from the last to the last but two of the second rank; and so on in a cross order: and the inference is as in the 18th definition. It is demonstrated in Prop. 23rd. of Book 5. AXIOMS. I. EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another.a II. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. III. A multiple of a greater magnitude is greater than the same multiple of a less. IV. That magnitude, of which a multiple is greater than the same 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 many others E, F, each of each; what multiple soever AB-is of E, the same multiple shall AB and CD together, be of E and F together. Equimultiples of magnitudes are multiples that contain them respectively the same number of times. 1. A G Because AB is the same multiple of E that CD is of F, as many magnitudes as there 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 CH, HD equal each of them to F: therefore the number of the magnitudes CH, HD shall be equal to the number of the others AG, GB: and because AG is equal to E, and CH to F, therefore H 2 Ax. AG and CH together are equal to E and F B C F together: for the same reason, because GB is D Therefore, if any magnitudes, how many soever, 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; 'For 'the same demonstration holds in any number of mag'nitudes, 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 the sixth is of F the fourth: then shall AG, the first many magnitudes in AB equal to C, as there are in DE equal to F: in like manner, as many as there are in BG equal to C, so many are there in EH equal to F: therefore as many as there are in the whole AG equal to C, so many are there in the whole DH equal to F: therefore AG is the same multiple of C that DH is of F; that is, AG, the first and fifth together, is the same multiple of C the second, that DH, the third and sixth together, is of F the fourth. If, therefore, the first be & the same multiple, &c. Q. E. D. B H cl D E K COR. From this it is plain, that if any number of magnitudes AB, BG, GH, be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; then the whole of the first, viz. AH, is the same multiple of C, that the whole of the last, viz. DL, is of F. PROP. III. THEOR. If the first be the same multiple of the second, which the third is of the fourth; and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth. Let A the first be the same multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken: then EF shall be the same multiple of B, that GH is of D. N * 2. 5. E A H L G བ Because EF is the same multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as there are in GH equal to C: F let EF be divided into the magnitudes EK, KF, each equal to A; and GH into GL, LH each K equal to C: therefore the number of the magnitudes EK, KF, are equal to the number of the others GL, LH: and because A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the same multiple of B, that GL is of D: for the same reason, KF is the same multiple of B, that LH is of D; and so, if there be more parts in EF, GH equal to A, C: then because EK the first is the same multiple of B the second, which GL the third is of D the fourth, and that KF the fifth is the same multiple of B the second, which LH the sixth is of D the fourth; therefore EF, the first together with the fifth, is the same multiple* of B the second, which GH, the third together with the sixth, is of D the fourth. If, therefore, the first, &c. If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz. 'the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth. Let A the first have to B the second, the same ratio which C the third has to D the fourth; and of A, C let there be taken any equimultiples whatever E, F; of B, that N is of D: and because, as A is to B, * 3. 5. so is C to D,* and of A and C have been taken Hyp. certain equimultiples K and L; and of B and D have been taken certain equimultiples M and N; if therefore K be greater than M, L is greater than N: if equal, equal; and if less, less;* but K, L are any ⚫ 5 Def. equimultiples whatever of E, F; and M, N any whatever of G, H: therefore as E is to G, so is F to H. *5 Def. Therefore, if the first, &c. Q. E. D. COR. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third shall have the same ratio to the second and fourth: and in like manner, the first and the third shall have the same ratio to any equimultiples whatever of the second and fourth. Let A the first have to B the second, the same ratio which C the third has to D the fourth, and of A and C let E and F be any equimultiples whatever; then E shall be to B, as F to D. Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonstrated, as before, that K is the same multiple • 5. 5. |