Harriot, Thomas, Mss. 6786

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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6786_f419" o="419" n="837"/>
          <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"> This page contains a fully worked example of Viète's method of zetesis, porismus, and exegesis.
                <lb/>
              There are several references to propositions from Euclid's
                <emph style="it">Elements</emph>
              ; in order of appearance:
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI20.html"> VI.20 </ref>
              ,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI12.html"> VI.12 </ref>
              ,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html"> VI.13 </ref>
              ,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX19.html"> IX.19 </ref>
              , and
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI22.html"> VI.22 </ref>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI20.html"> VI.20 </ref>
                Similara polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI12.html"> VI.12 </ref>
                To three given straight lines to find a fourth proportional. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html"> VI.13 </ref>
                To two given straight lines to find a mean proportional. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV9.html"> V.9 </ref>
                Magnitudes which have the same ratio to the same are equal to one another: and magnitudes to which the same has the same ratio are equal. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX19.html"> IX.19 </ref>
                Given three nubers, to investigate when it is possible to find a fourth proportional to them. </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI22.html"> VI.22 </ref>
                If four straight lines be proportional, the rectilinear figures similar and similarly described upon them will also be proportional; and, if the rectilineal figures similar and similarly described upon them be proportional, the straight lines will themselves also be proportional. </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> Problema. usus habetur in aciebus </head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data media
              <emph style="super">trium</emph>
            media proportionalium, et ratione extremarum: invenire
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given the mean of three mean proportionals and the ratio of the extremes, find the extremes. ]</s>
          </p>
          <head xml:space="preserve" xml:lang="lat"/>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit media proportionalium data
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            . et ratio extremarum ut
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et disponatur ita.
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mo/>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the mean of the given proportionals be
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            and the ratio of the extremes as
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            , and display them thus. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ponatur for prima termina
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            Ergo per canone proportionalium.
              <lb/>
            consequenter, et de hypothesi.
              <lb/>
            vel etiam per 12 symbolum
              <lb/>
            per resolutionem
              <lb/>
            per reductione a communi
              <lb/>
            multiplicatione
              <lb/>
            vel
              <emph style="super">etiam</emph>
            per parabolismum
              <lb/>
            Ergo Analogia per constitutione
              <lb/>
            Et sua munia implevit
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Put for the first term
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            Therefore by the rules of proportionals.
              <lb/>
            Consequently, and by hypothesis
              <lb/>
            Or also in symbols
              <lb/>
            By resolution
              <lb/>
            By reduction by ocmmon multiplication
              <lb/>
            Or also by parabolismus
              <lb/>
            Therefore the ratio by constitution.
              <lb/>
            And its role satisfies the zetesis. ]</s>
          </p>
          <head xml:space="preserve" xml:lang="lat"/>
          <p xml:lang="lat">
            <s xml:space="preserve"> Conclusio per zetetin illata, per interpretatione, ita enunciatur:
              <lb/>
            ut tertia trium propotionalium ad primam, ita quadratum secunda ad quadratum primam
              <lb/>
            cuius propositionis veritas manifeste patet per collorarium 20
              <emph style="super">a</emph>
            prop 6
              <emph style="super">ti</emph>
            elementorum.
              <lb/>
            si elementum non occurret, argumentatio recederet per vestigia zetetos, usque
              <lb/>
            ad quæsiti
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Conclusion from that zetesis, by interpretation, thus stated:
              <lb/>
            As the third of three proportinals to the first, so is the square of the second to the square of the first, the truth of which proposition is clearly manifest by the corollary to proposition 20 of the 6th book of the
              <emph style="it">Elements</emph>
            .
              <lb/>
            If the element does not occur, the argument goes back to the vestige of the zetetic, as far as the supposed thing sought. ]</s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> Exegesis: ad </head>
          <head xml:space="preserve" xml:lang="lat"> Exegesis: ad </head>
        </div>
      </text>
    </echo>
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