1the Length B D is to B C. Let A F be drawn parallel to the Ho
rizon, to which continue out D B, meeting it in F; and let F E be
a Mean-proportional between D F and F B; and draw E O parallel
to D C, and A O ſhall be a Mean-proportional between C A and
A B: But if we ſuppoſe the Time
along A B, to be as A B, the Time a-
101[Figure 101]
long F B ſhall be as F B. And the
Time along all A C, ſhall be as the
Mean-proportional A O; and along
all F D ſhall be F E: Wherefore the
Time along the remainder B C ſhall
be B O; and along the remainder
B D ſhall be B E. But as B E is to
B O, ſo is B D to B C: Therefore
the Times along B D and B C, after the Deſcent along A B and
F B, or which is the ſame, along the Common part A B, ſhall be to
one another as the Lengths B D and B C: But that the Time along
B D, is to the Time along B C, out of Reſt in B, as the Length
B D to B C, hath already been demonſtrated. Therefore the Times
of the Motions along different Planes whoſe Elevations are equal, are
to one another as the Lengths of the ſaid Planes, whether the Motion
be made along the ſame out of Reſt, or whether another Motion of
the ſame Altitude do precede thoſe Motions: Which was to be de
monſtrated.
rizon, to which continue out D B, meeting it in F; and let F E be
a Mean-proportional between D F and F B; and draw E O parallel
to D C, and A O ſhall be a Mean-proportional between C A and
A B: But if we ſuppoſe the Time
along A B, to be as A B, the Time a-
101[Figure 101]long F B ſhall be as F B. And the
Time along all A C, ſhall be as the
Mean-proportional A O; and along
all F D ſhall be F E: Wherefore the
Time along the remainder B C ſhall
be B O; and along the remainder
B D ſhall be B E. But as B E is to
B O, ſo is B D to B C: Therefore
the Times along B D and B C, after the Deſcent along A B and
F B, or which is the ſame, along the Common part A B, ſhall be to
one another as the Lengths B D and B C: But that the Time along
B D, is to the Time along B C, out of Reſt in B, as the Length
B D to B C, hath already been demonſtrated. Therefore the Times
of the Motions along different Planes whoſe Elevations are equal, are
to one another as the Lengths of the ſaid Planes, whether the Motion
be made along the ſame out of Reſt, or whether another Motion of
the ſame Altitude do precede thoſe Motions: Which was to be de
monſtrated.
THEOR. XI. PROP. XI.
If a Plane, along which a Motion is made out of
Reſt, be divided at pleaſure, the Time of
the Motion along the firſt part, is to the Time
of the Motion along the ſecond, as the ſaid
firſt part is to the exceſſe whereby the ſame
part ſhall be exceeded by the Mean-Propor
tional between the whole Plane and the ſame
firſt part.
Reſt, be divided at pleaſure, the Time of
the Motion along the firſt part, is to the Time
of the Motion along the ſecond, as the ſaid
firſt part is to the exceſſe whereby the ſame
part ſhall be exceeded by the Mean-Propor
tional between the whole Plane and the ſame
firſt part.
Let the Motion be along the whole Plane A B, ex quiete in A,
which let be divided at pleaſure in C; and let A F be a Mean
proportional between the whole B A and the firſt part A C;
C F ſhall be the exceſſe of the Mean proportional F A above the part
A C. I ſay the Time of the Motion along A C is to the Time of the
following Motion along C B, as A C to C F. Which is manifeſt;
which let be divided at pleaſure in C; and let A F be a Mean
proportional between the whole B A and the firſt part A C;
C F ſhall be the exceſſe of the Mean proportional F A above the part
A C. I ſay the Time of the Motion along A C is to the Time of the
following Motion along C B, as A C to C F. Which is manifeſt;