Harriot, Thomas, Mss. 6789

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page |< < (44r) of 1074 > >|
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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="0087.jpg" o="44r" n="87"/>
          <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 to Bernardus Tornius Florentinus is to his commentary
                <emph style="it">De motu locali</emph>
              of William Heytesbury (c. 1313–1372/3). The text of Tornius is found as the seventh treatise in William Heytesbury,
                <emph style="it">De sensu composito et diviso</emph>
                <ref id="tornius_1494"> (Tornius </ref>
              , 73v–77v.
                <lb/>
              The reference to Alvarus Thomas is to his
                <emph style="it">Liber de triplici motu</emph>
                <ref id="thomas_1509" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/YHKVZ7B4&tocMode=thumbs&viewMode=image&pn=71&start=71"> (Thomas </ref>
              . </s>
              <lb/>
              <s xml:space="preserve"> The texts of both Alvarus and Tornius are verbal, expressed in terms of repeated proportions. Harriot converts their results into arithmetic notation, and in doing so shows how to find sums of some infinite series that are not simple geometric progressions.
                <lb/>
              His first example (from Tornius)
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>4</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>6</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>8</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              . Harriot shows how this can be treated as the sum of infinitely many other series, each of which is a simple geometric progression, namely:
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                    <mo>=</mo>
                    <mn>1</mn>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                        <mn>7</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                      <mrow>
                        <mn>8</mn>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                    <mo>=</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <lb/>
                <math>
                  <mstyle>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                  </mstyle>
                </math>
              . The sums of these series form a new geometric progression
                <math>
                  <mstyle>
                    <mn>1</mn>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>9</mn>
                      </mrow>
                    </mfrac>
                    <mo>+</mo>
                    <mo>\</mo>
                    <mi>d</mi>
                    <mi>o</mi>
                    <mi>t</mi>
                    <mi>s</mi>
                    <mo>=</mo>
                    <mn>1</mn>
                    <mfrac>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                      <mrow>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              .
                <lb/>
              The other two series on the page (from Alvarus) are summed in the same </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve"> Bernardus Tornius Florentinus
              <lb/>
            in doctrinam Heytisberi de
              <lb/>
            motu locali. pag. 77
              <lb/>
            conclusio
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Bernard Tornius Florentinus, in his teaching on Heytesbury's De motu locali, page 77, third ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Alvarus Thomas.
              <lb/>
            2 conclusio. pag.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Alvarus Thomas, Conclusion 2, page ]</s>
          </p>
        </div>
      </text>
    </echo>
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