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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/851.jpg" pagenum="158"/>
            <p type="head">
              <s>THEOR. VI. PROP. VI.</s>
            </p>
            <p type="main">
              <s>If from the higheſt or loweſt part of a Circle,
                <lb/>
              erect upon the Horizon, certain Planes be
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              drawn inclined towards the Circumference,
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              the Times of the Deſcents along the ſame
                <lb/>
              ſhall be equal.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Circle be erect upon the Horizon G H, whoſe Diameter
                <lb/>
              recited upon the loweſt point, that is upon the contact with the
                <lb/>
              Horizon, let be F A, and from the higheſt point A let certain
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              Planes A B and A C incline towards the Circumference: I ſay that the
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              Times of the Deſcents along the ſame are equal. </s>
              <s>Let B D and C E be
                <lb/>
              two Perpendiculars let fall unto the Diameter; and let A I be a Mean­
                <lb/>
              Proportional between the Altitudes
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.851.1.jpg" xlink:href="040/01/851/1.jpg" number="93"/>
                <lb/>
                <emph type="italics"/>
              of the Planes E A and A D. </s>
              <s>And
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              becauſe the Rectangles F A E and
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              F A D are equal to the Squares of
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              A C and A B; And alſo becauſe
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              that as the Rectangle F A E, is to
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              the Rectangle F A D, ſo is E A to
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              A D. </s>
              <s>Therefore as the Square of
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              C A is to the Square of B A,
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              ſo is the Line E A to the Line
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              A D. </s>
              <s>But as the Line E A is to
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              D A, ſo is the Square of I A to the Square of A D: Therefore
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              the Squares of the Lines C A and A B are to each other as the Squares
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              of the Lines I A and A D: And therefore as the Line C A is to A B,
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              ſo is I A to A D: But in the precedent Propoſition it hath been demon­
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              ſtrated that the proportion of the Time of the Deſcent along A C to the
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              Time of the Deſcent by A B, is compounded of the proportions of C A
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              to A B, and of D A to A I, which is the ſame with the proportion of
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              B A to A C: Therefore the proportion of the Time of the Deſcent along
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              A C, to the Time of the Deſcent along A B, is compounded of the pro­
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              portions of C A to A B, and of B A to A C: Therefore the proporti­
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              on of thoſe Times is a proportion of equality: Therefore the Propoſition
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              is evident.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The ſame is another way demonſtrated from the Mechanicks: Name­
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              ly that in the enſuing Figure the Moveable paſſeth in equal Times along
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              C A and D A. </s>
              <s>For let B A be equal to the ſaid D A, ond let fall the
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              Perpendiculars B E and D F: It is manifeſt by the Elements of the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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