Harriot, Thomas, Mss. 6784

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431
431 (216) 		432
432 (216v)
433
433 (217) 		434
434 (217v)
435
435 (218) 		436
436 (218v)
437
437 (219) 		438
438 (219v)
439
439 (220) 		440
440 (220v)
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page |< < (241) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f241" o="241" n="481"/>
          <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 page to Proposition II.12 from
                <emph style="it">Conicorum libri quattuor</emph>
                <ref id="apollonius_1566"> (Apollonius </ref>
              . </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> De
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit datus angulus.
              <math>
                <mstyle>
                  <mi>x</mi>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et punctum.
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            oportet ducere lineam
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            ut sit
              <lb/>
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let there be a given angle
              <math>
                <mstyle>
                  <mi>x</mi>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            and a point
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            It is required to draw a line
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            so that
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            , a given line. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> fiat:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <math>
                <mstyle>
                  <mi>x</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            producta, in
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            per punctum
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            , ad angulum
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
              <lb/>
            fiat hyperbola:
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Inscribatur
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>k</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
              <lb/>
            agatur
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            , quæ secabit
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , in
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod:
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            ,
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Make
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , meeting with
              <math>
                <mstyle>
                  <mi>x</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            extended at
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Make
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Through the point
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            , at angle
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            , make the hyperbola
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Inscribe
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>k</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Make
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
              <lb/>
            Construct
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            , which will cut
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>m</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            , the given line. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> per. 12,2
              <emph style="super">i</emph>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            by II.12 of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Aliter.
              <lb/>
            Et continuetur ad utrasque partes.
              <lb/>
            usque ad
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>q</mi>
                  <mi>p</mi>
                </mstyle>
              </math>
            , quæ secabit
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod:
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Another way.
              <lb/>
            And continuing on both sides, as far as
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>q</mi>
                </mstyle>
              </math>
            , make
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>q</mi>
                  <mi>p</mi>
                </mstyle>
              </math>
            , which will cut
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quoniam,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mo>=</mo>
                  <mi>f</mi>
                  <mi>p</mi>
                </mstyle>
              </math>
              <lb/>
            erit:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>p</mi>
                  <mo>=</mo>
                  <mi>g</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            . propter asymptotes
              <lb/>
            sed:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>e</mi>
                  <mo>=</mo>
                  <mi>q</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            . ob parallelismum,
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            Ergo:
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Since
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mo>=</mo>
                  <mi>f</mi>
                  <mi>p</mi>
                </mstyle>
              </math>
            , therefore
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>p</mi>
                  <mo>=</mo>
                  <mi>g</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            on account of the asymptotes.
              <lb/>
            But
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>e</mi>
                  <mo>=</mo>
                  <mi>q</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            because
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            are parallel.
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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