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| author | casties |
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| date | Fri, 07 Dec 2012 17:05:22 +0100 |
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<?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"> <metadata> <dcterms:identifier>ECHO:XT0KZ8QC.xml</dcterms:identifier> <dcterms:creator>Harriot, Thomas</dcterms:creator> <dcterms:title xml:lang="en">Mss. 6784</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date> <dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language> <dcterms:rights>CC-BY-SA</dcterms:rights> <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> <echodir>/permanent/library/XT0KZ8QC</echodir> <log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log> </metadata> <text xml:lang="eng" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"> <pb file="add_6784_f001" o="1" n="1"/> <head xml:id="echoid-head1" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f001v" o="1v" n="2"/> <pb file="add_6784_f002" o="2" n="3"/> <head xml:id="echoid-head2" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head3" xml:space="preserve"> AB) </head> <pb file="add_6784_f002v" o="2v" n="4"/> <pb file="add_6784_f003" o="3" n="5"/> <head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f003v" o="3v" n="6"/> <pb file="add_6784_f004" o="4" n="7"/> <head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f004v" o="4v" n="8"/> <pb file="add_6784_f005" o="5" n="9"/> <head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f005v" o="5v" n="10"/> <pb file="add_6784_f006" o="6" n="11"/> <head xml:id="echoid-head7" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f006v" o="6v" n="12"/> <pb file="add_6784_f007" o="7" n="13"/> <head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head9" xml:space="preserve"> AB) </head> <pb file="add_6784_f007v" o="7v" n="14"/> <pb file="add_6784_f008" o="8" n="15"/> <head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f008v" o="8v" n="16"/> <pb file="add_6784_f009" o="9" n="17"/> <head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head12" xml:space="preserve"> 2.AB) </head> <pb file="add_6784_f009v" o="9v" n="18"/> <pb file="add_6784_f010" o="10" n="19"/> <head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head14" xml:space="preserve"> AC) </head> <pb file="add_6784_f010v" o="10v" n="20"/> <pb file="add_6784_f011" o="11" n="21"/> <head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head16" xml:space="preserve"> AC.1) </head> <pb file="add_6784_f011v" o="11v" n="22"/> <pb file="add_6784_f012" o="12" n="23"/> <head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f012v" o="12v" n="24"/> <pb file="add_6784_f013" o="13" n="25"/> <head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head19" xml:space="preserve"> 2.BC) </head> <pb file="add_6784_f013v" o="13v" n="26"/> <pb file="add_6784_f014" o="14" n="27"/> <head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <head xml:id="echoid-head21" xml:space="preserve"> 1.BC) </head> <pb file="add_6784_f014v" o="14v" n="28"/> <pb file="add_6784_f015" o="15" n="29"/> <head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f015v" o="15v" n="30"/> <pb file="add_6784_f016" o="16" n="31"/> <head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f016v" o="16v" n="32"/> <pb file="add_6784_f017" o="17" n="33"/> <head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat"> Pappus 171. ad resectione rationis </head> <pb file="add_6784_f017v" o="17v" n="34"/> <pb file="add_6784_f018" o="18" n="35"/> <head xml:id="echoid-head25" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f018v" o="18v" n="36"/> <pb file="add_6784_f019" o="19" n="37"/> <head xml:id="echoid-head26" xml:space="preserve" xml:lang="lat"> De resectione spatij, problema </head> <head xml:id="echoid-head27" xml:space="preserve"> a) </head> <pb file="add_6784_f019v" o="19v" n="38"/> <pb file="add_6784_f020" o="20" n="39"/> <head xml:id="echoid-head28" xml:space="preserve"> Poristike </head> <pb file="add_6784_f020v" o="20v" n="40"/> <pb file="add_6784_f021" o="21" n="41"/> <pb file="add_6784_f021v" o="21v" n="42"/> <pb file="add_6784_f022" o="22" n="43"/> <pb file="add_6784_f022v" o="22v" n="44"/> <pb file="add_6784_f023" o="23" n="45"/> <pb file="add_6784_f023v" o="23v" n="46"/> <pb file="add_6784_f024" o="24" n="47"/> <head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat"> De sectione rationis </head> <head xml:id="echoid-head30" xml:space="preserve"> b.1) </head> <pb file="add_6784_f024v" o="24v" n="48"/> <pb file="add_6784_f025" o="25" n="49"/> <head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat"> De sectione rationis </head> <head xml:id="echoid-head32" xml:space="preserve"> b.2) </head> <pb file="add_6784_f025v" o="25v" n="50"/> <pb file="add_6784_f026" o="26" n="51"/> <head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat"> De sectione rationis </head> <head xml:id="echoid-head34" xml:space="preserve"> b.3) </head> <pb file="add_6784_f026v" o="26v" n="52"/> <pb file="add_6784_f027" o="27" n="53"/> <head xml:id="echoid-head35" xml:space="preserve" xml:lang="lat"> De sectione rationis </head> <head xml:id="echoid-head36" xml:space="preserve"> b.4) </head> <pb file="add_6784_f027v" o="27v" n="54"/> <pb file="add_6784_f028" o="28" n="55"/> <head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat"> Lemma ad sectionem rationis <lb/> et spatij </head> <pb file="add_6784_f028v" o="28v" n="56"/> <pb file="add_6784_f029" o="29" n="57"/> <pb file="add_6784_f029v" o="29v" n="58"/> <pb file="add_6784_f030" o="30" n="59"/> <pb file="add_6784_f030v" o="30v" n="60"/> <pb file="add_6784_f031" o="31" n="61"/> <pb file="add_6784_f031v" o="31v" n="62"/> <pb file="add_6784_f032" o="32" n="63"/> <pb file="add_6784_f032v" o="32v" n="64"/> <pb file="add_6784_f033" o="33" n="65"/> <pb file="add_6784_f033v" o="33v" n="66"/> <pb file="add_6784_f034" o="34" n="67"/> <pb file="add_6784_f034v" o="34v" n="68"/> <pb file="add_6784_f035" o="35" n="69"/> <pb file="add_6784_f035v" o="35v" n="70"/> <pb file="add_6784_f036" o="36" n="71"/> <pb file="add_6784_f036v" o="36v" n="72"/> <pb file="add_6784_f037" o="37" n="73"/> <pb file="add_6784_f037v" o="37v" n="74"/> <pb file="add_6784_f038" o="38" n="75"/> <pb file="add_6784_f038v" o="38v" n="76"/> <pb file="add_6784_f039" o="39" n="77"/> <pb file="add_6784_f039v" o="39v" n="78"/> <pb file="add_6784_f040" o="40" n="79"/> <head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat"> De resectione rationis </head> <pb file="add_6784_f040v" o="40v" n="80"/> <div xml:id="echoid-div1" type="page_commentary" level="2" n="1"> <p> <s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s1" xml:space="preserve"> De infinitis <lb/>[<emph style="it">tr: On infinity </emph>]<lb/> </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s3" xml:space="preserve"> Maior et Maior rationum infinitum. <lb/> fit termini minores et minores; cum probuerit indivisibilibis <lb/> ratio tandem infinitum. <lb/>[<emph style="it">tr: A greater and greater infinite ratio. the terms are smaller and smaller; while from indivisibles there will eventually come an infinite ratio. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s4" xml:space="preserve"> HA ad IA non potest <lb/> esse maior BA ad BC. <lb/> terminis scilicet decrescentibus. <lb/>[<emph style="it">tr: HA to IA cannot be greater than BA to BC. the terms of course decreasing. </emph>]<lb/> </s> </p> <pb file="add_6784_f041" o="41" n="81"/> <pb file="add_6784_f041v" o="41v" n="82"/> <pb file="add_6784_f042" o="42" n="83"/> <pb file="add_6784_f042v" o="42v" n="84"/> <pb file="add_6784_f043" o="43" n="85"/> <pb file="add_6784_f043v" o="43v" n="86"/> <pb file="add_6784_f044" o="44" n="87"/> <pb file="add_6784_f044v" o="44v" n="88"/> <pb file="add_6784_f045" o="45" n="89"/> <pb file="add_6784_f045v" o="45v" n="90"/> <pb file="add_6784_f046" o="46" n="91"/> <pb file="add_6784_f046v" o="46v" n="92"/> <pb file="add_6784_f047" o="47" n="93"/> <pb file="add_6784_f047v" o="47v" n="94"/> <pb file="add_6784_f048" o="48" n="95"/> <pb file="add_6784_f048v" o="48v" n="96"/> <pb file="add_6784_f049" o="49" n="97"/> <pb file="add_6784_f049v" o="49v" n="98"/> <pb 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<pb file="add_6784_f079" o="79" n="157"/> <pb file="add_6784_f079v" o="79v" n="158"/> <pb file="add_6784_f080" o="80" n="159"/> <pb file="add_6784_f080v" o="80v" n="160"/> <pb file="add_6784_f081" o="81" n="161"/> <pb file="add_6784_f081v" o="81v" n="162"/> <pb file="add_6784_f082" o="82" n="163"/> <pb file="add_6784_f082v" o="82v" n="164"/> <pb file="add_6784_f083" o="83" n="165"/> <pb file="add_6784_f083v" o="83v" n="166"/> <pb file="add_6784_f084" o="84" n="167"/> <pb file="add_6784_f084v" o="84v" n="168"/> <pb file="add_6784_f085" o="85" n="169"/> <pb file="add_6784_f085v" o="85v" n="170"/> <pb file="add_6784_f086" o="86" n="171"/> <pb file="add_6784_f086v" o="86v" n="172"/> <pb file="add_6784_f087" o="87" n="173"/> <pb file="add_6784_f087v" o="87v" n="174"/> <pb file="add_6784_f088" o="88" n="175"/> <div xml:id="echoid-div3" type="page_commentary" level="2" n="3"> <p> <s xml:id="echoid-s7" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s7" xml:space="preserve"> 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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>. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f090" o="90" n="179"/> <pb file="add_6784_f090v" o="90v" n="180"/> <pb file="add_6784_f091" o="91" n="181"/> <pb file="add_6784_f091v" o="91v" n="182"/> <pb file="add_6784_f092" o="92" n="183"/> <pb file="add_6784_f092v" o="92v" n="184"/> <pb file="add_6784_f093" o="93" n="185"/> <pb file="add_6784_f093v" o="93v" n="186"/> <pb file="add_6784_f094" o="94" n="187"/> <pb file="add_6784_f094v" o="94v" n="188"/> <pb file="add_6784_f095" o="95" n="189"/> <pb file="add_6784_f095v" o="95v" n="190"/> <pb file="add_6784_f096" o="96" n="191"/> <pb file="add_6784_f096v" o="96v" n="192"/> <pb file="add_6784_f097" o="97" n="193"/> <pb file="add_6784_f097v" o="97v" n="194"/> <pb file="add_6784_f098" o="98" n="195"/> <pb file="add_6784_f098v" o="98v" n="196"/> <pb file="add_6784_f099" o="99" n="197"/> <pb file="add_6784_f099v" o="99v" n="198"/> <pb file="add_6784_f100" o="100" n="199"/> <pb file="add_6784_f100v" o="100v" n="200"/> <pb file="add_6784_f101" o="101" n="201"/> <pb file="add_6784_f101v" o="101v" n="202"/> <pb file="add_6784_f102" o="102" n="203"/> <pb file="add_6784_f102v" o="102v" n="204"/> <pb file="add_6784_f103" o="103" n="205"/> <pb file="add_6784_f103v" o="103v" n="206"/> <pb file="add_6784_f104" o="104" n="207"/> <pb file="add_6784_f104v" o="104v" n="208"/> <pb file="add_6784_f105" o="105" n="209"/> <p xml:lang="lat"> <s xml:id="echoid-s11" xml:space="preserve"> Graecia <lb/> prævenians. <lb/> excitans. <lb/> vocans. <lb/> operans. <lb/> provens. <lb/> comians. <lb/> cooperans. <lb/> adiunans. <lb/> concomitans. <lb/> subsequens. <lb/> prosequens. </s> </p> <pb file="add_6784_f105v" o="105v" n="210"/> <pb file="add_6784_f106" o="106" n="211"/> <pb file="add_6784_f106v" o="106v" n="212"/> <pb file="add_6784_f107" o="107" n="213"/> <pb file="add_6784_f107v" o="107v" n="214"/> <pb file="add_6784_f108" o="108" n="215"/> <pb file="add_6784_f108v" o="108v" n="216"/> <pb file="add_6784_f109" o="109" n="217"/> <pb file="add_6784_f109v" o="109v" n="218"/> <pb file="add_6784_f110" o="110" n="219"/> <pb file="add_6784_f110v" o="110v" n="220"/> <pb file="add_6784_f111" o="111" n="221"/> <pb file="add_6784_f111v" o="111v" n="222"/> <pb file="add_6784_f112" o="112" n="223"/> <pb file="add_6784_f112v" o="112v" n="224"/> <pb file="add_6784_f113" o="113" n="225"/> <pb file="add_6784_f113v" o="113v" n="226"/> <pb file="add_6784_f114" o="114" n="227"/> <pb file="add_6784_f114v" o="114v" n="228"/> <pb file="add_6784_f115" o="115" n="229"/> <pb file="add_6784_f115v" o="115v" n="230"/> <pb file="add_6784_f116" o="116" n="231"/> <pb file="add_6784_f116v" o="116v" n="232"/> <pb file="add_6784_f117" o="117" n="233"/> <pb file="add_6784_f117v" o="117v" n="234"/> <pb file="add_6784_f118" o="118" n="235"/> <pb file="add_6784_f118v" o="118v" n="236"/> <pb file="add_6784_f119" o="119" n="237"/> <pb file="add_6784_f119v" o="119v" n="238"/> <pb file="add_6784_f120" o="120" n="239"/> <pb file="add_6784_f120v" o="120v" n="240"/> <pb file="add_6784_f121" o="121" n="241"/> <pb file="add_6784_f121v" o="121v" n="242"/> <pb file="add_6784_f122" o="122" n="243"/> <pb file="add_6784_f122v" o="122v" n="244"/> <pb file="add_6784_f123" o="123" n="245"/> <div xml:id="echoid-div5" type="page_commentary" level="2" n="5"> <p> <s xml:id="echoid-s12" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s12" xml:space="preserve"> The references on this page are to Pappus, Book 7, and to Giambattista Benedetti, <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph> (1585). </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s14" xml:space="preserve"> sit triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi><mi>d</mi></mstyle></math> <lb/>[<emph style="it">tr: 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> </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s15" xml:space="preserve"> dico quod <lb/>[<emph style="it">tr: I say that </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s16" xml:space="preserve"> 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> <lb/>[<emph style="it">tr: 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> </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s17" xml:space="preserve"> Unde sequitur <lb/>[<emph style="it">tr: whence it follows </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s18" xml:space="preserve"> Vide, Pappum. lib. 7. prop: 122. pag. 235. <lb/> et: Jo: Baptistum Benedictum pag. 362. <lb/>[<emph style="it">tr: See Pappus, Book 7, Proposition 122, page 235; and Johan Baptista Benedictus, page 362 </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s19" xml:space="preserve"> verte <lb/>[<emph style="it">tr: turn over </emph>]<lb/> </s> </p> <pb file="add_6784_f123v" o="123v" n="246"/> <pb file="add_6784_f124" o="124" n="247"/> <pb file="add_6784_f124v" o="124v" n="248"/> <pb file="add_6784_f125" o="125" n="249"/> <pb file="add_6784_f125v" o="125v" n="250"/> <pb file="add_6784_f126" o="126" n="251"/> <pb file="add_6784_f126v" o="126v" n="252"/> <pb file="add_6784_f127" o="127" n="253"/> <head xml:id="echoid-head40" xml:space="preserve"> Lemma. 1. Appol. Bat. pag. 81. </head> <p xml:lang="lat"> <s xml:id="echoid-s20" xml:space="preserve"> Sit: <lb/> Dico quod: <lb/> nam in utraque analogia <lb/>[<emph style="it">tr: Let: <lb/> I say that: <lb/> for in the both ratios </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s21" xml:space="preserve"> Sed ita Snellius <lb/>[<emph style="it">tr: But it is thus in Snell. </emph>]<lb/> </s> </p> <pb file="add_6784_f127v" o="127v" n="254"/> <pb file="add_6784_f128" o="128" n="255"/> <pb file="add_6784_f128v" o="128v" n="256"/> <pb file="add_6784_f129" o="129" n="257"/> <pb file="add_6784_f129v" o="129v" n="258"/> <pb file="add_6784_f130" o="130" n="259"/> <pb file="add_6784_f130v" o="130v" n="260"/> <pb file="add_6784_f131" o="131" n="261"/> <pb file="add_6784_f131v" o="131v" n="262"/> <pb file="add_6784_f132" o="132" n="263"/> <pb file="add_6784_f132v" o="132v" n="264"/> <pb file="add_6784_f133" o="133" n="265"/> <pb file="add_6784_f133v" o="133v" n="266"/> <pb file="add_6784_f134" o="134" n="267"/> <pb file="add_6784_f134v" o="134v" n="268"/> <pb file="add_6784_f135" o="135" n="269"/> <pb file="add_6784_f135v" o="135v" n="270"/> <pb file="add_6784_f136" o="136" n="271"/> <pb file="add_6784_f136v" o="136v" n="272"/> <pb file="add_6784_f137" o="137" n="273"/> <pb file="add_6784_f137v" o="137v" n="274"/> <pb file="add_6784_f138" o="138" n="275"/> <pb file="add_6784_f138v" o="138v" n="276"/> <pb file="add_6784_f139" o="139" n="277"/> <pb file="add_6784_f139v" o="139v" n="278"/> <pb file="add_6784_f140" o="140" n="279"/> <pb file="add_6784_f140v" o="140v" n="280"/> <pb file="add_6784_f141" o="141" n="281"/> <pb file="add_6784_f141v" o="141v" n="282"/> <pb file="add_6784_f142" o="142" n="283"/> <pb file="add_6784_f142v" o="142v" n="284"/> <pb file="add_6784_f143" o="143" n="285"/> <pb file="add_6784_f143v" o="143v" n="286"/> <pb file="add_6784_f144" o="144" n="287"/> <pb file="add_6784_f144v" o="144v" n="288"/> <pb file="add_6784_f145" o="145" n="289"/> <pb file="add_6784_f145v" o="145v" n="290"/> <pb file="add_6784_f146" o="146" n="291"/> <pb file="add_6784_f146v" o="146v" n="292"/> <pb file="add_6784_f147" o="147" n="293"/> <pb file="add_6784_f147v" o="147v" n="294"/> <pb file="add_6784_f148" o="148" n="295"/> <pb file="add_6784_f148v" o="148v" n="296"/> <pb file="add_6784_f149" o="149" n="297"/> <pb file="add_6784_f149v" o="149v" n="298"/> <div xml:id="echoid-div6" type="page_commentary" level="2" n="6"> <p> <s xml:id="echoid-s22" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s22" xml:space="preserve"> This page contains symbolic versions of Euclid Book II, Propositions 12 and 13: <lb/> II.12.In obtuse-angle triangles the square on the side opposite the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. <lb/> II.13. In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat"> Aliter de 12. 2<emph style="super">i</emph> Euclidis <lb/> et 13. <lb/>[<emph style="it">tr: Another way for Euclid II.12 and 13. </emph>]<lb/> </head> <pb file="add_6784_f150" o="150" n="299"/> <pb file="add_6784_f150v" o="150v" n="300"/> <pb file="add_6784_f151" o="151" n="301"/> <pb file="add_6784_f151v" o="151v" n="302"/> <div xml:id="echoid-div7" type="page_commentary" level="2" n="7"> <p> <s xml:id="echoid-s24" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s24" xml:space="preserve"> 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>. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f152" o="152" n="303"/> <pb file="add_6784_f152v" o="152v" n="304"/> <pb file="add_6784_f153" o="153" n="305"/> <pb file="add_6784_f153v" o="153v" n="306"/> <pb file="add_6784_f154" o="154" n="307"/> <pb file="add_6784_f154v" o="154v" n="308"/> <div xml:id="echoid-div8" type="page_commentary" level="2" n="8"> <p> <s xml:id="echoid-s26" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s26" xml:space="preserve"> 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>. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f155" o="155" n="309"/> <pb file="add_6784_f155v" o="155v" n="310"/> <pb file="add_6784_f156" o="156" n="311"/> <pb file="add_6784_f156v" o="156v" n="312"/> <pb file="add_6784_f157" o="157" n="313"/> <pb file="add_6784_f157v" o="157v" n="314"/> <pb file="add_6784_f158" o="158" n="315"/> <pb file="add_6784_f158v" o="158v" n="316"/> <pb file="add_6784_f159" o="159" n="317"/> <pb file="add_6784_f159v" o="159v" n="318"/> <pb file="add_6784_f160" o="160" n="319"/> <head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat"> phys. lib.6. Cap. 1 <lb/>[<emph style="it">tr: Physics, Book 6, Chapter 1 </emph>]<lb/> </head> <pb file="add_6784_f160v" o="160v" n="320"/> <pb file="add_6784_f161" o="161" n="321"/> <pb file="add_6784_f161v" o="161v" n="322"/> <pb file="add_6784_f162" o="162" n="323"/> <pb file="add_6784_f162v" o="162v" n="324"/> <pb file="add_6784_f163" o="163" n="325"/> <pb file="add_6784_f163v" o="163v" n="326"/> <pb file="add_6784_f164" o="164" n="327"/> <head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat"> Arist. lib. 6. Cap. 2 <lb/>[<emph style="it">tr: Aristotle, Book 6, Chapter 2 </emph>]<lb/> </head> <pb file="add_6784_f164v" o="164v" n="328"/> <pb file="add_6784_f165" o="165" n="329"/> <pb file="add_6784_f165v" o="165v" n="330"/> <pb file="add_6784_f166" o="166" n="331"/> <div xml:id="echoid-div9" type="page_commentary" level="2" n="9"> <p> <s xml:id="echoid-s28" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s28" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head44" xml:space="preserve" xml:lang="lat"> Residuum 5<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: The rest of the working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. </emph>]<lb/> </head> <pb file="add_6784_f166v" o="166v" n="332"/> <pb file="add_6784_f167" o="167" n="333"/> <div xml:id="echoid-div10" type="page_commentary" level="2" n="10"> <p> <s xml:id="echoid-s30" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s30" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat"> 5<emph style="super">a</emph> operatio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: Working (5) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> </emph>]<lb/> </head> <pb file="add_6784_f167v" o="167v" n="334"/> <pb file="add_6784_f168" o="168" n="335"/> <div xml:id="echoid-div11" type="page_commentary" level="2" n="11"> <p> <s xml:id="echoid-s32" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s32" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f168v" o="168v" n="336"/> <pb file="add_6784_f169" o="169" n="337"/> <div xml:id="echoid-div12" type="page_commentary" level="2" n="12"> <p> <s xml:id="echoid-s34" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s34" xml:space="preserve"> Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s36" xml:space="preserve"> <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/> vel, cuivis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <lb/>[<emph style="it">tr: or, for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s37" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo><mi>c</mi></mstyle></math>. cuivis. <lb/>[<emph style="it">tr: any </emph>]<lb/> </s> </p> <pb file="add_6784_f169v" o="169v" n="338"/> <pb file="add_6784_f170" o="170" n="339"/> <div xml:id="echoid-div13" type="page_commentary" level="2" n="13"> <p> <s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s38" xml:space="preserve"> Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f170v" o="170v" n="340"/> <pb file="add_6784_f171" o="171" n="341"/> <pb file="add_6784_f171v" o="171v" n="342"/> <div xml:id="echoid-div14" type="page_commentary" level="2" n="14"> <p> <s xml:id="echoid-s40" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s40" xml:space="preserve"> Calculations relating to formula (3) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat"> Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. <lb/>[<emph style="it">tr: Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> </emph>]<lb/> </head> <pb file="add_6784_f172" o="172" n="343"/> <pb file="add_6784_f172v" o="172v" n="344"/> <div xml:id="echoid-div15" type="page_commentary" level="2" n="15"> <p> <s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s42" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat"> Residuum 3<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: The rest of the working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s44" xml:space="preserve"> Residuum 4<emph style="super">a</emph> operationis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: The rest of the working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. </emph>]<lb/> </s> </p> <pb file="add_6784_f173" o="173" n="345"/> <div xml:id="echoid-div16" type="page_commentary" level="2" n="16"> <p> <s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s45" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat"> 3<emph style="super">a</emph> operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: Working (3) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s47" xml:space="preserve"> 4<emph style="super">a</emph> operatio G. <lb/>[<emph style="it">tr: Working (4) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> </emph>]<lb/> </s> </p> <pb file="add_6784_f173v" o="173v" n="346"/> <pb file="add_6784_f174" o="174" n="347"/> <pb file="add_6784_f174v" o="174v" n="348"/> <pb file="add_6784_f175" o="175" n="349"/> <pb file="add_6784_f175v" o="175v" n="350"/> <pb file="add_6784_f176" o="176" n="351"/> <head xml:id="echoid-head49" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.11 De tactibus </head> <p xml:lang="lat"> <s xml:id="echoid-s48" xml:space="preserve"> cave <lb/>[<emph style="it">tr: beware </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s49" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. est centrum circuli <lb/> circumscribentis. <lb/> 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/> habet periferias æquales. <lb/>[<emph style="it">tr: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is the centre of the circumscribing circle. <lb/> 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. </emph>]<lb/> </s> </p> <pb file="add_6784_f176v" o="176v" n="352"/> <pb file="add_6784_f177" o="177" n="353"/> <head xml:id="echoid-head50" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.2 </head> <p xml:lang="lat"> <s xml:id="echoid-s50" xml:space="preserve"> <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/> <lb/>[...]<lb/> <lb/> cuius superficies ut sequitur. <lb/>[<emph style="it">tr: 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/> <lb/>[...]<lb/> <lb/> whose surface is as follows. </emph>]<lb/> </s> </p> <pb file="add_6784_f177v" o="177v" n="354"/> <pb file="add_6784_f178" o="178" n="355"/> <head xml:id="echoid-head51" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.3 </head> <p xml:lang="lat"> <s xml:id="echoid-s51" xml:space="preserve"> <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/> <lb/>[...]<lb/> <lb/> cuius superficies ut sequitur. <lb/>[<emph style="it">tr: 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/> <lb/>[...]<lb/> <lb/> whose surface is as follows. </emph>]<lb/> </s> </p> <pb file="add_6784_f178v" o="178v" n="356"/> <pb file="add_6784_f179" o="179" n="357"/> <head xml:id="echoid-head52" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.4 </head> <pb file="add_6784_f179v" o="179v" n="358"/> <pb file="add_6784_f180" o="180" n="359"/> <div xml:id="echoid-div17" type="page_commentary" level="2" n="17"> <p> <s xml:id="echoid-s52" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s52" xml:space="preserve"> The reference in the top right hand corner is to Viète, <emph style="it">Apollonius Gallus</emph> (1600), Problem IX. </s> <lb/> <quote xml:lang="lat"> Problema IX. <lb/> Datis duobus circulis, & puncto, per datum punctum circulum describere quem duo dati circuli contingat. </quote> <lb/> <quote> IX. Given two circles and a point, through the given point describe a circle that touches the two given circles. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head53" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5) </head> <p xml:lang="lat"> <s xml:id="echoid-s54" xml:space="preserve"> Vide: Appol. Gall. prob. 9. <lb/>[<emph style="it">tr: See Apollonius Gallus, Problem IX. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s55" xml:space="preserve"> Aberratur de modo contingendi <lb/> circulos posititios alias operatio bona <lb/> vide igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>. <lb/>[<emph style="it">tr: There is an error in the method of touching the supposed circles, othersie the working is good; therefore see shee <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>..5.2. </emph>]<lb/> [<emph style="it">Note: Sheet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2 is Add MS 6784, f. 181. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s56" xml:space="preserve"> radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> distantia centrorum <lb/>[<emph style="it">tr: radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> distance of the centres. </emph>]<lb/> </s> </p> <pb file="add_6784_f180v" o="180v" n="360"/> <pb file="add_6784_f181" o="181" n="361"/> <div xml:id="echoid-div18" type="page_commentary" level="2" n="18"> <p> <s xml:id="echoid-s57" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s57" xml:space="preserve"> A continuation of the work on Add MS 6784, f. 180. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head54" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.5.2<emph style="super">o</emph>) </head> <p xml:lang="lat"> <s xml:id="echoid-s59" xml:space="preserve"> Vide: Appol: Gall. prob. 9. <lb/> fig: 2. <lb/>[<emph style="it">tr: See Apollonius Gallus, Problem IX, figure 2. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s60" xml:space="preserve"> radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) minoris <lb/> posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) maioris <lb/> distantia centrorum <lb/>[<emph style="it">tr: radius of the smaller supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> of the greater supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> distance of the centres. </emph>]<lb/> </s> </p> <pb file="add_6784_f181v" o="181v" n="362"/> <pb file="add_6784_f182" o="182" n="363"/> <head xml:id="echoid-head55" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.6.) </head> <p xml:lang="lat"> <s xml:id="echoid-s61" xml:space="preserve"> radius circuli posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>) <lb/> posititij (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>) <lb/> distantia centrorum <lb/>[<emph style="it">tr: radius of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <lb/> of the supposed circle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <lb/> distance of the centres. </emph>]<lb/> </s> </p> <pb file="add_6784_f182v" o="182v" n="364"/> <pb file="add_6784_f183" o="183" n="365"/> <head xml:id="echoid-head56" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.1 </head> <p xml:lang="lat"> <s xml:id="echoid-s62" xml:space="preserve"> data <lb/> <lb/>[...]<lb/> <lb/> Quæritur: vel. <lb/>[<emph style="it">tr: given <lb/> <lb/>[...]<lb/> <lb/> Sought, either: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s63" xml:space="preserve"> <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/> <lb/>[...]<lb/> <lb/> cuius superficies ut sequitur. <lb/>[<emph style="it">tr: 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/> <lb/>[...]<lb/> <lb/> whose surface is as follows. </emph>]<lb/> </s> </p> <pb file="add_6784_f183v" o="183v" n="366"/> <pb file="add_6784_f184" o="184" n="367"/> <head xml:id="echoid-head57" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.3 </head> <p xml:lang="lat"> <s xml:id="echoid-s64" xml:space="preserve"> <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/> <lb/>[...]<lb/> <lb/> cuius superficies ut sequitur. <lb/>[<emph style="it">tr: 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/> <lb/>[...]<lb/> <lb/> whose surface is as follows. </emph>]<lb/> </s> </p> <pb file="add_6784_f184v" o="184v" n="368"/> <pb file="add_6784_f185" o="185" n="369"/> <head xml:id="echoid-head58" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi></mstyle></math>.2 </head> <p xml:lang="lat"> <s xml:id="echoid-s65" xml:space="preserve"> <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/> <lb/>[...]<lb/> <lb/> cuius superficies ut sequitur. <lb/>[<emph style="it">tr: 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/> <lb/>[...]<lb/> <lb/> whose surface is as follows. </emph>]<lb/> </s> </p> <pb file="add_6784_f185v" o="185v" n="370"/> <pb file="add_6784_f186" o="186" n="371"/> <head xml:id="echoid-head59" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi></mstyle></math>.1 De tactibus </head> <pb file="add_6784_f186v" o="186v" n="372"/> <pb file="add_6784_f187" o="187" n="373"/> <pb file="add_6784_f187v" o="187v" n="374"/> <pb file="add_6784_f188" o="188" n="375"/> <pb file="add_6784_f188v" o="188v" n="376"/> <pb file="add_6784_f189" o="189" n="377"/> <pb file="add_6784_f189v" o="189v" n="378"/> <pb file="add_6784_f190" o="190" n="379"/> <pb file="add_6784_f190v" o="190v" n="380"/> <pb file="add_6784_f191" o="191" n="381"/> <pb file="add_6784_f191v" o="191v" n="382"/> <pb file="add_6784_f192" o="192" n="383"/> <pb file="add_6784_f192v" o="192v" n="384"/> <pb file="add_6784_f193" o="193" n="385"/> <pb file="add_6784_f193v" o="193v" n="386"/> <pb file="add_6784_f194" o="194" n="387"/> <head xml:id="echoid-head60" xml:space="preserve"> 7. (o o) </head> <pb file="add_6784_f194v" o="194v" n="388"/> <pb file="add_6784_f195" o="195" n="389"/> <head xml:id="echoid-head61" xml:space="preserve"> De tactibus <lb/> Probl. 6 (. o -) </head> <pb file="add_6784_f195v" o="195v" n="390"/> <pb file="add_6784_f196" o="196" n="391"/> <pb file="add_6784_f196v" o="196v" n="392"/> <pb file="add_6784_f197" o="197" n="393"/> <head xml:id="echoid-head62" xml:space="preserve"> 6) De tactibus <lb/>[<emph style="it">tr: On touching </emph>]<lb/> </head> <p> <s xml:id="echoid-s66" xml:space="preserve"> problema. <lb/> Datis tribus circulis <lb/> sese mutuo contingentibus: <lb/> invenire quartum circulum <lb/> qui mutus tangetur in datis. <lb/>[<emph style="it">tr: Problem. <lb/> Given three circles, mutually touching, to find a fourth circle that is mutually touched by those given. </emph>]<lb/> </s> </p> <p> <s xml:id="echoid-s67" xml:space="preserve"> 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/> <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/> 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/> Agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> in continuum <lb/> <lb/>[...]<lb/> <lb/> Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> recta contingens <lb/> 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/> 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/> 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/> 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/> 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/> 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>. <lb/>[<emph style="it">tr: 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>, 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/> 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/> <lb/>[...]<lb/> <lb/> 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/> 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/> 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/> 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>. </emph>]<lb/> </s> </p> <p> <s xml:id="echoid-s68" xml:space="preserve"> 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/> 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/> describatur circulus. <lb/> Dico quod: ille est circulus quæsitus <lb/> et contingit tres datos <lb/> 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>. <lb/>[<emph style="it">tr: 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/> 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/> 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>. </emph>]<lb/> </s> </p> <pb file="add_6784_f197v" o="197v" n="394"/> <pb file="add_6784_f198" o="198" n="395"/> <head xml:id="echoid-head63" xml:space="preserve"> <emph style="st">6.)</emph> 7.) </head> <p xml:lang="lat"> <s xml:id="echoid-s69" xml:space="preserve"> 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/> <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/> 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/> 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>. <lb/>[<emph style="it">tr: 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>, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s70" xml:space="preserve"> Oportet invenire circulum <lb/> contingentem tres datos: <lb/> (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>) <lb/>[<emph style="it">tr: 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>). </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s71" xml:space="preserve"> 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/> et continuetur ad utraque partes <lb/> 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/> 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/> perpendicularis. <lb/> Continuetur ad partes contrarias <lb/> 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/> Tum primo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>s</mi></mstyle></math> <lb/> quæ secabit periferiam circuli <lb/> 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/> Secundo, agatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>k</mi></mstyle></math> <lb/> quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> productam in <lb/> puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> Ultimo, centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>, intervallo <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> describatur circulus. <lb/> Dico quod: ille est circulus quæsitus <lb/> et contingit tres datos in <lb/> 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>. <lb/>[<emph style="it">tr: 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 <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/> 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/> 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/> 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>, 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/> Second, there is drawn the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>l</mi></mstyle></math>, 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/> 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/> 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>. </emph>]<lb/> </s> </p> <head xml:id="echoid-head64" xml:space="preserve" xml:lang="lat"> Exegesis arithmetica <lb/> pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math> radio. <lb/>[<emph style="it">tr: Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>h</mi></mstyle></math>. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s72" xml:space="preserve"> Datorum circulorum radii <lb/> dati sunt, et centrorum <lb/> distantiæ. <lb/> 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/> data sunt. Inde perpendicularis <lb/> <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/> 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/> 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/> <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/> <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/> 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/> rectos et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> pro-<lb/> ducta concurret cum illa <lb/> 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/> 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/> 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/> cuius latera data sunt. et <lb/> antea datum fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>h</mi></mstyle></math>. ergo dantur <lb/> <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/> <lb/>[...]<lb/> <lb/> Ergo tota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> datur <lb/> <lb/>[...]<lb/> <lb/> Ergo datur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>p</mi></mstyle></math> <lb/> sed antea nota fuit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, <lb/> ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> datur <lb/> Quod quærebatur. <lb/>[<emph style="it">tr: The radii of the fiven circles are given, and the distances of their centres. <lb/> 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. 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>. 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>. 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>. 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/> 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>. 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>, 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/> <lb/>[...]<lb/> <lb/> Therefore the total <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>a</mi></mstyle></math> is given. <lb/> <lb/>[...]<lb/> <lb/> 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/> Which was sought. