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diff texts/XML/echo/en/Harriot_Add_MS_6784_XT0KZ8QC.xml @ 6:22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
| author | casties |
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| date | Fri, 07 Dec 2012 17:05:22 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/texts/XML/echo/en/Harriot_Add_MS_6784_XT0KZ8QC.xml Fri Dec 07 17:05:22 2012 +0100 @@ -0,0 +1,5489 @@ +<?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 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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 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+<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" 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+The reference on this page is to Willebrord Snell's +<emph style="it">Apollonius Batavus</emph> (1608). +</s> +</p> +</emph>] +<lb/><lb/></s></p></div> +<head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat"> +Diagrammata <lb/> +Snellij +<lb/>[<emph style="it">tr: +Snell's diagrams +</emph>]<lb/> +</head> +<pb file="add_6784_f067v" o="67v" n="134"/> +<pb file="add_6784_f068" o="68" n="135"/> +<pb file="add_6784_f068v" o="68v" n="136"/> +<pb file="add_6784_f069" o="69" n="137"/> +<pb file="add_6784_f069v" o="69v" n="138"/> +<pb file="add_6784_f070" o="70" n="139"/> +<pb file="add_6784_f070v" o="70v" n="140"/> +<pb file="add_6784_f071" o="71" n="141"/> +<pb file="add_6784_f071v" o="71v" n="142"/> +<pb file="add_6784_f072" o="72" n="143"/> +<pb file="add_6784_f072v" o="72v" n="144"/> +<pb file="add_6784_f073" o="73" n="145"/> +<pb file="add_6784_f073v" o="73v" n="146"/> +<pb file="add_6784_f074" o="74" n="147"/> +<pb file="add_6784_f074v" o="74v" n="148"/> +<pb 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file="add_6784_f089v" o="89v" n="178"/> +<div xml:id="echoid-div4" type="page_commentary" level="2" n="4"> +<p> +<s xml:id="echoid-s9" xml:space="preserve">[<emph style="it">Note: +<p> +<s xml:id="echoid-s9" 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>+</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 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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 file="add_6784_f221v" o="221v" n="442"/> +<pb file="add_6784_f222" o="222" n="443"/> +<pb file="add_6784_f222v" o="222v" n="444"/> +<pb file="add_6784_f223" o="223" n="445"/> +<pb file="add_6784_f223v" o="223v" n="446"/> +<pb file="add_6784_f224" o="224" n="447"/> +<pb file="add_6784_f224v" o="224v" n="448"/> +<pb file="add_6784_f225" o="225" n="449"/> +<pb file="add_6784_f225v" o="225v" n="450"/> +<pb file="add_6784_f226" o="226" n="451"/> +<pb file="add_6784_f226v" o="226v" n="452"/> +<pb file="add_6784_f227" o="227" n="453"/> +<pb file="add_6784_f227v" o="227v" n="454"/> +<pb file="add_6784_f228" o="228" n="455"/> +<pb file="add_6784_f228v" o="228v" n="456"/> +<pb file="add_6784_f229" o="229" n="457"/> +<pb file="add_6784_f229v" o="229v" n="458"/> +<pb file="add_6784_f230" o="230" n="459"/> +<pb file="add_6784_f230v" o="230v" n="460"/> +<pb file="add_6784_f231" o="231" n="461"/> +<pb file="add_6784_f231v" o="231v" n="462"/> +<pb file="add_6784_f232" 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n="485"/> +<pb file="add_6784_f243v" o="243v" n="486"/> +<pb 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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for operating with +positive quantities ('more') and negative quantites ('lesse'). +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> \ No newline at end of file