Harriot, Thomas, Mss. 6787

List of thumbnails

< >
691
691 (346v) 		692
692 (347)
693
693 (347v) 		694
694 (348)
695
695 (348v) 		696
696 (349)
697
697 (349v) 		698
698 (350)
699
699 (350v) 		700
700 (351)
< >
page |< < (375) of 1155 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f375" o="375" n="748"/>
          <head xml:space="preserve" xml:lang="lat"> 4.) De parabola
            <lb/>
          5. casuum:
            <lb/>
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          4) On the parabola.
            <lb/>
          Case 5; ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sunt alij
              <lb/>
            casus.
              <lb/>
            Vide dorsum
              <lb/>
            chart: b.)
              <lb/>
            de
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            There are other cases. See the back of sheet b) on the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Cum duplex
              <lb/>
            sit
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            , duæ
              <lb/>
            erunt para-
              <lb/>
            polæ et ver-
              <lb/>
            tex alterius
              <lb/>
            erit inter
              <math>
                <mstyle>
                  <mi>D</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Since there are two cases for
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            , there are two parabolas and the vertex of the other will be between
              <math>
                <mstyle>
                  <mi>D</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> problema.
              <lb/>
            Datis tribus punctis,
              <lb/>
            quorum duo sunt
              <lb/>
            in parabola, et
              <lb/>
            tertium in centroide:
              <lb/>
            Invenire
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Problem.
              <lb/>
            Given three points of which two are on the parabola and the third is at the focus, find the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sint tria data puncta
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            . Sint
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            , in
              <lb/>
            parabola, et
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            in
              <lb/>
            centroide.
              <lb/>
            Connectantur: et circa
              <lb/>
            tres lineas ut diametros,
              <lb/>
            fiunt tres circuli. Quorum duo, circa
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            semivicem secant
              <lb/>
            in punctis
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            , quæ inugantur et fit
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            perpendicularis
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Deinde centro
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . Intervallo
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , fiat circulus
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Itidem centro
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , intervallo
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            fiat circulum
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>b</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , qui secabit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            in
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Unde
              <math>
                <mstyle>
                  <mi>u</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/>
            Fiat
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                  <mo>=</mo>
                  <mi>u</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            . Agatur
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            usque ad
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            . Agatur
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            cum proeluctione
              <lb/>
            quae secabit circulum (circa
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            centrum), in
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            : et
              <lb/>
            circa
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            diametrum, in
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Agatur
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            quæ erit parallela et
              <lb/>
            aequalis
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            . Ita
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            . Bisecetur
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            in puncto
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            etiam est medium punctum inter
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the three given points be
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            . Let
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            be on the parabola and
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            the focus.
              <lb/>
            They are connected and around the three lines as diameters are created three circles. Of which two, around
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            in turn cut in the points
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>t</mi>
                </mstyle>
              </math>
            , which are joined and
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            is perpendicular to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Then with centre
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , radius
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            , create a circle
              <math>
                <mstyle>
                  <mi>u</mi>
                  <mi>b</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            , which cuts
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Whence
              <math>
                <mstyle>
                  <mi>u</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>
            .
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                  <mo>=</mo>
                  <mi>u</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            . Take
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            as far as
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            . Take
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            with its extension which will cut the circle with centre
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            ; and the circle on the diameter
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            at
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            . Take
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>h</mi>
                </mstyle>
              </math>
            which will be parallel and equal to
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            . Thus
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            is bisected at the point
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            is therefore the midpoint between
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Dico quod; punctum
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            est vertex
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
              <emph style="super">axis</emph>
            diameter; et
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>l</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            lineæ ordinatim appli-
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            I say that the point
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            is the vertex of the parabola:
              <math>
                <mstyle>
                  <mi>g</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            is the diametric axis; and
              <math>
                <mstyle>
                  <mi>h</mi>
                  <mi>b</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>k</mi>
                  <mi>l</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            , ordinates. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ad exegesin arithmeticam agatur
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            . et
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            perpendicularis
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>t</mi>
                  <mi>s</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            est quadrilaterum in circulo. Dantur omnes lineæ et per-
              <lb/>
            pendicularis
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            . Tum
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>s</mi>
                  <mo>:</mo>
                  <mi>s</mi>
                  <mi>y</mi>
                  <mo>:</mo>
                  <mi>c</mi>
                  <mi>a</mi>
                  <mo>:</mo>
                  <mi>a</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            . Deinde
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>a</mi>
                      <mi>c</mi>
                      <mo>+</mo>
                      <mi>a</mi>
                      <mi>k</mi>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mi>k</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            To show the arithmetic take
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            , and
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            perpendicular to
              <math>
                <mstyle>
                  <mi>t</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <lb/>
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>t</mi>
                  <mi>s</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            is a cyclic quadrilateral. There are given all the lines and perpendiculars
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            . Then
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>s</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>s</mi>
                  <mi>y</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            are proportionals. Thence
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>a</mi>
                      <mi>c</mi>
                      <mo>+</mo>
                      <mi>a</mi>
                      <mi>k</mi>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mi>k</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Problema igitur satis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            The problem is therefore satisfactorily ]</s>
          </p>
        </div>
      </text>
    </echo>
Impressum