Harriot, Thomas, Mss. 6784

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481 (241) 		482
482 (241v)
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484 (242v)
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485 (243) 		486
486 (243v)
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488 (244v)
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490 (245v)
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page |< < (234) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f234" o="234" n="467"/>
          <head xml:space="preserve" xml:lang="lat"> De
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> A puncto extra circulum
              <lb/>
            ducere lineam secantem:
              <lb/>
            ut partes, exterior et
              <lb/>
            interior, sint in ratione
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            From a point outside a circle draw a line cutting it, so that the external and internal parts are in a given ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit (
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ) punctum datum, extra
              <lb/>
            circulum
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            cuius centrum
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et per
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            centrum agatur recta
              <lb/>
            infinita
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            quæ secabit
              <lb/>
            periferium circuli dati in punctis
              <lb/>
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            . et fiat
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            be the given point outside the circle
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            whose centre is
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            , and through the centre
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            there is constructed an infinite line
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            which cuts the circumference of the given circle at the points
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            ; and make
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit ratio data ut
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            .
              <lb/>
            oportet a puncto
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ducere
              <lb/>
            rectum secantem, nempe
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
              <lb/>
            ut pars exterior
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ad partem
              <lb/>
            interiorem
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , sit ut
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the given ratio be as
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>a</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            .
              <lb/>
            It is required to draw a cutting line from the point
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , namely
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            so that the external part
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            to the internal part
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            is as
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> producatur
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            , et fiat
              <math>
                <mstyle>
                  <mi>r</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Deinde secetur
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            in puncto
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            ut
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            sit ut
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            . et fiat
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            æqualis
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            . Inde fit
              <lb/>
            ut partes
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            ; sint in ratione
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>r</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Extend
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            and make
              <math>
                <mstyle>
                  <mi>r</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Then
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is cut at point
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            so that
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            is as
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            , and make
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            equal to
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            . Thence the parts
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            will be in the ratio
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>r</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Porro
              <emph style="st">super
                <math>
                  <mstyle>
                    <mi>a</mi>
                    <mi>g</mi>
                  </mstyle>
                </math>
              </emph>
            fiat circulus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            super diametrum
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . quæ
              <lb/>
            secabit circulum datum in puncto
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            . Et agatur recta
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , quæ
              <lb/>
            secabitur a circulo dato in puncto
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod ratio
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , est ut
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Then make the circle
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            with diameter
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            , whcih will cut the given circle at the point
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            . And construct the line
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , which will be cut by the given circle at the point
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that the ratio
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            is as
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Centro
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            intervallo
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , fiat circulus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Continuetur
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            .
              <lb/>
            et connectantur
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            , puncta. [???]
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            , et
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Quoniam anguli
              <lb/>
            ad
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            sunt recti (sunt enim in semicirculos.)
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            sunt
              <lb/>
            parallelæ. Et inde triangula
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            sunt similia. Quare ut
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
              <lb/>
            ad
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , sic
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            etiam
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Et quoniamm
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            sunt æqualis
              <lb/>
            per constructionem, ob parallelismum circulorum
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            , et
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , erunt
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
              <lb/>
            et
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            etiam æquales. Sed
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            etiam æqualis est
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Igitur ut
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            ita
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            , etiam
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . Quare etiam alterne ut
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
              <lb/>
            ita
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . Sed ut
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            ita
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            supra per constructionem.
              <lb/>
            Ergo
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            est ut
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            ad
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            . Factum est igitur quod
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            With centre
              <math>
                <mstyle>
                  <mi>e</mi>
                </mstyle>
              </math>
            and radius
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            make the circle
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Extend
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            and connect the points
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            . [???]
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Since the angles at
              <math>
                <mstyle>
                  <mi>i</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            are straight lines (for they are in a semicircle)
              <math>
                <mstyle>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            are paralle. And hence the triangles
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            are similar. Whence as
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , so are
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            and also
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . And since
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            are equal by construction, on account of the parallelism of the circles
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            , the lines
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>d</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            are also equal. But
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            is also equal to
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Therefore as
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            is to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            so is
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . Whence also, alternatively, as
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            is to
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , so is
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . But as
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            is to
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            so is
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            above by construction. Therefore
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            is as
              <math>
                <mstyle>
                  <mi>p</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            to
              <math>
                <mstyle>
                  <mi>q</mi>
                  <mi>r</mi>
                </mstyle>
              </math>
            . Therefore it is done as was required. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Corollarium.
              <lb/>
            Patet quod angulus
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            est rectus et in semicirculo cuius
              <lb/>
            diameter
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Corollary
              <lb/>
            It is clear that the angle
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            is a right angle in a semicircle whose diameter is
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . </s>
          </p>
        </div>
      </text>
    </echo>
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