And commensurable proportions are those, which are to one another as number to number. As when to a proportion given, one proportion is duplicate, another triplicate, the duplicate proportion will be to the triplicate proportion as 2 to 3; but to the given proportion it will be as 2 to 1; and therefore I call those three proportions commensurable.
The proportion of a deficient figure to its complement.
2. A deficient figure, which is made by a quantity continually decreasing to nothing by proportions everywhere proportional and commensurable, is to its complement, as the proportion of the whole altitude to an altitude diminished in any time is to the proportion of the whole quantity, which describes the figure, to the same quantity diminished in the same time.
Let the quantity A B (in [fig. 1]), by its motion through the altitude A C, describe the complete figure A D; and again, let the same quantity, by decreasing continually to nothing in C, describe the deficient figure A B E F C, whose complement will be the figure B D C F E. Now let A B be supposed to be moved till it lie in G K, so that the altitude diminished be G C, and A B diminished be G E; and let the proportion of the whole altitude A C to the diminished altitude G C, be, for example, triplicate to the proportion of the whole quantity A B or G K to the diminished quantity G E. And in like manner, let H I be taken equal to G E, and let it be diminished to H F; and let the proportion of G C to H C be triplicate to that of H I to H F; and let the same be done in as many parts of the strait line A C as is possible; and a line be drawn through the points B, E, F and C. I say the deficient figure A B E F C is to its complement B D C F E as 3 to 1, or as the proportion of A C to G C is to the proportion of A B, that is, of G K to G E.
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 e by a less difference than any quantity that can be assigned. Again, let H I be drawn parallel and equal to G E, cutting C B in P, the crooked line in F, and O E in I, and the proportion of C g to C H will be triplicate to the proportion of H F to H P, and I F will be greater than the third part of P I. But again, setting H in h so near to g, as that the difference between C h and C g may be but as a point, the point P will also in p be so near to l, as that the difference between C p and C l will be but as a point; and drawing h p till it meet with B D in i, cutting the crooked line in f and having drawn e o parallel to B D, cutting D C in o, the parallelogram f o will differ less from the third part of the parallelogram g f, than by any quantity that can be given. And so it will be in all other spaces generated in the same manner. Wherefore the differences of the arithmetical and geometrical means, which are but as so many points B, e, f, &c. (seeing the whole figure is made up of so many indivisible spaces) will constitute a certain line, such as is the line B E F C, which will divide the complete figure A D into two parts, whereof one, namely, A B E F C, which I call a deficient figure, is triple to the other, namely, B D C F E, which I call the complement thereof. And whereas the proportion of the altitudes to one another is in this case everywhere triplicate to that of the decreasing quantities to one another; in the same manner, if the proportion of the altitudes had been everywhere quadruplicate to that of the decreasing quantities, it might have been demonstrated that the deficient figure had been quadruple to its complement; and so in any other proportion. Wherefore, a deficient figure, which is made, &c. which was to be demonstrated.
The same rule holdeth also in the diminution of the bases of cylinders, as is demonstrated in the [second article] of chapter XV.
The proportion of deficient figures to the parallelograms in which they are described, set forth in a table.