Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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<
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>PROPOSITION.</
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<
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>If to any Cone or portion of a Cone a Eigure con
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ſiſting of Cylinders of equal heights be inſcri
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bed and another circumſcribed; and if its Axis
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be ſo divided as that the part which lyeth be
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twixt the point of diviſion and the Vertex be
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triple to the reſt; the Center of Gravity of
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the inſcribed Figure ſhall be nearer to the Baſe
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of the Cone than that point of diviſion: and
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the Center of Gravity of the circumſcribed
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ſhall be nearer to the Vertex than that ſame
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point.</
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Take therefore a Cone, whoſe Axis is N M. </
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<
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>Let it be divided
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in S ſo, as that N S be triple to the remainder S M. </
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<
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>I ſay, that
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the Center of Gravity of any Figure inſcribed, as was ſaid, in
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a Cone doth conſiſt in the Axis N M, and approacheth nearer to the Baſe
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of the Cone than the point S: and that the Center of Gravity of the
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Circumſcribed is likewiſe in the Axis N M, and nearer to the Vertex
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than is S. </
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<
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>Let a Figure therefore be ſuppoſed to be inſcribed by the Cy
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linders whoſe Axis M C, C B, B E, E A are equal. </
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<
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>Firſt therefore
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the Cylinder whoſe Axis is M C hath
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to the Cylinder whoſe Axis is C B the
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ſame proportion as its Baſe hath to
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the Baſe of the other (for their Alti
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tudes are equal.) But this propor
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tion is the ſame with that which the
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Square C N hath to the Square N B.
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</
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<
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>And ſo we might prove, that the Cy
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linder whoſe Axis is C B hath to the
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Cylinder whoſe Axis is B E the ſame
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proportion, as the Square B N hath to
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the Square N E: and the Cylinder
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whoſe Axis is B E hath to the Cylin
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der whoſe Axis is E A the ſame pro
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portion that the Square E N hath to
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the Square N A. </
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<
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>But the Lines N C,
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N B, E N, and N A equally exceed one
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another, and their exceſs equalleth the
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leaſt, that is N A. </
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<
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>Therefore they are certain Magnitudes, to wit, in
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ſcribed Cylinders having conſequently to one another the ſame proporti
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on as the Squares of Lines that equally exceed one another, and the ex-
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</
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