1the given part C D. Take a Mean-proportional betwixt B C and C D,
to which ſuppoſe B A equal; and let C E be a third proportional be-
114[Figure 114]
tween B C and C A. I ſay, that E B is the Space that after
the Fall out of C ſhall be past in the ſame Time as the ſaid
C D is paſſed. For if we ſuppoſe the Time along C B
to be as C B; B A (that is the Mean-proportional betwixt
B C and C D) ſhall be the Time along C D. And becauſe
C A is the Mean proportional betwixt B C and C E, C A
ſhall be the Time along C E: But the whole B C is the
Time along the Whole C B: Therefore the part B A ſhall be
the Time along the part E B, after the Fall out of C: But
the ſaid B A was the Time along C D: Therefore C D and
E B ſhall be paſt in equal Times out of Reſt in C: Which
was to be done.
to which ſuppoſe B A equal; and let C E be a third proportional be-
114[Figure 114]tween B C and C A. I ſay, that E B is the Space that after
the Fall out of C ſhall be past in the ſame Time as the ſaid
C D is paſſed. For if we ſuppoſe the Time along C B
to be as C B; B A (that is the Mean-proportional betwixt
B C and C D) ſhall be the Time along C D. And becauſe
C A is the Mean proportional betwixt B C and C E, C A
ſhall be the Time along C E: But the whole B C is the
Time along the Whole C B: Therefore the part B A ſhall be
the Time along the part E B, after the Fall out of C: But
the ſaid B A was the Time along C D: Therefore C D and
E B ſhall be paſt in equal Times out of Reſt in C: Which
was to be done.
THEOR. XIV. PROP. XXI.
If along the Perpendicular a Fall be made ex quie
te, in which from the begining of the Motion
a part is taken at pleaſure, paſſed in any Time,
after which an Inflex Motion followeth along
any Plane however Inclined, the Space which
along that Plane is paſſed in a Time equal to
the Time of the Fall already made along the
Perpendicular ſhall be to the Space then paſ
ſed along the Perpendicular more than double,
and leſſe than triple.
te, in which from the begining of the Motion
a part is taken at pleaſure, paſſed in any Time,
after which an Inflex Motion followeth along
any Plane however Inclined, the Space which
along that Plane is paſſed in a Time equal to
the Time of the Fall already made along the
Perpendicular ſhall be to the Space then paſ
ſed along the Perpendicular more than double,
and leſſe than triple.
From the Horizon A E let fall a Perpendicular A B, along which
from the begining A let a Fall be made, of which let a part A C
be taken at pleaſure; then out of C let any Plane G be inclined at
pleaſure: along which after the Fall along A C let the Motion be con
tinued. I ſay, the Space paſſed by that Motion along C G in a Time
equall to the Time of the Fall along A C, is more than double, and leſs
than triple that ſame Space A C. For ſuppoſe C F equal to A C, and
extending out the Plane G C as far as the Horizon in E, and as C E
is to E F, ſo let F E be to E G. If therefore we ſuppoſe the Time of
from the begining A let a Fall be made, of which let a part A C
be taken at pleaſure; then out of C let any Plane G be inclined at
pleaſure: along which after the Fall along A C let the Motion be con
tinued. I ſay, the Space paſſed by that Motion along C G in a Time
equall to the Time of the Fall along A C, is more than double, and leſs
than triple that ſame Space A C. For ſuppoſe C F equal to A C, and
extending out the Plane G C as far as the Horizon in E, and as C E
is to E F, ſo let F E be to E G. If therefore we ſuppoſe the Time of
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