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comparison texts/XML/echo/en/Harriot_Add_MS_6784_XT0KZ8QC.xml @ 6:22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
| author | casties |
|---|---|
| date | Fri, 07 Dec 2012 17:05:22 +0100 |
| parents | |
| children |
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| 5:0d8b27aa70aa | 6:22d6a63640c6 |
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| 1 <?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: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"> | |
| 2 <metadata> | |
| 3 <dcterms:identifier>ECHO:XT0KZ8QC.xml</dcterms:identifier> | |
| 4 <dcterms:creator>Harriot, Thomas</dcterms:creator> | |
| 5 <dcterms:title xml:lang="en">Mss. 6784</dcterms:title> | |
| 6 <dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date> | |
| 7 <dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language> | |
| 8 <dcterms:rights>CC-BY-SA</dcterms:rights> | |
| 9 <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> | |
| 10 <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> | |
| 11 <echodir>/permanent/library/XT0KZ8QC</echodir> | |
| 12 <log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log> | |
| 13 </metadata> | |
| 14 | |
| 15 <text xml:lang="eng" type="free"> | |
| 16 <div xml:id="echoid-div1" type="section" level="1" n="1"> | |
| 17 <pb file="add_6784_f001" o="1" n="1"/> | |
| 18 <head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat"> | |
| 19 De resectione rationis | |
| 20 </head> | |
| 21 <pb file="add_6784_f001v" o="1v" n="2"/> | |
| 22 <pb file="add_6784_f002" o="2" n="3"/> | |
| 23 <head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat"> | |
| 24 De resectione rationis | |
| 25 </head> | |
| 26 <head xml:id="echoid-head3" xml:space="preserve"> | |
| 27 AB) | |
| 28 </head> | |
| 29 <pb file="add_6784_f002v" o="2v" n="4"/> | |
| 30 <pb file="add_6784_f003" o="3" n="5"/> | |
| 31 <head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat"> | |
| 32 De resectione rationis | |
| 33 </head> | |
| 34 <pb file="add_6784_f003v" o="3v" n="6"/> | |
| 35 <pb file="add_6784_f004" o="4" n="7"/> | |
| 36 <head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat"> | |
| 37 De resectione rationis | |
| 38 </head> | |
| 39 <pb file="add_6784_f004v" o="4v" n="8"/> | |
| 40 <pb file="add_6784_f005" o="5" n="9"/> | |
| 41 <head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat"> | |
| 42 De resectione rationis | |
| 43 </head> | |
| 44 <pb file="add_6784_f005v" o="5v" n="10"/> | |
| 45 <pb file="add_6784_f006" o="6" n="11"/> | |
| 46 <head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat"> | |
| 47 De resectione rationis | |
| 48 </head> | |
| 49 <pb file="add_6784_f006v" o="6v" n="12"/> | |
| 50 <pb file="add_6784_f007" o="7" n="13"/> | |
| 51 <head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat"> | |
| 52 De resectione rationis | |
| 53 </head> | |
| 54 <head xml:id="echoid-head9" xml:space="preserve"> | |
| 55 AB) | |
| 56 </head> | |
| 57 <pb file="add_6784_f007v" o="7v" n="14"/> | |
| 58 <pb file="add_6784_f008" o="8" n="15"/> | |
| 59 <head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat"> | |
| 60 De resectione rationis | |
| 61 </head> | |
| 62 <pb file="add_6784_f008v" o="8v" n="16"/> | |
| 63 <pb file="add_6784_f009" o="9" n="17"/> | |
| 64 <head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat"> | |
| 65 De resectione rationis | |
| 66 </head> | |
| 67 <head xml:id="echoid-head12" xml:space="preserve"> | |
| 68 2.AB) | |
| 69 </head> | |
| 70 <pb file="add_6784_f009v" o="9v" n="18"/> | |
| 71 <pb file="add_6784_f010" o="10" n="19"/> | |
| 72 <head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat"> | |
| 73 De resectione rationis | |
| 74 </head> | |
| 75 <head xml:id="echoid-head14" xml:space="preserve"> | |
| 76 AC) | |
| 77 </head> | |
| 78 <pb file="add_6784_f010v" o="10v" n="20"/> | |
| 79 <pb file="add_6784_f011" o="11" n="21"/> | |
| 80 <head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat"> | |
| 81 De resectione rationis | |
| 82 </head> | |
| 83 <head xml:id="echoid-head16" xml:space="preserve"> | |
| 84 AC.1) | |
| 85 </head> | |
| 86 <pb file="add_6784_f011v" o="11v" n="22"/> | |
| 87 <pb file="add_6784_f012" o="12" n="23"/> | |
| 88 <head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat"> | |
| 89 De resectione rationis | |
| 90 </head> | |
| 91 <pb file="add_6784_f012v" o="12v" n="24"/> | |
| 92 <pb file="add_6784_f013" o="13" n="25"/> | |
| 93 <head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat"> | |
| 94 De resectione rationis | |
| 95 </head> | |
| 96 <head xml:id="echoid-head19" xml:space="preserve"> | |
| 97 2.BC) | |
| 98 </head> | |
| 99 <pb file="add_6784_f013v" o="13v" n="26"/> | |
| 100 <pb file="add_6784_f014" o="14" n="27"/> | |
| 101 <head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat"> | |
| 102 De resectione rationis | |
| 103 </head> | |
| 104 <head xml:id="echoid-head21" xml:space="preserve"> | |
| 105 1.BC) | |
| 106 </head> | |
| 107 <pb file="add_6784_f014v" o="14v" n="28"/> | |
| 108 <pb file="add_6784_f015" o="15" n="29"/> | |
| 109 <head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat"> | |
| 110 De resectione rationis | |
| 111 </head> | |
| 112 <pb file="add_6784_f015v" o="15v" n="30"/> | |
| 113 <pb file="add_6784_f016" o="16" n="31"/> | |
| 114 <head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat"> | |
| 115 De resectione rationis | |
| 116 </head> | |
| 117 <pb file="add_6784_f016v" o="16v" n="32"/> | |
| 118 <pb file="add_6784_f017" o="17" n="33"/> | |
| 119 <head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat"> | |
| 120 Pappus 171. ad resectione rationis | |
| 121 </head> | |
| 122 <pb file="add_6784_f017v" o="17v" n="34"/> | |
| 123 <pb file="add_6784_f018" o="18" n="35"/> | |
| 124 <head xml:id="echoid-head25" xml:space="preserve" xml:lang="lat"> | |
| 125 De resectione rationis | |
| 126 </head> | |
| 127 <pb file="add_6784_f018v" o="18v" n="36"/> | |
| 128 <pb file="add_6784_f019" o="19" n="37"/> | |
| 129 <head xml:id="echoid-head26" xml:space="preserve" xml:lang="lat"> | |
| 130 De resectione spatij, problema | |
| 131 </head> | |
| 132 <head xml:id="echoid-head27" xml:space="preserve"> | |
| 133 a) | |
| 134 </head> | |
| 135 <pb file="add_6784_f019v" o="19v" n="38"/> | |
| 136 <pb file="add_6784_f020" o="20" n="39"/> | |
| 137 <head xml:id="echoid-head28" xml:space="preserve"> | |
| 138 Poristike | |
| 139 </head> | |
| 140 <pb file="add_6784_f020v" o="20v" n="40"/> | |
| 141 <pb file="add_6784_f021" o="21" n="41"/> | |
| 142 <pb file="add_6784_f021v" o="21v" n="42"/> | |
| 143 <pb file="add_6784_f022" o="22" n="43"/> | |
| 144 <pb file="add_6784_f022v" o="22v" n="44"/> | |
| 145 <pb file="add_6784_f023" o="23" n="45"/> | |
| 146 <pb file="add_6784_f023v" o="23v" n="46"/> | |
| 147 <pb file="add_6784_f024" o="24" n="47"/> | |
| 148 <head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat"> | |
| 149 De sectione rationis | |
| 150 </head> | |
| 151 <head xml:id="echoid-head30" xml:space="preserve"> | |
| 152 b.1) | |
| 153 </head> | |
| 154 <pb file="add_6784_f024v" o="24v" n="48"/> | |
| 155 <pb file="add_6784_f025" o="25" n="49"/> | |
| 156 <head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat"> | |
| 157 De sectione rationis | |
| 158 </head> | |
| 159 <head xml:id="echoid-head32" xml:space="preserve"> | |
| 160 b.2) | |
| 161 </head> | |
| 162 <pb file="add_6784_f025v" o="25v" n="50"/> | |
| 163 <pb file="add_6784_f026" o="26" n="51"/> | |
| 164 <head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat"> | |
| 165 De sectione rationis | |
| 166 </head> | |
| 167 <head xml:id="echoid-head34" xml:space="preserve"> | |
| 168 b.3) | |
| 169 </head> | |
| 170 <pb file="add_6784_f026v" o="26v" n="52"/> | |
| 171 <pb file="add_6784_f027" o="27" n="53"/> | |
| 172 <head xml:id="echoid-head35" xml:space="preserve" xml:lang="lat"> | |
| 173 De sectione rationis | |
| 174 </head> | |
| 175 <head xml:id="echoid-head36" xml:space="preserve"> | |
| 176 b.4) | |
| 177 </head> | |
| 178 <pb file="add_6784_f027v" o="27v" n="54"/> | |
| 179 <pb file="add_6784_f028" o="28" n="55"/> | |
| 180 <head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat"> | |
| 181 Lemma ad sectionem rationis <lb/> | |
| 182 et spatij | |
| 183 </head> | |
| 184 <pb file="add_6784_f028v" o="28v" n="56"/> | |
| 185 <pb file="add_6784_f029" o="29" n="57"/> | |
| 186 <pb file="add_6784_f029v" o="29v" n="58"/> | |
| 187 <pb file="add_6784_f030" o="30" n="59"/> | |
| 188 <pb file="add_6784_f030v" o="30v" n="60"/> | |
| 189 <pb file="add_6784_f031" o="31" n="61"/> | |
| 190 <pb file="add_6784_f031v" o="31v" n="62"/> | |
| 191 <pb file="add_6784_f032" o="32" n="63"/> | |
| 192 <pb file="add_6784_f032v" o="32v" n="64"/> | |
| 193 <pb file="add_6784_f033" o="33" n="65"/> | |
| 194 <pb file="add_6784_f033v" o="33v" n="66"/> | |
| 195 <pb file="add_6784_f034" o="34" n="67"/> | |
| 196 <pb file="add_6784_f034v" o="34v" n="68"/> | |
| 197 <pb file="add_6784_f035" o="35" n="69"/> | |
| 198 <pb file="add_6784_f035v" o="35v" n="70"/> | |
| 199 <pb file="add_6784_f036" o="36" n="71"/> | |
| 200 <pb file="add_6784_f036v" o="36v" n="72"/> | |
| 201 <pb file="add_6784_f037" o="37" n="73"/> | |
| 202 <pb file="add_6784_f037v" o="37v" n="74"/> | |
| 203 <pb file="add_6784_f038" o="38" n="75"/> | |
| 204 <pb file="add_6784_f038v" o="38v" n="76"/> | |
| 205 <pb file="add_6784_f039" o="39" n="77"/> | |
| 206 <pb file="add_6784_f039v" o="39v" n="78"/> | |
| 207 <pb file="add_6784_f040" o="40" n="79"/> | |
| 208 <head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat"> | |
| 209 De resectione rationis | |
| 210 </head> | |
| 211 <pb file="add_6784_f040v" o="40v" n="80"/> | |
| 212 <div xml:id="echoid-div1" type="page_commentary" level="2" n="1"> | |
| 213 <p> | |
| 214 <s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note: | |
| 215 <p> | |
| 216 <s xml:id="echoid-s1" xml:space="preserve"> | |
| 217 De infinitis | |
| 218 <lb/>[<emph style="it">tr: | |
| 219 On infinity | |
| 220 </emph>]<lb/> | |
| 221 </s> | |
| 222 </p> | |
| 223 </emph>] | |
| 224 <lb/><lb/></s></p></div> | |
| 225 <p xml:lang="lat"> | |
| 226 <s xml:id="echoid-s3" xml:space="preserve"> | |
| 227 Maior et Maior rationum infinitum. <lb/> | |
| 228 fit termini minores et minores; cum probuerit indivisibilibis <lb/> | |
| 229 ratio tandem infinitum. | |
| 230 <lb/>[<emph style="it">tr: | |
| 231 A greater and greater infinite ratio. | |
| 232 the terms are smaller and smaller; | |
| 233 while from indivisibles there will eventually come an infinite ratio. | |
| 234 </emph>]<lb/> | |
| 235 </s> | |
| 236 </p> | |
| 237 <p xml:lang="lat"> | |
| 238 <s xml:id="echoid-s4" xml:space="preserve"> | |
| 239 HA ad IA non potest <lb/> | |
| 240 esse maior BA ad BC. <lb/> | |
| 241 terminis scilicet decrescentibus. | |
| 242 <lb/>[<emph style="it">tr: | |
| 243 HA to IA cannot be greater than BA to BC. | |
| 244 the terms of course decreasing. | |
| 245 </emph>]<lb/> | |
| 246 </s> | |
| 247 </p> | |
| 248 <pb file="add_6784_f041" o="41" n="81"/> | |
| 249 <pb file="add_6784_f041v" o="41v" n="82"/> | |
| 250 <pb file="add_6784_f042" o="42" n="83"/> | |
| 251 <pb file="add_6784_f042v" o="42v" n="84"/> | |
| 252 <pb file="add_6784_f043" o="43" n="85"/> | |
| 253 <pb file="add_6784_f043v" o="43v" n="86"/> | |
| 254 <pb file="add_6784_f044" o="44" n="87"/> | |
| 255 <pb file="add_6784_f044v" o="44v" n="88"/> | |
| 256 <pb file="add_6784_f045" o="45" n="89"/> | |
| 257 <pb file="add_6784_f045v" o="45v" n="90"/> | |
| 258 <pb file="add_6784_f046" o="46" n="91"/> | |
| 259 <pb file="add_6784_f046v" o="46v" n="92"/> | |
| 260 <pb file="add_6784_f047" o="47" n="93"/> | |
| 261 <pb file="add_6784_f047v" o="47v" n="94"/> | |
| 262 <pb file="add_6784_f048" o="48" n="95"/> | |
| 263 <pb file="add_6784_f048v" o="48v" n="96"/> | |
| 264 <pb file="add_6784_f049" o="49" n="97"/> | |
| 265 <pb file="add_6784_f049v" o="49v" n="98"/> | |
| 266 <pb file="add_6784_f050" o="50" n="99"/> | |
| 267 <pb file="add_6784_f050v" o="50v" n="100"/> | |
| 268 <pb file="add_6784_f051" o="51" n="101"/> | |
| 269 <pb file="add_6784_f051v" o="51v" n="102"/> | |
| 270 <pb file="add_6784_f052" o="52" n="103"/> | |
| 271 <pb file="add_6784_f052v" o="52v" n="104"/> | |
| 272 <pb file="add_6784_f053" o="53" n="105"/> | |
| 273 <pb file="add_6784_f053v" o="53v" n="106"/> | |
| 274 <pb file="add_6784_f054" o="54" n="107"/> | |
| 275 <pb file="add_6784_f054v" o="54v" n="108"/> | |
| 276 <pb file="add_6784_f055" o="55" n="109"/> | |
| 277 <pb file="add_6784_f055v" o="55v" n="110"/> | |
| 278 <pb file="add_6784_f056" o="56" n="111"/> | |
| 279 <pb file="add_6784_f056v" o="56v" n="112"/> | |
| 280 <pb file="add_6784_f057" o="57" n="113"/> | |
| 281 <pb file="add_6784_f057v" o="57v" n="114"/> | |
| 282 <pb file="add_6784_f058" o="58" n="115"/> | |
| 283 <pb file="add_6784_f058v" o="58v" n="116"/> | |
| 284 <pb file="add_6784_f059" o="59" n="117"/> | |
| 285 <pb file="add_6784_f059v" o="59v" n="118"/> | |
| 286 <pb file="add_6784_f060" o="60" n="119"/> | |
| 287 <pb file="add_6784_f060v" o="60v" n="120"/> | |
| 288 <pb file="add_6784_f061" o="61" n="121"/> | |
| 289 <pb file="add_6784_f061v" o="61v" n="122"/> | |
| 290 <pb file="add_6784_f062" o="62" n="123"/> | |
| 291 <pb file="add_6784_f062v" o="62v" n="124"/> | |
| 292 <pb file="add_6784_f063" o="63" n="125"/> | |
| 293 <pb file="add_6784_f063v" o="63v" n="126"/> | |
| 294 <pb file="add_6784_f064" o="64" n="127"/> | |
| 295 <pb file="add_6784_f064v" o="64v" n="128"/> | |
| 296 <pb file="add_6784_f065" o="65" n="129"/> | |
| 297 <pb file="add_6784_f065v" o="65v" n="130"/> | |
| 298 <pb file="add_6784_f066" o="66" n="131"/> | |
| 299 <pb file="add_6784_f066v" o="66v" n="132"/> | |
| 300 <pb file="add_6784_f067" o="67" n="133"/> | |
| 301 <div xml:id="echoid-div2" type="page_commentary" level="2" n="2"> | |
| 302 <p> | |
| 303 <s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note: | |
| 304 <p> | |
| 305 <s xml:id="echoid-s5" xml:space="preserve"> | |
| 306 The reference on this page is to Willebrord Snell's | |
| 307 <emph style="it">Apollonius Batavus</emph> (1608). | |
| 308 </s> | |
| 309 </p> | |
| 310 </emph>] | |
| 311 <lb/><lb/></s></p></div> | |
| 312 <head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat"> | |
| 313 Diagrammata <lb/> | |
| 314 Snellij | |
| 315 <lb/>[<emph style="it">tr: | |
| 316 Snell's diagrams | |
| 317 </emph>]<lb/> | |
| 318 </head> | |
| 319 <pb file="add_6784_f067v" o="67v" n="134"/> | |
| 320 <pb file="add_6784_f068" o="68" n="135"/> | |
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| 358 <pb file="add_6784_f087" o="87" n="173"/> | |
| 359 <pb file="add_6784_f087v" o="87v" n="174"/> | |
| 360 <pb file="add_6784_f088" o="88" n="175"/> | |
| 361 <div xml:id="echoid-div3" type="page_commentary" level="2" n="3"> | |
| 362 <p> | |
| 363 <s xml:id="echoid-s7" xml:space="preserve">[<emph style="it">Note: | |
| 364 <p> | |
| 365 <s xml:id="echoid-s7" xml:space="preserve"> | |
| 366 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mo>-</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. | |
| 367 </s> | |
| 368 </p> | |
| 369 </emph>] | |
| 370 <lb/><lb/></s></p></div> | |
| 371 <pb file="add_6784_f088v" o="88v" n="176"/> | |
| 372 <pb file="add_6784_f089" o="89" n="177"/> | |
| 373 <pb file="add_6784_f089v" o="89v" n="178"/> | |
| 374 <div xml:id="echoid-div4" type="page_commentary" level="2" n="4"> | |
| 375 <p> | |
| 376 <s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note: | |
| 377 <p> | |
| 378 <s xml:id="echoid-s9" xml:space="preserve"> | |
| 379 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo>-</mo><mi>g</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo maxsize="1">)</mo></mstyle></math>. | |
| 380 </s> | |
| 381 </p> | |
| 382 </emph>] | |
| 383 <lb/><lb/></s></p></div> | |
| 384 <pb file="add_6784_f090" o="90" n="179"/> | |
| 385 <pb file="add_6784_f090v" o="90v" n="180"/> | |
| 386 <pb file="add_6784_f091" o="91" n="181"/> | |
| 387 <pb file="add_6784_f091v" o="91v" n="182"/> | |
| 388 <pb file="add_6784_f092" o="92" n="183"/> | |
| 389 <pb file="add_6784_f092v" o="92v" n="184"/> | |
| 390 <pb file="add_6784_f093" o="93" n="185"/> | |
| 391 <pb file="add_6784_f093v" o="93v" n="186"/> | |
| 392 <pb file="add_6784_f094" o="94" n="187"/> | |
| 393 <pb file="add_6784_f094v" o="94v" n="188"/> | |
| 394 <pb file="add_6784_f095" o="95" n="189"/> | |
| 395 <pb file="add_6784_f095v" o="95v" n="190"/> | |
| 396 <pb file="add_6784_f096" o="96" n="191"/> | |
| 397 <pb file="add_6784_f096v" o="96v" n="192"/> | |
| 398 <pb file="add_6784_f097" o="97" n="193"/> | |
| 399 <pb file="add_6784_f097v" o="97v" n="194"/> | |
| 400 <pb file="add_6784_f098" o="98" n="195"/> | |
| 401 <pb file="add_6784_f098v" o="98v" n="196"/> | |
| 402 <pb file="add_6784_f099" o="99" n="197"/> | |
| 403 <pb file="add_6784_f099v" o="99v" n="198"/> | |
| 404 <pb file="add_6784_f100" o="100" n="199"/> | |
| 405 <pb file="add_6784_f100v" o="100v" n="200"/> | |
| 406 <pb file="add_6784_f101" o="101" n="201"/> | |
| 407 <pb file="add_6784_f101v" o="101v" n="202"/> | |
| 408 <pb file="add_6784_f102" o="102" n="203"/> | |
| 409 <pb file="add_6784_f102v" o="102v" n="204"/> | |
| 410 <pb file="add_6784_f103" o="103" n="205"/> | |
| 411 <pb file="add_6784_f103v" o="103v" n="206"/> | |
| 412 <pb file="add_6784_f104" o="104" n="207"/> | |
| 413 <pb file="add_6784_f104v" o="104v" n="208"/> | |
| 414 <pb file="add_6784_f105" o="105" n="209"/> | |
| 415 <p xml:lang="lat"> | |
| 416 <s xml:id="echoid-s11" xml:space="preserve"> | |
| 417 Graecia <lb/> | |
| 418 prævenians. <lb/> | |
| 419 excitans. <lb/> | |
| 420 vocans. <lb/> | |
| 421 operans. <lb/> | |
| 422 provens. <lb/> | |
| 423 comians. <lb/> | |
| 424 cooperans. <lb/> | |
| 425 adiunans. <lb/> | |
| 426 concomitans. <lb/> | |
| 427 subsequens. <lb/> | |
| 428 prosequens. | |
| 429 </s> | |
| 430 </p> | |
| 431 <pb file="add_6784_f105v" o="105v" n="210"/> | |
| 432 <pb file="add_6784_f106" o="106" n="211"/> | |
| 433 <pb file="add_6784_f106v" o="106v" n="212"/> | |
| 434 <pb file="add_6784_f107" o="107" n="213"/> | |
| 435 <pb file="add_6784_f107v" o="107v" n="214"/> | |
| 436 <pb file="add_6784_f108" o="108" n="215"/> | |
| 437 <pb file="add_6784_f108v" o="108v" n="216"/> | |
| 438 <pb file="add_6784_f109" o="109" n="217"/> | |
| 439 <pb file="add_6784_f109v" o="109v" n="218"/> | |
| 440 <pb file="add_6784_f110" o="110" n="219"/> | |
| 441 <pb file="add_6784_f110v" o="110v" n="220"/> | |
| 442 <pb file="add_6784_f111" o="111" n="221"/> | |
| 443 <pb file="add_6784_f111v" o="111v" n="222"/> | |
| 444 <pb file="add_6784_f112" o="112" n="223"/> | |
| 445 <pb file="add_6784_f112v" o="112v" n="224"/> | |
| 446 <pb file="add_6784_f113" o="113" n="225"/> | |
| 447 <pb file="add_6784_f113v" o="113v" n="226"/> | |
| 448 <pb file="add_6784_f114" o="114" n="227"/> | |
| 449 <pb file="add_6784_f114v" o="114v" n="228"/> | |
| 450 <pb file="add_6784_f115" o="115" n="229"/> | |
| 451 <pb file="add_6784_f115v" o="115v" n="230"/> | |
| 452 <pb file="add_6784_f116" o="116" n="231"/> | |
| 453 <pb file="add_6784_f116v" o="116v" n="232"/> | |
| 454 <pb file="add_6784_f117" o="117" n="233"/> | |
| 455 <pb file="add_6784_f117v" o="117v" n="234"/> | |
| 456 <pb file="add_6784_f118" o="118" n="235"/> | |
| 457 <pb file="add_6784_f118v" o="118v" n="236"/> | |
| 458 <pb file="add_6784_f119" o="119" n="237"/> | |
| 459 <pb file="add_6784_f119v" o="119v" n="238"/> | |
| 460 <pb file="add_6784_f120" o="120" n="239"/> | |
| 461 <pb file="add_6784_f120v" o="120v" n="240"/> | |
| 462 <pb file="add_6784_f121" o="121" n="241"/> | |
| 463 <pb file="add_6784_f121v" o="121v" n="242"/> | |
| 464 <pb file="add_6784_f122" o="122" n="243"/> | |
| 465 <pb file="add_6784_f122v" o="122v" n="244"/> | |
| 466 <pb file="add_6784_f123" o="123" n="245"/> | |
| 467 <div xml:id="echoid-div5" type="page_commentary" level="2" n="5"> | |
| 468 <p> | |
| 469 <s xml:id="echoid-s12" xml:space="preserve">[<emph style="it">Note: | |
| 470 <p> | |
| 471 <s xml:id="echoid-s12" xml:space="preserve"> | |
| 472 The references on this page are to Pappus, Book 7, | |
| 473 and to Giambattista Benedetti, | |
| 474 <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585). | |
| 475 </s> | |
| 476 </p> | |
| 477 </emph>] | |
| 478 <lb/><lb/></s></p></div> | |
| 479 <p xml:lang="lat"> | |
| 480 <s xml:id="echoid-s14" xml:space="preserve"> | |
| 481 sit triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math> | |
| 482 <lb/>[<emph style="it">tr: | |
| 483 let there be a triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math> | |
| 484 </emph>]<lb/> | |
| 485 </s> | |
| 486 <lb/> | |
| 487 <s xml:id="echoid-s15" xml:space="preserve"> | |
| 488 dico quod | |
| 489 <lb/>[<emph style="it">tr: | |
| 490 I say that | |
| 491 </emph>]<lb/> | |
| 492 </s> | |
| 493 </p> | |
| 494 <p xml:lang="lat"> | |
| 495 <s xml:id="echoid-s16" xml:space="preserve"> | |
| 496 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math> perpendicularis ad, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> | |
| 497 <lb/>[<emph style="it">tr: | |
| 498 let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>e</mi></mstyle></math> be perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> | |
| 499 </emph>]<lb/> | |
| 500 </s> | |
| 501 </p> | |
| 502 <p xml:lang="lat"> | |
| 503 <s xml:id="echoid-s17" xml:space="preserve"> | |
| 504 Unde sequitur | |
| 505 <lb/>[<emph style="it">tr: | |
| 506 whence it follows | |
| 507 </emph>]<lb/> | |
| 508 </s> | |
| 509 </p> | |
| 510 <p xml:lang="lat"> | |
| 511 <s xml:id="echoid-s18" xml:space="preserve"> | |
| 512 Vide, Pappum. lib. 7. prop: 122. pag. 235. <lb/> | |
| 513 et: Jo: Baptistum Benedictum pag. 362. | |
| 514 <lb/>[<emph style="it">tr: | |
| 515 See Pappus, Book 7, Proposition 122, page 235; and Johan Baptista Benedictus, page 362 | |
| 516 </emph>]<lb/> | |
| 517 </s> | |
| 518 </p> | |
| 519 <p xml:lang="lat"> | |
| 520 <s xml:id="echoid-s19" xml:space="preserve"> | |
| 521 verte | |
| 522 <lb/>[<emph style="it">tr: | |
| 523 turn over | |
| 524 </emph>]<lb/> | |
| 525 </s> | |
| 526 </p> | |
| 527 <pb file="add_6784_f123v" o="123v" n="246"/> | |
| 528 <pb file="add_6784_f124" o="124" n="247"/> | |
| 529 <pb file="add_6784_f124v" o="124v" n="248"/> | |
| 530 <pb file="add_6784_f125" o="125" n="249"/> | |
| 531 <pb file="add_6784_f125v" o="125v" n="250"/> | |
| 532 <pb file="add_6784_f126" o="126" n="251"/> | |
| 533 <pb file="add_6784_f126v" o="126v" n="252"/> | |
| 534 <pb file="add_6784_f127" o="127" n="253"/> | |
| 535 <head xml:id="echoid-head40" xml:space="preserve"> | |
| 536 Lemma. 1. Appol. Bat. pag. 81. | |
| 537 </head> | |
| 538 <p xml:lang="lat"> | |
| 539 <s xml:id="echoid-s20" xml:space="preserve"> | |
| 540 Sit: <lb/> | |
| 541 Dico quod: <lb/> | |
| 542 nam in utraque analogia | |
| 543 <lb/>[<emph style="it">tr: | |
| 544 Let: <lb/> | |
| 545 I say that: <lb/> | |
| 546 for in the both ratios | |
| 547 </emph>]<lb/> | |
| 548 </s> | |
| 549 </p> | |
| 550 <p xml:lang="lat"> | |
| 551 <s xml:id="echoid-s21" xml:space="preserve"> | |
| 552 Sed ita Snellius | |
| 553 <lb/>[<emph style="it">tr: | |
| 554 But it is thus in Snell. | |
| 555 </emph>]<lb/> | |
| 556 </s> | |
| 557 </p> | |
| 558 <pb file="add_6784_f127v" o="127v" n="254"/> | |
| 559 <pb file="add_6784_f128" o="128" n="255"/> | |
| 560 <pb file="add_6784_f128v" o="128v" n="256"/> | |
| 561 <pb file="add_6784_f129" o="129" n="257"/> | |
| 562 <pb file="add_6784_f129v" o="129v" n="258"/> | |
| 563 <pb file="add_6784_f130" o="130" n="259"/> | |
| 564 <pb file="add_6784_f130v" o="130v" n="260"/> | |
| 565 <pb file="add_6784_f131" o="131" n="261"/> | |
| 566 <pb file="add_6784_f131v" o="131v" n="262"/> | |
| 567 <pb file="add_6784_f132" o="132" n="263"/> | |
| 568 <pb file="add_6784_f132v" o="132v" n="264"/> | |
| 569 <pb file="add_6784_f133" o="133" n="265"/> | |
| 570 <pb file="add_6784_f133v" o="133v" n="266"/> | |
| 571 <pb file="add_6784_f134" o="134" n="267"/> | |
| 572 <pb file="add_6784_f134v" o="134v" n="268"/> | |
| 573 <pb file="add_6784_f135" o="135" n="269"/> | |
| 574 <pb file="add_6784_f135v" o="135v" n="270"/> | |
| 575 <pb file="add_6784_f136" o="136" n="271"/> | |
| 576 <pb file="add_6784_f136v" o="136v" n="272"/> | |
| 577 <pb file="add_6784_f137" o="137" n="273"/> | |
| 578 <pb file="add_6784_f137v" o="137v" n="274"/> | |
| 579 <pb file="add_6784_f138" o="138" n="275"/> | |
| 580 <pb file="add_6784_f138v" o="138v" n="276"/> | |
| 581 <pb file="add_6784_f139" o="139" n="277"/> | |
| 582 <pb file="add_6784_f139v" o="139v" n="278"/> | |
| 583 <pb file="add_6784_f140" o="140" n="279"/> | |
| 584 <pb file="add_6784_f140v" o="140v" n="280"/> | |
| 585 <pb file="add_6784_f141" o="141" n="281"/> | |
| 586 <pb file="add_6784_f141v" o="141v" n="282"/> | |
| 587 <pb file="add_6784_f142" o="142" n="283"/> | |
| 588 <pb file="add_6784_f142v" o="142v" n="284"/> | |
| 589 <pb file="add_6784_f143" o="143" n="285"/> | |
| 590 <pb file="add_6784_f143v" o="143v" n="286"/> | |
| 591 <pb file="add_6784_f144" o="144" n="287"/> | |
| 592 <pb file="add_6784_f144v" o="144v" n="288"/> | |
| 593 <pb file="add_6784_f145" o="145" n="289"/> | |
| 594 <pb file="add_6784_f145v" o="145v" n="290"/> | |
| 595 <pb file="add_6784_f146" o="146" n="291"/> | |
| 596 <pb file="add_6784_f146v" o="146v" n="292"/> | |
| 597 <pb file="add_6784_f147" o="147" n="293"/> | |
| 598 <pb file="add_6784_f147v" o="147v" n="294"/> | |
| 599 <pb file="add_6784_f148" o="148" n="295"/> | |
| 600 <pb file="add_6784_f148v" o="148v" n="296"/> | |
| 601 <pb file="add_6784_f149" o="149" n="297"/> | |
| 602 <pb file="add_6784_f149v" o="149v" n="298"/> | |
| 603 <div xml:id="echoid-div6" type="page_commentary" level="2" n="6"> | |
| 604 <p> | |
| 605 <s xml:id="echoid-s22" xml:space="preserve">[<emph style="it">Note: | |
| 606 <p> | |
| 607 <s xml:id="echoid-s22" xml:space="preserve"> | |
| 608 This page contains symbolic versions of Euclid Book II, Propositions 12 and 13: <lb/> | |
| 609 II.12.In obtuse-angle triangles the square on the side opposite the obtuse angle | |
| 610 is greater than the sum of the squares on the sides containing the obtuse angle | |
| 611 by twice the rectangle contained by one of the sides about the obtuse angle, | |
| 612 namely that on which the perpendicular falls, and the straight line cut off outside | |
| 613 by the perpendicular towards the obtuse angle. <lb/> | |
| 614 II.13. In acute-angled triangles the square on the side opposite the acute angle | |
| 615 is less than the sum of the squares on the sides containing the acute angle | |
| 616 by twice the rectangle contained by one of the sides about the acute angle, | |
| 617 namely that on which the perpendicular falls, and the straight line cut off within | |
| 618 by the perpendicular towards the acute angle. | |
| 619 </s> | |
| 620 </p> | |
| 621 </emph>] | |
| 622 <lb/><lb/></s></p></div> | |
| 623 <head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat"> | |
| 624 Aliter de 12. 2<emph style="super">i</emph> Euclidis <lb/> | |
| 625 et 13. | |
| 626 <lb/>[<emph style="it">tr: | |
| 627 Another way for Euclid II.12 and 13. | |
| 628 </emph>]<lb/> | |
| 629 </head> | |
| 630 <pb file="add_6784_f150" o="150" n="299"/> | |
| 631 <pb file="add_6784_f150v" o="150v" n="300"/> | |
| 632 <pb file="add_6784_f151" o="151" n="301"/> | |
| 633 <pb file="add_6784_f151v" o="151v" n="302"/> | |
| 634 <div xml:id="echoid-div7" type="page_commentary" level="2" n="7"> | |
| 635 <p> | |
| 636 <s xml:id="echoid-s24" xml:space="preserve">[<emph style="it">Note: | |
| 637 <p> | |
| 638 <s xml:id="echoid-s24" xml:space="preserve"> | |
| 639 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mo>-</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo maxsize="1">)</mo></mstyle></math>. | |
| 640 </s> | |
| 641 </p> | |
| 642 </emph>] | |
| 643 <lb/><lb/></s></p></div> | |
| 644 <pb file="add_6784_f152" o="152" n="303"/> | |
| 645 <pb file="add_6784_f152v" o="152v" n="304"/> | |
| 646 <pb file="add_6784_f153" o="153" n="305"/> | |
| 647 <pb file="add_6784_f153v" o="153v" n="306"/> | |
| 648 <pb file="add_6784_f154" o="154" n="307"/> | |
| 649 <pb file="add_6784_f154v" o="154v" n="308"/> | |
| 650 <div xml:id="echoid-div8" type="page_commentary" level="2" n="8"> | |
| 651 <p> | |
| 652 <s xml:id="echoid-s26" xml:space="preserve">[<emph style="it">Note: | |
| 653 <p> | |
| 654 <s xml:id="echoid-s26" xml:space="preserve"> | |
| 655 Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>c</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>b</mi><mo maxsize="1">)</mo></mstyle></math>. | |
| 656 </s> | |
| 657 </p> | |
| 658 </emph>] | |
| 659 <lb/><lb/></s></p></div> | |
| 660 <pb file="add_6784_f155" o="155" n="309"/> | |
| 661 <pb file="add_6784_f155v" o="155v" n="310"/> | |
| 662 <pb file="add_6784_f156" o="156" n="311"/> | |
| 663 <pb file="add_6784_f156v" o="156v" n="312"/> | |
| 664 <pb file="add_6784_f157" o="157" n="313"/> | |
| 665 <pb file="add_6784_f157v" o="157v" n="314"/> | |
| 666 <pb file="add_6784_f158" o="158" n="315"/> | |
| 667 <pb file="add_6784_f158v" o="158v" n="316"/> | |
| 668 <pb file="add_6784_f159" o="159" n="317"/> | |
| 669 <pb file="add_6784_f159v" o="159v" n="318"/> | |
| 670 <pb file="add_6784_f160" o="160" n="319"/> | |
| 671 <head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat"> | |
| 672 phys. lib.6. Cap. 1 | |
| 673 <lb/>[<emph style="it">tr: | |
| 674 Physics, Book 6, Chapter 1 | |
| 675 </emph>]<lb/> | |
| 676 </head> | |
| 677 <pb file="add_6784_f160v" o="160v" n="320"/> | |
| 678 <pb file="add_6784_f161" o="161" n="321"/> | |
| 679 <pb file="add_6784_f161v" o="161v" n="322"/> | |
| 680 <pb file="add_6784_f162" o="162" n="323"/> | |
| 681 <pb file="add_6784_f162v" o="162v" n="324"/> | |
| 682 <pb file="add_6784_f163" o="163" n="325"/> | |
| 683 <pb file="add_6784_f163v" o="163v" n="326"/> | |
| 684 <pb file="add_6784_f164" o="164" n="327"/> | |
| 685 <head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat"> | |
| 686 Arist. lib. 6. Cap. 2 | |
| 687 <lb/>[<emph style="it">tr: | |
| 688 Aristotle, Book 6, Chapter 2 | |
| 689 </emph>]<lb/> | |
| 690 </head> | |
| 691 <pb file="add_6784_f164v" o="164v" n="328"/> | |
| 692 <pb file="add_6784_f165" o="165" n="329"/> | |
| 693 <pb file="add_6784_f165v" o="165v" n="330"/> | |
| 694 <pb file="add_6784_f166" o="166" n="331"/> | |
| 695 <div xml:id="echoid-div9" type="page_commentary" level="2" n="9"> | |
| 696 <p> | |
| 697 <s xml:id="echoid-s28" xml:space="preserve">[<emph style="it">Note: | |
| 698 <p> | |
| 699 <s xml:id="echoid-s28" xml:space="preserve"> | |
| 700 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 701 </s> | |
| 702 </p> | |
| 703 </emph>] | |
| 704 <lb/><lb/></s></p></div> | |
| 705 <head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat"> | |
| 706 Residuum 5<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 707 <lb/>[<emph style="it">tr: | |
| 708 The rest of the working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 709 </emph>]<lb/> | |
| 710 </head> | |
| 711 <pb file="add_6784_f166v" o="166v" n="332"/> | |
| 712 <pb file="add_6784_f167" o="167" n="333"/> | |
| 713 <div xml:id="echoid-div10" type="page_commentary" level="2" n="10"> | |
| 714 <p> | |
| 715 <s xml:id="echoid-s30" xml:space="preserve">[<emph style="it">Note: | |
| 716 <p> | |
| 717 <s xml:id="echoid-s30" xml:space="preserve"> | |
| 718 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 719 </s> | |
| 720 </p> | |
| 721 </emph>] | |
| 722 <lb/><lb/></s></p></div> | |
| 723 <head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat"> | |
| 724 5<emph style="super">a</emph> operatio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 725 <lb/>[<emph style="it">tr: | |
| 726 Working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> | |
| 727 </emph>]<lb/> | |
| 728 </head> | |
| 729 <pb file="add_6784_f167v" o="167v" n="334"/> | |
| 730 <pb file="add_6784_f168" o="168" n="335"/> | |
| 731 <div xml:id="echoid-div11" type="page_commentary" level="2" n="11"> | |
| 732 <p> | |
| 733 <s xml:id="echoid-s32" xml:space="preserve">[<emph style="it">Note: | |
| 734 <p> | |
| 735 <s xml:id="echoid-s32" xml:space="preserve"> | |
| 736 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 737 </s> | |
