Let the Horizontal Lines be A F the upper, and C D the low
er; between which let the Perpendicular A C, and inclined
Plane D F, be cut in B; and let A R be a Mean-Proportional
between the whole Perpendicular C A, and the upper part A B; and
let F S be a Mean-proportional between the whole Inclined Plane D F,
and the upper part B F. I ſay, that the Time of the Fall along the
whole Perpendicular A C hath the ſame proportion to the Time along
its upper part A B, with the lower of the Plane, that is, with B D,
as A C hath to the Mean-proporti
onal of the Perpendicular, that is
104[Figure 104]
A R, with S D, which is the ex
ceſſe of the whole Plane D F above
its Mean-proportional F S. Let a
Line be drawn from R to S, which
ſhall be parallel to the two Horizon
tal Lines. And becauſe the Time of
the Fall along all A C, is to the
Time along the part A B, as C A is
to the Mean proportional A R, if we ſuppoſe A C to be the Time of
the Fall along A C, A R ſhall be the Time of the Fall along A B,
and R C that along the remainder B C. For if the Time along A C
be ſuppoſed, as was done, to be A C it ſelf the Time along F D ſhall
be F D; and in like manner D S may be concluded to be the Time a
long B D, after F B, or after A B. The Time therefore along the
whole A C, is A R, with R C; And the Time along the inflected
Plane A B D, ſhall be A R, with S D: Which was to be proved.
er; between which let the Perpendicular A C, and inclined
Plane D F, be cut in B; and let A R be a Mean-Proportional
between the whole Perpendicular C A, and the upper part A B; and
let F S be a Mean-proportional between the whole Inclined Plane D F,
and the upper part B F. I ſay, that the Time of the Fall along the
whole Perpendicular A C hath the ſame proportion to the Time along
its upper part A B, with the lower of the Plane, that is, with B D,
as A C hath to the Mean-proporti
onal of the Perpendicular, that is
104[Figure 104]A R, with S D, which is the ex
ceſſe of the whole Plane D F above
its Mean-proportional F S. Let a
Line be drawn from R to S, which
ſhall be parallel to the two Horizon
tal Lines. And becauſe the Time of
the Fall along all A C, is to the
Time along the part A B, as C A is
to the Mean proportional A R, if we ſuppoſe A C to be the Time of
the Fall along A C, A R ſhall be the Time of the Fall along A B,
and R C that along the remainder B C. For if the Time along A C
be ſuppoſed, as was done, to be A C it ſelf the Time along F D ſhall
be F D; and in like manner D S may be concluded to be the Time a
long B D, after F B, or after A B. The Time therefore along the
whole A C, is A R, with R C; And the Time along the inflected
Plane A B D, ſhall be A R, with S D: Which was to be proved.
The ſame happeneth, if inſtead of the Perpendicular, another
Plane were taken, as ſuppoſe N O; and the Demonstration is the
ſame.
Plane were taken, as ſuppoſe N O; and the Demonstration is the
ſame.
PROBL I. PROP. XIII.
A Perpendicular being given, to Inflect a Plane
unto it, along which, when it hath the ſame
Elevation with the ſaid Perpendicular, it may
make a Motion after its Fall along the Per
pendicular in the ſame Time, as along the
ſame Perpendicular ex quiete.
unto it, along which, when it hath the ſame
Elevation with the ſaid Perpendicular, it may
make a Motion after its Fall along the Per
pendicular in the ſame Time, as along the
ſame Perpendicular ex quiete.
Let the Perpendicular given be A B, to which extended to C,
let the part B C be equal; and draw the Horizontal Lines
C E and A G. It is required from B to inflect a Plane reach
ing to the Horizon C E, along which a Motion, after the Fall out
let the part B C be equal; and draw the Horizontal Lines
C E and A G. It is required from B to inflect a Plane reach
ing to the Horizon C E, along which a Motion, after the Fall out
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