[Sidenote: How a plain deficient figure may be described in a parallelogram, so that it be to a triangle of the same base and altitude, as another deficient figure, plain or solid, twice taken, is to the same deficient figure, together with the complete figure, in which it is described.] 9. If any of these deficient figures, of which I have now spoken, as A B C D (in the 5th figure) be inscribed within the complete figure B E, having A D C E for its complement; and there be made upon C B produced the triangle A B I; and the parallelogram A B I K be completed; and there be drawn parallel to the strait line C I, any number of lines, as M F, cutting every one of them the crooked line of the deficient figure in D, and the strait lines A C, A B and A I in H, G, and L; and as G F is to G D, so G L be made to another, G N; and through all the points N there be drawn the line A N I: there will be a deficient figure A N I B, whose complement will be A N I K. I say, the figure A N I B is to the triangle A B I, as the deficient figure A B C D twice taken is to the same deficient figure together with the complete figure B E. For as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G M to G N, so is the magnitude of the figure A N I B to that of its complement A N I K, by the second article of this chapter. 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. [Sidenote: The transferring of certain properties of deficient figures described in a parallelogram to the proportions of spaces transmitted with several degrees of velocity.]