Harriot, Thomas, Mss. 6785

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f135" o="135" n="269"/>
          <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 examines Proposition VI from
                <emph style="it">Supplementum geometriæ</emph>
                <ref id="Viete_1593c" target="http://www.e-rara.ch/zut/content/pageview/2684113"> (Viète 1593c, Prop </ref>
              . See also Add MS 6785
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/KN1CRTZ2/&start=280&viewMode=image&pn=285"> f. </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio VI.
                  <lb/>
                Dato triangulo rectangulo, invenire aliud triangulum rectangulum majus, & aeque altum; ut quod fit sub differentia basium ipsorum & differentia hypotenusarum, aequale fit dato cuicumque recti-lineo.</s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given a right-angled triangle, to find another larger right-angled triangle, with equal height, so that the product of the difference of the bases and the difference of the hypotenuses is equal to a given </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> In prop: 6.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          From proposition 6 of the ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data
              <lb/>
            prima
              <lb/>
            quarta
              <lb/>
            quatuor
              <lb/>
            parallela
              <lb/>
            Quæsita
              <lb/>
            continue
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given
              <lb/>
            first
              <lb/>
            fourth
              <lb/>
            four
              <lb/>
            parallel
              <lb/>
            Sought
              <lb/>
            continued ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Conclusio ex inferiore
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Conclusion from the demonstration ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Demonstratio per compositionem.
              <lb/>
            Sint primo constructio quatuor proportionales
              <lb/>
            per 5
              <emph style="super">tam</emph>
            prop. […] Unde
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
              <lb/>
            est æqualis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            parallelæ. Iam fiat
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et ducatur recta
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
              <emph style="super">parallela
                <math>
                  <mstyle>
                    <mi>β</mi>
                    <mi>α</mi>
                  </mstyle>
                </math>
              </emph>
            et
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            sit parallela
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Ergo angulus
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>β</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>β</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            , angulo
              <math>
                <mstyle>
                  <mi>μ</mi>
                  <mi>D</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            , et tertius angulo tertio.
              <lb/>
            Ergo triangula
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            simila et æqualia. Et producta
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            transibit per
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            , alias
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            non sunt æquales. Sit producta
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            versus
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Et ducatur
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
              <emph style="super">parallela
                <math>
                  <mstyle>
                    <mi>α</mi>
                    <mi>γ</mi>
                  </mstyle>
                </math>
              </emph>
            . Sit inde
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>δ</mi>
                </mstyle>
              </math>
            parallela
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            . Ergo anguli
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>δ</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            , æqualis, et
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>ɛ</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>ɛ</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>δ</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>δ</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et æqualis
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>ɛ</mi>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>θ</mi>
                </mstyle>
              </math>
            . Et quia
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            æqualis inter parallelas, æqualis etiam
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>λ</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Conclusio igitur
              <lb/>
            facile colligitur et manifesta. vel triplex ut
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Demonstration by construction.
              <lb/>
            Let there be first constructed four proportionals by the 5th proposition.
              <lb/>
            Whence
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            are parallel.
              <lb/>
            Now construct
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            , and the line
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            is parallel to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Therefore the angule
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>β</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>β</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            to angle
              <math>
                <mstyle>
                  <mi>μ</mi>
                  <mi>D</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            , and the third angle to the third. Therefore the triangles
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            are similar and qual.
              <lb/>
            And
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>μ</mi>
                </mstyle>
              </math>
            produced will pass through
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            , otherwis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>α</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            are not equal. Let
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            be produced towars
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            .
              <lb/>
            And
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            is constrcuted parallel to
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>δ</mi>
                </mstyle>
              </math>
            be parallel to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            . Therefore angles
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>δ</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            are equal, and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>ɛ</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            ; and
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>ɛ</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>δ</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>δ</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            ; and
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>γ</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            ; and
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>α</mi>
                  <mi>ɛ</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>γ</mi>
                  <mi>θ</mi>
                </mstyle>
              </math>
            . And because
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>β</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>β</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            are equal between parallels,
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>λ</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>λ</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            are also equal.
              <lb/>
            Therefore the conclusion is easily gathered and shown, or three times, as ]</s>
          </p>
        </div>
      </text>
    </echo>
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