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_f087" o="87" n="173"/>
          <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"> On this page, Harriot works on Propositions 12 and 13 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>
              . Proposition 12 is mentioned explicitly at the top of the page. The work continues with Proposition 13 below the dividing line. </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio XII.
                  <lb/>
                Data media trium proportionalium et differentia extremarum, invenire </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given three proportionals and the difference of the extremes, to find the </s>
              </quote>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio XII.
                  <lb/>
                Data media trium proportionalium & adgregato extremarum, invenire </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given three proportionals and the sum of the extremes, to find the </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> In both of these propositions, Viète showed how the standard construction for three proportionals can lead to the given equation. Harriot works the other way round: beginning from an equation, he gives a construction that represents the same relationship geometrically. This is what he means by 'effectio æquationis' or 'the construction of an </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> In Effectiones Geometricas. prop. 12 ex 9 et
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          From Effectiones Geometricas, Proposition XII, from pages 9 and ]</head>
          <p xml:lang="">
            <s xml:space="preserve"> Data media trium proportionalium et differentia extremarum: invenire
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given the mean of three proportionals and thd difference of the extremes, find the ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Data.
              <lb/>
            Media.
              <lb/>
            Differentia.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given.
              <lb/>
            Mean.
              <lb/>
            Difference.
              <lb/>
            ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Data Media trium proportionalium et
              <lb/>
            aggregato extremarum: invenire
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given the mean of three proportionals and the sum of the extremes, find the ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Data.
              <lb/>
            Media.
              <lb/>
            Adgreg.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given.
              <lb/>
            Mean.
              <lb/>
            Sum.
              <lb/>
            ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Methodus ad exhibenda quæsita
              <lb/>
            in numeris.
              <lb/>
            Dimidium
              <lb/>
            Subtrahe
              <math>
                <mstyle>
                  <msqrt>
                    <mrow>
                      <mn>3</mn>
                      <mn>6</mn>
                    </mrow>
                  </msqrt>
                </mstyle>
              </math>
            id est
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
              <lb/>
            vel
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            . pro
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            . Adde pro
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
              <lb/>
            Multiplica
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
              <lb/>
            per
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            . et erit
              <lb/>
            Hoc est.
              <lb/>
            Cuius radix.
              <lb/>
            Ergo
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mn>6</mn>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                      <mn>3</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            est
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mn>4</mn>
                </mstyle>
              </math>
            . prima proportionalis.
              <lb/>
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                      <mn>3</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            plus
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            est
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mn>9</mn>
                </mstyle>
              </math>
            . tertia
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A method of showing the sought quantities in numbers.
              <lb/>
            Halve.
              <lb/>
            Subtract
              <math>
                <mstyle>
                  <msqrt>
                    <mrow>
                      <mn>3</mn>
                      <mn>6</mn>
                    </mrow>
                  </msqrt>
                </mstyle>
              </math>
            , that is
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , or
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            for
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Add for
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
              <lb/>
            Multiply
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>H</mi>
                </mstyle>
              </math>
            by
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            and it will be
              <lb/>
            That is
              <lb/>
            Whose root is
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            (
              <math>
                <mstyle>
                  <mn>6</mn>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                      <mn>3</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            ) minus
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            (
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                  <mi>I</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            ) is
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , or
              <math>
                <mstyle>
                  <mn>4</mn>
                </mstyle>
              </math>
            , the first proportional.
              <lb/>
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>1</mn>
                      <mn>3</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            plus
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mn>5</mn>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            is
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , or
              <math>
                <mstyle>
                  <mn>9</mn>
                </mstyle>
              </math>
            , the third proportional. </s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Brevius.
              <lb/>
            Et est accurate
              <lb/>
            modus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            More briefly.
              <lb/>
            And it is precisely the ancient ]</s>
          </p>
          <p xml:lang="">
            <s xml:space="preserve"> Poste.
              <lb/>
            Etsi modus operandi videtur specie quadam differe antiquo
              <lb/>
            consideranti tamen, et operanti per commpendium; est omnino
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Postscript.
              <lb/>
            Although the mode of operation seems in certain respects to differ from the ancient way, nevertheless examined, and carried out more briefly, it is exactly the same.</s>
          </p>
        </div>
      </text>
    </echo>
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