1let the Angle B A E be cut in two equal parts, and draw A F. I ſay,
that the Time along A B is to the Time along A E B, as A E is to A E F.
For in regard the Angle F A B is equal to the Angle F A E, and the An
gle E A G to the Angle A B F, the whole Angle G A F ſhall be equal to
the two Angles F A B, and A B F;
to which alſo the Angle G F A
126[Figure 126]
is equal: Therefore the Line G F
is equal to G A. And becauſe the
Rectangle B G E is equal to the
Square of G A, it ſhall likewiſe
be equal to the Square of G F, and
the three Lines B G, G F, and
G E ſhall be proportionals. And
if we ſuppoſe A E to be the Time
along A E, G E ſhall be the Time
along G E, and G F the Time along the whole G B, and E F the Time
along E B, after the Deſcent out of G, or out of A, along A E: The Time,
therefore, along A E, or along A B ſhall be to the Time along A E B, as
A E is to A E F: Which was to be determined.
that the Time along A B is to the Time along A E B, as A E is to A E F.
For in regard the Angle F A B is equal to the Angle F A E, and the An
gle E A G to the Angle A B F, the whole Angle G A F ſhall be equal to
the two Angles F A B, and A B F;
to which alſo the Angle G F A
126[Figure 126]is equal: Therefore the Line G F
is equal to G A. And becauſe the
Rectangle B G E is equal to the
Square of G A, it ſhall likewiſe
be equal to the Square of G F, and
the three Lines B G, G F, and
G E ſhall be proportionals. And
if we ſuppoſe A E to be the Time
along A E, G E ſhall be the Time
along G E, and G F the Time along the whole G B, and E F the Time
along E B, after the Deſcent out of G, or out of A, along A E: The Time,
therefore, along A E, or along A B ſhall be to the Time along A E B, as
A E is to A E F: Which was to be determined.
More briefly thus. Let G F be cut equal to G A: It is manifeſt
that G F is the Mean-proportional between B G, and G E. The reſt as
before.
that G F is the Mean-proportional between B G, and G E. The reſt as
before.
PROBL. XI. PROP. XXIX.
Any Horizontal Space being given upon the
end of which a Perpendicular is erected,
in which a part is taken equal to half of the
Space given in the Horizontal a Moveable fal
ling from that height, and turned along the
Horizon, ſhall paſſe the Horizontal Space to
gether with the Perpendicular in a ſhorter
Time than any other Space of the Perpendi
cular with the ſame Horizontal Space.
end of which a Perpendicular is erected,
in which a part is taken equal to half of the
Space given in the Horizontal a Moveable fal
ling from that height, and turned along the
Horizon, ſhall paſſe the Horizontal Space to
gether with the Perpendicular in a ſhorter
Time than any other Space of the Perpendi
cular with the ſame Horizontal Space.
Let there be an Horizontal Space in which let any Space be given
B C, and on B let there be a Perpendicular erected, in which let
B A be the half of the foreſaid B C. I ſay, that the Time in which
a Moveable let fall out of A paſſeth both the Spaces A B and B C is the
ſhorteſt of all Times in which the ſaid Space B C with a part of the
Perpendicular, whether greater or leſſer than the part A B, ſhall be paſ
ſed. Let a greater be taken, as in the ſirſt Figure, or leſſer, as in the
B C, and on B let there be a Perpendicular erected, in which let
B A be the half of the foreſaid B C. I ſay, that the Time in which
a Moveable let fall out of A paſſeth both the Spaces A B and B C is the
ſhorteſt of all Times in which the ſaid Space B C with a part of the
Perpendicular, whether greater or leſſer than the part A B, ſhall be paſ
ſed. Let a greater be taken, as in the ſirſt Figure, or leſſer, as in the