquo illorum quatuor foraminum, <lb/>circunuoluenda eſt Tabula, ita, vt extremitas lineę meridianę ſupra orizontem, ex <lb/>ęquo incidat inter duodecimam horam, & lineam fiducię, computum incipiendo à <lb/>duodecima hora, ſi vero dicta indicis hora fuerit ex ijs quę <reg norm="incipiunt" type="context">incipiũt</reg> à media nocte & <lb/>deſinunt poſtea in meridie, oportebit, acu mediante, circunuoluere Tabulam, quo-<lb/>uſque punctum extremum meridianæ ſub terra, medio loco exiſtat inter <reg norm="duodecimam" type="context">duodecimã</reg> <lb/>horam, & horam oſtenſam à linea fiducię. </s> + <s xml:space="preserve">Quo facto continuo videbis erectam <reg norm="cae- li" type="simple">cę-<lb/>li</reg> figuram. </s> + <s xml:space="preserve">& quia vidiſti loca planetarum in Ephemeridibus, videbis etiam <lb/>eorum loca accidentalia in domibus ſcilicetaccidentalibus, ſi aliquas fixarum in <lb/>Aranea deſiderabis, accipere poteris Ocu. ♉, cor. ♌, ſpi. ♍, Liram, Aquilam, & <lb/>Arcturum, dum locus fuerit capax. </s> + <s xml:space="preserve">Nec te moueat, quod oportebit lineam fiducię <lb/>ſupra gra. Solis quotidie collocare, quod nihil refert. </s> + <s xml:space="preserve">Nam oportet etiam quoti-<lb/>die cordam fuſo 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 Aſtrologia Iudiciaria.</head> + <head style="it" xml:space="preserve">Fu<unsure/>llustriſſ <var>.D.</var> Volfardo Aiſeſtain.</head> + <p> + <s xml:space="preserve"><emph style="sc">NIhil</emph> mihi gratius & iucundius afferri potuit tuis literis, quibus te cupi-<lb/>dum oſtendis ſciendi rationem, quare ego non vna methodo ad omnes <lb/>propoſitiones demonſtrandas vſus ſim, hoc eſt, </s> + <s xml:space="preserve">quare non omnia ea Eucl. <lb/>Theoremata citem in vnaquaque propoſitione, quę ad <reg norm="eam" type="context">eã</reg> demonſtrandam <reg norm="faciunt" type="context">faciũt</reg>, <lb/>quemadmodum in mea Gnomonica vidiſti me aliquando omiſiſſe. </s> + <s xml:space="preserve">Reſpondeo <reg norm="quod" type="simple">ꝙ</reg> <lb/>mathematicę demonſtrationes, hominibus Euclidis Elementa poſſidentibus, non in <lb/>digent aliqua citatione numerorum Theorematum ipſius Euclidis, & ſi aliquando <lb/>vſus ſum aliqua citatione eorundem, hoc feci propter conſuetudinem noſtri tempo <lb/>ris, vel etiam ad faciliorem intelligentiam illorum, quibus ſcribebam. </s> + <s xml:space="preserve">Sed omnia <lb/>quamuis minima citare, vt <reg norm="faciunt" type="context">faciũt</reg> nonnulli, mihi, nimis laborioſum, <reg norm="ſuperfluumque" type="simple">ſuperfluumq́;</reg> <lb/>videtur, preſertim ijs (vt dixi) qui memoria tenent prima Elementa. </s> + <s xml:space="preserve">Hęc igitur <lb/>eſt vna ratio. </s> + <s xml:space="preserve">Alia, quia multoties, ita coniuncta eſt ſpeculatio cum ipſa concluſio <lb/>ne, vt mihi ſępius viſum ſit ſuperfluum, aliquid de ipſa theoria ſcribere. </s> + <s xml:space="preserve">In iis <lb/>enim, quę dum puer eramſcripſi, videbis ſcrupuloſam illam methodum, ſed po-<lb/>ſtea, non niſi in arduis propoſitionibus me nihil eſſentiale prętermittere.</s> + </p> + <p> + <s xml:space="preserve">Circa vero id de quo me interrogas, ſcilicet, vtrum putem omnia vera eſſe, ea <lb/>quę ſcripta reperiuntur in libris Aſtrologæ iudiciarię. </s> + <s xml:space="preserve">Reſpondeo 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 falſa eſſe. </s> + <s xml:space="preserve">Nam illa multitudo partium, vt pars vitę, pars Hylech, pars <lb/>futurorum, & reliquę omnium domorum cœleſtium, ſalua parte fortunę, ſunt merę <lb/>nugę. </s> + <s xml:space="preserve">Idem dico de faciebus, ſiue decanis, de terminis, & de gradibus ipſis, vt pu-<lb/>ta azemenis, puteis, vacuis, fumoſis, & 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">Obſeruationes etiam complexionum ſeu inſluentiarum ipſorum Planetarum rectè <lb/>factæ ſunt, quę etiam à coloribus ipſorum Planetarum ferè iudicari poſſunt. </s> + <s xml:space="preserve">Con-<lb/>iunctiones <reg norm="aſpectusque" type="simple">aſpectusq́;</reg> ipſorum inuicem, ſimiliter mirabilia faciunt, & ex maiori par <lb/>te, ea, quę de iſtis ſcribuntur vera ſunt. </s> + <s xml:space="preserve">Reuolutiones annuę ſimiliter, cum Domino <lb/>anni. </s> + <s xml:space="preserve">Dominum verò orbis <reg norm="Diuiſoremque" type="simple">Diuiſoremq́;</reg> non approbo, nam hic pendet à termino, <lb/>ille verò ab hora. </s> + <s xml:space="preserve">Nouenarias autem Dodecathemoria, Alfridarias, & multa iis ſi <lb/>milia omnia nego. </s> + <s xml:space="preserve">Antiſcia, vera ſunt, ideſt influunt, malos tamen effectus, alia <lb/>plus alia verò minus, prout aliqua eorum ſunt tetragona, alia verò trigona, alia ma-<lb/>gna, alia parua, magna ſunt, vt Arietis cum Virgine, & Librę cum Piſcibus, parua ve <lb/>rò, <reg norm="debiliaque" type="simple">debiliaq́;</reg> Geminorum cum Cancro, & Sagittarij <reg norm="cum" type="context">cũ</reg> Capricorno. </s> + <s xml:space="preserve">Sed difuſius <lb/>hęc <reg norm="omnia" type="wordlist">oĩa</reg> videbis in meo illo particulari tractatu, de quo tibi aliàs dixi, in quo multa <lb/>videbis, quę omnia ab experientia, ex multis à me obſeruatis, comprobata ſunt, <lb/>quem quidem tractatum cum quibuſdam alijs meis ſpeculationibus in lucem prode <lb/>re cupio, ſi fieri poterit, antequam ad directionem mei Horoſcopi cum corpore <lb/>Martis Anęretę perueniam, quę quidem directio circa annum milleſimum quin-<lb/>genteſimum nonageſimum ſecundum 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 reſpiciunt, benignus lector corrigat, quiſciat multos errores in <lb/>editione irrepſiſſe, 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 ſuperficialis, erit ac</xhtml:td> + <xhtml:td xml:space="preserve">in vnitate, ſupreficialis 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 ſunt</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, ſint</xhtml:td> + <xhtml:td xml:space="preserve">tertiæ ſint</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">eſt</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">diſtinguendo</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">ſubſequens</xhtml:td> + <xhtml:td xml:space="preserve">ſubſequentem</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">ſpeculari</xhtml:td> + <xhtml:td xml:space="preserve">conſiderari</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> ſunt 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">addendũ</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">ſingulos itinere</xhtml:td> + <xhtml:td xml:space="preserve">ſingulos 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">ſumma</xhtml:td> + <xhtml:td xml:space="preserve">ſummam</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">deſideraremus</xhtml:td> + <xhtml:td xml:space="preserve">deſiderauerimus</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">diſpoſitis facit</xhtml:td> + <xhtml:td