| 738 </p> | |
| 739 </emph>] | |
| 740 <lb/><lb/></s></p></div> | |
| 741 <pb file="add_6784_f168v" o="168v" n="336"/> | |
| 742 <pb file="add_6784_f169" o="169" n="337"/> | |
| 743 <div xml:id="echoid-div12" type="page_commentary" level="2" n="12"> | |
| 744 <p> | |
| 745 <s xml:id="echoid-s34" xml:space="preserve">[<emph style="it">Note: | |
| 746 <p> | |
| 747 <s xml:id="echoid-s34" xml:space="preserve"> | |
| 748 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 749 </s> | |
| 750 </p> | |
| 751 </emph>] | |
| 752 <lb/><lb/></s></p></div> | |
| 753 <p xml:lang="lat"> | |
| 754 <s xml:id="echoid-s36" xml:space="preserve"> | |
| 755 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>d</mi></mstyle></math>. (si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi><mo>=</mo><mn>0</mn></mstyle></math>.) <lb/> | |
| 756 vel, cuivis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. | |
| 757 <lb/>[<emph style="it">tr: | |
| 758 or, for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> | |
| 759 </emph>]<lb/> | |
| 760 </s> | |
| 761 </p> | |
| 762 <p xml:lang="lat"> | |
| 763 <s xml:id="echoid-s37" xml:space="preserve"> | |
| 764 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>. cuivis. | |
| 765 <lb/>[<emph style="it">tr: | |
| 766 any | |
| 767 </emph>]<lb/> | |
| 768 </s> | |
| 769 </p> | |
| 770 <pb file="add_6784_f169v" o="169v" n="338"/> | |
| 771 <pb file="add_6784_f170" o="170" n="339"/> | |
| 772 <div xml:id="echoid-div13" type="page_commentary" level="2" n="13"> | |
| 773 <p> | |
| 774 <s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note: | |
| 775 <p> | |
| 776 <s xml:id="echoid-s38" xml:space="preserve"> | |
| 777 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 778 </s> | |
| 779 </p> | |
| 780 </emph>] | |
| 781 <lb/><lb/></s></p></div> | |
| 782 <pb file="add_6784_f170v" o="170v" n="340"/> | |
| 783 <pb file="add_6784_f171" o="171" n="341"/> | |
| 784 <pb file="add_6784_f171v" o="171v" n="342"/> | |
| 785 <div xml:id="echoid-div14" type="page_commentary" level="2" n="14"> | |
| 786 <p> | |
| 787 <s xml:id="echoid-s40" xml:space="preserve">[<emph style="it">Note: | |
| 788 <p> | |
| 789 <s xml:id="echoid-s40" xml:space="preserve"> | |
| 790 Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 791 </s> | |
| 792 </p> | |
| 793 </emph>] | |
| 794 <lb/><lb/></s></p></div> | |
| 795 <head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat"> | |
| 796 Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. | |
| 797 <lb/>[<emph style="it">tr: | |
| 798 Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> | |
| 799 </emph>]<lb/> | |
| 800 </head> | |
| 801 <pb file="add_6784_f172" o="172" n="343"/> | |
| 802 <pb file="add_6784_f172v" o="172v" n="344"/> | |
| 803 <div xml:id="echoid-div15" type="page_commentary" level="2" n="15"> | |
| 804 <p> | |
| 805 <s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note: | |
| 806 <p> | |
| 807 <s xml:id="echoid-s42" xml:space="preserve"> | |
| 808 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 809 </s> | |
| 810 </p> | |
| 811 </emph>] | |
| 812 <lb/><lb/></s></p></div> | |
| 813 <head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat"> | |
| 814 Residuum 3<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 815 <lb/>[<emph style="it">tr: | |
| 816 The rest of the working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 817 </emph>]<lb/> | |
| 818 </head> | |
| 819 <p xml:lang="lat"> | |
| 820 <s xml:id="echoid-s44" xml:space="preserve"> | |
| 821 Residuum 4<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 822 <lb/>[<emph style="it">tr: | |
| 823 The rest of the working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 824 </emph>]<lb/> | |
| 825 </s> | |
| 826 </p> | |
| 827 <pb file="add_6784_f173" o="173" n="345"/> | |
| 828 <div xml:id="echoid-div16" type="page_commentary" level="2" n="16"> | |
| 829 <p> | |
| 830 <s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note: | |
| 831 <p> | |
| 832 <s xml:id="echoid-s45" xml:space="preserve"> | |
| 833 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 834 </s> | |
| 835 </p> | |
| 836 </emph>] | |
| 837 <lb/><lb/></s></p></div> | |
| 838 <head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat"> | |
| 839 3<emph style="super">a</emph> operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 840 <lb/>[<emph style="it">tr: | |
| 841 Working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> | |
| 842 </emph>]<lb/> | |
| 843 </head> | |
| 844 <p xml:lang="lat"> | |
| 845 <s xml:id="echoid-s47" xml:space="preserve"> | |
| 846 4<emph style="super">a</emph> operatio G. | |
| 847 <lb/>[<emph style="it">tr: | |
| 848 Working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> | |
| 849 </emph>]<lb/> | |
| 850 </s> | |
| 851 </p> | |
| 852 <pb file="add_6784_f173v" o="173v" n="346"/> | |
| 853 <pb file="add_6784_f174" o="174" n="347"/> | |
| 854 <pb file="add_6784_f174v" o="174v" n="348"/> | |
| 855 <pb file="add_6784_f175" o="175" n="349"/> | |
| 856 <pb file="add_6784_f175v" o="175v" n="350"/> | |
| 857 <pb file="add_6784_f176" o="176" n="351"/> | |
| 858 <head xml:id="echoid-head49" xml:space="preserve"> | |
| 859 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.11 De tactibus | |
| 860 </head> | |
| 861 <p xml:lang="lat"> | |
| 862 <s xml:id="echoid-s48" xml:space="preserve"> | |
| 863 cave | |
| 864 <lb/>[<emph style="it">tr: | |
| 865 beware | |
| 866 </emph>]<lb/> | |
| 867 </s> | |
| 868 </p> | |
| 869 <p xml:lang="lat"> | |
| 870 <s xml:id="echoid-s49" xml:space="preserve"> | |
| 871 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. est centrum circuli <lb/> | |
| 872 circumscribentis. <lb/> | |
| 873 Tria traingula. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> | |
| 874 habet periferias æquales. | |
| 875 <lb/>[<emph style="it">tr: | |
| 876 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is the centre of the circumscribing circle. <lb/> | |
| 877 The three triangles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math> have equal circumferences. | |
| 878 </emph>]<lb/> | |
| 879 </s> | |
| 880 </p> | |
| 881 <pb file="add_6784_f176v" o="176v" n="352"/> | |
| 882 <pb file="add_6784_f177" o="177" n="353"/> | |
| 883 <head xml:id="echoid-head50" xml:space="preserve"> | |
| 884 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.2 | |
| 885 </head> | |
| 886 <p xml:lang="lat"> | |
| 887 <s xml:id="echoid-s50" xml:space="preserve"> | |
| 888 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, latera <lb/> | |
| 889 <lb/>[...]<lb/> <lb/> | |
| 890 cuius superficies ut sequitur. | |
| 891 <lb/>[<emph style="it">tr: | |
| 892 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, with sides: <lb/> | |
| 893 <lb/>[...]<lb/> <lb/> | |
| 894 whose surface is as follows. | |
| 895 </emph>]<lb/> | |
| 896 </s> | |
| 897 </p> | |
| 898 <pb file="add_6784_f177v" o="177v" n="354"/> | |
| 899 <pb file="add_6784_f178" o="178" n="355"/> | |
| 900 <head xml:id="echoid-head51" xml:space="preserve"> | |
| 901 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.3 | |
| 902 </head> | |
| 903 <p xml:lang="lat"> | |
| 904 <s xml:id="echoid-s51" xml:space="preserve"> | |
| 905 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, latera <lb/> | |
| 906 <lb/>[...]<lb/> <lb/> | |
| 907 cuius superficies ut sequitur. | |
| 908 <lb/>[<emph style="it">tr: | |
| 909 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>, with sides: <lb/> | |
| 910 <lb/>[...]<lb/> <lb/> | |
| 911 whose surface is as follows. | |
| 912 </emph>]<lb/> | |
| 913 </s> | |
| 914 </p> | |
| 915 <pb file="add_6784_f178v" o="178v" n="356"/> | |
| 916 <pb file="add_6784_f179" o="179" n="357"/> | |
| 917 <head xml:id="echoid-head52" xml:space="preserve"> | |
| 918 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.4 | |
| 919 </head> | |
| 920 <pb file="add_6784_f179v" o="179v" n="358"/> | |
| 921 <pb file="add_6784_f180" o="180" n="359"/> | |
| 922 <div xml:id="echoid-div17" type="page_commentary" level="2" n="17"> | |
| 923 <p> | |
| 924 <s xml:id="echoid-s52" xml:space="preserve">[<emph style="it">Note: | |
| 925 <p> | |
| 926 <s xml:id="echoid-s52" xml:space="preserve"> | |
| 927 The reference in the top right hand corner is to Viète, | |
| 928 <emph style="it">Apollonius Gallus</emph> (1600), Problem IX. | |
| 929 </s> | |
| 930 <lb/> | |
| 931 <quote xml:lang="lat"> | |
| 932 Problema IX. <lb/> | |
| 933 Datis duobus circulis, & puncto, per datum punctum circulum describere | |
| 934 quem duo dati circuli contingat. | |
| 935 </quote> | |
| 936 <lb/> | |
| 937 <quote> | |
| 938 IX. Given two circles and a point, through the given point describe a circle that touches the two given circles. | |
| 939 </quote> | |
| 940 </p> | |
| 941 </emph>] | |
| 942 <lb/><lb/></s></p></div> | |
| 943 <head xml:id="echoid-head53" xml:space="preserve"> | |
| 944 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5) | |
| 945 </head> | |
| 946 <p xml:lang="lat"> | |
| 947 <s xml:id="echoid-s54" xml:space="preserve"> | |
| 948 Vide: Appol. Gall. prob. 9. | |
| 949 <lb/>[<emph style="it">tr: | |
| 950 See Apollonius Gallus, Problem IX. | |
| 951 </emph>]<lb/> | |
| 952 </s> | |
| 953 </p> | |
| 954 <p xml:lang="lat"> | |
| 955 <s xml:id="echoid-s55" xml:space="preserve"> | |
| 956 Aberratur de modo contingendi <lb/> | |
| 957 circulos posititios alias operatio bona <lb/> | |
| 958 vide igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>. | |
| 959 <lb/>[<emph style="it">tr: | |
| 960 There is an error in the method of touching the supposed circles, othersie the working is good; | |
| 961 therefore see shee <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>..5.2. | |
| 962 </emph>]<lb/> | |
| 963 [<emph style="it">Note: | |
| 964 Sheet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2 is Add MS 6784, f. 181. | |
| 965 </emph>]<lb/> | |
| 966 </s> | |
| 967 </p> | |
| 968 <p xml:lang="lat"> | |
| 969 <s xml:id="echoid-s56" xml:space="preserve"> | |
| 970 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> | |
| 971 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> | |
| 972 distantia centrorum | |
| 973 <lb/>[<emph style="it">tr: | |
| 974 radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> | |
| 975 of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> | |
| 976 distance of the centres. | |
| 977 </emph>]<lb/> | |
| 978 </s> | |
| 979 </p> | |
| 980 <pb file="add_6784_f180v" o="180v" n="360"/> | |
| 981 <pb file="add_6784_f181" o="181" n="361"/> | |
| 982 <div xml:id="echoid-div18" type="page_commentary" level="2" n="18"> | |
| 983 <p> | |
| 984 <s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: | |
| 985 <p> | |
| 986 <s xml:id="echoid-s57" xml:space="preserve"> | |
| 987 A continuation of the work on Add MS 6784, f. 180. | |
| 988 </s> | |
| 989 </p> | |
| 990 </emph>] | |
| 991 <lb/><lb/></s></p></div> | |
| 992 <head xml:id="echoid-head54" xml:space="preserve"> | |
| 993 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>) | |
| 994 </head> | |
| 995 <p xml:lang="lat"> | |
| 996 <s xml:id="echoid-s59" xml:space="preserve"> | |
| 997 Vide: Appol: Gall. prob. 9. <lb/> | |
| 998 fig: 2. | |
| 999 <lb/>[<emph style="it">tr: | |
| 1000 See Apollonius Gallus, Problem IX, figure 2. | |
| 1001 </emph>]<lb/> | |
| 1002 </s> | |
| 1003 </p> | |
| 1004 <p xml:lang="lat"> | |
| 1005 <s xml:id="echoid-s60" xml:space="preserve"> | |
| 1006 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> | |
| 1007 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> | |
| 1008 distantia centrorum | |
| 1009 <lb/>[<emph style="it">tr: | |
| 1010 radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> | |
| 1011 of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> | |
| 1012 distance of the centres. | |
| 1013 </emph>]<lb/> | |
| 1014 </s> | |
| 1015 </p> | |
| 1016 <pb file="add_6784_f181v" o="181v" n="362"/> | |
| 1017 <pb file="add_6784_f182" o="182" n="363"/> | |
| 1018 <head xml:id="echoid-head55" xml:space="preserve"> | |
| 1019 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.6.) | |
| 1020 </head> | |
| 1021 <p xml:lang="lat"> | |
| 1022 <s xml:id="echoid-s61" xml:space="preserve"> | |
| 1023 radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) <lb/> | |
| 1024 posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) <lb/> | |
| 1025 distantia centrorum | |
| 1026 <lb/>[<emph style="it">tr: | |
| 1027 radius of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> | |
| 1028 of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> | |
| 1029 distance of the centres. | |
| 1030 </emph>]<lb/> | |
| 1031 </s> | |
| 1032 </p> | |
| 1033 <pb file="add_6784_f182v" o="182v" n="364"/> | |
| 1034 <pb file="add_6784_f183" o="183" n="365"/> | |
| 1035 <head xml:id="echoid-head56" xml:space="preserve"> | |
| 1036 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.1 | |
| 1037 </head> | |
| 1038 <p xml:lang="lat"> | |
| 1039 <s xml:id="echoid-s62" xml:space="preserve"> | |
| 1040 data <lb/> | |
| 1041 <lb/>[...]<lb/> <lb/> | |
| 1042 Quæritur: vel. | |
| 1043 <lb/>[<emph style="it">tr: | |
| 1044 given <lb/> | |
| 1045 <lb/>[...]<lb/> <lb/> | |
| 1046 Sought, either: | |
| 1047 </emph>]<lb/> | |
| 1048 </s> | |
| 1049 </p> | |
| 1050 <p xml:lang="lat"> | |
| 1051 <s xml:id="echoid-s63" xml:space="preserve"> | |
| 1052 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, latera <lb/> | |
| 1053 <lb/>[...]<lb/> <lb/> | |
| 1054 cuius superficies ut sequitur. | |
| 1055 <lb/>[<emph style="it">tr: | |
| 1056 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>d</mi></mstyle></math>, with sides: <lb/> | |
| 1057 <lb/>[...]<lb/> <lb/> | |
| 1058 whose surface is as follows. | |
| 1059 </emph>]<lb/> | |
| 1060 </s> | |
| 1061 </p> | |
| 1062 <pb file="add_6784_f183v" o="183v" n="366"/> | |
| 1063 <pb file="add_6784_f184" o="184" n="367"/> | |
| 1064 <head xml:id="echoid-head57" xml:space="preserve"> | |
| 1065 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.3 | |
| 1066 </head> | |
| 1067 <p xml:lang="lat"> | |
| 1068 <s xml:id="echoid-s64" xml:space="preserve"> | |
| 1069 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math>, latera <lb/> | |
| 1070 <lb/>[...]<lb/> <lb/> | |
| 1071 cuius superficies ut sequitur. | |
| 1072 <lb/>[<emph style="it">tr: | |
| 1073 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math>, with sides: <lb/> | |
| 1074 <lb/>[...]<lb/> <lb/> | |
| 1075 whose surface is as follows. | |
| 1076 </emph>]<lb/> | |
| 1077 </s> | |
| 1078 </p> | |
| 1079 <pb file="add_6784_f184v" o="184v" n="368"/> | |
| 1080 <pb file="add_6784_f185" o="185" n="369"/> | |
| 1081 <head xml:id="echoid-head58" xml:space="preserve"> | |
| 1082 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.2 | |
| 1083 </head> | |
| 1084 <p xml:lang="lat"> | |
| 1085 <s xml:id="echoid-s65" xml:space="preserve"> | |
| 1086 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>Δ</mo></mstyle></math>,<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, latera <lb/> | |
| 1087 <lb/>[...]<lb/> <lb/> | |
| 1088 cuius superficies ut sequitur. | |
| 1089 <lb/>[<emph style="it">tr: | |
| 1090 Triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>, with sides: <lb/> | |
| 1091 <lb/>[...]<lb/> <lb/> | |
| 1092 whose surface is as follows. | |
| 1093 </emph>]<lb/> | |
| 1094 </s> | |
| 1095 </p> | |
| 1096 <pb file="add_6784_f185v" o="185v" n="370"/> | |
| 1097 <pb file="add_6784_f186" o="186" n="371"/> | |
| 1098 <head xml:id="echoid-head59" xml:space="preserve"> | |
| 1099 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.1 De tactibus | |
| 1100 </head> | |
| 1101 <pb file="add_6784_f186v" o="186v" n="372"/> | |
| 1102 <pb file="add_6784_f187" o="187" n="373"/> | |
| 1103 <pb file="add_6784_f187v" o="187v" n="374"/> | |
| 1104 <pb file="add_6784_f188" o="188" n="375"/> | |
| 1105 <pb file="add_6784_f188v" o="188v" n="376"/> | |
| 1106 <pb file="add_6784_f189" o="189" n="377"/> | |
| 1107 <pb file="add_6784_f189v" o="189v" n="378"/> | |
| 1108 <pb file="add_6784_f190" o="190" n="379"/> | |
| 1109 <pb file="add_6784_f190v" o="190v" n="380"/> | |
| 1110 <pb file="add_6784_f191" o="191" n="381"/> | |
| 1111 <pb file="add_6784_f191v" o="191v" n="382"/> | |
| 1112 <pb file="add_6784_f192" o="192" n="383"/> | |
| 1113 <pb file="add_6784_f192v" o="192v" n="384"/> | |
| 1114 <pb file="add_6784_f193" o="193" n="385"/> | |
| 1115 <pb file="add_6784_f193v" o="193v" n="386"/> | |
| 1116 <pb file="add_6784_f194" o="194" n="387"/> | |
| 1117 <head xml:id="echoid-head60" xml:space="preserve"> | |
| 1118 7. (o o) | |
| 1119 </head> | |
| 1120 <pb file="add_6784_f194v" o="194v" n="388"/> | |
| 1121 <pb file="add_6784_f195" o="195" n="389"/> | |
| 1122 <head xml:id="echoid-head61" xml:space="preserve"> | |
| 1123 De tactibus <lb/> | |
| 1124 Probl. 6 (. o -) | |
| 1125 </head> | |
| 1126 <pb file="add_6784_f195v" o="195v" n="390"/> | |
| 1127 <pb file="add_6784_f196" o="196" n="391"/> | |
| 1128 <pb file="add_6784_f196v" o="196v" n="392"/> | |
| 1129 <pb file="add_6784_f197" o="197" n="393"/> | |
| 1130 <head xml:id="echoid-head62" xml:space="preserve"> | |
| 1131 6) De tactibus | |
| 1132 <lb/>[<emph style="it">tr: | |
| 1133 On touching | |
| 1134 </emph>]<lb/> | |
| 1135 </head> | |
| 1136 <p> | |
| 1137 <s xml:id="echoid-s66" xml:space="preserve"> | |
| 1138 problema. <lb/> | |
| 1139 Datis tribus circulis <lb/> | |
| 1140 sese mutuo contingentibus: <lb/> | |
| 1141 invenire quartum circulum <lb/> | |
| 1142 qui mutus tangetur in datis. | |
| 1143 <lb/>[<emph style="it">tr: | |
| 1144 Problem. <lb/> | |
| 1145 Given three circles, mutually touching, to find a fourth circle that is mutually touched by those given. | |
| 1146 </emph>]<lb/> | |
| 1147 </s> | |
| 1148 </p> | |
| 1149 <p> | |
| 1150 <s xml:id="echoid-s67" xml:space="preserve"> | |
| 1151 Sint tres dati circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <lb/> | |
| 1152 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>t</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, sese mutuo contingentes <lb/> | |
| 1153 in punctis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. cuius centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> | |
| 1154 Agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> in continuum <lb/> | |
| 1155 <lb/>[...]<lb/> <lb/> | |
| 1156 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> recta contingens <lb/> | |
| 1157 circulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> | |
| 1158 Agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> in continuum quæ secabit <lb/> | |
| 1159 circulum cuius centrum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> puncto. <lb/> | |
| 1160 fiat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi><mi>i</mi></mstyle></math> recta, parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1161 Et ad lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> productam sint per-<lb/> | |
| 1162 pendicularis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>l</mi></mstyle></math>. | |
| 1163 <lb/>[<emph style="it">tr: | |
| 1164 Let the three given circles be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>t</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, mutually touching at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, | |
| 1165 whose centres are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/> | |
| 1166 There is constructed the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math>. <lb/> | |
| 1167 <lb/>[...]<lb/> <lb/> | |
| 1168 There is constructed the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> touching the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> | |
| 1169 There is constructed the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> which will cut the circule whose centre is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> | |
| 1170 Let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>a</mi><mi>i</mi></mstyle></math>be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1171 And to the extended line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> let there be perpendiculars <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>i</mi><mi>l</mi></mstyle></math>. | |
| 1172 </emph>]<lb/> | |
| 1173 </s> | |
| 1174 </p> | |
| 1175 <p> | |
| 1176 <s xml:id="echoid-s68" xml:space="preserve"> | |
| 1177 Bissecetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> | |
| 1178 Centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>b</mi></mstyle></math>, <lb/> | |
| 1179 describatur circulus. <lb/> | |
| 1180 Dico quod: ille est circulus quæsitus <lb/> | |
| 1181 et contingit tres datos <lb/> | |
| 1182 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. | |
| 1183 <lb/>[<emph style="it">tr: | |
| 1184 Let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be bisected at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. <lb/> | |
| 1185 With centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>b</mi></mstyle></math>, there is drawn a circle. <lb/> | |
| 1186 I say that this is the circle sought, and that it touches the tree given circles at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. | |
| 1187 </emph>]<lb/> | |
| 1188 </s> | |
| 1189 </p> | |
| 1190 <pb file="add_6784_f197v" o="197v" n="394"/> | |
| 1191 <pb file="add_6784_f198" o="198" n="395"/> | |
| 1192 <head xml:id="echoid-head63" xml:space="preserve"> | |
| 1193 <emph style="st">6.)</emph> 7.) | |
| 1194 </head> | |
| 1195 <p xml:lang="lat"> | |
| 1196 <s xml:id="echoid-s69" xml:space="preserve"> | |
| 1197 Sint tres dati circuli, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <lb/> | |
| 1198 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math>, sese mutuo <lb/> | |
| 1199 contingentes in punctis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <lb/> | |
| 1200 cuius centra, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. | |
| 1201 <lb/>[<emph style="it">tr: | |
| 1202 Let there be three given circles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>r</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>u</mi><mi>e</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math>, mutually touching in the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, | |
| 1203 whose centres are at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. | |
| 1204 </emph>]<lb/> | |
| 1205 </s> | |
| 1206 </p> | |
| 1207 <p xml:lang="lat"> | |
| 1208 <s xml:id="echoid-s70" xml:space="preserve"> | |
| 1209 Oportet invenire circulum <lb/> | |
| 1210 contingentem tres datos: <lb/> | |
| 1211 (nempe, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>h</mi><mi>t</mi></mstyle></math>, cius centrum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>) | |
| 1212 <lb/>[<emph style="it">tr: | |
| 1213 One must find the circle touching the three given ones (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>h</mi><mi>t</mi></mstyle></math>, with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>). | |
| 1214 </emph>]<lb/> | |
| 1215 </s> | |
| 1216 </p> | |
| 1217 <p xml:lang="lat"> | |
| 1218 <s xml:id="echoid-s71" xml:space="preserve"> | |
| 1219 Per centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, agatur recta <lb/> | |
| 1220 et continuetur ad utraque partes <lb/> | |
| 1221 et fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> | |
| 1222 Et ad illam lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> <lb/> | |
| 1223 perpendicularis. <lb/> | |
| 1224 Continuetur ad partes contrarias <lb/> | |
| 1225 usque ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, et fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi><mo>=</mo><mi>s</mi><mi>a</mi></mstyle></math>. <lb/> | |
| 1226 Tum primo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math> <lb/> | |
| 1227 quæ secabit periferiam circuli <lb/> | |
| 1228 cuius centrum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> | |
| 1229 Secundo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> <lb/> | |
| 1230 quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> productam in <lb/> | |
| 1231 puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> | |
| 1232 Ultimo, centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, intervallo <lb/> | |
| 1233 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> describatur circulus. <lb/> | |
| 1234 Dico quod: ille est circulus quæsitus <lb/> | |
| 1235 et contingit tres datos in <lb/> | |
| 1236 punctis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. | |
| 1237 <lb/>[<emph style="it">tr: | |
| 1238 Through the centres <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, a line is drawn and continued on both sides, and so there are | |
| 1239 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> | |
| 1240 And to that line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> be perpendicular. <lb/> | |
| 1241 It is continued to both sides as far as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi><mo>=</mo><mi>s</mi><mi>a</mi></mstyle></math>. <lb/> | |
| 1242 Then, first, there is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>, | |
| 1243 which will cut the circumference of the circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> | |
| 1244 Second, there is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math>, | |
| 1245 which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> extended, in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> | |
| 1246 Finally, with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math>, there is drawn the required circle. <lb/> | |
| 1247 I say that this is the circle sought, and it touches the three given at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. | |
| 1248 </emph>]<lb/> | |
| 1249 </s> | |
| 1250 </p> | |
| 1251 <head xml:id="echoid-head64" xml:space="preserve" xml:lang="lat"> | |
| 1252 Exegesis arithmetica <lb/> | |
| 1253 pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> radio. | |
| 1254 <lb/>[<emph style="it">tr: | |
| 1255 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math>. | |
| 1256 </emph>]<lb/> | |
| 1257 </head> | |
| 1258 <p xml:lang="lat"> | |
| 1259 <s xml:id="echoid-s72" xml:space="preserve"> | |
| 1260 Datorum circulorum radii <lb/> | |
| 1261 dati sunt, et centrorum <lb/> | |
| 1262 distantiæ. <lb/> | |
| 1263 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>a</mi><mi>y</mi></mstyle></math> <lb/> | |
| 1264 data sunt. Inde perpendicularis <lb/> | |
| 1265 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>m</mi></mstyle></math>. Inde tota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math>. <lb/> | |
| 1266 Inde datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. Inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>. <lb/> | |
| 1267 Tum cum datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>a</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, datur <lb/> | |
| 1268 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>m</mi></mstyle></math> et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>. Et cum datur <lb/> | |
| 1269 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>, datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>s</mi></mstyle></math>. <lb/> | |
| 1270 Tum lineæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> ad angulos <lb/> | |
| 1271 rectos et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> pro-<lb/> | |
| 1272 ducta concurret cum illa <lb/> | |
| 1273 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> sunt <lb/> | |
| 1274 æquales. et triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>h</mi></mstyle></math> <lb/> | |
| 1275 simile est triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi><mi>h</mi></mstyle></math>, <lb/> | |
| 1276 cuius latera data sunt. et <lb/> | |
| 1277 antea datum fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. ergo dantur <lb/> | |
| 1278 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 1279 <lb/>[...]<lb/> <lb/> | |
| 1280 Ergo tota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> datur <lb/> | |
| 1281 <lb/>[...]<lb/> <lb/> | |
| 1282 Ergo datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> <lb/> | |
| 1283 sed antea nota fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <lb/> | |
| 1284 ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> datur <lb/> | |
| 1285 Quod quærebatur. | |
| 1286 <lb/>[<emph style="it">tr: | |
| 1287 The radii of the fiven circles are given, and the distances of their centres. <lb/> | |
| 1288 Therefore the sides of the triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>a</mi><mi>y</mi></mstyle></math> are given. | |
| 1289 Hence the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math>, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>m</mi></mstyle></math>. Hence the total, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>m</mi></mstyle></math>. | |
| 1290 Hence there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>a</mi></mstyle></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math>. | |
| 1291 Then since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi></mstyle></math> are given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>m</mi></mstyle></math> is given and thence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math>. | |
| 1292 And since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>x</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>θ</mi></mstyle></math> are given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>s</mi></mstyle></math> are given. <lb/> | |
| 1293 Then the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is at right angles to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> extended meets with it at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. | |
| 1294 The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> are equal. And the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>h</mi></mstyle></math> is similar to triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>s</mi><mi>h</mi></mstyle></math>, | |
| 1295 whose sides are given. And earlier <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math> was given. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> are given. <lb/> | |
| 1296 <lb/>[...]<lb/> <lb/> | |
| 1297 Therefore the total <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> is given. <lb/> | |
| 1298 <lb/>[...]<lb/> <lb/> | |
| 1299 Therfore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math>, but earlier <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> became known, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> is given. <lb/> | |
| 1300 Which was sought. | |
| 1301 </emph>]<lb/> | |
| 1302 </s> | |
| 1303 </p> | |
| 1304 <p xml:lang="lat"> | |
| 1305 <s xml:id="echoid-s73" xml:space="preserve"> | |
| 1306 Per doctrinam sinuum <lb/> | |
| 1307 opus abbreviatur, sed <lb/> | |
| 1308 alia method ut convenit. | |
| 1309 <lb/>[<emph style="it">tr: | |
| 1310 By the doctrine of sines, the work is shorter, but another method, as convenient. | |
| 1311 </emph>]<lb/> | |
| 1312 </s> | |
| 1313 </p> | |
| 1314 <pb file="add_6784_f198v" o="198v" n="396"/> | |
| 1315 <pb file="add_6784_f199" o="199" n="397"/> | |
| 1316 <pb file="add_6784_f199v" o="199v" n="398"/> | |
| 1317 <pb file="add_6784_f200" o="200" n="399"/> | |
| 1318 <head xml:id="echoid-head65" xml:space="preserve"> | |
| 1319 6.) | |
| 1320 </head> | |
| 1321 <head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat"> | |
| 1322 Arithmetica Exegesis <lb/> | |
| 1323 radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math> | |
| 1324 <lb/>[<emph style="it">tr: | |
| 1325 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>. | |
| 1326 </emph>]<lb/> | |
| 1327 </head> | |
| 1328 <p xml:lang="lat"> | |
| 1329 <s xml:id="echoid-s74" xml:space="preserve"> | |
| 1330 Datorum circulorum radij dati <lb/> | |
| 1331 sunt, et centrorum distantiæ <lb/> | |
| 1332 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <lb/> | |
| 1333 cum sit, ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>,</mo><mi>h</mi><mi>p</mi><mo>:</mo><mi>a</mi><mi>f</mi><mo>,</mo><mi>f</mi><mi>p</mi></mstyle></math>. <lb/> | |
| 1334 datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math> cui æqualis <lb/> | |
| 1335 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> contingens. | |
