BOOK v. PROP, xvii. xx.] MATHEMATICS. PL ANE GEOMETRY. 



583 



Let the two sets of magnitudes be 



A, B, C) , ., , ( A : B :: E : P 

 D! E, F | Buch that 1 D : E : : B : 

 it is to be proved that A : : : D : F. 



Of A, B, D take any equimultiples, mA, mB, mT>, and 

 of C, E, F, any equimultiples. nO, nE, 7iF ; then 



(Prop. VIII.) m.\ : mB : : nE : nF, ) ( mA, mB, nC 

 and (Prop. VI.) mD : nE : :mB : nC J ( mD, nE, nF. 



Hence the first of the three magnitudes, on the right, 

 are relate^ to the other three, as in Proposition XV. ; 

 .'. if mATnC, then mD7nF ; or if mD7"F, then mA 

 7nC. Consequently, (Prop. V.) A : C : : D : F; .'. if 

 there be three magnitudes, <Jrc. Q. E. D. 



COR. If to one of the above sets a fourth magnitude 

 P be annexed, and to the other set, a fourth magnitude 

 Q be prefixed, such as to furnish an additional proportion, 

 Q : D : : C : P ; then, from the hypotheses, and the 

 foregoing conclusion, namely, that 



it follows that A, C, P are related to Q, D, F, as in the 

 proposition ; and therefore that A : P : : Q : F. And 

 in this way may the proposition be extended to any 

 number of magnitudes. 



PROPOSITION XVII. THEOREM. 



If there be two sets of magnitudes, the number being 

 the same in each set, such that the first is to the second 

 in the one set, as the first to the second in the other set, 

 the second to the third in the former set, as the second 

 to the third in the latter set, and so on : then as the first 

 magnitude in the one set is to the last, so is the first 

 magnitude in the other set to the last. 



First, let there be three magnitudes in each set, 

 namely 



A, B, CM . ,,, ( A : B : : D : E 

 D, E, F j " l lt ( B : C : : E : F, 

 then it is to be proved that A : C : : D : F. 



Take any equimultiples, mA, mD of A, D ; and any 

 equimultiples 0, F, of C, F ; then (Props. VL and 

 VII.), 



B : mA : : E : mD \ f A, C, D, F 

 and B : nC : : E : nF ) ( mA, C, mD, nF. 



Let mA 7 "C, then (Ax. 4) a multiple pmA of mA, 

 may be taken so great that />m A will contain B oftener 

 than pnC will contain B. But /nuT) contains E as often 

 as pmA contains B (Def. V.) ; while pnf contains E 

 only at often as ;mC contains B, . . pmD contains E 

 oftener than pnF contains E, . . mD 7 "F ; so that if 

 mA 7 "C then mD 7 "F. In like manner it may be 

 proved, by interchanging A, C, with D, F, and also 

 interchanging B with E, that if mD 7 F, then mA 7 

 >iC ; . . A : C : : D : F. 



Next let there be four magnitudes in each set 

 namely, 



A, B, C, P ) . . , . f A : C : : D : F 

 D, E;F,QJ 8Uchthat lc:P::F:Q, 

 then it is to be proved that A : P : : D : Q. And this is 

 done as in the first case, since the three magnitudes 

 A, C, P are related, by the foregoing proportions, to the 

 three D, F, Q, as that case supposes ; . '. A : P : : D : Q. 

 And in like manner may the case for fire magnitudes be 

 deduced from this for/ur , and so on for any number 

 of magnitudes in each set ; .'. if there be two sett of 

 tintijiiitudti, (fee. Q. E. D. 



COR. If the consequents in one proportion be the 

 antecedents in another, a third proportion may be 

 formed, having the same antecedents as the first, and 

 the same consequents as the second ; thus if 

 A :B : :C :D 



and B : K : : D : L' 



then A : K : : C : L. 



Horn tt will be obserred that, in the last three proposition*, the 

 mafrnitudes la each set are nil of the tame kind; but that those in 

 one let, need not be of the tame kind as those in the other sets. 



PROPOSITION XVIII. THEOREM. 

 In any proportion, the sum of the first two terms is to 



their difference, as the sum of the other two is to their 

 difference. 



Let A : B : : C : D ; then A + B:A^B::C + D: 

 C c D. 



For (Prop. XIII.) A + B:A::C + D:C, 

 and (inverting Prop. XIV.) A : A </> B : : C : C on D. 



Consequently (Prop. XVII., Cor.) A + B : A oo B : : 

 C + D : C t D, . . in any proportion, &c. Q. E. D. 



PROPOSITION XIX. THEOREM. 



If the antecedents in one proportion be the same as 

 those in another, then the first antecedent is to the Sum 

 or difference of the first consequents, as the second ante- 

 cedent is to the sum or difference of the second 

 consequents. 



Let the proportions be A : B : : C : D ; and A : E : : 

 C : F ; it is to be proved that A:BE::C:DF. 

 Inverting the first of the given proportions (Prop. 



B : A : : D : CM /. (Prop. XVII., Cor.) 

 but A : E : : C : F j B : E : : D : F. 



Consequently (Props. XIII., XIV.) B : B & E : : D : 

 D j; F ; and, comparing this with the first of the given 

 proportions, 



A:BE::C:D;fcF; .'. if the antecedents, dkc. 

 Q. E. D. 



COR. If the terms are all homogeneous, then by alter- 

 nation, A:C::BE:DF;andB:D::BE: 

 D*F. 



PROPOSITION XX. THEOREM. 



In a proportion whose terms are homogeneous, the sum 

 of the greatest and least terms exceeds the sum of the 

 other two. 



Let A : B : : C : D ; and first let an antecedent, A, be 

 the greatest term ; then will D be the least (Prop. IX.) ; 

 and it is to be proved that (A + D) 7 (B + C.) 



By Prop. XIV., and inversion, A : A B : : C : C 

 D: but by hyp. A 7 0, /. (Prop. IV.) (A B) 

 7 (C D). To each of these unequals add B + D, 

 then (A + D) 7 (B + C). 



Next, let a consequent, B, be the greatest : then by 

 inverting the proportion, B : A : : D : C ; and since by 

 hyp. B is the greatest of these, C is the least, and .' . as 

 juat proved (B -f- C) 7 (A + D) ; .'. tn a proportion, 

 fee. Q. E. D. 



COR. If the proportion be A : B : : B : C, then (A 

 -f- C) 7 2 B ; that is, the sum of the extremes, in three 

 proportionals, exceeds twice the mean. 



The three propositions following, though of no appli- 

 cation in the Sixth Book of Euclid, will be found useful 

 in the consideration of incommensurable quantities 



PROPOSITION XXL THEOREM. 



If a magnitude measure each of two others, it will also 

 measure their sum and difference. 



Let A, B be any two magnitudes, and let C be a third 

 magnitude which measures each ; that is, such that A --- 

 mC and B = nC, m and n being whole numbers. Then 

 A + B = n,C -|- nC = (m + ) C ; also A </> B = mC 

 v> C = (m c/i n) C : but C is contained in the former 

 m -f- n times exactly, and in the latter m en n times 

 exactly, .'. C measures both A + B, and A t/> B ; .'.if 

 a magnitude, See. Q. E. D. 



COR. If C measure B, and also A + B and A B, 

 it must likewise measure A : for the sum of A B 

 and B is A, and the difference of A + B and B, is A ; 

 and as shown above, C measures both this sum and 

 difference. 



PROPOSITION XXIL PROBLEM. 



Two magnitudes of the same kind being given to find 

 their greatest common measure. 



Let the two given magnitudes be A, B : it is required 



