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="040/01/852.jpg" pagenum="159"/>
                <emph type="italics"/>
              Mechanicks: That the Moment of the Weight elevated upon the Plane
                <lb/>
              according to the Line A B C, is
                <lb/>
              to its total Moment, as B E to B A;
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.852.1.jpg" xlink:href="040/01/852/1.jpg" number="94"/>
                <lb/>
                <emph type="italics"/>
              And that the Moment of the ſame
                <lb/>
              Weight upon the Elevation A D,
                <lb/>
              is to its total Moment, as D F to
                <lb/>
              D A or B A: Therefore the Mo­
                <lb/>
              ment of the ſaid Weight upon the
                <lb/>
              Plane inclined according to D A,
                <lb/>
              is to the Moment upon the Plane
                <lb/>
              inclined according to A B C, as
                <lb/>
              the Line D F to the Line B E:
                <lb/>
              Therefore the Spaces which the
                <lb/>
              ſaid Weight ſhall paſſe in equal
                <lb/>
              Times along the Inclined Planes C A and D A, ſhall be to each other as
                <lb/>
              the Line B E to D F; by the ſecond Propoſition of the Firſt Book:
                <lb/>
              But as B E is to D F, ſo A C is demonſtrated to be to D A:
                <lb/>
              Therefore the ſame Moveable will in equal Times paſſe the Lines
                <lb/>
              C A and D A.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              And that C A is to D A as B E is to D F, is thus demonſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Draw a Line from C to D; and by D and B draw the Lines
                <lb/>
              D G L, (cutting C A in the point I) and B H, Parallels to A F:
                <lb/>
              And the Angle A D I ſhall be equal to the Angle D C A, for that
                <lb/>
              the parts L A and A D of the Circumference ſubtending them, are
                <lb/>
              equal, and the Angle D A C common to them both: Therefore of
                <lb/>
              the equiangled Triangles C A D and D A I, the ſides about the
                <lb/>
              equal Angles ſhall be proportional: And as C A is to A D, ſo is
                <lb/>
              D A to A I, that is B A to A I, or H A to A G; that is, B E to
                <lb/>
              D F: Which was to be proved.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Or elſe the ſame ſhall be demonſtrated more ſpeedily thus.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Vnto the Horizon A B, let a Circle be erect, whoſe Diameter is
                <lb/>
              perpendicular to the Horizon: and
                <lb/>
              from the higheſt Term D let a Plane
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.852.2.jpg" xlink:href="040/01/852/2.jpg" number="95"/>
                <lb/>
                <emph type="italics"/>
              at pleaſure D F, be inclined to the
                <lb/>
              Circumference. </s>
              <s>I ſay that the De­
                <lb/>
              ſcent along the Plane D F, and the
                <lb/>
              Fall along the Diameter B C, will
                <lb/>
              be paſſed by the ſame Moveable in
                <lb/>
              equal Times. </s>
              <s>For let F G be drawn
                <lb/>
              parallel to the Horizon A B, which
                <lb/>
              ſhall be perpendicular to the Diameter
                <lb/>
              D C, and let a Line conjoyn F and
                <lb/>
              C: and becauſe the Time of the Fall
                <lb/>
              along D C, is to the Time of the Fall along D G, as the Mean
                <lb/>
              Proportional between C D and D G, is to the ſaid D G; and the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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