1And ſince it was ſuppoſed that the Time along A C is as A C, C E ſhall
be the Time along E C; and E F the Time along the whole E O; and
the part C F that along the part C O: But C F is equal to the ſaid C A:
Therefore that is done which was required: For the Time C A is the
Time of the Fall along A C ex quiete in A; and C F (which is equal
to C A) is the Time along C O, after the Deſcent along E C, or after
the Fall along A C: Which was the Propoſition.
be the Time along E C; and E F the Time along the whole E O; and
the part C F that along the part C O: But C F is equal to the ſaid C A:
Therefore that is done which was required: For the Time C A is the
Time of the Fall along A C ex quiete in A; and C F (which is equal
to C A) is the Time along C O, after the Deſcent along E C, or after
the Fall along A C: Which was the Propoſition.
And here it is to be noted, that the ſame may happen if the preceding
Motion be not made along the Perpendicular, but along an Inclined Plane:
As in the following Figure, in which let the preceding Lation be made
along the inclined Plane A S beneath the Horizon A E: And the Demon
ſtration is the very ſame.
Motion be not made along the Perpendicular, but along an Inclined Plane:
As in the following Figure, in which let the preceding Lation be made
along the inclined Plane A S beneath the Horizon A E: And the Demon
ſtration is the very ſame.
SCHOLIUM.
If one obſerve well, it ſhall be manifeſt, that the leſſe the given
Line I R wanteth of being triple to the ſaid A C, the nearer
ſhall the Inflected Plane, along which the ſecond Motion is
to be made, which ſuppoſe to be C O, come to the Perpen
dicular, along which in a Time equal to A C a Space ſhall
be paſſed triple to A C.
Line I R wanteth of being triple to the ſaid A C, the nearer
ſhall the Inflected Plane, along which the ſecond Motion is
to be made, which ſuppoſe to be C O, come to the Perpen
dicular, along which in a Time equal to A C a Space ſhall
be paſſed triple to A C.
For in caſe I R were very near the triple of A C, I M ſhould be well
near equal to M N: And if, as I M is to M N by Conſtruction, ſo
A C is to C E, then it is evident that the ſaid C E will be found but
little bigger than C A, and, which followeth of conſequence, the point E
ſhall be found very near the point A, and C O to containe a very acute
118[Figure 118]
Angle with C S, and
almoſt to concur both in
one Line. And on the
contrary, if the ſaid I R
were but a very little
more than double the
ſaid A C, I M ſhould
be a very ſhort Line.
Hence it may happen
alſo that A C may come
to be very ſhort in reſpect of C E which ſhall be very long, and ſhall ap
proach very near the Horizontal Parallel drawn from C. And from
hence we may collect, that if in the preſent Figure after the Deſcent along
the inclined Plane A C, a Reflexion be made along the Horizontal Line,
as v. gr. C T, the Space along which the Moveable afterwards moved
in a Time equal to the Time of the Deſcent along A C would be exactly
double to the Space A C. And it appears that the like Diſcourſe may be
here applied: For it is apparent by what hath been ſaid, that ſince O E
near equal to M N: And if, as I M is to M N by Conſtruction, ſo
A C is to C E, then it is evident that the ſaid C E will be found but
little bigger than C A, and, which followeth of conſequence, the point E
ſhall be found very near the point A, and C O to containe a very acute
118[Figure 118]Angle with C S, and
almoſt to concur both in
one Line. And on the
contrary, if the ſaid I R
were but a very little
more than double the
ſaid A C, I M ſhould
be a very ſhort Line.
Hence it may happen
alſo that A C may come
to be very ſhort in reſpect of C E which ſhall be very long, and ſhall ap
proach very near the Horizontal Parallel drawn from C. And from
hence we may collect, that if in the preſent Figure after the Deſcent along
the inclined Plane A C, a Reflexion be made along the Horizontal Line,
as v. gr. C T, the Space along which the Moveable afterwards moved
in a Time equal to the Time of the Deſcent along A C would be exactly
double to the Space A C. And it appears that the like Diſcourſe may be
here applied: For it is apparent by what hath been ſaid, that ſince O E