1the ſame Space S. Again as the Space S is to the Space R, ſo let the
Time E be to the Time G: Therefore G is the Time in which B would
paſſe the Space R. And becauſe the proportion of C to G is compounded
of the proportions of C to E, and of E to G; And ſince the proportion
of C to E is the ſame with that of the Velocities of the Moveables A and
B contrarily taken; that is, with that of T and V; And the proportion
of E to G is the ſame with the proportion of the Spaces S and R: There
fore the Propoſition is demonſtrated.
Time E be to the Time G: Therefore G is the Time in which B would
paſſe the Space R. And becauſe the proportion of C to G is compounded
of the proportions of C to E, and of E to G; And ſince the proportion
of C to E is the ſame with that of the Velocities of the Moveables A and
B contrarily taken; that is, with that of T and V; And the proportion
of E to G is the ſame with the proportion of the Spaces S and R: There
fore the Propoſition is demonſtrated.
THEOR. VI. PROP. VI.
If two Moveables move with an Equable Motion,
the proportion of their Velocities ſhall be com
pounded of the proportion of the Spaces paſ
ſed, and of the proportion of the Times con
trarily taken.
the proportion of their Velocities ſhall be com
pounded of the proportion of the Spaces paſ
ſed, and of the proportion of the Times con
trarily taken.
Let A and B be the two Moveables moving with an Equable
Motion; and let the Spaces by them paſſed, be as V to T; and
let the Times be as S to R. I ſay that the proportion of the Ve
locity of the Moveable A, to that of the Velocity of B, ſhall be
compounded of the proportions of the Space V to the Space T, and
of the Time R to the Time S. Let C be the Velocity with which the
Moveable A paſſeth the Space V in the Time S: And let the Velocity C
be to the Velo-
80[Figure 80]
city E, as the
Space V is to
the Space T;
And E ſhall
be the Veloci
ty with which
the Moveable
B paſſeth the Space T in the Time S: Again, let the Velocity E be to the
other Velocity G, as the Time R is to the Time S; And G ſhall be the
Velocity with which the Moveable B paſſeth the Space T in the Time R.
We have therefore the Velocity C, wherewith the Moveable A paſſeth
the Space V in the Time S; and the Velocity G, wherewith the Move
able B paſſeth the Space T in the Time R: And the proportion of C to
G is compounded of the proportions of C to E and of E to G: But the
proportion of C to E, is ſuppoſed the ſame with that of the Space V to
the Space T; and the proportion of E to G, is the ſame with that of R
to S: Therefore the Propoſition is manifest.
Motion; and let the Spaces by them paſſed, be as V to T; and
let the Times be as S to R. I ſay that the proportion of the Ve
locity of the Moveable A, to that of the Velocity of B, ſhall be
compounded of the proportions of the Space V to the Space T, and
of the Time R to the Time S. Let C be the Velocity with which the
Moveable A paſſeth the Space V in the Time S: And let the Velocity C
be to the Velo-
80[Figure 80]city E, as the
Space V is to
the Space T;
And E ſhall
be the Veloci
ty with which
the Moveable
B paſſeth the Space T in the Time S: Again, let the Velocity E be to the
other Velocity G, as the Time R is to the Time S; And G ſhall be the
Velocity with which the Moveable B paſſeth the Space T in the Time R.
We have therefore the Velocity C, wherewith the Moveable A paſſeth
the Space V in the Time S; and the Velocity G, wherewith the Move
able B paſſeth the Space T in the Time R: And the proportion of C to
G is compounded of the proportions of C to E and of E to G: But the
proportion of C to E, is ſuppoſed the ſame with that of the Space V to
the Space T; and the proportion of E to G, is the ſame with that of R
to S: Therefore the Propoſition is manifest.