For (by art. 2, chapter XV.) the proportion of the complement B E F C D to the deficient figure A B E F C is all the proportions of D B to B A, O E to E G, Q F to F H, and of all the lines parallel to D B terminated in the line B E F C, to all the parallels to A B terminated in the same points of the line B E F C. And seeing the proportions of D B to O E, and of D B to Q F &c. are everywhere triplicate of the proportions of A B to G E, and of A B to H F &c. the proportions of H F to A B, and of G E to A B &c. (by art. 16, chap. XIII.), are triplicate of the proportions of Q F to D B, and of O E to D B &c. and therefore the deficient figure A B E F C, which is the aggregate of all the lines H F, G E, A B, &c. is triple to the complement B E F C D made of all the lines Q F, O E, D B, &c.; which was to be proved. It follows from hence, that the same complement B E F C D is 1⁄4 of the whole parallelogram. And by the same method may be calculated in all other deficient figures, generated as above declared, the proportion of the parallelogram to either of its parts; as that when the parallels increase from a point in the same proportion, the parallelogram will be divided into two equal triangles; when one increase is double to the other, it will be divided into a semiparabola and its complement, or into 2 and 1. The same construction standing, the same conclusion may otherwise be demonstrated thus. Let the strait line C B be drawn cutting G K in L, and through L let M N be drawn parallel to the strait line A C; wherefore the parallelograms G M and L D will be equal. Then let L K be divided into three equal parts, so that it may be to one of those parts in the same proportion which the proportion of A C to G C, or of G K to G L, hath to the proportion of G K to G E. Therefore L K will be to one of those three parts as the arithmetical proportion between G K and G L is to the arithmetical proportion between G K and the same G K wanting the third part of L K; and K E will be somewhat greater than a third of L K. Seeing now the altitude A G or M L is, by reason of the continual decrease, to be supposed less than any quantity that can be given; L K, which is intercepted between the diagonal B C and the side B D, will be also less than any quantity that can be given; and consequently, if G be put so near to A in _g_, as that the difference between C _g_ and C A be less than any quantity that can be assigned, the difference also between C _l_ (removing L to _l_) and C B, will be less than any quantity that can be assigned; and the line _g l_ being drawn and produced to the line B D in _k_, cutting the crooked line in _e_, the proportion of G _k_ to G _l_ will still be triplicate to the proportion of G _k_ to G _e_, and the difference between _k_ and _e_, the third part of _k l_, will be less than any quantity that can be given; and therefore the parallelogram _e_ D will differ from a third part of the parallelogram A