THEOR. XV. PROP. XXIV.
There being given between the ſame Horizontal
Parallels a Perpendicular and a Plane eleva
ted from its loweſt term, the Space that a
Moveable after the Fall along the Perpendi
cular paſſeth along the Elevated Plane in a
Time equal to the Time of the Fall, is greater
than that Perpendicular, but leſſe than double
the ſame.
Parallels a Perpendicular and a Plane eleva
ted from its loweſt term, the Space that a
Moveable after the Fall along the Perpendi
cular paſſeth along the Elevated Plane in a
Time equal to the Time of the Fall, is greater
than that Perpendicular, but leſſe than double
the ſame.
Between the ſame Horizontal Parallels B C and H G let there
be the Perpendicular A E; and let the Elevated Plane be E B,
along which after the Fall along the Perpendicular A E out of
the Term E let a Reflexion be made towards B. I ſay, that the Space,
along which the Moveable aſcendeth in a Time equal to the Time of the
Deſcent A E, is greater than A E, but leſſe than double the ſame A E.
Let E D be equal to A E, and as E B is to B D, ſo let D B be to B F. It
ſhall be proved, firſt that the point F is the Term at which the Moveable
with a Reflex Motion along E B arriveth in a Time equal to the Time
A E: And then, that E F is greater than E A, but leſſe than double the
ſame. If we ſuppoſe the Time of the Deſcent along A E to be as A E,
the Time of the Deſcent along B E, or Aſcent along E B ſhall be as the
ſame Line B E: And D B being a Mean-Proportional betwixt E B
and B F, and B E being the Time of Deſcent along the whole B E, B D
ſhall be the Time of the Deſcent along B F, and the Remaining part
D E the Time of the
122[Figure 122]
Deſcent along the Re
maining part F E: But
the Time along F E ex
quiete in B, and the
Time of the Aſcent a
long E F is the ſame, ſince that the Degree of Velocity in E was acqui
red along the Deſcent B E, or A E: Therefore the ſame Time D E ſhall
be that in which the Moveable after the Fall out of A along A E,
with a Reflex Motion along E B ſhall reach to the Mark F: But it hath
been ſuppoſed that E D is equal to the ſaid A E: Which was firſt to be
proved. And becauſe that as the whole E B is to the whole B D, ſo is the
part taken away D B to the part taken away B F, therefore, as the whole
E B is to the whole B D, ſo ſhall the Remainder E D be to D F:
But E B is greater than B D: Therefore E D is greater than D F, and
E F leſſe than double to D E or A E: Which was to be proved.
be the Perpendicular A E; and let the Elevated Plane be E B,
along which after the Fall along the Perpendicular A E out of
the Term E let a Reflexion be made towards B. I ſay, that the Space,
along which the Moveable aſcendeth in a Time equal to the Time of the
Deſcent A E, is greater than A E, but leſſe than double the ſame A E.
Let E D be equal to A E, and as E B is to B D, ſo let D B be to B F. It
ſhall be proved, firſt that the point F is the Term at which the Moveable
with a Reflex Motion along E B arriveth in a Time equal to the Time
A E: And then, that E F is greater than E A, but leſſe than double the
ſame. If we ſuppoſe the Time of the Deſcent along A E to be as A E,
the Time of the Deſcent along B E, or Aſcent along E B ſhall be as the
ſame Line B E: And D B being a Mean-Proportional betwixt E B
and B F, and B E being the Time of Deſcent along the whole B E, B D
ſhall be the Time of the Deſcent along B F, and the Remaining part
D E the Time of the
122[Figure 122]Deſcent along the Re
maining part F E: But
the Time along F E ex
quiete in B, and the
Time of the Aſcent a
long E F is the ſame, ſince that the Degree of Velocity in E was acqui
red along the Deſcent B E, or A E: Therefore the ſame Time D E ſhall
be that in which the Moveable after the Fall out of A along A E,
with a Reflex Motion along E B ſhall reach to the Mark F: But it hath
been ſuppoſed that E D is equal to the ſaid A E: Which was firſt to be
proved. And becauſe that as the whole E B is to the whole B D, ſo is the
part taken away D B to the part taken away B F, therefore, as the whole
E B is to the whole B D, ſo ſhall the Remainder E D be to D F:
But E B is greater than B D: Therefore E D is greater than D F, and
E F leſſe than double to D E or A E: Which was to be proved.
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