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. PROPOSITION I. THEOREM. 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. Then whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. А а в с н р EBecause 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 AĞ, GB; and because AG is equal to E, and CH to F, therefore AG and CH together are equal to Eand Ftogether: (1.ax. 2.) for the same reason, because GB is equal to E, and HD to F; GB and HD together are equal to E and F together: wherefore as many magnitudes as there are in AB equal to E, so many are there in AB, CD together, 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 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 others: "For the same demonstration holds in any number of magnitudes, which was here applied to two.' Q.E. D. PROPOSITION II. THEOREM. 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 together with the fifth, be the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth. A B G D E 1 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 manner, as many as there are in BG equal to C, so many are there in EH equal to F: 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 the second C, that DH, the third and sixth together, is of the fourth F. If therefore, the first be the same multiple, &c. Q.E.D. 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. А в а н р н к І. PROPOSITION III. THEOREM. 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 equi. multiples, 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 equimultiples EF, GH be taken. D-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: let EF be divided into the magnitudes EK, KF, each equal to A; and GH into GL, LH, each equal to C: therefore the number of the magnitudes EK, KF shall be 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 equal 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: therefore, because the first EK is the same multiple of the second B, which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth D; EF the first, together with the fifth, is the same multiple of the second B, (v. 2.) which GH the third, together with the sixth, is of the fourth D. If, therefore, the first, &c. Q.E.D. PROPOSITION IV. THEOREM. 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 the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H. Take of E and F any equimultiples whatever K, L, and of G, H any equimultiples whatever M, N :' then because E is the same multiple of A, that Fis of C; and of E and F have been taken equimultiples K, L: therefore K is the same multiple of A, that I is of C: (v. 3.) for the same reason, M is the same multiple of B, that N is of 'D. And because, as A is to B, so is C to D, (hyp.) and if equal, equal; if less, less : (v. def. 5.) and M, N any whatever of G, H; 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 the third C has to the fourth D. and of A and C let E and F be any equimultiples whatever. Then E shall be to B as F to D. and of B, D any equimultiples whatever G, H: then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: and because A is to B, as C is to D, (hyp.) and of A and C certain equimultiples have been taken viz., K and L; and of B and D certain equimultiples G, H; and if equal, equal ; if less, less : (v. def. 5.) and G, H any whatever of B, D; PROPOSITION V. THEOREM. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole. Let the magnitude AB be the same multiple of CD, that AE taken G A E в Take AG the same multiple of FD, that AE is of CF: therefore AE is the same multiple of CF, that EG is of CD: (v. 1.) but AE, by the hypothesis, is the same multiple of CF, that AB'is of CD; wherefore EG is equal to AB: (v. ax. 1.) and the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, (constr.) and that AG has been proved equal to EB; therefore AE is the same multiple of CF, that EB is of FD: but AE is the same multiple of CF that AB is of CD: (hyp.) therefore EB is the same multiple of FD, that AB is of CD. Therefore, if one magnitude, &c. Q. E. D. PROPOSITION VI. THEOREM. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two; the remainders are either equal to these others, or equimultiples of them. Let the two magnitudes AB, CD he equimultiples of the two E, F, and let AG, CH taken from the first two be equimultiples of the same E, F. Then the remainders GB, HD shall be either equal to E, F, or equimultiples of them. A G B E- Make CK equal to F: and because AG is the same multiple of E, that CH is of F: (hyp.) and that GB is equal to E, and CK to F; therefore AB is the same multiple of E, that KH is of F': but AB, by the hypothesis, is the same multiple of E, that CD is of F; therefore KH is the same multiple of F, that CD is of F: wherefore KH is equal to CD: (v. ax. 1.) take away the common magnitude CH, but KC is equal to F: (constr.) therefore HD is equal to F. Next let GB be a multiple of E. AGB K c__I_D FMake CR the same multiple of F, that GB is of E: and because AG is the same multiple of E, that CH is of F: (hyp.) and GB the same multiple of E, that CK is of F; therefore AB is the same multiple of E, that KH’is of F: (v. 2.) hut AB is the same multiple of E, that CD is of F; (hyp.) take away CH from both; therefore the remainder KC is equal to the remainder HD: and because GB is the same multiple of E, that KC is of F, (constr.) and that KC is equal to HD; If, therefore, two magnitudes, &c. Q.E.D. |