Harriot, Thomas, Mss. 6782

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page |< < (330) of 1011 > >|
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            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> This page is a continuation of Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/VWXURW4V&start=80&viewMode=image&pn=88"> f. </ref>
              .
                <lb/>
              Note 3 gives the triangular numbers in general algebraic
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mi>n</mi>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <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>1</mn>
                        <mo>×</mo>
                        <mn>2</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mi>n</mi>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
                        <mo>+</mo>
                        <mn>1</mn>
                        <mo maxsize="1">)</mo>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
                        <mo>+</mo>
                        <mn>2</mn>
                        <mo maxsize="1">)</mo>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                        <mo>×</mo>
                        <mn>2</mn>
                        <mo>×</mo>
                        <mn>3</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mfrac>
                      <mrow>
                        <mi>n</mi>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
                        <mo>+</mo>
                        <mn>1</mn>
                        <mo maxsize="1">)</mo>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
                        <mo>+</mo>
                        <mn>2</mn>
                        <mo maxsize="1">)</mo>
                        <mo maxsize="1">(</mo>
                        <mi>n</mi>
                        <mo>+</mo>
                        <mn>3</mn>
                        <mo maxsize="1">)</mo>
                      </mrow>
                      <mrow>
                        <mn>1</mn>
                        <mo>×</mo>
                        <mn>2</mn>
                        <mo>×</mo>
                        <mn>3</mn>
                        <mo>×</mo>
                        <mn>4</mn>
                      </mrow>
                    </mfrac>
                  </mstyle>
                </math>
              .
                <lb/>
              On the right these formula are given labels that in modern subscript notation would
                <math>
                  <mstyle>
                    <mrow>
                      <msub>
                        <mi>p</mi>
                        <mn>1</mn>
                      </msub>
                    </mrow>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mrow>
                      <msub>
                        <mi>p</mi>
                        <mn>2</mn>
                      </msub>
                    </mrow>
                  </mstyle>
                </math>
              ,
                <math>
                  <mstyle>
                    <mrow>
                      <msub>
                        <mi>p</mi>
                        <mn>3</mn>
                      </msub>
                    </mrow>
                  </mstyle>
                </math>
              , and so on.
                <lb/>
              In the fourth notation, in the lower half of the
                <emph style="it">p</emph>
              has been replaced by
                <emph style="it">v</emph>
              , and the terms have been multiplied out to give a one-line expression (or in Harriot's terms, an equation) instead of a fraction.
                <lb/>
              See also page 1 of the 'Magisteria' (Add MS
                <ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/HSPGZ0AE&start=210&viewMode=image&pn=215"> f. </ref>
              ). </s>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p xml:lang="lat">
            <s xml:space="preserve"> 3. Generalis notatio
              <lb/>
            triangularium
              <lb/>
            in notis
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            3. General notation for triangular numbers in general ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Melius ad continuam
              <lb/>
            additionem
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Better for continual addition of triangular ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> 4. Quarta notatio
              <lb/>
            per
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            4. Fourth notation, by means of ]</s>
          </p>
        </div>
      </text>
    </echo>
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