1ſecond, which let be E B. It is to be proved that the Time in which the
Spaces E B and B C are paſſed is longer than the Time in which A B
and B C are paſſed. Let the Time along A B be as A B; the ſame ſhall
be the Time of the Motion along the Horizontal Space B G; becauſe
B C is double to A B, and the Time along both the Spaces A B C ſhall
be double of O B A. Let B O
127[Figure 127]
be a Mean-proportional between
E B and B A. B O ſhall be the
Time of the Fall along E B.
Again, let the Horizontal Space
B D be double to the ſaid B E:
It is manifeſt that the Time of it
after the Fall E B is the ſame
B O. As D B is to B C, or as
E B is to B A, ſo let O B be to
B N: and in regard the Motion
along the Horizontal Plane is Equable, and O B being the Time along
B D after the Fall out of E, therefore N B ſhall be the Time along B C
after the Fall from the ſame Altitude E. Hence it is manifeſt, that O B,
together with B N is the Time along E B C; and becauſe the double of
B A is the Time along A B C; it remains to be proved, that O B, to
gether with B N is more than double B A. Now becauſe O B is a Mean
between E B and B A, the proportion of E B to B A is double the pro
portion of O B to B A: and, in regard that E B is to B A, as O B is to
B N, the proportion of O B to B N ſhall alſo be double the proportion of
O B to B A: But that proportion of O B to B N is compounded of the
proportions of O B to B A, and of A B to B N: therefore the proportion
of A B to B N is the ſame with that of O B to B A. Therefore B O,
B A, and B N are three continual Proportionals, and O B, together with
B N, are greater than double B A: Whereupon the Propoſition is ma
nifeſt.
Spaces E B and B C are paſſed is longer than the Time in which A B
and B C are paſſed. Let the Time along A B be as A B; the ſame ſhall
be the Time of the Motion along the Horizontal Space B G; becauſe
B C is double to A B, and the Time along both the Spaces A B C ſhall
be double of O B A. Let B O
127[Figure 127]be a Mean-proportional between
E B and B A. B O ſhall be the
Time of the Fall along E B.
Again, let the Horizontal Space
B D be double to the ſaid B E:
It is manifeſt that the Time of it
after the Fall E B is the ſame
B O. As D B is to B C, or as
E B is to B A, ſo let O B be to
B N: and in regard the Motion
along the Horizontal Plane is Equable, and O B being the Time along
B D after the Fall out of E, therefore N B ſhall be the Time along B C
after the Fall from the ſame Altitude E. Hence it is manifeſt, that O B,
together with B N is the Time along E B C; and becauſe the double of
B A is the Time along A B C; it remains to be proved, that O B, to
gether with B N is more than double B A. Now becauſe O B is a Mean
between E B and B A, the proportion of E B to B A is double the pro
portion of O B to B A: and, in regard that E B is to B A, as O B is to
B N, the proportion of O B to B N ſhall alſo be double the proportion of
O B to B A: But that proportion of O B to B N is compounded of the
proportions of O B to B A, and of A B to B N: therefore the proportion
of A B to B N is the ſame with that of O B to B A. Therefore B O,
B A, and B N are three continual Proportionals, and O B, together with
B N, are greater than double B A: Whereupon the Propoſition is ma
nifeſt.
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