* From or after
the Fall A B.
the Fall A B.
THEOR. XIII. PROP. XVI.
If the parts of an inclined Plane and Perpendicu
lar, the Times of whoſe Motions ex quiete are
equal, be joyned together at the ſame point, a
Moveable coming out of any ſublimer Height
ſhall ſooner paſſe the ſaid part of the inclined
Plane, than that part of the Perpendicular.
lar, the Times of whoſe Motions ex quiete are
equal, be joyned together at the ſame point, a
Moveable coming out of any ſublimer Height
ſhall ſooner paſſe the ſaid part of the inclined
Plane, than that part of the Perpendicular.
Let the Perpendicular be E B, and the Inclined Plane C E, joyned
at the ſame Point E, the Times of whoſe Motions from off Reſt in
E are equal, and in the Perpendicular continued out, let a ſublime
point A be taken at pleaſure, out of which the Moveables may be let
fall. I ſay, that the Inclined Plane E C ſhall be paſſed in a leſſe Time
than the Perpendicular E B, after the Fall A E. Draw a Line from C
to B, and having drawn the Horizontal Line A D continue out C E till
it meet the ſame in D; and let D F be a Mean-Proportional between
C D and D E; and let A G be a
108[Figure 108]
Mean-Proportional between B A and
A E; and draw F G and D G. And
becauſe the Time of the Motion along
E C and E B out of Reſt in E are
equal, the Angle C ſhall be a Right
Angle, by the ſecond Corollary of the
Sixth Propoſition; and A is a Right
Angle, and the Vertical Angles
at E are equal: Therefore the Tri
angles A E D and C E B are equian
gled, and the Sides about equal An
gles are Proportionals: Therefore as
B E is to E C, ſo is D E to E A.
Therefore the Rectangle B E A is
equal to the Rectangle C E D: And
becauſe the Rectangle C D E ex
ceedeth the Rectangle C E D, by the Square E D, and the Rectangle
B A E doth exceed the Rectangle B E A, by the Square E A: The
exceſſe of the Rectangle C D E above the Rectangle B A E, that is of
the Square F D above the Square A G ſhall be the ſame as the exceſſe
of the Square D E above the Square A E; which exceſs is the
Square D A: Therefore the Square F D is equal to the two Squares
G A and A D, to which the Square G D is alſo equal: Therefore the
at the ſame Point E, the Times of whoſe Motions from off Reſt in
E are equal, and in the Perpendicular continued out, let a ſublime
point A be taken at pleaſure, out of which the Moveables may be let
fall. I ſay, that the Inclined Plane E C ſhall be paſſed in a leſſe Time
than the Perpendicular E B, after the Fall A E. Draw a Line from C
to B, and having drawn the Horizontal Line A D continue out C E till
it meet the ſame in D; and let D F be a Mean-Proportional between
C D and D E; and let A G be a
108[Figure 108]Mean-Proportional between B A and
A E; and draw F G and D G. And
becauſe the Time of the Motion along
E C and E B out of Reſt in E are
equal, the Angle C ſhall be a Right
Angle, by the ſecond Corollary of the
Sixth Propoſition; and A is a Right
Angle, and the Vertical Angles
at E are equal: Therefore the Tri
angles A E D and C E B are equian
gled, and the Sides about equal An
gles are Proportionals: Therefore as
B E is to E C, ſo is D E to E A.
Therefore the Rectangle B E A is
equal to the Rectangle C E D: And
becauſe the Rectangle C D E ex
ceedeth the Rectangle C E D, by the Square E D, and the Rectangle
B A E doth exceed the Rectangle B E A, by the Square E A: The
exceſſe of the Rectangle C D E above the Rectangle B A E, that is of
the Square F D above the Square A G ſhall be the ſame as the exceſſe
of the Square D E above the Square A E; which exceſs is the
Square D A: Therefore the Square F D is equal to the two Squares
G A and A D, to which the Square G D is alſo equal: Therefore the