Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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L M, V C: they ſhall be ordinately applyed. </
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<
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>Divide therefore E B in
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to three equal parts, of which let the middlemoſt be Q Y: and divide
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this ſo in the point I that Q I may have the ſame proportion to I Y, as
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the Baſe whoſe Diameter is V C hath to the Baſe whoſe Diameter is
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L M; that is, that the Square V C hath to Square L M. </
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>It is to be de
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monſtrated that I is the Center of Gravity of the Fruſtrum L M C.
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>Draw the Line N S, by the by, equall to B R: and let S X be equal to
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E R: and unto N S and S X aſſume a third proportional S G: and as
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N G is to G S, ſo let B Q be to I O. </
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>And it nothing matters whether
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the point O fall above or below L M. </
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>And becauſe in the Section V R C
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the Lines L M and V C are ordinately
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applyed, it ſhall be that as the Square
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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. </
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<
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>Where
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fore as G Y is to Y I, ſo ſhall both N S and
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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
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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
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N S and double S X. </
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>Therefore N S, S X, and S G are three proporti
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onal Lines: And as S G is to G N, ſo is the aſſumed O I to two thirds
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of E B; that is, to N X: And as the compound of N S and double
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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. </
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<
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>By what therefore hath been
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above demonſtrated, thoſe Lines taken together are a third part of N S;
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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. </
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>And let it be the Cen
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ter of Gravity of the
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Fruſtrum
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L R M of the Conoid: Therefore the
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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
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Fruſtrum
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is to the
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proportion L R M, ſo is the Line A O to that which intervenes betwixt
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O and the ſaid point. </
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>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. </
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<
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>And becauſe that as the
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Fruſtum
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V L M C
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is to the proportion L R M, ſo is N G to G S: and as N G to G S, ſo is
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two thirds of E B to O I: and two thirds of E B is equal to the Line
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A O: it ſhall be that as the
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Fruſtum
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V L M O is to the proportion
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L R M, ſo is A O to O I. </
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<
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>It is manifeſt therefore that of the
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Fruſtum
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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-
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