Page:Kant's Prolegomena etc (1883).djvu/287

 mechanical construction. Complete similarity and equality, in so far as they can only be cognised in intuition, is congruity. All geometrical construction of complete identity rests on congruity. This congruity of two motions combined with a third (in short, the motu composito itself) can never take place, when the two former are presented in one and the same space, i.e. relative [space]. Hence all attempts to demonstrate the above proposition in its three cases, have always been mechanical solutions only, inasmuch, namely, as though moving causes by which a given motion was combined with another, were made to produce a third, the proofs that the former were the same as the latter, and as such, admitted of being presented in pure intuition a priori [were not given].

Observation 2.

When, for instance, a velocity $$AB$$ is termed double, nothing else can be understood thereby, but that it consists of two simple and equal [velocities] $$AB$$ and $$BC$$, (see Fig. 1). But if a double velocity be explained by saying that it is a motion by which a doubly great space is passed over in the same time, something is here assumed which is not necessarily implied, namely, that two equal velocities may be combined in the same way as two equal spaces, for it is not in itself obvious that a given velocity consists of smaller [velocities]; and in the same way that a rapidity consists of slownesses as a space does of smaller [spaces]. For the parts of the velocity are not outside one another, as the parts of the space; and if the former are to be considered as quantity, the conception of their quantity, as it is intensive, must be constructed in a different manner to that of the extensive quantity of space. But this construction is possible in no other way than by the mediate composition of two equal motions, one of which is that of the body, the other that of the relative space in the contrary direction, but which, for this reason, is completely identical with an equal motion of the body in the previous direction. For in the same direction two equal velocities would not admit of being compounded in one body, except through external moving causes; for