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s73" xml:space="preserve"> Per doctrinam sinuum <lb/> opus abbreviatur, sed <lb/> alia method ut convenit. <lb/>[<emph style="it">tr: By the doctrine of sines, the work is shorter, but another method, as convenient. </emph>]<lb/> </s> </p> <pb file="add_6784_f198v" o="198v" n="396"/> <pb file="add_6784_f199" o="199" n="397"/> <pb file="add_6784_f199v" o="199v" n="398"/> <pb file="add_6784_f200" o="200" n="399"/> <head xml:id="echoid-head65" xml:space="preserve"> 6.) </head> <head xml:id="echoid-head66" xml:space="preserve" xml:lang="lat"> Arithmetica Exegesis <lb/> radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math> <lb/>[<emph style="it">tr: Arithmetical exegesis, for radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s74" xml:space="preserve"> Datorum circulorum radij dati <lb/> sunt, et centrorum distantiæ <lb/> 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/> 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/> 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/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> contingens. <lb/>[<emph style="it">tr: The radii of given circles are given, and the distances of their centres. <lb/> 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>, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s75" xml:space="preserve"> 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/> Sunt igitur duo triangula <lb/> 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/> constituuntur super eandem <lb/> basim <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>z</mi></mstyle></math>. datur igitur verti-<lb/> cum distantia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math>. <lb/>[<emph style="it">tr: 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/> 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/> Therefore the vertical distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>b</mi></mstyle></math> is given. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s76" xml:space="preserve"> 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/> 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/> nota igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>n</mi></mstyle></math>. <lb/> 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/> circuli circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. <lb/> 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/> Tum: <lb/> Datur igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, cuius dimidium <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>y</mi></mstyle></math>, radius quæsitus. <lb/>[<emph style="it">tr: 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>, 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/> 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/> 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/> Then: <lb/> 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/> </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s77" xml:space="preserve"> Per Canonem triangulorum <lb/> alia methodo <emph style="super">ut covenit</emph>, operatio fit <lb/> brevior. <lb/>[<emph style="it">tr: By the Canons for triangles, there is another method, as convenient, which may be carried ore briefly. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s78" xml:space="preserve"> Nota. <lb/> 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/> fit etiam geometrica <lb/> 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>. <lb/>[<emph style="it">tr: Note. <lb/> 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>. </emph>]<lb/> </s> </p> <head xml:id="echoid-head67" xml:space="preserve" xml:lang="lat"> Arithmetica exegesis <lb/> radij <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math> <lb/> cæteris datis. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s79" xml:space="preserve"> Datorum circulorum radij dati <lb/> sunt, et centrorum distantiæ <lb/> 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/> Datur igitur perpendicularis <lb/> <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/> fit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>p</mi></mstyle></math>. <lb/> 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/> unde data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>. <lb/> 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/> nota; unde nota perpendicularis <lb/> <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/> 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/> 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/> Denique fiat: <lb/> Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>, quod <lb/> quærebatur. <lb/>[<emph style="it">tr: The radii of given circles are given, and the distances of their centres. <lb/> 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/> 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/> 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/> 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. 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/> 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/> 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/> Then let there be constructed: <lb/> 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. </emph>]<lb/> </s> </p> <head xml:id="echoid-head68" xml:space="preserve" xml:lang="lat"> Geometria exegesis <lb/> ipsius radii <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. <lb/>[<emph style="it">tr: Geometric exegesis, for the same radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>h</mi></mstyle></math>. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s80" xml:space="preserve"> Trium datorum circulorum <lb/> 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/> 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/> <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/> Ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi><mi>n</mi></mstyle></math>; quæ secabit circulum <lb/> 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/> Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>o</mi></mstyle></math>, quæ producta secabit <lb/> 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/> Agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>p</mi></mstyle></math> et producatur ad <lb/> utraque partes quæ secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <lb/> Tum fiat: <lb/> Datur igiture <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>a</mi></mstyle></math>, et centrum circuli <lb/> quæsiti. <lb/>[<emph style="it">tr: 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/> 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/> 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/> 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/> 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/> Then: <lb/> 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. </emph>]<lb/> </s> </p> <pb file="add_6784_f200v" o="200v" n="400"/> <pb file="add_6784_f201" o="201" n="401"/> <div xml:id="echoid-div19" type="page_commentary" level="2" n="19"> <p> <s xml:id="echoid-s81" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s81" xml:space="preserve"> The reference to Pappus is to Commandino's edition of Books III to VIII, <emph style="it">Mathematicae collecitones</emph> (1558). The proposition on page 48v–49 is Proposition IV.15 (not 13, as Harriot appears to have written). A diagram for this proposition appears on Add MS 6784, f. 202; this page shows only calculations of ratios. </s> <lb/> <quote xml:lang="lat"> Theorema XV. Propositio XV. <lb/> Iisdem positis describatur circulus HRT, qui & semicirculos iam dictos, & circulum LGH contingat in punctis HRT, atque a centris A P ad BC basim perpendiculares ducantur AM PN. Dico vt AM vna cum diametro circuli EGH ad diametrum ipsius, ita esse PN ad circuli HRT diametrum. </quote> <lb/> <quote> The same being supposed [as in Proposition 14], there is drawn the circle HRT, which touches both the semicircles 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 perpendiculars AM and PN. I say that as AM together with the diameter of the circle EGH is to that that diameter itself, so is PN to the diamter of the circle HRT. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head69" xml:space="preserve"> 5.) pappus. prop. 13. pag. 49. </head> <pb file="add_6784_f201v" o="201v" n="402"/> <pb file="add_6784_f202" o="202" n="403"/> <head xml:id="echoid-head70" xml:space="preserve"> 4.) </head> <p> <s xml:id="echoid-s83" xml:space="preserve"> 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/> contingant se in puncto <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/> 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/> oportet describere circulum <lb/> contingentem duos circulos <lb/> datos, et lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>. <lb/>[<emph style="it">tr: 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/> 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/> 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>. </emph>]<lb/> </s> </p> <p> <s xml:id="echoid-s84" xml:space="preserve"> <lb/>[...]<lb/> <lb/> 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/> 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/> <lb/>[...]<lb/> <lb/>[<emph style="it">tr: <lb/>[...]<lb/> <lb/> 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/> 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/> <lb/>[...]<lb/> </emph>]<lb/> </s> </p> <p> <s xml:id="echoid-s85" xml:space="preserve"> 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/> agatur ad angulos rectos, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>a</mi></mstyle></math>. <lb/> 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/> agatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>a</mi></mstyle></math>, quæ secabit periferi-<lb/> am minoris circuli in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi></mstyle></math>. <lb/> 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/> riam maioris circulam in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/> 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/> 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/> erit quæsitus. <lb/>[<emph style="it">tr: 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/> 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/> 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/> 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/> 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/> 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. </emph>]<lb/> </s> </p> <pb file="add_6784_f202v" o="202v" n="404"/> <pb file="add_6784_f203" o="203" n="405"/> <div xml:id="echoid-div20" type="page_commentary" level="2" n="20"> <p> <s xml:id="echoid-s86" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s86" xml:space="preserve"> The reference to Pappus is to Commandino's edition of Books III to VIII, <emph style="it">Mathematicae collecitones</emph> (1558). The proposition on page 47 is Proposition IV.14. Harriot's diagram is the same as the one given by Commandino except for his use of lower case letters. A second diagram for the same proposition appears on Add MS 6784, f. 204. </s> <lb/> <quote xml:lang="lat"> Theorema XIIII. Propositio XIIII. <lb/> Sint duo semicirculi BGC BED: & ipsos contingat circulus EFGH: a cuius centro A ad BC basim semicirculorum perpendicularis ducatur AM. Dico ut BM as eam, quæ ex centro circuli EFGH, ita esse in prima figura vtramque simul CB BD ad earum excessum CD; in secunda vero, & tertia figura, ita esse excessum CB BD ad vtramque ipsarum CB BD. </quote> <lb/> <quote> Let there be two semicircles BGC and BED, and their touching circle EFGH, from whose centre A to BC, the base of the semicircle, there is drawn the perpendicular AM. I say that as BM is to that line from the centre of the circle EFGH, inthe first figure will be CB and BD togher to their excess, CD; but in the second and third figure, it will be as the excess of CB over BD to both of CB and BD together. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head71" xml:space="preserve" xml:lang="lat"> pappus. pag. <lb/> 47. <lb/>[<emph style="it">tr: Pappus, page 47. </emph>]<lb/> </head> <pb file="add_6784_f203v" o="203v" n="406"/> <pb file="add_6784_f204" o="204" n="407"/> <div xml:id="echoid-div21" type="page_commentary" level="2" n="21"> <p> <s xml:id="echoid-s88" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s88" xml:space="preserve"> A further diagram for Pappus, <emph style="it">Mathematicae collectiones</emph>, Propostion IV.14. See also the previous folio, Add MS 6784, f. 203. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat"> pappus. pag. <lb/> 47. <lb/>[<emph style="it">tr: Pappus, page 47. </emph>]<lb/> </head> <pb file="add_6784_f204v" o="204v" n="408"/> <pb file="add_6784_f205" o="205" n="409"/> <div xml:id="echoid-div22" type="page_commentary" level="2" n="22"> <p> <s xml:id="echoid-s90" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s90" xml:space="preserve"> Further work on Pappus, Propostion IV.14. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head73" xml:space="preserve"> 2) pappus. pag. 47 </head> <pb file="add_6784_f205v" o="205v" n="410"/> <pb file="add_6784_f206" o="206" n="411"/> <div xml:id="echoid-div23" type="page_commentary" level="2" n="23"> <p> <s xml:id="echoid-s92" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s92" xml:space="preserve"> Further work on Pappus, Propostion IV.14. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head74" xml:space="preserve"> 3) pappus. <emph style="super">pag.</emph> 47 </head> <pb file="add_6784_f206v" o="206v" n="412"/> <pb file="add_6784_f207" o="207" n="413"/> <div xml:id="echoid-div24" type="page_commentary" level="2" n="24"> <p> <s xml:id="echoid-s94" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s94" xml:space="preserve"> Lists of variations of increasing (c) and decreasing (d) columns, together with other rough work for the 'Magisteria' (Add MS 6782, f. 107 to f. 146v). <lb/> This page is important because it carries a date, day, time, and year: June 28 (Sunday) 10.30am, 1618. </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s96" xml:space="preserve"> De causa reflexionis ad angulos æquales. <lb/>[<emph style="it">tr: On the cause of reflection at equal angles. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s97" xml:space="preserve"> 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/> ante mer: 1618 <lb/>[<emph style="it">tr: June 28 (Sunday) 10.30am 1618 </emph>]<lb/> </s> </p> <pb file="add_6784_f207v" o="207v" n="414"/> <div xml:id="echoid-div25" type="page_commentary" level="2" n="25"> <p> <s xml:id="echoid-s98" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s98" xml:space="preserve"> Further lists of variations of increasing (c) and decreasing (d) columns (see Add MS 6784, f. 413). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f208" o="208" n="415"/> <div xml:id="echoid-div26" type="page_commentary" level="2" n="26"> <p> <s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s100" xml:space="preserve"> Difference tables similar to those on pages 10 and 11 of the 'Magisteria' (Add MS 6782, f. 117 and f. 118). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f208v" o="208v" n="416"/> <div xml:id="echoid-div27" type="page_commentary" level="2" n="27"> <p> <s xml:id="echoid-s102" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s102" xml:space="preserve"> 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, similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f209" o="209" n="417"/> <pb file="add_6784_f209v" o="209v" n="418"/> <pb file="add_6784_f210" o="210" n="419"/> <div xml:id="echoid-div28" type="page_commentary" level="2" n="28"> <p> <s xml:id="echoid-s104" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s104" xml:space="preserve"> Rough working for page 15 of the 'Magisteria' (Add MS 6782, f. 122). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f210v" o="210v" n="420"/> <div xml:id="echoid-div29" type="page_commentary" level="2" n="29"> <p> <s xml:id="echoid-s106" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s106" xml:space="preserve"> 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, similar to those on page 14 of the 'Magisteria' (Add MS 6782, f. 121). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f211" o="211" n="421"/> <div xml:id="echoid-div30" type="page_commentary" level="2" n="30"> <p> <s xml:id="echoid-s108" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s108" xml:space="preserve"> An incomplete version of the difference table on page 9 of the 'Magisteria' (Add MS 6782, f. 116). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f211v" o="211v" n="422"/> <div xml:id="echoid-div31" type="page_commentary" level="2" n="31"> <p> <s xml:id="echoid-s110" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s110" xml:space="preserve"> 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; see page 16 of the 'Magisteria' (Add MS 6782, f. 123). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f212" o="212" n="423"/> <div xml:id="echoid-div32" type="page_commentary" level="2" n="32"> <p> <s xml:id="echoid-s112" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s112" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat"> Operatio. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. <lb/>[<emph style="it">tr: Working on <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math> </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s114" xml:space="preserve"> operatio. 1<emph style="super">a</emph> <lb/>[<emph style="it">tr: Working (1) </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s115" xml:space="preserve"> operatio. 2<emph style="super">a</emph> <lb/>[<emph style="it">tr: Working (2) </emph>]<lb/> </s> </p> <pb file="add_6784_f212v" o="212v" n="424"/> <pb file="add_6784_f213" o="213" n="425"/> <div xml:id="echoid-div33" type="page_commentary" level="2" n="33"> <p> <s xml:id="echoid-s116" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s116" xml:space="preserve"> Calculations relating to formula (5) on pages 27 to 34 of the 'Magisteria' (Add MS 6782, f. 134 to f. 142). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head76" xml:space="preserve" xml:lang="lat"> Residuum operationis. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi></mstyle></math>. </head> <p xml:lang="lat"> <s xml:id="echoid-s118" xml:space="preserve"> 2<emph style="super">a</emph> Working (2) </s> </p> <pb file="add_6784_f213v" o="213v" n="426"/> <pb file="add_6784_f214" o="214" n="427"/> <div xml:id="echoid-div34" type="page_commentary" level="2" n="34"> <p> <s xml:id="echoid-s119" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s119" xml:space="preserve"> General notation for triangular numbers. <lb/> See also page 2 of the 'Magisteria' (Add MS 6782, f. 109). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head77" xml:space="preserve" xml:lang="lat"> 3<emph style="super">a</emph> notatio triangularium per notas generales. <lb/>[<emph style="it">tr: 3rd notation for triangular numbers, in general symbols. </emph>]<lb/> </head> <pb file="add_6784_f214v" o="214v" n="428"/> <pb file="add_6784_f215" o="215" n="429"/> <pb file="add_6784_f215v" o="215v" n="430"/> <pb file="add_6784_f216" o="216" n="431"/> <div xml:id="echoid-div35" type="page_commentary" level="2" n="35"> <p> <s xml:id="echoid-s121" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s121" xml:space="preserve"> Square roots of binomes of the fifth and sixth kind by the general rule derived in Add MS 6788, f. 15 (and elsewhere). Here Harriot works with two types of fifth binome, (<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>), according to whether the difference between the squares of the two terms is a square or not. 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/> Similarly he distinguishes two types of sixth binomes, (<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>). 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/> In all cases the roots are cross-checked. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f216v" o="216v" n="432"/> <pb file="add_6784_f217" o="217" n="433"/> <div xml:id="echoid-div36" type="page_commentary" level="2" n="36"> <p> <s xml:id="echoid-s123" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s123" xml:space="preserve"> Square roots of binomes of the third and fourth kind (<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>), by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In both cases the roots are checked by multiplication. