1Time A. As B is to A, ſo let C D be to another Line, to which let C X
be equal that deſcendeth from C unto the Horizon: It is manifeſt that
the Plane C X is that along which the Moveable deſcendeth in the Gi
ven Time A. For it hath been demonſtrated, that the Time along the
inclined Plane hath the ſame proportion to the Time along its ^{*} Eleva-
tion, as the Length of the Plane hath to the Length of its Elevation,.
The Time, therefore, along C X is to the Time along C D, as C X is to
C D, that is, as the Time A is to the Time B: But the Time B is that
in which the Perpendicular is paſt ex quiete: Therefore the Time A is
that in which the Plane C X is paſſed.
be equal that deſcendeth from C unto the Horizon: It is manifeſt that
the Plane C X is that along which the Moveable deſcendeth in the Gi
ven Time A. For it hath been demonſtrated, that the Time along the
inclined Plane hath the ſame proportion to the Time along its ^{*} Eleva-
tion, as the Length of the Plane hath to the Length of its Elevation,.
The Time, therefore, along C X is to the Time along C D, as C X is to
C D, that is, as the Time A is to the Time B: But the Time B is that
in which the Perpendicular is paſt ex quiete: Therefore the Time A is
that in which the Plane C X is paſſed.
* Or Perpendi
cular.
cular.
PROBL. IX. PROP. XXIII.
A Space paſt ex quiete along the Perpendicular in
any Time being given, to inflect a Plane from
the loweſt term of that Space, along which,
after the Fall along the Perpendicular, a Space
equal to any Space given may be paſſed in the
ſame Time: which nevertheleſſe is more than
double, and leſſe than triple the Space paſſed
along the Perpendicular.
any Time being given, to inflect a Plane from
the loweſt term of that Space, along which,
after the Fall along the Perpendicular, a Space
equal to any Space given may be paſſed in the
ſame Time: which nevertheleſſe is more than
double, and leſſe than triple the Space paſſed
along the Perpendicular.
Along the Perpendicular A S, in the Time A C, let the Space
A C be paſt ex quiete in A; to which let I R be more than
double, and leſſe than triple. It is required from the Terme C
to inflect a Plane, along which a Moveable after the Fall along A C
may in the ſame Time A C paſſe a Space equal to the ſaid I R. Let
R N, and N M be equal to A C: And look what proportion the part
I M hath to M N, the ſame ſhall the Line A C have to another, equal
to which draw C E from C to
117[Figure 117]
the Horizon A E, which con
tinue out towards O, and take
C F, F G, and G O, equal to
the ſaid R N, N M, and M I.
I ſay, that the Time along the
inflected Plane C O, after the
Fall A G, is equal to the Time
A C out of Reſt in A. For in
regard that as O G is to G F,
ſo is F C to C E by Compoſition it ſhall be that as O F is to F G or F C,
ſo is F E to E C; and as one of the Antecedents is to one of the Con
ſequents, ſo are all to all; that is, the whole O E is to E F as F E to
E C: Therefore O E, E F, and E C are Continual Proportionals:
A C be paſt ex quiete in A; to which let I R be more than
double, and leſſe than triple. It is required from the Terme C
to inflect a Plane, along which a Moveable after the Fall along A C
may in the ſame Time A C paſſe a Space equal to the ſaid I R. Let
R N, and N M be equal to A C: And look what proportion the part
I M hath to M N, the ſame ſhall the Line A C have to another, equal
to which draw C E from C to
117[Figure 117]the Horizon A E, which con
tinue out towards O, and take
C F, F G, and G O, equal to
the ſaid R N, N M, and M I.
I ſay, that the Time along the
inflected Plane C O, after the
Fall A G, is equal to the Time
A C out of Reſt in A. For in
regard that as O G is to G F,
ſo is F C to C E by Compoſition it ſhall be that as O F is to F G or F C,
ſo is F E to E C; and as one of the Antecedents is to one of the Con
ſequents, ſo are all to all; that is, the whole O E is to E F as F E to
E C: Therefore O E, E F, and E C are Continual Proportionals:
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