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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/846.jpg" pagenum="153"/>
            <p type="head">
              <s>THEOREM.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The degrees of Velocity of a Moveable deſcending with a Natural
                <lb/>
              Motion from the ſame height along Planes in any manner inclined
                <lb/>
              at the arrival to the Horizon are alwaies equal, Impediments be­
                <lb/>
              ing removed.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Here we are in the firſt place to advertiſe you, that it having
                <lb/>
              been proved, that in any Inclination of the Plane the Move­
                <lb/>
              able from its receſſion from Quieſſence goeth encreaſing its Ve­
                <lb/>
              locity, or quantity of its
                <emph type="italics"/>
              Impetus,
                <emph.end type="italics"/>
              with the proportion of the
                <lb/>
              Time (according to the Definition which the Author giveth of
                <lb/>
              Motion naturally Accelerate) whereupon, as he hath by the pre­
                <lb/>
              cedent Propoſition demonſtrated, the Spaces paſſed are in dupli­
                <lb/>
              cate proportion to the Times, and, conſequently, to the degrees
                <lb/>
              of Velocity: look what the
                <emph type="italics"/>
              Impetus's
                <emph.end type="italics"/>
              were in that which was firſt
                <lb/>
              moved, ſuch proportionally ſhall be the degrees of Velocity gai­
                <lb/>
              ned in the ſame Time; ſeeing that both theſe and thoſe encreaſe
                <lb/>
              with the ſame proportion in the ſame Time.</s>
            </p>
            <p type="main">
              <s>Now let the inclined Plane be A B, its elevation above the Ho
                <lb/>
              rizon the Perpendicular A C, and the Horizontal Plane C B: and
                <lb/>
              becauſe, as was even now concluded, the
                <emph type="italics"/>
              Impetus
                <emph.end type="italics"/>
              of a Moveable
                <lb/>
              along the Perpendicular A C is to the
                <emph type="italics"/>
              Impetus
                <emph.end type="italics"/>
              of the ſame along
                <lb/>
              the inclined Plane A B, as A B is to A C, let there be taken in the
                <lb/>
              inclined Plane A B, A D a third proportional to A B and A C:
                <lb/>
              The
                <emph type="italics"/>
              Impetus,
                <emph.end type="italics"/>
              therefore, along A C is to the
                <emph type="italics"/>
              Impetus
                <emph.end type="italics"/>
              along A B,
                <lb/>
              that is along A D, as A C is to
                <lb/>
                <figure id="id.040.01.846.1.jpg" xlink:href="040/01/846/1.jpg" number="89"/>
                <lb/>
              A D: And therefore the Move­
                <lb/>
              able in the ſame Time that it
                <lb/>
              would paſs the Perpendicular
                <lb/>
              Space AC, ſhall likewiſe paſs the
                <lb/>
              Space A D, in the inclined Plane
                <lb/>
              A B, (the Moments being as
                <lb/>
              the Spaces:) And the degree of Velocity in C ſhall have the ſame
                <lb/>
              proportion to the degree of Velocity in D, as A C hath to A D:
                <lb/>
              But the degree of Velocity in B is to the ſame degree in D, as the
                <lb/>
              Time along A B is to the Time along AD, by the definition of
                <lb/>
              Accelerate Motion; And the Time along AB is to the Time along
                <lb/>
              A D, as the ſame A C, the Mean Proportional between B A and
                <lb/>
              A D, is to A D, by the laſt Corollary of the ſecond Propoſition:
                <lb/>
              Therefore the degrees of Velocity in B and in C have to the de­
                <lb/>
              gree in D, the ſame Proportion as A C hath to A D; and therefore
                <lb/>
              are equal: Which is the
                <emph type="italics"/>
              Theorem
                <emph.end type="italics"/>
              intended to be demonſtrated.</s>
            </p>
            <p type="main">
              <s>By this we may more concludingly prove the enſuing third </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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