1the Impetus, which it obtaineth in C, whoſe Meaſure is ſuppoſed to be
A C, Let A S be a Mean-proportional betwixt B A and A C. We will
demonſtrate that the Impetus in B is to the Impetus in C, as S A is to
A C. Let the Horizontal Line C D be double to the ſaid A C; and B E
double to B A. It appeareth by what hath been demonſtrated, That the
Cadent along A C being turned along the Horizon C D, and according
to the Impetus acquired in C, with an Equable Motion, ſhall paſs the
Space C D in a Time equal to that
in which the ſaid A C is paſſed
148[Figure 148]
with an Accelerate Motion; and
likewiſe that B E is paſſed in the
ſame time as A B: But the Time of
the Deſcent along A B is A S: There
fore the Horizontal Line B E is
paſſed in A S. As the Time S A is
to the Time A C, ſo let E B be to
B L. And becauſe the Motion by
B E is Equable, the Space B L ſhall be paſſed in the Time A C ac
cording to the Moment of Celerity in B: But in the ſame Time A C
the Space C D is paſſed, according to the Moment of Velocity in C:
the Moments of Velocity therefore are to one another as the Spaces
which according to the ſame Moments are paſſed in the ſame Time:
Therefore the Moment of Velocity in C is to the Moment of Celerity in
B, as D C is to B L. And becauſe as D C is to B E, ſo are their halfs,
to wit, C A to A B: but as E B is to B L, ſo is B A to A S: Therefore,
exæquali, as D C is to B L, ſo is C A to A S: that is, as the Moment
of Velocity in C is to the Moment of Velocity in B, ſo is C A to A S; that
is, the Time along C A to the Time along A B. I he manner of Meaſu
ring the Impetus, or the Moment of Velocity upon a Line along which it
makes a Motion of Deſcent is therefore manifeſt; which Impetus
is indeed ſuppoſed to encreaſe according to the Proportion of the
Time.
A C, Let A S be a Mean-proportional betwixt B A and A C. We will
demonſtrate that the Impetus in B is to the Impetus in C, as S A is to
A C. Let the Horizontal Line C D be double to the ſaid A C; and B E
double to B A. It appeareth by what hath been demonſtrated, That the
Cadent along A C being turned along the Horizon C D, and according
to the Impetus acquired in C, with an Equable Motion, ſhall paſs the
Space C D in a Time equal to that
in which the ſaid A C is paſſed
148[Figure 148]with an Accelerate Motion; and
likewiſe that B E is paſſed in the
ſame time as A B: But the Time of
the Deſcent along A B is A S: There
fore the Horizontal Line B E is
paſſed in A S. As the Time S A is
to the Time A C, ſo let E B be to
B L. And becauſe the Motion by
B E is Equable, the Space B L ſhall be paſſed in the Time A C ac
cording to the Moment of Celerity in B: But in the ſame Time A C
the Space C D is paſſed, according to the Moment of Velocity in C:
the Moments of Velocity therefore are to one another as the Spaces
which according to the ſame Moments are paſſed in the ſame Time:
Therefore the Moment of Velocity in C is to the Moment of Celerity in
B, as D C is to B L. And becauſe as D C is to B E, ſo are their halfs,
to wit, C A to A B: but as E B is to B L, ſo is B A to A S: Therefore,
exæquali, as D C is to B L, ſo is C A to A S: that is, as the Moment
of Velocity in C is to the Moment of Velocity in B, ſo is C A to A S; that
is, the Time along C A to the Time along A B. I he manner of Meaſu
ring the Impetus, or the Moment of Velocity upon a Line along which it
makes a Motion of Deſcent is therefore manifeſt; which Impetus
is indeed ſuppoſed to encreaſe according to the Proportion of the
Time.
But this, before we proceed any farther, is to be premoniſhed, that in
regard we are to ſpeak for the future of the Motion compounded of the
Equable Horizontal, and of the Naturally Accelerate downwards, (for
from this Mixtion reſults, and by it is deſigned the Line of the Project,
that is a Parabola;) it is neceſſary that we define ſome common meaſure
according to which we may meaſure the Velocity, Impetus, or Moment
of both the Motions. And ſeeing that of the Equable Motion the de
grees of Velocity are innumerable, of which you may not take any
promiſcuouſly, but one certain one which may be be compared and con
joyned with the Degree of Velocity naturally Accelerate. I can think of
no more eaſie way for the electing and determining of that, than by aſ
ſuming another of the ſame kind. And that I may the better expreſs
my meaning; Let A C be Perpendicular to the Horizon C B; and A C
regard we are to ſpeak for the future of the Motion compounded of the
Equable Horizontal, and of the Naturally Accelerate downwards, (for
from this Mixtion reſults, and by it is deſigned the Line of the Project,
that is a Parabola;) it is neceſſary that we define ſome common meaſure
according to which we may meaſure the Velocity, Impetus, or Moment
of both the Motions. And ſeeing that of the Equable Motion the de
grees of Velocity are innumerable, of which you may not take any
promiſcuouſly, but one certain one which may be be compared and con
joyned with the Degree of Velocity naturally Accelerate. I can think of
no more eaſie way for the electing and determining of that, than by aſ
ſuming another of the ſame kind. And that I may the better expreſs
my meaning; Let A C be Perpendicular to the Horizon C B; and A C