PROBL. II. PROP. V.
In the Axis of a given Parabola prolonged to find
a ſublime point out of which the Moveable
falling ſhall deſcribe the ſaid Parabola.
a ſublime point out of which the Moveable
falling ſhall deſcribe the ſaid Parabola.
Let the Parabola be A B, its Amplitude H B, and its prolonged
Axis H E; in which a Sublimity is to be found, out of which the
Moveable falling, and converting the Impetus conceived in A
along the Horizontal Line, deſcribeth the Parabola A B. Draw the
Horizontal Line A G, which ſhall be Parallel to B H, and ſuppoſing A F
equal to A H draw the Right Line F B, which toucheth the Parabola in
B, and cutteth the Horizontal Line A G in G; and unto F A and A G
let A E be a third Proportional. I ſay, that E is the ſublime Point re
quired, out of which the Moveable falling ex quiete in E, and the Im
petus conceived in A being converted along the Horizontal Line over
taking the Impetus of the Deſcent
154[Figure 154]
in H ex quiete in A, deſcribeth the
Parabola A B. For if we ſuppoſe
E A to be the Meaſure of the Time
of the Fall from E to A, and of
the Impetus acquired in A, A G
(that is a Mean-proportional be
tween E A and A F) ſhall be the
Time and the Impetus coming
from F to A, or from A to H. And
becauſe the Moveable coming out of
E in the Time E A with the Impetus acquired in A paſſeth in the Ho
rizontal Lation with an Equable Motion the double of E A; There
fore likewiſe moving with the ſame Impetus it ſhall in the Time A G
paſs the double of G A, to wit, the Mean-proportional B H (for the
Spaces paſſed with the ſame Equable Motion are to one another as the
Times of the ſaid Motions:) And along the Perpendicular A H ſhall
be paſſed with a Motion ex quiete in the ſame Time G A: Therefore
the Amplitude H B, and Altitude A H are paſſed by the Moveable in the
ſame Time: Therefore the Parabola A B ſhall be deſcribed by the
Deſcent of the Project coming from the Sublimity E: Which was re
quired.
Axis H E; in which a Sublimity is to be found, out of which the
Moveable falling, and converting the Impetus conceived in A
along the Horizontal Line, deſcribeth the Parabola A B. Draw the
Horizontal Line A G, which ſhall be Parallel to B H, and ſuppoſing A F
equal to A H draw the Right Line F B, which toucheth the Parabola in
B, and cutteth the Horizontal Line A G in G; and unto F A and A G
let A E be a third Proportional. I ſay, that E is the ſublime Point re
quired, out of which the Moveable falling ex quiete in E, and the Im
petus conceived in A being converted along the Horizontal Line over
taking the Impetus of the Deſcent
154[Figure 154]in H ex quiete in A, deſcribeth the
Parabola A B. For if we ſuppoſe
E A to be the Meaſure of the Time
of the Fall from E to A, and of
the Impetus acquired in A, A G
(that is a Mean-proportional be
tween E A and A F) ſhall be the
Time and the Impetus coming
from F to A, or from A to H. And
becauſe the Moveable coming out of
E in the Time E A with the Impetus acquired in A paſſeth in the Ho
rizontal Lation with an Equable Motion the double of E A; There
fore likewiſe moving with the ſame Impetus it ſhall in the Time A G
paſs the double of G A, to wit, the Mean-proportional B H (for the
Spaces paſſed with the ſame Equable Motion are to one another as the
Times of the ſaid Motions:) And along the Perpendicular A H ſhall
be paſſed with a Motion ex quiete in the ſame Time G A: Therefore
the Amplitude H B, and Altitude A H are paſſed by the Moveable in the
ſame Time: Therefore the Parabola A B ſhall be deſcribed by the
Deſcent of the Project coming from the Sublimity E: Which was re
quired.
COROLLARY.
Hence it appeareth that the half of the Baſe or Amplitude of the
Semiparabola (which is the fourth part of the Amplitude of
the whole Parabola) is a Mean-proportional betwixt its Al
titude and the Sublimity out of which the Moveable falling
deſcribeth it.
Semiparabola (which is the fourth part of the Amplitude of
the whole Parabola) is a Mean-proportional betwixt its Al
titude and the Sublimity out of which the Moveable falling
deſcribeth it.
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