xml:space="preserve">diſpoſitis, 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">diſpoſitis, facit</xhtml:td> + <xhtml:td xml:space="preserve">diſpoſitis, 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">ſufficeret</xhtml:td> + <xhtml:td xml:space="preserve">ſuffecißet</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 ſecundam</xhtml:td> + <xhtml:td xml:space="preserve">quorum ſecundum</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">ſecundæ</xhtml:td> + <xhtml:td xml:space="preserve">ſecundi</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">diſtantius</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">ſit</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, ſi quis</xhtml:td> + <xhtml:td xml:space="preserve">patebit, quòd ſi 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">paſtam</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">ꝓponit</reg> <reg norm="non" type="context">nõ</reg> <reg norm="concludit" type="context">cõcludit</reg> melius <reg norm="autem" type="wordlist">aũt</reg></xhtml:td> + <xhtml:td xml:space="preserve">proponit, <reg norm="non" type="context">nõ</reg> concludit, melius <reg norm="autem" type="wordlist">aũt</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 ſuſtineantur</xhtml:td> + <xhtml:td xml:space="preserve">materiales ſuſtineantur</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">exiſtente, 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">ſufficere</xhtml:td> + <xhtml:td xml:space="preserve">ſufficeret</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">verſa</xhtml:td> + <xhtml:td xml:space="preserve">verſam</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">ſit</xhtml:td> + <xhtml:td xml:space="preserve">ſint</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">ſcindenda</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">oppeſitus</xhtml:td> + <xhtml:td xml:space="preserve">oppoſitum</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">ſestineret</xhtml:td> + <xhtml:td xml:space="preserve">ſustineret</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">ſuffieiant</xhtml:td> + <xhtml:td xml:space="preserve">ſufficiat</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"> ſubſeſquialter<unsure/> + </xhtml:td> + <xhtml:td xml:space="preserve">ſubſeſquialterum</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">philoſophi ſupra</xhtml:td> + <xhtml:td xml:space="preserve">philoſophi, proximè ſupra</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 menſe</xhtml:td> + <xhtml:td xml:space="preserve">contingeret vt menſe</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 conſiderans</xhtml:td> + <xhtml:td xml:space="preserve">regionis) in quo non conſiderauit</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 poſuerimus, 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 ipſos</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">phyloſophiæ</xhtml:td> + <xhtml:td xml:space="preserve">philoſophiæ</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">ſunt 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, creſcat</xhtml:td> + <xhtml:td xml:space="preserve">diffundatur, & creſcat</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 ſer</xhtml:td> + <xhtml:td xml:space="preserve">aliquando me ſer</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">ſoli, quod punctum</xhtml:td> + <xhtml:td xml:space="preserve">ſolis, 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 ſit futurum, vt</xhtml:td> + <xhtml:td xml:space="preserve">neceßarium ſit, 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 ſi velimus</xhtml:td> + <xhtml:td xml:space="preserve">ſi 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"> poteſt 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">ſit circulus</xhtml:td> + <xhtml:td xml:space="preserve">ſit 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 ſi 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">deſignabimus</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">ſi quæ</xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="ſique" type="simple">ſiq;</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">ſupradicta, minuta</xhtml:td> + <xhtml:td xml:space="preserve">ſupradicta 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 cœlo</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">ſit & ob id</xhtml:td> + <xhtml:td xml:space="preserve">ſit, 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> + <xhtml:td xml:space="preserve">in numero</xhtml:td> + <xhtml:td xml:space="preserve">numero</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">cum non videat ſi</xhtml:td> + <xhtml:td xml:space="preserve">cum non videat quod ſi</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">ſuo centro</xhtml:td> + <xhtml:td xml:space="preserve">ſuis centris</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">39</xhtml:td> + <xhtml:td xml:space="preserve">mille milliaria</xhtml:td> + <xhtml:td xml:space="preserve">mille milliariorum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">255</xhtml:td> + <xhtml:td xml:space="preserve">41</xhtml:td> + <xhtml:td xml:space="preserve">t<unsure/>recent a mille</xhtml:td> + <xhtml:td xml:space="preserve">tercenties mille</xhtml:td> + </xhtml:tr> + </xhtml:table> + <pb n="440" file="0440" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0440"/> + <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">257</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve">quem</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">257</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">quem</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">257</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <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 huiuſmodi ſitu, 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">ſupponuntur</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 ſit</xhtml:td> + <xhtml:td xml:space="preserve">a.K. ſit</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">ſemidiameter eße vnius</xhtml:td> + <xhtml:td xml:space="preserve">ſemidiameter 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 vbieſt <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 ſeu 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 ſub 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">eſt</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">ſatile</xhtml:td> + <xhtml:td xml:space="preserve">ſatilis</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">circunſcriptibilis</xhtml:td> + <xhtml:td xml:space="preserve">circunſcribentis</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="ſolidiſque" type="simple">ſolidiſq;</reg> + </xhtml:td> + <xhtml:td xml:space="preserve">anguli contingentiæ <reg norm="ſolidique" type="simple">ſolidiq;</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> + <xhtml:td xml:space="preserve">diſtincta, procedendo</xhtml:td> + <xhtml:td xml:space="preserve">diſtincta ſint, 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">eaſdem</xhtml:td> + <xhtml:td xml:space="preserve">eoſdem</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> + </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">quotieſcunque</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">ſi conſtitueremus</xhtml:td> + <xhtml:td xml:space="preserve">ſi nos conſtitueremus</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">ſimili ad</xhtml:td> + <xhtml:td xml:space="preserve">ſimili. 