Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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For let A B, B C, B D, and B E be four proportional Lines. </
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<
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>And
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as B E is to E A, ſo let F G be to 3/4 of A C. </
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<
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>And as the Line equal
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to A B and to double B C and to triple B D is to the Line equal
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to the quadruples of A B, B C, and B D, ſo let H G be to A C. </
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<
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>It is
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to be proved, that H F is a fourth part of A B. </
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>Foraſmuch therefore
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as A B, B C, B D, and B E
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are proportionals, A C,
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C D, and D E ſhall be in
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the ſame proportion: And
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as the quadruple of the ſaid
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A B, B C, and B D is to
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A B with the double of B C and triple of B D, ſo is the quadruple of
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A C, C D, and D E; that is, the quadruple of A E; to A C with the
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double of C D, and triple of D E. </
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<
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>And ſo is A C to H G. </
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<
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>Therefore
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as the triple of A E is to A C, with the double of C D and triple of
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D E, ſo is 3/4 of A C to H G. </
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>And as the triple of A E is to the triple of
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E B, ſo is 3/4 A C to G F: Therefore, by the Converſe of the twenty
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fourth of the fifth, As triple A E is to A C with double C D and tri
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ple D B, ſo is 3/4 of A C to H F: And as the quadruple of A E is to A C
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with the double of C D and triple of D B; that is, to A B with C B and
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B D, ſo is A C to H F. And, by Permutation, as the quadruple of A E
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is to A C, ſo is A B with C B and B D to H F. </
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>And as A C is to A E, ſo
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is A B to A B with C B and B D. Therefore,
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ex æquali,
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by Perturbed
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proportion, as quadruple A E is to A E, ſo is A B to H F. </
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>Wherefore it
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is manifeſt that H F is the fourth part of A B.
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<
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>PROPOSITION.</
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<
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>The Center of Gravity of the
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Fruſtum
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of any Py
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ramid or Cone, cut equidiſtant to the Plane
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of the Baſe, is in the Axis, and doth ſo divide
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the ſame, that the part towards the leſſer Baſe
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is to the remainder, as the triple of the greater
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Baſe, with the double of the mean Space be
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twixt the greater and leſſer Baſe, together
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with the leſſer Baſe is to the triple of the leſſer
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Baſe, together with the ſame double of the
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mean Space, as alſo of the greater Baſe.</
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