1822.] C.’s Reply to D. 215 
the momentum of either one before the stroke.” That bodies 
act with a force equal to their momentum, is a maxim which D. 
has repeatedly and triumphantly quoted, and momentum is the 
quantity of motion ina given direction. Itis also quite clear that 
neither of the balls can themselves act in a direction opposite to 
that in which they are moving. The utmost intensity of force, 
therefore, with which either of the balls can ‘act, is its own 
momentum ; and that only in the direction towards which it 
moves. The acting force is necessarily the same at the time of 
the collision as before ; and consequently at the mstant of colli- 
sion each ball acts with a force equal to its own momentum in the 
direction towards which it moves; and as both balls are moving 
in opposite directions, they each act with a force equal to their 
own momentum in a direction opposite to the direction of the 
other ball. The intensity of the collision, therefore, is the sum 
of the momenta of the two, but the force in each direction is the 
momentum of each one ; and consequently “‘ the intensity of the 
stroke ‘as felt by each body in a direction opposite to that in 
which it was moving,” is equal to the momentum of one ball, and 
not the momenta of two ; forif they acted in each direction with 
a force equal to the momenta of two balls, it is evident the 
whole force would be doubled by the collision, which is impos- 
sible. 
D. professes to demonstrate the proposition from the principles 
admitted in the whole theory, and he commences by stating 
truly, that “ By the old theory, if a hard body A, having the 
velocity of a, strike another hard equal body A’ at rest, the 
Aa Aa@,, 
ca A= alt 
This he properly treats as the intensity of the stroke, and uses it 
as such in his reasoning. But in the same argument in which 
he uses this as correct, he states, and assumes that he has 
proved, that “ when one of the bodies is at rest,” ‘ the intensity 
of the stroke on each is equal to the momentum of the moving 
body.” I have already shown that the latter statement is not 
true ; but if it were, the former could not beso; and the reason- 
ing can little deserve the term of strict mathematical induction, 
which assumes in its support as true two propositions quite 
inconsistent with each other; namely, that the intensity of the 
motion communicated to A’ by the impulse is 
stroke is equal to ee or half the momentum of A, and also equal 
to the momentum of the moving body, or the whole momentum 
of A. It is, however, worthy of the corollory which he founds 
upon it, but which has already been sufficiently refuted. 
“Hence,” he says, “the two equal bodies after the impulse 
recede towards the parts whence they came with the same 
momenta they had before they met.” 
“ In the theory of motion rightly understood,” says Maclaurin, 
