1the proportion of the Time C to the Time D. Let the Space paſſed by the
Moveable E, with the Velocity A in the Time C, be G: And as the
Velocity A is to the Velocity B,
78[Figure 78]
ſo let G be to I: And as the
Time C is to the Time D, ſo
let I be to L: It is manifeſt,
that I is the Space paſſed by F
in the ſame Time in which E
paſſeth thorow G; ſeeing that
the Spaces G and I are as the
Velocities A and B; and ſeeing that as the Time C is to the Time D, ſo
is I unto L; and ſince that I is the Space paſſed by the Moveable F in the
Time C: Therefore L ſhall be the Space that F paſſeth in the Time D,
with the Velocity B: But the proportion of G to L, is compounded of the
proportions of G to I, and of I to L; that is, of the proportions of the
Velocity A to the Velocity B, and of the Time C to the Time D:
Therefore the Propoſition is demonſtrated.
Moveable E, with the Velocity A in the Time C, be G: And as the
Velocity A is to the Velocity B,
78[Figure 78]ſo let G be to I: And as the
Time C is to the Time D, ſo
let I be to L: It is manifeſt,
that I is the Space paſſed by F
in the ſame Time in which E
paſſeth thorow G; ſeeing that
the Spaces G and I are as the
Velocities A and B; and ſeeing that as the Time C is to the Time D, ſo
is I unto L; and ſince that I is the Space paſſed by the Moveable F in the
Time C: Therefore L ſhall be the Space that F paſſeth in the Time D,
with the Velocity B: But the proportion of G to L, is compounded of the
proportions of G to I, and of I to L; that is, of the proportions of the
Velocity A to the Velocity B, and of the Time C to the Time D:
Therefore the Propoſition is demonſtrated.
THEOR. V. PROP. V.
If two Moveables move with an Equable Motion,
but with unequal Velocities, and if the Spaces
paſſed be alſo unequal, the Times ſhall be to
each other in a proportion compounded of the
proportion of the Spaces, and of the proporti
on of the Velocities contrarily taken.
but with unequal Velocities, and if the Spaces
paſſed be alſo unequal, the Times ſhall be to
each other in a proportion compounded of the
proportion of the Spaces, and of the proporti
on of the Velocities contrarily taken.
Let A and B be the two Moveables, and let the Velocity of A be to
the Velocity of B, as V to T, and let the Spaces paſſed, be as S to
R. I ſay the proportion of the Time in which A is moved to the
Time in which B is moved, ſhall be compounded of the proportions of the
Velocity T to the Velocity V, and of the Space S to the Space R. Let C be
the Time of the Motion A;
79[Figure 79]
and as the Velocity T is to
the Velocity V, ſo let the
Time C be to the Time E:
And for aſmuch as C is the
Time in which A with
the Velocity V paſſeth the
Space S; and that the
Time C is to the Time E, as the Velocity T of the Moveable B is to the
Velocity V, E ſhall be the Time in which the Moveable B would paſſe
the Velocity of B, as V to T, and let the Spaces paſſed, be as S to
R. I ſay the proportion of the Time in which A is moved to the
Time in which B is moved, ſhall be compounded of the proportions of the
Velocity T to the Velocity V, and of the Space S to the Space R. Let C be
the Time of the Motion A;
79[Figure 79]and as the Velocity T is to
the Velocity V, ſo let the
Time C be to the Time E:
And for aſmuch as C is the
Time in which A with
the Velocity V paſſeth the
Space S; and that the
Time C is to the Time E, as the Velocity T of the Moveable B is to the
Velocity V, E ſhall be the Time in which the Moveable B would paſſe