Harriot, Thomas, Mss. 6784

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page |< < (136) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f136" o="136" n="271"/>
          <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"> The problem pursued in this and many other folios in Add MS 6784 is 'the determinate section', as set out in Pappus,
                <emph style="it">Mathematicae collectiones</emph>
                <ref id="pappus_1588"> (Pappus </ref>
              , Book 7. For a statement of the problem see Add MS 6784
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=240&viewMode=image&pn=247"> f. </ref>
              . </s>
              <lb/>
              <s xml:space="preserve"> There is a particular reference on this folio to page 214 of Commandino's edition of Pappus. Page 214 is part of Commandino's lengthy commentary to Proposition 85, also denoted Lemma XI.</s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Problema V. Propos. LXXXV.
                  <lb/>
                Semicirculo positione dato
                  <math>
                    <mstyle>
                      <mi>A</mi>
                      <mi>B</mi>
                      <mi>C</mi>
                    </mstyle>
                  </math>
                , & dato puncto
                  <math>
                    <mstyle>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                , describere per
                  <math>
                    <mstyle>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                semicirculum, qualis est
                  <math>
                    <mstyle>
                      <mi>D</mi>
                      <mi>E</mi>
                      <mi>F</mi>
                    </mstyle>
                  </math>
                , ita vt ducatur contingens
                  <math>
                    <mstyle>
                      <mi>B</mi>
                      <mi>C</mi>
                    </mstyle>
                  </math>
                , fiat
                  <math>
                    <mstyle>
                      <mi>A</mi>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                ipsi
                  <math>
                    <mstyle>
                      <mi>B</mi>
                      <mi>E</mi>
                    </mstyle>
                  </math>
                æqualis. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given a semicircle
                  <math>
                    <mstyle>
                      <mi>A</mi>
                      <mi>B</mi>
                      <mi>C</mi>
                    </mstyle>
                  </math>
                and a point
                  <math>
                    <mstyle>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                , draw through
                  <math>
                    <mstyle>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                a semicircle
                  <math>
                    <mstyle>
                      <mi>D</mi>
                      <mi>E</mi>
                      <mi>F</mi>
                    </mstyle>
                  </math>
                , so that when the tangent
                  <math>
                    <mstyle>
                      <mi>B</mi>
                      <mi>C</mi>
                    </mstyle>
                  </math>
                is drawn,
                  <math>
                    <mstyle>
                      <mi>A</mi>
                      <mi>D</mi>
                    </mstyle>
                  </math>
                is equal to
                  <math>
                    <mstyle>
                      <mi>B</mi>
                      <mi>E</mi>
                    </mstyle>
                  </math>
                . </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve"> Vide Commandinum in pappo pag. </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> De Sectione Determinata.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On a determinate section. ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit data recta linea
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            ; et in ea puncta,
              <math>
                <mstyle>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>L</mi>
                </mstyle>
              </math>
            : oportet ipsam
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            ita
              <lb/>
            secare in puncto
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            : ut </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> 1. Constructio:
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            media proportionalis
              <lb/>
            inter
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            . Fiat
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            æqualis
              <lb/>
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            ; Dividatur
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            bisariam
              <lb/>
            in
              <math>
                <mstyle>
                  <mi>N</mi>
                </mstyle>
              </math>
            : et fiat
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>N</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Connectantur
              <math>
                <mstyle>
                  <mi>O</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>P</mi>
                </mstyle>
              </math>
            . et lineæ
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
              <lb/>
            fiat æqualis
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod punctum
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            est punctum quæ-
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            1. Construction
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            be a mean proportional between
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            . Make
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            is bisected at
              <math>
                <mstyle>
                  <mi>N</mi>
                </mstyle>
              </math>
            . And make
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>N</mi>
                </mstyle>
              </math>
            . Connect
              <math>
                <mstyle>
                  <mi>O</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>P</mi>
                </mstyle>
              </math>
            , and make
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            equal to the line
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
              <lb/>
            fiat æqualis
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that the point
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            is the point sought </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> 2. Constructio:
              <lb/>
            Sint
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>R</mi>
                </mstyle>
              </math>
            , et
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>V</mi>
                </mstyle>
              </math>
            æquales. Sit
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>S</mi>
                </mstyle>
              </math>
            media
              <lb/>
            proportionalis inter
              <math>
                <mstyle>
                  <mi>R</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            . Dividatur
              <math>
                <mstyle>
                  <mi>V</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
              <lb/>
            bisariam in
              <math>
                <mstyle>
                  <mi>T</mi>
                </mstyle>
              </math>
            . Centro
              <math>
                <mstyle>
                  <mi>T</mi>
                </mstyle>
              </math>
            , describatur periferia
              <lb/>
              <math>
                <mstyle>
                  <mi>V</mi>
                  <mi>X</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            . Erigitur perpendicularis
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>X</mi>
                </mstyle>
              </math>
            . Fiat
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>Y</mi>
                </mstyle>
              </math>
              <lb/>
            æqualis
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>S</mi>
                </mstyle>
              </math>
            . Et per
              <math>
                <mstyle>
                  <mi>Y</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>S</mi>
                </mstyle>
              </math>
            ducatur linea ad
              <lb/>
              <math>
                <mstyle>
                  <mi>Z</mi>
                </mstyle>
              </math>
            . Et fiat
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>Y</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            est punctum
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            2. Construction
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>R</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>V</mi>
                </mstyle>
              </math>
            be equal. Let
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>S</mi>
                </mstyle>
              </math>
            be the mean proportional between
              <math>
                <mstyle>
                  <mi>R</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>V</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            is bisected at
              <math>
                <mstyle>
                  <mi>T</mi>
                </mstyle>
              </math>
            . With centre
              <math>
                <mstyle>
                  <mi>T</mi>
                </mstyle>
              </math>
            , there is drawn the circumference
              <math>
                <mstyle>
                  <mi>V</mi>
                  <mi>X</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            . There is erected the perpendicular
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>X</mi>
                </mstyle>
              </math>
            . Make
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>Y</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>S</mi>
                </mstyle>
              </math>
            . And thorugh
              <math>
                <mstyle>
                  <mi>Y</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>S</mi>
                </mstyle>
              </math>
            there is drawn the line to
              <math>
                <mstyle>
                  <mi>Z</mi>
                </mstyle>
              </math>
            . And make
              <math>
                <mstyle>
                  <mi>T</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>Y</mi>
                  <mi>Z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            is the point sought. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">3
              <emph style="super">i</emph>
            . Constructio:
              <lb/>
            fiat
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>l</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
              <lb/>
            […]
              <lb/>
            bisecetur
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
              <lb/>
            […]
              <lb/>
            Dico quod
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            est punctum
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            3rd Construction
              <lb/>
            make
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>l</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
              <lb/>
              <lb/>
            bisect
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
              <lb/>
              <lb/>
            I say that
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            is the point sought. </s>
          </p>
        </div>
      </text>
    </echo>
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