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_f019" o="19" n="37"/>
          <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 an area', 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 spacium contineant dato spacio æquale.</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 on them, which contain an area equal to a given area.</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>e</mi>
                  </mstyle>
                </math>
              , the point
                <math>
                  <mstyle>
                    <mi>o</mi>
                  </mstyle>
                </math>
              , and an area of size
                <math>
                  <mstyle>
                    <mi>x</mi>
                    <mi>x</mi>
                  </mstyle>
                </math>
              , construct the line
                <math>
                  <mstyle>
                    <mi>a</mi>
                    <mi>o</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>
                    <mi>x</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>e</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=30&viewMode=image&pn=37"> f. </ref>
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=40&viewMode=image&pn=45"> f. </ref>
              , demonstrates the use of Viète's concepts of zetetic, exegetic, and poristic. There is also a direct reference on Add MS 6784
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=30&viewMode=image&pn=37"> f. </ref>
              to Viète,
                <emph style="it">Effectionum geometricarum</emph>
              ,
                <ref id="Viete_1593b" target="http://www.e-rara.ch/zut/content/pageview/2684103"> (Viète 1593b, Props 11, </ref>
              . </s>
              <lb/>
              <s xml:space="preserve"> The three pages Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=40&viewMode=image&pn=41"> f. </ref>
              to
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=40&viewMode=image&pn=45"> f. </ref>
              appear to have been written out for Thomas Aylesbury in December 1608 (see Add MS 6784
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=40&viewMode=image&pn=43"> f. </ref>
              ), perhaps as a 'worked example' of Viète's methods. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> a) De resectione spatij,
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On the cutting off of an area, ]</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>e</mi>
                </mstyle>
              </math>
            , lineæ 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>
                  <mi>X</mi>
                </mstyle>
              </math>
            ,
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given infinite lines
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            , endpoints
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            , a point
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            and an area
              <math>
                <mstyle>
                  <mi>X</mi>
                  <mi>X</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quæsitum:
              <lb/>
            Ducere lineam
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>o</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            :
              <lb/>
            ut
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Sought:
              <lb/>
            To draw a line
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>o</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            so that: </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Zetetike
              <lb/>
            Sit iam factum:
              <lb/>
            ut 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>e</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Zetetic.
              <lb/>
            Let it be already done, so that
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            is 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>e</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> ergo si
              <lb/>
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            fiat
              <lb/>
            æqualis
              <lb/>
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            ,
              <lb/>
            problema
              <lb/>
            absoluitur.
              <lb/>
            dupliciter
              <lb/>
            enim
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore if
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            becomes equal to
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , the problem is solved, for it is twofold. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Vieta de effect
              <lb/>
            prop. 11. &
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Viète, De effectionum, propositions 11 and ]</s>
          </p>
        </div>
      </text>
    </echo>
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