| 1336 <lb/>[<emph style="it">tr: | |
| 1337 The radii of given circles are given, and the distances of their centres. <lb/> | |
| 1338 Therefore the sides of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>p</mi><mi>a</mi></mstyle></math>, and since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>p</mi><mo>=</mo><mi>a</mi><mi>f</mi><mo>:</mo><mi>f</mi><mi>p</mi></mstyle></math>, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi></mstyle></math>, | |
| 1339 and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>h</mi></mstyle></math>, which is equal to the angent <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math>. | |
| 1340 </emph>]<lb/> | |
| 1341 </s> | |
| 1342 </p> | |
| 1343 <p xml:lang="lat"> | |
| 1344 <s xml:id="echoid-s75" xml:space="preserve"> | |
| 1345 Ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math> datis, datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1346 Sunt igitur duo triangula <lb/> | |
| 1347 datorum laterum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1348 constituuntur super eandem <lb/> | |
| 1349 basim <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. datur igitur verti-<lb/> | |
| 1350 cum distantia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math>. | |
| 1351 <lb/>[<emph style="it">tr: | |
| 1352 From <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>z</mi></mstyle></math>, given, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1353 Therefore there are two triangles with given sides <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>p</mi><mi>z</mi></mstyle></math>, constructed on the same base <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. <lb/> | |
| 1354 Therefore the vertical distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math> is given. | |
| 1355 </emph>]<lb/> | |
| 1356 </s> | |
| 1357 </p> | |
| 1358 <p xml:lang="lat"> | |
| 1359 <s xml:id="echoid-s76" xml:space="preserve"> | |
| 1360 Ex triangulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>z</mi></mstyle></math> datorum laterum <lb/> | |
| 1361 datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> perpendicularis <lb/> | |
| 1362 nota igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> | |
| 1363 fiunt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>λ</mi></mstyle></math>, æquales radio <lb/> | |
| 1364 circuli circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> | |
| 1365 Dantur, igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>λ</mi></mstyle></math>. <lb/> | |
| 1366 Tum: <lb/> | |
| 1367 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, cuius dimidium <lb/> | |
| 1368 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>, radius quæsitus. | |
| 1369 <lb/>[<emph style="it">tr: | |
| 1370 From the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>z</mi></mstyle></math> with given sides there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>, | |
| 1371 and the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> is known, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> | |
| 1372 There are constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>n</mi><mi>λ</mi></mstyle></math>, equal to the radius of the circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> | |
| 1373 Therefore there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>λ</mi></mstyle></math>. <lb/> | |
| 1374 Then: <lb/> | |
| 1375 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, whose half, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>, is the sought radius. <lb/> | |
| 1376 </emph>]<lb/> | |
| 1377 </s> | |
| 1378 </p> | |
| 1379 <p xml:lang="lat"> | |
| 1380 <s xml:id="echoid-s77" xml:space="preserve"> | |
| 1381 Per Canonem triangulorum <lb/> | |
| 1382 alia methodo <emph style="super">ut covenit</emph>, operatio fit <lb/> | |
| 1383 brevior. | |
| 1384 <lb/>[<emph style="it">tr: | |
| 1385 By the Canons for triangles, there is another method, as convenient, which may be carried ore briefly. | |
| 1386 </emph>]<lb/> | |
| 1387 </s> | |
| 1388 </p> | |
| 1389 <p xml:lang="lat"> | |
| 1390 <s xml:id="echoid-s78" xml:space="preserve"> | |
| 1391 Nota. <lb/> | |
| 1392 per puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>η</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi></mstyle></math> <lb/> | |
| 1393 fit etiam geometrica <lb/> | |
| 1394 constructio, loco <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>. | |
| 1395 <lb/>[<emph style="it">tr: | |
| 1396 Note. <lb/> | |
| 1397 Through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>η</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi></mstyle></math> there may also be carried out a geometric construction, instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>q</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math>. | |
| 1398 </emph>]<lb/> | |
| 1399 </s> | |
| 1400 </p> | |
| 1401 <head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat"> | |
| 1402 Arithmetica exegesis <lb/> | |
| 1403 radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> <lb/> | |
| 1404 cæteris datis. | |
| 1405 <lb/>[<emph style="it">tr: | |
| 1406 Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, given the rest. | |
| 1407 </emph>]<lb/> | |
| 1408 </head> | |
| 1409 <p xml:lang="lat"> | |
| 1410 <s xml:id="echoid-s79" xml:space="preserve"> | |
| 1411 Datorum circulorum radij dati <lb/> | |
| 1412 sunt, et centrorum distantiæ <lb/> | |
| 1413 Ergo lateri trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, <lb/> | |
| 1414 Datur igitur perpendicularis <lb/> | |
| 1415 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, et linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>. Unde nota <lb/> | |
| 1416 fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi></mstyle></math>. <lb/> | |
| 1417 Cum data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>o</mi></mstyle></math> <lb/> | |
| 1418 unde data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>. <lb/> | |
| 1419 Tum, trianguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>o</mi></mstyle></math> latera sunt <lb/> | |
| 1420 nota; unde nota perpendicularis <lb/> | |
| 1421 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi></mstyle></math>. Et linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>u</mi></mstyle></math>, cui æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 1422 Dantur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>h</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 1423 Dantur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>f</mi></mstyle></math>. <lb/> | |
| 1424 Denique fiat: <lb/> | |
| 1425 Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, quod <lb/> | |
| 1426 quærebatur. | |
| 1427 <lb/>[<emph style="it">tr: | |
| 1428 The radii of given circles are given, and the distances of their centres. <lb/> | |
| 1429 Therefore the sides of the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>p</mi><mi>y</mi></mstyle></math>. <lb/> | |
| 1430 Therefore there is given the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi><mi>n</mi></mstyle></math>. Whence there is known <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi></mstyle></math>. <lb/> | |
| 1431 Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>o</mi></mstyle></math> are given, there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>. <lb/> | |
| 1432 Then the sides of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi><mi>o</mi></mstyle></math> are known, whence the perpendicular <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>u</mi></mstyle></math> is known. | |
| 1433 And the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>u</mi></mstyle></math>, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 1434 Therefore there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 1435 Thereofre there are given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>f</mi></mstyle></math>. <lb/> | |
| 1436 Then let there be constructed: <lb/> | |
| 1437 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, which was sought. | |
| 1438 </emph>]<lb/> | |
| 1439 </s> | |
| 1440 </p> | |
| 1441 <head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat"> | |
| 1442 Geometria exegesis <lb/> | |
| 1443 ipsius radii <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. | |
| 1444 <lb/>[<emph style="it">tr: | |
| 1445 Geometric exegesis, for the same radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. | |
| 1446 </emph>]<lb/> | |
| 1447 </head> | |
| 1448 <p xml:lang="lat"> | |
| 1449 <s xml:id="echoid-s80" xml:space="preserve"> | |
| 1450 Trium datorum circulorum <lb/> | |
| 1451 centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, connectantur. <lb/> | |
| 1452 per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> acta <lb/> | |
| 1453 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> faciat angulos rectos cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> | |
| 1454 Ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>; quæ secabit circulum <lb/> | |
| 1455 circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> | |
| 1456 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, quæ producta secabit <lb/> | |
| 1457 eandem circulum circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> | |
| 1458 Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> et producatur ad <lb/> | |
| 1459 utraque partes quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> | |
| 1460 in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> | |
| 1461 Tum fiat: <lb/> | |
| 1462 Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, et centrum circuli <lb/> | |
| 1463 quæsiti. | |
| 1464 <lb/>[<emph style="it">tr: | |
| 1465 Let the centres of the given circles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>, be connected. <lb/> | |
| 1466 Through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>z</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math> let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be constructed; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> makes a right angle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/> | |
| 1467 Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>, which cuts the circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. <lb/> | |
| 1468 Let there be constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, which extended sill cut the same circle about <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <lb/> | |
| 1469 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> be constructed and extended on both sides, which will cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> | |
| 1470 Then: <lb/> | |
| 1471 Therefore there is given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, and the centre of the circle sought. | |
| 1472 </emph>]<lb/> | |
| 1473 </s> | |
| 1474 </p> | |
| 1475 <pb file="add_6784_f200v" o="200v" n="400"/> | |
| 1476 <pb file="add_6784_f201" o="201" n="401"/> | |
| 1477 <div xml:id="echoid-div19" type="page_commentary" level="2" n="19"> | |
| 1478 <p> | |
| 1479 <s xml:id="echoid-s81" xml:space="preserve">[<emph style="it">Note: | |
| 1480 <p> | |
| 1481 <s xml:id="echoid-s81" xml:space="preserve"> | |
| 1482 The reference to Pappus is to Commandino's edition of Books III to VIII, | |
| 1483 <emph style="it">Mathematicae collecitones</emph> (1558). | |
| 1484 The proposition on page 48v–49 is Proposition IV.15 (not 13, as Harriot appears to have written). | |
| 1485 A diagram for this proposition appears on Add MS 6784, f. 202; | |
| 1486 this page shows only calculations of ratios. | |
| 1487 </s> | |
| 1488 <lb/> | |
| 1489 <quote xml:lang="lat"> | |
| 1490 Theorema XV. Propositio XV. <lb/> | |
| 1491 Iisdem positis describatur circulus HRT, qui & semicirculos iam dictos, & circulum LGH contingat | |
| 1492 in punctis HRT, atque a centris A P ad BC basim perpendiculares ducantur AM PN. Dico vt AM vna cum | |
| 1493 diametro circuli EGH ad diametrum ipsius, ita esse PN ad circuli HRT diametrum. | |
| 1494 </quote> | |
| 1495 <lb/> | |
| 1496 <quote> | |
| 1497 The same being supposed [as in Proposition 14], there is drawn the circle HRT, which touches both the semicircles | |
| 1498 already given and the circle LGH, in the points H, R, T. And from the centres A and P to the base there are drawn | |
| 1499 perpendiculars AM and PN. I say that as AM together with the diameter of the circle EGH is to that that diameter itself, | |
| 1500 so is PN to the diamter of the circle HRT. | |
| 1501 </quote> | |
| 1502 </p> | |
| 1503 </emph>] | |
| 1504 <lb/><lb/></s></p></div> | |
| 1505 <head xml:id="echoid-head69" xml:space="preserve"> | |
| 1506 5.) pappus. prop. 13. pag. 49. | |
| 1507 </head> | |
| 1508 <pb file="add_6784_f201v" o="201v" n="402"/> | |
| 1509 <pb file="add_6784_f202" o="202" n="403"/> | |
| 1510 <head xml:id="echoid-head70" xml:space="preserve"> | |
| 1511 4.) | |
| 1512 </head> | |
| 1513 <p> | |
| 1514 <s xml:id="echoid-s83" xml:space="preserve"> | |
| 1515 Sint duo circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>d</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math> <lb/> | |
| 1516 contingant se in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> | |
| 1517 sit recta per centra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi><mi>p</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> | |
| 1518 oportet describere circulum <lb/> | |
| 1519 contingentem duos circulos <lb/> | |
| 1520 datos, et lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 1521 <lb/>[<emph style="it">tr: | |
| 1522 Let there be two circles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>e</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi><mi>c</mi></mstyle></math> touching in the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> | |
| 1523 Let the line through the centre be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi><mi>p</mi><mi>c</mi><mi>d</mi></mstyle></math>. <lb/> | |
| 1524 One must draw the circle touching the two given circles and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 1525 </emph>]<lb/> | |
| 1526 </s> | |
| 1527 </p> | |
| 1528 <p> | |
| 1529 <s xml:id="echoid-s84" xml:space="preserve"> | |
| 1530 <lb/>[...]<lb/> <lb/> | |
| 1531 Jungantur puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> | |
| 1532 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>d</mi></mstyle></math>. <lb/> | |
| 1533 <lb/>[...]<lb/> | |
| 1534 <lb/>[<emph style="it">tr: | |
| 1535 <lb/>[...]<lb/> <lb/> | |
| 1536 Let the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> be joined. <lb/> | |
| 1537 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mi>k</mi></mstyle></math> be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>d</mi></mstyle></math>. <lb/> | |
| 1538 <lb/>[...]<lb/> | |
| 1539 </emph>]<lb/> | |
| 1540 </s> | |
| 1541 </p> | |
| 1542 <p> | |
| 1543 <s xml:id="echoid-s85" xml:space="preserve"> | |
| 1544 Bisecetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math>, puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. <lb/> | |
| 1545 agatur ad angulos rectos, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi></mstyle></math>. <lb/> | |
| 1546 fiat, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi><mo>=</mo><mi>m</mi><mi>k</mi></mstyle></math>. <lb/> | |
| 1547 agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>a</mi></mstyle></math>, quæ secabit periferi-<lb/> | |
| 1548 am minoris circuli in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> | |
| 1549 agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>a</mi><mi>g</mi></mstyle></math>, quæ secabit perife-<lb/> | |
| 1550 riam maioris circulam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> | |
| 1551 Dico quod: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi><mo>=</mo><mi>a</mi><mi>g</mi><mo>=</mo><mi>a</mi><mi>e</mi></mstyle></math>. <lb/> | |
| 1552 et ideo, circulus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>, <lb/> | |
| 1553 erit quæsitus. | |
| 1554 <lb/>[<emph style="it">tr: | |
| 1555 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>l</mi></mstyle></math> be bisected at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. <lb/> | |
| 1556 There is constructed at right angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi></mstyle></math>. <lb/> | |
| 1557 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi><mo>=</mo><mi>m</mi><mi>k</mi></mstyle></math>. <lb/> | |
| 1558 There is constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>a</mi></mstyle></math>, which will cut the circumference of the smaller circle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> | |
| 1559 There is constructed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>a</mi><mi>g</mi></mstyle></math>, which will cut the circumference of the larger circle at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> | |
| 1560 I say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>m</mi><mo>=</mo><mi>a</mi><mi>g</mi><mo>=</mo><mi>a</mi><mi>e</mi></mstyle></math>, and therfore the circle through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> will be the one required. | |
| 1561 </emph>]<lb/> | |
| 1562 </s> | |
| 1563 </p> | |
| 1564 <pb file="add_6784_f202v" o="202v" n="404"/> | |
| 1565 <pb file="add_6784_f203" o="203" n="405"/> | |
| 1566 <div xml:id="echoid-div20" type="page_commentary" level="2" n="20"> | |
| 1567 <p> | |
| 1568 <s xml:id="echoid-s86" xml:space="preserve">[<emph style="it">Note: | |
| 1569 <p> | |
| 1570 <s xml:id="echoid-s86" xml:space="preserve"> | |
| 1571 The reference to Pappus is to Commandino's edition of Books III to VIII, | |
| 1572 <emph style="it">Mathematicae collecitones</emph> (1558). | |
| 1573 The proposition on page 47 is Proposition IV.14. | |
| 1574 Harriot's diagram is the same as the one given by Commandino except for his use of lower case letters. | |
| 1575 A second diagram for the same proposition appears on Add MS 6784, f. 204. | |
| 1576 </s> | |
| 1577 <lb/> | |
| 1578 <quote xml:lang="lat"> | |
| 1579 Theorema XIIII. Propositio XIIII. <lb/> | |
| 1580 Sint duo semicirculi BGC BED: & ipsos contingat circulus EFGH: a cuius centro A ad BC basim semicirculorum | |
| 1581 perpendicularis ducatur AM. Dico ut BM as eam, quæ ex centro circuli EFGH, | |
| 1582 ita esse in prima figura vtramque simul CB BD ad earum excessum CD; | |
| 1583 in secunda vero, & tertia figura, ita esse excessum CB BD ad vtramque ipsarum CB BD. | |
| 1584 </quote> | |
| 1585 <lb/> | |
| 1586 <quote> | |
| 1587 Let there be two semicircles BGC and BED, and their touching circle EFGH, from whose centre A to BC, | |
| 1588 the base of the semicircle, there is drawn the perpendicular AM. | |
| 1589 I say that as BM is to that line from the centre of the circle EFGH, | |
| 1590 inthe first figure will be CB and BD togher to their excess, CD; | |
| 1591 but in the second and third figure, it will be as the excess of CB over BD to both of CB and BD together. | |
| 1592 </quote> | |
| 1593 </p> | |
| 1594 </emph>] | |
| 1595 <lb/><lb/></s></p></div> | |
| 1596 <head xml:id="echoid-head71" xml:space="preserve" xml:lang="lat"> | |
| 1597 pappus. pag. <lb/> | |
| 1598 47. | |
| 1599 <lb/>[<emph style="it">tr: | |
| 1600 Pappus, page 47. | |
| 1601 </emph>]<lb/> | |
| 1602 </head> | |
| 1603 <pb file="add_6784_f203v" o="203v" n="406"/> | |
| 1604 <pb file="add_6784_f204" o="204" n="407"/> | |
| 1605 <div xml:id="echoid-div21" type="page_commentary" level="2" n="21"> | |
| 1606 <p> | |
| 1607 <s xml:id="echoid-s88" xml:space="preserve">[<emph style="it">Note: | |
| 1608 <p> | |
| 1609 <s xml:id="echoid-s88" xml:space="preserve"> | |
| 1610 A further diagram for Pappus, <emph style="it">Mathematicae collectiones</emph>, Propostion IV.14. | |
| 1611 See also the previous folio, Add MS 6784, f. 203. | |
| 1612 </s> | |
| 1613 </p> | |
| 1614 </emph>] | |
| 1615 <lb/><lb/></s></p></div> | |
| 1616 <head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat"> | |
| 1617 pappus. pag. <lb/> | |
| 1618 47. | |
| 1619 <lb/>[<emph style="it">tr: | |
| 1620 Pappus, page 47. | |
| 1621 </emph>]<lb/> | |
| 1622 </head> | |
| 1623 <pb file="add_6784_f204v" o="204v" n="408"/> | |
| 1624 <pb file="add_6784_f205" o="205" n="409"/> | |
| 1625 <div xml:id="echoid-div22" type="page_commentary" level="2" n="22"> | |
| 1626 <p> | |
| 1627 <s xml:id="echoid-s90" xml:space="preserve">[<emph style="it">Note: | |
| 1628 <p> | |
| 1629 <s xml:id="echoid-s90" xml:space="preserve"> | |
| 1630 Further work on Pappus, Propostion IV.14. | |
| 1631 </s> | |
| 1632 </p> | |
| 1633 </emph>] | |
| 1634 <lb/><lb/></s></p></div> | |
| 1635 <head xml:id="echoid-head73" xml:space="preserve"> | |
| 1636 2) pappus. pag. 47 | |
| 1637 </head> | |
| 1638 <pb file="add_6784_f205v" o="205v" n="410"/> | |
| 1639 <pb file="add_6784_f206" o="206" n="411"/> | |
| 1640 <div xml:id="echoid-div23" type="page_commentary" level="2" n="23"> | |
| 1641 <p> | |
| 1642 <s xml:id="echoid-s92" xml:space="preserve">[<emph style="it">Note: | |
| 1643 <p> | |
| 1644 <s xml:id="echoid-s92" xml:space="preserve"> | |
| 1645 Further work on Pappus, Propostion IV.14. | |
| 1646 </s> | |
| 1647 </p> | |
| 1648 </emph>] | |
| 1649 <lb/><lb/></s></p></div> | |
| 1650 <head xml:id="echoid-head74" xml:space="preserve"> | |
| 1651 3) pappus. <emph style="super">pag.</emph> 47 | |
| 1652 </head> | |
| 1653 <pb file="add_6784_f206v" o="206v" n="412"/> | |
| 1654 <pb file="add_6784_f207" o="207" n="413"/> | |
| 1655 <div xml:id="echoid-div24" type="page_commentary" level="2" n="24"> | |
| 1656 <p> | |
| 1657 <s xml:id="echoid-s94" xml:space="preserve">[<emph style="it">Note: | |
| 1658 <p> | |
| 1659 <s xml:id="echoid-s94" xml:space="preserve"> | |
| 1660 Lists of variations of increasing (c) and decreasing (d) columns, | |
| 1661 together with other rough work for the 'Magisteria' (Add MS 6782, f. 107 to f. 146v). <lb/> | |
| 1662 This page is important because it carries a date, day, time, and year: June 28 (Sunday) 10.30am, 1618. | |
| 1663 </s> | |
| 1664 </p> | |
| 1665 </emph>] | |
| 1666 <lb/><lb/></s></p></div> | |
| 1667 <p xml:lang="lat"> | |
| 1668 <s xml:id="echoid-s96" xml:space="preserve"> | |
| 1669 De causa reflexionis ad angulos æquales. | |
| 1670 <lb/>[<emph style="it">tr: | |
| 1671 On the cause of reflection at equal angles. | |
| 1672 </emph>]<lb/> | |
| 1673 </s> | |
| 1674 </p> | |
| 1675 <p xml:lang="lat"> | |
| 1676 <s xml:id="echoid-s97" xml:space="preserve"> | |
| 1677 June 28. .ho: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><mn>1</mn><mn>0</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math> <lb/> | |
| 1678 ante mer: 1618 | |
| 1679 <lb/>[<emph style="it">tr: | |
| 1680 June 28 (Sunday) 10.30am 1618 | |
| 1681 </emph>]<lb/> | |
| 1682 </s> | |
| 1683 </p> | |
| 1684 <pb file="add_6784_f207v" o="207v" n="414"/> | |
| 1685 <div xml:id="echoid-div25" type="page_commentary" level="2" n="25"> | |
| 1686 <p> | |
| 1687 <s xml:id="echoid-s98" xml:space="preserve">[<emph style="it">Note: | |
| 1688 <p> | |
| 1689 <s xml:id="echoid-s98" xml:space="preserve"> | |
| 1690 Further lists of variations of increasing (c) and decreasing (d) columns (see Add MS 6784, f. 413). | |
| 1691 </s> | |
| 1692 </p> | |
| 1693 </emph>] | |
| 1694 <lb/><lb/></s></p></div> | |
| 1695 <pb file="add_6784_f208" o="208" n="415"/> | |
| 1696 <div xml:id="echoid-div26" type="page_commentary" level="2" n="26"> | |
| 1697 <p> | |
| 1698 <s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note: | |
| 1699 <p> | |
| 1700 <s xml:id="echoid-s100" xml:space="preserve"> | |
| 1701 Difference tables similar to those on pages 10 and 11 of the 'Magisteria' (Add MS 6782, f. 117 and f. 118). | |
| 1702 </s> | |
| 1703 </p> | |
| 1704 </emph>] | |
| 1705 <lb/><lb/></s></p></div> | |
| 1706 <pb file="add_6784_f208v" o="208v" n="416"/> | |
| 1707 <div xml:id="echoid-div27" type="page_commentary" level="2" n="27"> | |
| 1708 <p> | |
| 1709 <s xml:id="echoid-s102" xml:space="preserve">[<emph style="it">Note: | |
| 1710 <p> | |
| 1711 <s xml:id="echoid-s102" xml:space="preserve"> | |
| 1712 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> column of a difference table, | |
| 1713 similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). | |
| 1714 </s> | |
| 1715 </p> | |
| 1716 </emph>] | |
| 1717 <lb/><lb/></s></p></div> | |
| 1718 <pb file="add_6784_f209" o="209" n="417"/> | |
| 1719 <pb file="add_6784_f209v" o="209v" n="418"/> | |
| 1720 <pb file="add_6784_f210" o="210" n="419"/> | |
| 1721 <div xml:id="echoid-div28" type="page_commentary" level="2" n="28"> | |
| 1722 <p> | |
| 1723 <s xml:id="echoid-s104" xml:space="preserve">[<emph style="it">Note: | |
| 1724 <p> | |
| 1725 <s xml:id="echoid-s104" xml:space="preserve"> | |
| 1726 Rough working for page 15 of the 'Magisteria' (Add MS 6782, f. 122). | |
| 1727 </s> | |
| 1728 </p> | |
| 1729 </emph>] | |
| 1730 <lb/><lb/></s></p></div> | |
| 1731 <pb file="add_6784_f210v" o="210v" n="420"/> | |
| 1732 <div xml:id="echoid-div29" type="page_commentary" level="2" n="29"> | |
| 1733 <p> | |
| 1734 <s xml:id="echoid-s106" xml:space="preserve">[<emph style="it">Note: | |
| 1735 <p> | |
| 1736 <s xml:id="echoid-s106" xml:space="preserve"> | |
| 1737 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> columns of a difference table, | |
| 1738 similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). | |
| 1739 </s> | |
| 1740 </p> | |
| 1741 </emph>] | |
| 1742 <lb/><lb/></s></p></div> | |
| 1743 <pb file="add_6784_f211" o="211" n="421"/> | |
| 1744 <div xml:id="echoid-div30" type="page_commentary" level="2" n="30"> | |
| 1745 <p> | |
| 1746 <s xml:id="echoid-s108" xml:space="preserve">[<emph style="it">Note: | |
| 1747 <p> | |
| 1748 <s xml:id="echoid-s108" xml:space="preserve"> | |
| 1749 An incomplete version of the difference table on page 9 of the 'Magisteria' (Add MS 6782, f. 116). | |
| 1750 </s> | |
| 1751 </p> | |
| 1752 </emph>] | |
| 1753 <lb/><lb/></s></p></div> | |
| 1754 <pb file="add_6784_f211v" o="211v" n="422"/> | |
| 1755 <div xml:id="echoid-div31" type="page_commentary" level="2" n="31"> | |
| 1756 <p> | |
| 1757 <s xml:id="echoid-s110" xml:space="preserve">[<emph style="it">Note: | |
| 1758 <p> | |
| 1759 <s xml:id="echoid-s110" xml:space="preserve"> | |
| 1760 Formulae for entries in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> columns of a difference table; | |
| 1761 see page 16 of the 'Magisteria' (Add MS 6782, f. 123). | |
| 1762 </s> | |
| 1763 </p> | |
| 1764 </emph>] | |
| 1765 <lb/><lb/></s></p></div> | |
| 1766 <pb file="add_6784_f212" o="212" n="423"/> | |
| 1767 <div xml:id="echoid-div32" type="page_commentary" level="2" n="32"> | |
| 1768 <p> | |
| 1769 <s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note: | |
| 1770 <p> | |
| 1771 <s xml:id="echoid-s112" xml:space="preserve"> | |
| 1772 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 1773 </s> | |
| 1774 </p> | |
| 1775 </emph>] | |
| 1776 <lb/><lb/></s></p></div> | |
| 1777 <head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat"> | |
| 1778 Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 1779 <lb/>[<emph style="it">tr: | |
| 1780 Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> | |
| 1781 </emph>]<lb/> | |
| 1782 </head> | |
| 1783 <p xml:lang="lat"> | |
| 1784 <s xml:id="echoid-s114" xml:space="preserve"> | |
| 1785 operatio. 1<emph style="super">a</emph> | |
| 1786 <lb/>[<emph style="it">tr: | |
| 1787 Working (1) | |
| 1788 </emph>]<lb/> | |
| 1789 </s> | |
| 1790 </p> | |
| 1791 <p xml:lang="lat"> | |
| 1792 <s xml:id="echoid-s115" xml:space="preserve"> | |
| 1793 operatio. 2<emph style="super">a</emph> | |
| 1794 <lb/>[<emph style="it">tr: | |
| 1795 Working (2) | |
| 1796 </emph>]<lb/> | |
| 1797 </s> | |
| 1798 </p> | |
| 1799 <pb file="add_6784_f212v" o="212v" n="424"/> | |
| 1800 <pb file="add_6784_f213" o="213" n="425"/> | |
| 1801 <div xml:id="echoid-div33" type="page_commentary" level="2" n="33"> | |
| 1802 <p> | |
| 1803 <s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note: | |
| 1804 <p> | |
| 1805 <s xml:id="echoid-s116" xml:space="preserve"> | |
| 1806 Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). | |
| 1807 </s> | |
| 1808 </p> | |
| 1809 </emph>] | |
| 1810 <lb/><lb/></s></p></div> | |
| 1811 <head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat"> | |
| 1812 Residuum operationis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. | |
| 1813 </head> | |
| 1814 <p xml:lang="lat"> | |
| 1815 <s xml:id="echoid-s118" xml:space="preserve"> | |
| 1816 2<emph style="super">a</emph> | |
| 1817 Working (2) | |
| 1818 </s> | |
| 1819 </p> | |
| 1820 <pb file="add_6784_f213v" o="213v" n="426"/> | |
| 1821 <pb file="add_6784_f214" o="214" n="427"/> | |
| 1822 <div xml:id="echoid-div34" type="page_commentary" level="2" n="34"> | |
| 1823 <p> | |
| 1824 <s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note: | |
| 1825 <p> | |
| 1826 <s xml:id="echoid-s119" xml:space="preserve"> | |
| 1827 General notation for triangular numbers. <lb/> | |
| 1828 See also page 2 of the 'Magisteria' (Add MS 6782, f. 109). | |
| 1829 </s> | |
| 1830 </p> | |
| 1831 </emph>] | |
| 1832 <lb/><lb/></s></p></div> | |
| 1833 <head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat"> | |
| 1834 3<emph style="super">a</emph> notatio triangularium per notas generales. | |
| 1835 <lb/>[<emph style="it">tr: | |
| 1836 3rd notation for triangular numbers, in general symbols. | |
| 1837 </emph>]<lb/> | |
| 1838 </head> | |
| 1839 <pb file="add_6784_f214v" o="214v" n="428"/> | |
| 1840 <pb file="add_6784_f215" o="215" n="429"/> | |
| 1841 <pb file="add_6784_f215v" o="215v" n="430"/> | |
| 1842 <pb file="add_6784_f216" o="216" n="431"/> | |
| 1843 <div xml:id="echoid-div35" type="page_commentary" level="2" n="35"> | |
| 1844 <p> | |
| 1845 <s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note: | |
| 1846 <p> | |
| 1847 <s xml:id="echoid-s121" xml:space="preserve"> | |
| 1848 Square roots of binomes of the fifth and sixth kind | |
| 1849 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). | |
| 1850 Here Harriot works with two types of fifth binome, | |
| 1851 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mi>b</mi></mstyle></math>), | |
| 1852 according to whether the difference between the squares of the two terms is a square or not. | |
| 1853 Elsewhere he refers to these as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʹ</mo></mstyle></math> and bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mo>ʺ</mo></mstyle></math>. <lb/> | |
| 1854 Similarly he distinguishes two types of sixth binomes, | |
| 1855 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>f</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>c</mi></mrow></msqrt></mstyle></math>). | |
| 1856 Elsewhere he refers to these as bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʹ</mo></mstyle></math> and bin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mo>ʺ</mo></mstyle></math>. <lb/> | |
| 1857 In all cases the roots are cross-checked. | |
| 1858 </s> | |
| 1859 </p> | |
| 1860 </emph>] | |
| 1861 <lb/><lb/></s></p></div> | |
| 1862 <pb file="add_6784_f216v" o="216v" n="432"/> | |
| 1863 <pb file="add_6784_f217" o="217" n="433"/> | |
| 1864 <div xml:id="echoid-div36" type="page_commentary" level="2" n="36"> | |
| 1865 <p> | |
| 1866 <s xml:id="echoid-s123" xml:space="preserve">[<emph style="it">Note: | |
| 1867 <p> | |
| 1868 <s xml:id="echoid-s123" xml:space="preserve"> | |
| 1869 Square roots of binomes of the third and fourth kind | |
| 1870 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>d</mi></mrow></msqrt></mstyle></math>), | |
| 1871 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). | |
| 1872 In both cases the roots are checked by multiplication. | |
| 1873 </s> | |
| 1874 </p> | |
| 1875 </emph>] | |
| 1876 <lb/><lb/></s></p></div> | |
| 1877 <pb file="add_6784_f217v" o="217v" n="434"/> | |
| 1878 <pb file="add_6784_f218" o="218" n="435"/> | |
| 1879 <div xml:id="echoid-div37" type="page_commentary" level="2" n="37"> | |
| 1880 <p> | |
| 1881 <s xml:id="echoid-s125" xml:space="preserve">[<emph style="it">Note: | |
| 1882 <p> | |
| 1883 <s xml:id="echoid-s125" xml:space="preserve"> | |
