Let A B be a Perpendicular between the Horizontal Parallels A O
and B C; and let F E be greater than B A, but leſſe than double
the ſame. It is required between the ſaid Parallels from the point
B to raiſe a Plane, along which the Moveable after the Fall from A to
B may with a Reflex Motion in a Time equal to the Time of the Fall
along A B paſſe a Space aſcending equal to the ſaid E F. Suppoſe E D
equall to A B, the Remaining Part D F ſhall be leſſe, for that the whole
E F is leſſe than double to A B: Let D I be equal to D F, and as E I is
to I D, ſo let D F be to another Space F X, and out of B let the Right-
124[Figure 124]
Line B O be reflected, equal to E X. I ſay, that the Plane along B O
is that along which after the Fall A B a Moveable in a Time equal
to the Time of the Fall along A B paſſeth aſcending a Space equal to
the given Space E F. Suppoſe B R and R S equal to the ſaid E D and
D F. And becauſe that as E I is to I D, ſo is D F to F X; therefore,
by Compoſition, as E D is to D I, ſo ſhall D X be to X F; that is, as
E D is to D F, ſo ſhall D X be to X F, and E X to X D; that is, as
B O is to O R, ſo ſhall R O be to O S: And if we ſuppoſe the Time
along A B to be A B, the Time along O B ſhall be the ſame O B, and
R O the Time along O S, and the Remaining Part B R the Time along
the Remaining Part S B, deſcending from O to B: But the Time of
the Deſcent along S B from Rest in O, is equal to the Time of the
Aſcent from B to S after the Fall A B: Therefore B O is the Plane ele
vated from B, along which after the Fall along A B the Space B S
equal to the given Space E F is paſſed in the Time B R or B A: Which
was required to be done.
and B C; and let F E be greater than B A, but leſſe than double
the ſame. It is required between the ſaid Parallels from the point
B to raiſe a Plane, along which the Moveable after the Fall from A to
B may with a Reflex Motion in a Time equal to the Time of the Fall
along A B paſſe a Space aſcending equal to the ſaid E F. Suppoſe E D
equall to A B, the Remaining Part D F ſhall be leſſe, for that the whole
E F is leſſe than double to A B: Let D I be equal to D F, and as E I is
to I D, ſo let D F be to another Space F X, and out of B let the Right-
124[Figure 124]Line B O be reflected, equal to E X. I ſay, that the Plane along B O
is that along which after the Fall A B a Moveable in a Time equal
to the Time of the Fall along A B paſſeth aſcending a Space equal to
the given Space E F. Suppoſe B R and R S equal to the ſaid E D and
D F. And becauſe that as E I is to I D, ſo is D F to F X; therefore,
by Compoſition, as E D is to D I, ſo ſhall D X be to X F; that is, as
E D is to D F, ſo ſhall D X be to X F, and E X to X D; that is, as
B O is to O R, ſo ſhall R O be to O S: And if we ſuppoſe the Time
along A B to be A B, the Time along O B ſhall be the ſame O B, and
R O the Time along O S, and the Remaining Part B R the Time along
the Remaining Part S B, deſcending from O to B: But the Time of
the Deſcent along S B from Rest in O, is equal to the Time of the
Aſcent from B to S after the Fall A B: Therefore B O is the Plane ele
vated from B, along which after the Fall along A B the Space B S
equal to the given Space E F is paſſed in the Time B R or B A: Which
was required to be done.