Harriot, Thomas, Mss. 6784

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page |< < (40) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f040" o="40" n="79"/>
          <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 cutting-off of a ratio', 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=40&viewMode=image&pn=47"> f. </ref>
              . </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> De resectione
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          The cutting off of a ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sint dati linea
              <math>
                <mstyle/>
              </math>
            AB
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            AD
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>t</mi>
                  <mi>p</mi>
                  <mi>u</mi>
                  <mi>n</mi>
                  <mi>c</mi>
                  <mi>t</mi>
                  <mi>u</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            C
              <math>
                <mstyle>
                  <mo>.</mo>
                </mstyle>
              </math>
              <lb/>
            oportet resecare
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            in ratione data
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Sit iam factum, ut:
              <lb/>
            tum in datam
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let there be given lines
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            and a point
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            .
              <lb/>
            It is required to cut off
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            in the fiven ratio
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            . Let it be already done, so that:
              <lb/>
            then in the given ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Synthesis.
              <lb/>
            A puncto
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            agatur
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            parallela
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
              <lb/>
            et fiat
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            parallela
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
              <lb/>
            æqualis
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            . et per puncta
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>K</mi>
                </mstyle>
              </math>
            ducatur
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
              <lb/>
            usque ad
              <math>
                <mstyle>
                  <mi>I</mi>
                </mstyle>
              </math>
            . quæ secabit
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>H</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Demonstratio
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Synthesis
              <lb/>
            From the point
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            there is constructed
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            , and make
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            . And through the points
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>K</mi>
                </mstyle>
              </math>
            there is drawn
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>K</mi>
                </mstyle>
              </math>
            as far as
              <math>
                <mstyle>
                  <mi>I</mi>
                </mstyle>
              </math>
            , which will cut
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>H</mi>
                </mstyle>
              </math>
            .
              <lb/>
            The demonstration is ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quatuor possunt esse resectiones ab uno
              <lb/>
            puncto dato et duabus solummodo lineis
              <lb/>
            datis, secundum quantitatem rationes datæ.
              <lb/>
            Sed numquam punctibus duabus, nisi in uno casu
              <lb/>
            rationis æqualitatis, scilicet, cum
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            sit æqualis
              <lb/>
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            There are four possible sections from one given point and just two given lines, according to the size of the given ratio.
              <lb/>
            But never from two points, unless in the case of equal ratio, namely, if
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Et sic de
              <lb/>
            cæteris.
              <lb/>
            Nullum vide in
              <lb/>
            hac propositione
              <lb/>
            difficultatem.
              <lb/>
            De Infinitis
              <lb/>
            speculatio ut
              <lb/>
            altera pagina
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            And so for the rest.
              <lb/>
            There is nothing of difficulty in this proposition.
              <lb/>
            On consideration of infinity as in the other page, ]
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            The other page mentioned here is the reverse of this one, Add MS
              <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=80&viewMode=image&pn=80"> f. </ref>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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