Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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Let the Altitude G F of the Parabola F H have the ſame proporti
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on to the Altitude C B of the Parabola B D, as the Sublimity B A
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hath to the Sublimity F E. </
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>I ſay, that the Amplitude H G is equal
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to the Amplitude D C. </
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>For ſince the firſt G F hath the ſame propor
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tion to the ſecond C B, as the third B A hath to the fourth F E; There
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fore, the Rectangle
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G F E of the firſt and
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fourth, ſhall be equal to
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the Rectangle C B A
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of the ſecond and
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third: Therefore the
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Squares that are equal
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to theſe Rectangles ſhall
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be equal to one another:
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But the Square of half of G H is equal to the Rectangle G F E; and
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the Square of half of C D is equal to the Rectangle C B A: There
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fore theſe Squares, and their Sides, and the doubles of their Sides ſhall
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be equal: But theſe are the Amplitudes G H and C D: Therefore the
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Propoſition is manifeſt.
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<
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>LEMMA
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pro ſequenti.
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>If a Right Line be cut according to any proportion, the Squares
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of the Mean-proportionals between the whole and the two
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parts are equal to the Square of the whole.</
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Let A B be cut according to any proportion in C. </
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>I ſay, that the
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Squares of the Mean-proportional Lines between the whole A B and
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the parts A C and C B, being taken together are equal to the Square of
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the whole A B. </
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<
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>And this appeareth, a Semi-
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circle being deſcribed upon the whole Line
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B A, and from C a Perpendicular being ere
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cted C D, and Lines being drawn from D to
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A, and from D to B. </
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<
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>For D A is the Mean
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proportional betwixt A B and A C; and D B is the Mean-proporti
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onal between A B and B C: And the Squares of the Lines D A and
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D B taken together are equal to the Square of the whole Line A B,
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the Angle A D B in the Semicircle being a Right-Angle: Therefore
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the Propoſition is manifest.
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