Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/961.jpg" pagenum="268"/>
            <p type="main">
              <s>
                <emph type="italics"/>
              For let A B, B C, B D, and B E be four proportional Lines. </s>
              <s>And
                <lb/>
              as B E is to E A, ſo let F G be to 3/4 of A C. </s>
              <s>And as the Line equal
                <lb/>
              to A B and to double B C and to triple B D is to the Line equal
                <lb/>
              to the quadruples of A B, B C, and B D, ſo let H G be to A C. </s>
              <s>It is
                <lb/>
              to be proved, that H F is a fourth part of A B. </s>
              <s>Foraſmuch therefore
                <lb/>
              as A B, B C, B D, and B E
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.961.1.jpg" xlink:href="040/01/961/1.jpg" number="178"/>
                <lb/>
                <emph type="italics"/>
              are proportionals, A C,
                <lb/>
              C D, and D E ſhall be in
                <lb/>
              the ſame proportion: And
                <lb/>
              as the quadruple of the ſaid
                <lb/>
              A B, B C, and B D is to
                <lb/>
              A B with the double of B C and triple of B D, ſo is the quadruple of
                <lb/>
              A C, C D, and D E; that is, the quadruple of A E; to A C with the
                <lb/>
              double of C D, and triple of D E. </s>
              <s>And ſo is A C to H G. </s>
              <s>Therefore
                <lb/>
              as the triple of A E is to A C, with the double of C D and triple of
                <lb/>
              D E, ſo is 3/4 of A C to H G. </s>
              <s>And as the triple of A E is to the triple of
                <lb/>
              E B, ſo is 3/4 A C to G F: Therefore, by the Converſe of the twenty
                <lb/>
              fourth of the fifth, As triple A E is to A C with double C D and tri­
                <lb/>
              ple D B, ſo is 3/4 of A C to H F: And as the quadruple of A E is to A C
                <lb/>
              with the double of C D and triple of D B; that is, to A B with C B and
                <lb/>
              B D, ſo is A C to H F. And, by Permutation, as the quadruple of A E
                <lb/>
              is to A C, ſo is A B with C B and B D to H F. </s>
              <s>And as A C is to A E, ſo
                <lb/>
              is A B to A B with C B and B D. Therefore,
                <emph.end type="italics"/>
              ex æquali,
                <emph type="italics"/>
              by Perturbed
                <lb/>
              proportion, as quadruple A E is to A E, ſo is A B to H F. </s>
              <s>Wherefore it
                <lb/>
              is manifeſt that H F is the fourth part of A B.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>The Center of Gravity of the
                <emph type="italics"/>
              Fruſtum
                <emph.end type="italics"/>
              of any Py­
                <lb/>
              ramid or Cone, cut equidiſtant to the Plane
                <lb/>
              of the Baſe, is in the Axis, and doth ſo divide
                <lb/>
              the ſame, that the part towards the leſſer Baſe
                <lb/>
              is to the remainder, as the triple of the greater
                <lb/>
              Baſe, with the double of the mean Space be­
                <lb/>
              twixt the greater and leſſer Baſe, together
                <lb/>
              with the leſſer Baſe is to the triple of the leſſer
                <lb/>
              Baſe, together with the ſame double of the
                <lb/>
              mean Space, as alſo of the greater Baſe.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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