Harriot, Thomas, Mss. 6787

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page |< < (227) of 1155 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f227" o="227" n="452"/>
          <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 four pages devoted to Proposition 14 from Chapter XIX of
                <emph style="it">Variorum responsorum liber VIII</emph>
                <ref id="Viete_1593d" target="http://www.e-rara.ch/zut/content/pageview/2684276"> (Viete 1593d, Chapter 19, Prop </ref>
              , a lengthy chapter on plane and spherical triangles. </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> XIV.
                  <lb/>
                Prothechidion.
                  <lb/>
                Data duorum maximorum in sphæra circulorum inclinatione, quorum unus secatur a tertio per alterius polos, arguitur quanta fit maxima differentia suarum a nodo longitudinum.
                  <lb/>
                Et contra. Ex maxima differentia longitudinum a nodo, arguitur quanta fit circulum </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given the inclination of two great circles on a sphere, one of which is cut by a third through the pole of the other, there is to be found the greatest difference in their longitudes from the node. Conversely, from the greatest difference of longitudes from the node, there may be found the inclination of the </s>
              </quote>
              <s xml:space="preserve"> In the crossed out sentence halfway down the page there are references to Finck and Clavius.
                <lb/>
              The reference to Thomas Finck is to his
                <emph style="it">Geometriae rotundi libri XIIII</emph>
                <ref id="finck_1583"> (Finck </ref>
              .
                <lb/>
              The reference to Clavius is to his
                <emph style="it">Triangula rectilinea, atque sphaerica</emph>
                <ref id="clavius_1586" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fmpiwg%2Fonline%2Fpermanent%2Flibrary%2FYC97H42F&tocMode=thumbs&viewMode=images&start=351&pn=457"> (Clavius 1586, </ref>
              . </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve"> Vieta. lib. 8. resp. pag. 35. prop. 14.
            <foreign xml:lang="gre">proch?on</foreign>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          Viète, Responsorum liber VIII, page 35, Proposition ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ista propositio est utilis in calcu-
              <lb/>
            lationibus astronimicis. per eam cog-
              <lb/>
            noscitur maxima differentia inter
              <lb/>
            numerationes per Eclipticam et proprios
              <lb/>
            circulos planetorum, et ubi est &c.
              <lb/>
            Etiam:
              <lb/>
            æquationibus
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            This proposition is useful in astronomical calculations. By it may be known the maximum difference between observations by the ecliptic and the nearest orbits of planets, and where it is.
              <lb/>
            Also, the equations of the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit triangulum rectangulum
              <lb/>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , angulus rectus
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Quæritur
              <lb/>
            Maxima differentia inter
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            et
              <lb/>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Nam arcus
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            in diversis
              <lb/>
            positionibus inter
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            , facit diversis
              <lb/>
            differentias longitudinum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            et
              <lb/>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            be a right-angled triangle with right angle at
              <math>
                <mstyle>
                  <mi>C</mi>
                </mstyle>
              </math>
            . There is sought the maximum differene between
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . For the arc
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            in various positions between
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            makes various differences of longitude
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> polo
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            .
              <emph style="super">et</emph>
            per punctum
              <math>
                <mstyle>
                  <mi>B</mi>
                </mstyle>
              </math>
            describatur
              <lb/>
            parallelus
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            . Ergo:
              <lb/>
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            est differentia inter
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
            sit diameter paralleli
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            et
              <lb/>
            sit perpendcularis illi,
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Taking the pole
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            , there is drawn through
              <math>
                <mstyle>
                  <mi>B</mi>
                </mstyle>
              </math>
            parallel
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            is the difference between
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            . Let the diameter parallel to it be
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            and the perpendicuar to it
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <emph style="st"> Dico quando
                <emph style="super">*</emph>
                <math>
                  <mstyle>
                    <mi>D</mi>
                    <mi>M</mi>
                  </mstyle>
                </math>
              linea est æqualis sinui
                <math>
                  <mstyle>
                    <mi>D</mi>
                    <mi>C</mi>
                  </mstyle>
                </math>
              , hoc est
                <math>
                  <mstyle>
                    <mi>D</mi>
                    <mi>P</mi>
                  </mstyle>
                </math>
              . Tum
                <math>
                  <mstyle>
                    <mi>M</mi>
                    <mi>O</mi>
                  </mstyle>
                </math>
              erit æqualis
                <lb/>
              semidiametro sphæræ, scilicet
                <math>
                  <mstyle>
                    <mi>H</mi>
                    <mi>F</mi>
                  </mstyle>
                </math>
              . et
                <math>
                  <mstyle>
                    <mi>D</mi>
                    <mi>C</mi>
                  </mstyle>
                </math>
              erit differentia quæsita maxima.
                <lb/>
              Pro demonstratione nota diagramma in Finkio pag. 393. et Clavium </emph>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I say that when the line
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            is equal to the sine
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , that is,
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            , then
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            will be equal to the semidiameter of a spehre, namely
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            will be the sought maximum difference. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Hic notabo solummodo proportiones in Vieta.
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            , sinus arcus
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , hoc est anguli
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            erit sinus versus [???].
              <lb/>
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            est arcus similis
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . Sit
              <math>
                <mstyle>
                  <mi>K</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>N</mi>
                  <mi>O</mi>
                  <mo>=</mo>
                  <mi>D</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            .
              <emph style="st">ex</emph>
              <emph style="super">in</emph>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Here I have noted only the proportions in Viète. Let
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            , be the sine of arc
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , that is of angle
              <math>
                <mstyle>
                  <mi>A</mi>
                </mstyle>
              </math>
            . The versed sine will be
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            . The arc
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            is similar to
              <math>
                <mstyle>
                  <mi>G</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>K</mi>
                  <mi>L</mi>
                </mstyle>
              </math>
            be equal to
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>N</mi>
                  <mi>O</mi>
                  <mo>=</mo>
                  <mi>D</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            . The rest from the diagram. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> *
              <lb/>
            Dico quando
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            habet
              <lb/>
              <emph style="st">ratio</emph>
            ad
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            sinum
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            :
              <lb/>
            eandem rationem quam
              <lb/>
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            vel
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            ad
              <lb/>
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            . Tum
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            erit
              <lb/>
            differentia maxima.
              <lb/>
            Vel melius ita:
              <lb/>
            Quando
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            sit
              <lb/>
            parallela
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            . hoc
              <lb/>
            est quando
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>H</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            est
              <lb/>
            rectus angulus.
              <lb/>
            Demonstratio
              <lb/>
            Habetur in alia
              <lb/>
            charta
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I say that when
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            has the same ratio to
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>P</mi>
                </mstyle>
              </math>
            , the sine of
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            as
              <math>
                <mstyle>
                  <mi>O</mi>
                  <mi>M</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>M</mi>
                  <mi>D</mi>
                </mstyle>
              </math>
            or
              <math>
                <mstyle>
                  <mi>L</mi>
                  <mi>I</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>I</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            , then
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            will be the maximum difference.
              <lb/>
            Or better thus: when
              <math>
                <mstyle>
                  <mi>H</mi>
                  <mi>O</mi>
                </mstyle>
              </math>
            is a right angle.
              <lb/>
            The demonstration is to be found in another sheet, ]</s>
          </p>
        </div>
      </text>
    </echo>
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