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<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> 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xml:space="preserve">ipſæ</xhtml:td> + <xhtml:td xml:space="preserve">ipſi</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">337</xhtml:td> + <xhtml:td xml:space="preserve">42</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">338</xhtml:td> + <xhtml:td xml:space="preserve">14</xhtml:td> + <xhtml:td xml:space="preserve">protracta</xhtml:td> + <xhtml:td xml:space="preserve">protracto</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">338</xhtml:td> + <xhtml:td xml:space="preserve">26</xhtml:td> + <xhtml:td xml:space="preserve">ratiotinare</xhtml:td> + <xhtml:td xml:space="preserve">apta</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">340</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">Alius modus</xhtml:td> + <xhtml:td xml:space="preserve">Alium modum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">341</xhtml:td> + <xhtml:td xml:space="preserve">9</xhtml:td> + <xhtml:td xml:space="preserve">æquale</xhtml:td> + <xhtml:td xml:space="preserve">æqualem</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">343</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">videbitur</xhtml:td> + <xhtml:td xml:space="preserve">videri</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">344</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve">hoc eſt</xhtml:td> + <xhtml:td xml:space="preserve">hoc est quod</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">344</xhtml:td> + <xhtml:td xml:space="preserve">@1</xhtml:td> + <xhtml:td xml:space="preserve">punctus</xhtml:td> + <xhtml:td xml:space="preserve">punctum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">345</xhtml:td> + <xhtml:td xml:space="preserve">6</xhtml:td> + <xhtml:td xml:space="preserve">ei</xhtml:td> + <xhtml:td xml:space="preserve">cum altitudine</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">345</xhtml:td> + <xhtml:td xml:space="preserve">19</xhtml:td> + <xhtml:td xml:space="preserve">ipſum repertum</xhtml:td> + <xhtml:td xml:space="preserve">ipſum, vt repertum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">347</xhtml:td> + <xhtml:td xml:space="preserve">16</xhtml:td> + <xhtml:td xml:space="preserve">reflectit</xhtml:td> + <xhtml:td xml:space="preserve">reflectitur</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">352</xhtml:td> + <xhtml:td xml:space="preserve">38</xhtml:td> + <xhtml:td xml:space="preserve">opinauerunt</xhtml:td> + <xhtml:td xml:space="preserve">opinati ſune</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <xhtml:td xml:space="preserve">parabolem</xhtml:td> + <xhtml:td xml:space="preserve">parabolam</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">15</xhtml:td> + <xhtml:td xml:space="preserve">hæc</xhtml:td> + <xhtml:td xml:space="preserve">hic</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">353</xhtml:td> + <xhtml:td xml:space="preserve">18</xhtml:td> + <xhtml:td xml:space="preserve">parallelam</xhtml:td> + <xhtml:td xml:space="preserve">parallela</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">356</xhtml:td> + <xhtml:td xml:space="preserve">15</xhtml:td> + <xhtml:td xml:space="preserve">muri, &</xhtml:td> 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<xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">364</xhtml:td> + <xhtml:td xml:space="preserve">23</xhtml:td> + <xhtml:td xml:space="preserve">diameter</xhtml:td> + <xhtml:td xml:space="preserve">diametrum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">367</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">prolung ando</xhtml:td> + <xhtml:td xml:space="preserve">produεendo</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">369</xhtml:td> + <xhtml:td xml:space="preserve">2</xhtml:td> + <xhtml:td xml:space="preserve">æqualem</xhtml:td> + <xhtml:td xml:space="preserve">æquale</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">370</xhtml:td> + <xhtml:td xml:space="preserve">4</xhtml:td> + <xhtml:td xml:space="preserve">ſinus</xhtml:td> + <xhtml:td xml:space="preserve">ſinum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">370</xhtml:td> + <xhtml:td xml:space="preserve">30</xhtml:td> + <xhtml:td xml:space="preserve">ſumma</xhtml:td> + <xhtml:td xml:space="preserve">ſummam</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">372</xhtml:td> + <xhtml:td xml:space="preserve">1</xhtml:td> + <xhtml:td xml:space="preserve">cæterarum</xhtml:td> + <xhtml:td xml:space="preserve">cæteris</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">372</xhtml:td> + <xhtml:td xml:space="preserve">36</xhtml:td> + <xhtml:td xml:space="preserve">qui qui</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">378</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">hæc</xhtml:td> + <xhtml:td xml:space="preserve">hæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">382</xhtml:td> + <xhtml:td xml:space="preserve">41</xhtml:td> + <xhtml:td xml:space="preserve">prolung atis</xhtml:td> + <xhtml:td xml:space="preserve">productis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">3</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + <xhtml:td xml:space="preserve">qui</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">13</xhtml:td> + <xhtml:td xml:space="preserve">ginabis</xhtml:td> + <xhtml:td xml:space="preserve">ginaberis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383</xhtml:td> + <xhtml:td xml:space="preserve">20</xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="triangulique" type="simple">trianguliq;</reg> + </xhtml:td> + <xhtml:td xml:space="preserve"><reg norm="triangulaque" type="simple">triangulaq́;</reg></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">383 <lb/>384</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">in prima parabola obliqua ex-<lb/>tremitatem inferiorem diametri <lb/>ſignabis charactere <var>.d.</var> in ſecunda <lb/>vero charactere <var>.r.