Harriot, Thomas, Mss. 6784

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          <pb file="add_6784_f362" o="362" n="723"/>
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            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> In the preceding folio, Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/XT0KZ8QC/&start=720&viewMode=image&pn=721"> f. </ref>
              , Harriot derived a formula for the sum of a finite geometric progression based on Euclid
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV12.html"/>
              . Here he gives an alternative derivation based on Euclid
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX35.html"/>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX35.html"/>
                If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it. </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> 2.) 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. 9. pr:
              <lb/>
            [
              <emph style="bf">Translation: </emph>
              <emph style="it">Elements</emph>
            Book IX, Proposition 35 </s>
            <lb/>
            <s xml:space="preserve"> Si sint quotlibet numeri deinceps proportionales, detrahuntur autem
              <lb/>
            de secundo et ultimo æquales ipsi primo: erit quemadmodum
              <lb/>
            secundi excessus ad primum, ita ultima excessus ad omnes qui ultimum
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If there are as many numbers as we please in proportion, and the first is subtracted from the second and the last, then just as the difference of the second is to the first, so is the difference of the last to all before the last.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Progressio
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            An increasing ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> In notis universalibus: sit
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            , primus:
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            , secundus:
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            , ultimus:
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            ,
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            In general notation, let
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            be the first term;
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            the second term;
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            the last term;
              <math>
                <mstyle>
                  <mi>o</mi>
                </mstyle>
              </math>
            the sum. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Progressio
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            A decreasing ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> In notis universalis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            In general notation we ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Vel: in notis magis universalis.
              <lb/>
            sit
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            , primus terminus rationis.
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            , secundus.
              <lb/>
              <math>
                <mstyle>
                  <mi>M</mi>
                </mstyle>
              </math>
            , maxumus terminus progressionis
              <lb/>
              <math>
                <mstyle>
                  <mi>m</mi>
                </mstyle>
              </math>
            , minimus.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Or, in more general notation, let
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            be the first term of the ratio,
              <math>
                <mstyle>
                  <mi>s</mi>
                </mstyle>
              </math>
            the second,
              <math>
                <mstyle>
                  <mi>M</mi>
                </mstyle>
              </math>
            the greatest term of the progression,
              <math>
                <mstyle>
                  <mi>m</mi>
                </mstyle>
              </math>
            the least. Then: </s>
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