Page:Kant's Prolegomena etc (1883).djvu/284

 Demonstration.

First Case.—Two motions in the same line and direction arrive at the same time in one and the same point.



Let two velocities, $$AB$$ and $$ab$$, be presented as contained in one velocity of the motion. Let these velocities be assumed, for the time, as equal, $$AB = ab$$; in this case I assert they cannot be presented at once in the same point, in one and the same space (whether absolute or relative). For, because the lines $$AB$$ and $$ab$$, denoting the velocities, are properly spaces, passed over in equal times, the composition of these spaces $$AB$$ and $$ab = BC$$, and, therefore, the Line $$AC$$, as the sum of the spaces, cannot but express the sum of both velocities. But the parts $$AB$$ and $$BC$$ do not, individually, present the velocity $$= ab$$; for they are not passed over in the same time as $$ab$$. Thus, the double line $$AC$$, which is traversed in the same time as the line $$ab$$, does not represent the double velocity of the latter, as was required. Hence the composition of two velocities in one direction in the same space does not admit of being sensuously presented.

On the contrary, if the body $$A$$ be presented as moved in absolute space with the velocity $$AB$$, and I give to the relative space, a velocity $$ab = AB$$ in addition, in the contrary direction $$ba = CB$$; this is the same as though I distributed the latter velocity to the body in the direction $$AB$$ (axiom 1). But the body moves itself, in this case, in the same time through the sum of the lines $$AB$$ and $$BC = 2 ab$$, in which it would have traversed the line $$ab = AB$$ only, and yet its velocity is conceived as the sum of the two equal velocities $$AB$$ and $$ab$$, which is what was required.

Second Case.—Two motions in exactly contrary directions are united in one and the same point.