BOOK v. PROP, vi. x.] MATHEMATICS. PLANE GEOMETRY. 581 



contained in some multiple mB of its consequent oftener 

 than C is contained in mD (Def. VI.) Therefore, p being 

 any whole number, A must be contained oftener in p 

 than C is contained in pmD. 



Let nA be the greatest multiple of A that does not 

 exceed pmB ; then nA is not > pmB, and A ia contained 

 in nA, and in pmB the same number of times, namely, 

 n times. Therefore C is contained in pmD less than n 

 times. But C is contained in nC exactly n times, . C 

 >pmD. Consequently, of the four magnitudes, equi- 

 multiples of the antecedents, and equimultiples of the 

 consequents, may be taken, as below : 



A : B : : C : D 

 nA proB nG pmD 



such that nC > pmD, and yet nA not > pmB : but, by 

 hypothesis, this is impossible : therefore the magnitudes 

 cannot be other than proportionals ; . ' . if four magni- 

 tude*, i-c. Q. E.D. 



PROPOSITION VI. THEOREM. 

 In any proportion, if like multiples of the antecedents, 

 and like multiples of the consequents, be taken, the 

 results will form a proportion. 



Of the four proportionals, let equimultiples of ante- 

 cedents and of consequents be taken as below, m, n, p, 

 and 5 being any whole numbers : 

 A : B : : C : D 

 mA nB mC nD 

 pmA 3iB pmC ynD 



Then (Prop. III.) if pmA > <piB, it must follow that 

 pmCxjnD; or if pmC>qnD, that pmA > 5718. But 

 pmA, pmC, are any equimultiples of mA, mC ; and 

 7B//"D, are any equimultiples of nB,nD. Consequently 

 (Prop. V.), 



mA : nB : : mC : nD. 



Therefore, in any proposition, Ac. Q. E. D 



XOTK Either m or n mr of count bt unit ; u also in erery cut 

 where the multiple! are unroll ietrd. 



In Proposition V. it was demonstrated that "If four 

 magnitudes be such that whatever equimultiples of the 

 antecedents, and whatever equimultiples of the conse- 

 quents be taken, the multiple of one antecedent cannot 

 be greater than that of its consequent, without the mul- 

 tiple of the other antecedent being greater also than its 

 consequent, the four magnitudes are proportionals." It 

 is now proved that these multiples themselves are alt-o 

 proportionals. Consequently (Prop. IV.), according as 

 the first multiple is greater than, less than, or equal to 

 tip- second, so will the third multiple be greater than, 

 leas than, or equal to the fourth multiple. And this is 

 the condition which constitutes Euclid's criterion of pro- 

 portional magnitudes, as embodied in his celebrated fifth 

 definition, which, in the version of Playfair, is expressed 

 as follows : 



Euclid's Definition of Proportionals (Def. V.) 



If there be four magnitudes, and if any equimultiples 

 whatsoever be taken of the first and third, and any 

 i-.piimultiples whatsoever of the second and fourth, and 

 if according as the multiple of the first be greater than 

 the multiple of the second, equal to it, or less, the 

 multiple of the third also is greater than the multiple of 

 the fourth, equal to it, or less ; then the four magnitudes 

 are proportionals. This definition of Euclid is deduced 

 hero as a theorem ; but in what follows it will be referred 

 to as "Def. V., page 573." 



PROPOSITION VII. THEOREM. 

 The terms of any proportion form also a proportion 

 when they are taken INVERSELY ; that is, the second 

 1 term is to the first as the fourth is to the third, or the 

 second has the same ratio to the first that the fourth has 

 t< the third. 



Let the proportion be A : B : : C : D ; then also 

 B : A : : D : C. 



For (Prop. VI.) mA : nB : : mC : nD. 



And (Prop. IV.) according as nB is greater than, less 



than, or equal to m\, so is nD greater than, less than, or 

 equal to mC ; and m, n are any whole numbers what- 

 ever. But when this is the case in reference to four 

 magnitudes B, A, D, C, they are proportionals. (Def. 

 V., page 573). 



/. B : A : : D : C. 



Hence the terms of any proportion, <fec. Q. E. D. 



COR. Therefore a consequent is contained as often in 

 a multiple of its antecedent, as the other consequent is 

 contained in a like multiple of its antecedent, but not 

 of teiier. (Def. VI.) 



PROPOSITION VIII. THEOREM. 



In any proportion, if equimultiples of the first two 

 terms be taken, and also equimultiples of the last two, 

 the results will form a proportion. 



Let the proportion be A : B : : C : D ; then also 

 mA : mB : : nC : nD 



By Def. VI., A : B : : mA : mB ; and C : D : : C : 

 nD 



But A:B::C:D; .-. (Prop. II.) C : D : : mA : 

 mB 



and therefore, by the same prop. , mA : mB : : nC : ?vD ; 

 . . in any proportion, <fcc. Q E. D. 



PROPOSITION IX. THEOREM. 



In a proportion consisting of homogeneous magnitudes 

 that is, magnitudes all of the same kind if one ante- 

 cedent be greater than the other, the consequent of the 

 former will be greater than the consequent of the 

 latter. 



Let the magnitudes forming the proportion A : B : : 

 C : 1) be all of the same kind, and let A > C; then 

 also B > D. For let A C = P ; then (Ax. 4) there 

 exists some number m, such that mP > D, and con- 

 sequently such that mA contains D oftener than mC 

 contains D. 



But by taking the terms of the proportion inversely 

 (Prop. VII.) 



B:A::D:C 

 mA mC 



/. m.V does not contain B oftener than mC contains 1) 

 (Def. VI.), .'. mA contains D oftener than it contains 

 H, therefore B > D .'. in a proportion, <tc. Q. E. D. 



COR. I. In a proportion consisting of homogeneous 

 magnitudes, if one consequent be greater than the other, 

 the antecedent of the former will be greater than that of 

 the latter. 



This follows from the present proposition by inversion ; 

 and, consequently, in a proportion whose terms are all 

 homogeneous, if one antecedent be greater than, less 

 than, or equal to the other antecedent, the consequent of 

 the former will be greater than, less than, or equal to 

 the consequent of the latter, and conversely. 



COR. II. Therefore (Prop. II.) if two proportions 

 have three corresponding terms in each equal, each to 

 each, the fourth terms will be equal. 



Nont. It most be carefully observed that Proposition IX., as also 

 X. and XI. following, apply only when the magnitudes are all four 

 of the name kind. There can be no such relation as that implied in 

 the word ratio between things of different kinds. In the other 

 propositions hitherto discussed, it is necessary only that the first 

 and second of the magnitudes be of the same kind, and that the 

 third and fourth be aluo of the same kind. The latter pair, how- 

 erer, may differ in kind from the former pair: one pair may be 

 lines, or numbers, and the other pair surfaces, or solids : but no 

 ralin can exist between hcterpgmcoui quantities, or quantities 

 unlike in kind. 



PROPOSITION X. THEOREM. 



If any number of homogeneous magnitudes be pro- 

 portionals, then as one antecedent is to its consequent, 

 10 is the sum of all the antecedents to the sum of all the 

 consequents. 



First, let there be four proportionals, and let any 

 equimultiples of the antecedents, and any equal multi- 

 ales of the consequents, be taken thus : 

 A:B : :C:D 

 mA nB mC nD 

 It is to be proved that 



A:B::A + C:BfD 



