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

Table of figures

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      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/950.jpg" pagenum="257"/>
                <emph type="italics"/>
              their Centers and the ſaid points, may be leſſer than any other Lines.
                <lb/>
              </s>
              <s>To expreſs the ſame in other terms, we have reduced it to an impoſſibi­
                <lb/>
              lity, that the Center of the Conoid ſhould not fall betwixt the Centers of
                <lb/>
              the inſcribed and circumſcribed Figures.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>Suppoſing three proportional Lines, and that
                <lb/>
              what proportion the leaſt hath to the exceſs
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              by which the greateſt exceeds the leaſt, the
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              ſame ſhould a Line given have to two thirds of
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              the exceſs by which the greateſt exceeds the
                <lb/>
              middlemoſt: and moreover, that what pro­
                <lb/>
              portion that compounded of the greateſt, and
                <lb/>
              of double the middlemoſt, hath unto that com­
                <lb/>
              pounded of the triple of the greateſt and mid­
                <lb/>
              dlemoſt, the ſame hath another Line given, to
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              the exceſs by which the greateſt exceeds the
                <lb/>
              middle one; both the given Lines taken toge­
                <lb/>
              ther ſhall be a third part of the greateſt of the
                <lb/>
              proportional Lines.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let A B, B C, and B F, be three proportional Lines; and what
                <lb/>
              proportion B F hath to F A, the ſame let M S have to two thirds
                <lb/>
              of C A. </s>
              <s>And what proportion that compounded of A B and the
                <lb/>
              double of B C hath to that compounded of the triple of both A B and
                <lb/>
              B C, the ſame let another, to wit S N, have to A C. </s>
              <s>Becauſe therefore
                <lb/>
              that A B, B C, and C F,
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.950.1.jpg" xlink:href="040/01/950/1.jpg" number="171"/>
                <lb/>
                <emph type="italics"/>
              are proportionals, A G
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              and C F ſhall, for the ſame
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              reaſon, be likewiſe ſo.
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              </s>
              <s>Therefore, as A B is to
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              B C, ſo is A C to C F:
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              and as the triple of A B is to the triple of B C, ſo is A C to C F:
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              Therefore, what proportion the triple of A B with the triple of B C
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              hath to the triple of C B, the ſame ſhall A C have to a Line leſs than
                <lb/>
              C F. </s>
              <s>Let it be C O. </s>
              <s>Wherefore by Compoſition and by Converſion of
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              proportion, O A ſhall have to A C, the ſame proportion, as triple A B
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              with Sextuple B C, hath to triple A B with triple B C. </s>
              <s>But A C hath
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              to S N the ſame proportion, that triple A B with triple B C hath to A B
                <lb/>
              with double B C: Therefore,
                <emph.end type="italics"/>
              ex equali,
                <emph type="italics"/>
              O A to NS ſhall have the
                <lb/>
              ſame proportion, as triple A B with Sexcuple B C hath to A B with
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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