| 1884 Square roots of binomes of the first and second kind | |
| 1885 (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi><mo maxsize="1">)</mo></mrow></msqrt><mo>+</mo><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math>), | |
| 1886 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). | |
| 1887 In both cases the roots are checked by multiplication. | |
| 1888 </s> | |
| 1889 </p> | |
| 1890 </emph>] | |
| 1891 <lb/><lb/></s></p></div> | |
| 1892 <pb file="add_6784_f218v" o="218v" n="436"/> | |
| 1893 <pb file="add_6784_f219" o="219" n="437"/> | |
| 1894 <div xml:id="echoid-div38" type="page_commentary" level="2" n="38"> | |
| 1895 <p> | |
| 1896 <s xml:id="echoid-s127" xml:space="preserve">[<emph style="it">Note: | |
| 1897 <p> | |
| 1898 <s xml:id="echoid-s127" xml:space="preserve"> | |
| 1899 Square roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>b</mi><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mrow></msqrt></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt></mstyle></math>, | |
| 1900 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. | |
| 1901 The numerical examples in Add MS 6783, f. 360v, f. 361, and Add MS 6782, f. 228, | |
| 1902 are closely related to the work on this page. | |
| 1903 </s> | |
| 1904 </p> | |
| 1905 </emph>] | |
| 1906 <lb/><lb/></s></p></div> | |
| 1907 <p xml:lang="lat"> | |
| 1908 <s xml:id="echoid-s129" xml:space="preserve"> | |
| 1909 Nam: eius quadratum | |
| 1910 <lb/>[<emph style="it">tr: | |
| 1911 For: its square | |
| 1912 </emph>]<lb/> | |
| 1913 </s> | |
| 1914 </p> | |
| 1915 <p xml:lang="lat"> | |
| 1916 <s xml:id="echoid-s130" xml:space="preserve"> | |
| 1917 Quia: duo quad: | |
| 1918 <lb/>[<emph style="it">tr: | |
| 1919 Because: two squares | |
| 1920 </emph>]<lb/> | |
| 1921 </s> | |
| 1922 </p> | |
| 1923 <p xml:lang="lat"> | |
| 1924 <s xml:id="echoid-s131" xml:space="preserve"> | |
| 1925 Et: duo rectang: | |
| 1926 <lb/>[<emph style="it">tr: | |
| 1927 And: two rectangles | |
| 1928 </emph>]<lb/> | |
| 1929 </s> | |
| 1930 </p> | |
| 1931 <pb file="add_6784_f219v" o="219v" n="438"/> | |
| 1932 <pb file="add_6784_f220" o="220" n="439"/> | |
| 1933 <div xml:id="echoid-div39" type="page_commentary" level="2" n="39"> | |
| 1934 <p> | |
| 1935 <s xml:id="echoid-s132" xml:space="preserve">[<emph style="it">Note: | |
| 1936 <p> | |
| 1937 <s xml:id="echoid-s132" xml:space="preserve"> | |
| 1938 Square roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mo>+</mo><mi>c</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mn>2</mn><mi>b</mi><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mrow></msqrt><mo>+</mo><mn>2</mn><mi>b</mi><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>2</mn><mi>b</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>c</mi></mrow></msqrt><mo>+</mo><mn>4</mn><mi>d</mi><mi>c</mi></mstyle></math>, | |
| 1939 by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. | |
| 1940 </s> | |
| 1941 </p> | |
| 1942 </emph>] | |
| 1943 <lb/><lb/></s></p></div> | |
| 1944 <pb file="add_6784_f220v" o="220v" n="440"/> | |
| 1945 <pb file="add_6784_f221" o="221" n="441"/> | |
| 1946 <head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat"> | |
| 1947 Examinatio æquationis per numeros | |
| 1948 <lb/>[<emph style="it">tr: | |
| 1949 An examination of an equation in numbers | |
| 1950 </emph>]<lb/> | |
| 1951 </head> | |
| 1952 <p xml:lang="lat"> | |
| 1953 <s xml:id="echoid-s134" xml:space="preserve"> | |
| 1954 et ita est (ut supra) | |
| 1955 <lb/>[<emph style="it">tr: | |
| 1956 and so it is (as above) | |
| 1957 </emph>]<lb/> | |
| 1958 </s> | |
| 1959 </p> | |
| 1960 <p xml:lang="lat"> | |
| 1961 <s xml:id="echoid-s135" xml:space="preserve"> | |
| 1962 et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> | |
| 1963 <lb/>[<emph style="it">tr: | |
| 1964 and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> | |
| 1965 </emph>]<lb/> | |
| 1966 </s> | |
| 1967 <lb/> | |
| 1968 <s xml:id="echoid-s136" xml:space="preserve"> | |
| 1969 et ita est (ut infra) | |
| 1970 <lb/>[<emph style="it">tr: | |
| 1971 and so it is (as below) | |
| 1972 </emph>]<lb/> | |
| 1973 </s> | |
| 1974 </p> | |
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| 1976 <pb file="add_6784_f222" o="222" n="443"/> | |
| 1977 <pb file="add_6784_f222v" o="222v" n="444"/> | |
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| 1988 <pb file="add_6784_f228" o="228" n="455"/> | |
| 1989 <pb file="add_6784_f228v" o="228v" n="456"/> | |
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| 1991 <pb file="add_6784_f229v" o="229v" n="458"/> | |
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| 2009 <pb file="add_6784_f238v" o="238v" n="476"/> | |
| 2010 <pb file="add_6784_f239" o="239" n="477"/> | |
| 2011 <pb file="add_6784_f239v" o="239v" n="478"/> | |
| 2012 <pb file="add_6784_f240" o="240" n="479"/> | |
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| 2019 <pb file="add_6784_f243v" o="243v" n="486"/> | |
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| 2024 <pb file="add_6784_f246" o="246" n="491"/> | |
| 2025 <head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat"> | |
| 2026 3. | |
| 2027 </head> | |
| 2028 <p xml:lang="lat"> | |
| 2029 <s xml:id="echoid-s137" xml:space="preserve"> | |
| 2030 In Achille | |
| 2031 <lb/>[<emph style="it">tr: | |
| 2032 On Achilles | |
| 2033 </emph>]<lb/> | |
| 2034 </s> | |
| 2035 <lb/> | |
| 2036 <s xml:id="echoid-s138" xml:space="preserve"> | |
| 2037 vel per æquationem rationum | |
| 2038 <lb/>[<emph style="it">tr: | |
| 2039 or by the equality of ratios | |
| 2040 </emph>]<lb/> | |
| 2041 </s> | |
| 2042 </p> | |
| 2043 <p xml:lang="lat"> | |
| 2044 <s xml:id="echoid-s139" xml:space="preserve"> | |
| 2045 Aliter | |
| 2046 <lb/>[<emph style="it">tr: | |
| 2047 Another way | |
| 2048 </emph>]<lb/> | |
| 2049 </s> | |
| 2050 <lb/> | |
| 2051 <s xml:id="echoid-s140" xml:space="preserve"> | |
| 2052 Sit ratio motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mo>=</mo><mi>c</mi><mi>o</mi></mstyle></math> | |
| 2053 <lb/>[<emph style="it">tr: | |
| 2054 Let the ratio of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>o</mi></mstyle></math>. | |
| 2055 </emph>]<lb/> | |
| 2056 </s> | |
| 2057 <lb/> | |
| 2058 <s xml:id="echoid-s141" xml:space="preserve"> | |
| 2059 Tempus Tempus | |
| 2060 <lb/>[<emph style="it">tr: | |
| 2061 Time; Time | |
| 2062 </emph>]<lb/> | |
| 2063 </s> | |
| 2064 </p> | |
| 2065 <p xml:lang="lat"> | |
| 2066 <s xml:id="echoid-s142" xml:space="preserve"> | |
| 2067 Aliter | |
| 2068 <lb/>[<emph style="it">tr: | |
| 2069 Another way | |
| 2070 </emph>]<lb/> | |
| 2071 </s> | |
| 2072 </p> | |
| 2073 <pb file="add_6784_f246v" o="246v" n="492"/> | |
| 2074 <pb file="add_6784_f247" o="247" n="493"/> | |
| 2075 <head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat"> | |
| 2076 4. | |
| 2077 </head> | |
| 2078 <pb file="add_6784_f247v" o="247v" n="494"/> | |
| 2079 <pb file="add_6784_f248" o="248" n="495"/> | |
| 2080 <head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat"> | |
| 2081 5. | |
| 2082 </head> | |
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| 2230 <div xml:id="echoid-div40" type="page_commentary" level="2" n="40"> | |
| 2231 <p> | |
| 2232 <s xml:id="echoid-s143" xml:space="preserve">[<emph style="it">Note: | |
| 2233 <p> | |
| 2234 <s xml:id="echoid-s143" xml:space="preserve"> | |
| 2235 The verses on this page describe the rules for operating with | |
| 2236 positive quantities ('more') and negative quantites ('lesse'). | |
| 2237 The first verse sets out the rules for multiplication. | |
| 2238 The second and third verses deal with subtraction of a negative quantity from a negative quantity, | |
| 2239 where the result may be either positive or negative. <lb/> | |
| 2240 Like folios Add MS 6784, f. 323, f. 324, which follow soon after it, this one appears to be based on Viète, | |
| 2241 <emph style="it">In artem analyticen isagoge</emph>, 1591, | |
| 2242 in this case on Chapter IV, Praeceptum II and Praeceptum III. | |
| 2243 </s> | |
| 2244 </p> | |
| 2245 </emph>] | |
| 2246 <lb/><lb/></s></p></div> | |
| 2247 <p> | |
| 2248 <s xml:id="echoid-s145" xml:space="preserve"> | |
| 2249 If more by more must needes make more <lb/> | |
| 2250 Then lesse by more makes lesse of more <lb/> | |
| 2251 And lesse by lesse makes lesse of lesse <lb/> | |
| 2252 If more be more and lesse be lesse. | |
| 2253 </s> | |
| 2254 </p> | |
| 2255 <p> | |
| 2256 <s xml:id="echoid-s146" xml:space="preserve"> | |
| 2257 Yet lesse of lesse makes lesse or more <lb/> | |
| 2258 The which is best keep both in store <lb/> | |
| 2259 If lesse of lesse thou <emph style="super">you</emph> wilt make lesse <lb/> | |
| 2260 Then pull <emph style="super">bate</emph> the same from that is lesse | |
| 2261 </s> | |
| 2262 </p> | |
| 2263 <p> | |
| 2264 <s xml:id="echoid-s147" xml:space="preserve"> | |
| 2265 But if the same thou <emph style="super">you</emph> wilt make more <lb/> | |
| 2266 Then add the same <emph style="super">to it</emph> to that is <emph style="super">the sign of</emph> more <lb/> | |
| 2267 The signe <emph style="super">rule</emph> of more is best to use <lb/> | |
| 2268 Except some <emph style="super">Yet for some</emph> cause | |
| 2269 the <emph style="super">do</emph> other choose <emph style="super">then it refuse</emph> <lb/> | |
| 2270 For <emph style="super">So</emph> <emph style="super">Yet</emph> both are one, for both are true <lb/> | |
| 2271 of this inough and so adew. | |
| 2272 </s> | |
| 2273 </p> | |
| 2274 <pb file="add_6784_f322" o="322" n="643"/> | |
| 2275 <div xml:id="echoid-div41" type="page_commentary" level="2" n="41"> | |
| 2276 <p> | |
| 2277 <s xml:id="echoid-s148" xml:space="preserve">[<emph style="it">Note: | |
| 2278 <p> | |
| 2279 <s xml:id="echoid-s148" xml:space="preserve"> | |
| 2280 This page shows several examples of additions and subtractions using letters. | |
| 2281 Note that here such operations are only carried out between quantities of the same dimension. | |
| 2282 </s> | |
| 2283 </p> | |
| 2284 </emph>] | |
| 2285 <lb/><lb/></s></p></div> | |
| 2286 <head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat"> | |
| 2287 1) Operationes logisticæ, in notis | |
| 2288 <lb/>[<emph style="it">tr: | |
| 2289 The operations of arithmetic in symbols. | |
| 2290 </emph>]<lb/> | |
| 2291 </head> | |
| 2292 <p xml:lang="lat"> | |
| 2293 <s xml:id="echoid-s150" xml:space="preserve"> | |
| 2294 adde | |
| 2295 <lb/>[<emph style="it">tr: | |
| 2296 add | |
| 2297 </emph>]<lb/> | |
| 2298 </s> | |
| 2299 <lb/> | |
| 2300 <s xml:id="echoid-s151" xml:space="preserve"> | |
| 2301 summa | |
| 2302 <lb/>[<emph style="it">tr: | |
| 2303 sum | |
| 2304 </emph>]<lb/> | |
| 2305 </s> | |
| 2306 </p> | |
| 2307 <p xml:lang="lat"> | |
| 2308 <s xml:id="echoid-s152" xml:space="preserve"> | |
| 2309 subduce | |
| 2310 <lb/>[<emph style="it">tr: | |
| 2311 subtract | |
| 2312 </emph>]<lb/> | |
| 2313 </s> | |
| 2314 <lb/> | |
| 2315 <s xml:id="echoid-s153" xml:space="preserve"> | |
| 2316 reliqua | |
| 2317 <lb/>[<emph style="it">tr: | |
| 2318 remainder | |
| 2319 </emph>]<lb/> | |
| 2320 </s> | |
| 2321 </p> | |
| 2322 <pb file="add_6784_f322v" o="322v" n="644"/> | |
| 2323 <pb file="add_6784_f323" o="323" n="645"/> | |
| 2324 <div xml:id="echoid-div42" type="page_commentary" level="2" n="42"> | |
| 2325 <p> | |
| 2326 <s xml:id="echoid-s154" xml:space="preserve">[<emph style="it">Note: | |
| 2327 <p> | |
| 2328 <s xml:id="echoid-s154" xml:space="preserve"> | |
| 2329 This page shows examples of multiplication and division using letters. | |
| 2330 </s> | |
| 2331 </p> | |
| 2332 </emph>] | |
| 2333 <lb/><lb/></s></p></div> | |
| 2334 <head xml:id="echoid-head83" xml:space="preserve"> | |
| 2335 2) | |
| 2336 </head> | |
| 2337 <p xml:lang="lat"> | |
| 2338 <s xml:id="echoid-s156" xml:space="preserve"> | |
| 2339 multip. | |
| 2340 <lb/>[<emph style="it">tr: | |
| 2341 multiply | |
| 2342 </emph>]<lb/> | |
| 2343 </s> | |
| 2344 <lb/> | |
| 2345 <s xml:id="echoid-s157" xml:space="preserve"> | |
| 2346 in | |
| 2347 <lb/>[<emph style="it">tr: | |
| 2348 by | |
| 2349 </emph>]<lb/> | |
| 2350 </s> | |
| 2351 <lb/> | |
| 2352 <s xml:id="echoid-s158" xml:space="preserve"> | |
| 2353 facta | |
| 2354 <lb/>[<emph style="it">tr: | |
| 2355 product | |
| 2356 </emph>]<lb/> | |
| 2357 </s> | |
| 2358 </p> | |
| 2359 <p xml:lang="lat"> | |
| 2360 <s xml:id="echoid-s159" xml:space="preserve"> | |
| 2361 applica | |
| 2362 <lb/>[<emph style="it">tr: | |
| 2363 divide | |
| 2364 </emph>]<lb/> | |
| 2365 </s> | |
| 2366 <lb/> | |
| 2367 <s xml:id="echoid-s160" xml:space="preserve"> | |
| 2368 ad | |
| 2369 <lb/>[<emph style="it">tr: | |
| 2370 by | |
| 2371 </emph>]<lb/> | |
| 2372 </s> | |
| 2373 <lb/> | |
| 2374 <s xml:id="echoid-s161" xml:space="preserve"> | |
| 2375 orta | |
| 2376 <lb/>[<emph style="it">tr: | |
| 2377 result | |
| 2378 </emph>]<lb/> | |
| 2379 </s> | |
| 2380 </p> | |
| 2381 <p xml:lang="lat"> | |
| 2382 <s xml:id="echoid-s162" xml:space="preserve"> | |
| 2383 manifestum <lb/> | |
| 2384 per præcog-<lb/> | |
| 2385 nitam genera-<lb/> | |
| 2386 tionem. | |
| 2387 <lb/>[<emph style="it">tr: | |
| 2388 evident from the previously learned constructions | |
| 2389 </emph>]<lb/> | |
| 2390 </s> | |
| 2391 </p> | |
| 2392 <pb file="add_6784_f323v" o="323v" n="646"/> | |
| 2393 <pb file="add_6784_f324" o="324" n="647"/> | |
| 2394 <div xml:id="echoid-div43" type="page_commentary" level="2" n="43"> | |
| 2395 <p> | |
| 2396 <s xml:id="echoid-s163" xml:space="preserve">[<emph style="it">Note: | |
| 2397 <p> | |
| 2398 <s xml:id="echoid-s163" xml:space="preserve"> | |
| 2399 The examples of division on this page are taken directly from Viète, | |
| 2400 <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter IV, end of Praeceptum IV, | |
| 2401 but Harriot has re-written the examples in his own symbolic notation. | |
| 2402 </s> | |
| 2403 </p> | |
| 2404 </emph>] | |
| 2405 <lb/><lb/></s></p></div> | |
| 2406 <head xml:id="echoid-head84" xml:space="preserve"> | |
| 2407 3) | |
| 2408 </head> | |
| 2409 <pb file="add_6784_f324v" o="324v" n="648"/> | |
| 2410 <pb file="add_6784_f325" o="325" n="649"/> | |
| 2411 <head xml:id="echoid-head85" xml:space="preserve"> | |
| 2412 4) | |
| 2413 </head> | |
| 2414 <div xml:id="echoid-div44" type="page_commentary" level="2" n="44"> | |
| 2415 <p> | |
| 2416 <s xml:id="echoid-s165" xml:space="preserve">[<emph style="it">Note: | |
| 2417 <p> | |
| 2418 <s xml:id="echoid-s165" xml:space="preserve"> | |
| 2419 The terminology and examples on this page are taken directly from Viète, | |
| 2420 <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter V, | |
| 2421 but Harriot has re-written the examples in his own symbolic notation. | |
| 2422 </s> | |
| 2423 </p> | |
| 2424 </emph>] | |
| 2425 <lb/><lb/></s></p></div> | |
| 2426 <p xml:lang=""> | |
| 2427 <s xml:id="echoid-s167" xml:space="preserve"> | |
| 2428 Sit: | |
| 2429 <lb/>[<emph style="it">tr: | |
| 2430 Let: | |
| 2431 </emph>]<lb/> | |
| 2432 </s> | |
| 2433 <lb/> | |
| 2434 <s xml:id="echoid-s168" xml:space="preserve"> | |
| 2435 Dico quod: per Antithesin. | |
| 2436 <lb/>[<emph style="it">tr: | |
| 2437 I say that, by antihesis: | |
| 2438 </emph>]<lb/> | |
| 2439 </s> | |
| 2440 <lb/> | |
| 2441 <s xml:id="echoid-s169" xml:space="preserve"> | |
| 2442 Quoniam: | |
| 2443 <lb/>[<emph style="it">tr: | |
| 2444 Because: | |
| 2445 </emph>]<lb/> | |
| 2446 </s> | |
| 2447 <lb/> | |
| 2448 <s xml:id="echoid-s170" xml:space="preserve"> | |
| 2449 Adde utrolique. | |
| 2450 <lb/>[<emph style="it">tr: | |
| 2451 Add to each side. | |
| 2452 </emph>]<lb/> | |
| 2453 </s> | |
| 2454 <lb/> | |
| 2455 <s xml:id="echoid-s171" xml:space="preserve"> | |
| 2456 Ergo: | |
| 2457 <lb/>[<emph style="it">tr: | |
| 2458 Therefore: | |
| 2459 </emph>]<lb/> | |
| 2460 </s> | |
| 2461 </p> | |
| 2462 <p xml:lang=""> | |
| 2463 <s xml:id="echoid-s172" xml:space="preserve"> | |
| 2464 Secundo: sit, | |
| 2465 <lb/>[<emph style="it">tr: | |
| 2466 Second, let: | |
| 2467 </emph>]<lb/> | |
| 2468 </s> | |
| 2469 <lb/> | |
| 2470 <s xml:id="echoid-s173" xml:space="preserve"> | |
| 2471 Dico quod: | |
| 2472 <lb/>[<emph style="it">tr: | |
| 2473 I say that: | |
| 2474 </emph>]<lb/> | |
| 2475 </s> | |
| 2476 <lb/> | |
| 2477 <s xml:id="echoid-s174" xml:space="preserve"> | |
| 2478 Quoniam: | |
| 2479 <lb/>[<emph style="it">tr: | |
| 2480 Because: | |
| 2481 </emph>]<lb/> | |
| 2482 </s> | |
| 2483 <lb/> | |
| 2484 <s xml:id="echoid-s175" xml:space="preserve"> | |
| 2485 Adde utrolique. | |
| 2486 <lb/>[<emph style="it">tr: | |
| 2487 Add to each side. | |
| 2488 </emph>]<lb/> | |
| 2489 </s> | |
| 2490 <lb/> | |
| 2491 <s xml:id="echoid-s176" xml:space="preserve"> | |
| 2492 Ergo. | |
| 2493 <lb/>[<emph style="it">tr: | |
| 2494 Therefore. | |
| 2495 </emph>]<lb/> | |
| 2496 </s> | |
| 2497 <lb/> | |
| 2498 <s xml:id="echoid-s177" xml:space="preserve"> | |
| 2499 Et ita. | |
| 2500 <lb/>[<emph style="it">tr: | |
| 2501 And thus. | |
| 2502 </emph>]<lb/> | |
| 2503 </s> | |
| 2504 <lb/> | |
| 2505 <s xml:id="echoid-s178" xml:space="preserve"> | |
| 2506 Sit. | |
| 2507 <lb/>[<emph style="it">tr: | |
| 2508 Let. | |
| 2509 </emph>]<lb/> | |
| 2510 </s> | |
| 2511 <lb/> | |
| 2512 <s xml:id="echoid-s179" xml:space="preserve"> | |
| 2513 Dico quod. per Hypobibasmum. | |
| 2514 <lb/>[<emph style="it">tr: | |
| 2515 I say that, by hypobibasmus. | |
| 2516 </emph>]<lb/> | |
| 2517 </s> | |
| 2518 </p> | |
| 2519 <p xml:lang=""> | |
| 2520 <s xml:id="echoid-s180" xml:space="preserve"> | |
| 2521 Sit. | |
| 2522 <lb/>[<emph style="it">tr: | |
| 2523 Let. | |
| 2524 </emph>]<lb/> | |
| 2525 </s> | |
| 2526 <lb/> | |
| 2527 <s xml:id="echoid-s181" xml:space="preserve"> | |
| 2528 Dico quod: per Parabolismum. | |
| 2529 <lb/>[<emph style="it">tr: | |
| 2530 I say that, by parabolismus. | |
| 2531 </emph>]<lb/> | |
| 2532 </s> | |
| 2533 <lb/> | |
| 2534 <s xml:id="echoid-s182" xml:space="preserve"> | |
| 2535 Vel, sit: | |
| 2536 <lb/>[<emph style="it">tr: | |
| 2537 Or, let: | |
| 2538 </emph>]<lb/> | |
| 2539 </s> | |
| 2540 <lb/> | |
| 2541 <s xml:id="echoid-s183" xml:space="preserve"> | |
| 2542 dico quod. | |
| 2543 <lb/>[<emph style="it">tr: | |
| 2544 I say that. | |
| 2545 </emph>]<lb/> | |
| 2546 </s> | |
| 2547 </p> | |
| 2548 <pb file="add_6784_f325v" o="325v" n="650"/> | |
| 2549 <pb file="add_6784_f326" o="326" n="651"/> | |
| 2550 <pb file="add_6784_f326v" o="326v" n="652"/> | |
| 2551 <pb file="add_6784_f327" o="327" n="653"/> | |
| 2552 <pb file="add_6784_f327v" o="327v" n="654"/> | |
| 2553 <pb file="add_6784_f328" o="328" n="655"/> | |
| 2554 <pb file="add_6784_f328v" o="328v" n="656"/> | |
| 2555 <pb file="add_6784_f329" o="329" n="657"/> | |
| 2556 <pb file="add_6784_f329v" o="329v" n="658"/> | |
| 2557 <pb file="add_6784_f330" o="330" n="659"/> | |
| 2558 <pb file="add_6784_f330v" o="330v" n="660"/> | |
| 2559 <pb file="add_6784_f331" o="331" n="661"/> | |
| 2560 <div xml:id="echoid-div45" type="page_commentary" level="2" n="45"> | |
| 2561 <p> | |
| 2562 <s xml:id="echoid-s184" xml:space="preserve">[<emph style="it">Note: | |
| 2563 <p> | |
| 2564 <s xml:id="echoid-s184" xml:space="preserve"> | |
| 2565 The reference to Apollonius is to pages 5 and 6 of Commandino's edition, | |
| 2566 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). | |
| 2567 There are also references at the bottom of the page to | |
| 2568 Viète an Cardano. | |
| 2569 </s> | |
| 2570 <lb/> | |
| 2571 <s xml:id="echoid-s185" xml:space="preserve"> | |
| 2572 The reference to Viète is to <emph style="it">Apollonius Gallus</emph>, Appendix 2, Problem V. | |
| 2573 </s> | |
| 2574 <lb/> | |
| 2575 <quote xml:lang="lat"> | |
| 2576 V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant rationem. | |
| 2577 </quote> | |
| 2578 <lb/> | |
| 2579 <quote> | |
| 2580 Given a triangle, find a point from which there may be drawn three straight lines | |
| 2581 to the vertices of the given triangle, keeping a fixed ratio. | |
| 2582 </quote> | |
| 2583 <lb/> | |
| 2584 <s xml:id="echoid-s186" xml:space="preserve"> | |
| 2585 The reference to Cardano is to his <emph style="it">Opus novum de proportionibus</emph>. | |
| 2586 The relevant Propositions are 154 (though mistakenly described in the <emph style="it">Opus novum</emph> as 144) | |
| 2587 and 160. | |
| 2588 </s> | |
| 2589 <lb/> | |
| 2590 <quote xml:lang="lat"> | |
| 2591 Propositio centesimaquadragesimaquarta <lb/> | |
| 2592 Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant | |
| 2593 proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto | |
| 2594 extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ | |
| 2595 seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, | |
| 2596 ipsæ erunt in proportione mediæ ad adiectam. <lb/> | |
| 2597 Hæc propositio est admirabilis: ... | |
| 2598 </quote> | |
| 2599 <lb/> | |
| 2600 <quote xml:lang="lat"> | |
| 2601 Propositio centesimasexagesima <lb/> | |
| 2602 Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus datis. | |
| 2603 </quote> | |
| 2604 </p> | |
| 2605 </emph>] | |
| 2606 <lb/><lb/></s></p></div> | |
| 2607 <head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat"> | |
| 2608 5. Appolonius. pag. 5. 6. | |
| 2609 <lb/>[<emph style="it">tr: | |
| 2610 Apollonius, pages 5, 6. | |
| 2611 </emph>]<lb/> | |
| 2612 </head> | |
| 2613 <p xml:lang="lat"> | |
| 2614 <s xml:id="echoid-s188" xml:space="preserve"> | |
| 2615 Quæsitum: <lb/> | |
| 2616 ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> | |
| 2617 erit; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi></mstyle></math>. | |
| 2618 <lb/>[<emph style="it">tr: | |
| 2619 Sought: <lb/> | |
| 2620 Wherever a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is placed on the circumference, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>b</mi><mo>=</mo><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>:</mo><mi>k</mi><mi>b</mi></mstyle></math>. | |
| 2621 </emph>]<lb/> | |
| 2622 </s> | |
| 2623 </p> | |
| 2624 <p xml:lang="lat"> | |
| 2625 <s xml:id="echoid-s189" xml:space="preserve"> | |
| 2626 sint data puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> | |
| 2627 Data ratio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/> | |
| 2628 producatur, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, versus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> | |
| 2629 <lb/>[<emph style="it">tr: | |
| 2630 Let the given points be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, the given ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> be produced towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. | |
| 2631 </emph>]<lb/> | |
| 2632 </s> | |
| 2633 </p> | |
| 2634 <p xml:lang="lat"> | |
| 2635 <s xml:id="echoid-s190" xml:space="preserve"> | |
| 2636 Dico quod: | |
| 2637 <lb/>[<emph style="it">tr: | |
| 2638 I say that: | |
| 2639 </emph>]<lb/> | |
| 2640 </s> | |
| 2641 </p> | |
| 2642 <p xml:lang="lat"> | |
| 2643 <s xml:id="echoid-s191" xml:space="preserve"> | |
| 2644 Inde: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> maior, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math><lb/> | |
| 2645 minor, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> <lb/> | |
| 2646 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mo>=</mo><mi>g</mi></mstyle></math> <lb/> | |
| 2647 fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>h</mi></mstyle></math> periferia <lb/> | |
| 2648 sumatur quovis puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> | |
| 2649 Ducantur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math>. | |
| 2650 <lb/>[<emph style="it">tr: | |
| 2651 Whence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math>, less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi><mo>=</mo><mi>g</mi></mstyle></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>h</mi></mstyle></math> be the circumference, | |
| 2652 taking any point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. Let there be drawn <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>f</mi></mstyle></math>. | |
| 2653 </emph>]<lb/> | |
| 2654 </s> | |
| 2655 </p> | |
| 2656 <p xml:lang="lat"> | |
| 2657 <s xml:id="echoid-s192" xml:space="preserve"> | |
| 2658 * Ducantur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math>, parallela, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. <lb/> | |
| 2659 ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> | |
| 2660 erit; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>b</mi></mstyle></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>: vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>b</mi></mstyle></math>. | |
| 2661 <lb/>[<emph style="it">tr: | |
| 2662 Taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math> parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, wherever the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is placed on the circumference, | |
| 2663 then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi><mo>:</mo><mi>h</mi><mi>b</mi><mo>=</mo><mi>c</mi><mo>:</mo><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>k</mi><mo>:</mo><mi>k</mi><mi>b</mi></mstyle></math>. | |
| 2664 </emph>]<lb/> | |
| 2665 </s> | |
| 2666 </p> | |
| 2667 <p xml:lang="lat"> | |
| 2668 <s xml:id="echoid-s193" xml:space="preserve"> | |
| 2669 Corollaria. <lb/> | |
| 2670 Hinc a tribus punctis sive sint in recta <lb/> | |
| 2671 vel non; possunt duci tres lineæ ad unum <lb/> | |
| 2672 punctum, <emph style="st">ut s</emph> et erunt in data ratione. | |
| 2673 <lb/>[<emph style="it">tr: | |
| 2674 Corollary <lb/> | |
| 2675 Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, | |
| 2676 and they will be in the given ratio. | |
| 2677 </emph>]<lb/> | |
| 2678 </s> | |
| 2679 </p> | |
| 2680 <p xml:lang="lat"> | |
| 2681 <s xml:id="echoid-s194" xml:space="preserve"> | |
| 2682 vide vertam <lb/> | |
| 2683 in Apolonio gallo <lb/> | |
| 2684 et card: de prop. pag. 145. 162. | |
| 2685 <lb/>[<emph style="it">tr: | |
| 2686 see over, in <emph style="it">Apollonius Gallus</emph>, | |
| 2687 and Cardano, <emph style="it">De proportionibus</emph>, pages 145, 162. | |
| 2688 </emph>]<lb/> | |
| 2689 </s> | |
| 2690 </p> | |
| 2691 <pb file="add_6784_f331v" o="331v" n="662"/> | |
| 2692 <pb file="add_6784_f332" o="332" n="663"/> | |
| 2693 <div xml:id="echoid-div46" type="page_commentary" level="2" n="46"> | |
| 2694 <p> | |
| 2695 <s xml:id="echoid-s195" xml:space="preserve">[<emph style="it">Note: | |
| 2696 <p> | |
| 2697 <s xml:id="echoid-s195" xml:space="preserve"> | |
| 2698 The reference is to pages 5 and 6 of Commandino's edition of Apollonius, | |
| 2699 <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). | |
| 2700 </s> | |
| 2701 </p> | |
| 2702 </emph>] | |
| 2703 <lb/><lb/></s></p></div> | |
| 2704 <head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat"> | |
| 2705 Ad appolonium. pa. 5. 6. | |
| 2706 <lb/>[<emph style="it">tr: | |
| 2707 On Apollonius, pages 5, 6 | |
| 2708 </emph>]<lb/> | |
| 2709 </head> | |
| 2710 <p xml:lang="lat"> | |
| 2711 <s xml:id="echoid-s197" xml:space="preserve"> | |
| 2712 Data puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, in linea, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> <lb/> | |
| 2713 Invenire lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> | |
| 2714 ita ut sit: <lb/> | |
| 2715 Sit factum: <lb/> | |
| 2716 Tum: | |
| 2717 <lb/>[<emph style="it">tr: | |
| 2718 Given a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> in a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, find the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math> so that: <lb/> | |
| 2719 Let it be done, then: | |
| 2720 </emph>]<lb/> | |
| 2721 </s> | |
| 2722 </p> | |
| 2723 <p xml:lang="lat"> | |
| 2724 <s xml:id="echoid-s198" xml:space="preserve"> | |
| 2725 Aliter <lb/> | |
| 2726 <lb/>[...]<lb/> <lb/> | |
| 2727 sed idem ut supra | |
| 2728 <lb/>[<emph style="it">tr: | |
| 2729 Another way <lb/> | |
| 2730 <lb/>[...]<lb/> <lb/> | |
| 2731 but the same as above | |
| 2732 </emph>]<lb/> | |
| 2733 </s> | |
| 2734 </p> | |
| 2735 <p xml:lang="lat"> | |
| 2736 <s xml:id="echoid-s199" xml:space="preserve"> | |
| 2737 Invenire <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math> | |
| 2738 <lb/>[<emph style="it">tr: | |
| 2739 To find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math>. | |
| 2740 </emph>]<lb/> | |
| 2741 </s> | |
| 2742 </p> | |
| 2743 <pb file="add_6784_f332v" o="332v" n="664"/> | |
| 2744 <pb file="add_6784_f333" o="333" n="665"/> | |
| 2745 <pb file="add_6784_f333v" o="333v" n="666"/> | |
| 2746 <pb file="add_6784_f334" o="334" n="667"/> | |
| 2747 <pb file="add_6784_f334v" o="334v" n="668"/> | |
| 2748 <pb file="add_6784_f335" o="335" n="669"/> | |
| 2749 <pb file="add_6784_f335v" o="335v" n="670"/> | |
| 2750 <pb file="add_6784_f336" o="336" n="671"/> | |
| 2751 <pb file="add_6784_f336v" o="336v" n="672"/> | |
| 2752 <pb file="add_6784_f337" o="337" n="673"/> | |
| 2753 <pb file="add_6784_f337v" o="337v" n="674"/> | |
| 2754 <pb file="add_6784_f338" o="338" n="675"/> | |
| 2755 <pb file="add_6784_f338v" o="338v" n="676"/> | |
| 2756 <pb file="add_6784_f339" o="339" n="677"/> | |
| 2757 <pb file="add_6784_f339v" o="339v" n="678"/> | |
| 2758 <pb file="add_6784_f340" o="340" n="679"/> | |
| 2759 <pb file="add_6784_f340v" o="340v" n="680"/> | |
| 2760 <pb file="add_6784_f341" o="341" n="681"/> | |
| 2761 <pb file="add_6784_f341v" o="341v" n="682"/> | |
| 2762 <pb file="add_6784_f342" o="342" n="683"/> | |
| 2763 <pb file="add_6784_f342v" o="342v" n="684"/> | |
| 2764 <pb file="add_6784_f343" o="343" n="685"/> | |
| 2765 <pb file="add_6784_f343v" o="343v" n="686"/> | |
| 2766 <pb file="add_6784_f344" o="344" n="687"/> | |
| 2767 <pb file="add_6784_f344v" o="344v" n="688"/> | |
| 2768 <pb file="add_6784_f345" o="345" n="689"/> | |
| 2769 <pb file="add_6784_f345v" o="345v" n="690"/> | |
| 2770 <pb file="add_6784_f346" o="346" n="691"/> | |
| 2771 <pb file="add_6784_f346v" o="346v" n="692"/> | |
| 2772 <pb file="add_6784_f347" o="347" n="693"/> | |
| 2773 <pb file="add_6784_f347v" o="347v" n="694"/> | |
| 2774 <pb file="add_6784_f348" o="348" n="695"/> | |
| 2775 <div xml:id="echoid-div47" type="page_commentary" level="2" n="47"> | |
| 2776 <p> | |
| 2777 <s xml:id="echoid-s200" xml:space="preserve">[<emph style="it">Note: | |
| 2778 <p> | |
| 2779 <s xml:id="echoid-s200" xml:space="preserve"> | |
| 2780 On this page Harriot investigates Proposition 18 from Viète's | |
| 2781 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 2782 </s> | |
| 2783 <lb/> | |
| 2784 <quote xml:lang="lat"> | |
| 2785 Proposition XVIII. <lb/> | |
| 2786 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, | |
| 2787 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: | |
| 2788 triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine | |
| 2789 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ | |
| 2790 basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. | |
| 2791 </quote> | |
| 2792 <lb/> | |
| 2793 <quote> | |
| 2794 If two triangles are each isosceles, equal to one another in theri legs, | |
| 2795 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 2796 then three times the product of the square of the common leg and half the base of the first | |
| 2797 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, | |
| 2798 when reduced by the cube of the same half base thus decreased or increased, | |
| 2799 is equal to the product of the second base and the square of the common leg. | |
| 2800 </quote> | |
| 2801 <lb/> | |
| 2802 <s xml:id="echoid-s201" xml:space="preserve"> | |
| 2803 For Harriot's statement of Propostion 18, and a geometric version of the proof, see Add MS 6784, f. 349. | |
| 2804 Here he works the proposition algebraically. | |
| 2805 </s> | |
| 2806 <lb/> | |
| 2807 <s xml:id="echoid-s202" xml:space="preserve"> | |
| 2808 This page also refers to Proposition 17 from the <emph style="it">Supplementum</emph>, | |
| 2809 (see MS 6784, f. 350). | |
| 2810 </s> | |
| 2811 </p> | |
| 2812 </emph>] | |
| 2813 <lb/><lb/></s></p></div> | |
| 2814 <head xml:id="echoid-head88" xml:space="preserve"> | |
| 2815 prop. 18. Supplementi. | |
| 2816 <lb/>[<emph style="it">tr: | |
| 2817 Proposition 18 from the Supplementum | |
| 2818 </emph>]<lb/> | |
| 2819 </head> | |
| 2820 <pb file="add_6784_f348v" o="348v" n="696"/> | |
| 2821 <pb file="add_6784_f349" o="349" n="697"/> | |
| 2822 <div xml:id="echoid-div48" type="page_commentary" level="2" n="48"> | |
| 2823 <p> | |
| 2824 <s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note: | |
| 2825 <p> | |
| 2826 <s xml:id="echoid-s204" xml:space="preserve"> | |
| 2827 On this page Harriot investigates Proposition 18 from Viète's | |
| 2828 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 2829 </s> | |
| 2830 <lb/> | |
| 2831 <quote xml:lang="lat"> | |
| 2832 Proposition XVIII. <lb/> | |
| 2833 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, | |
| 2834 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: | |
| 2835 triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine | |
| 2836 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ | |
| 2837 basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. | |
| 2838 </quote> | |
| 2839 <lb/> | |
| 2840 <quote> | |
| 2841 If two triangles are each isosceles, equal to one another in theri legs, | |
| 2842 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 2843 then three times the product of the square of the common leg and half the base of the first | |
| 2844 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, | |
| 2845 when reduced by the cube of the same half base thus decreased or increased, | |
| 2846 is equal to the product of the second base and the square of the common leg. | |
| 2847 </quote> | |
| 2848 <lb/> | |
| 2849 <s xml:id="echoid-s205" xml:space="preserve"> | |
| 2850 This page refers to several previous propositions from the <emph style="it">Supplementum</emph>, | |
| 2851 namely Proposition 12 and 14b (Add MS 6784, f. 353), | |
| 2852 Proposition 16 (add MS 6784, f. 351) and Proposition 17 (add MS 6784, f. 350). | |
| 2853 </s> | |
| 2854 </p> | |
| 2855 </emph>] | |
| 2856 <lb/><lb/></s></p></div> | |
| 2857 <head xml:id="echoid-head89" xml:space="preserve"> | |
| 2858 prop. 18. Supplementi. | |
| 2859 <lb/>[<emph style="it">tr: | |
| 2860 Proposition 18 from the Supplementum | |
| 2861 </emph>]<lb/> | |
| 2862 </head> | |
| 2863 <p xml:lang="lat"> | |
| 2864 <s xml:id="echoid-s207" xml:space="preserve"> | |
| 2865 Si duo triangula fuerint aequicrura singula, et ipsa alterum alteri cruribus aequalia; angulus <lb/> | |
| 2866 autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum <lb/> | |
| 2867 sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine <lb/> | |
| 2868 ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem <lb/> | |
| 2869 dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem <lb/> | |
| 2870 cruris quadrato. | |
| 2871 <lb/>[<emph style="it">tr: | |
| 2872 If two triangles are each isosceles, equal to one another in their legs, | |
| 2873 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 2874 then three times the product of the square of the common leg and half the base of the first | |
| 2875 decreased or increased by a length whose square is equal to three times the square of the altitude of the first, | |
| 2876 when reduced by the cube of the same half base thus decreased or increased, | |
| 2877 is equal to the product of the second base and the square of the common leg. | |
| 2878 </emph>]<lb/> | |
| 2879 </s> | |
| 2880 </p> | |
| 2881 <p xml:lang="lat"> | |
| 2882 <s xml:id="echoid-s208" xml:space="preserve"> | |
| 2883 Sit triangulum primum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, secundum <lb/> | |
| 2884 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi><mi>E</mi></mstyle></math>. quorum crura et anguli sint <lb/> | |
| 2885 ut exigit propositio. Et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math> dupla <lb/> | |
| 2886 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. Tum quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> erit triplum quadrati <lb/> | |
| 2887 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> Â <lb/> | |
| 2888 Dico | |
| 2889 <lb/>[<emph style="it">tr: | |
| 2890 Let the first triangle be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> and the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi><mi>E</mi></mstyle></math>, whose sides and angles are as specified in the proposition. | |
| 2891 And let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi></mstyle></math> be twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. Then the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>F</mi></mstyle></math> is three times the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. | |
| 2892 </emph>]<lb/> | |
| 2893 </s> | |
| 2894 </p> | |
| 2895 <p xml:lang="lat"> | |
| 2896 <s xml:id="echoid-s209" xml:space="preserve"> | |
| 2897 Nam: <lb/> | |
| 2898 per 15,p <lb/>[...]<lb/> Hoc est, in notis proportionalium quas notum 12,p <lb/> | |
| 2899 1<emph style="super">o</emph>. Ducantur omnia per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> <lb/> | |
| 2900 <lb/>[...]<lb/> <lb/> | |
| 2901 Hoc est in notis 12,p. | |
| 2902 <lb/>[<emph style="it">tr: | |
| 2903 For by Proposition 15 | |
| 2904 <lb/>[...]<lb/> that is, in the notation for proportionals noted in Proposition 12, <lb/> | |
| 2905 1. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. <lb/> | |
| 2906 <lb/>[...]<lb/> <lb/> | |
| 2907 That is, in the notation of Proposition 12 | |
| 2908 </emph>]<lb/> | |
| 2909 </s> | |
| 2910 </p> | |
| 2911 <p xml:lang="lat"> | |
| 2912 <s xml:id="echoid-s210" xml:space="preserve"> | |
| 2913 2<emph style="super">o</emph>. Ducantur omnia per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>G</mi></mstyle></math> <lb/> | |
| 2914 <lb/>[...]<lb/> <lb/> | |
| 2915 Hoc est in notis 12,p. | |
| 2916 <lb/>[<emph style="it">tr: | |
| 2917 2. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>G</mi></mstyle></math>. <lb/> | |
| 2918 <lb/>[...]<lb/> <lb/> | |
| 2919 That is, in the notation of Proposition 12 | |
| 2920 </emph>]<lb/> | |
| 2921 </s> | |
| 2922 </p> | |
| 2923 <p xml:lang="lat"> | |
| 2924 <s xml:id="echoid-s211" xml:space="preserve"> | |
| 2925 Deinde per 16.p <lb/> | |
| 2926 Hoc est in notis 12,p. <lb/> | |
| 2927 Sed: per consect: 14.p <lb/> | |
| 2928 Ergo patet propositum | |
| 2929 <lb/>[<emph style="it">tr: | |
| 2930 Thence by Proposition 16, <lb/> | |
| 2931 That is, in the notation of Proposition 12 <lb/> | |
| 2932 But by the consequence of Proposition 14, <lb/> | |
| 2933 Thus the propostion is shown. | |
| 2934 </emph>]<lb/> | |
| 2935 </s> | |
| 2936 </p> | |
| 2937 <p xml:lang="lat"> | |
| 2938 <s xml:id="echoid-s212" xml:space="preserve"> | |
| 2939 Cum 16<emph style="super">a</emph> et 17<emph style="super">a</emph> prop. basis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> notabatur (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) | |
| 2940 ideo eius partes <lb/> | |
| 2941 Scilicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> alijs vocalibus notandæ sunt. pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> nota (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>) <lb/> | |
| 2942 et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>). <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> servent easdem notas quas ibi <lb/> | |
| 2943 habuerunt. Videlicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math>, (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>). <lb/> | |
| 2944 Propositum igitur simplicibus notis ita significatur: | |
| 2945 <lb/>[<emph style="it">tr: | |
| 2946 Since in Propositions 16 adn 17, the base <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, therefore its parts, | |
| 2947 namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> may be denoted by other names; | |
| 2948 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> put the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math> the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. | |
| 2949 For <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> use the same notation as they had there, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>=</mo><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi><mo>=</mo><mi>c</mi></mstyle></math>. <lb/> | |
| 2950 In simple notation the proposition may therefore be written: | |
| 2951 </emph>]<lb/> | |
| 2952 </s> | |
| 2953 </p> | |
| 2954 <p xml:lang="lat"> | |
| 2955 <s xml:id="echoid-s213" xml:space="preserve"> | |
| 2956 igitur: <lb/> | |
| 2957 Quando æquatio est sub ista <lb/> | |
| 2958 forma: <lb/> | |
| 2959 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> erit duplex vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. vel. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>. | |
| 2960 <lb/>[<emph style="it">tr: | |
| 2961 When the equation is in this form, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is twofold, either <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>C</mi></mstyle></math>. | |
| 2962 </emph>]<lb/> | |
| 2963 </s> | |
| 2964 </p> | |
| 2965 <pb file="add_6784_f349v" o="349v" n="698"/> | |
| 2966 <pb file="add_6784_f350" o="350" n="699"/> | |
| 2967 <div xml:id="echoid-div49" type="page_commentary" level="2" n="49"> | |
| 2968 <p> | |
| 2969 <s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note: | |
| 2970 <p> | |
| 2971 <s xml:id="echoid-s214" xml:space="preserve"> | |
| 2972 On this page Harriot investigates Proposition 17 from Viète's | |
| 2973 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 2974 </s> | |
| 2975 <lb/> | |
| 2976 <quote xml:lang="lat"> | |
| 2977 Proposition XVII. <lb/> | |
| 2978 Si duo triangula fuerint aequicrura singula, & ipsa alterumalteria cruribus aequalia, | |
| 2979 angulus autem, quem is qui est ad basin secundi relinquit e duobus rectis, | |
| 2980 sit triplus anguli qui est ad basin primi: solidum triplum sub base primi & cruris communis quadrato, | |
| 2981 minus cubo e base primi, aequale est solido sub base secundi & cruris communis quadrato. | |
| 2982 </quote> | |
| 2983 <lb/> | |
| 2984 <quote> | |
| 2985 If two triangles are each isosceles, both with equal legs, | |
| 2986 and moreover the angle at the base of the second subtracted from two right angles is | |
| 2987 three times the angle at the base of the first, | |
| 2988 then three times the product of the base of the first and the square of the common side, | |
| 2989 minus the cube of the first base, is equal to the product of the second base and the square of the common side. | |
| 2990 </quote> | |
| 2991 <lb/> | |
| 2992 <s xml:id="echoid-s215" xml:space="preserve"> | |
| 2993 The working contains reference to three propositions from Euclid's <emph style="it">Elements</emph>. | |
| 2994 </s> | |
| 2995 <lb/> | |
| 2996 <quote> | |
| 2997 II.6 If a straight line be bisected and produced to any point, | |
| 2998 the rectangle contained by the whole line so increased, and the part produced, | |
| 2999 together with the square of half the line, is equal to the square of the line made up of the half, | |
| 3000 and the produced part. | |
| 3001 </quote> | |
| 3002 <lb/> | |
| 3003 <quote> | |
| 3004 III.36 If from a point without a circle two straight lines be drawn to it, | |
| 3005 one of which is a tangent to the circle, and the other cuts it; | |
| 3006 the rectangle under the whole cutting line and the external segment is equal to the square of the tangent. | |
| 3007 </quote> | |
| 3008 <lb/> | |
| 3009 <quote> | |
| 3010 I. 47 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides. | |
| 3011 </quote> | |
| 3012 </p> | |
| 3013 </emph>] | |
| 3014 <lb/><lb/></s></p></div> | |
| 3015 <head xml:id="echoid-head90" xml:space="preserve"> | |
| 3016 prop. 17. Supplementi. | |
| 3017 <lb/>[<emph style="it">tr: | |
| 3018 Proposition 17 from the Supplementum | |
| 3019 </emph>]<lb/> | |
| 3020 </head> | |
| 3021 <p xml:lang="lat"> | |
| 3022 <s xml:id="echoid-s217" xml:space="preserve"> | |
| 3023 Si duo triangula fuerint aequicrura singula, <lb/> | |
| 3024 et ipsa alterumalteria cruribus aequalia; angulus <lb/> | |
| 3025 autem, quem is qui est ad basin secundi relinquit <lb/> | |
| 3026 e duobus rectis, sit triplus anguli qui est ad basin <lb/> | |
| 3027 <emph style="st">secundi</emph> <emph style="super">primi</emph>. Solidum triplum sub base primi et cruris <lb/> | |
| 3028 communis quadrato, minus cubo e base primi: aequale <lb/> | |
| 3029 est solido sub base secundiet cruris communis <lb/> | |
| 3030 quadrato. | |
| 3031 <lb/>[<emph style="it">tr: | |
| 3032 If two triangles are each isosceles, the legs of one equal to the legs of the other, | |
| 3033 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 3034 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, | |
| 3035 is equal to the product of the second base and the square of the same side. | |
| 3036 </emph>]<lb/> | |
| 3037 </s> | |
| 3038 </p> | |
| 3039 <p xml:lang="lat"> | |
| 3040 <s xml:id="echoid-s218" xml:space="preserve"> | |
| 3041 per 6,2 el. <lb/> | |
| 3042 per 36,3 el. <lb/> | |
| 3043 per 47,1 el. <lb/> | |
| 3044 <lb/>[...]<lb/> <lb/> | |
| 3045 quia parallogramma æquialta <lb/> | |
| 3046 et sunt ut bases. <lb/> | |
| 3047 <lb/>[...]<lb/> <lb/> | |
| 3048 vel per notas <lb/> | |
| 3049 simplices <lb/> | |
| 3050 Hæque Resoluatur Analogia, erit: <lb/> | |
| 3051 Propositum | |
| 3052 <lb/>[<emph style="it">tr: | |
| 3053 by Elements II.6 <lb/> | |
| 3054 by Elements III.35 <lb/> | |
| 3055 by Elements I.47 <lb/> | |
| 3056 <lb/>[...]<lb/> <lb/> | |
| 3057 because the parallelograms are of equal height and are as the bases. <lb/> | |
| 3058 <lb/>[...]<lb/> <lb/> | |
| 3059 or in simple notation <lb/> | |
| 3060 And this ratio is resolved, hence the proposition: | |
| 3061 </emph>]<lb/> | |
| 3062 </s> | |
| 3063 </p> | |
| 3064 <pb file="add_6784_f350v" o="350v" n="700"/> | |
| 3065 <pb file="add_6784_f351" o="351" n="701"/> | |
| 3066 <div xml:id="echoid-div50" type="page_commentary" level="2" n="50"> | |
| 3067 <p> | |
| 3068 <s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note: | |
| 3069 <p> | |
| 3070 <s xml:id="echoid-s219" xml:space="preserve"> | |
| 3071 On this page Harriot investigates Proposition 16 from Viète's | |
| 3072 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 3073 </s> | |
| 3074 <lb/> | |
| 3075 <quote xml:lang="lat"> | |
| 3076 Proposition XVI. <lb/> | |
| 3077 Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, | |
| 3078 angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: | |
| 3079 cubus ex base primi, minus triplo solido sub base primi & cruris communis quadrato, | |
| 3080 aequalis est solido sub base secundi & ejusdem cruris quadrato. | |
| 3081 </quote> | |
| 3082 <lb/> | |
| 3083 <quote> | |
| 3084 If two triangles are each isosceles, the legs of one equal to the legs of the other, | |
| 3085 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 3086 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, | |
| 3087 is equal to the product of the second base and the square of the same side. | |
| 3088 </quote> | |
| 3089 <lb/> | |
| 3090 <s xml:id="echoid-s220" xml:space="preserve"> | |
| 3091 The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.5. | |
| 3092 </s> | |
| 3093 <lb/> | |
| 3094 <quote> | |
| 3095 II.5 If a straight line be divided into two equal parts and also into two unequal parts, | |
| 3096 the rectangle contained by the unequal parts, | |
| 3097 together with the square of the line between the points of section, | |
| 3098 is equal to the square of half that line. | |
| 3099 </quote> | |
| 3100 </p> | |
| 3101 </emph>] | |
| 3102 <lb/><lb/></s></p></div> | |
| 3103 <head xml:id="echoid-head91" xml:space="preserve"> | |
| 3104 prop. 16. Supplementi. | |
| 3105 <lb/>[<emph style="it">tr: | |
| 3106 Proposition 16 from the Supplementum | |
| 3107 </emph>]<lb/> | |
| 3108 </head> | |
| 3109 <p xml:lang="lat"> | |
| 3110 <s xml:id="echoid-s222" xml:space="preserve"> | |
| 3111 Si duo triangula fuerint aequicrura singula, <lb/> | |
| 3112 et ipsa alterum alteri cruribus aequalia: angulus <lb/> | |
| 3113 autem qui est ad basin secundi sit triplus <lb/> | |
| 3114 anguli qui est ad basin primi. Cubus ex <lb/> | |
| 3115 base primi, minus triplo solido sub base primi <lb/> | |
| 3116 et cruris communis quadrato, aequalis <lb/> | |
| 3117 est solido sub base secundi et ejusdem <lb/> | |
| 3118 cruris quadrato. | |
| 3119 <lb/>[<emph style="it">tr: | |
| 3120 If two triangles are each isosceles, the legs of one equal to the legs of the other, | |
| 3121 and moreover the angle at the base of the second is three times the angle at the base of the first, | |
| 3122 then the cube of the first base, minus three times the product of the base of the first and the square of the common side, | |
| 3123 is equal to the product of the second base and the square of the same side. | |
| 3124 </emph>]<lb/> | |
| 3125 </s> | |
| 3126 </p> | |
| 3127 <p xml:lang="lat"> | |
| 3128 <s xml:id="echoid-s223" xml:space="preserve"> | |
| 3129 per 5,2 el. <lb/> | |
| 3130 <lb/>[...]<lb/> <lb/> | |
| 3131 Quia parallogramma æquialta <lb/> | |
| 3132 et sunt ut bases. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>H</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>D</mi></mstyle></math>. <lb/> | |
| 3133 <lb/>[...]<lb/> <lb/> | |
| 3134 vel per notas <lb/> | |
| 3135 simplices <lb/> | |
| 3136 Resoluatur analogia et erit: <lb/> | |
| 3137 Propositum | |
| 3138 <lb/>[<emph style="it">tr: | |
| 3139 by Elements II.5 <lb/> | |
| 3140 <lb/>[...]<lb/> <lb/> | |
| 3141 Because the parallelograms are of equal height and are as the bases <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>H</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>D</mi></mstyle></math>. <lb/> | |
| 3142 <lb/>[...]<lb/> <lb/> | |
| 3143 or in simple notation <lb/> | |
| 3144 The ratio is resolved, and hence the proposition: | |
| 3145 </emph>]<lb/> | |
| 3146 </s> | |
| 3147 </p> | |
| 3148 <pb file="add_6784_f351v" o="351v" n="702"/> | |
| 3149 <pb file="add_6784_f352" o="352" n="703"/> | |
| 3150 <div xml:id="echoid-div51" type="page_commentary" level="2" n="51"> | |
| 3151 <p> | |
| 3152 <s xml:id="echoid-s224" xml:space="preserve">[<emph style="it">Note: | |
| 3153 <p> | |
| 3154 <s xml:id="echoid-s224" xml:space="preserve"> | |
| 3155 On this page Harriot investigates Proposition 15 from Viète's | |
| 3156 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 3157 </s> | |
| 3158 <lb/> | |
| 3159 <quote xml:lang="lat"> | |
| 3160 Proposition XV. <lb/> | |
| 3161 Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra centrum; | |
| 3162 & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, | |
| 3163 linea recta faciens cum diametro angulum æqualem trienti recti; | |
| 3164 a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: | |
| 3165 triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, | |
| 3166 quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est proportionalis. | |
| 3167 </quote> | |
| 3168 <lb/> | |
| 3169 <quote> | |
| 3170 If from the circumference of a circle there fall two perpendiculars onto the diameter, | |
| 3171 one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn | |
| 3172 from the point of incidence of the other perpendicular a straight line making an angle equal to | |
| 3173 one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, | |
| 3174 there is drawn another line to the angle of the semicircle, then three times the square of it | |
| 3175 is equal to the square of the perpendicular which falls off-centre | |
| 3176 and the squares of the segments of the diameter between which the perpendicular is the mean proportional. | |
| 3177 </quote> | |
| 3178 <lb/> | |
| 3179 <s xml:id="echoid-s225" xml:space="preserve"> | |
| 3180 The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.4. | |
| 3181 </s> | |
| 3182 <lb/> | |
| 3183 <quote> | |
| 3184 II.4 If a straight line be divided into any two parts, | |
| 3185 the square of the whole line is equal to the squares of the parts, | |
| 3186 together with twice the rectangle contained by the parts. | |
| 3187 </quote> | |
| 3188 </p> | |
| 3189 </emph>] | |
| 3190 <lb/><lb/></s></p></div> | |
| 3191 <head xml:id="echoid-head92" xml:space="preserve"> | |
| 3192 prop. 15. Supplementi | |
| 3193 <lb/>[<emph style="it">tr: | |
| 3194 Proposition 15 from the Supplementum | |
| 3195 </emph>]<lb/> | |
| 3196 </head> | |
| 3197 <p xml:lang="lat"> | |
| 3198 <s xml:id="echoid-s227" xml:space="preserve"> | |
| 3199 Si e circumferential circuli cadant in <lb/> | |
| 3200 diametrum perpendiculares duæ; una in <lb/> | |
| 3201 centro; altera extra centrum: et ad per-<lb/> | |
| 3202 pendicularem in centro agatur ex puncto <lb/> | |
| 3203 incidentiæ perpendicularis alterius, linea <lb/> | |
| 3204 recta faciens cum diametro angulum æqualem <lb/> | |
| 3205 trienti recti, a puncto autem quo acta illa secat <lb/> | |
| 3206 perpendiculare in centro, ducatur alia <lb/> | |
| 3207 linea recta ad angulum semicirculi; Triplum <lb/> | |
| 3208 quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, <lb/> | |
| 3209 quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est <lb/> | |
| 3210 proportionalis. | |
| 3211 <lb/>[<emph style="it">tr: | |
| 3212 If from the circumference of a circle there fall two perpendiculars onto the diameter, | |
| 3213 one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn | |
| 3214 from the point of incidence of the other perpendicular a straight line making an angle equal to | |
| 3215 one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, | |
| 3216 there is drawn another line to the angle of the semicircle, then three times the square of it | |
| 3217 is equal to the square of the perpendicular which falls off-centre | |
| 3218 and the squares of the segments of the diameter between which the perpendicular is the mean proportional. | |
| 3219 </emph>]<lb/> | |
| 3220 </s> | |
| 3221 <lb/> | |
| 3222 <s xml:id="echoid-s228" xml:space="preserve"> | |
| 3223 Sit diameter circuli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math>, a cuius circumferentia cadat perpendiculariter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math> et fit <lb/> | |
| 3224 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> minus segmentum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> maius, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> verum centro. Sed et cadat quoque e circumferentia <lb/> | |
| 3225 perpendiculariter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math>, et ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> ducatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> ita ut angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi><mi>E</mi></mstyle></math> sit æqualis trienti <lb/> | |
| 3226 recti, unde fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> dupla ipsius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math>; et iungatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. Dico triplum quadratum ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> <lb/> | |
| 3227 æquari quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math>, una cum quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> et quadrato ex <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. | |
| 3228 <lb/>[<emph style="it">tr: | |
| 3229 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>C</mi></mstyle></math> be the diameter of a circle, from whose circumference there falls perpendicularly <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math>, | |
| 3230 and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be the lesser segment, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> the greater, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> the centre. | |
| 3231 But there also falls perpendicularly from the circumference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math>, and from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> there is drawn a line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> | |
| 3232 so that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>B</mi><mi>E</mi></mstyle></math> is equal to a third of a right angle, whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> is twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> is joined. | |
| 3233 I say that three times the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math> is equal to the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>B</mi></mstyle></math> | |
| 3234 together with the square on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and the squareon <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. | |
| 3235 </emph>]<lb/> | |
| 3236 </s> | |
| 3237 <lb/> | |
| 3238 <s xml:id="echoid-s229" xml:space="preserve"> | |
| 3239 Etiam <lb/> | |
| 3240 per 4,2 El. <lb/> | |
| 3241 <lb/>[...]<lb/> | |
| 3242 Addatur utrovisque <lb/> | |
| 3243 <lb/>[...]<lb/> | |
| 3244 Ergo <lb/> | |
| 3245 propositum | |
| 3246 <lb/>[<emph style="it">tr: | |
| 3247 Also by Elements II.4 <lb/> | |
| 3248 <lb/>[...]<lb/> <lb/> | |
| 3249 Hence the proposition | |
| 3250 </emph>]<lb/> | |
| 3251 </s> | |
| 3252 </p> | |
| 3253 <head xml:id="echoid-head93" xml:space="preserve"> | |
| 3254 Hinc tale Consectarium potest efferri | |
| 3255 <lb/>[<emph style="it">tr: | |
| 3256 Here a Consequence of this kind may be inferred | |
| 3257 </emph>]<lb/> | |
| 3258 </head> | |
| 3259 <p xml:lang="lat"> | |
| 3260 <s xml:id="echoid-s230" xml:space="preserve"> | |
| 3261 Datis tribus continue proportionalibus: invenire lineam cuius <lb/> | |
| 3262 quadratum sit tertia pars adgregati quadratorum e tribus <lb/> | |
| 3263 proportionalibus. | |
| 3264 <lb/>[<emph style="it">tr: | |
| 3265 Given three continued proportionals, | |
| 3266 find a line whose square is a third of the sum of the squares of all three proportionals. | |
| 3267 </emph>]<lb/> | |
| 3268 </s> | |
| 3269 </p> | |
| 3270 <pb file="add_6784_f352v" o="352v" n="704"/> | |
| 3271 <pb file="add_6784_f353" o="353" n="705"/> | |
| 3272 <div xml:id="echoid-div52" type="page_commentary" level="2" n="52"> | |
| 3273 <p> | |
| 3274 <s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note: | |
| 3275 <p> | |
| 3276 <s xml:id="echoid-s231" xml:space="preserve"> | |
| 3277 On this page Harriot investigates Propositions 12, 13, and 14 from Viète's | |
| 3278 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 3279 </s> | |
| 3280 <lb/> | |
| 3281 <quote xml:lang="lat"> | |
| 3282 Proposition XII. <lb/> | |
| 3283 Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, | |
| 3284 minus solido quod fit sub eadem composita & adgregato quadratorum a tribus, | |
| 3285 æqualis est solido sub eadem composita & quadrato secundæ. | |
| 3286 </quote> | |
| 3287 <lb/> | |
| 3288 <quote> | |
| 3289 If there are three proportional lines, the cube of the sum of the two extremes, | |
| 3290 minus the product of that sum and the sum of squares of all three, | |
| 3291 is equal to the product of the sum and the square of the second. | |
| 3292 </quote> | |
| 3293 <lb/> | |
| 3294 <quote xml:lang="lat"> | |
| 3295 Proposition XIII. <lb/> | |
| 3296 Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregato quadratorum a tribus, | |
| 3297 minus cubo e prima, æquale est solido sub eadem prima & adgregato quadratorum secundæ & tertiæ. | |
| 3298 </quote> | |
| 3299 <lb/> | |
| 3300 <quote> | |
| 3301 If there are three proportional lines, the product of the first and the sum of squares of all three, | |
| 3302 minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. | |
| 3303 </quote> | |
| 3304 <lb/> | |
| 3305 <quote xml:lang="lat"> | |
| 3306 Proposition XIV. <lb/> | |
| 3307 Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregatum quadratorum a tribus, | |
| 3308 minus cubo e tertia, æquale est solido sub eadem tertia & adgregato quadratorum primæ & secundæ. | |
| 3309 </quote> | |
| 3310 <lb/> | |
| 3311 <quote> | |
| 3312 If there are three proportional lines, the product of the first and the sum of squares of all three, | |
| 3313 minus the cube of the third, is equal to the product of the third and the sum of the first and second. | |
| 3314 </quote> | |
| 3315 <lb/> | |
| 3316 <s xml:id="echoid-s232" xml:space="preserve"> | |
| 3317 The 'Consectarium' appears verbally in Viete's proposition; Harriot has re-written it in symbolic notation. | |
| 3318 </s> | |
| 3319 </p> | |
| 3320 </emph>] | |
| 3321 <lb/><lb/></s></p></div> | |
| 3322 <head xml:id="echoid-head94" xml:space="preserve"> | |
| 3323 prop. 12. Supplementi | |
| 3324 <lb/>[<emph style="it">tr: | |
| 3325 Proposition 12 from the Supplementum | |
| 3326 </emph>]<lb/> | |
| 3327 </head> | |
| 3328 <p xml:lang="lat"> | |
| 3329 <s xml:id="echoid-s234" xml:space="preserve"> | |
| 3330 Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, <lb/> | |
| 3331 minus solido quod fit sub eadem composita et adgregato quadratorum a tribus: <lb/> | |
| 3332 æqualis est solido sub eadem composita et quadrato secundæ. | |
| 3333 <lb/>[<emph style="it">tr: | |
| 3334 If there are three proportional lines, the cube of the sum of the two extremes, | |
| 3335 minus the product of that sum and the sum of squares of all three, | |
| 3336 is equal to the product of the sum and the square of the second. | |
| 3337 </emph>]<lb/> | |
| 3338 </s> | |
| 3339 <lb/> | |
| 3340 <s xml:id="echoid-s235" xml:space="preserve"> | |
| 3341 Sint 3 continue proportionales <lb/> | |
| 3342 utrinque addatur <lb/> | |
| 3343 <lb/>[...]<lb/> <lb/> | |
| 3344 Fiant solida ab extremis et etiam a medijs, et inde: <lb/> | |
| 3345 propositum | |
| 3346 <lb/>[<emph style="it">tr: | |
| 3347 let there be three continued proportionals <lb/> | |
| 3348 add to each side <lb/> | |
| 3349 <lb/>[...]<lb/> <lb/> | |
| 3350 There may be made solids from the extremes and also form the means, and hence the proposition: | |
| 3351 </emph>]<lb/> | |
| 3352 </s> | |
| 3353 </p> | |
| 3354 <p xml:lang="lat"> | |
| 3355 <s xml:id="echoid-s236" xml:space="preserve"> | |
| 3356 Prop. 13. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregato <lb/> | |
| 3357 quadratorum tribus, minus cubo e prima: æquale est solido sub eadem <lb/> | |
| 3358 prima et adgregato quadratorum secundæ et tertiæ. | |
| 3359 <lb/>[<emph style="it">tr: | |
| 3360 Proposition 13. If there are three proportional lines, the product of the first and the sum of squares of all three, | |
| 3361 minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. | |
| 3362 </emph>]<lb/> | |
| 3363 </s> | |
| 3364 <lb/> | |
| 3365 <s xml:id="echoid-s237" xml:space="preserve"> | |
| 3366 Sint tres continue proportionales <lb/> | |
| 3367 <lb/>[...]<lb/> <lb/> | |
| 3368 Resoluatur Analogia et erit: <lb/> | |
| 3369 Propositum | |
| 3370 <lb/>[<emph style="it">tr: | |
| 3371 Let there be three continued proportionals <lb/> | |
| 3372 <lb/>[...]<lb/> <lb/> | |
| 3373 The ratio is resolved, and hence the proposition: | |
| 3374 </emph>]<lb/> | |
| 3375 </s> | |
| 3376 </p> | |
| 3377 <p xml:lang="lat"> | |
| 3378 <s xml:id="echoid-s238" xml:space="preserve"> | |
| 3379 Prop. 14. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregatum quadratorum <lb/> | |
| 3380 a tribus minus cubo e tertia: æquale est solido sub eadem tertia et adgregato <lb/> | |
| 3381 quadratorum primæ et secundæ. | |
| 3382 <lb/>[<emph style="it">tr: | |
| 3383 Proposition 14. If there are three proportional lines, the product of the first and the sum of squares of all three, | |
| 3384 minus the cube of the third, is equal to the product of the third and the sum of the first and second. | |
| 3385 </emph>]<lb/> | |
| 3386 </s> | |
| 3387 <lb/> | |
| 3388 <s xml:id="echoid-s239" xml:space="preserve"> | |
| 3389 Sint tres continue proportionales <lb/> | |
| 3390 <lb/>[...]<lb/> <lb/> | |
| 3391 Resoluatur Analogia et erit: <lb/> | |
| 3392 Propositum | |
| 3393 <lb/>[<emph style="it">tr: | |
| 3394 Let there be three continued proportionals <lb/> | |
| 3395 <lb/>[...]<lb/> <lb/> | |
| 3396 The ratio is resolved, and hence the proposition: | |
| 3397 </emph>]<lb/> | |
| 3398 </s> | |
| 3399 </p> | |
| 3400 <head xml:id="echoid-head95" xml:space="preserve"> | |
| 3401 Consectarium | |
| 3402 <lb/>[<emph style="it">tr: | |
