Harriot, Thomas, Mss. 6785

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            <p>
              <s xml:space="preserve">[
                <emph style="bf">Commentary:</emph>
              </s>
            </p>
            <p>
              <s xml:space="preserve"> Harriot refers to Euclid,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV19.html"/>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV19.html"/>
                If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole. </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <p>
            <s xml:space="preserve"> Of 3 magnitudes in continuall proportion: there differences being
              <lb/>
            given; to find the </s>
          </p>
          <p>
            <s xml:space="preserve"> ffirst let us seeke out the theoreme delivering the proper
              <lb/>
            effection whereby it should be performed, by </s>
          </p>
          <p>
            <s xml:space="preserve"> The planes of the proportionalles
              <lb/>
            being noted, let the difference
              <lb/>
            betwixt the first & second be
              <lb/>
            called
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            : & betwixt the second &
              <lb/>
            third
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            </s>
          </p>
          <p>
            <s xml:space="preserve"> Then suppose the first proportionall to be
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . then the second
              <lb/>
            wilbe
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                </mstyle>
              </math>
            . &amp the third
              <math>
                <mstyle>
                  <mi>a</mi>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mo>+</mo>
                  <mi>c</mi>
                </mstyle>
              </math>
            </s>
          </p>
          <p>
            <s xml:space="preserve"> Now seing the square of the second is æquall to the oblong
              <lb/>
            of the extremes. let the multiplications be performed
              <lb/>
            as here </s>
          </p>
          <p>
            <s xml:space="preserve"> Then according to the art
              <lb/>
            let the æquation be [???]
              <lb/>
            & it wilbe </s>
          </p>
          <p>
            <s xml:space="preserve"> The square of the first difference, is æquall to the obling, made of
              <lb/>
            the difference of the differences, & the first </s>
            <lb/>
            <s xml:space="preserve"> or: The first difference is a meane proportionall betwixt the difference
              <lb/>
            of the differences &ampa; the first </s>
            <lb/>
            <s xml:space="preserve"> Wherefour the rule is:
              <lb/>
            Square the first difference; & divide by the difference of the
              <lb/>
            differences, & the quotient wilbe the first proportionall
              <lb/>
            The second & third
              <emph style="super">proportionalls</emph>
            are threfour known by their </s>
          </p>
          <p>
            <s xml:space="preserve"> What proportion therefore in Euclide wilbe the element whereby to demonstrate
              <lb/>
            the problem by composition is easily manifest. that is to say, the 19th of the 5th.
              <lb/>
            </s>
            <lb/>
            <s xml:space="preserve"> From it will issue these two </s>
            <lb/>
            <s xml:space="preserve"> If these be 3 magnitudes in continuall proportion; as the first hath
              <lb/>
            to the second; so hath the first difference, to the second </s>
            <lb/>
            <s xml:space="preserve"> And: The first difference is a meane proportionall
              <lb/>
            betwixt the first proportionall & the difference of
              <lb/>
            </s>
          </p>
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