Harriot, Thomas, Mss. 6784

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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f361" o="361" n="721"/>
          <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 folio, Harriot derives the sum of a finite geometric progression, using Euclid
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV12.html"/>
              and its numerical counterpart,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/propVII12.html"/>
              . He then extends his result to an infinite (decreasing) progression, by arguing that the final term must be infnitely small, that is, nothing.
                <lb/>
              </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV12.html"/>
                If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/propVII12.html"/>
                If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents. </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> 1.) De progressione
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On geometric ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
            <lb/>
            <s xml:space="preserve"> el. 5. pr:
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <emph style="it">Elements</emph>
            , Book 5, Proposition 12. </s>
            <lb/>
            <s xml:space="preserve"> el. 7. pr.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <emph style="it">Elements</emph>
            , Book 7, Proposition 12. </s>
            <lb/>
            <s xml:space="preserve"> Si sint magnitudines quotcunque proportionales, Quemadmodum
              <lb/>
            se habuerit una antecedentium ad unam consequentium: Ita
              <lb/>
            se habebunt omnes antecedentes ad omnes
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If any number of magnitudes are proportional, then just as as one antecedent is to its consequent, so will the sum of the antecedents be to the sum of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sint continue proportionales.
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the continued proportionals be
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> In notis universalibus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            In general notation we ]</s>
            <lb/>
            <s xml:space="preserve">
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            . primum.
              <emph style="st">p</emph>
            . primus terminus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            . first term.
              <emph style="st">p</emph>
            . first term of the ratio. </s>
            <lb/>
            <s xml:space="preserve">
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            . secunda.
              <emph style="st">s</emph>
            .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            . second.
              <emph style="st">s</emph>
            . second. </s>
            <lb/>
            <s xml:space="preserve">
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            . last. </s>
            <lb/>
            <s xml:space="preserve">
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            .
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            . all. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ergo; si,
              <emph style="st">p</emph>
            >
              <emph style="st">s</emph>
            ut in progressi
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore if
              <emph style="st">p</emph>
            >
              <emph style="st">s</emph>
            are in a decreasing progression: </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ergo; si,
              <emph style="st">p</emph>
            &lt;
              <emph style="st">s</emph>
            ut in progressi
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore if
              <emph style="st">p</emph>
            >
              <emph style="st">s</emph>
            are in an increasing progression: </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> De
              <emph style="st">infinitis</emph>
            progressionibus
              <lb/>
            decrescentibus in
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            For a progression descreasing ]</s>
            <lb/>
            <s xml:space="preserve"> Cum progressio decrescit et
              <lb/>
            numerus terminorum sit infinitus;
              <lb/>
            ultimus terminus est infinite
              <lb/>
            minimus hoc est nullius
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Since the progression decreases and the number of terms is infinite, the last term is infnitely small, that is, of no quantity.</s>
            <lb/>
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            ]</s>
          </p>
        </div>
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