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

Table of figures

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              <s>
                <pb xlink:href="040/01/944.jpg" pagenum="251"/>
                <emph type="italics"/>
              to the Cylinder Y Z as S Y to R Z, that is, as Y A to A Z: and, by the
                <lb/>
              ſame reaſon, the Cylinder whoſe Axis is Z Y is to that whoſe Axis is
                <lb/>
              Z V, as Z A is to A V. </s>
              <s>The ſaid Cylinders, therefore, are to one ano­
                <lb/>
              ther as the Lines D A, A Y; Z A, A V: But theſe are equally exceed­
                <lb/>
              ing to one another, and the exceſs is equal to the leaſt, ſo that A Z is
                <lb/>
              double to A V; and A Y is triple the
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.944.1.jpg" xlink:href="040/01/944/1.jpg" number="167"/>
                <lb/>
                <emph type="italics"/>
              ſame; and D A Quadruple. </s>
              <s>Thoſe
                <lb/>
              Cylinders, therefore, are certain Mag­
                <lb/>
              nitudes in order equally exceeding one
                <lb/>
              another, whoſe exceſs is equal to the
                <lb/>
              leaſt of them, and is the Line X M,
                <lb/>
              in which they are ſuſpended at equal
                <lb/>
              diſtances (for that each of the Cy­
                <lb/>
              linders hath its Center of Gravity in
                <lb/>
              the miaſt of the Axis.) Wherefore,
                <lb/>
              by what hath been above demonſtra­
                <lb/>
              ted, the Center of Gravity of the Mag­
                <lb/>
              nitude compounded of them all divi­
                <lb/>
              deth the Line X M ſo, that the part
                <lb/>
              towards X is double to the reſt. </s>
              <s>Divide it, therefore, and, let X
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              be
                <lb/>
              double
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              M: therefore is
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              the Center of Gravity of the inſcribed Fi­
                <lb/>
              gure. </s>
              <s>Divide A V in two equal parts in
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
              :
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              X ſhall be double to
                <lb/>
              M E: But X
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              is double to
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              M: Wherefore
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              E ſhall be triple E
                <emph.end type="italics"/>
                <foreign lang="grc">α.</foreign>
                <emph type="italics"/>
              But
                <emph.end type="italics"/>
                <lb/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              E is triple E N: It is manifeſt, therefore, that E N is greater than
                <lb/>
              E X; and for that cauſe
                <emph.end type="italics"/>
                <foreign lang="grc">α,</foreign>
                <emph type="italics"/>
              which is the Center of Gravity of the in­
                <lb/>
              ſcribed Figure, cometh nearer to the Baſe of the Conoid than N. </s>
              <s>And
                <lb/>
              becauſe that as A E is to E N, ſo is the part taken away
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              E to the part
                <lb/>
              taken away E
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
              :
                <emph type="italics"/>
              and the remaining part ſhall be to the remaming part,
                <lb/>
              that is, A
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              to N
                <emph.end type="italics"/>
                <foreign lang="grc">α,</foreign>
                <emph type="italics"/>
              as A E to E N. Therefore
                <emph.end type="italics"/>
                <foreign lang="grc">α</foreign>
                <emph type="italics"/>
              N is the third part of
                <lb/>
              A
                <emph.end type="italics"/>
                <foreign lang="grc">ε,</foreign>
                <emph type="italics"/>
              and the ſixt part of A V. </s>
              <s>And in the ſame manner the Cylinders of
                <lb/>
              the circumſcribed Figure may be demonſtrated to be equally exceeding
                <lb/>
              one another, and the exceſs to me equal to the least; and that they have
                <lb/>
              their Centers of Gravity at equal diſtances in the Line
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              M. </s>
              <s>If therefore
                <emph.end type="italics"/>
                <lb/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              M be divided in
                <emph.end type="italics"/>
                <foreign lang="grc">π,</foreign>
                <emph type="italics"/>
              ſo as that
                <emph.end type="italics"/>
                <foreign lang="grc">ε π</foreign>
                <emph type="italics"/>
              be double to the remaining part
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              M;
                <emph.end type="italics"/>
                <lb/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              ſhall be the Center of Gravity of the whole circumſcribed Magnitude.
                <lb/>
              </s>
              <s>And ſince
                <emph.end type="italics"/>
                <foreign lang="grc">ε π</foreign>
                <emph type="italics"/>
              is double to
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              M; and A
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              leſs than double EM: (for
                <lb/>
              that they are equal:) the whole A E ſhall be leſs than triple E
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              : Where­
                <lb/>
              fore E
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              ſhall be greater than E N. And, ſince
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              M is triple to M
                <emph.end type="italics"/>
                <foreign lang="grc">π,</foreign>
                <lb/>
                <emph type="italics"/>
              and M E with twice
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              A is likewiſe triple to M E: the whole A E with
                <lb/>
              A
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              ſhall be triple to E
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              : But A E is triple to E N: Wherefore the
                <lb/>
              remaining part A
                <emph.end type="italics"/>
                <foreign lang="grc">ε</foreign>
                <emph type="italics"/>
              ſhall be triple to the remaining part
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              N. </s>
              <s>Therefore
                <lb/>
              N
                <emph.end type="italics"/>
                <foreign lang="grc">π</foreign>
                <emph type="italics"/>
              is the ſixth part of A V. </s>
              <s>And theſe are the things that were to be
                <lb/>
              demonſtrated.
                <emph.end type="italics"/>
              </s>
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