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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/736.jpg" pagenum="44"/>
              if I ſhall ſhew you what proportion the Superficies of equall Cy­
                <lb/>
              linders have to one another, we ſhall obtain our deſire. </s>
              <s>I ſay there­
                <lb/>
              fore, that</s>
            </p>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Superficies of Equal Cylinders, their Baſes being
                <lb/>
              ſubſtracted, are to one another in ſubduple proportion
                <lb/>
              of their lengths.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Take two equall Cylinders, the heights of which let be A B,
                <lb/>
              and C D: and let the Line E be a Mean-proportional
                <lb/>
              between them. </s>
              <s>I ſay, the Superficies of the Cylinder A B,
                <lb/>
              the Baſes ſubſtracted, hath the ſame proportion to the Superficies
                <lb/>
              of the Cylinder C D, the Baſes in like manner ſubſtracted, as the
                <lb/>
              Line A B hath to the Line E, which is ſubduple of the proportion
                <lb/>
              of A B to C D. </s>
              <s>Cut the part of the Cylinder A B in F, and let the
                <lb/>
              height A F be equal to C D: And becauſe the Baſes of equal Cy­
                <lb/>
              linders anſwer Reciprocally to their heights, the Circle, Baſe of
                <lb/>
              the Cylinder C D, to the Circle, Baſe of the
                <lb/>
                <figure id="id.040.01.736.1.jpg" xlink:href="040/01/736/1.jpg" number="61"/>
                <lb/>
              Cylinder A B, ſhall be as the height B A to
                <lb/>
              D C: And becauſe Circles are to one ano­
                <lb/>
              ther as the Squares of their Diameters, the
                <lb/>
              ſaid Squares ſhall have the ſame proportion,
                <lb/>
              that B A hath to C D: But as B A, is to
                <lb/>
              C D, ſo is the Square B A to the Square of
                <lb/>
              E: Therefore thoſe four Squares are Pro­
                <lb/>
              portionals: And therefore their Sides ſhall
                <lb/>
              be Proportionals. </s>
              <s>And as the Line A B is to
                <lb/>
              E, ſo is the Diameter of the Circle C to the
                <lb/>
              Diameter of the Circle A: But as are the
                <lb/>
              Diameters, ſo are the Circumferences; and
                <lb/>
              as are the Circumferences, ſo likewiſe are the Superficies of Cylin­
                <lb/>
              ders equal in Height. </s>
              <s>Therefore as the Line A B is to E, ſo is the
                <lb/>
              Superficies of the Cylinder C D to the Superficies of the Cylinder
                <lb/>
              A F. </s>
              <s>Becauſe therefore the height A F to the height A B, is as the
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              Superficies A F to the Superficies A B: And as is the height A B
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              to the Line E, ſo is the Superficies C D to the Superficies A F:
                <lb/>
              Therefore by Perturbation of Proportion as the height A F is to
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              E, ſo is the Superficies C D to the Superficies A B: And, by Con­
                <lb/>
              verſion, as the Superficies of the Cylinder A B is to the Superficies
                <lb/>
              of the Cylinder C D, ſo is the Line E to the Line A F; that is, to
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              the Line C D: or as A B to E: Which is in ſubduple proportion
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              of A B to C D: Which is that which was to be proved.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum