THEOR. V. PROP. V.
The proportion of the Times of the Deſcents
along Planes that have different Inclinations
and Lengths, and the Elivations unequal, is
compounded of the proportion of the Lengths
of thoſe Planes, and of the ſubduple proporti
on of their Elevations Reciprocally taken.
along Planes that have different Inclinations
and Lengths, and the Elivations unequal, is
compounded of the proportion of the Lengths
of thoſe Planes, and of the ſubduple proporti
on of their Elevations Reciprocally taken.
Let A B and A C be Planes inclined after different manners,
whoſe Lengths are unequal, as alſo their Elevations. I ſay,
the proportion of the Time of the Deſcent along A C to the
Time along A B, is compounded of the proportion of the ſaid A C
to A B, and of the ſubduple proportion of their Elevation Recipro
cally taken. For let the Perpendicular A D be drawn, with which
let the Horizontal Lines B G and C D interſect, and let A L be a
Mean-proportional between C A and A E; and from the point L let
a Parallel be drawn to the Horizon interſecting
92[Figure 92]
the Plane A C in F; and A F ſhall be a Mean
proportional between C A and A E. And becauſe
the Time along A C is to the Time along A E, as
the Line F A to A E; and the Time along A E is
to the Time along A B, as the ſaid A E to the ſaid
A B: It is manifeſt that the Time along A C is to
the Time along A B, as A F to A B. It remaineth,
therefore, to be demonſtrated, that the proportion
of A F to A B is compounded of the proportion of
C A to A B, and of the proportion of G A to A L;
which is the ſubduple proportion of the Elevati
ons D A and A G Reciprocally taken. But that is manifeſt, C A
being put between F A and A B: For the proportion of F A to A C
is the ſame as that of L A to A D, or of G A to A L; which is ſub
duple of the proportion of the Elevations G A and A D; and the
proportion of C A to A B is the proportion of the Lengths: Therefore
the Propoſition is manifeſt.
whoſe Lengths are unequal, as alſo their Elevations. I ſay,
the proportion of the Time of the Deſcent along A C to the
Time along A B, is compounded of the proportion of the ſaid A C
to A B, and of the ſubduple proportion of their Elevation Recipro
cally taken. For let the Perpendicular A D be drawn, with which
let the Horizontal Lines B G and C D interſect, and let A L be a
Mean-proportional between C A and A E; and from the point L let
a Parallel be drawn to the Horizon interſecting
92[Figure 92]the Plane A C in F; and A F ſhall be a Mean
proportional between C A and A E. And becauſe
the Time along A C is to the Time along A E, as
the Line F A to A E; and the Time along A E is
to the Time along A B, as the ſaid A E to the ſaid
A B: It is manifeſt that the Time along A C is to
the Time along A B, as A F to A B. It remaineth,
therefore, to be demonſtrated, that the proportion
of A F to A B is compounded of the proportion of
C A to A B, and of the proportion of G A to A L;
which is the ſubduple proportion of the Elevati
ons D A and A G Reciprocally taken. But that is manifeſt, C A
being put between F A and A B: For the proportion of F A to A C
is the ſame as that of L A to A D, or of G A to A L; which is ſub
duple of the proportion of the Elevations G A and A D; and the
proportion of C A to A B is the proportion of the Lengths: Therefore
the Propoſition is manifeſt.
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