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_f024" o="24" n="47"/>
          <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>
              , Book 7. In Commandino's edition of 1588, the problem is stated on page 158v
                <ref id="pappus_1588"> (Pappus </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Per datum punctum rectam lineam ducere secantem a duabus rectis lineis positione datis ad data in ipsis puncta lineas, quæ proportionem habeant eandem datæ proportioni.</s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Through a given point, draw a line cutting off line segments from two lines given in position, to points given in them, which have a ratio the same as a given ratio.</s>
              </quote>
              <lb/>
              <s xml:space="preserve"> Using Harriot's lettering, the problem may be stated as follows. Given the lines
                <math>
                  <mstyle>
                    <mi>u</mi>
                    <mi>e</mi>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mi>t</mi>
                    <mi>q</mi>
                  </mstyle>
                </math>
              , the point
                <math>
                  <mstyle>
                    <mi>o</mi>
                  </mstyle>
                </math>
              , and the ratio
                <math>
                  <mstyle>
                    <mi>x</mi>
                    <mo>:</mo>
                    <mi>z</mi>
                  </mstyle>
                </math>
              , construct the line
                <math>
                  <mstyle>
                    <mi>o</mi>
                    <mi>a</mi>
                    <mi>y</mi>
                  </mstyle>
                </math>
              so that
                <math>
                  <mstyle>
                    <mi>u</mi>
                    <mi>a</mi>
                    <mo>:</mo>
                    <mi>t</mi>
                    <mi>y</mi>
                    <mo>=</mo>
                    <mi>x</mi>
                    <mo>:</mo>
                    <mi>z</mi>
                  </mstyle>
                </math>
              . There are many variations of the problem according to the relative positions of
                <math>
                  <mstyle>
                    <mi>u</mi>
                    <mi>e</mi>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mi>t</mi>
                    <mi>q</mi>
                  </mstyle>
                </math>
              , and
                <math>
                  <mstyle>
                    <mi>o</mi>
                  </mstyle>
                </math>
              , which Harriot explored in this and other folios. </s>
              <lb/>
              <s xml:space="preserve"> This example, in Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=40&viewMode=image&pn=47"> f. </ref>
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=50&viewMode=image&pn=55"> f. </ref>
              , demonstrates the use of Viète's concepts of zetetic, poristic, analysis, and synthesis. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> b.1) De sectione
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On the cutting off of a ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data:
              <lb/>
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            . lineaæ infinitæ
              <lb/>
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            . termini.
              <lb/>
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            . punctum.
              <lb/>
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given:
              <lb/>
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            , infinite lines.
              <lb/>
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            , endpoints.
              <lb/>
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            , a point.
              <lb/>
              <math>
                <mstyle>
                  <mi>x</mi>
                  <mo>:</mo>
                  <mi>z</mi>
                </mstyle>
              </math>
            , a ratio. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quæsitum:
              <lb/>
            Ducere lineam
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>a</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
              <lb/>
            Ut fiat:
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>a</mi>
                  <mo>,</mo>
                  <mi>t</mi>
                  <mi>y</mi>
                  <mo>:</mo>
                  <mi>x</mi>
                  <mo>,</mo>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Hoc fit si ducetur
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , vel
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , vel
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Quæritur igitur una, nempe
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Sought:
              <lb/>
            To draw the ine
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>a</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            so that
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>a</mi>
                  <mo>:</mo>
                  <mi>t</mi>
                  <mi>y</mi>
                  <mo>=</mo>
                  <mi>x</mi>
                  <mo>:</mo>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            This is done if there are drawn
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , or
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , or
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Therefore, one is sought, namely
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Interpretatio
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Interpretation of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Zetetice 1
              <emph style="super">a</emph>
              <lb/>
            Sit iam factum.
              <lb/>
            Et agatur
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>a</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat:
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            parall:
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            . parall.
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            agatur recta
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            quæ secabit
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Zetetic 1.
              <lb/>
            Let it be already done, and the line
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>a</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            constructed.
              <lb/>
            Then make
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            The line
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            is constructed which cuts
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            .
              <lb/>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ergo ad determinatum sectione
              <lb/>
            eductam est. datur igitur
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore the determination of the section is brought out. Therefore
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            is given. </s>
          </p>
        </div>
      </text>
    </echo>
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