Harriot, Thomas, Mss. 6784

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page |< < (331) of 862 > >|
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      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f331" o="331" n="661"/>
          <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 Apollonius is to pages 5 and 6 of Commandino's
                <emph style="it">Apollonii Pergaei conicorum libri quattuor</emph>
                <ref id="apollonius_1566"> (Apollonius </ref>
              . There are also references at the bottom of the page to Viète and Cardano. </s>
              <lb/>
              <s xml:space="preserve"> The reference to Viète is to
                <emph style="it">Apollonius Gallus</emph>
              , Appendix II, Problem 5
                <ref id="Viete_1600a" target="http://www.e-rara.ch/zut/content/pageview/2684213"> (Viete 1600a, Appendix II, Prob </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given a triangle, find a point from which there may be drawn three straight lines to the vertices of the given triangle, keeping a fixed ratio.</s>
              </quote>
              <lb/>
              <s xml:space="preserve"> The reference to Cardano is to his
                <emph style="it">Opus novum de proportionibus</emph>
              . The relevant Propositions are
                <ref id="cardano_1570a" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/archimedes/carda_propo_015_la_1570&start=161&viewMode=image&pn=164"> Prop </ref>
              (though mistakenly described in the
                <emph style="it">Opus novum</emph>
              as 144) and
                <ref id="cardano_1570a" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/archimedes/carda_propo_015_la_1570&start=181&viewMode=image&pn=181"> Prop </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio centesimaquadragesimaquarta
                  <lb/>
                Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, ipsæ erunt in proportione mediæ ad adiectam.
                  <lb/>
                Hæc propositio est admirabilis: </s>
              </quote>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio centesimasexagesima
                  <lb/>
                Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> 5. Appolonius. pag. 5.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Apollonius, pages 5, ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quæsitum:
              <lb/>
            ubicunque signatur in periferia punctum
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
              <lb/>
            erit;
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            :
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            : vel
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Sought:
              <lb/>
            Wherever a point
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is placed on the circumference, then
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                  <mo>:</mo>
                  <mi>h</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>c</mi>
                  <mo>:</mo>
                  <mi>d</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>k</mi>
                  <mo>:</mo>
                  <mi>k</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sint data puncta
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <lb/>
            Data ratio.
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <lb/>
            producatur,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , versus,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the given points be
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the given ratio
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mo>:</mo>
                  <mi>d</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            be produced towards
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Dico
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I say ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Inde:
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            maior, quam
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            minor, quam
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            fiat
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>k</mi>
                  <mo>=</mo>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            fiat
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            periferia
              <lb/>
            sumatur quovis puncta
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
              <lb/>
            Ducantur:
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Whence,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            is greater than
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            , less than
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>k</mi>
                  <mo>=</mo>
                  <mi>g</mi>
                </mstyle>
              </math>
            , let
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            be the circumference, taking any point
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Let there be drawn
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> * Ducantur
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            , parallela,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            .
              <lb/>
            ubicunque signatur in periferia punctum
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
              <lb/>
            erit;
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            :
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            : vel
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Taking
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            parallel to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , wherever the point
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            is placed on the circumference, then
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                  <mo>:</mo>
                  <mi>h</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>c</mi>
                  <mo>:</mo>
                  <mi>d</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>k</mi>
                  <mo>:</mo>
                  <mi>k</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Corollaria.
              <lb/>
            Hinc a tribus punctis sive sint in recta
              <lb/>
            vel non; possunt duci tres lineæ ad unum
              <lb/>
            punctum,
              <emph style="st">ut s</emph>
            et erunt in data
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Corollary
              <lb/>
            Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, and they will be in the given ratio.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> vide vertam
              <lb/>
            in Apolonio gallo
              <lb/>
            et card: de prop. pag. 145.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            see over, in
              <emph style="it">Apollonius Gallus</emph>
            , and Cardano,
              <emph style="it">De proportionibus</emph>
            , pages 145, 162. </s>
          </p>
        </div>
      </text>
    </echo>
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