Page:Kant's Prolegomena etc (1883).djvu/286

 angle, enclosed by the latter, by $$BAC$$ (it matters not whether it be a right angle, as in this case, or any other angle). If these two motions are to occur, at the same time, in the directions $$AB$$ and $$AC$$, and indeed in the same space, they would not be aide to occur, at the same time, in both these lines $$AB$$ and $$AC$$, but only in lines running parallel to these. It would have, therefore, to be assumed, that one of these motions effected a change in the other (namely, the deviation from the given course), although the directions remained the same on either side. But this is contrary to the assumption of the proposition, which indicates by the word composition, that both the given motions are contained in a third, and must therefore be one with this, and not that, by one changing the other, a third is produced.

On the other hand, let the motion $$AC$$ be taken as proceeding in absolute space, but instead of the motion $$AB$$, the motion of the relative space in the opposite direction. Let the line $$AC$$ be divided into three equal parts, $$AE, EF, FG$$. Now, while the body $$A$$ in absolute space passes over the line $$AE$$, the relative space, and therewith the point $$E$$, passes over the space $$Ee = MA$$; while the body passes over the two parts together $$= AF$$, the relative space and therewith the point $$F$$, describes the line $$Ff = NA$$; while, finally, the body passes over the whole line $$AC$$, the relative space, and therewith the point $$C$$ describes the line $$Cc = BA$$. All this is the same as though the body $$A$$ had passed over in these three divisions of time, the lines $$Em$$, $$Fn$$ and $$CD=AM, AN, AB$$, and in the whole time in which it passes over $$AC$$, had passed over the line $$CD = AB$$. It is therefore at the last moment in the point $$D$$, and in the whole time gradually in all points of the diagonal line $$AD$$, which expresses the direction as well as the velocity of the compound motion.

Observation 1.

Geometrical construction demands that one quantity should be identical with the other, or two quantities in composition, with a third, not that they should pronuceproduce [sic] the third as causes, which would be mechanical con-