Harriot, Thomas, Mss. 6784

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    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f188" o="188" n="375"/>
          <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"> The problem pursued in this and the seven following folios is 'on tangencies', as set out in Pappus,
                <emph style="it">Mathematicae collectiones</emph>
              , Book 7. In Commandino's edition of 1588
                <ref id="pappus_1588"> (Pappus </ref>
              , the problem is stated on page 159–159v. </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Punctis, & rectis lineis, & circulis quibuscumque duobus datis circulum describere magnitudine datum, qui per datum punctum, vel data puncta transeat; contingat autem vnamquamque datarum </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Given any two points, lines, or circles in position, to draw a circle of a given size, which passes through any given point or points and touches any of the given lines.</s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> 1.) Ad locum de tactibus (.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On the place of ]</head>
          <head xml:space="preserve" xml:lang="lat"> De parabola.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On the parabola. ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit curvum
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            parabola.
              <lb/>
            cuius axis
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            , producta.
              <lb/>
            recta,
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            .
              <lb/>
            fiat,
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>a</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mi>b</mi>
                      <mi>x</mi>
                    </mrow>
                    <mrow>
                      <mn>4</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            .
              <lb/>
            et sit quælibet linea ordinatim
              <lb/>
            applicata
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <lb/>
            agatur
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod:
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the curve
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            be a parabola, with axis
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            extended,
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>x</mi>
                </mstyle>
              </math>
            a straight line.
              <lb/>
            Make
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>a</mi>
                  <mo>=</mo>
                  <mfrac>
                    <mrow>
                      <mi>b</mi>
                      <mi>x</mi>
                    </mrow>
                    <mrow>
                      <mn>4</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            and let
              <math>
                <mstyle>
                  <mi>c</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            be any ordinate. Construct
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            . I say that
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                  <mo>=</mo>
                  <mi>a</mi>
                  <mi>d</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Corollarium. 1
              <emph style="super">m</emph>
            .
              <lb/>
            Hinc patet quod curva parabolæ
              <lb/>
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            , est locus centrorum omnium
              <lb/>
            circulorum qui possunt contingere
              <lb/>
            lineam
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            , et punctum
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Huiusmodi punctum
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            in variis
              <lb/>
            scriptis nostris, usurpandimus
              <lb/>
            appellate, parabolæ
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Corollary 1.
              <lb/>
            Here it is clear that the parabolic curve
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>d</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            is the locus of the centres of all the circles that can touch the line
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            and the point
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            .
              <lb/>
            In this way, the point
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , in my various writings, I may call the centroid of the parabola. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Corollarium. 2
              <emph style="super">m</emph>
            .
              <lb/>
            Patet etiam pro descriptione
              <lb/>
            parabolæ: si
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            dividatur in partes
              <lb/>
            æquales, et continuentur in producta
              <lb/>
            erit prima centrali æqualis numero
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
            . 2
              <emph style="super">a</emph>
            centralis
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                </mstyle>
              </math>
            .
              <lb/>
            3
              <emph style="super">a</emph>
            centralis
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>3</mn>
                </mstyle>
              </math>
            . &c.
              <lb/>
            ut si
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            dividatur in 3 partes
              <lb/>
            æquales, erit ad prima centralis
              <lb/>
            3 + 1, hoc est 4.
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            , 5.
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                </mstyle>
              </math>
            , 6.
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                </mstyle>
              </math>
            , 7
              <lb/>
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>l</mi>
                </mstyle>
              </math>
            , 8. &
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Corollary 2.
              <lb/>
            The description of the parabola is also clear. If
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            is divided into equal parts and they are continued in extension, the first from the centre will be equal in numbers to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
            , the second from the centre to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>2</mn>
                </mstyle>
              </math>
            , the third to
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mn>3</mn>
                </mstyle>
              </math>
            , etc. so if
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            is divided into 3 equal parts, the first from the centre will be 3 + 1, that is, 4,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>f</mi>
                  <mo>=</mo>
                  <mn>5</mn>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>g</mi>
                  <mo>=</mo>
                  <mn>6</mn>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>i</mi>
                  <mo>=</mo>
                  <mn>7</mn>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>l</mi>
                  <mo>=</mo>
                  <mn>8</mn>
                </mstyle>
              </math>
            , etc. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Corollarium. 3
              <emph style="super">m</emph>
            .
              <lb/>
            Sit
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            angulus rectus et
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
              <lb/>
            et sit
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            parallela
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            . fiat
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>m</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            centro
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            , et intervallo,
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
              <lb/>
            agatur periferia, et secabit
              <lb/>
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            in puncto
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
              <lb/>
            agatur
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            perpendic:
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Dico quod:
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>q</mi>
                  <mo>=</mo>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            Et punctum
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            est in
              <lb/>
            parabola. Ut patet
              <lb/>
            ex
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Corollary 3.
              <lb/>
            Let
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            be a right angle and
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>b</mi>
                  <mo>=</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
            , and let
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            be parallel to
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Make
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>m</mi>
                  <mo>=</mo>
                  <mi>b</mi>
                  <mi>e</mi>
                </mstyle>
              </math>
            .
              <lb/>
            With centre
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            and radius
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mi>m</mi>
                </mstyle>
              </math>
            construct a circumference, and it willl cut
              <math>
                <mstyle>
                  <mi>e</mi>
                  <mi>f</mi>
                </mstyle>
              </math>
            at the point
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            . Construct
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>q</mi>
                </mstyle>
              </math>
            perpendicular to
              <math>
                <mstyle>
                  <mi>o</mi>
                  <mi>t</mi>
                </mstyle>
              </math>
            .
              <lb/>
            I say that
              <math>
                <mstyle>
                  <mi>f</mi>
                  <mi>q</mi>
                  <mo>=</mo>
                  <mi>f</mi>
                  <mi>a</mi>
                </mstyle>
              </math>
            . And the point
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            is on the parabola. As is clear from the suppositions. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve">
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            turn ]</s>
          </p>
        </div>
      </text>
    </echo>
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