</var></xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">384</xhtml:td> + <xhtml:td xml:space="preserve">29</xhtml:td> + <xhtml:td xml:space="preserve">maior proportio</xhtml:td> + <xhtml:td xml:space="preserve">maiorem proportionem</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">384</xhtml:td> + <xhtml:td xml:space="preserve">37</xhtml:td> + <xhtml:td xml:space="preserve">proportio halebit</xhtml:td> + <xhtml:td xml:space="preserve">proportionem habebis</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">385</xhtml:td> + <xhtml:td xml:space="preserve">5</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.c.b.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.e.b.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">22</xhtml:td> + <xhtml:td xml:space="preserve">b.e.</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">29</xhtml:td> + <xhtml:td xml:space="preserve">poſitum</xhtml:td> + <xhtml:td xml:space="preserve">poſitus</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">31</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">33</xhtml:td> + <xhtml:td xml:space="preserve">& proportion alitate</xhtml:td> + <xhtml:td xml:space="preserve">& ex proportionalitate</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">34</xhtml:td> + <xhtml:td xml:space="preserve">antecedens in</xhtml:td> + <xhtml:td xml:space="preserve">antecedens <var>.A.B.</var> in</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">386</xhtml:td> + <xhtml:td xml:space="preserve">40</xhtml:td> + <xhtml:td xml:space="preserve">quod</xhtml:td> + <xhtml:td xml:space="preserve">quæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">2</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.B.</var>)</xhtml:td> + <xhtml:td xml:space="preserve">ß)</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">8</xhtml:td> + <xhtml:td xml:space="preserve">b. cum duplo <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.d.</var> cum duplo <var>.b.e.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">45</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve">b.c.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">45</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.e.</var></xhtml:td> + <xhtml:td xml:space="preserve"> <var>.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">388</xhtml:td> + <xhtml:td xml:space="preserve">46</xhtml:td> + <xhtml:td xml:space="preserve">ad <var>.e.b.</var></xhtml:td> + <xhtml:td xml:space="preserve">ad <var>.d.b.</var> cum <var>.e.b.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">390</xhtml:td> + <xhtml:td xml:space="preserve">25</xhtml:td> + <xhtml:td xml:space="preserve">diuiſa</xhtml:td> + <xhtml:td xml:space="preserve">diuiſas</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">392</xhtml:td> + <xhtml:td xml:space="preserve">3</xhtml:td> + <xhtml:td xml:space="preserve">ſit paraboles <var>.a.</var></xhtml:td> + <xhtml:td xml:space="preserve">ſit paraboles <var>.a.b.c.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">392</xhtml:td> + <xhtml:td xml:space="preserve">27</xhtml:td> + <xhtml:td xml:space="preserve">aliqua reliquarum</xhtml:td> + <xhtml:td xml:space="preserve">aliquis reliquorum</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">393 <lb/>395 <lb/>397</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">in parabola vbi ſub <var>.g.</var><unsure/> eſt <var>.i.</var> pone <var>.r.</var> <lb/>& ſupra ſolidum minus dicatur <lb/>cubus minor (leantur <lb/>duo vero lineæ ſine characterib. de</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">18</xhtml:td> + <xhtml:td xml:space="preserve">vt <var>.y.</var> ad <var>.m.n.</var></xhtml:td> + <xhtml:td xml:space="preserve">vt p. ad <var>.m.n.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">33</xhtml:td> + <xhtml:td xml:space="preserve">x.o.</xhtml:td> + <xhtml:td xml:space="preserve">n.o.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">394</xhtml:td> + <xhtml:td xml:space="preserve">34</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.i.k.</var> ad <var>.i.k.</var> ad <var>f.g.</var></xhtml:td> + <xhtml:td xml:space="preserve">i.k. ad <var>.f.g.</var></xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">395</xhtml:td> + <xhtml:td xml:space="preserve">11</xhtml:td> + <xhtml:td xml:space="preserve">quintas</xhtml:td> + <xhtml:td xml:space="preserve">quintæ</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">396</xhtml:td> + <xhtml:td xml:space="preserve">1</xhtml:td> + <xhtml:td xml:space="preserve"> <var>.d.b.</var></xhtml:td> + <xhtml:td xml:space="preserve">g.b.</xhtml:td> + </xhtml:tr> + <xhtml:tr> + <xhtml:td xml:space="preserve">397</xhtml:td> + <xhtml:td xml:space="preserve"> </xhtml:td> + <xhtml:td xml:space="preserve">Vbirubrica dicit. Defenſio nostra <lb/>contra Antonium Bergam, & <lb/>Alexandrum Piccol.</xhtml:td> + <xhtml:td xml:space="preserve">dicatur. contra Anto. Bergam & <lb/>Alex Piccol atq, defenſio 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">ſemidiametrum</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">ſuis</xhtml:td> + <xhtml:td xml:space="preserve">ſuæ</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 ſupræ</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 ſors</xhtml:td> + <xhtml:td xml:space="preserve">qua ſorte</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 ſupra</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">fineſtræ</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 𐆑 (U+10191; D800+DD91). </s> + <s xml:space="preserve">Das gleiche Zeichen innerhalb eines Wortes: vorher𐆑nachher. </s> + <s xml:space="preserve">Das Zeichen wird testweise zu X normalisiert. </s> + <s xml:space="preserve">Das Zeichen 𐆒 (U+10192; D800+DD92) wird dagegen nicht normalisiert: vorher𐆒nachher. </s> + </p> + </div> + </div> + </text> +</echo> \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/Goerz_2008.xml Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,2563 @@ +<?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/>rÍ</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/>dÍ</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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/examples/generateId.xsl Wed Nov 09 15:27:46 2011 +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-xml/lib/saxon.txt Wed Nov 09 15:27:46 2011 +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-xml/src/constants.properties Wed Nov 09 15:27:46 2011 +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/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/exception/ApplicationException.java Wed Nov 09 15:27:46 2011 +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); + } + +} +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/util/Util.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,21 @@ +package de.mpg.mpiwg.berlin.mpdl.util; + +import java.io.File; +import java.io.FileInputStream; +import java.io.IOException; +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) { + } + return props; + } + +}
