1For the Time along A C is to the Time along all
A B, as A C to the Mean-proportional A F: There-
102[Figure 102]
fore, by Diviſion, the Time along A C, ſhall be to
the Time along the remainder C B as A C to C F:
If therefore the Time along A C be ſuppoſed to be
the ſaid A C, the Time along C B ſhall be C F:
Which was the Propoſition.
A B, as A C to the Mean-proportional A F: There-
102[Figure 102]fore, by Diviſion, the Time along A C, ſhall be to
the Time along the remainder C B as A C to C F:
If therefore the Time along A C be ſuppoſed to be
the ſaid A C, the Time along C B ſhall be C F:
Which was the Propoſition.
But if the Motion be not made along the continu
ate Plane A C B, but by the inflected Plane A C D
until it come to the Horizon B D, to which from F a Parallel is
drawn F E. It ſhall in like manner be
103[Figure 103]
demonſtrated, that the Time along
A C is to the Time along the reflected
Plane C D, as A C is to C E. For
the Time along A C is to the Time a
long C B, as A C is to C F: But the
Time along C B, after A C hath been
demonſtrated to be to the Time along
C D, after the ſaid Deſoent along
A C, as C B is to C D; that is, as
C F to C E: Therefore, ex æquali, the Time along A C ſhall be to
the Time along C D, as the Line A C to C E.
ate Plane A C B, but by the inflected Plane A C D
until it come to the Horizon B D, to which from F a Parallel is
drawn F E. It ſhall in like manner be
103[Figure 103]demonſtrated, that the Time along
A C is to the Time along the reflected
Plane C D, as A C is to C E. For
the Time along A C is to the Time a
long C B, as A C is to C F: But the
Time along C B, after A C hath been
demonſtrated to be to the Time along
C D, after the ſaid Deſoent along
A C, as C B is to C D; that is, as
C F to C E: Therefore, ex æquali, the Time along A C ſhall be to
the Time along C D, as the Line A C to C E.
THEOR. XII. PROP. XII.
If the Perpendicular and Plane Inclined at plea
ſure, be cut between the ſame Horizontal
Lines, and Mean-Proportionals between
them and the parts of them contained betwixt
the common Section and upper Horizontal
Line be given; the Time of the Motion a
long the Perpendicular ſhall have the ſame
proportion to the Time of the Motion along
the upper part of the Perpendicular, and af
terwards along the lower part of the interſe
cted Plane, as the Length of the whole Per
pendicular hath to the Line compounded of
the Mean-Proportional given upon the Per
pendicular, and of the exceſſe by which the
whole Plane exceeds its Mean-Proporttonal.
ſure, be cut between the ſame Horizontal
Lines, and Mean-Proportionals between
them and the parts of them contained betwixt
the common Section and upper Horizontal
Line be given; the Time of the Motion a
long the Perpendicular ſhall have the ſame
proportion to the Time of the Motion along
the upper part of the Perpendicular, and af
terwards along the lower part of the interſe
cted Plane, as the Length of the whole Per
pendicular hath to the Line compounded of
the Mean-Proportional given upon the Per
pendicular, and of the exceſſe by which the
whole Plane exceeds its Mean-Proporttonal.
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