Harriot, Thomas, Mss. 6785

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f096" o="96" n="191"/>
          <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 contains further work on Propostion 12 from
                <emph style="it">Effectionum geometricarum canonica recensio</emph>
                <ref id="Viete_1593b" target="http://www.e-rara.ch/zut/content/pageview/2684103"> (Viète 1593b, Prop </ref>
              . At the end, Harriot makes the same observation as on Add MS 6785
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/KN1CRTZ2/&start=180&viewMode=image&pn=187"> f. </ref>
              , that the method of solving the equation is essentially the same as the 'ancient' method, that is, the traditional method taught in every algebra text. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> Alia operatio per
            <emph style="st">[???]</emph>
            <emph style="super">solam</emph>
          proportionem [???] ad illam [???] quod
            <lb/>
          prop. 12. effectionum geometricarum
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Another method using a single proportion [???] to that [???] done in Proposition 12 of the
            <emph style="it">Effectionum geomtericarum</emph>
          . </head>
          <p xml:lang="">
            <s xml:space="preserve"> Dico quod:
              <lb/>
            Nam:
              <lb/>
            per const:
              <lb/>
            et per invers:
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I say that:
              <lb/>
            For:
              <lb/>
            By constrcution
              <lb/>
            And by ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> A lemma.
              <lb/>
            Ergo.
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            , vel maior,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            mminor,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            æqualis,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            sit
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            maior
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            :
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            maior
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ; et
              <lb/>
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            maior
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            et
              <lb/>
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            maior,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            quod contra hypothesin
              <lb/>
            Ergo
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            non maior
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            sit
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            minor
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            :
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            minor
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ; et
              <lb/>
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            minor
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , et
              <lb/>
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , minor,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            quod contra hypothesin
              <lb/>
            Ergo
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            non minor
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Ergo:
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A. Lemma.
              <lb/>
            Therefore,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is either greater than, less than, or equal to
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Suppose
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is greater than
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ; then
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is greater than
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is greater than
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is greater than
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , which is against the hypothesis.
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is not greater than
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Suppose
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is less than
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ; then
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is less than
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is less than
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>h</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is less than
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , which is against the hypothesis.
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is not less than
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> praxis ista per compendium
              <lb/>
            eadem omnino est cum
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            The practice of this more briefly is exactly the same as the ancient ]</s>
          </p>
        </div>
      </text>
    </echo>
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