1to R E: Therefore C R, R E, and R A are Proportionals. Farther
more, becauſe as B A is to A C, ſo E A is ſuppoſed to be to A R, and,
106[Figure 106]
in regard of the likeneſſe of the Triangles,
as B A is to A C, ſo is X A to A R: There
fore, as E A is to A R, ſo is X A to A R:
Therefore E A and X A are equal. Now if
we underſtand the Time along R A to be as
R A, the Time along R C ſhall be R E, the
Mean-Proportional between C R and R A:
And A E ſhall be the Time along A C after
R A or after X A: But the Time along X A
is X A, ſo long as R A is the Time along R
A: But it hath been proved that X A and
A E are equal: Therefore the Propoſition is proved.
more, becauſe as B A is to A C, ſo E A is ſuppoſed to be to A R, and,
106[Figure 106]in regard of the likeneſſe of the Triangles,
as B A is to A C, ſo is X A to A R: There
fore, as E A is to A R, ſo is X A to A R:
Therefore E A and X A are equal. Now if
we underſtand the Time along R A to be as
R A, the Time along R C ſhall be R E, the
Mean-Proportional between C R and R A:
And A E ſhall be the Time along A C after
R A or after X A: But the Time along X A
is X A, ſo long as R A is the Time along R
A: But it hath been proved that X A and
A E are equal: Therefore the Propoſition is proved.
PROBL. III. PROP. XV.
A Perpendicular and a Plane inflected to it being
given, to find a part in the Perpendicular ex
tended downwards which ſhall be paſſed in the
ſame. Time as the inflected Plane after the Fall
along the given Perpendicular.
given, to find a part in the Perpendicular ex
tended downwards which ſhall be paſſed in the
ſame. Time as the inflected Plane after the Fall
along the given Perpendicular.
Let the Perpendicular be A B, and the Plane Inſlected to it B C. It
is required in the Perpendicular extended downwards to find a
part which from the Fall out of A ſhall be paſt in the ſame Time as
B C is paſſed from the ſame Fall out of A. Draw the Horizontal Line
A D, with which let C B meet extended to D; and let D E be a Mean
proportional between C D and D B;
107[Figure 107]
and let B F be equal to B E; and let
A G be a third Proportional to B A and
A F. I ſay, B G is the Space that after
the Fall A B ſhall be paſt in the ſame
Time, as the Plane B C ſhall be paſt af
ter the ſame Fall. For if we ſuppoſe
the Time along A B to be as A B, the
Time along D B ſhall be as D B: And
becauſe D E is the Mean-proportional
between B D and D C, the ſame D E
ſhall be the Time along the whole D C, and B E the Time along the Part
or Remainder B C ex quiete, in D, or ^{*} ex caſu A B: And it may in
like manner be proved, that B F is the Time along B G, after the ſame
Fall: But B F is equal to B E: Which was the Propoſition to be proved.
is required in the Perpendicular extended downwards to find a
part which from the Fall out of A ſhall be paſt in the ſame Time as
B C is paſſed from the ſame Fall out of A. Draw the Horizontal Line
A D, with which let C B meet extended to D; and let D E be a Mean
proportional between C D and D B;
107[Figure 107]and let B F be equal to B E; and let
A G be a third Proportional to B A and
A F. I ſay, B G is the Space that after
the Fall A B ſhall be paſt in the ſame
Time, as the Plane B C ſhall be paſt af
ter the ſame Fall. For if we ſuppoſe
the Time along A B to be as A B, the
Time along D B ſhall be as D B: And
becauſe D E is the Mean-proportional
between B D and D C, the ſame D E
ſhall be the Time along the whole D C, and B E the Time along the Part
or Remainder B C ex quiete, in D, or ^{*} ex caſu A B: And it may in
like manner be proved, that B F is the Time along B G, after the ſame
Fall: But B F is equal to B E: Which was the Propoſition to be proved.
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