Harriot, Thomas, Mss. 6787

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f161" o="161" n="320"/>
          <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"> Here Harriot examines a statement that appears as Proposition VI of the 'Dati sexti', Chapter XIX of Viète's
                <emph style="it">Variorum responsorum liber VIII</emph>
                <ref id="Viete_1593d" target="http://www.e-rara.ch/zut/content/pageview/2684285"> (Viete 1593d, Chapter 19, DATI SEXTI, Prop </ref>
              . See also Add MS 6782
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/HSPGZ0AE&start=870&viewMode=image&pn=878"> f. </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> VI.
                  <lb/>
                Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulares peripheriæ.</s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> The reference to Pitiscus is
                <emph style="it">Trigonometria: sive de solutione triangulorum tractarus brevis et perpsicuus</emph>
              (1595). </s>
              <lb/>
              <s xml:space="preserve"> The reference to Lansberg is to
                <emph style="it">Triangulorum geometriae libri quatuor</emph>
              (1591). </s>
              <lb/>
              <s xml:space="preserve"> The reference to Regiomontanus is to
                <emph style="it">De triangulis omnimodis libri quinque</emph>
                <ref id="regiomontanus_1533" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/KR7R1SZG/pageimg&start=111&viewMode=images&pn=120&mode=imagepath"> (Regiomontanus [1464], 1533, 1561, Prop IV.31)</ref>
              . </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data differentia.)
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            differentia
              <emph style="super">duarum</emph>
            peripheriam.
              <lb/>
            ratio sinuum quæsitarum peripheriarum ut
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            […]
              <lb/>
            sinus
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            arcus
              <lb/>
            Datur igitur
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            . nam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                  <mo>+</mo>
                  <mi>e</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Datur etiam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            . nam sinus complementi est
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            vel dimidij arcus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Cætera ut supra. et habetur
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            sinus
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            arcus.
              <lb/>
            Tum:
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . arcus minor
              <lb/>
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . arcus maior quæsitis
              <lb/>
            Tum etiam
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            arcus maior </s>
          </p>
          <head xml:space="preserve" xml:lang="lat"> 12.) Data summa vel differentia duarum periferiarum,
            <lb/>
          quarum sinus datam habeant rationem: dantur singulares
            <lb/>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> per Tangentes exhibit
              <lb/>
            Vieta in responsis pag. 37.
              <lb/>
            Pitiscus pag. 92.
              <lb/>
            Lansbergis. pag.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Shown by tangents
              <lb/>
            by Viète in Responsorum, page 37,
              <lb/>
            Pitiscus, page 92,
              <lb/>
            Lansberg, page ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Quæ modo usui accomodatior est, quam per
              <lb/>
            sinus solos, quando tangentibus ut
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Which method of use is more convenient than by sines alone, when by tangents, as one ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sed quando non licet Tangentibus uti, modus per solos sinus (etsi laboriosior)
              <lb/>
            adhibendus est. Exhibatur a Regiomontano lib. 4. prop. 31. de triangulis
              <lb/>
            Modus ille hic apponitur paucis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            But when one does not want to use tangents, the method by sines alone (though more laborious) is shown. It is given by Regiomontanus, Book IV, Proposition 31 of De triangulis. That method set out here is explained a ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data summa.)
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            summa duarum peripheriam.
              <lb/>
            ratio sinuum quæsitarum peripheriarum ut
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat:
              <lb/>
            Datur ergo
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            , nam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>e</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            est sinus dimidij arcus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Datur etiam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            . nam sinus complementi est
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            vel dimidij arcus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            […]
              <lb/>
            sinus
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            arcus
              <lb/>
            Tum:
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>+</mo>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            arcus maior
              <lb/>
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            arcus minor quæsita
              <lb/>
            Tum etiam:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . arcus maior
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given the sum.)
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            be the sum of the two arcs, and the ratio of the two sines of the sought arcs as
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            construct:
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            is given, for
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>e</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            is the sine of half the arc
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Also
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            is gien, for the sine of the complement is
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , or half the arc
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            the sine of arc
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            Then
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>+</mo>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the greater arc.
              <lb/>
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            the lesser arc sought.
              <lb/>
            Then also
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                  <mo>-</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the greater arc sought. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Data differentia.)
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            differentia
              <emph style="super">duarum</emph>
            peripheriam.
              <lb/>
            ratio sinuum quæsitarum peripheriarum ut
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
            […]
              <lb/>
            sinus
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            arcus
              <lb/>
            Datur igitur
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            . nam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                  <mo>+</mo>
                  <mi>e</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Datur etiam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            . nam sinus complementi est
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            vel dimidij arcus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Cætera ut supra. et habetur
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            sinus
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            arcus.
              <lb/>
            Tum:
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . arcus minor
              <lb/>
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . arcus maior quæsitis
              <lb/>
            Tum etiam
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            arcus maior
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given the difference.)
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            be the difference of the two sought arcs, and the ratio of the sines of the sought arcs
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            sine of the arc
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            is given, for
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>c</mi>
                  <mo>+</mo>
                  <mi>e</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>d</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Also
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>o</mi>
                </mstyle>
              </math>
            is given, for the sine of the complement is
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , or half the arc
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            The rest as above. And we have
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            the sine of arc
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Then
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the lesser arc.
              <lb/>
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>g</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the greater arc sought.
              <lb/>
            Then also
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            , the greater arc sought. </s>
          </p>
        </div>
      </text>
    </echo>
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