Harriot, Thomas, Mss. 6785

List of thumbnails

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page |< < (191) of 882 > >|
    <echo version="1.0RC">
      <text xml:lang="eng" type="free">
        <div type="section" level="1" n="1">
          <pb file="add_6785_f191" o="191" n="381"/>
          <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 demonstration on this page relies on Euclid,
                <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV12.html"/>
              . </s>
              <lb/>
              <quote>
                <s xml:space="preserve">
                  <ref target="http://aleph0.clarku.edu/~djoyce/java/elements/bookV/propV12.html"/>
                If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents. </s>
              </quote>
              <s xml:space="preserve">]</s>
            </p>
          </div>
          <head xml:space="preserve" xml:lang="lat"> De Continue proportionalibus. Et
            <lb/>
          [
            <emph style="bf">Translation: </emph>
          On continued propotions. An ]</head>
          <p xml:lang="lat">
            <s xml:space="preserve"> Si fuerint quantitates continue proportionales; Erit ut terminus
              <lb/>
            rationis maior, ad terminum rationis minorem: ita differentia
              <lb/>
            compositæ ex omnibus et minimæ, ad differentiae compositæ ex
              <lb/>
            omnibus et
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            If there are quantities in continued proportion: as the greater term of the ratio is to the lesser term of the ratio, so will be the difference between the sum and the least to the difference between the sum and the greatest.</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sint continue proportionales.
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let there be continued proportionals
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>d</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>f</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>g</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>h</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            . </s>
            <lb/>
            <s xml:space="preserve"> Differentiæ compositæ ex omnibus
              <lb/>
            et
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            The difference between the sum and the ]</s>
            <lb/>
            <s xml:space="preserve"> Hoc est, omnes
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            That is, all before ]</s>
            <lb/>
            <s xml:space="preserve"> Differentæ compositæ ex omnibus
              <lb/>
            et
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            The difference between the sum and the ]</s>
            <s xml:space="preserve"> Hoc est, omnis
              <lb/>
            consequentes, quæ ita sub ante-
              <lb/>
            cedentibus
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            That is, all the consequents, which are thus placed under the ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Inde manifesta Theorematis demonstratio; quia ut
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ita (per
              <lb/>
            synthetica) omnes antecedentes ad omnes consequentes. (per 12. pr. li.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Thus the demonstration of the theorem is clear; because as
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            is to
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            so, by construction, are all the antecedents to all the consequents (by Book 5, Proposition 12). </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Datis
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            .
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            , dabitur compositæ ex omnibus
              <lb/>
            quæ significetur per
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Given
              <math>
                <mstyle>
                  <mi>b</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            , ...,
              <math>
                <mstyle>
                  <mi>k</mi>
                </mstyle>
              </math>
            , there is given the sum of all of them, denoted by
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Sit tota composita ducenda per primam
              <lb/>
            quantitatem affirmatam et secundum negatum:
              <lb/>
            facta erit
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>c</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            . cæteræ partes
              <lb/>
            intermediæ factæ eliduntur quoniam æquales
              <lb/>
            affirmatæ et negatæ scilicet
              <math>
                <mstyle>
                  <mo>-</mo>
                  <mi>c</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            et
              <math>
                <mstyle>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Let the total be multiplied by the first quantity taken positively and the second taken negatively: it will make
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>c</mi>
                  <mi>k</mi>
                </mstyle>
              </math>
            . The other intermediate terms will be destroyed because of equal positive and negative quantities like
              <math>
                <mstyle>
                  <mo>-</mo>
                  <mi>c</mi>
                  <mi>b</mi>
                </mstyle>
              </math>
            and
              <math>
                <mstyle>
                  <mo>+</mo>
                  <mi>b</mi>
                  <mi>c</mi>
                </mstyle>
              </math>
            . </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Ergo
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>b</mi>
                      <mi>b</mi>
                      <mo>-</mo>
                      <mi>c</mi>
                      <mi>k</mi>
                    </mrow>
                    <mrow>
                      <mi>b</mi>
                      <mo>-</mo>
                      <mi>c</mi>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mi>a</mi>
                </mstyle>
              </math>
            . compositæ ex omnibus. patet igitur
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Therefore
              <math>
                <mstyle>
                  <mfrac>
                    <mrow>
                      <mi>b</mi>
                      <mi>b</mi>
                      <mo>-</mo>
                      <mi>c</mi>
                      <mi>k</mi>
                    </mrow>
                    <mrow>
                      <mi>b</mi>
                      <mo>-</mo>
                      <mi>c</mi>
                    </mrow>
                  </mfrac>
                  <mo>=</mo>
                  <mi>a</mi>
                </mstyle>
              </math>
            is the sum of all of them. The demonstration is therefore clear. </s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> alia Notatio
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            another notation for ]</s>
          </p>
          <p xml:lang="lat">
            <s xml:space="preserve"> Hinc in infinitis progressionibus cum
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            hoc est ultima quantitatis in infinitum
              <lb/>
            abeat, tres isti termini sint proport[ionales] videlicet:
              <lb/>
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            co[mpositæ om]nibus.
              <lb/>
            [
              <emph style="bf">Translation: </emph>
            Hence, in an infinite progression, since
              <math>
                <mstyle>
                  <mi>u</mi>
                </mstyle>
              </math>
            , that is, the last quantity, disappears to infinity, these three terms are clearly proportional:
              <math>
                <mstyle>
                  <mi>b</mi>
                  <mo>-</mo>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>c</mi>
                </mstyle>
              </math>
            ,
              <math>
                <mstyle>
                  <mi>a</mi>
                </mstyle>
              </math>
            the sum of all. </s>
          </p>
        </div>
      </text>
    </echo>
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