1Line D F is equal to D G, and the Angle D G F is equal to the An
gle D F G, and the Angle E G F is leſſc than the Angle E F G, and
the oppoſite Side E F leſſe than the Side E G. Now if we ſuppoſe
the Time of the Fall along A E to be as A E, the Time by D E ſhall
be as D E; and A G being a Mean-Proportional between B A and A E,
A G ſhall be the Time along the whole A B, and the part E G ſhall be
the Time along the Part E B ex quiete in A. And it may in like man
ner be proved that E F is the Time along E C after the Deſcent D E, or
after the Fall A E: But E F is proved to be leſſer than E G: Therefore
the Propoſition is proved.
gle D F G, and the Angle E G F is leſſc than the Angle E F G, and
the oppoſite Side E F leſſe than the Side E G. Now if we ſuppoſe
the Time of the Fall along A E to be as A E, the Time by D E ſhall
be as D E; and A G being a Mean-Proportional between B A and A E,
A G ſhall be the Time along the whole A B, and the part E G ſhall be
the Time along the Part E B ex quiete in A. And it may in like man
ner be proved that E F is the Time along E C after the Deſcent D E, or
after the Fall A E: But E F is proved to be leſſer than E G: Therefore
the Propoſition is proved.
COROLLARY.
By this and the precedent it appears, that the Space that is paſ
ſed along the Perpendicular after the Fall from above in the
ſame Time in which the Inclined Plane is paſt, is leſſe than
that which is paſt in the ſame Time as in the Inclined, no fall
from above preceding, yet greater than the ſaid Inclined
Plane.
ſed along the Perpendicular after the Fall from above in the
ſame Time in which the Inclined Plane is paſt, is leſſe than
that which is paſt in the ſame Time as in the Inclined, no fall
from above preceding, yet greater than the ſaid Inclined
Plane.
For it having been proved, but now, that of the Moveables coming
from the ſublime Term A the Time of the Converſion along E C is
ſhorter than the Time of the Progreſſion along E B; It is manifeſt that
the Space that is paſt along E B in a Time equal to the Time along E C
is leſs than the whole Space E B. And that the ſame Space along the
Perpendicular is greater than E C is mani
feſted by reaſſuming the Figure of the pre-
109[Figure 109]
cedent Propoſition, in which the part of the
Perpendicular B G hath been demonſtrated
to be paſſed in the ſame Time as B C after
the Fall A B: But that B G is greater than
B C is thus collected. Becauſe B E and F B
are equal, and B A leſſer than B D, F B,
hath greater proportion to B A, than E B
hath to B D: And, by Compoſition, F A
hath greater proportion to A B, than E D
to D B: But as F A is to A B, ſo is G F
to F B, (for A F is the Mean-Proportional
between B A and A G:) And in like man
ner, as E D is to B D, ſo is C E to E B: Therefore G B hath greater
proportion to B F, than C B hath to B E: Therefore G B is greater
than B C.
from the ſublime Term A the Time of the Converſion along E C is
ſhorter than the Time of the Progreſſion along E B; It is manifeſt that
the Space that is paſt along E B in a Time equal to the Time along E C
is leſs than the whole Space E B. And that the ſame Space along the
Perpendicular is greater than E C is mani
feſted by reaſſuming the Figure of the pre-
109[Figure 109]cedent Propoſition, in which the part of the
Perpendicular B G hath been demonſtrated
to be paſſed in the ſame Time as B C after
the Fall A B: But that B G is greater than
B C is thus collected. Becauſe B E and F B
are equal, and B A leſſer than B D, F B,
hath greater proportion to B A, than E B
hath to B D: And, by Compoſition, F A
hath greater proportion to A B, than E D
to D B: But as F A is to A B, ſo is G F
to F B, (for A F is the Mean-Proportional
between B A and A G:) And in like man
ner, as E D is to B D, ſo is C E to E B: Therefore G B hath greater
proportion to B F, than C B hath to B E: Therefore G B is greater
than B C.
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib