Harriot, Thomas, Mss. 6785

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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f144" o="144" n="287"/>
          <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 is the first of a set of twenty-one pages exploring propositions from
                <emph style="it">Supplementum geometriæ</emph>
                <ref id="Viete_1593c" target="http://www.e-rara.ch/zut/content/pageview/2684110"> (Viète </ref>
              . In the
                <emph style="it">Effectionum geometricarum</emph>
                <ref id="Viete_1593b" target="http://www.e-rara.ch/zut/content/pageview/2684099"> (Viète </ref>
              , which preceded it, Viète gave Euclidean constructions to demonstrate relationships between proportional lines, and showed that they corresponded to quadratic, or sometimes quartic, equations. This, however, gave him only a limited range of constructions, or equations, insufficient for the requirements of the analytic art by which he meant to leave no problem unsolved (
                <foreign xml:lang="lat">nulla non problema solvere</foreign>
              ,
                <emph style="it">In artem analyticen isagoge</emph>
              ,
                <ref id="viete_1591" target="http://gallica.bnf.fr/ark:/12148/bpt6k108865t/f17.image"> (Viète </ref>
              , final sentence). The
                <emph style="it">Supplementum geometricarum</emph>
              was intended to remedy this shortcoming (
                <foreign xml:lang="lat">defecta Geometriæ</foreign>
              ) by offering constructions that went beyond the limitations of ruler and compass. Thus the first statement of the book is: </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> A quovis puncto ad duas quavis lineas rectam ducere, interceptam ab iis præfinito possibili quocumque intersegmento.</s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> To draw a straight from any point to any two straight lines, the intercept between them being any possible predefined distance.</s>
              </quote>
              <lb/>
              <s xml:space="preserve"> Such constructions are sometimes knownn as
                <foreign xml:lang="lat">neusis</foreign>
              constructions.
                <lb/>
              On this page, Harriot examines Propositions 3 and
                <ref id="Viete_1593c" target="http://www.e-rara.ch/zut/content/pageview/2684112"> (Viète 1593c, Props 3, </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio III.
                  <lb/>
                Si duae lineae rectae a puncto extra circulum eductae ipsum secent, pars autem exterior primae fit proportionalis inter partem exteriorem secundae & partem interiorem ejusdem: erit quoque pars exterior secundae proportionalis inter partem exteriorem primae & partem interiorem </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> If two straight lines drawn from a point outside a circle cut it in such a way that the external part of the first is a proportional between the external and internal parts of the second, the external part of the second will be a proportional between the external and internal parts of the </s>
              </quote>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio IV.
                  <lb/>
                Si duae lineae rectae a puncto extra circulum eductae ipsum secent quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus: exteriores partes permutatim sumptae, erunt continue proportionales inter partes </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> If two straight lines drawn from a point outside a circle cut it, and moreover the product of the external parts is equal to that of the internal parts, the external parts taken in turn will be continued proportionals between the internal </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> There is also a reference to Euclid,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV19.html"/>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV19.html"/>
                If a whole be to a whole as a part subtracted be to a part subtracted, then the remainder is also to the remainder as the whole is to the whole </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> In prop: 3am
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          From the 3rd proposition of the ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> sunt partes ablatæ a
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            et in eadem ratione
              <lb/>
            partes reliquæ sunt
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            quæ per 19,5 sunt
              <emph style="super">etiam</emph>
            in eadem ratione.
              <lb/>
            Ergo ut supra.
              <lb/>
            Ergo si
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            sit media proportionalis inter
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            erit inter
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            the parts are taken from
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            in the same ratio.
              <lb/>
            the remaining parts are
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , which by [Euclid's
              <emph style="it">Elements</emph>
            ] V.19 are also in the same ratio.
              <lb/>
            Therefore as above.
              <lb/>
            Therefore if
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            is the mean proportional between
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , then
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            will be between
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> In
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          From the ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> continue proportionales ut supra
              <lb/>
            […]
              <lb/>
            Et per synæresin
              <lb/>
            Ex ratione constructionis
              <lb/>
            […]
              <lb/>
            Ergo
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . continuæ
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            continued proportionals as above
              <lb/>
              <lb/>
            And by synæresis
              <lb/>
            By reason of the construction
              <lb/>
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            are continued proportionals. </s>
          </p>
        </div>
      </text>
    </echo>
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