Binary file software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/.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/xml/general/Constants.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,35 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.general; + +import java.net.URL; +import java.util.Properties; + +import de.mpg.mpiwg.berlin.mpdl.util.Util; + +public class Constants { + public static String DEFAULT_LANGUAGE = "en"; + 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 getDocumentDir() { + if (properties != null) + return properties.getProperty("docDir"); + else + return "no properties file"; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/test/TestLocal.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,38 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.test; + +import java.net.URL; + +import de.mpg.mpiwg.berlin.mpdl.exception.ApplicationException; +import de.mpg.mpiwg.berlin.mpdl.xml.xquery.XQueryEvaluator; + +public class TestLocal { + private XQueryEvaluator xQueryEvaluator; + + public TestLocal() { + init(); + } + + public static void main(String[] args) throws ApplicationException { + try { + TestLocal test = new TestLocal(); + test.xqueries(); + } catch (Exception e) { + e.printStackTrace(); + } + } + + private void init() { + xQueryEvaluator = new XQueryEvaluator(); + } + + private void xqueries() 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"); + String bla = ""; + } catch (Exception e) { + throw new ApplicationException(e); + } + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/tmp/GetFragmentContentHandler.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,93 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.tmp; + +import org.xml.sax.*; + +public class GetFragmentContentHandler implements ContentHandler { + private String ms1Name; + private String ms2Name; + private int ms1Pos; + private int ms2Pos; + + private int msCounter = 1; + private boolean startFragment = false; + private String fragment = ""; + + public GetFragmentContentHandler(String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) { + this.ms1Name = ms1Name; + this.ms2Name = ms2Name; + this.ms1Pos = ms1Pos; + this.ms2Pos = ms2Pos; + } + + public void startDocument() throws SAXException { + fragment += "<?xml version=\"1.0\"?>\n"; + } + + public void endDocument() throws SAXException { + } + + public void characters(char[] c, int start, int length) throws SAXException { + if (startFragment) { + char[] cCopy = new char[length]; + System.arraycopy(c, start, cCopy, 0, length); + String charactersStr = String.valueOf(cCopy); + if (charactersStr != null && ! (charactersStr.trim().equals(""))) { + fragment += charactersStr; + } + } + } + + public void ignorableWhitespace(char[] c, int start, int length) throws SAXException { + } + + public void processingInstruction(String target, String data) throws SAXException { + } + + public void setDocumentLocator(org.xml.sax.Locator arg1) { + } + + 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 (localName.equals(ms1Name)) { + if (msCounter == ms1Pos) { + startFragment = true; + } + if (msCounter == ms2Pos) { + startFragment = false; + throw new SAXException("end"); + } + msCounter++; + } + if (startFragment) { + fragment += "<" + name; + int attrSize = attrs.getLength(); + String attrString = ""; + for (int i=0; i<attrSize; i++) { + String attrQName = attrs.getQName(i); + String attrValue = attrs.getValue(i); + // attrValue = StringUtilEscapeChars.forXML(attrValue); + attrString = attrString + " " + attrQName + "=\"" + attrValue + "\""; + } + fragment += attrString + ">"; + } + } + + public void endElement(String uri, String localName, String name) throws SAXException { + if (startFragment) { + fragment += "</" + name + ">"; + } + } + + public void startPrefixMapping(String prefix, String uri) throws SAXException { + } + + public String getFragment() { + return fragment; + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/tmp/XmlExamples.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,259 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.tmp; + +import java.io.BufferedInputStream; +import java.io.File; +import java.io.FileInputStream; +import java.io.IOException; +import java.io.InputStream; +import java.io.StringReader; +import java.io.StringWriter; +import java.util.Date; + +import javax.xml.transform.OutputKeys; +import javax.xml.transform.Transformer; +import javax.xml.transform.TransformerConfigurationException; +import javax.xml.transform.TransformerException; +import javax.xml.transform.TransformerFactory; +import javax.xml.transform.stream.StreamResult; +import javax.xml.transform.stream.StreamSource; + +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.xml.transform.BasicTransformer; + +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.XQueryEvaluator; +import net.sf.saxon.s9api.XQueryExecutable; +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 XmlExamples { + private XsltTransformer xsltTransformer; + private Processor processor; + private boolean saxonFast = false; + + public static void main(String[] args) { + try { + XmlExamples xmlUtil = new XmlExamples(); + xmlUtil.saxonFast = true; + // xmlUtil.init(); + // xmlUtil.testXQuery(); + // xmlUtil.testGetFragment(); + xmlUtil.testMpdlTransformer(); + // xmlUtil.performanceTestGetFragment(); + // xmlUtil.performanceTestGetFragment("text/Goerz_2008.xml"); + } catch (Exception e) { + e.printStackTrace(); + } + } + + private void init() { + try { + StreamSource xslStreamSource = new StreamSource("examples/getFragment.xsl"); + if (saxonFast) { + xslStreamSource = new StreamSource("examples/getFragmentSaxonFast.xsl"); + } + processor = new Processor(true); + XsltCompiler xsltCompiler = processor.newXsltCompiler(); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltTransformer = xsltExecutable.load(); + } catch (SaxonApiException e) { + e.printStackTrace(); + } + } + + private void testMpdlTransformer() { + try { + // String fragment = getFragment("examples/Benedetti_1585.xml", "pb", 300, "pb", 301); + BasicTransformer mpdlTransformer = new BasicTransformer(); + String srcUrl = "http://mpdl-test.mpiwg-berlin.mpg.de:30030/mpdl/page-query-result.xql?document=/echo/la/Benedetti_1585.xml&mode=text&pn=13&export=pureXml"; + String xslUrl = "http://mpdl-test.mpiwg-berlin.mpg.de:30030/mpdl/presentation/pageHtml.xsl"; + String outputPropertiesStr = "method=xhtml media-type=text/html omit-xml-declaration=no indent=no encoding=utf-8"; + mpdlTransformer.transform(srcUrl, xslUrl, "", outputPropertiesStr); + // xslUrl = "http://mpdl-test.mpiwg-berlin.mpg.de:30030/mpdl/presentation/pageXml.xsl"; + // mpdlTransformer.transformByTransformerFactory(srcUrl, xslUrl, "", outputPropertiesStr); + } catch (ApplicationException e) { + e.printStackTrace(); + } + } + + private void testGetFragment() { + getFragment("examples/Benedetti_1585.xml", "pb", 300, "pb", 302); + // getFragmentByJavax("examples/Benedetti_1585.xml", "pb", 300, "pb", 301); + /* + getFragment("examples/Test_1789.xml", "pb", 1, "pb", 