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

Table of figures

< >
< >
page |< < of 701 > >|
    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/857.jpg" pagenum="164"/>
            <p type="head">
              <s>THEOR. IX. PROP. IX.</s>
            </p>
            <p type="main">
              <s>If two Planes be inclined at pleaſure from a point
                <lb/>
              in a Line parallel to the Horizon, and be inter­
                <lb/>
              ſected by a Line which may make Angles Al­
                <lb/>
              ternately equal to the Angles contained be­
                <lb/>
              tween the ſaid Planes and Horizontal Parallel,
                <lb/>
              the Motion along the parts cut off by the ſaid
                <lb/>
              Line, ſhall be performed in equal Times.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              From off the point C of the Horizontal Line X, let any two Planes
                <lb/>
              be inclined at pleaſure C D and C E, and in any point of the
                <lb/>
              Line C D make the Angle C D F equal to the Angle X C E:
                <lb/>
              and let the Line D F cut the Plane C E in F, in ſuch a manner that
                <lb/>
              the Angles C D F and C F D may be equal to the Angles X C E, L C D
                <lb/>
              Alternately taken. </s>
              <s>I ſay, that
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.857.1.jpg" xlink:href="040/01/857/1.jpg" number="99"/>
                <lb/>
                <emph type="italics"/>
              the Times of the Deſcents along
                <lb/>
              C D and C F are equal. </s>
              <s>And
                <lb/>
              that (the Angle C D F being
                <lb/>
              ſuppoſed equal to the Angle
                <lb/>
              X C E) the Angle C F D is
                <lb/>
              equal to the Angle D C L, is
                <lb/>
              manifeſt. </s>
              <s>For the Common An­
                <lb/>
              gle D C F being taken from the
                <lb/>
              three Angles of the Triangle
                <lb/>
              C D F equal to two Right An­
                <lb/>
              gles, to which are equal all the Angles made with to the Line L X
                <lb/>
              at the point C, there remains in the Triangle two Angles C D F and
                <lb/>
              C F D, equal to the two Angles X C E and L C D: But it was ſup­
                <lb/>
              poſed that C D F is equal to the Angle X C E: Therefore the remaining
                <lb/>
              Angle C F D is equal to the remaining angle D C L. </s>
              <s>Let the Plane
                <lb/>
              C E be ſuppoſed equal to the Plane C D, and from the points D and
                <lb/>
              E raiſe the Perpendiculars D A and E B, unto the Horizontal Paral­
                <lb/>
              lel X L; and from C unto D F let fall the Perpendicular C G. </s>
              <s>And
                <lb/>
              becauſe the Angle C D G is equal to the Angle E C B; and becauſe
                <lb/>
              D G C and C B E are Right Angles; The Triangles C D G and
                <lb/>
              C B E ſhall be equiangled: And as D C is to C G, ſo let C E be
                <lb/>
              to E B: But D C is equal to C E: Therefore C G ſhall be equal to
                <lb/>
              E B. </s>
              <s>And inregard that of the Triangles D A C and C G F, the An­
                <lb/>
              gles C and A are equal to the Angles F and G: Therefore as C D is to
                <lb/>
              D A, ſo ſhall F C be to C G; and Alternately, as D C is to C F, ſo
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum