Harriot, Thomas, Mss. 6784

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f140" o="140" n="279"/>
          <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"> There is a reference on this and the four following folios to page 212 of Commandino's edition of
                <emph style="it">Mathematicae collectiones</emph>
                <ref id="pappus_1588"> (Pappus </ref>
              . Page 212 contains 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"> Pappus. 212. </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> Hic habetur usus determinatæ
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Here is found the use of a determinate ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> 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,
              <lb/>
            qualis est
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , ita ut 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>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            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>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Constructio:
              <lb/>
            Centro
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            , intervallo
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            , describitur periferia
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>I</mi>
                  <mi>Q</mi>
                </mstyle>
              </math>
            ; Et inscribatur
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
              <lb/>
            æqualis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            . A puncto
              <math>
                <mstyle>
                  <mi>I</mi>
                </mstyle>
              </math>
            ducatur
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            perpendicularis ad
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dividatur
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            bisariam in puncto
              <math>
                <mstyle>
                  <mi>K</mi>
                </mstyle>
              </math>
            . fiat
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>K</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dividatur
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            bisariam 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>
            vel
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>N</mi>
                </mstyle>
              </math>
            . Sit
              <math>
                <mstyle>
                  <mi>P</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
              <lb/>
            perpendicularis ad
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . et ducatur
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            , cui fiat æqualis
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            . Dico quod
              <lb/>
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            est centrum semicirculo quæsiti. qui describatur et sit
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . a puncto
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
            ducatur contingens
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            et producatur ad
              <math>
                <mstyle>
                  <mi>B</mi>
                </mstyle>
              </math>
            . Dico quod
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            æqualis
              <lb/>
            est
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Construction
              <lb/>
            With centre
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            and radius
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            , there isdrawn the circumference
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>I</mi>
                  <mi>Q</mi>
                </mstyle>
              </math>
            . And there is inscribed
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            . From the point
              <math>
                <mstyle>
                  <mi>I</mi>
                </mstyle>
              </math>
            there is drawn
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            perpendicular to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . The line
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            is bisected at the point
              <math>
                <mstyle>
                  <mi>K</mi>
                </mstyle>
              </math>
            . Make
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>K</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . The line
              <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>
            euqal to
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>N</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>N</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>P</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            be perpendicular to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            is drawn, to which make
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            equal. I say that
              <math>
                <mstyle>
                  <mi>G</mi>
                </mstyle>
              </math>
            is the centre of the semicircle sought, which is drawn, and is
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . From the point
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            there is drawn the tangent
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            extended to
              <math>
                <mstyle>
                  <mi>B</mi>
                </mstyle>
              </math>
            . I say that
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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