Book V. the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. VI. N. B. · When four magnitudes are proportionals, it is usually VII. the fifth definition) the multiple of the first is greater than VIII. IX. X. have to the third the duplicate ratio of that which it has to the XI. See N. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it Definition A, to wit, of compound ratio. the first is said to have to the last of them the ratio com- magnitude. For example, if A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D: And if A has to B the same ratio which E has to F; and B to C Book V. the same ratio that G has to H; and C to D the same that K has to L; then, by this definition, A is said to have to D the ratio compounded of ralios which are the same with the ratios of E to F, G to H, and K to L: and the same thing is to be understood when it is more briefly expressed, by saying A has to D the ratio compounded of the ratios of E to F, G to H, and K to L. In like manner, the same things being supposed, if M has to N the same ratio which A has to D; then, for shortness' sake, XII. one another, as also the consequents to one another. < nify certain ways of changing either the order or magnitude XIII. this word is used when there are four proportionals, and it is See N. XIV. it is inferred, that the second is to the first as the fourth to the XV. and it is inferred, that the first, together with the second, is to XVI. is inferred, that the excess of the first above the second is to XVII. Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the Q Book V. second, as the third to its excess above the fourth. Prop. E, book 5. XVIII. when there is any number of magnitudes more than two, and which arise from the different order in which the magnitudes Sare taken two and two.' XIX. when the first magnitude is to the second of the first rank, XX. lity, in perturbate or disorderly proportion* ; this term is used AXIOMS. 1. EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another. 4 Prop. lib. 2. Archimedis de sphæra et cylindro, II. Book V. Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another. III. IV. multiple of another, is greater than that other magnitude. PROP. I. THEOR. IF any number of magnitudes be equimultiples of as many, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other. Let any number of magnitudes, AB, CD be equimultiples of as may others E, F, each of each ; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB, equal to E, so many are there in CD, equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into A CH, HD, equal each of them to F: the number therefore of the 'magnitudes CH, HD shall G. be equal to the number of the others AG, GB: and because AG is equal to E, and CH to E F, therefore AG and CH together are equal B to a E and F together: for the same reason, a Ax.2. because GB is equal to E, and HD to F; GB 5. C and HD together are equal to E and F together. Wherefore, as many magnitudes as are in AB equal to E, so many are there in AB, CD to. H! gether equal to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple is AB and CD together of E and F D together. Therefore, if any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other: • For the same demonstration Book v. holds in any number of magnitudes, which was here applied to two.' Q. E. D: PROP. II. THEOR. IF the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth. Let AB the first, be the same multiple of C the second, that DE the third is of F the fourth; and BG the fifth, the same multiple of C the second, that EH D A E. B Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal to C, as there are in DE equal to F: in like с н F D E K н с L F |