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f217v" o="217v" n="434"/> <pb file="add_6784_f218" o="218" n="435"/> <div xml:id="echoid-div37" type="page_commentary" level="2" n="37"> <p> <s xml:id="echoid-s125" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s125" xml:space="preserve"> Square roots of binomes of the first and second kind (<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>), by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In both cases the roots are checked by multiplication. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f218v" o="218v" n="436"/> <pb file="add_6784_f219" o="219" n="437"/> <div xml:id="echoid-div38" type="page_commentary" level="2" n="38"> <p> <s xml:id="echoid-s127" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s127" xml:space="preserve"> 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>, by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. The numerical examples in Add MS 6783, f. 360v, f. 361, and Add MS 6782, f. 228, are closely related to the work on this page. </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s129" xml:space="preserve"> Nam: eius quadratum <lb/>[<emph style="it">tr: For: its square </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s130" xml:space="preserve"> Quia: duo quad: <lb/>[<emph style="it">tr: Because: two squares </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s131" xml:space="preserve"> Et: duo rectang: <lb/>[<emph style="it">tr: And: two rectangles </emph>]<lb/> </s> </p> <pb file="add_6784_f219v" o="219v" n="438"/> <pb file="add_6784_f220" o="220" n="439"/> <div xml:id="echoid-div39" type="page_commentary" level="2" n="39"> <p> <s xml:id="echoid-s132" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s132" xml:space="preserve"> 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>, by the general rule derived in Add MS 6788, f. 15 (and elsewhere). In each case, the root is checked by multiplication. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f220v" o="220v" n="440"/> <pb file="add_6784_f221" o="221" n="441"/> <head xml:id="echoid-head78" xml:space="preserve" xml:lang="lat"> Examinatio æquationis per numeros <lb/>[<emph style="it">tr: An examination of an equation in numbers </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s134" xml:space="preserve"> et ita est (ut supra) <lb/>[<emph style="it">tr: and so it is (as above) </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s135" xml:space="preserve"> et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> <lb/>[<emph style="it">tr: and for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s136" xml:space="preserve"> et ita est (ut infra) <lb/>[<emph style="it">tr: and so it is (as below) </emph>]<lb/> </s> </p> <pb 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file="add_6784_f244" o="244" n="487"/> <pb file="add_6784_f244v" o="244v" n="488"/> <pb file="add_6784_f245" o="245" n="489"/> <pb file="add_6784_f245v" o="245v" n="490"/> <pb file="add_6784_f246" o="246" n="491"/> <head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat"> 3. </head> <p xml:lang="lat"> <s xml:id="echoid-s137" xml:space="preserve"> In Achille <lb/>[<emph style="it">tr: On Achilles </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s138" xml:space="preserve"> vel per æquationem rationum <lb/>[<emph style="it">tr: or by the equality of ratios </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s139" xml:space="preserve"> Aliter <lb/>[<emph style="it">tr: Another way </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s140" xml:space="preserve"> Sit ratio motus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math>, ad <math 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The first verse sets out the rules for multiplication. The second and third verses deal with subtraction of a negative quantity from a negative quantity, where the result may be either positive or negative. <lb/> Like folios Add MS 6784, f. 323, f. 324, which follow soon after it, this one appears to be based on Viète, <emph style="it">In artem analyticen isagoge</emph>, 1591, in this case on Chapter IV, Praeceptum II and Praeceptum III. </s> </p> </emph>] <lb/><lb/></s></p></div> <p> <s xml:id="echoid-s145" xml:space="preserve"> If more by more must needes make more <lb/> Then lesse by more makes lesse of more <lb/> And lesse by lesse makes lesse of lesse <lb/> If more be more and lesse be lesse. </s> </p> <p> <s xml:id="echoid-s146" xml:space="preserve"> Yet lesse of lesse makes lesse or more <lb/> The which is best keep both in store <lb/> If lesse of lesse thou <emph style="super">you</emph> wilt make lesse <lb/> Then pull <emph style="super">bate</emph> the same from that is lesse </s> </p> <p> <s xml:id="echoid-s147" xml:space="preserve"> But if the same thou <emph style="super">you</emph> wilt make more <lb/> Then add the same <emph style="super">to it</emph> to that is <emph style="super">the sign of</emph> more <lb/> The signe <emph style="super">rule</emph> of more is best to use <lb/> Except some <emph style="super">Yet for some</emph> cause the <emph style="super">do</emph> other choose <emph style="super">then it refuse</emph> <lb/> For <emph style="super">So</emph> <emph style="super">Yet</emph> both are one, for both are true <lb/> of this inough and so adew. </s> </p> <pb file="add_6784_f322" o="322" n="643"/> <div xml:id="echoid-div41" type="page_commentary" level="2" n="41"> <p> <s xml:id="echoid-s148" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s148" xml:space="preserve"> This page shows several examples of additions and subtractions using letters. Note that here such operations are only carried out between quantities of the same dimension. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat"> 1) Operationes logisticæ, in notis <lb/>[<emph style="it">tr: The operations of arithmetic in symbols. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s150" xml:space="preserve"> adde <lb/>[<emph style="it">tr: add </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s151" xml:space="preserve"> summa <lb/>[<emph style="it">tr: sum </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s152" xml:space="preserve"> subduce <lb/>[<emph style="it">tr: subtract </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s153" xml:space="preserve"> reliqua <lb/>[<emph style="it">tr: remainder </emph>]<lb/> </s> </p> <pb file="add_6784_f322v" o="322v" n="644"/> <pb file="add_6784_f323" o="323" n="645"/> <div xml:id="echoid-div42" type="page_commentary" level="2" n="42"> <p> <s xml:id="echoid-s154" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s154" xml:space="preserve"> This page shows examples of multiplication and division using letters. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head83" xml:space="preserve"> 2) </head> <p xml:lang="lat"> <s xml:id="echoid-s156" xml:space="preserve"> multip. <lb/>[<emph style="it">tr: multiply </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s157" xml:space="preserve"> in <lb/>[<emph style="it">tr: by </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s158" xml:space="preserve"> facta <lb/>[<emph style="it">tr: product </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s159" xml:space="preserve"> applica <lb/>[<emph style="it">tr: divide </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s160" xml:space="preserve"> ad <lb/>[<emph style="it">tr: by </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s161" xml:space="preserve"> orta <lb/>[<emph style="it">tr: result </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s162" xml:space="preserve"> manifestum <lb/> per præcog-<lb/> nitam genera-<lb/> tionem. <lb/>[<emph style="it">tr: evident from the previously learned constructions </emph>]<lb/> </s> </p> <pb file="add_6784_f323v" o="323v" n="646"/> <pb file="add_6784_f324" o="324" n="647"/> <div xml:id="echoid-div43" type="page_commentary" level="2" n="43"> <p> <s xml:id="echoid-s163" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s163" xml:space="preserve"> The examples of division on this page are taken directly from Viète, <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter IV, end of Praeceptum IV, but Harriot has re-written the examples in his own symbolic notation. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head84" xml:space="preserve"> 3) </head> <pb file="add_6784_f324v" o="324v" n="648"/> <pb file="add_6784_f325" o="325" n="649"/> <head xml:id="echoid-head85" xml:space="preserve"> 4) </head> <div xml:id="echoid-div44" type="page_commentary" level="2" n="44"> <p> <s xml:id="echoid-s165" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s165" xml:space="preserve"> The terminology and examples on this page are taken directly from Viète, <emph style="it">In artem analyticen isagoge</emph>, 1591, Chapter V, but Harriot has re-written the examples in his own symbolic notation. </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang=""> <s xml:id="echoid-s167" xml:space="preserve"> Sit: <lb/>[<emph style="it">tr: Let: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s168" xml:space="preserve"> Dico quod: per Antithesin. <lb/>[<emph style="it">tr: I say that, by antihesis: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s169" xml:space="preserve"> Quoniam: <lb/>[<emph style="it">tr: Because: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s170" xml:space="preserve"> Adde utrolique. <lb/>[<emph style="it">tr: Add to each side. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s171" xml:space="preserve"> Ergo: <lb/>[<emph style="it">tr: Therefore: </emph>]<lb/> </s> </p> <p xml:lang=""> <s xml:id="echoid-s172" xml:space="preserve"> Secundo: sit, <lb/>[<emph style="it">tr: Second, let: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s173" xml:space="preserve"> Dico quod: <lb/>[<emph style="it">tr: I say that: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s174" xml:space="preserve"> Quoniam: <lb/>[<emph style="it">tr: Because: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s175" xml:space="preserve"> Adde utrolique. <lb/>[<emph style="it">tr: Add to each side. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s176" xml:space="preserve"> Ergo. <lb/>[<emph style="it">tr: Therefore. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s177" xml:space="preserve"> Et ita. <lb/>[<emph style="it">tr: And thus. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s178" xml:space="preserve"> Sit. <lb/>[<emph style="it">tr: Let. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s179" xml:space="preserve"> Dico quod. per Hypobibasmum. <lb/>[<emph style="it">tr: I say that, by hypobibasmus. </emph>]<lb/> </s> </p> <p xml:lang=""> <s xml:id="echoid-s180" xml:space="preserve"> Sit. <lb/>[<emph style="it">tr: Let. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s181" xml:space="preserve"> Dico quod: per Parabolismum. <lb/>[<emph style="it">tr: I say that, by parabolismus. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s182" xml:space="preserve"> Vel, sit: <lb/>[<emph style="it">tr: Or, let: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s183" xml:space="preserve"> dico quod. <lb/>[<emph style="it">tr: I say that. </emph>]<lb/> </s> </p> <pb file="add_6784_f325v" o="325v" n="650"/> <pb file="add_6784_f326" o="326" n="651"/> <pb file="add_6784_f326v" o="326v" n="652"/> <pb file="add_6784_f327" o="327" n="653"/> <pb file="add_6784_f327v" o="327v" n="654"/> <pb file="add_6784_f328" o="328" n="655"/> <pb file="add_6784_f328v" o="328v" n="656"/> <pb file="add_6784_f329" o="329" n="657"/> <pb file="add_6784_f329v" o="329v" n="658"/> <pb file="add_6784_f330" o="330" n="659"/> <pb file="add_6784_f330v" o="330v" n="660"/> <pb file="add_6784_f331" o="331" n="661"/> <div xml:id="echoid-div45" type="page_commentary" level="2" n="45"> <p> <s xml:id="echoid-s184" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s184" xml:space="preserve"> The reference to Apollonius is to pages 5 and 6 of Commandino's edition, <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). There are also references at the bottom of the page to Viète an Cardano. </s> <lb/> <s xml:id="echoid-s185" xml:space="preserve"> The reference to Viète is to <emph style="it">Apollonius Gallus</emph>, Appendix 2, Problem V. </s> <lb/> <quote xml:lang="lat"> V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant rationem. </quote> <lb/> <quote> Given a triangle, find a point from which there may be drawn three straight lines to the vertices of the given triangle, keeping a fixed ratio. </quote> <lb/> <s xml:id="echoid-s186" xml:space="preserve"> The reference to Cardano is to his <emph style="it">Opus novum de proportionibus</emph>. The relevant Propositions are 154 (though mistakenly described in the <emph style="it">Opus novum</emph> as 144) and 160. </s> <lb/> <quote xml:lang="lat"> Propositio centesimaquadragesimaquarta <lb/> Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, ipsæ erunt in proportione mediæ ad adiectam. <lb/> Hæc propositio est admirabilis: ... </quote> <lb/> <quote xml:lang="lat"> Propositio centesimasexagesima <lb/> Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus datis. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat"> 5. Appolonius. pag. 5. 6. <lb/>[<emph style="it">tr: Apollonius, pages 5, 6. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s188" xml:space="preserve"> Quæsitum: <lb/> ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> 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>. <lb/>[<emph style="it">tr: Sought: <lb/> 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s189" xml:space="preserve"> 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/> 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/> 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> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s190" xml:space="preserve"> Dico quod: <lb/>[<emph style="it">tr: I say that: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s191" xml:space="preserve"> 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/> minor, quam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi></mstyle></math> <lb/> 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/> fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>h</mi></mstyle></math> periferia <lb/> sumatur quovis puncta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> 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>. <lb/>[<emph style="it">tr: 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, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s192" xml:space="preserve"> * 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/> ubicunque signatur in periferia punctum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> <lb/> 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>. <lb/>[<emph style="it">tr: 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, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s193" xml:space="preserve"> Corollaria. <lb/> Hinc a tribus punctis sive sint in recta <lb/> vel non; possunt duci tres lineæ ad unum <lb/> punctum, <emph style="st">ut s</emph> et erunt in data ratione. <lb/>[<emph style="it">tr: Corollary <lb/> Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, and they will be in the given ratio. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s194" xml:space="preserve"> vide vertam <lb/> in Apolonio gallo <lb/> et card: de prop. pag. 145. 162. <lb/>[<emph style="it">tr: see over, in <emph style="it">Apollonius Gallus</emph>, and Cardano, <emph style="it">De proportionibus</emph>, pages 145, 162. </emph>]<lb/> </s> </p> <pb file="add_6784_f331v" o="331v" n="662"/> <pb file="add_6784_f332" o="332" n="663"/> <div xml:id="echoid-div46" type="page_commentary" level="2" n="46"> <p> <s xml:id="echoid-s195" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s195" xml:space="preserve"> The reference is to pages 5 and 6 of Commandino's edition of Apollonius, <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph> (1566). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat"> Ad appolonium. pa. 5. 6. <lb/>[<emph style="it">tr: On Apollonius, pages 5, 6 </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s197" xml:space="preserve"> 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/> Invenire lineam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> <lb/> ita ut sit: <lb/> Sit factum: <lb/> Tum: <lb/>[<emph style="it">tr: 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/> Let it be done, then: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s198" xml:space="preserve"> Aliter <lb/> <lb/>[...]<lb/> <lb/> sed idem ut supra <lb/>[<emph style="it">tr: Another way <lb/> <lb/>[...]<lb/> <lb/> but the same as above </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s199" xml:space="preserve"> Invenire <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math> <lb/>[<emph style="it">tr: To find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>k</mi></mstyle></math>. </emph>]<lb/> </s> </p> <pb file="add_6784_f332v" o="332v" n="664"/> <pb file="add_6784_f333" o="333" n="665"/> <pb file="add_6784_f333v" o="333v" n="666"/> <pb file="add_6784_f334" o="334" n="667"/> <pb file="add_6784_f334v" o="334v" n="668"/> <pb file="add_6784_f335" o="335" n="669"/> <pb file="add_6784_f335v" o="335v" n="670"/> <pb file="add_6784_f336" o="336" n="671"/> <pb file="add_6784_f336v" o="336v" n="672"/> <pb file="add_6784_f337" o="337" n="673"/> <pb file="add_6784_f337v" o="337v" n="674"/> <pb file="add_6784_f338" o="338" n="675"/> <pb file="add_6784_f338v" o="338v" n="676"/> <pb file="add_6784_f339" o="339" n="677"/> <pb file="add_6784_f339v" o="339v" n="678"/> <pb file="add_6784_f340" o="340" n="679"/> <pb file="add_6784_f340v" o="340v" n="680"/> <pb file="add_6784_f341" o="341" n="681"/> <pb file="add_6784_f341v" o="341v" n="682"/> <pb file="add_6784_f342" o="342" n="683"/> <pb file="add_6784_f342v" o="342v" n="684"/> <pb file="add_6784_f343" o="343" n="685"/> <pb file="add_6784_f343v" o="343v" n="686"/> <pb file="add_6784_f344" o="344" n="687"/> <pb file="add_6784_f344v" o="344v" n="688"/> <pb file="add_6784_f345" o="345" n="689"/> <pb file="add_6784_f345v" o="345v" n="690"/> <pb file="add_6784_f346" o="346" n="691"/> <pb file="add_6784_f346v" o="346v" n="692"/> <pb file="add_6784_f347" o="347" n="693"/> <pb file="add_6784_f347v" o="347v" n="694"/> <pb file="add_6784_f348" o="348" n="695"/> <div xml:id="echoid-div47" type="page_commentary" level="2" n="47"> <p> <s xml:id="echoid-s200" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s200" xml:space="preserve"> On this page Harriot investigates Proposition 18 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XVIII. <lb/> Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. </quote> <lb/> <quote> If two triangles are each isosceles, equal to one another in theri legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg. </quote> <lb/> <s xml:id="echoid-s201" xml:space="preserve"> For Harriot's statement of Propostion 18, and a geometric version of the proof, see Add MS 6784, f. 349. Here he works the proposition algebraically. </s> <lb/> <s xml:id="echoid-s202" xml:space="preserve"> This page also refers to Proposition 17 from the <emph style="it">Supplementum</emph>, (see MS 6784, f. 350). