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

Table of figures

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              <s>
                <pb xlink:href="040/01/952.jpg" pagenum="259"/>
                <emph type="italics"/>
              L M, V C: they ſhall be ordinately applyed. </s>
              <s>Divide therefore E B in­
                <lb/>
              to three equal parts, of which let the middlemoſt be Q Y: and divide
                <lb/>
              this ſo in the point I that Q I may have the ſame proportion to I Y, as
                <lb/>
              the Baſe whoſe Diameter is V C hath to the Baſe whoſe Diameter is
                <lb/>
              L M; that is, that the Square V C hath to Square L M. </s>
              <s>It is to be de­
                <lb/>
              monſtrated that I is the Center of Gravity of the Fruſtrum L M C.
                <lb/>
              </s>
              <s>Draw the Line N S, by the by, equall to B R: and let S X be equal to
                <lb/>
              E R: and unto N S and S X aſſume a third proportional S G: and as
                <lb/>
              N G is to G S, ſo let B Q be to I O. </s>
              <s>And it nothing matters whether
                <lb/>
              the point O fall above or below L M. </s>
              <s>And becauſe in the Section V R C
                <lb/>
              the Lines L M and V C are ordinately
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.952.1.jpg" xlink:href="040/01/952/1.jpg" number="172"/>
                <lb/>
                <emph type="italics"/>
              applyed, it ſhall be that as the Square
                <lb/>
              V C is to the Square L M, ſo is the Line
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              B R to R E: And as the Square V C is
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              to the Square L M, ſo is Q I to I Y: and
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              as B R is to R E, ſo is N S to S X: There­
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              fore Q I is to I Y, as R S is to S X. </s>
              <s>Where­
                <lb/>
              fore as G Y is to Y I, ſo ſhall both N S and
                <lb/>
              S X be to S X: and as E B is to Y I, ſo
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              ſhall the compound of triple N S and tri­
                <lb/>
              ple S X be to S X: But as E B is to B Y,
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              ſo is the compound of triple N S and S X
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              both together to the compound of N S and S X: Therefore, as E B is to
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              B I, ſo is the compound of triple N S and triple S X to the compound of
                <lb/>
              N S and double S X. </s>
              <s>Therefore N S, S X, and S G are three proporti­
                <lb/>
              onal Lines: And as S G is to G N, ſo is the aſſumed O I to two thirds
                <lb/>
              of E B; that is, to N X: And as the compound of N S and double
                <lb/>
              S X is to the compound of triple N S and triple S X, ſo is another aſſu­
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              med Line I B to B E; that is, to N X. </s>
              <s>By what therefore hath been
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              above demonſtrated, thoſe Lines taken together are a third part of N S;
                <lb/>
              that is, of R B: Therefore R B is triple to B O: Wherefore O ſhall
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              be the Center of Gravity of the Conoid v R C. </s>
              <s>And let it be the Cen­
                <lb/>
              ter of Gravity of the
                <emph.end type="italics"/>
              Fruſtrum
                <emph type="italics"/>
              L R M of the Conoid: Therefore the
                <lb/>
              Center of Gravity of V L M C is in the Line O B, and in that point
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              which ſo terminates it, that as V L M C of the
                <emph.end type="italics"/>
              Fruſtrum
                <emph type="italics"/>
              is to the
                <lb/>
              proportion L R M, ſo is the Line A O to that which intervenes betwixt
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              O and the ſaid point. </s>
              <s>And becauſe R O is two thirds of R B; and
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              R A two thirds of R E; the remaining part A O ſhall be two thirds
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              of the remaining part E B. </s>
              <s>And becauſe that as the
                <emph.end type="italics"/>
              Fruſtum
                <emph type="italics"/>
              V L M C
                <lb/>
              is to the proportion L R M, ſo is N G to G S: and as N G to G S, ſo is
                <lb/>
              two thirds of E B to O I: and two thirds of E B is equal to the Line
                <lb/>
              A O: it ſhall be that as the
                <emph.end type="italics"/>
              Fruſtum
                <emph type="italics"/>
              V L M O is to the proportion
                <lb/>
              L R M, ſo is A O to O I. </s>
              <s>It is manifeſt therefore that of the
                <emph.end type="italics"/>
              Fruſtum
                <lb/>
                <emph type="italics"/>
              V L M C the Center of Gravity is the point I, and ſo divideth the Axis,
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              [as?] that the part towards the leſſer Baſe is to the part towards the grea-
                <emph.end type="italics"/>
              </s>
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