| 3403 Consequence | |
| 3404 </emph>]<lb/> | |
| 3405 </head> | |
| 3406 <p xml:lang="lat"> | |
| 3407 <s xml:id="echoid-s240" xml:space="preserve"> | |
| 3408 Quia æquantur æqualibus <lb/> | |
| 3409 ex antecedente consectario. | |
| 3410 <lb/>[<emph style="it">tr: | |
| 3411 Because equals are equated to equals, by the preceding conclusion. | |
| 3412 </emph>]<lb/> | |
| 3413 </s> | |
| 3414 </p> | |
| 3415 <pb file="add_6784_f353v" o="353v" n="706"/> | |
| 3416 <pb file="add_6784_f354" o="354" n="707"/> | |
| 3417 <div xml:id="echoid-div53" type="page_commentary" level="2" n="53"> | |
| 3418 <p> | |
| 3419 <s xml:id="echoid-s241" xml:space="preserve">[<emph style="it">Note: | |
| 3420 <p> | |
| 3421 <s xml:id="echoid-s241" xml:space="preserve"> | |
| 3422 On this page Harriot investigates Propositions 10 and 11 from Viète's | |
| 3423 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 3424 </s> | |
| 3425 <lb/> | |
| 3426 <quote xml:lang="lat"> | |
| 3427 Proposition X. <lb/> | |
| 3428 Si fuerint tres lineæ rectæ proportionales: est ut prima ad tertiam, | |
| 3429 ita adgregatum quadratorum primæ & secundæ ad adgregatum quadratorum secundæ & tertiæ. | |
| 3430 </quote> | |
| 3431 <lb/> | |
| 3432 <quote> | |
| 3433 If there are three proportional lines, as the first is to the third, | |
| 3434 so is the sum of squares of the first and second to the sum of squares of the second and third. | |
| 3435 </quote> | |
| 3436 <lb/> | |
| 3437 <quote xml:lang="lat"> | |
| 3438 Proposition XI. <lb/> | |
| 3439 Si fuerint tres lineæ rectæ proportionales: est ut prima ad adgregatum primae & tertiæ, | |
| 3440 ita quadratum secundæ ad adgregatum quadratorum secundæ & tertiæ. | |
| 3441 </quote> | |
| 3442 <lb/> | |
| 3443 <quote> | |
| 3444 If there are three proportional lines, as the first is to the sum of the first and third, | |
| 3445 so is the square of the second to the sum of squares of the second and third. | |
| 3446 </quote> | |
| 3447 <lb/> | |
| 3448 <s xml:id="echoid-s242" xml:space="preserve"> | |
| 3449 There are two references to Euclid's <emph style="it">Elements</emph>, Proposition VI.20. | |
| 3450 </s> | |
| 3451 <lb/> | |
| 3452 <quote> | |
| 3453 VI.20 Similar polygons my be divided into the same number of similar triangles, | |
| 3454 each similar pair of which are proportional to the polygons; | |
| 3455 and the polygons are to each other in the duplicate ratio of their homologous sides. | |
| 3456 </quote> | |
| 3457 <lb/> | |
| 3458 <s xml:id="echoid-s243" xml:space="preserve"> | |
| 3459 The 'Consectarium' appears verbally in Viete's proposition; Harriot has reinterpreted it symbolically. | |
| 3460 </s> | |
| 3461 </p> | |
| 3462 </emph>] | |
| 3463 <lb/><lb/></s></p></div> | |
| 3464 <head xml:id="echoid-head96" xml:space="preserve"> | |
| 3465 prop. 10. Supplementi | |
| 3466 <lb/>[<emph style="it">tr: | |
| 3467 Proposition 10 from the Supplementum | |
| 3468 </emph>]<lb/> | |
| 3469 </head> | |
| 3470 <p xml:lang="lat"> | |
| 3471 <s xml:id="echoid-s245" xml:space="preserve"> | |
| 3472 Si fuerint tres lineæ rectæ proportionales: Est ut prima ad tertiam, ita adgregatum <lb/> | |
| 3473 quadratorum primæ et secundæ ad adgregatum quadratorum secundæ et tertiæ. | |
| 3474 <lb/>[<emph style="it">tr: | |
| 3475 If there are three proportional lines, as the first is to the third, | |
| 3476 so is the sum of squares of the first and second to the sum of squares of the second and third. | |
| 3477 </emph>]<lb/> | |
| 3478 </s> | |
| 3479 <lb/> | |
| 3480 <s xml:id="echoid-s246" xml:space="preserve"> | |
| 3481 sint tres proportionales <lb/> | |
| 3482 continue <lb/> | |
| 3483 consequetur <lb/> | |
| 3484 vel <lb/> | |
| 3485 Et per synæresin <lb/> | |
| 3486 Et per 20,6 Euclid <lb/> | |
| 3487 Ergo pro conclusione | |
| 3488 <lb/>[<emph style="it">tr: | |
| 3489 let there be three continued proportionals <lb/> | |
| 3490 consequently <lb/> | |
| 3491 or <lb/> | |
| 3492 And by synæresis <lb/> | |
| 3493 And by Euclid VI.20 <lb/> | |
| 3494 Therefore in conclusion | |
| 3495 </emph>]<lb/> | |
| 3496 </s> | |
| 3497 </p> | |
| 3498 <head xml:id="echoid-head97" xml:space="preserve"> | |
| 3499 prop. 11. | |
| 3500 <lb/>[<emph style="it">tr: | |
| 3501 Proposition 11 | |
| 3502 </emph>]<lb/> | |
| 3503 </head> | |
| 3504 <p xml:lang="lat"> | |
| 3505 <s xml:id="echoid-s247" xml:space="preserve"> | |
| 3506 Si fuerint tres lineæ rectæ proportionales, est ut prima ad adgregatum primae et <lb/> | |
| 3507 tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ et tertiæ. | |
| 3508 <lb/>[<emph style="it">tr: | |
| 3509 If there are three proportional lines, as the first is to the sum of the first and third, | |
| 3510 so is the square of the second to the sum of squares of the second and third. | |
| 3511 </emph>]<lb/> | |
| 3512 </s> | |
| 3513 <lb/> | |
| 3514 <s xml:id="echoid-s248" xml:space="preserve"> | |
| 3515 sint tres proportionales <lb/> | |
| 3516 per 20,6 El <lb/> | |
| 3517 Et per Synæresin <lb/> | |
| 3518 Concluditur | |
| 3519 <lb/>[<emph style="it">tr: | |
| 3520 let there be three proportionals <lb/> | |
| 3521 by Elements VI.20 <lb/> | |
| 3522 And by synæresin <lb/> | |
| 3523 It may be concluded. | |
| 3524 </emph>]<lb/> | |
| 3525 </s> | |
| 3526 </p> | |
| 3527 <head xml:id="echoid-head98" xml:space="preserve"> | |
| 3528 Consectarium | |
| 3529 <lb/>[<emph style="it">tr: | |
| 3530 Consequence | |
| 3531 </emph>]<lb/> | |
| 3532 </head> | |
| 3533 <p xml:lang="lat"> | |
| 3534 <s xml:id="echoid-s249" xml:space="preserve"> | |
| 3535 Itaque si fuerint tres lineæ rectæ proportionales, tria solida ab ijs <lb/> | |
| 3536 effecta æqualia sunt. | |
| 3537 per 10<emph style="super">am</emph> conculsionem <lb/> | |
| 3538 per 11<emph style="super">am</emph> conclu. <lb/> | |
| 3539 <lb/>[...]<lb/> <lb/> | |
| 3540 Dua prima solida sunt æqualia, quia unum factum est ab extremis analogia 10<emph style="super">am</emph> <lb/> | |
| 3541 et alterum a modijs. | |
| 3542 Tertium est factum a modijs <emph style="st">inferioris</emph> analogia 11<emph style="super">am</emph>, <lb/> | |
| 3543 cuius extremæ sunt eædem <emph style="st">superioris</emph> <emph style="super">analogia 10am</emph>, | |
| 3544 et illo æquale. | |
| 3545 <lb/>[<emph style="it">tr: | |
| 3546 Therefore if there are three lines in proportion, three solids constructed from them are equal. <lb/> | |
| 3547 by the conclusion of the 10th <lb/> | |
| 3548 by the conclusion of the 11th <lb/> | |
| 3549 <lb/>[...]<lb/> <lb/> | |
| 3550 The two first solids are equal, because one is made from the extremes of the ratio of the 10th, | |
| 3551 and the other by the method <lb/> | |
| 3552 The third is made by the method of the ratio of the 11th, whose extremes are the same as in the ratio of the 10th, | |
| 3553 and is equal to that one. | |
| 3554 </emph>]<lb/> | |
| 3555 </s> | |
| 3556 </p> | |
| 3557 <pb file="add_6784_f354v" o="354v" n="708"/> | |
| 3558 <pb file="add_6784_f355" o="355" n="709"/> | |
| 3559 <div xml:id="echoid-div54" type="page_commentary" level="2" n="54"> | |
| 3560 <p> | |
| 3561 <s xml:id="echoid-s250" xml:space="preserve">[<emph style="it">Note: | |
| 3562 <p> | |
| 3563 <s xml:id="echoid-s250" xml:space="preserve"> | |
| 3564 On this page Harriot examines a particular case arising from Proposition VII of Viète's | |
| 3565 <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. | |
| 3566 The same proposition is the subject of Chapter V of Viète's | |
| 3567 <emph style="it">Variorum responsorum libri VIII</emph>, which was also published in 1593. | |
| 3568 </s> | |
| 3569 <lb/> | |
| 3570 <quote xml:lang="lat"> | |
| 3571 Caput V <lb/> | |
| 3572 Propositio <lb/> | |
| 3573 Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. | |
| 3574 </quote> | |
| 3575 <lb/> | |
| 3576 <quote> | |
| 3577 Construct four lines in continued proportion, whose extremes are in double ratio. | |
| 3578 </quote> | |
| 3579 <lb/> | |
| 3580 <s xml:id="echoid-s251" xml:space="preserve"> | |
| 3581 The text in the <emph style="it">Variorum</emph> refers to the <emph style="it">Supplementum</emph>, | |
| 3582 indicating that the <emph style="it">Supplementum</emph> was written first. | |
| 3583 </s> | |
| 3584 </p> | |
| 3585 </emph>] | |
| 3586 <lb/><lb/></s></p></div> | |
| 3587 <head xml:id="echoid-head99" xml:space="preserve"> | |
| 3588 Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag. 4. | |
| 3589 <lb/>[<emph style="it">tr: | |
| 3590 On a corollary to Proposition 7 of the Supplement. | |
| 3591 Also Chapter 5, Variorum liber responsorum, page 4. | |
| 3592 </emph>]<lb/> | |
| 3593 </head> | |
| 3594 <p xml:lang="lat"> | |
| 3595 <s xml:id="echoid-s253" xml:space="preserve"> | |
| 3596 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> prima proportionalium, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> ea <lb/> | |
| 3597 cuius quadratum est triplum quadrati <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/> | |
| 3598 Tum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> est dupla ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>; et per assumptum <lb/> | |
| 3599 ex poristicis in alia charta demonstratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> <lb/> | |
| 3600 erit quarta proportionalis. Per propositione <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi></mstyle></math> est secunda et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> tertia. <lb/> | |
| 3601 Sed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> est æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> propter similitudine triangulorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi><mi>B</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>C</mi></mstyle></math>, et <lb/> | |
| 3602 analogiam precedentam ut sequitur. <lb/> | |
| 3603 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>.</mo><mi>E</mi><mi>A</mi><mo>.</mo><mi>E</mi><mi>G</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo></mstyle></math> Analogia precedens. <lb/> | |
| 3604 <lb/>[...]<lb/> <lb/> | |
| 3605 Et per similitudi-<lb/> | |
| 3606 num Δ<emph style="super">orum</emph>. | |
| 3607 <lb/>[...]<lb/> <lb/> | |
| 3608 Ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>E</mi><mo>.</mo><mi>F</mi><mi>B</mi><mo>.</mo><mi>A</mi><mi>C</mi><mo>.</mo></mstyle></math> continue proportionales. | |
| 3609 <lb/>[<emph style="it">tr: | |
| 3610 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be the first proportional, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> that whose square is three times the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/> | |
| 3611 Then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> is twice <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>; and by taking it from the proof demonstrated in the other sheet, | |
| 3612 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> will be the fourth proportional. By the proposition <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi></mstyle></math> is the second and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> the third. <lb/> | |
| 3613 But <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>G</mi></mstyle></math> because of similar triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>C</mi></mstyle></math>, and <lb/> | |
| 3614 the precding ratio, as follows. <lb/> | |
| 3615 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>:</mo><mi>E</mi><mi>A</mi><mo>:</mo><mi>E</mi><mi>G</mi><mo>:</mo><mi>A</mi><mi>C</mi></mstyle></math> preceding ratio. <lb/> | |
| 3616 <lb/>[...]<lb/> <lb/> | |
| 3617 And by similar triangles. <lb/> | |
| 3618 <lb/>[...]<lb/> <lb/> | |
| 3619 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>E</mi><mo>:</mo><mi>F</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>C</mi></mstyle></math> are continued proportionals. | |
| 3620 </emph>]<lb/> | |
| 3621 [<emph style="it">Note: | |
| 3622 The other sheet mentioned in this paragraph appears to be Add MS 6784, f. 356. | |
| 3623 </emph>]<lb/> | |
| 3624 </s> | |
| 3625 </p> | |
| 3626 <p xml:lang="lat"> | |
| 3627 <s xml:id="echoid-s254" xml:space="preserve"> | |
| 3628 Datis igitur extremis in ratione dupla, mediæ ita compendiosæ <lb/> | |
| 3629 inveniuntur. | |
| 3630 <lb/>[<emph style="it">tr: | |
| 3631 Therefore given the extremes in double ratio, the mean is briefly found. | |
| 3632 </emph>]<lb/> | |
| 3633 </s> | |
| 3634 <lb/> | |
| 3635 <s xml:id="echoid-s255" xml:space="preserve"> | |
| 3636 Sit maxima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> bisariam divisa in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> et intervallo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> describatur <lb/> | |
| 3637 circulus. Et sit <emph style="st">prima</emph> <emph style="super">minima</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> inscripta | |
| 3638 et producta ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/> | |
| 3639 Ducatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> ita ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math> sit æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. et acta fit linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math>. <lb/> | |
| 3640 Quatuor igitur continue proportionales ex supra demonstratis sunt. | |
| 3641 <lb/>[<emph style="it">tr: | |
| 3642 Let the maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> be cut in half at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> and with radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>C</mi></mstyle></math> there is described a circle. | |
| 3643 And let the minimum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> be inscribed and produced to the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. | |
| 3644 Construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, and let the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>B</mi></mstyle></math> be joined. <lb/> | |
| 3645 Therefore there are the four continued proportionals that were demonstrated above. | |
| 3646 </emph>]<lb/> | |
| 3647 </s> | |
| 3648 </p> | |
| 3649 <pb file="add_6784_f355v" o="355v" n="710"/> | |
| 3650 <pb file="add_6784_f356" o="356" n="711"/> | |
| 3651 <div xml:id="echoid-div55" type="page_commentary" level="2" n="55"> | |
| 3652 <p> | |
| 3653 <s xml:id="echoid-s256" xml:space="preserve">[<emph style="it">Note: | |
| 3654 <p> | |
| 3655 <s xml:id="echoid-s256" xml:space="preserve"> | |
| 3656 On this page Harriot examines a particular case arising from Proposition VII of Viète's | |
| 3657 <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. | |
| 3658 </s> | |
| 3659 </p> | |
| 3660 </emph>] | |
| 3661 <lb/><lb/></s></p></div> | |
| 3662 <head xml:id="echoid-head100" xml:space="preserve"> | |
| 3663 prop. 7. Supplementi de corrollario | |
| 3664 <lb/>[<emph style="it">tr: | |
| 3665 Proposition 7 of the Supplement, on a corollary | |
| 3666 </emph>]<lb/> | |
| 3667 </head> | |
| 3668 <p xml:lang="lat"> | |
| 3669 <s xml:id="echoid-s258" xml:space="preserve"> | |
| 3670 Sint 4<emph style="super">or</emph> proportionales <lb/> | |
| 3671 in specie. <lb/> | |
| 3672 Si quarta sit dupla ad prima, erit: <lb/> | |
| 3673 <lb/>[...]<lb/> <lb/> | |
| 3674 Ergo quatuor proportionales <lb/> | |
| 3675 quarum extremæ sunt in <lb/> | |
| 3676 ratione dupla erunt | |
| 3677 <lb/>[<emph style="it">tr: | |
| 3678 Let there be 4 proportionals in general form. <lb/> | |
| 3679 If the fourth is twice the firs, then: <lb/> | |
| 3680 <lb/>[...]<lb/> <lb/> | |
| 3681 Therefore the four proportionals whose extremes are in double ratio will be | |
| 3682 </emph>]<lb/> | |
| 3683 </s> | |
| 3684 </p> | |
| 3685 <p xml:lang="lat"> | |
| 3686 <s xml:id="echoid-s259" xml:space="preserve"> | |
| 3687 Tunc fac <lb/>[...]<lb/> et nota quadratorum differentiam. | |
| 3688 <lb/>[<emph style="it">tr: | |
| 3689 Then make [the square of the first and second and the square of the third and fourth], | |
| 3690 and note the difference of the squares.</emph>]<lb/> | |
| 3691 </s> | |
| 3692 </p> | |
| 3693 <p xml:lang="lat"> | |
| 3694 <s xml:id="echoid-s260" xml:space="preserve"> | |
| 3695 Differentia quadratorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi></mstyle></math><lb/> | |
| 3696 Hoc est triplum quadratum primæ proportionalis. | |
| 3697 <lb/>[<emph style="it">tr: | |
| 3698 The difference of the squares is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>b</mi></mstyle></math>. <lb/> | |
| 3699 This is three times the square of the first proportional. | |
| 3700 </emph>]<lb/> | |
| 3701 </s> | |
| 3702 </p> | |
| 3703 <pb file="add_6784_f356v" o="356v" n="712"/> | |
| 3704 <pb file="add_6784_f357" o="357" n="713"/> | |
| 3705 <div xml:id="echoid-div56" type="page_commentary" level="2" n="56"> | |
| 3706 <p> | |
| 3707 <s xml:id="echoid-s261" xml:space="preserve">[<emph style="it">Note: | |
| 3708 <p> | |
| 3709 <s xml:id="echoid-s261" xml:space="preserve"> | |
| 3710 This page investigates the proposition that is the subject of Chapter V of Viète's | |
| 3711 <emph style="it">Variorum responsorum libri VIII</emph>. | |
| 3712 It appears to be a continuation of Add MS 6784, f. 355. | |
| 3713 </s> | |
| 3714 <lb/> | |
| 3715 <quote xml:lang="lat"> | |
| 3716 Caput V <lb/> | |
| 3717 Propositio <lb/> | |
| 3718 Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. | |
| 3719 </quote> | |
| 3720 <lb/> | |
| 3721 <quote> | |
| 3722 Construct four lines in continued proportion, whose extremes are in double ratio. | |
| 3723 </quote> | |
| 3724 </p> | |
| 3725 </emph>] | |
| 3726 <lb/><lb/></s></p></div> | |
| 3727 <head xml:id="echoid-head101" xml:space="preserve"> | |
| 3728 In Cap. 5. Resp. lib. 8. pag. 4. | |
| 3729 <lb/>[<emph style="it">tr: | |
| 3730 Chapter 5, Variorum liber responsorum, page 4. | |
| 3731 </emph>]<lb/> | |
| 3732 </head> | |
| 3733 <pb file="add_6784_f357v" o="357v" n="714"/> | |
| 3734 <pb file="add_6784_f358" o="358" n="715"/> | |
| 3735 <div xml:id="echoid-div57" type="page_commentary" level="2" n="57"> | |
| 3736 <p> | |
| 3737 <s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note: | |
| 3738 <p> | |
| 3739 <s xml:id="echoid-s263" xml:space="preserve"> | |
| 3740 On this page Harriot examines Proposition VII from Viète's | |
| 3741 <emph style="it">Supplementum geometriæ</emph> (1593). | |
| 3742 </s> | |
| 3743 <lb/> | |
| 3744 <quote xml:lang="lat"> | |
| 3745 Propositio VII. <lb/> | |
| 3746 Data è tribus propositis lineis rectis proportionalibus prima, | |
| 3747 & ea cujus quadratum æquale fit ei quo differt quadratum compositae ex secunda & tertia | |
| 3748 Ã quadrato compositæ ex secunda & prima, invenire secundam & tertiam proprtionales. | |
| 3749 </quote> | |
| 3750 <lb/> | |
| 3751 <quote> | |
| 3752 Given the first of three proposed proportional straight lines, | |
| 3753 and another whose square is equal to the difference between the square of the sum of the second and third, | |
| 3754 and the square of the sum of the second and first, find the second and third proportionals. | |
| 3755 </quote> | |
| 3756 </p> | |
| 3757 </emph>] | |
| 3758 <lb/><lb/></s></p></div> | |
| 3759 <head xml:id="echoid-head102" xml:space="preserve"> | |
| 3760 prop. 7. Supplementi | |
| 3761 <lb/>[<emph style="it">tr: | |
| 3762 Proposition 7 of the Supplement | |
| 3763 </emph>]<lb/> | |
| 3764 </head> | |
| 3765 <p xml:lang="lat"> | |
| 3766 <s xml:id="echoid-s265" xml:space="preserve"> | |
| 3767 Data e tribus propositis lineis rectis proportionalibus prima et ea <lb/> | |
| 3768 cujus quadratum aequale fit ei quo differt quadratum compositae ex <lb/> | |
| 3769 secunda et tertia a quadrato compositæ ex secunda et prima: invenire <lb/> | |
| 3770 secundam et tertiam proprtionales. | |
| 3771 <lb/>[<emph style="it">tr: | |
| 3772 Given the first of three proposed proportional straight lines, | |
| 3773 and another whose square is equal to the difference between the square of the sum of the second and third, | |
| 3774 and the square of the sum of the second and first, find the second and third proportionals. | |
| 3775 </emph>]<lb/> | |
| 3776 </s> | |
| 3777 </p> | |
| 3778 <p xml:lang="lat"> | |
| 3779 <s xml:id="echoid-s266" xml:space="preserve"> | |
| 3780 Data prima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> <lb/> | |
| 3781 Et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> | |
| 3782 <lb/>[<emph style="it">tr: | |
| 3783 The first given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> and the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. | |
| 3784 </emph>]<lb/> | |
| 3785 </s> | |
| 3786 </p> | |
| 3787 <p xml:lang="lat"> | |
| 3788 <s xml:id="echoid-s267" xml:space="preserve"> | |
| 3789 Tum tres proportionales <lb/> | |
| 3790 erunt. | |
| 3791 <lb/>[<emph style="it">tr: | |
| 3792 Then the three proportionals will be: | |
| 3793 </emph>]<lb/> | |
| 3794 </s> | |
| 3795 </p> | |
| 3796 <pb file="add_6784_f358v" o="358v" n="716"/> | |
| 3797 <pb file="add_6784_f359" o="359" n="717"/> | |
| 3798 <head xml:id="echoid-head103" xml:space="preserve"> | |
| 3799 a) Achilles | |
| 3800 </head> | |
| 3801 <p xml:lang="lat"> | |
| 3802 <s xml:id="echoid-s268" xml:space="preserve"> | |
| 3803 Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, Achilles. <lb/> | |
| 3804 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. | |
| 3805 <lb/>[<emph style="it">tr: | |
| 3806 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be Achilles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> the tortoise. | |
| 3807 </emph>]<lb/> | |
| 3808 </s> | |
| 3809 </p> | |
| 3810 <p xml:lang="lat"> | |
| 3811 <s xml:id="echoid-s269" xml:space="preserve"> | |
| 3812 Sit ratio motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <lb/> | |
| 3813 ut: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <lb/> | |
| 3814 nempe: 10 ad 1. | |
| 3815 <lb/>[<emph style="it">tr: | |
| 3816 Let the ratio of the motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, namely, 1 to 10. | |
| 3817 </emph>]<lb/> | |
| 3818 </s> | |
| 3819 </p> | |
| 3820 <p xml:lang="lat"> | |
| 3821 <s xml:id="echoid-s270" xml:space="preserve"> | |
| 3822 Et sit distantia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. nempe 1 <foreign xml:lang="fr">mille</foreign> pases. | |
| 3823 <lb/>[<emph style="it">tr: | |
| 3824 And let the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, namely, one thousand pases. | |
| 3825 </emph>]<lb/> | |
| 3826 </s> | |
| 3827 </p> | |
| 3828 <p xml:lang="lat"> | |
| 3829 <s xml:id="echoid-s271" xml:space="preserve"> | |
| 3830 Et sit motus utriusque in eadem linea et ad easdem partes, nempe <lb/> | |
| 3831 ab <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3832 <lb/>[<emph style="it">tr: | |
| 3833 And suppose the motion of both is in the same line and in the same direction, | |
| 3834 namely, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3835 </emph>]<lb/> | |
| 3836 </s> | |
| 3837 </p> | |
| 3838 <p xml:lang="lat"> | |
| 3839 <s xml:id="echoid-s272" xml:space="preserve"> | |
| 3840 Quæritur ex datis punctum ubi Achilles comprehendet testudinem. | |
| 3841 <lb/>[<emph style="it">tr: | |
| 3842 From what is given there is sought the point where Achilles catches up with the tortoise. | |
| 3843 </emph>]<lb/> | |
| 3844 </s> | |
| 3845 </p> | |
| 3846 <p xml:lang="lat"> | |
| 3847 <s xml:id="echoid-s273" xml:space="preserve"> | |
| 3848 Quæestio solvitur exhibendo summam infinitæ progressionis decrescentis <lb/> | |
| 3849 ut sequitur: (species summa infinitæ progressionis decrescentis <lb/> | |
| 3850 ut in doctrinam de <reg norm="progressionis" type="abbr">prog</reg>: | |
| 3851 <reg norm="geometricæ" type="abbr">geom</reg>: est:) | |
| 3852 <lb/>[<emph style="it">tr: | |
| 3853 The problem is solved by producing the sum of an infinite decreasing progression as follows: | |
| 3854 (the case of the sum of an infinite decreasing progression as in the teaching of geometric porgressions is:) | |
| 3855 </emph>]<lb/> | |
| 3856 </s> | |
| 3857 </p> | |
| 3858 <p xml:lang="lat"> | |
| 3859 <s xml:id="echoid-s274" xml:space="preserve"> | |
| 3860 Alia progressiones. | |
| 3861 <lb/>[<emph style="it">tr: | |
| 3862 Other progressions. | |
| 3863 </emph>]<lb/> | |
| 3864 </s> | |
| 3865 </p> | |
| 3866 <p xml:lang="lat"> | |
| 3867 <s xml:id="echoid-s275" xml:space="preserve"> | |
| 3868 (ut Archimedes de <lb/> | |
| 3869 quad: parab: pr: 23) | |
| 3870 <lb/>[<emph style="it">tr: | |
| 3871 (as Archimedes in the quadrature of the parabola, proposition 23) | |
| 3872 </emph>]<lb/> | |
| 3873 </s> | |
| 3874 </p> | |
| 3875 <pb file="add_6784_f359v" o="359v" n="718"/> | |
| 3876 <pb file="add_6784_f360" o="360" n="719"/> | |
| 3877 <head xml:id="echoid-head104" xml:space="preserve"> | |
| 3878 b) Achilles | |
| 3879 </head> | |
| 3880 <p xml:lang="lat"> | |
| 3881 <s xml:id="echoid-s276" xml:space="preserve"> | |
| 3882 Sit (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>), Achilles. <lb/> | |
| 3883 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. | |
| 3884 <lb/>[<emph style="it">tr: | |
| 3885 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> be Achilles, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> the tortoise. | |
| 3886 </emph>]<lb/> | |
| 3887 </s> | |
| 3888 </p> | |
| 3889 <p xml:lang="lat"> | |
| 3890 <s xml:id="echoid-s277" xml:space="preserve"> | |
| 3891 Sit velocitas motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, ad velocitatem motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, <lb/> | |
| 3892 ut: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. | |
| 3893 <lb/>[<emph style="it">tr: | |
| 3894 Let the speed of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> to the speed of motion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. | |
| 3895 </emph>]<lb/> | |
| 3896 </s> | |
| 3897 </p> | |
| 3898 <p xml:lang="lat"> | |
| 3899 <s xml:id="echoid-s278" xml:space="preserve"> | |
| 3900 Sit distantia inter (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) et (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>). <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. | |
| 3901 <lb/>[<emph style="it">tr: | |
| 3902 Let the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. | |
| 3903 </emph>]<lb/> | |
| 3904 </s> | |
| 3905 </p> | |
| 3906 <p xml:lang="lat"> | |
| 3907 <s xml:id="echoid-s279" xml:space="preserve"> | |
| 3908 Et sit motus utriusque in eadem linea et ad easdem partes, nempe ab (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>), et (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>) <lb/> | |
| 3909 versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3910 <lb/>[<emph style="it">tr: | |
| 3911 And let the mtion of both be in the same line and the same direction, | |
| 3912 namely from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3913 </emph>]<lb/> | |
| 3914 </s> | |
| 3915 </p> | |
| 3916 <p xml:lang="lat"> | |
| 3917 <s xml:id="echoid-s280" xml:space="preserve"> | |
| 3918 Quæritur ex datis punctum ubi Achilles comprehendet testudinem. | |
| 3919 <lb/>[<emph style="it">tr: | |
| 3920 From what is given there is sought the point where Achilles catches up with the tortoise. | |
| 3921 </emph>]<lb/> | |
| 3922 </s> | |
| 3923 </p> | |
| 3924 <p xml:lang="lat"> | |
| 3925 <s xml:id="echoid-s281" xml:space="preserve"> | |
| 3926 Ponatur illud punctum esse <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>w</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. | |
| 3927 <lb/>[<emph style="it">tr: | |
| 3928 Suppose this point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>, and let the distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi><mi>w</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. | |
| 3929 </emph>]<lb/> | |
| 3930 </s> | |
| 3931 <lb/> | |
| 3932 <s xml:id="echoid-s282" xml:space="preserve"> | |
| 3933 Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. Et inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3934 <lb/>[<emph style="it">tr: | |
| 3935 Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> is found; and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. | |
| 3936 </emph>]<lb/> | |
| 3937 </s> | |
| 3938 </p> | |
| 3939 <p xml:lang="lat"> | |
| 3940 <s xml:id="echoid-s283" xml:space="preserve"> | |
| 3941 In numeris sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 10. <lb/> | |
| 3942 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 2. <lb/> | |
| 3943 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. 2. mill | |
| 3944 <lb/>[<emph style="it">tr: | |
| 3945 In numbers let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>=</mo><mn>1</mn><mn>0</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>=</mo><mn>2</mn></mstyle></math> miles | |
| 3946 </emph>]<lb/> | |
| 3947 </s> | |
| 3948 </p> | |
| 3949 <p xml:lang="lat"> | |
| 3950 <s xml:id="echoid-s284" xml:space="preserve"> | |
| 3951 Aliter. | |
| 3952 <lb/>[<emph style="it">tr: | |
| 3953 Another way. | |
| 3954 </emph>]<lb/> | |
| 3955 </s> | |
| 3956 </p> | |
| 3957 <p xml:lang="lat"> | |
| 3958 <s xml:id="echoid-s285" xml:space="preserve"> | |
| 3959 Aliter 2<emph style="super">o</emph>. <lb/> | |
| 3960 Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math> et sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. | |
| 3961 <lb/>[<emph style="it">tr: | |
| 3962 A second way. <lb/> | |
| 3963 There is sought <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>w</mi></mstyle></math>, and suppose it is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi></mstyle></math>. | |
| 3964 </emph>]<lb/> | |
| 3965 </s> | |
| 3966 </p> | |
| 3967 <p xml:lang="lat"> | |
| 3968 <s xml:id="echoid-s286" xml:space="preserve"> | |
| 3969 Exemplum de duabus [¿]numeribus[?]. | |
| 3970 <lb/>[<emph style="it">tr: | |
| 3971 An example from two numbers. | |
| 3972 </emph>]<lb/> | |
| 3973 </s> | |
| 3974 </p> | |
| 3975 <pb file="add_6784_f360v" o="360v" n="720"/> | |
| 3976 <pb file="add_6784_f361" o="361" n="721"/> | |
| 3977 <div xml:id="echoid-div58" type="page_commentary" level="2" n="58"> | |
| 3978 <p> | |
| 3979 <s xml:id="echoid-s287" xml:space="preserve">[<emph style="it">Note: | |
| 3980 <p> | |
| 3981 <s xml:id="echoid-s287" xml:space="preserve"> | |
| 3982 On this folio, Harriot derives the sum of a finite geometric progression, | |
| 3983 using Euclid V.12 and its numerical counterpoart, Euclid VII.12. | |
| 3984 He then extends his result to an infinite (decreasing) progression, | |
| 3985 by arguing that the final term must be infnitely small, that is, nothing. <lb/> | |
| 3986 Euclid V.12: If any number of magnitudes be proportional, | |
| 3987 as one of the antecedents is to one of the consequents, | |
| 3988 so will all the antecedents be to all the consequents. <lb/> | |
| 3989 Euclid VII.12: If there be as many numbers as we please in proportion, then, | |
| 3990 as one of the antecedents is to one of the consequents, | |
| 3991 so are all the antecedents to all the consequents. | |
| 3992 </s> | |
| 3993 </p> | |
| 3994 </emph>] | |
| 3995 <lb/><lb/></s></p></div> | |
| 3996 <head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat"> | |
| 3997 1.) De progressione geometrica. | |
| 3998 <lb/>[<emph style="it">tr: | |
| 3999 On geometric porgressions | |
| 4000 </emph>]<lb/> | |
| 4001 </head> | |
| 4002 <p xml:lang="lat"> | |
| 4003 <s xml:id="echoid-s289" xml:space="preserve"> | |
| 4004 Theorema. | |
| 4005 <lb/>[<emph style="it">tr: | |
| 4006 Theorem | |
| 4007 </emph>]<lb/> | |
| 4008 </s> | |
| 4009 <lb/> | |
| 4010 <s xml:id="echoid-s290" xml:space="preserve"> | |
| 4011 el. 5. pr: 12. | |
| 4012 <lb/>[<emph style="it">tr: | |
| 4013 <emph style="it">Elements</emph>, Book 5, Proposition 12. | |
| 4014 </emph>]<lb/> | |
| 4015 </s> | |
| 4016 <lb/> | |
| 4017 <s xml:id="echoid-s291" xml:space="preserve"> | |
| 4018 el. 7. pr. 12. | |
| 4019 <lb/>[<emph style="it">tr: | |
| 4020 <emph style="it">Elements</emph>, Book 7, Proposition 12. | |
| 4021 </emph>]<lb/> | |
| 4022 </s> | |
| 4023 <lb/> | |
| 4024 <s xml:id="echoid-s292" xml:space="preserve"> | |
| 4025 Si sint magnitudines quotcunque proportionales, Quemadmodum <lb/> | |
| 4026 se habuerit una antecedentium ad unam consequentium: Ita <lb/> | |
