Harriot, Thomas, Mss. 6784

List of thumbnails

< >
601
601 (301) 		602
602 (301v)
603
603 (302) 		604
604 (302v)
605
605 (303) 		606
606 (303v)
607
607 (304) 		608
608 (304v)
609
609 (305) 		610
610 (305v)
< >
page |< < (355) of 862 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6784_f355" o="355" n="709"/>
          <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"> On this page Harriot examines a particular case arising from Proposition VII of
                <emph style="it">Supplementum geometriæ</emph>
                <ref id="Viete_1593c" target="http://www.e-rara.ch/zut/content/pageview/2684114"> (Viète 1593c, Prop </ref>
              , when the fourth proportional is twice the first. The same proposition is the subject of Chapter V of Viète's
                <emph style="it">Variorum responsorum libri VIII</emph>
                <ref id="Viete_1593d" target="http://www.e-rara.ch/zut/content/pageview/2684223"> (Viete 1593d, Chapter </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Caput V
                  <lb/>
                Propositio
                  <lb/>
                Describere quatuor lineas rectas continue proportionales, quarum extremæ sint in ratione </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Construct four lines in continued proportion, whose extremes are in double </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> The
                <emph style="it">Variorum</emph>
              refers to the
                <emph style="it">Supplementum</emph>
              , indicating that the
                <emph style="it">Supplementum</emph>
              was written first. </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve"> Ad Corollorium prop. 7. Supplementi. Et ad cap. 5. Resp. lib. 8. pag.
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On a corollary to Proposition 7 of the Supplement. Also Chapter 5, Variorum liber responsorum, page 4.</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            prima proportionalium, et
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            ea
              <lb/>
            cuius quadratum est triplum quadrati
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Tum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            est dupla ad
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            ; et per assumptum
              <lb/>
            ex poristicis in alia charta demonstratum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
              <lb/>
            erit quarta proportionalis. Per propositione
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            est secunda et
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            tertia.
              <lb/>
            Sed
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            est æqualis
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            propter similitudine triangulorum
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , et
              <lb/>
            analogiam precedentam ut sequitur.
              <lb/>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mo>.</mo>
                  <mi>E</mi>
                  <mi>A</mi>
                  <mo>.</mo>
                  <mi>E</mi>
                  <mi>G</mi>
                  <mo>.</mo>
                  <mi>A</mi>
                  <mi>C</mi>
                  <mo>.</mo>
                </mstyle>
              </math>
            Analogia precedens.
              <lb/>
            […]
              <lb/>
            Et per similitudi-
              <lb/>
            num Δ
              <emph style="super">orum</emph>
            […]
              <lb/>
            Ergo.
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mo>.</mo>
                  <mi>A</mi>
                  <mi>E</mi>
                  <mo>.</mo>
                  <mi>F</mi>
                  <mi>B</mi>
                  <mo>.</mo>
                  <mi>A</mi>
                  <mi>C</mi>
                  <mo>.</mo>
                </mstyle>
              </math>
            continue
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            be the first proportional, and
              <math>
                <mstyle>
                  <mi>B</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            that whose square is three times the square of
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Then
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            is twice
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            ; and by taking it from the proof demonstrated in the other sheet,
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            will be the fourth proportional. By the proposition
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>A</mi>
                </mstyle>
              </math>
            is the second and
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            the third.
              <lb/>
            But
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>G</mi>
                </mstyle>
              </math>
            because of similar triangles
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            , and
              <lb/>
            the precding ratio, as follows.
              <lb/>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mo>:</mo>
                  <mi>E</mi>
                  <mi>A</mi>
                  <mo>:</mo>
                  <mi>E</mi>
                  <mi>G</mi>
                  <mo>:</mo>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            preceding ratio.
              <lb/>
              <lb/>
            And by similar triangles.
              <lb/>
              <lb/>
            Therefore
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                  <mo>:</mo>
                  <mi>A</mi>
                  <mi>E</mi>
                  <mo>:</mo>
                  <mi>F</mi>
                  <mi>B</mi>
                  <mo>:</mo>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            are continued proportionals.
              <lb/>
            [
              <emph style="bf">Commentary: </emph>
            The other sheet mentioned in this paragraph appears to be Add MS
              <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Flibrary%2FXT0KZ8QC%2F&viewMode=image&pn=705&start=710"> f. </ref>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Datis igitur extremis in ratione dupla, mediæ ita compendiosæ
              <lb/>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore given the extremes in double ratio, the mean is briefly ]</s>
            <lb/>
            <s xml:space="preserve"> Sit maxima
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            bisariam divisa in puncto
              <math>
                <mstyle>
                  <mi>D</mi>
                </mstyle>
              </math>
            et intervallo
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            describatur
              <lb/>
            circulus. Et sit
              <emph style="st">prima</emph>
              <emph style="super">minima</emph>
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            inscripta et producta ad partes
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Ducatur
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            ita ut
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            sit æqualis
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            . et acta fit linea
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            .
              <lb/>
            Quatuor igitur continue proportionales ex supra demonstratis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the maximum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            be cut in half at the point
              <math>
                <mstyle>
                  <mi>D</mi>
                </mstyle>
              </math>
            and with radius
              <math>
                <mstyle>
                  <mi>D</mi>
                  <mi>C</mi>
                </mstyle>
              </math>
            there is described a circle. And let the minimum
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            be inscribed and produced to the point
              <math>
                <mstyle>
                  <mi>E</mi>
                </mstyle>
              </math>
            . Construct
              <math>
                <mstyle>
                  <mi>C</mi>
                  <mi>E</mi>
                </mstyle>
              </math>
            so that
              <math>
                <mstyle>
                  <mi>E</mi>
                  <mi>F</mi>
                </mstyle>
              </math>
            is equal to
              <math>
                <mstyle>
                  <mi>A</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            , and let the line
              <math>
                <mstyle>
                  <mi>F</mi>
                  <mi>B</mi>
                </mstyle>
              </math>
            be joined.
              <lb/>
            Therefore there are the four continued proportionals that were demonstrated ]</s>
          </p>
        </div>
      </text>
    </echo>
Impressum