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head88" xml:space="preserve"> prop. 18. Supplementi. <lb/>[<emph style="it">tr: Proposition 18 from the Supplementum </emph>]<lb/> </head> <pb file="add_6784_f348v" o="348v" n="696"/> <pb file="add_6784_f349" o="349" n="697"/> <div xml:id="echoid-div48" type="page_commentary" level="2" n="48"> <p> <s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s204" xml:space="preserve"> On this page Harriot investigates Proposition 18 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XVIII. <lb/> Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: triplum solidum sub quadrato cruris communis & dimidia base primi multata continuatave longitudine ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi & ejusdem cruris quadrato. </quote> <lb/> <quote> If two triangles are each isosceles, equal to one another in theri legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg. </quote> <lb/> <s xml:id="echoid-s205" xml:space="preserve"> This page refers to several previous propositions from the <emph style="it">Supplementum</emph>, namely Proposition 12 and 14b (Add MS 6784, f. 353), Proposition 16 (add MS 6784, f. 351) and Proposition 17 (add MS 6784, f. 350). </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head89" xml:space="preserve"> prop. 18. Supplementi. <lb/>[<emph style="it">tr: Proposition 18 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s207" xml:space="preserve"> Si duo triangula fuerint aequicrura singula, et ipsa alterum alteri cruribus aequalia; angulus <lb/> autem qui est ad basin secundi sit triplus anguli qui est ad basin primi. Triplum solidum <lb/> sub quadrato cruris communis, et dimidia base primi multata continuatave longitudine <lb/> ejus cujus quadratum æquale est triplo quadrato altitudinis primi, cum multabitur ejusdem <lb/> dimidiæ basis multatæ continuatve cubo, æquale est solido sub base secundi et ejusdem <lb/> cruris quadrato. <lb/>[<emph style="it">tr: If two triangles are each isosceles, equal to one another in their legs, and moreover the angle at the base of the second is three times the angle at the base of the first, then three times the product of the square of the common leg and half the base of the first decreased or increased by a length whose square is equal to three times the square of the altitude of the first, when reduced by the cube of the same half base thus decreased or increased, is equal to the product of the second base and the square of the common leg. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s208" xml:space="preserve"> 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/> <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/> 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/> <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/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>  <lb/> Dico <lb/>[<emph style="it">tr: 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. 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s209" xml:space="preserve"> Nam: <lb/> per 15,p <lb/>[...]<lb/> Hoc est, in notis proportionalium quas notum 12,p <lb/> 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/> <lb/>[...]<lb/> <lb/> Hoc est in notis 12,p. <lb/>[<emph style="it">tr: For by Proposition 15 <lb/>[...]<lb/> that is, in the notation for proportionals noted in Proposition 12, <lb/> 1. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>G</mi></mstyle></math>. <lb/> <lb/>[...]<lb/> <lb/> That is, in the notation of Proposition 12 </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s210" xml:space="preserve"> 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/> <lb/>[...]<lb/> <lb/> Hoc est in notis 12,p. <lb/>[<emph style="it">tr: 2. Multiply everything by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>G</mi></mstyle></math>. <lb/> <lb/>[...]<lb/> <lb/> That is, in the notation of Proposition 12 </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s211" xml:space="preserve"> Deinde per 16.p <lb/> Hoc est in notis 12,p. <lb/> Sed: per consect: 14.p <lb/> Ergo patet propositum <lb/>[<emph style="it">tr: Thence by Proposition 16, <lb/> That is, in the notation of Proposition 12 <lb/> But by the consequence of Proposition 14, <lb/> Thus the propostion is shown. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s212" xml:space="preserve"> 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>) ideo eius partes <lb/> 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/> 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/> 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/> Propositum igitur simplicibus notis ita significatur: <lb/>[<emph style="it">tr: 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, 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; 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>. 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/> In simple notation the proposition may therefore be written: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s213" xml:space="preserve"> igitur: <lb/> Quando æquatio est sub ista <lb/> forma: <lb/> <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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <pb file="add_6784_f349v" o="349v" n="698"/> <pb file="add_6784_f350" o="350" n="699"/> <div xml:id="echoid-div49" type="page_commentary" level="2" n="49"> <p> <s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s214" xml:space="preserve"> On this page Harriot investigates Proposition 17 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XVII. <lb/> Si duo triangula fuerint aequicrura singula, & ipsa alterumalteria cruribus aequalia, angulus autem, quem is qui est ad basin secundi relinquit e duobus rectis, sit triplus anguli qui est ad basin primi: solidum triplum sub base primi & cruris communis quadrato, minus cubo e base primi, aequale est solido sub base secundi & cruris communis quadrato. </quote> <lb/> <quote> If two triangles are each isosceles, both with equal legs, and moreover the angle at the base of the second subtracted from two right angles is three times the angle at the base of the first, then three times the product of the base of the first and the square of the common side, minus the cube of the first base, is equal to the product of the second base and the square of the common side. </quote> <lb/> <s xml:id="echoid-s215" xml:space="preserve"> The working contains reference to three propositions from Euclid's <emph style="it">Elements</emph>. </s> <lb/> <quote> II.6 If a straight line be bisected and produced to any point, the rectangle contained by the whole line so increased, and the part produced, together with the square of half the line, is equal to the square of the line made up of the half, and the produced part. </quote> <lb/> <quote> III.36 If from a point without a circle two straight lines be drawn to it, one of which is a tangent to the circle, and the other cuts it; the rectangle under the whole cutting line and the external segment is equal to the square of the tangent. </quote> <lb/> <quote> I. 47 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head90" xml:space="preserve"> prop. 17. Supplementi. <lb/>[<emph style="it">tr: Proposition 17 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s217" xml:space="preserve"> Si duo triangula fuerint aequicrura singula, <lb/> et ipsa alterumalteria cruribus aequalia; angulus <lb/> autem, quem is qui est ad basin secundi relinquit <lb/> e duobus rectis, sit triplus anguli qui est ad basin <lb/> <emph style="st">secundi</emph> <emph style="super">primi</emph>. Solidum triplum sub base primi et cruris <lb/> communis quadrato, minus cubo e base primi: aequale <lb/> est solido sub base secundiet cruris communis <lb/> quadrato. <lb/>[<emph style="it">tr: If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, 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, is equal to the product of the second base and the square of the same side. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s218" xml:space="preserve"> per 6,2 el. <lb/> per 36,3 el. <lb/> per 47,1 el. <lb/> <lb/>[...]<lb/> <lb/> quia parallogramma æquialta <lb/> et sunt ut bases. <lb/> <lb/>[...]<lb/> <lb/> vel per notas <lb/> simplices <lb/> Hæque Resoluatur Analogia, erit: <lb/> Propositum <lb/>[<emph style="it">tr: by Elements II.6 <lb/> by Elements III.35 <lb/> by Elements I.47 <lb/> <lb/>[...]<lb/> <lb/> because the parallelograms are of equal height and are as the bases. <lb/> <lb/>[...]<lb/> <lb/> or in simple notation <lb/> And this ratio is resolved, hence the proposition: </emph>]<lb/> </s> </p> <pb file="add_6784_f350v" o="350v" n="700"/> <pb file="add_6784_f351" o="351" n="701"/> <div xml:id="echoid-div50" type="page_commentary" level="2" n="50"> <p> <s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s219" xml:space="preserve"> On this page Harriot investigates Proposition 16 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XVI. <lb/> Si duo triangula fuerint aequicrura singula, & ipsa alterum alteri cruribus aequalia, angulus autem qui est ad basin secundi sit triplus anguli qui est ad basin primi: cubus ex base primi, minus triplo solido sub base primi & cruris communis quadrato, aequalis est solido sub base secundi & ejusdem cruris quadrato. </quote> <lb/> <quote> If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, 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, is equal to the product of the second base and the square of the same side. </quote> <lb/> <s xml:id="echoid-s220" xml:space="preserve"> The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.5. </s> <lb/> <quote> II.5 If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half that line. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head91" xml:space="preserve"> prop. 16. Supplementi. <lb/>[<emph style="it">tr: Proposition 16 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s222" xml:space="preserve"> Si duo triangula fuerint aequicrura singula, <lb/> et ipsa alterum alteri cruribus aequalia: angulus <lb/> autem qui est ad basin secundi sit triplus <lb/> anguli qui est ad basin primi. Cubus ex <lb/> base primi, minus triplo solido sub base primi <lb/> et cruris communis quadrato, aequalis <lb/> est solido sub base secundi et ejusdem <lb/> cruris quadrato. <lb/>[<emph style="it">tr: If two triangles are each isosceles, the legs of one equal to the legs of the other, and moreover the angle at the base of the second is three times the angle at the base of the first, 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, is equal to the product of the second base and the square of the same side. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s223" xml:space="preserve"> per 5,2 el. <lb/> <lb/>[...]<lb/> <lb/> Quia parallogramma æquialta <lb/> 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/> <lb/>[...]<lb/> <lb/> vel per notas <lb/> simplices <lb/> Resoluatur analogia et erit: <lb/> Propositum <lb/>[<emph style="it">tr: by Elements II.5 <lb/> <lb/>[...]<lb/> <lb/> 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/> <lb/>[...]<lb/> <lb/> or in simple notation <lb/> The ratio is resolved, and hence the proposition: </emph>]<lb/> </s> </p> <pb file="add_6784_f351v" o="351v" n="702"/> <pb file="add_6784_f352" o="352" n="703"/> <div xml:id="echoid-div51" type="page_commentary" level="2" n="51"> <p> <s xml:id="echoid-s224" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s224" xml:space="preserve"> On this page Harriot investigates Proposition 15 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XV. <lb/> Si e circumferential circuli cadant in diametrum perpendiculares duæ, una in centro, altera extra centrum; & ad perpendicularem in centro agatur ex puncto incidentiæ perpendicularis alterius, linea recta faciens cum diametro angulum æqualem trienti recti; a puncto autem quo acta illa secat perpendiculare in centro, ducatur alia linea recta ad angulum semicirculi: triplum quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est proportionalis. </quote> <lb/> <quote> If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean proportional. </quote> <lb/> <s xml:id="echoid-s225" xml:space="preserve"> The working contains a reference to Euclid's <emph style="it">Elements</emph>, Proposition II.4. </s> <lb/> <quote> II.4 If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts, together with twice the rectangle contained by the parts. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head92" xml:space="preserve"> prop. 15. Supplementi <lb/>[<emph style="it">tr: Proposition 15 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s227" xml:space="preserve"> Si e circumferential circuli cadant in <lb/> diametrum perpendiculares duæ; una in <lb/> centro; altera extra centrum: et ad per-<lb/> pendicularem in centro agatur ex puncto <lb/> incidentiæ perpendicularis alterius, linea <lb/> recta faciens cum diametro angulum æqualem <lb/> trienti recti, a puncto autem quo acta illa secat <lb/> perpendiculare in centro, ducatur alia <lb/> linea recta ad angulum semicirculi; Triplum <lb/> quadratum huius, æquale est tam quadrato perpendicularis quae incidit extra centrum, <lb/> quam quadratis segmentorum diametri, inter quæ perpendicularis illa media est <lb/> proportionalis. <lb/>[<emph style="it">tr: If from the circumference of a circle there fall two perpendiculars onto the diameter, one to the centre, the other off-centre; and to the perpendicular to the centre there is drawn from the point of incidence of the other perpendicular a straight line making an angle equal to one-third of a right angle to the diameter; moreover from the point where that line cuts the perpendicular to the centre, there is drawn another line to the angle of the semicircle, then three times the square of it is equal to the square of the perpendicular which falls off-centre and the squares of the segments of the diameter between which the perpendicular is the mean proportional. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s228" xml:space="preserve"> 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/> <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/> 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/> 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/> 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>. <lb/>[<emph style="it">tr: 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>, 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. 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> 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. 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> 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>. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s229" xml:space="preserve"> Etiam <lb/> per 4,2 El. <lb/> <lb/>[...]<lb/> Addatur utrovisque <lb/> <lb/>[...]<lb/> Ergo <lb/> propositum <lb/>[<emph style="it">tr: Also by Elements II.4 <lb/> <lb/>[...]<lb/> <lb/> Hence the proposition </emph>]<lb/> </s> </p> <head xml:id="echoid-head93" xml:space="preserve"> Hinc tale Consectarium potest efferri <lb/>[<emph style="it">tr: Here a Consequence of this kind may be inferred </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s230" xml:space="preserve"> Datis tribus continue proportionalibus: invenire lineam cuius <lb/> quadratum sit tertia pars adgregati quadratorum e tribus <lb/> proportionalibus. <lb/>[<emph style="it">tr: Given three continued proportionals, find a line whose square is a third of the sum of the squares of all three proportionals. </emph>]<lb/> </s> </p> <pb file="add_6784_f352v" o="352v" n="704"/> <pb file="add_6784_f353" o="353" n="705"/> <div xml:id="echoid-div52" type="page_commentary" level="2" n="52"> <p> <s xml:id="echoid-s231" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s231" xml:space="preserve"> On this page Harriot investigates Propositions 12, 13, and 14 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition XII. <lb/> Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, minus solido quod fit sub eadem composita & adgregato quadratorum a tribus, æqualis est solido sub eadem composita & quadrato secundæ. </quote> <lb/> <quote> If there are three proportional lines, the cube of the sum of the two extremes, minus the product of that sum and the sum of squares of all three, is equal to the product of the sum and the square of the second. </quote> <lb/> <quote xml:lang="lat"> Proposition XIII. <lb/> Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregato quadratorum a tribus, minus cubo e prima, æquale est solido sub eadem prima & adgregato quadratorum secundæ & tertiæ. </quote> <lb/> <quote> If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. </quote> <lb/> <quote xml:lang="lat"> Proposition XIV. <lb/> Si fuerint tres lineæ rectæ proportionales: solidum sub prima & adgregatum quadratorum a tribus, minus cubo e tertia, æquale est solido sub eadem tertia & adgregato quadratorum primæ & secundæ. </quote> <lb/> <quote> If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the third, is equal to the product of the third and the sum of the first and second. </quote> <lb/> <s xml:id="echoid-s232" xml:space="preserve"> The 'Consectarium' appears verbally in Viete's proposition; Harriot has re-written it in symbolic notation. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head94" xml:space="preserve"> prop. 12. Supplementi <lb/>[<emph style="it">tr: Proposition 12 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s234" xml:space="preserve"> Si fuerint tres lineæ rectæ proportionales: cubus compositæ e duabus extremis, <lb/> minus solido quod fit sub eadem composita et adgregato quadratorum a tribus: <lb/> æqualis est solido sub eadem composita et quadrato secundæ. <lb/>[<emph style="it">tr: If there are three proportional lines, the cube of the sum of the two extremes, minus the product of that sum and the sum of squares of all three, is equal to the product of the sum and the square of the second. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s235" xml:space="preserve"> Sint 3 continue proportionales <lb/> utrinque addatur <lb/> <lb/>[...]<lb/> <lb/> Fiant solida ab extremis et etiam a medijs, et inde: <lb/> propositum <lb/>[<emph style="it">tr: let there be three continued proportionals <lb/> add to each side <lb/> <lb/>[...]<lb/> <lb/> There may be made solids from the extremes and also form the means, and hence the proposition: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s236" xml:space="preserve"> Prop. 13. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregato <lb/> quadratorum tribus, minus cubo e prima: æquale est solido sub eadem <lb/> prima et adgregato quadratorum secundæ et tertiæ. <lb/>[<emph style="it">tr: Proposition 13. If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the first, is equal to the product of the first and the sum of squares of the second and third. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s237" xml:space="preserve"> Sint tres continue proportionales <lb/> <lb/>[...]<lb/> <lb/> Resoluatur Analogia et erit: <lb/> Propositum <lb/>[<emph style="it">tr: Let there be three continued proportionals <lb/> <lb/>[...]<lb/> <lb/> The ratio is resolved, and hence the proposition: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s238" xml:space="preserve"> Prop. 14. Si fuerint tres lineæ rectæ proportionales: solidum sub prima et adgregatum quadratorum <lb/> a tribus minus cubo e tertia: æquale est solido sub eadem tertia et adgregato <lb/> quadratorum primæ et secundæ. <lb/>[<emph style="it">tr: Proposition 14. If there are three proportional lines, the product of the first and the sum of squares of all three, minus the cube of the third, is equal to the product of the third and the sum of the first and second. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s239" xml:space="preserve"> Sint tres continue proportionales <lb/> <lb/>[...]<lb/> <lb/> Resoluatur Analogia et erit: <lb/> Propositum <lb/>[<emph style="it">tr: Let there be three continued proportionals <lb/> <lb/>[...]