| 4027 se habebunt omnes antecedentes ad omnes consequentes. | |
| 4028 <lb/>[<emph style="it">tr: | |
| 4029 If any number of magnitudes are proportional, | |
| 4030 then just as as one antecedent is to its consequent, | |
| 4031 so will the sum of the antecedents be to the sum of the consequents. | |
| 4032 </emph>]<lb/> | |
| 4033 </s> | |
| 4034 </p> | |
| 4035 <p xml:lang="lat"> | |
| 4036 <s xml:id="echoid-s293" xml:space="preserve"> | |
| 4037 Sint continue proportionales. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. | |
| 4038 <lb/>[<emph style="it">tr: | |
| 4039 Let the continued proportionals be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. | |
| 4040 </emph>]<lb/> | |
| 4041 </s> | |
| 4042 </p> | |
| 4043 <p xml:lang="lat"> | |
| 4044 <s xml:id="echoid-s294" xml:space="preserve"> | |
| 4045 In notis universalibus sit. | |
| 4046 <lb/>[<emph style="it">tr: | |
| 4047 In general notation we have | |
| 4048 </emph>]<lb/> | |
| 4049 </s> | |
| 4050 <lb/> | |
| 4051 <s xml:id="echoid-s295" xml:space="preserve"> | |
| 4052 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primum. <emph style="st">p</emph>. primus terminus rationis. | |
| 4053 <lb/>[<emph style="it">tr: | |
| 4054 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. first term. <emph style="st">p</emph>. first term of the ratio. | |
| 4055 </emph>]<lb/> | |
| 4056 </s> | |
| 4057 <lb/> | |
| 4058 <s xml:id="echoid-s296" xml:space="preserve"> | |
| 4059 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. secunda. <emph style="st">s</emph>. secundus. | |
| 4060 <lb/>[<emph style="it">tr: | |
| 4061 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. second. <emph style="st">s</emph>. second. | |
| 4062 </emph>]<lb/> | |
| 4063 </s> | |
| 4064 <lb/> | |
| 4065 <s xml:id="echoid-s297" xml:space="preserve"> | |
| 4066 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. ultima. | |
| 4067 <lb/>[<emph style="it">tr: | |
| 4068 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. last. | |
| 4069 </emph>]<lb/> | |
| 4070 </s> | |
| 4071 <lb/> | |
| 4072 <s xml:id="echoid-s298" xml:space="preserve"> | |
| 4073 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes. | |
| 4074 <lb/>[<emph style="it">tr: | |
| 4075 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. all. | |
| 4076 </emph>]<lb/> | |
| 4077 </s> | |
| 4078 </p> | |
| 4079 <p xml:lang="lat"> | |
| 4080 <s xml:id="echoid-s299" xml:space="preserve"> | |
| 4081 Ergo; si, <emph style="st">p</emph> > <emph style="st">s</emph> ut in progressi decrescente: | |
| 4082 <lb/>[<emph style="it">tr: | |
| 4083 Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in a decreasing progression: | |
| 4084 </emph>]<lb/> | |
| 4085 </s> | |
| 4086 </p> | |
| 4087 <p xml:lang="lat"> | |
| 4088 <s xml:id="echoid-s300" xml:space="preserve"> | |
| 4089 Ergo; si, <emph style="st">p</emph> < <emph style="st">s</emph> ut in progressi crescente: | |
| 4090 <lb/>[<emph style="it">tr: | |
| 4091 Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in an increasing progression: | |
| 4092 </emph>]<lb/> | |
| 4093 </s> | |
| 4094 </p> | |
| 4095 <p xml:lang="lat"> | |
| 4096 <s xml:id="echoid-s301" xml:space="preserve"> | |
| 4097 De <emph style="st">infinitis</emph> progressionibus <lb/> | |
| 4098 decrescentibus in infinitum: | |
| 4099 <lb/>[<emph style="it">tr: | |
| 4100 For a progression descreasing indefinitely: | |
| 4101 </emph>]<lb/> | |
| 4102 </s> | |
| 4103 <lb/> | |
| 4104 <s xml:id="echoid-s302" xml:space="preserve"> | |
| 4105 Cum progressio decrescit et <lb/> | |
| 4106 numerus terminorum sit infinitus; <lb/> | |
| 4107 ultimus terminus est infinite <lb/> | |
| 4108 minimus hoc est nullius quantiatis. | |
| 4109 <lb/>[<emph style="it">tr: | |
| 4110 Since the progression decreases and the number of terms is infinite, the last term is infnitely small, | |
| 4111 that is, of no quantity. | |
| 4112 </emph>]<lb/> | |
| 4113 </s> | |
| 4114 <lb/> | |
| 4115 <s xml:id="echoid-s303" xml:space="preserve"> | |
| 4116 Ideo: | |
| 4117 <lb/>[<emph style="it">tr: | |
| 4118 Therefore. | |
| 4119 </emph>]<lb/> | |
| 4120 </s> | |
| 4121 </p> | |
| 4122 <pb file="add_6784_f361v" o="361v" n="722"/> | |
| 4123 <pb file="add_6784_f362" o="362" n="723"/> | |
| 4124 <div xml:id="echoid-div59" type="page_commentary" level="2" n="59"> | |
| 4125 <p> | |
| 4126 <s xml:id="echoid-s304" xml:space="preserve">[<emph style="it">Note: | |
| 4127 <p> | |
| 4128 <s xml:id="echoid-s304" xml:space="preserve"> | |
| 4129 In the preceding folio, f. 361, Harriot derived a formula for the sum of a finite geometric progression | |
| 4130 based on Euclid V.12. Here he gives an alternative derivation based on Euclid IX. 35. <lb/> | |
| 4131 Euclid IX. 35: If as many numbers as we please be in continued proportion, | |
| 4132 and there be subtracted from the second and the last numbers equal to the first, | |
| 4133 then as the excess of the second is to the first, | |
| 4134 so will the excess of the last be to all those before it. | |
| 4135 </s> | |
| 4136 </p> | |
| 4137 </emph>] | |
| 4138 <lb/><lb/></s></p></div> | |
| 4139 <head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat"> | |
| 4140 2.) De progressione geometrica. | |
| 4141 <lb/>[<emph style="it">tr: | |
| 4142 On geometric porgressions | |
| 4143 </emph>]<lb/> | |
| 4144 </head> | |
| 4145 <p xml:lang="lat"> | |
| 4146 <s xml:id="echoid-s306" xml:space="preserve"> | |
| 4147 Theoremata. | |
| 4148 <lb/>[<emph style="it">tr: | |
| 4149 Theorem | |
| 4150 </emph>]<lb/> | |
| 4151 </s> | |
| 4152 <lb/> | |
| 4153 <s xml:id="echoid-s307" xml:space="preserve"> | |
| 4154 el. 9. pr: 35. | |
| 4155 <lb/>[<emph style="it">tr: | |
| 4156 <emph style="it">Elements</emph> Book IX, Proposition 35 | |
| 4157 </emph>]<lb/> | |
| 4158 </s> | |
| 4159 <lb/> | |
| 4160 <s xml:id="echoid-s308" xml:space="preserve"> | |
| 4161 Si sint quotlibet numeri deinceps proportionales, detrahuntur autem <lb/> | |
| 4162 de secundo et ultimo æquales ipsi primo: erit quemadmodum <lb/> | |
| 4163 secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum <lb/> | |
| 4164 antecedunt. | |
| 4165 <lb/>[<emph style="it">tr: | |
| 4166 If there are as many numbers as we please in proportion, | |
| 4167 and the first is subtracted from the second and the last, | |
| 4168 then just as the difference of the second is to the first, | |
| 4169 so is the difference of the last to all before the last. | |
| 4170 </emph>]<lb/> | |
| 4171 </s> | |
| 4172 </p> | |
| 4173 <p xml:lang="lat"> | |
| 4174 <s xml:id="echoid-s309" xml:space="preserve"> | |
| 4175 Progressio crescens: | |
| 4176 <lb/>[<emph style="it">tr: | |
| 4177 An increasing progression: | |
| 4178 </emph>]<lb/> | |
| 4179 </s> | |
| 4180 </p> | |
| 4181 <p xml:lang="lat"> | |
| 4182 <s xml:id="echoid-s310" xml:space="preserve"> | |
| 4183 In notis universalibus: sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, primus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, secundus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, ultimus: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, omnes. | |
| 4184 <lb/>[<emph style="it">tr: | |
| 4185 In general notation, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> be the first term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the second term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> the last term; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> the sum. | |
| 4186 </emph>]<lb/> | |
| 4187 </s> | |
| 4188 </p> | |
| 4189 <p xml:lang="lat"> | |
| 4190 <s xml:id="echoid-s311" xml:space="preserve"> | |
| 4191 Progressio decrescens: | |
| 4192 <lb/>[<emph style="it">tr: | |
| 4193 A decreasing progression: | |
| 4194 </emph>]<lb/> | |
| 4195 </s> | |
| 4196 </p> | |
| 4197 <p xml:lang="lat"> | |
| 4198 <s xml:id="echoid-s312" xml:space="preserve"> | |
| 4199 In notis universalis erit: | |
| 4200 <lb/>[<emph style="it">tr: | |
| 4201 In general notation we have: | |
| 4202 </emph>]<lb/> | |
| 4203 </s> | |
| 4204 </p> | |
| 4205 <p xml:lang="lat"> | |
| 4206 <s xml:id="echoid-s313" xml:space="preserve"> | |
| 4207 Vel: in notis magis universalis. <lb/> | |
| 4208 sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, primus terminus rationis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>, secundus. <lb/> | |
| 4209 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>, maxumus terminus progressionis <lb/> | |
| 4210 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, minimus. Tum: | |
| 4211 <lb/>[<emph style="it">tr: | |
| 4212 Or, in more general notation, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> be the first term of the ratio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> the second, | |
| 4213 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> the greatest term of the progression, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> the least. Then: | |
| 4214 </emph>]<lb/> | |
| 4215 </s> | |
| 4216 </p> | |
| 4217 <pb file="add_6784_f362v" o="362v" n="724"/> | |
| 4218 <pb file="add_6784_f363" o="363" n="725"/> | |
| 4219 <div xml:id="echoid-div60" type="page_commentary" level="2" n="60"> | |
| 4220 <p> | |
| 4221 <s xml:id="echoid-s314" xml:space="preserve">[<emph style="it">Note: | |
| 4222 <p> | |
| 4223 <s xml:id="echoid-s314" xml:space="preserve"> | |
| 4224 In this folio Harriot repeats statements that are to be found in Viete, | |
| 4225 <emph style="it">Variorum responsorum</emph>, Chapter XVII (1646, 397–398). <lb/> | |
| 4226 Harriot's letters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>, <emph style="st">M</emph>, <emph style="st">m</emph> | |
| 4227 correspond to Viete's <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>X</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>. <lb/> | |
| 4228 Harriot's final comments refer to the final sentence of Viete's penultimate paragraph (1646, 398): <lb/> | |
| 4229 <foreign xml:lang="lat"> | |
| 4230 Et ut differentia terminorum rationis ad terminorum rationis majorem, | |
| 4231 ita maxima ad compositam ex ombnibus plus cremento. | |
| 4232 </foreign> <lb/> | |
| 4233 <lb/>[<emph style="it">tr: | |
| 4234 As the difference in the terms of the ratio is to the greater term of the ratio, | |
| 4235 so is the the greatest term of the progression to the sum plus an increment. | |
| 4236 </emph>]<lb/> | |
| 4237 </s> | |
| 4238 </p> | |
| 4239 </emph>] | |
| 4240 <lb/><lb/></s></p></div> | |
| 4241 <head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat"> | |
| 4242 3.) De progressione geometrica. (ut Vieta in var: resp.) | |
| 4243 <lb/>[<emph style="it">tr: | |
| 4244 On geometric progressions (as Viete in <emph style="it">Variorum responsorum</emph>) | |
| 4245 </emph>]<lb/> | |
| 4246 </head> | |
| 4247 <p xml:lang="lat"> | |
| 4248 <s xml:id="echoid-s316" xml:space="preserve"> | |
| 4249 Crescente. | |
| 4250 <lb/>[<emph style="it">tr: | |
| 4251 Increasing. | |
| 4252 </emph>]<lb/> | |
| 4253 </s> | |
| 4254 <s xml:id="echoid-s317" xml:space="preserve"> | |
| 4255 decrescente. | |
| 4256 <lb/>[<emph style="it">tr: | |
| 4257 Decreasing. | |
| 4258 </emph>]<lb/> | |
| 4259 </s> | |
| 4260 </p> | |
| 4261 <p xml:lang="lat"> | |
| 4262 <s xml:id="echoid-s318" xml:space="preserve"> | |
| 4263 <emph style="st">m</emph>. minor terminus rationis. | |
| 4264 <lb/>[<emph style="it">tr: | |
| 4265 Let <emph style="st">m</emph> be the lesser terms of the ratio. | |
| 4266 </emph>]<lb/> | |
| 4267 </s> | |
| 4268 <lb/> | |
| 4269 <s xml:id="echoid-s319" xml:space="preserve"> | |
| 4270 <emph style="st">M</emph>. Maior terminus rationis. | |
| 4271 <lb/>[<emph style="it">tr: | |
| 4272 Let <emph style="st">M</emph> be the greater terms of the ratio. | |
| 4273 </emph>]<lb/> | |
| 4274 </s> | |
| 4275 <lb/> | |
| 4276 <s xml:id="echoid-s320" xml:space="preserve"> | |
| 4277 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>. maximus terminus progressionis. | |
| 4278 <lb/>[<emph style="it">tr: | |
| 4279 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the greatest term of the progression. | |
| 4280 </emph>]<lb/> | |
| 4281 </s> | |
| 4282 <lb/> | |
| 4283 <s xml:id="echoid-s321" xml:space="preserve"> | |
| 4284 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. minimus terminus progressionis. | |
| 4285 <lb/>[<emph style="it">tr: | |
| 4286 Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the least term of the progression. | |
| 4287 </emph>]<lb/> | |
| 4288 </s> | |
| 4289 <lb/> | |
| 4290 <s xml:id="echoid-s322" xml:space="preserve"> | |
| 4291 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes, id est summa omnium | |
| 4292 <lb/>[<emph style="it">tr: | |
| 4293 <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> is all, that is the sum of all. | |
| 4294 </emph>]<lb/> | |
| 4295 </s> | |
| 4296 </p> | |
| 4297 <p xml:lang="lat"> | |
| 4298 <s xml:id="echoid-s323" xml:space="preserve"> | |
| 4299 ita Vieta post δεδόμενα <lb/> | |
| 4300 in respons: pag. 29. | |
| 4301 <lb/>[<emph style="it">tr: | |
| 4302 thus Viete after δεδόμενα in | |
| 4303 <emph style="it">Variorum Responsorum</emph> page 29. | |
| 4304 </emph>]<lb/> | |
| 4305 </s> | |
| 4306 </p> | |
| 4307 <p xml:lang="lat"> | |
| 4308 <s xml:id="echoid-s324" xml:space="preserve"> | |
| 4309 apud Vieta dicitur crementum. | |
| 4310 <lb/>[<emph style="it">tr: | |
| 4311 in Viete this is said to be the increment. | |
| 4312 </emph>]<lb/> | |
| 4313 </s> | |
| 4314 </p> | |
| 4315 <pb file="add_6784_f363v" o="363v" n="726"/> | |
| 4316 <pb file="add_6784_f364" o="364" n="727"/> | |
| 4317 <div xml:id="echoid-div61" type="page_commentary" level="2" n="61"> | |
| 4318 <p> | |
| 4319 <s xml:id="echoid-s325" xml:space="preserve">[<emph style="it">Note: | |
| 4320 <p> | |
| 4321 <s xml:id="echoid-s325" xml:space="preserve"> | |
| 4322 On this folio an expression that looks like <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mo>=</mo><mi>s</mi></mstyle></math> is to be read as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mi>p</mi><mo>-</mo><mi>s</mi><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>. | |
| 4323 </s> | |
| 4324 </p> | |
| 4325 </emph>] | |
| 4326 <lb/><lb/></s></p></div> | |
| 4327 <head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat"> | |
| 4328 De progressionibus. <lb/> | |
| 4329 finitis & infinitis. | |
| 4330 <lb/>[<emph style="it">tr: | |
| 4331 On finite and infinite progressions | |
| 4332 </emph>]<lb/> | |
| 4333 </head> | |
| 4334 <p xml:lang="lat"> | |
| 4335 <s xml:id="echoid-s327" xml:space="preserve"> | |
| 4336 linea infinite <emph style="super">longa</emph>quælibet = æqualis alicui, plano <lb/> | |
| 4337 solido. <lb/> | |
| 4338 longo-solido. <lb/> | |
| 4339 plano-solido. <lb/> | |
| 4340 solido-solido. &c. | |
| 4341 <lb/>[<emph style="it">tr: | |
| 4342 An infinite line of any length is equal to some plane, or solid, or solid-length, or solid-plane, or solid-solid, etc. | |
| 4343 </emph>]<lb/> | |
| 4344 </s> | |
| 4345 </p> | |
| 4346 <p xml:lang="lat"> | |
| 4347 <s xml:id="echoid-s328" xml:space="preserve"> | |
| 4348 linea infinite brevis quælibet = æqualis alicui, puncto. <lb/> | |
| 4349 linea. <lb/> | |
| 4350 puncto-plano. <lb/> | |
| 4351 puncto-solido. &c. <lb/> | |
| 4352 <lb/>[<emph style="it">tr: | |
| 4353 Any infinitely short line is equal to some line-point, or plane-point, or solid-point, etc. | |
| 4354 </emph>]<lb/> | |
| 4355 </s> | |
| 4356 <lb/> | |
| 4357 <s xml:id="echoid-s329" xml:space="preserve"> | |
| 4358 Quælibet punctum terminat progressionem. | |
| 4359 <lb/>[<emph style="it">tr: | |
| 4360 Whatever point terminates the progression. | |
| 4361 </emph>]<lb/> | |
| 4362 </s> | |
| 4363 </p> | |
| 4364 <p xml:lang="lat"> | |
| 4365 <s xml:id="echoid-s330" xml:space="preserve"> | |
| 4366 infinite numero puncta = lineæ <lb/> | |
| 4367 plano. <lb/> | |
| 4368 solido. &c. | |
| 4369 <lb/>[<emph style="it">tr: | |
| 4370 an infinite number of points equal a line, or plane, or solid, etc. | |
| 4371 </emph>]<lb/> | |
| 4372 </s> | |
| 4373 <lb/> | |
| 4374 <s xml:id="echoid-s331" xml:space="preserve"> | |
| 4375 linea signata <lb/> | |
| 4376 terminat <lb/> | |
| 4377 progressionem. <lb/> | |
| 4378 ita planum signatum. | |
| 4379 <lb/>[<emph style="it">tr: | |
| 4380 a designated line terminates the progression; similarly a designated plane, | |
| 4381 </emph>]<lb/> | |
| 4382 </s> | |
| 4383 </p> | |
| 4384 <p xml:lang="lat"> | |
| 4385 <s xml:id="echoid-s332" xml:space="preserve"> | |
| 4386 hæc & alia huius generis <lb/> | |
| 4387 consideranda. | |
| 4388 <lb/>[<emph style="it">tr: | |
| 4389 these and others of this kind may be considered. | |
| 4390 </emph>]<lb/> | |
| 4391 </s> | |
| 4392 </p> | |
| 4393 <pb file="add_6784_f364v" o="364v" n="728"/> | |
| 4394 <pb file="add_6784_f365" o="365" n="729"/> | |
| 4395 <pb file="add_6784_f365v" o="365v" n="730"/> | |
| 4396 <pb file="add_6784_f366" o="366" n="731"/> | |
| 4397 <pb file="add_6784_f366v" o="366v" n="732"/> | |
| 4398 <pb file="add_6784_f367" o="367" n="733"/> | |
| 4399 <pb file="add_6784_f367v" o="367v" n="734"/> | |
| 4400 <pb file="add_6784_f368" o="368" n="735"/> | |
| 4401 <pb file="add_6784_f368v" o="368v" n="736"/> | |
| 4402 <pb file="add_6784_f369" o="369" n="737"/> | |
| 4403 <div xml:id="echoid-div62" type="page_commentary" level="2" n="62"> | |
| 4404 <p> | |
| 4405 <s xml:id="echoid-s333" xml:space="preserve">[<emph style="it">Note: | |
| 4406 <p> | |
| 4407 <s xml:id="echoid-s333" xml:space="preserve"> | |
| 4408 This page contains a symbolic version of Euclid Book II, Proposition 11: <lb/> | |
| 4409 II.11. To cut a given straight line so that the rectangle contained by the whole | |
| 4410 and one of the segments equals the square on the remaining segment. | |
| 4411 </s> | |
| 4412 </p> | |
| 4413 </emph>] | |
| 4414 <lb/><lb/></s></p></div> | |
| 4415 <head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat"> | |
| 4416 propositiones 2<emph style="super">i</emph> Euclidis | |
| 4417 <lb/>[<emph style="it">tr: | |
| 4418 Propositions from the second book of Euclid | |
| 4419 </emph>]<lb/> | |
| 4420 </head> | |
| 4421 <pb file="add_6784_f369v" o="369v" n="738"/> | |
| 4422 <pb file="add_6784_f370" o="370" n="739"/> | |
| 4423 <pb file="add_6784_f370v" o="370v" n="740"/> | |
| 4424 <pb file="add_6784_f371" o="371" n="741"/> | |
| 4425 <pb file="add_6784_f371v" o="371v" n="742"/> | |
| 4426 <pb file="add_6784_f372" o="372" n="743"/> | |
| 4427 <pb file="add_6784_f372v" o="372v" n="744"/> | |
| 4428 <pb file="add_6784_f373" o="373" n="745"/> | |
| 4429 <pb file="add_6784_f373v" o="373v" n="746"/> | |
| 4430 <pb file="add_6784_f374" o="374" n="747"/> | |
| 4431 <pb file="add_6784_f374v" o="374v" n="748"/> | |
| 4432 <pb file="add_6784_f375" o="375" n="749"/> | |
| 4433 <pb file="add_6784_f375v" o="375v" n="750"/> | |
| 4434 <pb file="add_6784_f376" o="376" n="751"/> | |
| 4435 <pb file="add_6784_f376v" o="376v" n="752"/> | |
| 4436 <pb file="add_6784_f377" o="377" n="753"/> | |
| 4437 <pb file="add_6784_f377v" o="377v" n="754"/> | |
| 4438 <pb file="add_6784_f378" o="378" n="755"/> | |
| 4439 <pb file="add_6784_f378v" o="378v" n="756"/> | |
| 4440 <pb file="add_6784_f379" o="379" n="757"/> | |
| 4441 <pb file="add_6784_f379v" o="379v" n="758"/> | |
| 4442 <pb file="add_6784_f380" o="380" n="759"/> | |
| 4443 <pb file="add_6784_f380v" o="380v" n="760"/> | |
| 4444 <pb file="add_6784_f381" o="381" n="761"/> | |
| 4445 <pb file="add_6784_f381v" o="381v" n="762"/> | |
| 4446 <pb file="add_6784_f382" o="382" n="763"/> | |
| 4447 <pb file="add_6784_f382v" o="382v" n="764"/> | |
| 4448 <pb file="add_6784_f383" o="383" n="765"/> | |
| 4449 <pb file="add_6784_f383v" o="383v" n="766"/> | |
| 4450 <pb file="add_6784_f384" o="384" n="767"/> | |
| 4451 <pb file="add_6784_f384v" o="384v" n="768"/> | |
| 4452 <pb file="add_6784_f385" o="385" n="769"/> | |
| 4453 <pb file="add_6784_f385v" o="385v" n="770"/> | |
| 4454 <pb file="add_6784_f386" o="386" n="771"/> | |
| 4455 <pb file="add_6784_f386v" o="386v" n="772"/> | |
| 4456 <pb file="add_6784_f387" o="387" n="773"/> | |
| 4457 <pb file="add_6784_f387v" o="387v" n="774"/> | |
| 4458 <pb file="add_6784_f388" o="388" n="775"/> | |
| 4459 <pb file="add_6784_f388v" o="388v" n="776"/> | |
| 4460 <pb file="add_6784_f389" o="389" n="777"/> | |
| 4461 <pb file="add_6784_f389v" o="389v" n="778"/> | |
| 4462 <pb file="add_6784_f390" o="390" n="779"/> | |
| 4463 <pb file="add_6784_f390v" o="390v" n="780"/> | |
| 4464 <pb file="add_6784_f391" o="391" n="781"/> | |
| 4465 <pb file="add_6784_f391v" o="391v" n="782"/> | |
| 4466 <pb file="add_6784_f392" o="392" n="783"/> | |
| 4467 <pb file="add_6784_f392v" o="392v" n="784"/> | |
| 4468 <pb file="add_6784_f393" o="393" n="785"/> | |
| 4469 <pb file="add_6784_f393v" o="393v" n="786"/> | |
| 4470 <pb file="add_6784_f394" o="394" n="787"/> | |
| 4471 <pb file="add_6784_f394v" o="394v" n="788"/> | |
| 4472 <pb file="add_6784_f395" o="395" n="789"/> | |
| 4473 <pb file="add_6784_f395v" o="395v" n="790"/> | |
| 4474 <pb file="add_6784_f396" o="396" n="791"/> | |
| 4475 <pb file="add_6784_f396v" o="396v" n="792"/> | |
| 4476 <pb file="add_6784_f397" o="397" n="793"/> | |
| 4477 <pb file="add_6784_f397v" o="397v" n="794"/> | |
| 4478 <head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat"> | |
| 4479 1.) De reductione æquationum | |
| 4480 <lb/>[<emph style="it">tr: | |
| 4481 On the reduction of equations | |
| 4482 </emph>]<lb/> | |
| 4483 </head> | |
| 4484 <pb file="add_6784_f398" o="398" n="795"/> | |
| 4485 <head xml:id="echoid-head111" xml:space="preserve"> | |
| 4486 3.) | |
| 4487 </head> | |
| 4488 <pb file="add_6784_f398v" o="398v" n="796"/> | |
| 4489 <pb file="add_6784_f399" o="399" n="797"/> | |
| 4490 <pb file="add_6784_f399v" o="399v" n="798"/> | |
| 4491 <pb file="add_6784_f400" o="400" n="799"/> | |
| 4492 <pb file="add_6784_f400v" o="400v" n="800"/> | |
| 4493 <head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat"> | |
| 4494 1)B) De reductione æquationum | |
| 4495 <lb/>[<emph style="it">tr: | |
| 4496 On the reduction of equations | |
| 4497 </emph>]<lb/> | |
| 4498 </head> | |
| 4499 <pb file="add_6784_f401" o="401" n="801"/> | |
| 4500 <pb file="add_6784_f401v" o="401v" n="802"/> | |
| 4501 <div xml:id="echoid-div63" type="page_commentary" level="2" n="63"> | |
| 4502 <p> | |
| 4503 <s xml:id="echoid-s335" xml:space="preserve">[<emph style="it">Note: | |
| 4504 <p> | |
| 4505 <s xml:id="echoid-s335" xml:space="preserve"> | |
| 4506 Here Harriot solves the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mo>=</mo><mn>6</mn><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> (in modern notation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mn>5</mn><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></math>) | |
| 4507 for the roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>+</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>-</mo><msqrt><mrow><mo>-</mo><mn>1</mn></mrow></msqrt></mstyle></math>. He then checks by multiplication | |
| 4508 that these valus do indeed satisfy the equation. | |
| 4509 </s> | |
| 4510 </p> | |
| 4511 </emph>] | |
| 4512 <lb/><lb/></s></p></div> | |
| 4513 <pb file="add_6784_f402" o="402" n="803"/> | |
| 4514 <div xml:id="echoid-div64" type="page_commentary" level="2" n="64"> | |
| 4515 <p> | |
| 4516 <s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note: | |
| 4517 <p> | |
| 4518 <s xml:id="echoid-s337" xml:space="preserve"> | |
| 4519 Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>0</mn><mo>+</mo><mn>4</mn><mo maxsize="1">)</mo></mstyle></math> up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mn>2</mn><mn>0</mn><mo>+</mo><mn>4</mn><mrow><msup><mo maxsize="1">)</mo><mn>5</mn></msup></mrow></mstyle></math> following the pattern laid out in Add MS 6782, f. 276. <lb/> | |
| 4520 A calculation below each box gives the sum of the figures contained in it. | |
| 4521 </s> | |
| 4522 </p> | |
| 4523 </emph>] | |
| 4524 <lb/><lb/></s></p></div> | |
| 4525 <pb file="add_6784_f402v" o="402v" n="804"/> | |
| 4526 <div xml:id="echoid-div65" type="page_commentary" level="2" n="65"> | |
| 4527 <p> | |
| 4528 <s xml:id="echoid-s339" xml:space="preserve">[<emph style="it">Note: | |
| 4529 <p> | |
| 4530 <s xml:id="echoid-s339" xml:space="preserve"> | |
| 4531 The calculations from the previous page (Add MS 6784, f. 402) are checked by root extractions | |
| 4532 </s> | |
| 4533 </p> | |
| 4534 </emph>] | |
| 4535 <lb/><lb/></s></p></div> | |
| 4536 <p> | |
| 4537 <s xml:id="echoid-s341" xml:space="preserve"> | |
| 4538 The extraction <lb/> | |
| 4539 of the roots. | |
| 4540 </s> | |
| 4541 </p> | |
| 4542 <pb file="add_6784_f403" o="403" n="805"/> | |
| 4543 <pb file="add_6784_f403v" o="403v" n="806"/> | |
| 4544 <pb file="add_6784_f404" o="404" n="807"/> | |
| 4545 <div xml:id="echoid-div66" type="page_commentary" level="2" n="66"> | |
| 4546 <p> | |
| 4547 <s xml:id="echoid-s342" xml:space="preserve">[<emph style="it">Note: | |
| 4548 <p> | |
| 4549 <s xml:id="echoid-s342" xml:space="preserve"> | |
| 4550 Third, fourth, and fifth powers of (20 + 4). <lb/> | |
| 4551 The binomial coefficients 3, 3 and 4, 6, 4 and 5, 10, 10, 5, | |
| 4552 appear amongst the numbers in the rightmost column. | |
| 4553 </s> | |
| 4554 </p> | |
| 4555 </emph>] | |
| 4556 <lb/><lb/></s></p></div> | |
| 4557 <pb file="add_6784_f404v" o="404v" n="808"/> | |
| 4558 <p> | |
| 4559 <s xml:id="echoid-s344" xml:space="preserve"> | |
| 4560 The doctrine of Algebraycall nombers is but <lb/> | |
| 4561 the doctrined of such continuall proportionalles of <lb/> | |
| 4562 which a unite is the first. | |
| 4563 </s> | |
| 4564 </p> | |
| 4565 <p> | |
| 4566 <s xml:id="echoid-s345" xml:space="preserve"> | |
| 4567 A unite being the first of continuall proportionalles; the second is <lb/> | |
| 4568 called a roote: because the third wilbe always a square: & the fourth <lb/> | |
| 4569 <emph style="st">third</emph> a cube, as Euclide demonstrateth. | |
| 4570 </s> | |
| 4571 <s xml:id="echoid-s346" xml:space="preserve"> | |
| 4572 The names of the other proportionalles <lb/> | |
| 4573 following are all compounded of squares, or cubes or both according <lb/> | |
| 4574 to Diophantus & others which follow him. | |
| 4575 </s> | |
| 4576 <s xml:id="echoid-s347" xml:space="preserve"> | |
| 4577 Some or other of the most parte of the later <lb/> | |
| 4578 writers gave the name of surdsolidus, of which the first or simple sursolid <lb/> | |
| 4579 is the sixt proportionall. &c. | |
| 4580 </s> | |
| 4581 </p> | |
| 4582 <p> | |
| 4583 <s xml:id="echoid-s348" xml:space="preserve"> | |
| 4584 Any nomber may be <emph style="super">any</emph> terme proportinall in a continuall progression <lb/> | |
| 4585 from a unite. | |
| 4586 </s> | |
| 4587 <s xml:id="echoid-s349" xml:space="preserve"> | |
| 4588 If the nomber terme be the second, the third is gotten by <lb/> | |
| 4589 multiplying the nomber into him self. | |
| 4590 </s> | |
| 4591 <s xml:id="echoid-s350" xml:space="preserve"> | |
| 4592 & the fourth by multiplying the <lb/> | |
| 4593 third by the second & so forth. | |
| 4594 </s> | |
| 4595 <s xml:id="echoid-s351" xml:space="preserve"> | |
| 4596 as also <emph style="super">by</emph> the doctrine of progression <lb/> | |
| 4597 any terme that is found another may be gotten compendiously <lb/> | |
| 4598 without continuall multiplications. | |
| 4599 </s> | |
| 4600 </p> | |
| 4601 <p> | |
| 4602 <s xml:id="echoid-s352" xml:space="preserve"> | |
| 4603 If a nomber that is known & designed to be the third, fourth, <lb/> | |
| 4604 or fifth or any other proportinall of another denomination: the <lb/> | |
| 4605 doctrine to find the second is that which is called the extraction <lb/> | |
| 4606 of the roote, which is taught in these papers. | |
| 4607 </s> | |
| 4608 </p> | |
| 4609 <p> | |
| 4610 <s xml:id="echoid-s353" xml:space="preserve"> | |
| 4611 The second proportionall is also called the first dignity, & the third the <lb/> | |
| 4612 second dignity, & the fourth the third dignity &c. | |
| 4613 </s> | |
| 4614 </p> | |
| 4615 <p> | |
| 4616 <s xml:id="echoid-s354" xml:space="preserve"> | |
| 4617 The third is also called the first power; the 4th the second power &c. | |
| 4618 </s> | |
| 4619 </p> | |
| 4620 <p> | |
| 4621 <s xml:id="echoid-s355" xml:space="preserve"> | |
| 4622 The first proportionall <lb/> | |
| 4623 is a unite. | |
| 4624 </s> | |
| 4625 </p> | |
| 4626 <p> | |
| 4627 <s xml:id="echoid-s356" xml:space="preserve"> | |
| 4628 The first dignity is <lb/> | |
| 4629 the second proportionall, <lb/> | |
| 4630 called a roote. | |
| 4631 </s> | |
| 4632 </p> | |
| 4633 <p> | |
| 4634 <s xml:id="echoid-s357" xml:space="preserve"> | |
| 4635 The first power is the <lb/> | |
| 4636 third proportionall <lb/> | |
| 4637 <emph style="st">called a square</emph> <lb/> | |
| 4638 or second Dignity <lb/> | |
| 4639 called a square. | |
| 4640 </s> | |
| 4641 </p> | |
| 4642 <p> | |
| 4643 <s xml:id="echoid-s358" xml:space="preserve"> | |
| 4644 The first solid is the <lb/> | |
| 4645 fourth proprtionall: <lb/> | |
| 4646 The third dignity: & <lb/> | |
| 4647 The second power, <lb/> | |
| 4648 called a cube. | |
| 4649 </s> | |
| 4650 </p> | |
| 4651 <p> | |
| 4652 <s xml:id="echoid-s359" xml:space="preserve"> | |
| 4653 The pythagoreans <lb/> | |
| 4654 did call 4 the first solid <lb/> | |
| 4655 as Boethius relateth. | |
| 4656 </s> | |
| 4657 <lb/> | |
| 4658 <s xml:id="echoid-s360" xml:space="preserve"> | |
| 4659 The nomber serveth to be, because pyramides are prime solids <lb/> | |
| 4660 & 4 amongst nombers is the first pyramide. | |
| 4661 </s> | |
| 4662 </p> | |
| 4663 <pb file="add_6784_f405" o="405" n="809"/> | |
| 4664 <pb file="add_6784_f405v" o="405v" n="810"/> | |
| 4665 <pb file="add_6784_f406" o="406" n="811"/> | |
| 4666 <pb file="add_6784_f406v" o="406v" n="812"/> | |
| 4667 <pb file="add_6784_f407" o="407" n="813"/> | |
| 4668 <div xml:id="echoid-div67" type="page_commentary" level="2" n="67"> | |
| 4669 <p> | |
| 4670 <s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note: | |
| 4671 <p> | |
| 4672 <s xml:id="echoid-s361" xml:space="preserve"> | |
| 4673 Here Harriot demonstrates that multiplication by 9 increases the number of digits by one | |
| 4674 as far as the 21st power but not at the 22nd power. | |
| 4675 Thus the number of digits alone is no guide to the size of the root. | |
| 4676 </s> | |
| 4677 </p> | |
| 4678 </emph>] | |
| 4679 <lb/><lb/></s></p></div> | |
| 4680 <p> | |
| 4681 <s xml:id="echoid-s363" xml:space="preserve"> | |
| 4682 An induction to prove that <lb/> | |
| 4683 to pricke the second figure for <lb/> | |
| 4684 the extraction of square rootes <lb/> | |
| 4685 & the third for cubes & 4th <lb/> | |
| 4686 for biquadrates etc. according <lb/> | |
| 4687 to the nomber of figures that <lb/> | |
| 4688 the greatest figure 9 doth <lb/> | |
| 4689 produce is no rule. | |
| 4690 </s> | |
| 4691 <s xml:id="echoid-s364" xml:space="preserve"> | |
| 4692 for we <lb/> | |
| 4693 may see how it breaketh in <lb/> | |
| 4694 the 22th <emph style="st">proportionall</emph> dignity & so <lb/> | |
| 4695 forwarde. | |
| 4696 </s> | |
| 4697 <s xml:id="echoid-s365" xml:space="preserve"> | |
| 4698 but the true case <lb/> | |
| 4699 of such pricking appeareth <lb/> | |
| 4700 out <emph style="super">of</emph> the speciosa genesis which <lb/> | |
| 4701 is in an other paper arranged. | |
| 4702 </s> | |
| 4703 </p> | |
| 4704 <pb file="add_6784_f407v" o="407v" n="814"/> | |
| 4705 <pb file="add_6784_f408" o="408" n="815"/> | |
| 4706 <div xml:id="echoid-div68" type="page_commentary" level="2" n="68"> | |
| 4707 <p> | |
| 4708 <s xml:id="echoid-s366" xml:space="preserve">[<emph style="it">Note: | |
| 4709 <p> | |
| 4710 <s xml:id="echoid-s366" xml:space="preserve"> | |
| 4711 Calculation of powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>a</mi></mstyle></math> to show how the digits of a three-digit number are distributed in the sum. | |
| 4712 </s> | |
| 4713 </p> | |
| 4714 </emph>] | |
| 4715 <lb/><lb/></s></p></div> | |
| 4716 <p> | |
| 4717 <s xml:id="echoid-s368" xml:space="preserve"> | |
| 4718 If the roote to be extracted be three figures <lb/> | |
| 4719 the two first as one may here see are to be had <lb/> | |
