Harriot, Thomas, Mss. 6787

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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f372" o="372" n="742"/>
          <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"> This page refers to Proposition 21 of Book I of Apollonius, as edited by Commandino
                <emph style="it">Conicorum libri quattuor</emph>
                <ref id="apollonius_1566"> (Apollonius </ref>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve"> I.21 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of the transverse side of the figure, as the upright side of the figure is to the transverse, and to each other as the areas contained by the straight lines cut off, as we have </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> De
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On the ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"/>
            <lb/>
            <s xml:space="preserve"> Si quotlibet lineis ordinatim applicatis ad diametrum circuli, aliæ numero
              <lb/>
            et quantitate æquales similiter applicentur ad similes partes lineæ maioris
              <emph style="st">vel</emph>
              <lb/>
            minoris
              <emph style="super">de</emph>
            data diametro circuli: termini illarum linearum sunt in
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If any number of ordinate lines are dropped to the diameter of a circle, that number of other lines, equal in quantity and similarly applied to similar parts of lines greater or less than the given diameter of a circe, then the ends of those lines are on an ]</s>
            <lb/>
            <s xml:space="preserve"> Sit diameter circuli
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            . lineæ ordinatim applicatæ
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . sit etiam
              <lb/>
            linea maior quam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mi>x</mi>
                </mstyle>
              </math>
            cui applicentur ad angulos rectos
              <math>
                <mstyle>
                  <mo>α</mo>
                  <mo>β</mo>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>ε</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            æquales
              <lb/>
              <emph style="super">lineis</emph>
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <emph style="st">Et sint partes lineæ
                <math>
                  <mstyle>
                    <mo>δ</mo>
                    <mi>x</mi>
                  </mstyle>
                </math>
              videlicet [???] quotlibet </emph>
              <lb/>
              <emph style="st">[???]</emph>
              <emph style="super">sed ita</emph>
            fiat ud
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mi>x</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ita
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mo>β</mo>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Quod etiam fit
              <lb/>
            si utraque lineæ
              <emph style="super">
                <math>
                  <mstyle>
                    <mi>d</mi>
                    <mi>c</mi>
                  </mstyle>
                </math>
              et
                <math>
                  <mstyle>
                    <mo>δ</mo>
                    <mi>x</mi>
                  </mstyle>
                </math>
              </emph>
            similter dividantur et ad utraque
              <emph style="st">æquales</emph>
            similes et
              <lb/>
            similiter sitas æquales lineæ ordinatim applicentur.
              <lb/>
            Dico quod puncta
              <math>
                <mstyle>
                  <mo>α</mo>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>ε</mo>
                </mstyle>
              </math>
            termini linearum
              <math>
                <mstyle>
                  <mo>α</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>ε</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            sunt in ellipsi.
              <lb/>
            Quoniam ex hypothesi
              <lb/>
            […] Ergo: per 21, prop: primi Apollonij, puncta
              <math>
                <mstyle>
                  <mo>α</mo>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>ε</mo>
                </mstyle>
              </math>
              <lb/>
            sunt in ellipsi. quod demonstrare
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the diameter of the circle be
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , and the ordinate lines
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ; also let the line greater than
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            be
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mi>x</mi>
                </mstyle>
              </math>
            , to which are applied at right angles
              <math>
                <mstyle>
                  <mo>α</mo>
                  <mo>β</mo>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mo>ε</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            euqal to the lines
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            in the circle. But thus, as
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mi>x</mi>
                </mstyle>
              </math>
            is to
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            so is
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mo>β</mo>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Which also happens if the two lines
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>δ</mo>
                  <mi>x</mi>
                </mstyle>
              </math>
            are similarly divided and to both parts, similar and similarly situated, equal ordinate lines are applied.
              <lb/>
            I say that the points
              <math>
                <mstyle>
                  <mo>α</mo>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mo>ε</mo>
                </mstyle>
              </math>
            , the ends of the lines
              <math>
                <mstyle>
                  <mo>α</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mo>ε</mo>
                  <mo>θ</mo>
                </mstyle>
              </math>
            , are on an ellipse.
              <lb/>
            Because from the hypothesis:
              <lb/>
            Therefore, by Proposition 21 of the first Book of Apollonius, the points
              <math>
                <mstyle>
                  <mo>α</mo>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>ε</mo>
                </mstyle>
              </math>
            are on an ellipse; which was to be demonstrated. </s>
          </p>
        </div>
      </text>
    </echo>
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