<lb/> <lb/> The ratio is resolved, and hence the proposition: </emph>]<lb/> </s> </p> <head xml:id="echoid-head95" xml:space="preserve"> Consectarium <lb/>[<emph style="it">tr: Consequence </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s240" xml:space="preserve"> Quia æquantur æqualibus <lb/> ex antecedente consectario. <lb/>[<emph style="it">tr: Because equals are equated to equals, by the preceding conclusion. </emph>]<lb/> </s> </p> <pb file="add_6784_f353v" o="353v" n="706"/> <pb file="add_6784_f354" o="354" n="707"/> <div xml:id="echoid-div53" type="page_commentary" level="2" n="53"> <p> <s xml:id="echoid-s241" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s241" xml:space="preserve"> On this page Harriot investigates Propositions 10 and 11 from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Proposition X. <lb/> Si fuerint tres lineæ rectæ proportionales: est ut prima ad tertiam, ita adgregatum quadratorum primæ & secundæ ad adgregatum quadratorum secundæ & tertiæ. </quote> <lb/> <quote> If there are three proportional lines, as the first is to the third, so is the sum of squares of the first and second to the sum of squares of the second and third. </quote> <lb/> <quote xml:lang="lat"> Proposition XI. <lb/> Si fuerint tres lineæ rectæ proportionales: est ut prima ad adgregatum primae & tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ & tertiæ. </quote> <lb/> <quote> If there are three proportional lines, as the first is to the sum of the first and third, so is the square of the second to the sum of squares of the second and third. </quote> <lb/> <s xml:id="echoid-s242" xml:space="preserve"> There are two references to Euclid's <emph style="it">Elements</emph>, Proposition VI.20. </s> <lb/> <quote> VI.20 Similar polygons my be divided into the same number of similar triangles, each similar pair of which are proportional to the polygons; and the polygons are to each other in the duplicate ratio of their homologous sides. </quote> <lb/> <s xml:id="echoid-s243" xml:space="preserve"> The 'Consectarium' appears verbally in Viete's proposition; Harriot has reinterpreted it symbolically. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head96" xml:space="preserve"> prop. 10. Supplementi <lb/>[<emph style="it">tr: Proposition 10 from the Supplementum </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s245" xml:space="preserve"> Si fuerint tres lineæ rectæ proportionales: Est ut prima ad tertiam, ita adgregatum <lb/> quadratorum primæ et secundæ ad adgregatum quadratorum secundæ et tertiæ. <lb/>[<emph style="it">tr: If there are three proportional lines, as the first is to the third, so is the sum of squares of the first and second to the sum of squares of the second and third. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s246" xml:space="preserve"> sint tres proportionales <lb/> continue <lb/> consequetur <lb/> vel <lb/> Et per synæresin <lb/> Et per 20,6 Euclid <lb/> Ergo pro conclusione <lb/>[<emph style="it">tr: let there be three continued proportionals <lb/> consequently <lb/> or <lb/> And by synæresis <lb/> And by Euclid VI.20 <lb/> Therefore in conclusion </emph>]<lb/> </s> </p> <head xml:id="echoid-head97" xml:space="preserve"> prop. 11. <lb/>[<emph style="it">tr: Proposition 11 </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s247" xml:space="preserve"> Si fuerint tres lineæ rectæ proportionales, est ut prima ad adgregatum primae et <lb/> tertiæ, ita quadratum secundæ ad adgregatum quadratorum secundæ et tertiæ. <lb/>[<emph style="it">tr: If there are three proportional lines, as the first is to the sum of the first and third, so is the square of the second to the sum of squares of the second and third. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s248" xml:space="preserve"> sint tres proportionales <lb/> per 20,6 El <lb/> Et per Synæresin <lb/> Concluditur <lb/>[<emph style="it">tr: let there be three proportionals <lb/> by Elements VI.20 <lb/> And by synæresin <lb/> It may be concluded. </emph>]<lb/> </s> </p> <head xml:id="echoid-head98" xml:space="preserve"> Consectarium <lb/>[<emph style="it">tr: Consequence </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s249" xml:space="preserve"> Itaque si fuerint tres lineæ rectæ proportionales, tria solida ab ijs <lb/> effecta æqualia sunt. per 10<emph style="super">am</emph> conculsionem <lb/> per 11<emph style="super">am</emph> conclu. <lb/> <lb/>[...]<lb/> <lb/> Dua prima solida sunt æqualia, quia unum factum est ab extremis analogia 10<emph style="super">am</emph> <lb/> et alterum a modijs. Tertium est factum a modijs <emph style="st">inferioris</emph> analogia 11<emph style="super">am</emph>, <lb/> cuius extremæ sunt eædem <emph style="st">superioris</emph> <emph style="super">analogia 10am</emph>, et illo æquale. <lb/>[<emph style="it">tr: Therefore if there are three lines in proportion, three solids constructed from them are equal. <lb/> by the conclusion of the 10th <lb/> by the conclusion of the 11th <lb/> <lb/>[...]<lb/> <lb/> The two first solids are equal, because one is made from the extremes of the ratio of the 10th, and the other by the method <lb/> 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, and is equal to that one. </emph>]<lb/> </s> </p> <pb file="add_6784_f354v" o="354v" n="708"/> <pb file="add_6784_f355" o="355" n="709"/> <div xml:id="echoid-div54" type="page_commentary" level="2" n="54"> <p> <s xml:id="echoid-s250" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s250" xml:space="preserve"> On this page Harriot examines a particular case arising from Proposition VII of Viète's <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. The same proposition is the subject of Chapter V of Viète's <emph style="it">Variorum responsorum libri VIII</emph>, which was also published in 1593. </s> <lb/> <quote xml:lang="lat"> Caput V <lb/> Propositio <lb/> Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. </quote> <lb/> <quote> Construct four lines in continued proportion, whose extremes are in double ratio. </quote> <lb/> <s xml:id="echoid-s251" xml:space="preserve"> The text in the <emph style="it">Variorum</emph> refers to the <emph style="it">Supplementum</emph>, indicating that the <emph style="it">Supplementum</emph> was written first. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head99" xml:space="preserve"> Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag. 4. <lb/>[<emph style="it">tr: On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page 4. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s253" xml:space="preserve"> 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/> cuius quadratum est triplum quadrati <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/> 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/> 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/> 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/> 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/> analogiam precedentam ut sequitur. <lb/> <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/> <lb/>[...]<lb/> <lb/> Et per similitudi-<lb/> num Δ<emph style="super">orum</emph>. <lb/>[...]<lb/> <lb/> 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. <lb/>[<emph style="it">tr: 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/> 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, <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/> 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/> the precding ratio, as follows. <lb/> <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/> <lb/>[...]<lb/> <lb/> And by similar triangles. <lb/> <lb/>[...]<lb/> <lb/> 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. </emph>]<lb/> [<emph style="it">Note: The other sheet mentioned in this paragraph appears to be Add MS 6784, f. 356. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s254" xml:space="preserve"> Datis igitur extremis in ratione dupla, mediæ ita compendiosæ <lb/> inveniuntur. <lb/>[<emph style="it">tr: Therefore given the extremes in double ratio, the mean is briefly found. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s255" xml:space="preserve"> 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/> 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 et producta ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/> 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/> Quatuor igitur continue proportionales ex supra demonstratis sunt. <lb/>[<emph style="it">tr: 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. 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>. 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/> Therefore there are the four continued proportionals that were demonstrated above. </emph>]<lb/> </s> </p> <pb file="add_6784_f355v" o="355v" n="710"/> <pb file="add_6784_f356" o="356" n="711"/> <div xml:id="echoid-div55" type="page_commentary" level="2" n="55"> <p> <s xml:id="echoid-s256" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s256" xml:space="preserve"> On this page Harriot examines a particular case arising from Proposition VII of Viète's <emph style="it">Supplementum geometriæ</emph> (1593), when the fourth proportional is twice the first. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head100" xml:space="preserve"> prop. 7. Supplementi de corrollario <lb/>[<emph style="it">tr: Proposition 7 of the Supplement, on a corollary </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s258" xml:space="preserve"> Sint 4<emph style="super">or</emph> proportionales <lb/> in specie. <lb/> Si quarta sit dupla ad prima, erit: <lb/> <lb/>[...]<lb/> <lb/> Ergo quatuor proportionales <lb/> quarum extremæ sunt in <lb/> ratione dupla erunt <lb/>[<emph style="it">tr: Let there be 4 proportionals in general form. <lb/> If the fourth is twice the firs, then: <lb/> <lb/>[...]<lb/> <lb/> Therefore the four proportionals whose extremes are in double ratio will be </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s259" xml:space="preserve"> Tunc fac <lb/>[...]<lb/> et nota quadratorum differentiam. <lb/>[<emph style="it">tr: Then make [the square of the first and second and the square of the third and fourth], and note the difference of the squares.</emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s260" xml:space="preserve"> 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/> Hoc est triplum quadratum primæ proportionalis. <lb/>[<emph style="it">tr: 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/> This is three times the square of the first proportional. </emph>]<lb/> </s> </p> <pb file="add_6784_f356v" o="356v" n="712"/> <pb file="add_6784_f357" o="357" n="713"/> <div xml:id="echoid-div56" type="page_commentary" level="2" n="56"> <p> <s xml:id="echoid-s261" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s261" xml:space="preserve"> This page investigates the proposition that is the subject of Chapter V of Viète's <emph style="it">Variorum responsorum libri VIII</emph>. It appears to be a continuation of Add MS 6784, f. 355. </s> <lb/> <quote xml:lang="lat"> Caput V <lb/> Propositio <lb/> Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione dupla. </quote> <lb/> <quote> Construct four lines in continued proportion, whose extremes are in double ratio. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head101" xml:space="preserve"> In Cap. 5. Resp. lib. 8. pag. 4. <lb/>[<emph style="it">tr: Chapter 5, Variorum liber responsorum, page 4. </emph>]<lb/> </head> <pb file="add_6784_f357v" o="357v" n="714"/> <pb file="add_6784_f358" o="358" n="715"/> <div xml:id="echoid-div57" type="page_commentary" level="2" n="57"> <p> <s xml:id="echoid-s263" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s263" xml:space="preserve"> On this page Harriot examines Proposition VII from Viète's <emph style="it">Supplementum geometriæ</emph> (1593). </s> <lb/> <quote xml:lang="lat"> Propositio VII. <lb/> Data è tribus propositis lineis rectis proportionalibus prima, & ea cujus quadratum æquale fit ei quo differt quadratum compositae ex secunda & tertia à quadrato compositæ ex secunda & prima, invenire secundam & tertiam proprtionales. </quote> <lb/> <quote> Given the first of three proposed proportional straight lines, and another whose square is equal to the difference between the square of the sum of the second and third, and the square of the sum of the second and first, find the second and third proportionals. </quote> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head102" xml:space="preserve"> prop. 7. Supplementi <lb/>[<emph style="it">tr: Proposition 7 of the Supplement </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s265" xml:space="preserve"> Data e tribus propositis lineis rectis proportionalibus prima et ea <lb/> cujus quadratum aequale fit ei quo differt quadratum compositae ex <lb/> secunda et tertia a quadrato compositæ ex secunda et prima: invenire <lb/> secundam et tertiam proprtionales. <lb/>[<emph style="it">tr: Given the first of three proposed proportional straight lines, and another whose square is equal to the difference between the square of the sum of the second and third, and the square of the sum of the second and first, find the second and third proportionals. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s266" xml:space="preserve"> Data prima <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> <lb/> Et recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s267" xml:space="preserve"> Tum tres proportionales <lb/> erunt. <lb/>[<emph style="it">tr: Then the three proportionals will be: </emph>]<lb/> </s> </p> <pb file="add_6784_f358v" o="358v" n="716"/> <pb file="add_6784_f359" o="359" n="717"/> <head xml:id="echoid-head103" xml:space="preserve"> a) Achilles </head> <p xml:lang="lat"> <s xml:id="echoid-s268" xml:space="preserve"> Sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, Achilles. <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s269" xml:space="preserve"> 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/> 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/> nempe: 10 ad 1. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s270" xml:space="preserve"> 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. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s271" xml:space="preserve"> Et sit motus utriusque in eadem linea et ad easdem partes, nempe <lb/> 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>. <lb/>[<emph style="it">tr: And suppose the motion of both is in the same line and in the same direction, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s272" xml:space="preserve"> Quæritur ex datis punctum ubi Achilles comprehendet testudinem. <lb/>[<emph style="it">tr: From what is given there is sought the point where Achilles catches up with the tortoise. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s273" xml:space="preserve"> Quæestio solvitur exhibendo summam infinitæ progressionis decrescentis <lb/> ut sequitur: (species summa infinitæ progressionis decrescentis <lb/> ut in doctrinam de <reg norm="progressionis" type="abbr">prog</reg>: <reg norm="geometricæ" type="abbr">geom</reg>: est:) <lb/>[<emph style="it">tr: The problem is solved by producing the sum of an infinite decreasing progression as follows: (the case of the sum of an infinite decreasing progression as in the teaching of geometric porgressions is:) </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s274" xml:space="preserve"> Alia progressiones. <lb/>[<emph style="it">tr: Other progressions. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s275" xml:space="preserve"> (ut Archimedes de <lb/> quad: parab: pr: 23) <lb/>[<emph style="it">tr: (as Archimedes in the quadrature of the parabola, proposition 23) </emph>]<lb/> </s> </p> <pb file="add_6784_f359v" o="359v" n="718"/> <pb file="add_6784_f360" o="360" n="719"/> <head xml:id="echoid-head104" xml:space="preserve"> b) Achilles </head> <p xml:lang="lat"> <s xml:id="echoid-s276" xml:space="preserve"> Sit (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>), Achilles. <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>t</mi></mstyle></math>, testudo. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s277" xml:space="preserve"> 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/> 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/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s278" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s279" xml:space="preserve"> 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/> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. <lb/>[<emph style="it">tr: And let the mtion of both be in the same line and the same direction, 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s280" xml:space="preserve"> Quæritur ex datis punctum ubi Achilles comprehendet testudinem. <lb/>[<emph style="it">tr: From what is given there is sought the point where Achilles catches up with the tortoise. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s281" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s282" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s283" xml:space="preserve"> In numeris sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. 10. <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. 2. <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. 2. mill <lb/>[<emph style="it">tr: 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 </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s284" xml:space="preserve"> Aliter. <lb/>[<emph style="it">tr: Another way. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s285" xml:space="preserve"> Aliter 2<emph style="super">o</emph>. <lb/> 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>. <lb/>[<emph style="it">tr: A second way. <lb/> 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s286" xml:space="preserve"> Exemplum de duabus [¿]numeribus[?]. <lb/>[<emph style="it">tr: An example from two numbers. </emph>]<lb/> </s> </p> <pb file="add_6784_f360v" o="360v" n="720"/> <pb file="add_6784_f361" o="361" n="721"/> <div xml:id="echoid-div58" type="page_commentary" level="2" n="58"> <p> <s xml:id="echoid-s287" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s287" xml:space="preserve"> On this folio, Harriot derives the sum of a finite geometric progression, using Euclid V.12 and its numerical counterpoart, Euclid VII.12. He then extends his result to an infinite (decreasing) progression, by arguing that the final term must be infnitely small, that is, nothing. <lb/> Euclid V.12: If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. <lb/> Euclid VII.12: If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head105" xml:space="preserve" xml:lang="lat"> 1.) De progressione geometrica. <lb/>[<emph style="it">tr: On geometric porgressions </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s289" xml:space="preserve"> Theorema. <lb/>[<emph style="it">tr: Theorem </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s290" xml:space="preserve"> el. 5. pr: 12. <lb/>[<emph style="it">tr: <emph style="it">Elements</emph>, Book 5, Proposition 12. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s291" xml:space="preserve"> el. 7. pr. 12. <lb/>[<emph style="it">tr: <emph style="it">Elements</emph>, Book 7, Proposition 12. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s292" xml:space="preserve"> Si sint magnitudines quotcunque proportionales, Quemadmodum <lb/> se habuerit una antecedentium ad unam consequentium: Ita <lb/> se habebunt omnes antecedentes ad omnes consequentes. <lb/>[<emph style="it">tr: If any number of magnitudes are proportional, then just as as one antecedent is to its consequent, so will the sum of the antecedents be to the sum of the consequents. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s293" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s294" xml:space="preserve"> In notis universalibus sit. <lb/>[<emph style="it">tr: In general notation we have </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s295" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math>. primum. <emph style="st">p</emph>. primus terminus rationis. <lb/>[<emph style="it">tr: <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. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s296" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. secunda. <emph style="st">s</emph>. secundus. <lb/>[<emph style="it">tr: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. second. <emph style="st">s</emph>. second. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s297" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. ultima. <lb/>[<emph style="it">tr: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>. last. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s298" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes. <lb/>[<emph style="it">tr: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. all. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s299" xml:space="preserve"> Ergo; si, <emph style="st">p</emph> > <emph style="st">s</emph> ut in progressi decrescente: <lb/>[<emph style="it">tr: Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in a decreasing progression: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s300" xml:space="preserve"> Ergo; si, <emph style="st">p</emph> < <emph style="st">s</emph> ut in progressi crescente: <lb/>[<emph style="it">tr: Therfore if <emph style="st">p</emph> > <emph style="st">s</emph> are in an increasing progression: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s301" xml:space="preserve"> De <emph style="st">infinitis</emph> progressionibus <lb/> decrescentibus in infinitum: <lb/>[<emph style="it">tr: For a progression descreasing indefinitely: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s302" xml:space="preserve"> Cum progressio decrescit et <lb/> numerus terminorum sit infinitus; <lb/> ultimus terminus est infinite <lb/> minimus hoc est nullius quantiatis. <lb/>[<emph style="it">tr: Since the progression decreases and the number of terms is infinite, the last term is infnitely small, that is, of no quantity. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s303" xml:space="preserve"> Ideo: <lb/>[<emph style="it">tr: Therefore. </emph>]<lb/> </s> </p> <pb file="add_6784_f361v" o="361v" n="722"/> <pb file="add_6784_f362" o="362" n="723"/> <div xml:id="echoid-div59" type="page_commentary" level="2" n="59"> <p> <s xml:id="echoid-s304" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s304" xml:space="preserve"> In the preceding folio, f. 361, Harriot derived a formula for the sum of a finite geometric progression based on Euclid V.12. Here he gives an alternative derivation based on Euclid IX. 35. <lb/> Euclid IX. 35: If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat"> 2.) De progressione geometrica. <lb/>[<emph style="it">tr: On geometric porgressions </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s306" xml:space="preserve"> Theoremata. <lb/>[<emph style="it">tr: Theorem </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s307" xml:space="preserve"> el. 9. pr: 35. <lb/>[<emph style="it">tr: <emph style="it">Elements</emph> Book IX, Proposition 35 </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s308" xml:space="preserve"> Si sint quotlibet numeri deinceps proportionales, detrahuntur autem <lb/> de secundo et ultimo æquales ipsi primo: erit quemadmodum <lb/> secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum <lb/> antecedunt. <lb/>[<emph style="it">tr: If there are as many numbers as we please in proportion, and the first is subtracted from the second and the last, then just as the difference of the second is to the first, so is the difference of the last to all before the last. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s309" xml:space="preserve"> Progressio crescens: <lb/>[<emph style="it">tr: An increasing progression: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s310" xml:space="preserve"> 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. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s311" xml:space="preserve"> Progressio decrescens: <lb/>[<emph style="it">tr: A decreasing progression: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s312" xml:space="preserve"> In notis universalis erit: <lb/>[<emph style="it">tr: In general notation we have: </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s313" xml:space="preserve"> Vel: in notis magis universalis. <lb/> 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/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>, maxumus terminus progressionis <lb/> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>, minimus. Tum: <lb/>[<emph style="it">tr: 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, <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: </emph>]<lb/> </s> </p> <pb file="add_6784_f362v" o="362v" n="724"/> <pb file="add_6784_f363" o="363" n="725"/> <div xml:id="echoid-div60" type="page_commentary" level="2" n="60"> <p> <s xml:id="echoid-s314" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s314" xml:space="preserve"> In this folio Harriot repeats statements that are to be found in Viete, <emph style="it">Variorum responsorum</emph>, Chapter XVII (1646, 397–398). <lb/> 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> 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/> Harriot's final comments refer to the final sentence of Viete's penultimate paragraph (1646, 398): <lb/> <foreign xml:lang="lat"> Et ut differentia terminorum rationis ad terminorum rationis majorem, ita maxima ad compositam ex ombnibus plus cremento. </foreign> <lb/> <lb/>[<emph style="it">tr: As the difference in the terms of the ratio is to the greater term of the ratio, so is the the greatest term of the progression to the sum plus an increment. </emph>]<lb/> </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head107" xml:space="preserve" xml:lang="lat"> 3.) De progressione geometrica. (ut Vieta in var: resp.) <lb/>[<emph style="it">tr: On geometric progressions (as Viete in <emph style="it">Variorum responsorum</emph>) </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s316" xml:space="preserve"> Crescente. <lb/>[<emph style="it">tr: Increasing. </emph>]<lb/> </s> <s xml:id="echoid-s317" xml:space="preserve"> decrescente. <lb/>[<emph style="it">tr: Decreasing. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s318" xml:space="preserve"> <emph style="st">m</emph>. minor terminus rationis. <lb/>[<emph style="it">tr: Let <emph style="st">m</emph> be the lesser terms of the ratio. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s319" xml:space="preserve"> <emph style="st">M</emph>. Maior terminus rationis. <lb/>[<emph style="it">tr: Let <emph style="st">M</emph> be the greater terms of the ratio. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s320" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>. maximus terminus progressionis. <lb/>[<emph style="it">tr: Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the greatest term of the progression. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s321" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math>. minimus terminus progressionis. <lb/>[<emph style="it">tr: Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math> be the least term of the progression. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s322" xml:space="preserve"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math>. omnes, id est summa omnium <lb/>[<emph style="it">tr: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi></mstyle></math> is all, that is the sum of all. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s323" xml:space="preserve"> ita Vieta post δεδόμενα <lb/> in respons: pag. 29. <lb/>[<emph style="it">tr: thus Viete after δεδόμενα in <emph style="it">Variorum Responsorum</emph> page 29. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s324" xml:space="preserve"> apud Vieta dicitur crementum. <lb/>[<emph style="it">tr: in Viete this is said to be the increment. </emph>]<lb/> </s> </p> <pb file="add_6784_f363v" o="363v" n="726"/> <pb file="add_6784_f364" o="364" n="727"/> <div xml:id="echoid-div61" type="page_commentary" level="2" n="61"> <p> <s xml:id="echoid-s325" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s325" xml:space="preserve"> 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>. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head108" xml:space="preserve" xml:lang="lat"> De progressionibus. <lb/> finitis & infinitis. <lb/>[<emph style="it">tr: On finite and infinite progressions </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s327" xml:space="preserve"> linea infinite <emph style="super">longa</emph>quælibet = æqualis alicui, plano <lb/> solido. <lb/> longo-solido. <lb/> plano-solido. <lb/> solido-solido. &c. <lb/>[<emph style="it">tr: An infinite line of any length is equal to some plane, or solid, or solid-length, or solid-plane, or solid-solid, etc. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s328" xml:space="preserve"> linea infinite brevis quælibet = æqualis alicui, puncto. <lb/> linea. <lb/> puncto-plano. <lb/> puncto-solido. &c. <lb/> <lb/>[<emph style="it">tr: Any infinitely short line is equal to some line-point, or plane-point, or solid-point, etc. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s329" xml:space="preserve"> Quælibet punctum terminat progressionem. <lb/>[<emph style="it">tr: Whatever point terminates the progression. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s330" xml:space="preserve"> infinite numero puncta = lineæ <lb/> plano. <lb/> solido. &c. <lb/>[<emph style="it">tr: an infinite number of points equal a line, or plane, or solid, etc. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s331" xml:space="preserve"> linea signata <lb/> terminat <lb/> progressionem. <lb/> ita planum signatum. <lb/>[<emph style="it">tr: a designated line terminates the progression; similarly a designated plane, </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s332" xml:space="preserve"> hæc & alia huius generis <lb/> consideranda. <lb/>[<emph style="it">tr: these and others of this kind may be considered. </emph>]<lb/> </s> </p> <pb file="add_6784_f364v" o="364v" n="728"/> <pb file="add_6784_f365" o="365" n="729"/> <pb file="add_6784_f365v" o="365v" n="730"/> <pb file="add_6784_f366" o="366" n="731"/> <pb file="add_6784_f366v" o="366v" n="732"/> <pb file="add_6784_f367" o="367" n="733"/> <pb file="add_6784_f367v" o="367v" n="734"/> <pb file="add_6784_f368" o="368" n="735"/> <pb file="add_6784_f368v" o="368v" n="736"/> <pb file="add_6784_f369" o="369" n="737"/> <div xml:id="echoid-div62" type="page_commentary" level="2" n="62"> <p> <s xml:id="echoid-s333" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s333" xml:space="preserve"> This page contains a symbolic version of Euclid Book II, Proposition 11: <lb/> II.11. To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head109" xml:space="preserve" xml:lang="lat"> propositiones 2<emph style="super">i</emph> Euclidis <lb/>[<emph style="it">tr: Propositions from the second book of Euclid </emph>]<lb/> </head> <pb file="add_6784_f369v" o="369v" n="738"/> <pb file="add_6784_f370" o="370" n="739"/> <pb file="add_6784_f370v" o="370v" n="740"/> <pb file="add_6784_f371" o="371" n="741"/> <pb file="add_6784_f371v" o="371v" n="742"/> <pb file="add_6784_f372" o="372" n="743"/> <pb file="add_6784_f372v" o="372v" n="744"/> <pb file="add_6784_f373" o="373" n="745"/> <pb file="add_6784_f373v" o="373v" n="746"/> <pb file="add_6784_f374" o="374" n="747"/> <pb file="add_6784_f374v" o="374v" n="748"/> <pb file="add_6784_f375" o="375" n="749"/> <pb file="add_6784_f375v" o="375v" n="750"/> <pb file="add_6784_f376" o="376" n="751"/> <pb file="add_6784_f376v" o="376v" n="752"/> <pb file="add_6784_f377" o="377" n="753"/> <pb file="add_6784_f377v" o="377v" n="754"/> <pb file="add_6784_f378" o="378" n="755"/> <pb file="add_6784_f378v" o="378v" n="756"/> <pb file="add_6784_f379" o="379" n="757"/> <pb file="add_6784_f379v" o="379v" n="758"/> <pb file="add_6784_f380" o="380" n="759"/> <pb file="add_6784_f380v" o="380v" n="760"/> <pb file="add_6784_f381" o="381" n="761"/> <pb file="add_6784_f381v" o="381v" n="762"/> <pb file="add_6784_f382" o="382" n="763"/> <pb file="add_6784_f382v" o="382v" n="764"/> <pb file="add_6784_f383" o="383" n="765"/> <pb file="add_6784_f383v" o="383v" n="766"/> <pb file="add_6784_f384" o="384" n="767"/> <pb file="add_6784_f384v" o="384v" n="768"/> <pb file="add_6784_f385" o="385" n="769"/> <pb file="add_6784_f385v" o="385v" n="770"/> <pb file="add_6784_f386" o="386" n="771"/> <pb file="add_6784_f386v" o="386v" n="772"/> <pb file="add_6784_f387" o="387" n="773"/> <pb file="add_6784_f387v" o="387v" n="774"/> <pb file="add_6784_f388" o="388" n="775"/> <pb file="add_6784_f388v" o="388v" n="776"/> <pb file="add_6784_f389" o="389" n="777"/> <pb file="add_6784_f389v" o="389v" n="778"/> <pb file="add_6784_f390" o="390" n="779"/> <pb file="add_6784_f390v" o="390v" n="780"/> <pb file="add_6784_f391" o="391" n="781"/> <pb file="add_6784_f391v" o="391v" n="782"/> <pb file="add_6784_f392" o="392" n="783"/> <pb file="add_6784_f392v" o="392v" n="784"/> <pb file="add_6784_f393" o="393" n="785"/> <pb file="add_6784_f393v" o="393v" n="786"/> <pb file="add_6784_f394" o="394" n="787"/> <pb file="add_6784_f394v" o="394v" n="788"/> <pb file="add_6784_f395" o="395" n="789"/> <pb file="add_6784_f395v" o="395v" n="790"/> <pb file="add_6784_f396" o="396" n="791"/> <pb file="add_6784_f396v" o="396v" n="792"/> <pb file="add_6784_f397" o="397" n="793"/> <pb file="add_6784_f397v" o="397v" n="794"/> <head xml:id="echoid-head110" xml:space="preserve" xml:lang="lat"> 1.) De reductione æquationum <lb/>[<emph style="it">tr: On the reduction of equations </emph>]<lb/> </head> <pb file="add_6784_f398" o="398" n="795"/> <head xml:id="echoid-head111" xml:space="preserve"> 3.) </head> <pb file="add_6784_f398v" o="398v" n="796"/> <pb file="add_6784_f399" o="399" n="797"/> <pb file="add_6784_f399v" o="399v" n="798"/> <pb file="add_6784_f400" o="400" n="799"/> <pb file="add_6784_f400v" o="400v" n="800"/> <head xml:id="echoid-head112" xml:space="preserve" xml:lang="lat"> 1)B) De reductione æquationum <lb/>[<emph style="it">tr: On the reduction of equations </emph>]<lb/> </head> <pb file="add_6784_f401" o="401" n="801"/> <pb file="add_6784_f401v" o="401v" n="802"/> <div xml:id="echoid-div63" type="page_commentary" level="2" n="63"> <p> <s xml:id="echoid-s335" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s335" xml:space="preserve"> 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>) 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 that these valus do indeed satisfy the equation. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f402" o="402" n="803"/> <div xml:id="echoid-div64" type="page_commentary" level="2" n="64"> <p> <s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s337" xml:space="preserve"> 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/> A calculation below each box gives the sum of the figures contained in it. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f402v" o="402v" n="804"/> <div xml:id="echoid-div65" type="page_commentary" level="2" n="65"> <p> <s xml:id="echoid-s339" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s339" xml:space="preserve"> The calculations from the previous page (Add MS 6784, f. 402) are checked by root extractions </s> </p> </emph>] <lb/><lb/></s></p></div> <p> <s xml:id="echoid-s341" xml:space="preserve"> The extraction <lb/> of the roots. </s> </p> <pb file="add_6784_f403" o="403" n="805"/> <pb file="add_6784_f403v" o="403v" n="806"/> <pb file="add_6784_f404" o="404" n="807"/> <div xml:id="echoid-div66" type="page_commentary" level="2" n="66"> <p> <s xml:id="echoid-s342" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s342" xml:space="preserve"> Third, fourth, and fifth powers of (20 + 4). <lb/> The binomial coefficients 3, 3 and 4, 6, 4 and 5, 10, 10, 5, appear amongst the numbers in the rightmost column. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f404v" o="404v" n="808"/> <p> <s xml:id="echoid-s344" xml:space="preserve"> The doctrine of Algebraycall nombers is but <lb/> the doctrined of such continuall proportionalles of <lb/> which a unite is the first. </s> </p> <p> <s xml:id="echoid-s345" xml:space="preserve"> A unite being the first of continuall proportionalles; the second is <lb/> called a roote: because the third wilbe always a square: & the fourth <lb/> <emph style="st">third</emph> a cube, as Euclide demonstrateth. </s> <s xml:id="echoid-s346" xml:space="preserve"> The names of the other proportionalles <lb/> following are all compounded of squares, or cubes or both according <lb/> to Diophantus & others which follow him. </s> <s xml:id="echoid-s347" xml:space="preserve"> Some or other of the most parte of the later <lb/> writers gave the name of surdsolidus, of which the first or simple sursolid <lb/> is the sixt proportionall. &c. </s> </p> <p> <s xml:id="echoid-s348" xml:space="preserve"> Any nomber may be <emph style="super">any</emph> terme proportinall in a continuall progression <lb/> from a unite. </s> <s xml:id="echoid-s349" xml:space="preserve"> If the nomber terme be the second, the third is gotten by <lb/> multiplying the nomber into him self. </s> <s xml:id="echoid-s350" xml:space="preserve"> & the fourth by multiplying the <lb/> third by the second & so forth. </s> <s xml:id="echoid-s351" xml:space="preserve"> as also <emph style="super">by</emph> the doctrine of progression <lb/> any terme that is found another may be gotten compendiously <lb/> without continuall multiplications. </s> </p> <p> <s xml:id="echoid-s352" xml:space="preserve"> If a nomber that is known & designed to be the third, fourth, <lb/> or fifth or any other proportinall of another denomination: the <lb/> doctrine to find the second is that which is called the extraction <lb/> of the roote, which is taught in these papers. </s> </p> <p> <s xml:id="echoid-s353" xml:space="preserve"> The second proportionall is also called the first dignity, & the third the <lb/> second dignity, & the fourth the third dignity &c. </s> </p> <p> <s xml:id="echoid-s354" xml:space="preserve"> The third is also called the first power; the 4th the second power &c. </s> </p> <p> <s xml:id="echoid-s355" xml:space="preserve"> The first proportionall <lb/> is a unite. </s> </p> <p> <s xml:id="echoid-s356" xml:space="preserve"> The first dignity is <lb/> the second proportionall, <lb/> called a roote. </s> </p> <p> <s xml:id="echoid-s357" xml:space="preserve"> The first power is the <lb/> third proportionall <lb/> <emph style="st">called a square</emph> <lb/> or second Dignity <lb/> called a square. </s> </p> <p> <s xml:id="echoid-s358" xml:space="preserve"> The first solid is the <lb/> fourth proprtionall: <lb/> The third dignity: & <lb/> The second power, <lb/> called a cube. </s> </p> <p> <s xml:id="echoid-s359" xml:space="preserve"> The pythagoreans <lb/> did call 4 the first solid <lb/> as Boethius relateth. </s> <lb/> <s xml:id="echoid-s360" xml:space="preserve"> The nomber serveth to be, because pyramides are prime solids <lb/> & 4 amongst nombers is the first pyramide. </s> </p> <pb file="add_6784_f405" o="405" n="809"/> <pb file="add_6784_f405v" o="405v" n="810"/> <pb file="add_6784_f406" o="406" n="811"/> <pb file="add_6784_f406v" o="406v" n="812"/> <pb file="add_6784_f407" o="407" n="813"/> <div xml:id="echoid-div67" type="page_commentary" level="2" n="67"> <p> <s xml:id="echoid-s361" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s361" xml:space="preserve"> Here Harriot demonstrates that multiplication by 9 increases the number of digits by one as far as the 21st power but not at the 22nd power. Thus the number of digits alone is no guide to the size of the root. </s> </p> </emph>] <lb/><lb/></s></p></div> <p> <s xml:id="echoid-s363" xml:space="preserve"> An induction to prove that <lb/> to pricke the second figure for <lb/> the extraction of square rootes <lb/> & the third for cubes & 4th <lb/> for biquadrates etc. according <lb/> to the nomber of figures that <lb/> the greatest figure 9 doth <lb/> produce is no rule. </s> <s xml:id="echoid-s364" xml:space="preserve"> for we <lb/> may see how it breaketh in <lb/> the 22th <emph style="st">proportionall</emph> dignity & so <lb/> forwarde. </s> <s xml:id="echoid-s365" xml:space="preserve"> but the true case <lb/> of such pricking appeareth <lb/> out <emph style="super">of</emph> the speciosa genesis which <lb/> is in an other paper arranged. </s> </p> <pb file="add_6784_f407v" o="407v" n="814"/> <pb file="add_6784_f408" o="408" n="815"/> <div xml:id="echoid-div68" type="page_commentary" level="2" n="68"> <p> <s xml:id="echoid-s366" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s366" xml:space="preserve"> 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. </s> </p> </emph>] <lb/><lb/></s></p></div> <p> <s xml:id="echoid-s368" xml:space="preserve"> If the roote to be extracted be three figures <lb/> the two first as one may here see are to be had <lb/> according to the generall rule, the next is <lb/> also to be gotten really after the same manner <lb/> that <emph style="super">is</emph> supposing the two first to be as one, & that <lb/> which foloweth, the second; although in appearance <lb/> & expressing by wordes it seems otherwise. </s> </p> <pb file="add_6784_f408v" o="408v" n="816"/> <pb file="add_6784_f409" o="409" n="817"/> <div xml:id="echoid-div69" type="page_commentary" level="2" n="69"> <p> <s xml:id="echoid-s369" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s369" xml:space="preserve"> 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>. