Harriot, Thomas, Mss. 6785

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page |< < (94) of 882 > >|
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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f094" o="94" n="187"/>
          <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 on this page is from Propositions 12 and 14 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, Props 12, </ref>
              . Harriot does not mention the
                <emph style="it">Effectionum</emph>
              explicitly here but the notation is essentially Viète's, except reduced to lower case letters. </s>
              <lb/>
              <s xml:space="preserve"> Note Harriot's use of the
                <math>
                  <mstyle>
                    <mo>=</mo>
                  </mstyle>
                </math>
              symbol for what we now write as
                <math>
                  <mstyle>
                    <mo>±</mo>
                  </mstyle>
                </math>
              .
                <lb/>
              Note also that once he has arrived at an equation, he regards the problem as solved. The rest is merely 'mechanicen', or practical calculation.</s>
              <lb/>
              <s xml:space="preserve"> Harriot's source for the method of Diophantus must have been the edition of Wilhelm
                <emph style="it">Diophanti Alexandrini rerum arithmeticarum libri sex</emph>
                <ref id="diophantus_1575"> (Diophantus </ref>
              . Mahomet was by now all that was remembered of the name of Muhammad ibn Musa al-Khwarizmi. His name appears in this form in Bombelli's
                <emph style="it">Algebra</emph>
                <ref id="bombelli_1579"> (Bombelli 1572, </ref>
              , for example. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> Data media trium proportionalium et differentia extremarum:
            <lb/>
          invenire
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Given the mean of three proportionals and thd difference of the extremes, find the ]</head>
          <p xml:lang="">
            <s xml:space="preserve"> Sit media data
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            . et differentia
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Et ponatur unus terminus ignotus
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Ergo alter erit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            . hoc est
              <lb/>
            vel
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>-</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Ergo: Resolutio.
              <lb/>
            Ergo per Mechanicen.
              <lb/>
            Hoc est Minor.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the given mean be
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            and the difference
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            , and denote one of the unkonwn extrmes by
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Therefore the other will be
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>±</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            , that is, either
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>-</mo>
                  <mi>b</mi>
                  <mn>4</mn>
                  <mi>o</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            a + b
              <math>
                <mstyle>
                  <mo>.</mo>
                </mstyle>
              </math>
              <lb/>
            Hence the solution.
              <lb/>
            Therefore by calculation
              <lb/>
            That is, the lesser extreme
              <lb/>
            the ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Et emitandas applicationes in fine mechanicas, melius est
              <lb/>
            ponere vel notare in principio, dimidium differentiæ
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            . tum differentia
              <lb/>
            tota erit
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mo>,</mo>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Ergo per resolutionem
              <lb/>
            solutio fit
              <lb/>
            Ergo per
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            To force out the divisions in the final calculation, it is better to put or denote from the beginning half the difference
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            , then the total difference will be
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mi>c</mi>
                </mstyle>
              </math>
            . Hence from the solution, the equation will be.
              <lb/>
            Therefore by ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Mechanicum secundum Diophantum
              <lb/>
            et Mahometen.
              <lb/>
            Adde utrique parte æquationis
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            […]
              <lb/>
            Et per
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Calculation according to Diohantus and Mahomet.
              <lb/>
            Add
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            to each side of the equation.
              <lb/>
              <lb/>
            And by ]</s>
          </p>
        </div>
      </text>
    </echo>
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