Harriot, Thomas, Mss. 6784

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611
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613
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615
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618 (309v)
619
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620 (310v)
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page |< < (349) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f349" o="349" n="697"/>
          <div type="page_commentary" level="0" n="0">
            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> On this page Harriot investigates Proposition 18 from
                <emph style="it">Supplementum geometriæ</emph>
                <ref id="Viete_1593c" target="http://www.e-rara.ch/zut/content/pageview/2684121"> (Viète 1593c, Prop </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> 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 </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> 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.</s>
              </quote>
              <lb/>
              <s 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
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=700&viewMode=image&pn=705"> f. </ref>
              ), Proposition 16 (add MS 6784
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=700&viewMode=image&pn=701"> f. </ref>
              ) and Proposition 17 (add MS 6784
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=690&viewMode=image&pn=699"> f. </ref>
              ). </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve"> prop. 18.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Proposition 18 from the ]</head>
          <p xml:lang="lat">
            <s 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
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            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.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit triangulum primum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , secundum
              <lb/>
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            . quorum crura et anguli sint
              <lb/>
            ut exigit propositio. Et sit
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            dupla
              <lb/>
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . Tum quadratum
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            erit triplum quadrati
              <lb/>
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
             
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the first triangle be
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            and the second
              <math>
                <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>
                <mstyle>
                  <mi>G</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            be twice
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . Then the square of
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            is three times the square of
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Nam:
              <lb/>
            per 15,p […] Hoc est, in notis proportionalium quas notum 12,p
              <lb/>
            1
              <emph style="super">o</emph>
            . Ducantur omnia per
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
              <lb/>
            […]
              <lb/>
            Hoc est in notis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            For by Proposition 15 that is, in the notation for proportionals noted in Proposition 12,
              <lb/>
            1. Multiply everything by
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            That is, in the notation of Proposition ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">2
              <emph style="super">o</emph>
            . Ducantur omnia per
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
              <lb/>
            […]
              <lb/>
            Hoc est in notis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            2. Multiply everything by
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            That is, in the notation of Proposition ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Deinde per 16.p
              <lb/>
            Hoc est in notis 12,p.
              <lb/>
            Sed: per consect: 14.p
              <lb/>
            Ergo patet
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            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 ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Cum 16
              <emph style="super">a</emph>
            et 17
              <emph style="super">a</emph>
            prop. basis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            notabatur (
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ) ideo eius partes
              <lb/>
            Scilicet
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            alijs vocalibus notandæ sunt. pro
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            nota (
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            )
              <lb/>
            et pro
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , (
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            ).
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            servent easdem notas quas ibi
              <lb/>
            habuerunt. Videlicet
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            , (
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ) et
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            , (
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ).
              <lb/>
            Propositum igitur simplicibus notis ita
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Since in Propositions 16 adn 17, the base
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            is denoted by
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , therefore its parts, namely
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            may be denoted by other names; for
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            put the letter
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            and for
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            the letter
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            . For
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            use the same notation as they had there, namely
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                  <mo>=</mo>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            In simple notation the proposition may therefore be ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> igitur:
              <lb/>
            Quando æquatio est sub ista
              <lb/>
            forma:
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            erit duplex vel.
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            . vel.
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            When the equation is in this form,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            is twofold, either
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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