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_f167" o="167" n="332"/>
          <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 refers to Propositions 48 and 49 of Book III of Apollonius, as edited by Commandino
                <emph style="it">Conicorum libri quattuor</emph>
                <ref id="apollonius_1566"> (Apollonius </ref>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve"> III.48 With the same things being so, it must be shown that the straight lines drawn from the point of contact to the points produced by the application make equal angles with the </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"/>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit ellipsis
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>f</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            :
              <lb/>
            cuius axis
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            centroides puncta
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>w</mi>
                </mstyle>
              </math>
            .
              <lb/>
            diametroides, recta
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
              <lb/>
            centrum,
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <lb/>
            circulus circa axim,
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>e</mi>
                  <mi>d</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            .
              <lb/>
            circulus circa diametroides,
              <math>
                <mstyle>
                  <mi>z</mi>
                  <mi>w</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            recta contingens ellipsin in
              <lb/>
            puncto
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            , fit
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <lb/>
            perpendicularis a centroide
              <math>
                <mstyle>
                  <mi>w</mi>
                </mstyle>
              </math>
              <lb/>
            ad illam fit
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
              <lb/>
            per 49.3 conicorum
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>e</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            est angulus
              <lb/>
            rectus
              <lb/>
            ergo punctum
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            in
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let there be an ellipse
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>f</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            with axis
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            , centroids at points
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>w</mi>
                </mstyle>
              </math>
            , diametroid the line
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            , centre
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            . The circle about the axis is
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>e</mi>
                  <mi>d</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            ; the circle about the diametroid is
              <math>
                <mstyle>
                  <mi>z</mi>
                  <mi>w</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ; the line touching the ellipse at the point
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            is
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            . Perpendicular to it from the centroid
              <math>
                <mstyle>
                  <mi>w</mi>
                </mstyle>
              </math>
            , construct
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            . By Proposition III.49 of the Conics,
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>e</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            is a right angle. Therefore, the point
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            is on the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> hinc sequitur
              <lb/>
            Si
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            producatur ad periferium in
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
              <lb/>
            et ducatur
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            parallela ad
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
              <lb/>
            continget etiam
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            is produced to the perpiphery at
              <math>
                <mstyle>
                  <mi>p</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            is taken paralle to
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , it will also touch the ellipse. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Si puncta
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            in periferia
              <lb/>
            connectantur
              <lb/>
            linea
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            transibit per alterum
              <lb/>
            centroides
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If the points
              <math>
                <mstyle>
                  <mi>x</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            in the periphery are joined, the line
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            will pass through the other centroid,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Hinc. conclusio
              <lb/>
            Si circa
              <emph style="super">axim</emph>
            ellipseos describatur circulus
              <lb/>
            et in circulo inscribatur parallelogrammum
              <lb/>
            ita ut duo latera transeant per centroides:
              <lb/>
            reliqua duo contingent ellipsin.
              <lb/>
            et si duo latera contingent ellipsin; reliqua
              <lb/>
            duo transibunt per centroides.
              <lb/>
            Ita etam:
              <lb/>
            Si circa axim Hyperboles &
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Hence, the conclusion.
              <lb/>
            If around the axis of an ellipse there is described a circle, and in the circle there is inscribed a parallelogram so that two sides pass thorugh the centroids, the other two are tangents to the ellipse. And if two sides are tangents to the ellipse, the other two will pass throug the centroids.
              <lb/>
            Thus also: if around the axis of a hyperbola, ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Alia conclusiones
              <lb/>
            iisdem positis.
              <lb/>
            per 48.3. conicorum.
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            faciunt æquale angulos ad
              <lb/>
            contingentem.
              <lb/>
            Si
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            agantur
              <lb/>
            parallelæ ad contingentes:
              <lb/>
            puncta
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            sunt in peri-
              <lb/>
            feria cuius diameter
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Other conclusions form the same assumptions.
              <lb/>
            By Proposition III.48 of the Cinics,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            make equal angles to the tangent.
              <lb/>
            If
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            are taken parallel to the tangents, the points
              <math>
                <mstyle>
                  <mi>z</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            are on the circumference whose diamter is
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Conveniat
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>z</mi>
                </mstyle>
              </math>
            cum
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            cum
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>v</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod:
              <lb/>
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>t</mi>
                  <mo>=</mo>
                  <mi>v</mi>
                  <mi>a</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>-</mo>
                  <mi>f</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
              <lb/>
            nam:
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>z</mi>
                </mstyle>
              </math>
            meet with
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            with
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>v</mi>
                </mstyle>
              </math>
            . I say that:
              <lb/>
              <math>
                <mstyle>
                  <mi>w</mi>
                  <mi>t</mi>
                  <mo>=</mo>
                  <mi>v</mi>
                  <mi>a</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>-</mo>
                  <mi>f</mi>
                  <mi>w</mi>
                </mstyle>
              </math>
              <lb/>
            for the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Dico
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I also ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Dico
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I also ]</s>
          </p>
        </div>
      </text>
    </echo>
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