| 4720 according to the generall rule, the next is <lb/> | |
| 4721 also to be gotten really after the same manner <lb/> | |
| 4722 that <emph style="super">is</emph> supposing the two first to be as one, & that <lb/> | |
| 4723 which foloweth, the second; although in appearance <lb/> | |
| 4724 & expressing by wordes it seems otherwise. | |
| 4725 </s> | |
| 4726 </p> | |
| 4727 <pb file="add_6784_f408v" o="408v" n="816"/> | |
| 4728 <pb file="add_6784_f409" o="409" n="817"/> | |
| 4729 <div xml:id="echoid-div69" type="page_commentary" level="2" n="69"> | |
| 4730 <p> | |
| 4731 <s xml:id="echoid-s369" xml:space="preserve">[<emph style="it">Note: | |
| 4732 <p> | |
| 4733 <s xml:id="echoid-s369" xml:space="preserve"> | |
| 4734 Powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>–</mo><mi>c</mi><mo maxsize="1">)</mo></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mo maxsize="1">)</mo></mstyle></math>. | |
| 4735 </s> | |
| 4736 </p> | |
| 4737 </emph>] | |
| 4738 <lb/><lb/></s></p></div> | |
| 4739 <pb file="add_6784_f409v" o="409v" n="818"/> | |
| 4740 <pb file="add_6784_f410" o="410" n="819"/> | |
| 4741 <pb file="add_6784_f410v" o="410v" n="820"/> | |
| 4742 <pb file="add_6784_f411" o="411" n="821"/> | |
| 4743 <div xml:id="echoid-div70" type="page_commentary" level="2" n="70"> | |
| 4744 <p> | |
| 4745 <s xml:id="echoid-s371" xml:space="preserve">[<emph style="it">Note: | |
| 4746 <p> | |
| 4747 <s xml:id="echoid-s371" xml:space="preserve"> | |
| 4748 Here and on folio Add MS 6784, f. 412, Harriot shows that the product of two or three unequal parts | |
| 4749 is always less than the product of the same number of equal parts. | |
| 4750 </s> | |
| 4751 </p> | |
| 4752 </emph>] | |
| 4753 <lb/><lb/></s></p></div> | |
| 4754 <head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat"> | |
| 4755 1<emph style="super">o</emph>. de bisectione. | |
| 4756 <lb/>[<emph style="it">tr: | |
| 4757 1. on bisection | |
| 4758 </emph>]<lb/> | |
| 4759 </head> | |
| 4760 <p xml:lang="lat"> | |
| 4761 <s xml:id="echoid-s373" xml:space="preserve"> | |
| 4762 Sit: tota linea. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math>. <lb/> | |
| 4763 vel duæ æquales partes. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>b</mi></mstyle></math>.<lb/> | |
| 4764 magnitudo facta ab illis <lb/> | |
| 4765 erit quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. | |
| 4766 <lb/>[<emph style="it">tr: | |
| 4767 Let the total line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math> <lb/> | |
| 4768 or two equal parts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>b</mi></mstyle></math>, <lb/> | |
| 4769 the size of their product will be the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. | |
| 4770 </emph>]<lb/> | |
| 4771 </s> | |
| 4772 <lb/> | |
| 4773 <s xml:id="echoid-s374" xml:space="preserve"> | |
| 4774 Sint inæquales partes. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/> | |
| 4775 et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math> | |
| 4776 <lb/>[<emph style="it">tr: | |
| 4777 Let there be unequal parts <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>. | |
| 4778 </emph>]<lb/> | |
| 4779 </s> | |
| 4780 <lb/> | |
| 4781 <s xml:id="echoid-s375" xml:space="preserve"> | |
| 4782 magnitudo facta: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>b</mi></mstyle></math>. | |
| 4783 <lb/>[<emph style="it">tr: | |
| 4784 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>c</mi><mo><</mo><mi>b</mi><mi>b</mi></mstyle></math>. | |
| 4785 </emph>]<lb/> | |
| 4786 </s> | |
| 4787 </p> | |
| 4788 <p xml:lang="lat"> | |
| 4789 <s xml:id="echoid-s376" xml:space="preserve"> | |
| 4790 Si linea dividatur utcunque in tot <lb/> | |
| 4791 partes inæquales, quot æquales: <lb/> | |
| 4792 Magnitudo facta ab inæquali-<lb/> | |
| 4793 bus, minor est illa quæ facta <lb/> | |
| 4794 ab æqualibus. | |
| 4795 <lb/>[<emph style="it">tr: | |
| 4796 If a line is divided in any way into as many unequal parts as equal parts, | |
| 4797 the size of the product of the unequal parts is less than the product of the equal parts. | |
| 4798 </emph>]<lb/> | |
| 4799 </s> | |
| 4800 </p> | |
| 4801 <p xml:lang="lat"> | |
| 4802 <s xml:id="echoid-s377" xml:space="preserve"> | |
| 4803 vel: | |
| 4804 <lb/>[<emph style="it">tr: | |
| 4805 or: | |
| 4806 </emph>]<lb/> | |
| 4807 </s> | |
| 4808 <lb/> | |
| 4809 <s xml:id="echoid-s378" xml:space="preserve"> | |
| 4810 Si aggregatum linearum inæqualium æqueretur <lb/> | |
| 4811 aggregato tot æqualium: Magnitudo facta &c. | |
| 4812 <lb/>[<emph style="it">tr: | |
| 4813 If the sum of the unnequal lines is equal to the sum of as many equals, the size of the product etc. | |
| 4814 </emph>]<lb/> | |
| 4815 </s> | |
| 4816 </p> | |
| 4817 <p xml:lang="lat"> | |
| 4818 <s xml:id="echoid-s379" xml:space="preserve"> | |
| 4819 etiam: | |
| 4820 <lb/>[<emph style="it">tr: | |
| 4821 also: | |
| 4822 </emph>]<lb/> | |
| 4823 </s> | |
| 4824 <lb/> | |
| 4825 <s xml:id="echoid-s380" xml:space="preserve"> | |
| 4826 plana facta ab inæqualibus <lb/> | |
| 4827 minora sunt quaduratis <lb/> | |
| 4828 facta ab æqualibus. | |
| 4829 <lb/>[<emph style="it">tr: | |
| 4830 planes made from unequals are less than squares made from equals. | |
| 4831 </emph>]<lb/> | |
| 4832 </s> | |
| 4833 </p> | |
| 4834 <head xml:id="echoid-head114" xml:space="preserve" xml:lang="lat"> | |
| 4835 2<emph style="it">o</emph>. De sectione in tres partes. | |
| 4836 <lb/>[<emph style="it">tr: | |
| 4837 2. On sectioning into three parts. | |
| 4838 </emph>]<lb/> | |
| 4839 </head> | |
| 4840 <p xml:lang="lat"> | |
| 4841 <s xml:id="echoid-s381" xml:space="preserve"> | |
| 4842 Casus primus | |
| 4843 <lb/>[<emph style="it">tr: | |
| 4844 First case. | |
| 4845 </emph>]<lb/> | |
| 4846 </s> | |
| 4847 <lb/> | |
| 4848 <s xml:id="echoid-s382" xml:space="preserve"> | |
| 4849 Sint tres inæquales partes. | |
| 4850 <lb/>[<emph style="it">tr: | |
| 4851 Let there be three unequalparts. | |
| 4852 </emph>]<lb/> | |
| 4853 </s> | |
| 4854 <lb/> | |
| 4855 <s xml:id="echoid-s383" xml:space="preserve"> | |
| 4856 magnitudo facta: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math> | |
| 4857 <lb/>[<emph style="it">tr: | |
| 4858 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>. | |
| 4859 </emph>]<lb/> | |
| 4860 </s> | |
| 4861 </p> | |
| 4862 <p xml:lang="lat"> | |
| 4863 <s xml:id="echoid-s384" xml:space="preserve"> | |
| 4864 Tres æquales partes. | |
| 4865 <lb/>[<emph style="it">tr: | |
| 4866 Three equal parts. | |
| 4867 </emph>]<lb/> | |
| 4868 </s> | |
| 4869 <lb/> | |
| 4870 <s xml:id="echoid-s385" xml:space="preserve"> | |
| 4871 magnitudo facta <lb/> | |
| 4872 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math> | |
| 4873 <lb/>[<emph style="it">tr: | |
| 4874 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mi>b</mi><mi>c</mi><mi>c</mi></mstyle></math>. | |
| 4875 </emph>]<lb/> | |
| 4876 </s> | |
| 4877 </p> | |
| 4878 <p xml:lang="lat"> | |
| 4879 <s xml:id="echoid-s386" xml:space="preserve"> | |
| 4880 Casus 2<emph style="super">a</emph>. | |
| 4881 <lb/>[<emph style="it">tr: | |
| 4882 Case 2. | |
| 4883 </emph>]<lb/> | |
| 4884 </s> | |
| 4885 <lb/> | |
| 4886 <s xml:id="echoid-s387" xml:space="preserve"> | |
| 4887 Sint tres inæquales partes. | |
| 4888 <lb/>[<emph style="it">tr: | |
| 4889 Let there be three unequal parts. | |
| 4890 </emph>]<lb/> | |
| 4891 </s> | |
| 4892 <lb/> | |
| 4893 <s xml:id="echoid-s388" xml:space="preserve"> | |
| 4894 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4895 <lb/>[<emph style="it">tr: | |
| 4896 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4897 </emph>]<lb/> | |
| 4898 </s> | |
| 4899 </p> | |
| 4900 <p xml:lang="lat"> | |
| 4901 <s xml:id="echoid-s389" xml:space="preserve"> | |
| 4902 Tres æquales partes. | |
| 4903 <lb/>[<emph style="it">tr: | |
| 4904 Three equal parts. | |
| 4905 </emph>]<lb/> | |
| 4906 </s> | |
| 4907 <lb/> | |
| 4908 <s xml:id="echoid-s390" xml:space="preserve"> | |
| 4909 magnitudo facta <lb/> | |
| 4910 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4911 <lb/>[<emph style="it">tr: | |
| 4912 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4913 </emph>]<lb/> | |
| 4914 </s> | |
| 4915 </p> | |
| 4916 <pb file="add_6784_f411v" o="411v" n="822"/> | |
| 4917 <div xml:id="echoid-div71" type="page_commentary" level="2" n="71"> | |
| 4918 <p> | |
| 4919 <s xml:id="echoid-s391" xml:space="preserve">[<emph style="it">Note: | |
| 4920 <p> | |
| 4921 <s xml:id="echoid-s391" xml:space="preserve"> | |
| 4922 Note the combinations of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> (greater than), <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>l</mi></mstyle></math> (less than), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math> (equals), | |
| 4923 and of the symbols <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo><</mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>></mo></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo></mstyle></math> in the lower part of the page. | |
| 4924 </s> | |
| 4925 </p> | |
| 4926 </emph>] | |
| 4927 <lb/><lb/></s></p></div> | |
| 4928 <pb file="add_6784_f412" o="412" n="823"/> | |
| 4929 <div xml:id="echoid-div72" type="page_commentary" level="2" n="72"> | |
| 4930 <p> | |
| 4931 <s xml:id="echoid-s393" xml:space="preserve">[<emph style="it">Note: | |
| 4932 <p> | |
| 4933 <s xml:id="echoid-s393" xml:space="preserve"> | |
| 4934 The continuation of Add MS 6784, f. 411. | |
| 4935 </s> | |
| 4936 </p> | |
| 4937 </emph>] | |
| 4938 <lb/><lb/></s></p></div> | |
| 4939 <p xml:lang="lat"> | |
| 4940 <s xml:id="echoid-s395" xml:space="preserve"> | |
| 4941 Casus 3<emph style="super">a</emph>. | |
| 4942 <lb/>[<emph style="it">tr: | |
| 4943 Case 3. | |
| 4944 </emph>]<lb/> | |
| 4945 </s> | |
| 4946 <lb/> | |
| 4947 <s xml:id="echoid-s396" xml:space="preserve"> | |
| 4948 Sint tres inæquales partes. | |
| 4949 <lb/>[<emph style="it">tr: | |
| 4950 Let there be three unequal parts. | |
| 4951 </emph>]<lb/> | |
| 4952 </s> | |
| 4953 <lb/> | |
| 4954 <s xml:id="echoid-s397" xml:space="preserve"> | |
| 4955 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4956 <lb/>[<emph style="it">tr: | |
| 4957 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4958 </emph>]<lb/> | |
| 4959 </s> | |
| 4960 </p> | |
| 4961 <p xml:lang="lat"> | |
| 4962 <s xml:id="echoid-s398" xml:space="preserve"> | |
| 4963 Tres æquales partes. | |
| 4964 <lb/>[<emph style="it">tr: | |
| 4965 Three equal parts. | |
| 4966 </emph>]<lb/> | |
| 4967 </s> | |
| 4968 <lb/> | |
| 4969 <s xml:id="echoid-s399" xml:space="preserve"> | |
| 4970 magnitudo facta <lb/> | |
| 4971 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4972 <lb/>[<emph style="it">tr: | |
| 4973 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mi>c</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>c</mi></mstyle></math>. | |
| 4974 </emph>]<lb/> | |
| 4975 </s> | |
| 4976 </p> | |
| 4977 <p xml:lang="lat"> | |
| 4978 <s xml:id="echoid-s400" xml:space="preserve"> | |
| 4979 Casus 4<emph style="super">a</emph>. | |
| 4980 <lb/>[<emph style="it">tr: | |
| 4981 Case 4. | |
| 4982 </emph>]<lb/> | |
| 4983 </s> | |
| 4984 <lb/> | |
| 4985 <s xml:id="echoid-s401" xml:space="preserve"> | |
| 4986 Sint tres inæquales partes. | |
| 4987 <lb/>[<emph style="it">tr: | |
| 4988 Let there be three unequal parts. | |
| 4989 </emph>]<lb/> | |
| 4990 </s> | |
| 4991 <lb/> | |
| 4992 <s xml:id="echoid-s402" xml:space="preserve"> | |
| 4993 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. | |
| 4994 <lb/>[<emph style="it">tr: | |
| 4995 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. | |
| 4996 </emph>]<lb/> | |
| 4997 </s> | |
| 4998 </p> | |
| 4999 <p xml:lang="lat"> | |
| 5000 <s xml:id="echoid-s403" xml:space="preserve"> | |
| 5001 Tres æquales partes. | |
| 5002 <lb/>[<emph style="it">tr: | |
| 5003 Three equal parts. | |
| 5004 </emph>]<lb/> | |
| 5005 </s> | |
| 5006 <lb/> | |
| 5007 <s xml:id="echoid-s404" xml:space="preserve"> | |
| 5008 magnitudo facta <lb/> | |
| 5009 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. | |
| 5010 <lb/>[<emph style="it">tr: | |
| 5011 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>d</mi><mi>c</mi></mstyle></math>. | |
| 5012 </emph>]<lb/> | |
| 5013 </s> | |
| 5014 </p> | |
| 5015 <p xml:lang="lat"> | |
| 5016 <s xml:id="echoid-s405" xml:space="preserve"> | |
| 5017 Casus 5<emph style="super">a</emph>. <lb/> | |
| 5018 et ultimus. | |
| 5019 <lb/>[<emph style="it">tr: | |
| 5020 Case 5, and last. | |
| 5021 </emph>]<lb/> | |
| 5022 </s> | |
| 5023 <lb/> | |
| 5024 <s xml:id="echoid-s406" xml:space="preserve"> | |
| 5025 Sint tres inæquales partes. | |
| 5026 <lb/>[<emph style="it">tr: | |
| 5027 Let there be three unequal parts. | |
| 5028 </emph>]<lb/> | |
| 5029 </s> | |
| 5030 <lb/> | |
| 5031 <s xml:id="echoid-s407" xml:space="preserve"> | |
| 5032 magnitudo facta. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. | |
| 5033 <lb/>[<emph style="it">tr: | |
| 5034 the size of the product is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. | |
| 5035 </emph>]<lb/> | |
| 5036 </s> | |
| 5037 </p> | |
| 5038 <p xml:lang="lat"> | |
| 5039 <s xml:id="echoid-s408" xml:space="preserve"> | |
| 5040 Tres æquales partes. | |
| 5041 <lb/>[<emph style="it">tr: | |
| 5042 Three equal parts. | |
| 5043 </emph>]<lb/> | |
| 5044 </s> | |
| 5045 <lb/> | |
| 5046 <s xml:id="echoid-s409" xml:space="preserve"> | |
| 5047 magnitudo facta <lb/> | |
| 5048 quæ cubus. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mo>,</mo><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. | |
| 5049 <lb/>[<emph style="it">tr: | |
| 5050 the size of the product which is a cube is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>+</mo><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>-</mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>></mo><mi>b</mi><mi>b</mi><mi>b</mi><mo>-</mo><mn>3</mn><mi>b</mi><mi>b</mi><mi>d</mi><mo>-</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. | |
| 5051 </emph>]<lb/> | |
| 5052 </s> | |
| 5053 </p> | |
| 5054 <p xml:lang="lat"> | |
| 5055 <s xml:id="echoid-s410" xml:space="preserve"> | |
| 5056 nam: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mo>,</mo><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mo>,</mo><mi>b</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi><mo>.</mo></mstyle></math> | |
| 5057 <lb/>[<emph style="it">tr: | |
| 5058 for: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>b</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>c</mi><mo>></mo><mi>d</mi><mi>d</mi><mi>d</mi><mo>+</mo><mn>9</mn><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math>. | |
| 5059 </emph>]<lb/> | |
| 5060 </s> | |
| 5061 </p> | |
| 5062 <pb file="add_6784_f412v" o="412v" n="824"/> | |
| 5063 <pb file="add_6784_f413" o="413" n="825"/> | |
| 5064 <pb file="add_6784_f413v" o="413v" n="826"/> | |
| 5065 <pb file="add_6784_f414" o="414" n="827"/> | |
| 5066 <div xml:id="echoid-div73" type="page_commentary" level="2" n="73"> | |
| 5067 <p> | |
| 5068 <s xml:id="echoid-s411" xml:space="preserve">[<emph style="it">Note: | |
| 5069 <p> | |
| 5070 <s xml:id="echoid-s411" xml:space="preserve"> | |
| 5071 Combinations of small numbers; see also Add MS 6784, f. 424. | |
| 5072 </s> | |
| 5073 </p> | |
| 5074 </emph>] | |
| 5075 <lb/><lb/></s></p></div> | |
| 5076 <pb file="add_6784_f414v" o="414v" n="828"/> | |
| 5077 <pb file="add_6784_f415" o="415" n="829"/> | |
| 5078 <div xml:id="echoid-div74" type="page_commentary" level="2" n="74"> | |
| 5079 <p> | |
| 5080 <s xml:id="echoid-s413" xml:space="preserve">[<emph style="it">Note: | |
| 5081 <p> | |
| 5082 <s xml:id="echoid-s413" xml:space="preserve"> | |
| 5083 This page summarizes in shorthand some rules that are written out in full in Harriot's treatise on cubic equations, | |
| 5084 on Add MS 6782, f. 186. <lb/> | |
| 5085 The abbreviations 'co:l' and 'co:pl' stand for 'longitudinal coefficient' and 'plane coefficient' respectively. | |
| 5086 In an equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>c</mi><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>f</mi></mstyle></math>, the longitudinal coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> | |
| 5087 and the plane coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. | |
| 5088 Below the diagram Harriot has set out the different conditions under which such an equation can have three real roots, | |
| 5089 not necessarily distinct. The same sets of roots are also listed in Add MS 6783, f. 281. <lb/> | |
| 5090 The relevant equations are worked in full in sheets marked C, D, E, F, G | |
| 5091 (Add MS 6782, f. 315, f. 315v, f. 317, f. 318, f. 319), and also in Add MS 6783, f. 185. <lb/> | |
| 5092 </s> | |
| 5093 </p> | |
| 5094 </emph>] | |
| 5095 <lb/><lb/></s></p></div> | |
| 5096 <pb file="add_6784_f415v" o="415v" n="830"/> | |
| 5097 <pb file="add_6784_f416" o="416" n="831"/> | |
| 5098 <head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat"> | |
| 5099 Ad generationes sequentium specierum æquationum | |
| 5100 <lb/>[<emph style="it">tr: | |
| 5101 On the generation of the following types of equation. | |
| 5102 </emph>]<lb/> | |
| 5103 </head> | |
| 5104 <p xml:lang="lat"> | |
| 5105 <s xml:id="echoid-s415" xml:space="preserve"> | |
| 5106 Æquatio <emph style="st">substantiva</emph> <lb/> | |
| 5107 parabolica. | |
| 5108 <lb/>[<emph style="it">tr: | |
| 5109 Parabolic equation | |
| 5110 </emph>]<lb/> | |
| 5111 </s> | |
| 5112 </p> | |
| 5113 <p xml:lang="lat"> | |
| 5114 <s xml:id="echoid-s416" xml:space="preserve"> | |
| 5115 Æquatio <emph style="st">adiectiva</emph> <emph style="super">hyperbolica</emph> <lb/> | |
| 5116 <emph style="st">sive additiva</emph>. | |
| 5117 Hyperbolic equation | |
| 5118 </s> | |
| 5119 </p> | |
| 5120 <p xml:lang="lat"> | |
| 5121 <s xml:id="echoid-s417" xml:space="preserve"> | |
| 5122 Æquatio <emph style="st">ablativa</emph> <emph style="super">elliptica</emph> <lb/> | |
| 5123 sive Bombellica. | |
| 5124 <lb/>[<emph style="it">tr: | |
| 5125 Elliptic, or Bombelli's, equation | |
| 5126 </emph>]<lb/> | |
| 5127 </s> | |
| 5128 </p> | |
| 5129 <p xml:lang="lat"> | |
| 5130 <s xml:id="echoid-s418" xml:space="preserve"> | |
| 5131 Ergo æquatio <emph style="st">nullitatis</emph> <emph style="st">prima</emph> <lb/> | |
| 5132 <emph style="st">sive [???]</emph> <lb/> | |
| 5133 <emph style="st">sive</emph> primitiva. | |
| 5134 <lb/>[<emph style="it">tr: | |
| 5135 Therefore the equation is primitive. | |
| 5136 </emph>]<lb/> | |
| 5137 </s> | |
| 5138 <lb/> | |
| 5139 <s xml:id="echoid-s419" xml:space="preserve"> | |
| 5140 Ergo verum quod proponebatur. | |
| 5141 <lb/>[<emph style="it">tr: | |
| 5142 Therefore what was proposed is true. | |
| 5143 </emph>]<lb/> | |
| 5144 </s> | |
| 5145 </p> | |
| 5146 <p xml:lang="lat"> | |
| 5147 <s xml:id="echoid-s420" xml:space="preserve"> | |
| 5148 Ad resolutiones sequentium specierum æquationum | |
| 5149 <lb/>[<emph style="it">tr: | |
| 5150 On solving the following types of equation | |
| 5151 </emph>]<lb/> | |
| 5152 </s> | |
| 5153 <lb/> | |
| 5154 <s xml:id="echoid-s421" xml:space="preserve"> | |
| 5155 æquatio parabolica. | |
| 5156 <lb/>[<emph style="it">tr: | |
| 5157 parabolic equation | |
| 5158 </emph>]<lb/> | |
| 5159 </s> | |
| 5160 <lb/> | |
| 5161 <s xml:id="echoid-s422" xml:space="preserve"> | |
| 5162 æquatio hyperbolica. | |
| 5163 <lb/>[<emph style="it">tr: | |
| 5164 hyperbolic equation | |
| 5165 </emph>]<lb/> | |
| 5166 </s> | |
| 5167 <lb/> | |
| 5168 <s xml:id="echoid-s423" xml:space="preserve"> | |
| 5169 æquatio elliptica. | |
| 5170 <lb/>[<emph style="it">tr: | |
| 5171 elliptic equation | |
| 5172 </emph>]<lb/> | |
| 5173 </s> | |
| 5174 </p> | |
| 5175 <p xml:lang="lat"> | |
| 5176 <s xml:id="echoid-s424" xml:space="preserve"> | |
| 5177 rinus. | |
| 5178 </s> | |
| 5179 <lb/> | |
| 5180 <s xml:id="echoid-s425" xml:space="preserve"> | |
| 5181 prærinus. | |
| 5182 </s> | |
| 5183 <lb/> | |
| 5184 <s xml:id="echoid-s426" xml:space="preserve"> | |
| 5185 prinus. | |
| 5186 </s> | |
| 5187 <lb/> | |
| 5188 <s xml:id="echoid-s427" xml:space="preserve"> | |
| 5189 prino. | |
| 5190 </s> | |
| 5191 <lb/> | |
| 5192 <s xml:id="echoid-s428" xml:space="preserve"> | |
| 5193 prinatus. prinatio. | |
| 5194 </s> | |
| 5195 <lb/> | |
| 5196 <s xml:id="echoid-s429" xml:space="preserve"> | |
| 5197 prinatimus. | |
| 5198 </s> | |
| 5199 </p> | |
| 5200 <pb file="add_6784_f416v" o="416v" n="832"/> | |
| 5201 <pb file="add_6784_f417" o="417" n="833"/> | |
| 5202 <pb file="add_6784_f417v" o="417v" n="834"/> | |
| 5203 <pb file="add_6784_f418" o="418" n="835"/> | |
| 5204 <pb file="add_6784_f418v" o="418v" n="836"/> | |
| 5205 <pb file="add_6784_f419" o="419" n="837"/> | |
| 5206 <pb file="add_6784_f419v" o="419v" n="838"/> | |
| 5207 <pb file="add_6784_f420" o="420" n="839"/> | |
| 5208 <pb file="add_6784_f420v" o="420v" n="840"/> | |
| 5209 <pb file="add_6784_f421" o="421" n="841"/> | |
| 5210 <pb file="add_6784_f421v" o="421v" n="842"/> | |
| 5211 <pb file="add_6784_f422" o="422" n="843"/> | |
| 5212 <pb file="add_6784_f422v" o="422v" n="844"/> | |
| 5213 <pb file="add_6784_f423" o="423" n="845"/> | |
| 5214 <div xml:id="echoid-div75" type="page_commentary" level="2" n="75"> | |
| 5215 <p> | |
| 5216 <s xml:id="echoid-s430" xml:space="preserve">[<emph style="it">Note: | |
| 5217 <p> | |
| 5218 <s xml:id="echoid-s430" xml:space="preserve"> | |
| 5219 The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>3</mn><mi>a</mi><mi>b</mi><mi>b</mi></mstyle></math> evaluated for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>3</mn><mi>b</mi></mstyle></math>, ... , <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mn>7</mn><mi>b</mi></mstyle></math>. | |
| 5220 The resulting coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>b</mi></mstyle></math> are listed in the table at the bottom of the page. | |
| 5221 Columns to the right list successive differences as far as the constant difference 6. | |
| 5222 The table has also been extrapolated upwards, giving rise to negative values in the first three columns. | |
| 5223 There is an error in the first column, however, which reading upwards should be: | |
| 5224 322, 110, 52, 18, 2, - 2, 0, 2, .... | |
| 5225 </s> | |
| 5226 </p> | |
| 5227 </emph>] | |
| 5228 <lb/><lb/></s></p></div> | |
| 5229 <pb file="add_6784_f423v" o="423v" n="846"/> | |
| 5230 <pb file="add_6784_f424" o="424" n="847"/> | |
| 5231 <div xml:id="echoid-div76" type="page_commentary" level="2" n="76"> | |
| 5232 <p> | |
| 5233 <s xml:id="echoid-s432" xml:space="preserve">[<emph style="it">Note: | |
| 5234 <p> | |
| 5235 <s xml:id="echoid-s432" xml:space="preserve"> | |
| 5236 Note various combinations of small numbers in the lower part of the page (see also Add MS 6784, f. 414). | |
| 5237 </s> | |
| 5238 </p> | |
| 5239 </emph>] | |
| 5240 <lb/><lb/></s></p></div> | |
| 5241 <pb file="add_6784_f424v" o="424v" n="848"/> | |
| 5242 <pb file="add_6784_f425" o="425" n="849"/> | |
| 5243 <pb file="add_6784_f425v" o="425v" n="850"/> | |
| 5244 <pb file="add_6784_f426" o="426" n="851"/> | |
| 5245 <pb file="add_6784_f426v" o="426v" n="852"/> | |
| 5246 <pb file="add_6784_f427" o="427" n="853"/> | |
| 5247 <pb file="add_6784_f427v" o="427v" n="854"/> | |
| 5248 <pb file="add_6784_f428" o="428" n="855"/> | |
| 5249 <div xml:id="echoid-div77" type="page_commentary" level="2" n="77"> | |
| 5250 <p> | |
| 5251 <s xml:id="echoid-s434" xml:space="preserve">[<emph style="it">Note: | |
| 5252 <p> | |
| 5253 <s xml:id="echoid-s434" xml:space="preserve"> | |
| 5254 Sums of some infinite geometric progressions. | |
| 5255 </s> | |
| 5256 </p> | |
| 5257 </emph>] | |
| 5258 <lb/><lb/></s></p></div> | |
| 5259 <pb file="add_6784_f428v" o="428v" n="856"/> | |
| 5260 <div xml:id="echoid-div78" type="page_commentary" level="2" n="78"> | |
| 5261 <p> | |
| 5262 <s xml:id="echoid-s436" xml:space="preserve">[<emph style="it">Note: | |
| 5263 <p> | |
| 5264 <s xml:id="echoid-s436" xml:space="preserve"> | |
| 5265 Triangles and circles filled with rectilinear figures (rectangles or triangles), | |
| 5266 in a way that can in principle be continued indefinitely. | |
| 5267 </s> | |
| 5268 </p> | |
| 5269 </emph>] | |
| 5270 <lb/><lb/></s></p></div> | |
| 5271 <pb file="add_6784_f429" o="429" n="857"/> | |
| 5272 <head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat"> | |
| 5273 De infinitis. Ex ratione motus, temporis et spatij. | |
| 5274 <lb/>[<emph style="it">tr: | |
| 5275 On infinity. From the ratio of motion, time and space. | |
| 5276 </emph>]<lb/> | |
| 5277 </head> | |
| 5278 <p xml:lang="lat"> | |
| 5279 <s xml:id="echoid-s438" xml:space="preserve"> | |
| 5280 Vide <reg norm="Aristotle" type="abbr">Arist</reg>. lib. 6. tret. 23. <lb/> | |
| 5281 proclum de motu lib. 1. pro. 14. | |
| 5282 <lb/>[<emph style="it">tr: | |
| 5283 See Aristotle, Book 6, Treatise 23. <lb/> | |
| 5284 Proclus, <emph style="it">De motu</emph>, Book 1, Proposition 14. | |
| 5285 </emph>]<lb/> | |
| 5286 </s> | |
| 5287 </p> | |
| 5288 <p xml:lang="lat"> | |
| 5289 <s xml:id="echoid-s439" xml:space="preserve"> | |
| 5290 1. <lb/> | |
| 5291 Moveatur A corpus <lb/> | |
| 5292 per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium in <lb/> | |
| 5293 tempore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> atque sit <lb/> | |
| 5294 ille motus uniformis. | |
| 5295 <lb/>[<emph style="it">tr: | |
| 5296 Let a body A be moved through a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in a time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> and let that motion is uniform. | |
| 5297 </emph>]<lb/> | |
| 5298 </s> | |
| 5299 </p> | |
| 5300 <!-- text in first column --> | |
| 5301 <p xml:lang="lat"> | |
| 5302 <s xml:id="echoid-s440" xml:space="preserve"> | |
| 5303 infinite <lb/> | |
| 5304 maximum | |
| 5305 <lb/>[<emph style="it">tr: | |
| 5306 infinite maximum | |
| 5307 </emph>]<lb/> | |
| 5308 </s> | |
| 5309 </p> | |
| 5310 <p xml:lang="lat"> | |
| 5311 <s xml:id="echoid-s441" xml:space="preserve"> | |
| 5312 minimum | |
| 5313 <lb/>[<emph style="it">tr: | |
| 5314 minimum | |
| 5315 </emph>]<lb/> | |
| 5316 </s> | |
| 5317 </p> | |
| 5318 <p xml:lang="lat"> | |
| 5319 <s xml:id="echoid-s442" xml:space="preserve"> | |
| 5320 indivisibile | |
| 5321 <lb/>[<emph style="it">tr: | |
| 5322 an indivisible | |
| 5323 </emph>]<lb/> | |
| 5324 </s> | |
| 5325 </p> | |
| 5326 <p xml:lang="lat"> | |
| 5327 <s xml:id="echoid-s443" xml:space="preserve"> | |
| 5328 punctum | |
| 5329 <lb/>[<emph style="it">tr: | |
| 5330 a point | |
| 5331 </emph>]<lb/> | |
| 5332 </s> | |
| 5333 </p> | |
| 5334 <!-- text in second column --> | |
| 5335 <p xml:lang="lat"> | |
| 5336 <s xml:id="echoid-s444" xml:space="preserve"> | |
| 5337 aliquod <lb/> | |
| 5338 infinite <lb/> | |
| 5339 maximum | |
| 5340 <lb/>[<emph style="it">tr: | |
| 5341 infinite maximum | |
| 5342 </emph>]<lb/> | |
| 5343 </s> | |
| 5344 </p> | |
| 5345 <p xml:lang="lat"> | |
| 5346 <s xml:id="echoid-s445" xml:space="preserve"> | |
| 5347 minimum <lb/> | |
| 5348 eadem <lb/> | |
| 5349 ratione | |
| 5350 <lb/>[<emph style="it">tr: | |
| 5351 minimum in the same ratio | |
| 5352 </emph>]<lb/> | |
| 5353 </s> | |
| 5354 </p> | |
| 5355 <p xml:lang="lat"> | |
| 5356 <s xml:id="echoid-s446" xml:space="preserve"> | |
| 5357 Indivisibile <lb/> | |
| 5358 eadem <lb/> | |
| 5359 ratione | |
| 5360 <lb/>[<emph style="it">tr: | |
| 5361 An indivisble in the same ratio | |
| 5362 </emph>]<lb/> | |
| 5363 </s> | |
| 5364 </p> | |
| 5365 <p xml:lang="lat"> | |
| 5366 <s xml:id="echoid-s447" xml:space="preserve"> | |
| 5367 Indivisibile <lb/> | |
| 5368 sed non punctum <lb/> | |
| 5369 vel instans ut alia <lb/> | |
| 5370 ratione inferetur. | |
| 5371 <lb/>[<emph style="it">tr: | |
| 5372 And indivisble but not a point or an instant that can be inferred from the other ratio. | |
| 5373 </emph>]<lb/> | |
| 5374 </s> | |
| 5375 </p> | |
| 5376 <p xml:lang="lat"> | |
| 5377 <s xml:id="echoid-s448" xml:space="preserve"> | |
| 5378 2. <lb/> | |
| 5379 Moveatur A corpus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium <lb/> | |
| 5380 in tempore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> atque sit ille <lb/> | |
| 5381 motus uniformis. | |
| 5382 <lb/>[<emph style="it">tr: | |
| 5383 Let a body A be moved thorugh a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> in time <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>d</mi><mi>e</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and let that motion be uniform. | |
| 5384 </emph>]<lb/> | |
| 5385 </s> | |
| 5386 </p> | |
| 5387 <!-- text in first column --> | |
| 5388 <p xml:lang="lat"> | |
| 5389 <s xml:id="echoid-s449" xml:space="preserve"> | |
| 5390 indivisibile | |
| 5391 <lb/>[<emph style="it">tr: | |
| 5392 an indivisible | |
| 5393 </emph>]<lb/> | |
| 5394 </s> | |
| 5395 </p> | |
| 5396 <p xml:lang="lat"> | |
| 5397 <s xml:id="echoid-s450" xml:space="preserve"> | |
| 5398 punctum | |
| 5399 <lb/>[<emph style="it">tr: | |
| 5400 a point | |
| 5401 </emph>]<lb/> | |
| 5402 </s> | |
| 5403 </p> | |
| 5404 <!-- text in second column --> | |
| 5405 <p xml:lang="lat"> | |
| 5406 <s xml:id="echoid-s451" xml:space="preserve"> | |
| 5407 Indivisibile <lb/> | |
| 5408 eadem ratione | |
| 5409 <lb/>[<emph style="it">tr: | |
| 5410 An indivisble in the same ratio | |
| 5411 </emph>]<lb/> | |
| 5412 </s> | |
| 5413 </p> | |
| 5414 <p xml:lang="lat"> | |
| 5415 <s xml:id="echoid-s452" xml:space="preserve"> | |
| 5416 Indivisibile quod <lb/> | |
| 5417 dimidium est <lb/> | |
| 5418 Indivisibilis ex <lb/> | |
| 5419 priori argumentatione. | |
| 5420 <lb/>[<emph style="it">tr: | |
| 5421 An indivisble whose half is indivisble by the previous argument. | |
| 5422 </emph>]<lb/> | |
| 5423 </s> | |
| 5424 </p> | |
| 5425 <p xml:lang="lat"> | |
| 5426 <s xml:id="echoid-s453" xml:space="preserve"> | |
| 5427 Ergo etiam: | |
| 5428 <lb/>[<emph style="it">tr: | |
| 5429 Therefore also | |
| 5430 </emph>]<lb/> | |
| 5431 </s> | |
| 5432 </p> | |
| 5433 <p xml:lang="lat"> | |
| 5434 <s xml:id="echoid-s454" xml:space="preserve"> | |
| 5435 Indivisibile quod <lb/> | |
| 5436 dimidium est <lb/> | |
| 5437 Indivisibilis ex <lb/> | |
| 5438 priori argumentatione. | |
| 5439 <lb/>[<emph style="it">tr: | |
| 5440 An indivisble whose half is indivisble by the previous argument. | |
| 5441 </emph>]<lb/> | |
| 5442 </s> | |
| 5443 </p> | |
| 5444 <p xml:lang="lat"> | |
| 5445 <s xml:id="echoid-s455" xml:space="preserve"> | |
| 5446 punctum | |
| 5447 <lb/>[<emph style="it">tr: | |
| 5448 a point | |
| 5449 </emph>]<lb/> | |
| 5450 </s> | |
| 5451 </p> | |
| 5452 <p xml:lang="lat"> | |
| 5453 <s xml:id="echoid-s456" xml:space="preserve"> | |
| 5454 punctum | |
| 5455 <lb/>[<emph style="it">tr: | |
| 5456 a point | |
| 5457 </emph>]<lb/> | |
| 5458 </s> | |
| 5459 </p> | |
| 5460 <p xml:lang="lat"> | |
| 5461 <s xml:id="echoid-s457" xml:space="preserve"> | |
| 5462 Ergo punctum quod ponebatur esse <lb/> | |
| 5463 indivisbile, alia ratione inferetur <lb/> | |
| 5464 Divisibile, et sic in infinitum. | |
| 5465 <lb/>[<emph style="it">tr: | |
| 5466 Therefore a point that can be supposed indivisble, is inferred from the other ratio to be divisible, | |
| 5467 and thus infinitely. | |
| 5468 </emph>]<lb/> | |
| 5469 </s> | |
| 5470 </p> | |
| 5471 <pb file="add_6784_f429v" o="429v" n="858"/> | |
| 5472 <pb file="add_6784_f430" o="430" n="859"/> | |
| 5473 <div xml:id="echoid-div79" type="page_commentary" level="2" n="79"> | |
| 5474 <p> | |
| 5475 <s xml:id="echoid-s458" xml:space="preserve">[<emph style="it">Note: | |
| 5476 <p> | |
| 5477 <s xml:id="echoid-s458" xml:space="preserve"> | |
| 5478 Triangles transformed to spirals. <lb/> | |
| 5479 See also Add MS 6785, f. 437 and Add MS 6784, f. 246, f. 247, f. 248. | |
| 5480 </s> | |
| 5481 </p> | |
| 5482 </emph>] | |
| 5483 <lb/><lb/></s></p></div> | |
| 5484 <pb file="add_6784_f430v" o="430v" n="860"/> | |
| 5485 <pb file="add_6784_f431" o="431" n="861"/> | |
| 5486 <pb file="add_6784_f431v" o="431v" n="862"/> | |
| 5487 </div> | |
| 5488 </text> | |
| 5489 </echo> |