Harriot, Thomas, Mss. 6787

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page |< < (246) of 1155 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6787_f246" o="246" n="490"/>
          <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"> Propositions on triangular numbers; this sheet was probably placed here deliberately because of the importance of triangular numbers in Harriot's method of interpolation
                <ref id="beery_and_stedall_2009"> Beery and Stedall </ref>
              .) </s>
              <lb/>
              <s xml:space="preserve"> The refrence to Viète is to his
                <emph style="it">Variorum responsorum liber VII</emph>
              , Chapter IX, Proposition 14
                <ref id="Viete_1593d" target="http://www.e-rara.ch/zut/content/pageview/2684241"> (Viete 1593d, Chapter 9, Prop </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Propositio XIV.
                  <lb/>
                Si fuerint lineae quotcunque æqualiter sese excedentes, fit autem prima excessui æqualis: octuplum ejus quod fit sub minima & composita ex omibus, adjunctum minimae quadrato, æquatur quadrato compositæ ex minima & extrema </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> If there are any number of lines exceeding each other, and moreover the first differences are equal, then eight times the product of the least and the sum of all, added to the square of the least, is equal to the square of the sum of the least and twice the </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> The reference to Stevin is to his
                <emph style="it">L'arithmétique ... aussi l'algebre</emph>
                <ref id="stevin_1585a"> (Stevin </ref>
              . Pages 558–642 contain Stevin's treatment of the 'Quatriesme livre d'algebre de Diophante d'Alexandrie'. On page 634, Stevin has the following Theorem. </s>
              <lb/>
              <quote xml:lang="fre">
                <s xml:space="preserve"> Nombre triangulaire multiplié par 8, & plus 1 faict quarré a sa racine </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> A triangular number multiplied by 8, plus 1, makes a square commensurable with its </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> The reference to Maurolico is to his
                <emph style="it">Arithmeticorum libri duo</emph>
                <ref id="maurolico_1575" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/XFWC6D23/pageimg&start=341&viewMode=images&pn=344&mode=imagepath"> (Maurolico 1575, Prop </ref>
              . </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Omnis triangulus octuplatus cum unitate, conficit sequentis imparis </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Eight times any triangular number, plus one, makes the square of the next odd </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> As an example, Maurolico gave
                <math>
                  <mstyle>
                    <mn>8</mn>
                    <mo>×</mo>
                    <mn>1</mn>
                    <mn>5</mn>
                    <mo>+</mo>
                    <mn>1</mn>
                    <mo>=</mo>
                    <mn>1</mn>
                    <mn>2</mn>
                    <mn>1</mn>
                  </mstyle>
                </math>
              ; note that 15 is the fifth triangular number, 11 is the sixth odd number. </s>
              <lb/>
              <s xml:space="preserve"> There is also a reference lower down to Maurolico's Proposition
                <ref id="maurolico_1575" target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/mpiwg/online/permanent/library/XFWC6D23/pageimg&start=321&viewMode=images&pn=326&mode=imagepath"> (Maurolico 1575, Prop </ref>
              : </s>
              <lb/>
              <quote xml:lang="lat">
                <s xml:space="preserve"> Omnis numerus triangulo deinceps: aequatur quadratis lateris trianguli </s>
              </quote>
              <lb/>
              <quote>
                <s xml:space="preserve"> Every triangular number joined with the preceding triangular number makes the square of its </s>
              </quote>
              <lb/>
              <s xml:space="preserve"> Maurolico's example is
                <math>
                  <mstyle>
                    <mn>1</mn>
                    <mn>5</mn>
                    <mo>+</mo>
                    <mn>1</mn>
                    <mn>0</mn>
                    <mo>=</mo>
                    <mn>2</mn>
                    <mn>5</mn>
                  </mstyle>
                </math>
              . </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat">
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> plutarchus platonica
              <lb/>
            quæstione 4
              <emph style="super">a</emph>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Plutarch on Plato, question ]</s>
            <lb/>
            <s xml:space="preserve"> Vieta resp. pa.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Viete,
              <emph style="it">Liber variorum responsorum</emph>
            , page 15. </s>
            <lb/>
            <s xml:space="preserve"> Maurolicus pr. 54.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Maurolico,
              <emph style="it">Arithmetica</emph>
            , Proposition 54. </s>
            <lb/>
            <s xml:space="preserve"> Stevin . pag. 634. arith.
              <lb/>
            in
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Stevin, page 634,
              <emph style="it">Arithmetic</emph>
            , on Diophantus. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Trianguli numeri octuplum, plus quadrato unitatis: æqualie est quadrato
              <lb/>
            facto a duplo latere trianguli plus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Eight times a trinagular number plus the square of one is equal to the square of double the side of the triangular number, plus one.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit latus trianguli
              <math>
                <mstyle>
                  <mi>n</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the side of the triangle be
              <math>
                <mstyle>
                  <mi>n</mi>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> Tum triangulus erit
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>n</mi>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>+</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Then the triangular number will be
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>n</mi>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>+</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            </s>
            <lb/>
            <s xml:space="preserve"> Unde propositio cum demonstratione in notis logisticis ita se
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Whence we have the proposition with its demonstration in arithmetic notation ]</s>
          </p>
          <head xml:space="preserve" xml:lang="lat">
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Marol. pr.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Maurolico, Proposition ]</s>
            <lb/>
            <s xml:space="preserve"> Omnis numerus triangulus, plus triangulo deinceps
              <emph style="st">priori</emph>
            : æquatur
              <lb/>
            quadrato lateris trianguli
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Every triangular number, plus the triangular numebr following, is equal to the square of the side of the greater triangular number.</s>
            <lb/>
            <s xml:space="preserve"> Sit latus maioris trianguli.
              <math>
                <mstyle>
                  <mi>n</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the side of the greater triangular number be
              <math>
                <mstyle>
                  <mi>n</mi>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> latus minoris triang:
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            the side of the smaller triangular number is
              <math>
                <mstyle>
                  <mi>n</mi>
                  <mo>-</mo>
                  <mn>1</mn>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> Ergo: maior triangulus.
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>n</mi>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>+</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore the greater triangular number is
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>n</mi>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>+</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> Minor triangulus.
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>-</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                      <mi>n</mi>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            the smaller triangular number is
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mo maxsize="1">(</mo>
                      <mi>n</mi>
                      <mo>-</mo>
                      <mn>1</mn>
                      <mo maxsize="1">)</mo>
                      <mi>n</mi>
                    </mrow>
                    <mrow>
                      <mn>2</mn>
                    </mrow>
                  </mfrac>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> Unde propositio cum demonstration in notis logisticis ita se
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Whence we have the proposition with its demonstration in arithmetic notation ]</s>
          </p>
        </div>
      </text>
    </echo>
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