Let the Altitude G F of the Parabola F H have the ſame proporti
on to the Altitude C B of the Parabola B D, as the Sublimity B A
hath to the Sublimity F E. I ſay, that the Amplitude H G is equal
to the Amplitude D C. For ſince the firſt G F hath the ſame propor
tion to the ſecond C B, as the third B A hath to the fourth F E; There
fore, the Rectangle
158[Figure 158]
G F E of the firſt and
fourth, ſhall be equal to
the Rectangle C B A
of the ſecond and
third: Therefore the
Squares that are equal
to theſe Rectangles ſhall
be equal to one another:
But the Square of half of G H is equal to the Rectangle G F E; and
the Square of half of C D is equal to the Rectangle C B A: There
fore theſe Squares, and their Sides, and the doubles of their Sides ſhall
be equal: But theſe are the Amplitudes G H and C D: Therefore the
Propoſition is manifeſt.
on to the Altitude C B of the Parabola B D, as the Sublimity B A
hath to the Sublimity F E. I ſay, that the Amplitude H G is equal
to the Amplitude D C. For ſince the firſt G F hath the ſame propor
tion to the ſecond C B, as the third B A hath to the fourth F E; There
fore, the Rectangle
158[Figure 158]G F E of the firſt and
fourth, ſhall be equal to
the Rectangle C B A
of the ſecond and
third: Therefore the
Squares that are equal
to theſe Rectangles ſhall
be equal to one another:
But the Square of half of G H is equal to the Rectangle G F E; and
the Square of half of C D is equal to the Rectangle C B A: There
fore theſe Squares, and their Sides, and the doubles of their Sides ſhall
be equal: But theſe are the Amplitudes G H and C D: Therefore the
Propoſition is manifeſt.
LEMMA pro ſequenti.
If a Right Line be cut according to any proportion, the Squares
of the Mean-proportionals between the whole and the two
parts are equal to the Square of the whole.
of the Mean-proportionals between the whole and the two
parts are equal to the Square of the whole.
Let A B be cut according to any proportion in C. I ſay, that the
Squares of the Mean-proportional Lines between the whole A B and
the parts A C and C B, being taken together are equal to the Square of
the whole A B. And this appeareth, a Semi-
159[Figure 159]
circle being deſcribed upon the whole Line
B A, and from C a Perpendicular being ere
cted C D, and Lines being drawn from D to
A, and from D to B. For D A is the Mean
proportional betwixt A B and A C; and D B is the Mean-proporti
onal between A B and B C: And the Squares of the Lines D A and
D B taken together are equal to the Square of the whole Line A B,
the Angle A D B in the Semicircle being a Right-Angle: Therefore
the Propoſition is manifest.
Squares of the Mean-proportional Lines between the whole A B and
the parts A C and C B, being taken together are equal to the Square of
the whole A B. And this appeareth, a Semi-
159[Figure 159]circle being deſcribed upon the whole Line
B A, and from C a Perpendicular being ere
cted C D, and Lines being drawn from D to
A, and from D to B. For D A is the Mean
proportional betwixt A B and A C; and D B is the Mean-proporti
onal between A B and B C: And the Squares of the Lines D A and
D B taken together are equal to the Square of the whole Line A B,
the Angle A D B in the Semicircle being a Right-Angle: Therefore
the Propoſition is manifest.
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