Let be one of these motions, and the other in the opposite direction, the velocity of which we assume here to be equal to that of the first; in this case the very idea of representing two such motions, at the same time, in one and the same space, and in one and the same point, in short, the case of such a composition of motions would itself be impossible, which is contrary to the assumption.
On the other hand, let the motion be conceived as in absolute space, and instead of the motion in the same absolute space, let the contrary motion of the relative space [be conceived] with the same velocity, which (according to axiom 1) is equal to the motion , and may thus be entirely substituted for it; in this case two exactly opposite and equal motions of the same point, at the same time, may be very well presented. Now, as the relative space is moved with the same velocity in the same direction with the point , this point, or the body, present therein, does not change its place in respect of the relative space; i.e., a body moved in two exactly contrary directions with equal velocity, rests, or generally expressed, its motion is equal to the difference of the velocities in the direction of the greater (which admits of being easily deduced from what has already been demonstrated).
Third Case.—Two motions of the same point are presented as combined according to directions that enclose an angle.
The two given motions are and , whose velocity and directions are expressed by these lines, but the