2); + getFragment("examples/Goerz_2008.xml", "pb", 58, "pb", 59); + getFragment("examples/Benedetti_1585.xml", "lb", 385, "lb", 387); + getFragment("examples/Benedetti_1585.xml", "pb", 300, "pb", 302); + getFragment("examples/Benedetti_1585.xml", "pb", 303, "pb", 304); + getFragment("examples/Goerz_2008.xml", "pb", 58, "pb", 59); + */ + } + + private void performanceTestGetFragment() { + for (int i=1; i<=100; i++) { + String fileName = "/Users/jwillenborg/java/eclipse-workspace/XmlUtil/examples/tmp/Test-" + i + ".xml"; + // String fileName = "examples/Test.xml"; + Date before = new Date(); + for (int j=300; j<=400; j++) { + getFragment(fileName, "pb", j, "pb", j+1); + } + Date after = new Date(); + System.out.println(i+ ". Needed time: " + (after.getTime() - before.getTime()) + " ms"); + } + } + + private void testXQuery() { + // String xqueryStr = "declare namespace TEI='http://www.tei-c.org/ns/1.0'; //TEI:pb[@n=\"1\"]/following::*[. << //TEI:pb[@n=\"2\"]]"; + // String xqueryStr = "declare namespace TEI='http://www.tei-c.org/ns/1.0'; (//TEI:pb[@n=\"1\"]/following::*) intersect (//TEI:pb[@n=\"2\"]/preceding::*)"; + String xqueryStr = "declare namespace TEI='http://www.tei-c.org/ns/1.0'; //TEI:pb[@n=\"1\"]"; + xquery("examples/Goerz_2008.xml", xqueryStr); + } + + private void xquery(String xmlFileName, String xqueryStr) { + try { + StreamSource xmlDoc = new StreamSource(xmlFileName); + XQueryCompiler xQueryCompiler = processor.newXQueryCompiler(); + XQueryExecutable xQueryExecutable = xQueryCompiler.compile(xqueryStr); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + XQueryEvaluator xQueryEvaluator = xQueryExecutable.load(); + Date before = new Date(); + xQueryEvaluator.setSource(xmlDoc); + XdmValue result = xQueryEvaluator.evaluate(); + // xQueryEvaluator.run(serializer); + // System.out.println(serializer.getOutputDestination().toString()); + Date after = new Date(); + System.out.println("Needed time: " + (after.getTime() - before.getTime()) + " ms"); + } catch (SaxonApiException e) { + e.printStackTrace(); + } + } + + private String getFragment(String xmlFileName, String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) { + String pageFragment = null; + try { + StreamSource xmlDoc = new StreamSource(new StringReader("<?xml version=\"1.0\"?><bla/>")); + if (! saxonFast) + xmlDoc = new StreamSource(xmlFileName); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.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"); + 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); + xsltTransformer.setParameter(xmlFileNameQName, xmlFileNameXdmValue); + xsltTransformer.setParameter(ms1NameQName, ms1NameXdmValue); + xsltTransformer.setParameter(ms1PositionQName, ms1PositionXdmValue); + xsltTransformer.setParameter(ms2NameQName, ms2NameXdmValue); + xsltTransformer.setParameter(ms2PositionQName, ms2PositionXdmValue); + xsltTransformer.transform(); // needs some time for bigger documents + pageFragment = serializer.getOutputDestination().toString(); + } catch (SaxonApiException e) { + e.printStackTrace(); + } + return pageFragment; + } + + private void getFragmentByJavax(String xmlFileName, String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) { + try { + StreamSource xslStreamSource = new StreamSource("examples/getFragment.xsl"); + Transformer transformer = TransformerFactory.newInstance().newTransformer(xslStreamSource); + transformer.setOutputProperty(OutputKeys.INDENT, "yes"); + StreamResult result = new StreamResult(new StringWriter()); + StreamSource xmlStreamSource = new StreamSource(xmlFileName); + transformer.setParameter("ms1Name", ms1Name); + transformer.setParameter("ms1Position", new Integer(ms1Pos)); + transformer.setParameter("ms2Name", ms2Name); + transformer.setParameter("ms2Position", new Integer(ms2Pos)); + transformer.transform(xmlStreamSource, result); + } catch (TransformerConfigurationException e) { + e.printStackTrace(); + } catch (TransformerException e) { + e.printStackTrace(); + } + } + + private void performanceTestGetFragment(String xmlFileName) { + try { + // init + StreamSource xslStreamSource = new StreamSource("examples/getFragment.xsl"); + Processor processor = new Processor(true); + XsltCompiler xsltCompiler = processor.newXsltCompiler(); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + XsltTransformer xsltTransformer = xsltExecutable.load(); + StreamSource xmlDoc = new StreamSource(xmlFileName); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.setDestination(serializer); + for (int i=1; i<=100; i++) { + QName ms1NameQName = new QName("ms1Name"); + XdmValue ms1NameXdmValue = new XdmAtomicValue("pb"); + QName ms1PositionQName = new QName("ms1Position"); + XdmValue ms1PositionXdmValue = new XdmAtomicValue(i); + QName ms2NameQName = new QName("ms2Name"); + XdmValue ms2NameXdmValue = new XdmAtomicValue("pb"); + QName ms2PositionQName = new QName("ms2Position"); + XdmValue ms2PositionXdmValue = new XdmAtomicValue(i+1); + xsltTransformer.setParameter(ms1NameQName, ms1NameXdmValue); + xsltTransformer.setParameter(ms1PositionQName, ms1PositionXdmValue); + xsltTransformer.setParameter(ms2NameQName, ms2NameXdmValue); + xsltTransformer.setParameter(ms2PositionQName, ms2PositionXdmValue); + Date before = new Date(); + xsltTransformer.transform(); // needs some time for bigger documents + Date after = new Date(); + System.out.println(i+ ". Needed time: " + (after.getTime() - before.getTime()) + " ms"); + } + } catch (SaxonApiException e) { + e.printStackTrace(); + } + } + + private void getFragmentByContentHandler(String xmlFileName, String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) { + Date before = new Date(); + File inputFile = new File(xmlFileName); + GetFragmentContentHandler getFragmentContentHandler = new GetFragmentContentHandler(ms1Name, ms1Pos, ms2Name, ms2Pos); + try { + XMLReader xmlParser = new SAXParser(); + xmlParser.setContentHandler(getFragmentContentHandler); + InputStream inputStream = new FileInputStream(inputFile); + BufferedInputStream bufferedInputStream = new BufferedInputStream(inputStream); + InputSource input = new InputSource(bufferedInputStream); + xmlParser.parse(input); + bufferedInputStream.close(); + String fragment = getFragmentContentHandler.getFragment(); + } catch (SAXException e) { + Date after = new Date(); + System.out.println(". Needed time: " + (after.getTime() - before.getTime()) + " ms"); + e.printStackTrace(); + } catch (IOException e) { + Date after = new Date(); + System.out.println(". Needed time: " + (after.getTime() - before.getTime()) + " ms"); + e.printStackTrace(); + } + } + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/BasicTransformer.