But, by the same article, as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G F to G D, that is, by construction, of G L to G N, so is the figure A B C D to its complement A D C E.
And by composition, as the proportion of G M to G L, together with that of G L to G N, is to the proportion of G M to G L, so is the complete figure B E to the deficient figure A B C D.
And by conversion, as the proportion of G M to G L is to both the proportions of G M to G L and of G L to G N, that is, to the proportion of G M to G N, which is the proportion compounded of both, so is the deficient figure A B C D to the complete figure B E.
But it was, as the proportion of G M to G L to that of G M to G N, so the figure A N I B to its complement A N I K. And therefore, A B C D. B E :: A N I B. A N I K are proportionals. And by composition, A B C D + B E. A B C D :: B K. A N I B are proportionals.
And by doubling the consequents, A B C D + B E. 2 A B C D :: B K. 2 A N I B are proportionals.
And by taking the halves of the third and the fourth, A B C D + B E. 2 A B C D :: A B I. A N I B are also proportionals; which was to be proved.
The transferring of certain properties of deficient figures described in a parallelogram to the proportions of spaces transmitted with several degrees of velocity.
10. From what has been said of deficient figures described in a parallelogram, may be found out what proportions spaces, transmitted with accelerated motion in determined times, have to the times themselves, according as the moved body is accelerated in the several times with one or more degrees of velocity.
For let the parallelogram A B C D, in the [6th figure], and in it the three-sided figure D E B C be described; and let F G be drawn anywhere parallel to the base, cutting the diagonal B D in H, and the crooked line B E D in E; and let the proportion of B C to B F be, for example, triplicate to that of F G to F E; whereupon the figure D E B C will be triple to its complement B E D A; and in like manner, I E being drawn parallel to B C, the three-sided figure E K B F will be triple to its complement B K E I. Wherefore the parts of the deficient figure cut off from the vertex by strait lines parallel to the base, namely, D E B C and E K B F, will be to one another as the parallelograms A C and I F; that is, in proportion compounded of the proportions of the altitudes and bases. Seeing therefore the proportion of the altitude B C to the altitude B F is triplicate to the proportion of the base D C to the base F E, the figure D E B C to the figure E K B F will be quadruplicate to the proportion of the same D C to F E. And by the same method, may be found out what proportion any of the said three-sided figures has to any part of the same, cut off from the vertex by a strait line parallel to the base.
Now as the said figures are understood to be described by the continual decreasing of the base, as of C D, for example, till it end in a point, as in B; so also they may be understood to be described by the continual increasing of a point, as of B, till it acquire any magnitude, as that of C D.