Harriot, Thomas, Mss. 6787

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f425" o="425" n="848"/>
          <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 reference towards the end of this page is to Giambattista
                <emph style="it">Diversarum speculationum mathematicarum et physicarum liber</emph>
                <ref id="benedetti_1585" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/163127KK&tocMode=concordance&viewMode=images&pn=38&start=31"> (Benedetti 1585, </ref>
              , though the theorem on page 26 is actually Theorem 41, not Theorem 45.
                <lb/>
              There is also a reference to Proposition 13 from
                <emph style="it">Effectionum geometricarum canonica recensio</emph>
                <ref id="Viete_1593b" target="http://www.e-rara.ch/zut/content/pageview/2684104"> (Viète 1593b, Prop </ref>
              , which explains how to find two quantities from their geometric mean and their sum. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data in partibus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given in parts of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ex Methodo
              <lb/>
            Adde [???]
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
              <lb/>
            vel illa æqualium
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>b</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
              <lb/>
            qui in centro
              <lb/>
            sed in
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            By the method: add
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            or its equal
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>b</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            which in the centre is but in the beginning .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> tum quæritur
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            et inde
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            . per Theor. 45. pag. 26. Joh. Baptistæ
              <lb/>
            de
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            then there is sought
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            and hence
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            , by Theorem 45, page 26, Johan Baptista de Benedictis </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Vel per
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Or by ]</s>
            <lb/>
            <s xml:space="preserve"> Sit
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            . tum
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            erit
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            . hoc multiplicatum per
              <math>
                <mstyle>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            faciet
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mi>r</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            quod æquale erit rectangulo
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            , hoc est 78,545,532.
              <lb/>
            Forma æquationis ita erit
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mi>r</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>q</mi>
                  <mo>=</mo>
                  <mn>7</mn>
                  <mn>8</mn>
                  <mo>,</mo>
                  <mn>5</mn>
                  <mn>4</mn>
                  <mn>5</mn>
                  <mo>,</mo>
                  <mn>5</mn>
                  <mn>3</mn>
                  <mn>2</mn>
                </mstyle>
              </math>
            .
              <lb/>
            Et duplis erit responsum,
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                  <mo>=</mo>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            . then
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            will be
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            . This multiplied by
              <math>
                <mstyle>
                  <mn>1</mn>
                  <mi>r</mi>
                </mstyle>
              </math>
            makes
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mi>r</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            which is equal to the rectangle of
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            , that is 78,545,532.
              <lb/>
            Thus the form of the equation will be
              <math>
                <mstyle>
                  <mn>2</mn>
                  <mn>0</mn>
                  <mo>,</mo>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mn>0</mn>
                  <mi>r</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                  <mi>q</mi>
                  <mo>=</mo>
                  <mn>7</mn>
                  <mn>8</mn>
                  <mo>,</mo>
                  <mn>5</mn>
                  <mn>4</mn>
                  <mn>5</mn>
                  <mo>,</mo>
                  <mn>5</mn>
                  <mn>3</mn>
                  <mn>2</mn>
                </mstyle>
              </math>
            .
              <lb/>
            And it will be twice the answer, ]</s>
            <lb/>
            <s xml:space="preserve"> Habetur
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            alias per 13 prop. Geom. Effect.
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>n</mi>
                </mstyle>
              </math>
            can be had otherwise by Proposition 13 of Viète,
              <emph style="it">Effectionum geometricarum</emph>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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