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,255 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.transform; + +import java.io.BufferedWriter; +import java.io.FileOutputStream; +import java.io.IOException; +import java.io.OutputStreamWriter; +import java.io.StringReader; +import java.io.StringWriter; +import java.io.Writer; + +import javax.xml.transform.OutputKeys; +import javax.xml.transform.Transformer; +import javax.xml.transform.TransformerConfigurationException; +import javax.xml.transform.TransformerException; +import javax.xml.transform.TransformerFactory; +import javax.xml.transform.stream.StreamResult; +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 BasicTransformer { + private Processor processor; + private XsltCompiler xsltCompiler; + private XsltTransformer xsltTransformer; + + public static void main(String[] args) { + try { + BasicTransformer t = new BasicTransformer(); + String xmlDocFileName = "file:/Users/jwillenborg/texts/mpdl/documents/echo/la/Aristoteles_1585_XSY685ZD.xml"; + String xslDocFileName = "/Users/jwillenborg/java/mpiwg-mpdl-xml/examples/generateId.xsl"; + String destFileName = "/Users/jwillenborg/java/mpiwg-mpdl-xml/examples/Aristoteles_1585_XSY685ZD-Ids.xml"; + // t.transform(xmlDocFileName, xslDocFileName, null, "encoding=utf-8 indent=yes", destFileName); + t.transform(xmlDocFileName, xslDocFileName, null, null); + // String result = t.transform(xmlDocFileName, xslDocFileName, null, null); + // ByteArrayInputStream is = new ByteArrayInputStream(result.getBytes("utf-8")); + // FileUtil.getInstance().saveInputStreamToLocalFile(is, destFileName); + } catch (Exception e) { + e.printStackTrace(); + } + } + + public BasicTransformer() { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + } + + public void transform(String srcUrl, String xslUrl, String parametersStr, String outputPropertiesStr, String destFileName) throws ApplicationException { + try { + StreamSource xslStreamSource = new StreamSource(xslUrl); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltTransformer = xsltExecutable.load(); + StreamSource xmlDoc = new StreamSource(srcUrl); + Serializer serializer = new Serializer(); + String encoding = "utf-8"; // default + if (outputPropertiesStr != null) { + String enc = getEncoding(outputPropertiesStr); + if (enc != null && ! enc.isEmpty()) + encoding = enc; + } + Writer writer = new BufferedWriter(new OutputStreamWriter(new FileOutputStream(destFileName), encoding)); + serializer.setOutputWriter(writer); + 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 + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String transform(String srcUrl, 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(srcUrl); + 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; + if (str.matches("encoding=([^ ]+)$")) { + encoding = outputPropertiesStr.toLowerCase().replaceAll("encoding=([^ ]+)$", "$1"); + } else if (str.matches("encoding=(.+) .*")) { + encoding = outputPropertiesStr.toLowerCase().replaceAll("encoding=(.+) .*", "$1"); + } + return encoding; + } + + private void setParameters(XsltTransformer xsltTransformer, String parametersStr) { + if (parametersStr != null && ! parametersStr.equals("")) { + String[] parameters = parametersStr.split(" "); + for (int i=0; i<parameters.length; i++) { + String param = parameters[i]; + int index = param.indexOf("="); + String key = param.substring(0, index); + String value = param.substring(index + 1); + if (key != null && value != null && ! key.isEmpty() && !value.isEmpty()) { + QName keyQName = new QName(key); + XdmValue valueXdmValue = new XdmAtomicValue(value); + xsltTransformer.setParameter(keyQName, valueXdmValue); + } + } + } + } + + private void setOutputProperties(Serializer serializer, String outputPropertiesStr) { + if (outputPropertiesStr != null && ! outputPropertiesStr.equals("")) { + String[] outputProps = outputPropertiesStr.split(" "); + for (int i=0; i<outputProps.length; i++) { + String prop = outputProps[i]; + int index = prop.indexOf("="); + String key = prop.substring(0, index).toLowerCase(); + String value = prop.substring(index + 1); + if (key != null && value != null && ! key.isEmpty() && !value.isEmpty()) { + if (key.equals("method")) + serializer.setOutputProperty(Serializer.Property.METHOD, value); + else if (key.equals("indent")) + serializer.setOutputProperty(Serializer.Property.INDENT, value); + else if (key.equals("media-type")) + serializer.setOutputProperty(Serializer.Property.MEDIA_TYPE, value); + else if (key.equals("omit-xml-declaration")) + serializer.setOutputProperty(Serializer.Property.OMIT_XML_DECLARATION, value); + else if (key.equals("encoding")) + serializer.setOutputProperty(Serializer.Property.ENCODING, value); + else if (key.equals("cdata-section-elements")) + serializer.setOutputProperty(Serializer.Property.CDATA_SECTION_ELEMENTS, value); + else if (key.equals("version")) + serializer.setOutputProperty(Serializer.Property.VERSION, value); + else if (key.equals("doctype-system")) + serializer.setOutputProperty(Serializer.Property.DOCTYPE_SYSTEM, value); + else if (key.equals("doctype-public")) + serializer.setOutputProperty(Serializer.Property.DOCTYPE_PUBLIC, value); + else if (key.equals("standalone")) + serializer.setOutputProperty(Serializer.Property.STANDALONE, value); + } + } + } + } + + private void setOutputProperties(Transformer transformer, String outputPropertiesStr) { + if (outputPropertiesStr != null && ! outputPropertiesStr.equals("")) { + String[] outputProps = outputPropertiesStr.split(" "); + for (int i=0; i<outputProps.length; i++) { + String prop = outputProps[i]; + int index = prop.indexOf("="); + String key = prop.substring(0, index).toLowerCase(); + String value = prop.substring(index + 1); + if (key != null && value != null && ! key.isEmpty() && !value.isEmpty()) { + if (key.equals("method")) + transformer.setOutputProperty(OutputKeys.METHOD, value); + if (key.equals("indent")) + transformer.setOutputProperty(OutputKeys.INDENT, value); + if (key.equals("media-type")) + transformer.setOutputProperty(OutputKeys.MEDIA_TYPE, value); + if (key.equals("omit-xml-declaration")) + transformer.setOutputProperty(OutputKeys.OMIT_XML_DECLARATION, value); + if (key.equals("encoding")) + transformer.setOutputProperty(OutputKeys.ENCODING, value); + else if (key.equals("cdata-section-elements")) + transformer.setOutputProperty(OutputKeys.CDATA_SECTION_ELEMENTS, value); + else if (key.equals("version")) + transformer.setOutputProperty(OutputKeys.VERSION, value); + else if (key.equals("doctype-system")) + transformer.setOutputProperty(OutputKeys.DOCTYPE_SYSTEM, value); + else if (key.equals("doctype-public")) + transformer.setOutputProperty(OutputKeys.DOCTYPE_PUBLIC, value); + else if (key.equals("standalone")) + transformer.setOutputProperty(OutputKeys.STANDALONE, value); + } + } + } + } + + private void getFragment(String xmlFileName, String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) throws ApplicationException { + try { + StreamSource xmlDoc = new StreamSource(new StringReader("<?xml version=\"1.0\"?