9272 Theorems and Problems 
the velocity common to Aand B. Now F 
and f are magnitudes by az. Ist: therefore 
they may be divided into equal parts, as well 
as the masses a and b to which they belong: 
but a and 6 have been divided into masses, 
each of which is equal to ¢; and each of 
these masses moves with the velocity common. 
to A and B; therefore g denotes the vis viva 
of each of them by axiom 2d. Hence it fol- 
lows that a and F are equimultiples of ¢ and 
g, aand F being divided into an equal number 
of parts; for the same reason 6 and / are 
equimultiples of the same magnitudes; con- 
sequently as F: f:: a:b by Euclid V. 4. 
Q. E. D. 
Coroxtiary 1. If two bodies be acted on, 
for the same or equal intervals of time, by 
motive or retarding forces, which are as their 
masses ; the vires vive acquired or lost by 
them are also as their masses, or as the 
momenta acquired or lost by them. For the 
accelerative forces are equal in this case by 
dynamics, and the times being equal, the 
vélocities are equal; therefore as F': f:: a:6 
by the proposition: But when the velocities 
are equal, the momenta are as the masses: 
Hence as F: f:: M:m. 
Cor. 2. Bodies, which ascend or descend 
for equal times near ihe surface of the earth, 