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f409v" o="409v" n="818"/> <pb file="add_6784_f410" o="410" n="819"/> <pb file="add_6784_f410v" o="410v" n="820"/> <pb file="add_6784_f411" o="411" n="821"/> <div xml:id="echoid-div70" type="page_commentary" level="2" n="70"> <p> <s xml:id="echoid-s371" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s371" xml:space="preserve"> Here and on folio Add MS 6784, f. 412, Harriot shows that the product of two or three unequal parts is always less than the product of the same number of equal parts. </s> </p> </emph>] <lb/><lb/></s></p></div> <head xml:id="echoid-head113" xml:space="preserve" xml:lang="lat"> 1<emph style="super">o</emph>. de bisectione. <lb/>[<emph style="it">tr: 1. on bisection </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s373" xml:space="preserve"> Sit: tota linea. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math>. <lb/> 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/> magnitudo facta ab illis <lb/> erit quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math>. <lb/>[<emph style="it">tr: Let the total line be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>b</mi></mstyle></math> <lb/> 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/> 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>. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s374" xml:space="preserve"> Sint inæquales partes. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/> et: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s375" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s376" xml:space="preserve"> Si linea dividatur utcunque in tot <lb/> partes inæquales, quot æquales: <lb/> Magnitudo facta ab inæquali-<lb/> bus, minor est illa quæ facta <lb/> ab æqualibus. <lb/>[<emph style="it">tr: If a line is divided in any way into as many unequal parts as equal parts, the size of the product of the unequal parts is less than the product of the equal parts. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s377" xml:space="preserve"> vel: <lb/>[<emph style="it">tr: or: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s378" xml:space="preserve"> Si aggregatum linearum inæqualium æqueretur <lb/> aggregato tot æqualium: Magnitudo facta &c. <lb/>[<emph style="it">tr: If the sum of the unnequal lines is equal to the sum of as many equals, the size of the product etc. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s379" xml:space="preserve"> etiam: <lb/>[<emph style="it">tr: also: </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s380" xml:space="preserve"> plana facta ab inæqualibus <lb/> minora sunt quaduratis <lb/> facta ab æqualibus. <lb/>[<emph style="it">tr: planes made from unequals are less than squares made from equals. </emph>]<lb/> </s> </p> <head xml:id="echoid-head114" xml:space="preserve" xml:lang="lat"> 2<emph style="it">o</emph>. De sectione in tres partes. <lb/>[<emph style="it">tr: 2. On sectioning into three parts. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s381" xml:space="preserve"> Casus primus <lb/>[<emph style="it">tr: First case. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s382" xml:space="preserve"> Sint tres inæquales partes. <lb/>[<emph style="it">tr: Let there be three unequalparts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s383" xml:space="preserve"> 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> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s384" xml:space="preserve"> Tres æquales partes. <lb/>[<emph style="it">tr: Three equal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s385" xml:space="preserve"> magnitudo facta <lb/> 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> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s386" xml:space="preserve"> Casus 2<emph style="super">a</emph>. <lb/>[<emph style="it">tr: Case 2. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s387" xml:space="preserve"> Sint tres inæquales partes. <lb/>[<emph style="it">tr: Let there be three unequal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s388" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s389" xml:space="preserve"> Tres æquales partes. <lb/>[<emph style="it">tr: Three equal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s390" xml:space="preserve"> magnitudo facta <lb/> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <pb file="add_6784_f411v" o="411v" n="822"/> <div xml:id="echoid-div71" type="page_commentary" level="2" n="71"> <p> <s xml:id="echoid-s391" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s391" xml:space="preserve"> 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), 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. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f412" o="412" n="823"/> <div xml:id="echoid-div72" type="page_commentary" level="2" n="72"> <p> <s xml:id="echoid-s393" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s393" xml:space="preserve"> The continuation of Add MS 6784, f. 411. </s> </p> </emph>] <lb/><lb/></s></p></div> <p xml:lang="lat"> <s xml:id="echoid-s395" xml:space="preserve"> Casus 3<emph style="super">a</emph>. <lb/>[<emph style="it">tr: Case 3. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s396" xml:space="preserve"> Sint tres inæquales partes. <lb/>[<emph style="it">tr: Let there be three unequal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s397" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s398" xml:space="preserve"> Tres æquales partes. <lb/>[<emph style="it">tr: Three equal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s399" xml:space="preserve"> magnitudo facta <lb/> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s400" xml:space="preserve"> Casus 4<emph style="super">a</emph>. <lb/>[<emph style="it">tr: Case 4. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s401" xml:space="preserve"> Sint tres inæquales partes. <lb/>[<emph style="it">tr: Let there be three unequal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s402" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s403" xml:space="preserve"> Tres æquales partes. <lb/>[<emph style="it">tr: Three equal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s404" xml:space="preserve"> magnitudo facta <lb/> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s405" xml:space="preserve"> Casus 5<emph style="super">a</emph>. <lb/> et ultimus. <lb/>[<emph style="it">tr: Case 5, and last. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s406" xml:space="preserve"> Sint tres inæquales partes. <lb/>[<emph style="it">tr: Let there be three unequal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s407" xml:space="preserve"> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s408" xml:space="preserve"> Tres æquales partes. <lb/>[<emph style="it">tr: Three equal parts. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s409" xml:space="preserve"> magnitudo facta <lb/> 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>. <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s410" xml:space="preserve"> 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> <lb/>[<emph style="it">tr: 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>. </emph>]<lb/> </s> </p> <pb file="add_6784_f412v" o="412v" n="824"/> <pb file="add_6784_f413" o="413" n="825"/> <pb file="add_6784_f413v" o="413v" n="826"/> <pb file="add_6784_f414" o="414" n="827"/> <div xml:id="echoid-div73" type="page_commentary" level="2" n="73"> <p> <s xml:id="echoid-s411" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s411" xml:space="preserve"> Combinations of small numbers; see also Add MS 6784, f. 424. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f414v" o="414v" n="828"/> <pb file="add_6784_f415" o="415" n="829"/> <div xml:id="echoid-div74" type="page_commentary" level="2" n="74"> <p> <s xml:id="echoid-s413" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s413" xml:space="preserve"> This page summarizes in shorthand some rules that are written out in full in Harriot's treatise on cubic equations, on Add MS 6782, f. 186. <lb/> The abbreviations 'co:l' and 'co:pl' stand for 'longitudinal coefficient' and 'plane coefficient' respectively. 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> and the plane coefficient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. Below the diagram Harriot has set out the different conditions under which such an equation can have three real roots, not necessarily distinct. The same sets of roots are also listed in Add MS 6783, f. 281. <lb/> The relevant equations are worked in full in sheets marked C, D, E, F, G (Add MS 6782, f. 315, f. 315v, f. 317, f. 318, f. 319), and also in Add MS 6783, f. 185. <lb/> </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f415v" o="415v" n="830"/> <pb file="add_6784_f416" o="416" n="831"/> <head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat"> Ad generationes sequentium specierum æquationum <lb/>[<emph style="it">tr: On the generation of the following types of equation. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s415" xml:space="preserve"> Æquatio <emph style="st">substantiva</emph> <lb/> parabolica. <lb/>[<emph style="it">tr: Parabolic equation </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s416" xml:space="preserve"> Æquatio <emph style="st">adiectiva</emph> <emph style="super">hyperbolica</emph> <lb/> <emph style="st">sive additiva</emph>. Hyperbolic equation </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s417" xml:space="preserve"> Æquatio <emph style="st">ablativa</emph> <emph style="super">elliptica</emph> <lb/> sive Bombellica. <lb/>[<emph style="it">tr: Elliptic, or Bombelli's, equation </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s418" xml:space="preserve"> Ergo æquatio <emph style="st">nullitatis</emph> <emph style="st">prima</emph> <lb/> <emph style="st">sive [???]</emph> <lb/> <emph style="st">sive</emph> primitiva. <lb/>[<emph style="it">tr: Therefore the equation is primitive. </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s419" xml:space="preserve"> Ergo verum quod proponebatur. <lb/>[<emph style="it">tr: Therefore what was proposed is true. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s420" xml:space="preserve"> Ad resolutiones sequentium specierum æquationum <lb/>[<emph style="it">tr: On solving the following types of equation </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s421" xml:space="preserve"> æquatio parabolica. <lb/>[<emph style="it">tr: parabolic equation </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s422" xml:space="preserve"> æquatio hyperbolica. <lb/>[<emph style="it">tr: hyperbolic equation </emph>]<lb/> </s> <lb/> <s xml:id="echoid-s423" xml:space="preserve"> æquatio elliptica. <lb/>[<emph style="it">tr: elliptic equation </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s424" xml:space="preserve"> rinus. </s> <lb/> <s xml:id="echoid-s425" xml:space="preserve"> prærinus. </s> <lb/> <s xml:id="echoid-s426" xml:space="preserve"> prinus. </s> <lb/> <s xml:id="echoid-s427" xml:space="preserve"> prino. </s> <lb/> <s xml:id="echoid-s428" xml:space="preserve"> prinatus. prinatio. </s> <lb/> <s xml:id="echoid-s429" xml:space="preserve"> prinatimus. </s> </p> <pb file="add_6784_f416v" o="416v" n="832"/> <pb file="add_6784_f417" o="417" n="833"/> <pb file="add_6784_f417v" o="417v" n="834"/> <pb file="add_6784_f418" o="418" n="835"/> <pb file="add_6784_f418v" o="418v" n="836"/> <pb file="add_6784_f419" o="419" n="837"/> <pb file="add_6784_f419v" o="419v" n="838"/> <pb file="add_6784_f420" o="420" n="839"/> <pb file="add_6784_f420v" o="420v" n="840"/> <pb file="add_6784_f421" o="421" n="841"/> <pb file="add_6784_f421v" o="421v" n="842"/> <pb file="add_6784_f422" o="422" n="843"/> <pb file="add_6784_f422v" o="422v" n="844"/> <pb file="add_6784_f423" o="423" n="845"/> <div xml:id="echoid-div75" type="page_commentary" level="2" n="75"> <p> <s xml:id="echoid-s430" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s430" xml:space="preserve"> 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>. 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. Columns to the right list successive differences as far as the constant difference 6. The table has also been extrapolated upwards, giving rise to negative values in the first three columns. There is an error in the first column, however, which reading upwards should be: 322, 110, 52, 18, 2, - 2, 0, 2, .... </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f423v" o="423v" n="846"/> <pb file="add_6784_f424" o="424" n="847"/> <div xml:id="echoid-div76" type="page_commentary" level="2" n="76"> <p> <s xml:id="echoid-s432" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s432" xml:space="preserve"> Note various combinations of small numbers in the lower part of the page (see also Add MS 6784, f. 414). </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f424v" o="424v" n="848"/> <pb file="add_6784_f425" o="425" n="849"/> <pb file="add_6784_f425v" o="425v" n="850"/> <pb file="add_6784_f426" o="426" n="851"/> <pb file="add_6784_f426v" o="426v" n="852"/> <pb file="add_6784_f427" o="427" n="853"/> <pb file="add_6784_f427v" o="427v" n="854"/> <pb file="add_6784_f428" o="428" n="855"/> <div xml:id="echoid-div77" type="page_commentary" level="2" n="77"> <p> <s xml:id="echoid-s434" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s434" xml:space="preserve"> Sums of some infinite geometric progressions. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f428v" o="428v" n="856"/> <div xml:id="echoid-div78" type="page_commentary" level="2" n="78"> <p> <s xml:id="echoid-s436" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s436" xml:space="preserve"> Triangles and circles filled with rectilinear figures (rectangles or triangles), in a way that can in principle be continued indefinitely. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f429" o="429" n="857"/> <head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat"> De infinitis. Ex ratione motus, temporis et spatij. <lb/>[<emph style="it">tr: On infinity. From the ratio of motion, time and space. </emph>]<lb/> </head> <p xml:lang="lat"> <s xml:id="echoid-s438" xml:space="preserve"> Vide <reg norm="Aristotle" type="abbr">Arist</reg>. lib. 6. tret. 23. <lb/> proclum de motu lib. 1. pro. 14. <lb/>[<emph style="it">tr: See Aristotle, Book 6, Treatise 23. <lb/> Proclus, <emph style="it">De motu</emph>, Book 1, Proposition 14. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s439" xml:space="preserve"> 1. <lb/> Moveatur A corpus <lb/> per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium in <lb/> tempore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi></mstyle></math> atque sit <lb/> ille motus uniformis. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <!-- text in first column --> <p xml:lang="lat"> <s xml:id="echoid-s440" xml:space="preserve"> infinite <lb/> maximum <lb/>[<emph style="it">tr: infinite maximum </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s441" xml:space="preserve"> minimum <lb/>[<emph style="it">tr: minimum </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s442" xml:space="preserve"> indivisibile <lb/>[<emph style="it">tr: an indivisible </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s443" xml:space="preserve"> punctum <lb/>[<emph style="it">tr: a point </emph>]<lb/> </s> </p> <!-- text in second column --> <p xml:lang="lat"> <s xml:id="echoid-s444" xml:space="preserve"> aliquod <lb/> infinite <lb/> maximum <lb/>[<emph style="it">tr: infinite maximum </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s445" xml:space="preserve"> minimum <lb/> eadem <lb/> ratione <lb/>[<emph style="it">tr: minimum in the same ratio </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s446" xml:space="preserve"> Indivisibile <lb/> eadem <lb/> ratione <lb/>[<emph style="it">tr: An indivisble in the same ratio </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s447" xml:space="preserve"> Indivisibile <lb/> sed non punctum <lb/> vel instans ut alia <lb/> ratione inferetur. <lb/>[<emph style="it">tr: And indivisble but not a point or an instant that can be inferred from the other ratio. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s448" xml:space="preserve"> 2. <lb/> Moveatur A corpus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> spatium <lb/> 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/> motus uniformis. <lb/>[<emph style="it">tr: 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. </emph>]<lb/> </s> </p> <!-- text in first column --> <p xml:lang="lat"> <s xml:id="echoid-s449" xml:space="preserve"> indivisibile <lb/>[<emph style="it">tr: an indivisible </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s450" xml:space="preserve"> punctum <lb/>[<emph style="it">tr: a point </emph>]<lb/> </s> </p> <!-- text in second column --> <p xml:lang="lat"> <s xml:id="echoid-s451" xml:space="preserve"> Indivisibile <lb/> eadem ratione <lb/>[<emph style="it">tr: An indivisble in the same ratio </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s452" xml:space="preserve"> Indivisibile quod <lb/> dimidium est <lb/> Indivisibilis ex <lb/> priori argumentatione. <lb/>[<emph style="it">tr: An indivisble whose half is indivisble by the previous argument. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s453" xml:space="preserve"> Ergo etiam: <lb/>[<emph style="it">tr: Therefore also </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s454" xml:space="preserve"> Indivisibile quod <lb/> dimidium est <lb/> Indivisibilis ex <lb/> priori argumentatione. <lb/>[<emph style="it">tr: An indivisble whose half is indivisble by the previous argument. </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s455" xml:space="preserve"> punctum <lb/>[<emph style="it">tr: a point </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s456" xml:space="preserve"> punctum <lb/>[<emph style="it">tr: a point </emph>]<lb/> </s> </p> <p xml:lang="lat"> <s xml:id="echoid-s457" xml:space="preserve"> Ergo punctum quod ponebatur esse <lb/> indivisbile, alia ratione inferetur <lb/> Divisibile, et sic in infinitum. <lb/>[<emph style="it">tr: Therefore a point that can be supposed indivisble, is inferred from the other ratio to be divisible, and thus infinitely. </emph>]<lb/> </s> </p> <pb file="add_6784_f429v" o="429v" n="858"/> <pb file="add_6784_f430" o="430" n="859"/> <div xml:id="echoid-div79" type="page_commentary" level="2" n="79"> <p> <s xml:id="echoid-s458" xml:space="preserve">[<emph style="it">Note: <p> <s xml:id="echoid-s458" xml:space="preserve"> Triangles transformed to spirals. <lb/> See also Add MS 6785, f. 437 and Add MS 6784, f. 246, f. 247, f. 248. </s> </p> </emph>] <lb/><lb/></s></p></div> <pb file="add_6784_f430v" o="430v" n="860"/> <pb file="add_6784_f431" o="431" n="861"/> <pb file="add_6784_f431v" o="431v" n="862"/> </div> </text> </echo>