><bla/>")); + xmlDoc = new StreamSource(xmlFileName); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + serializer.setOutputProperty(Serializer.Property.INDENT, "yes"); + xsltTransformer.setSource(xmlDoc); // needs some time for bigger documents + xsltTransformer.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"); + 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); + xsltTransformer.setParameter(xmlFileNameQName, xmlFileNameXdmValue); + xsltTransformer.setParameter(ms1NameQName, ms1NameXdmValue); + xsltTransformer.setParameter(ms1PositionQName, ms1PositionXdmValue); + xsltTransformer.setParameter(ms2NameQName, ms2NameXdmValue); + xsltTransformer.setParameter(ms2PositionQName, ms2PositionXdmValue); + xsltTransformer.transform(); // needs some time for bigger documents + // String pageFragment = serializer.getOutputDestination().toString(); + // System.out.println(pageFragment); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } + } + + private String transformByTransformerFactory(String xmlString, String xslFileName, String parametersStr, String outputPropertiesStr) throws ApplicationException { + String resultString = null; + try { + StreamSource xslSource = new StreamSource(xslFileName); + Transformer transformer = TransformerFactory.newInstance(net.sf.saxon.TransformerFactoryImpl.class.getName(), null).newTransformer(xslSource); + setOutputProperties(transformer, outputPropertiesStr); + StreamResult result = new StreamResult(new StringWriter()); + StreamSource source = new StreamSource(xmlString); + transformer.transform(source, result); + resultString = result.getWriter().toString(); + } catch (TransformerConfigurationException e) { + throw new ApplicationException(e); + } catch (TransformerException e) { + throw new ApplicationException(e); + } + return resultString; + } + + + +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/FragmentTransformer.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,94 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.transform; + +import java.io.IOException; +import java.io.StringWriter; +import java.net.URL; +import java.util.Date; + +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 FragmentTransformer { + private Processor processor; + 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(); + } + + private void init() throws ApplicationException { + try { + processor = new Processor(false); + xsltCompiler = processor.newXsltCompiler(); + URL getFragmentXslUrl = FragmentTransformer.class.getResource("getFragment.xsl"); + StreamSource xslStreamSource = new StreamSource(getFragmentXslUrl.openStream()); + XsltExecutable xsltExecutable = xsltCompiler.compile(xslStreamSource); + xsltFragmentTransformer = xsltExecutable.load(); + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } catch (IOException e) { + throw new ApplicationException(e); + } + } + + public String getFragment(String xmlFileName, String ms1Name, int ms1Pos, String ms2Name, int ms2Pos) 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"); + 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"); + 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(xmlFileNameQName, xmlFileNameXdmValue); + 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; + } +}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/software/mpdl-services/mpiwg-mpdl-xml/src/de/mpg/mpiwg/berlin/mpdl/xml/transform/getFragment.xsl Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,58 @@ +<?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> + +<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::*"/> + +<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:when> + <xsl:otherwise><xsl:apply-templates/></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: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/transform/getFragmentSaxonFast.xsl Wed Nov 09 15:27:46 2011 +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>
--- /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/XQueryEvaluator.java Wed Nov 09 15:27:46 2011 +0100 @@ -0,0 +1,120 @@ +package de.mpg.mpiwg.berlin.mpdl.xml.xquery; + +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.SaxonApiException; +import net.sf.saxon.s9api.Serializer; +import net.sf.saxon.s9api.XQueryCompiler; +import net.sf.saxon.s9api.XQueryExecutable; +import net.sf.saxon.s9api.XdmItem; +import net.sf.saxon.s9api.XdmSequenceIterator; +import net.sf.saxon.s9api.XdmValue; + +public class XQueryEvaluator { + private Processor processor; + private XQueryCompiler xQueryCompiler; + + public XQueryEvaluator() { + processor = new Processor(false); + xQueryCompiler = processor.newXQueryCompiler(); + } + + public XdmValue evaluate(String srcXmlStr, String xqueryStr) throws ApplicationException { + XdmValue result = (XdmValue) evaluate(srcXmlStr, xqueryStr, null); + return result; + } + + public XdmValue evaluate(URL srcUrl, String xqueryStr) throws ApplicationException { + XdmValue result = (XdmValue) evaluate(srcUrl, xqueryStr, null); + return result; + } + + public String evaluateAsString(String srcXmlStr, String xqueryStr) throws ApplicationException { + Object result = evaluate(srcXmlStr, xqueryStr, "asString"); + return (String) result; + } + + public String evaluateAsString(URL srcUrl, String xqueryStr) throws ApplicationException { + Object result = evaluate(srcUrl, xqueryStr, "asString"); + return (String) result; + } + + public String evaluateAsStringValue(String srcXmlStr, String xqueryStr) throws ApplicationException { + XdmValue val = (XdmValue) evaluate(srcXmlStr, xqueryStr, null); + return toStringValue(val); + } + + public String evaluateAsStringValue(URL srcUrl, String xqueryStr) throws ApplicationException { + XdmValue val = (XdmValue) evaluate(srcUrl, xqueryStr, null); + return toStringValue(val); + } + + public Object evaluate(String srcXmlStr, String xqueryStr, String optionsStr) throws ApplicationException { + try { + StringReader srcXmlStrReader = new StringReader(srcXmlStr); + StreamSource xmlDoc = new StreamSource(srcXmlStrReader); + XQueryExecutable xQueryExecutable = xQueryCompiler.compile(xqueryStr); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + 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); + } + return result; + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } + } + + public Object evaluate(URL srcUrl, String xqueryStr, String optionsStr) throws ApplicationException { + try { + StreamSource xmlDoc = new StreamSource(srcUrl.toString()); + XQueryExecutable xQueryExecutable = xQueryCompiler.compile(xqueryStr); + Serializer serializer = new Serializer(); + serializer.setOutputWriter(new StringWriter()); + 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); + } + return result; + } catch (SaxonApiException e) { + throw new ApplicationException(e); + } + } + + private String toString(XdmValue xdmValue) { + String result = ""; + XdmSequenceIterator iter = xdmValue.iterator(); + 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(); + } +}