MATHEMATICS. PLANE GEOMETRY. [BOOK v. PROP, xi.xvi. 



By l'i -..;.. III. if mA > nB, then mC > nl) ; con- 

 equently, if mA > B, then aluo (mA + mC) > (nB 



And' cooTOrtely, if (mA + mC) > (nB -(- nD), thon 

 mA > nB. For, from the firnt ptOBOrWB, if this were 

 nut the caae, neither could mC > nD ; and, consequently, 

 neither could (mA + mC) > (nB + D) ; . . if thu latter 

 condition hare place, to, of necessity, must the condition 

 mA > nB. 



.-. (Prop. V.) A : B : : A + C:B + D. 



Next let there be aix proportionals, 

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



It has already been proved, in reference to the last 

 four, that 



C:D: :C + E:D + F 

 But C : D : : A : B. 



.-., (Prop. II.) A : B : :C + E:D + 



Consequently, by the first case above, 

 A:B::A+C + E:B 



And in a similar way as the proof for six proportionals 

 has been derived from that for four, so may the proof 

 for eight be derived from that for six ; and so on for any 

 number of proportionals ; . . if any number of homo- 

 geneout magnitudes, to. Q . E. 1). 



PROPOSITION XL THBOKEM. 



If the terms of a proportion are all of the same kind, 

 they also form a proportion when taken ALTKRNATKI.Y ; 

 that is, the first is to the third as the second is to the 

 fourth. Let the homogeneous proportionals be 



A : B : : C : D ; then also A : C : : B : D. 



For let any equimultiples of A, B, and any equimultiples 

 of C, D, be taken and arranged as below, 



A:B::C:D; A : C : : B : D 

 mA mB nC nD ; iA nC wiB nD 



Then (Prop VI.) mA : mB : : nC : nD ; . . (Prop. IX.) 

 if in A. > nC, then mB > nD ; and if m!3 > nD, then 

 mA > wC. 



.'. (Prop V.) A : C : : B : D. 



Hence, if the terms of a projwrtion, &o. Q. E. D. 



PROPOSITION XII. THEOREM. 



If in any proportion an antecedent be a multiple or 

 sulnnultiple of its consequent, the other antecedent will 

 be a like multiple or submultiple of its consequent. 



Let A, B, C, D be four proportionals, such that A = 

 mB, then will C = mD. For since 



A: B : :C: D 



.*. (Prop. VI.) A : mB : : C : iD ; but A = mB, .-. 

 C-,D(Prop. IV.) 



Again, let the proportionals be such that B = mA ; 

 then will D = mC. 



For by inversion (Prop. VII.) B : A : : D : C 

 and (Prop. VI.) B : mA : : D : mC, but B = mA .-. 

 D = ,C. 



. . if in any proportion, <tc. Q. E D. 



COR. Wlu-ii the proportionals are homogeneous, if one 

 antecedent be a multiple of thu other, the consequent 

 of thu former will be a like multiple of the consequent 

 of the hitter. 



PROPOSITION XIII. THEOREM. 



In any proportion, the sum of an antecedent and its 

 consequent is to either term, as the sum of tho other 

 lent and consequent ia to the like term. 



Let the proportion be A : B : C : D 

 then A+B : A : C + D : 

 also A+B : B : C + D : D 

 For A cannot be contained oftener in mB than C is con- 

 tained in mD (Def. VI), .'. A cannot be contained oftener 

 in m(A + B) than C is contained in m(C + D). Nor can 

 C be contained oftener in m(C 4- D) than A U contained 

 in m(A + B) ; for if it could, C would be contained in 



mD oftener than A in mB, which is impossible (Def. VI.) 

 /. A: A + B:: C: C + D. 



Again, by inversion, B : A : : D : C ; therefore, as 

 just proved, B :A + B ::D :C+D; consequently 

 inverting the two proportions now deduced, 

 A + B : A : : C + D : C, and A + B:B::C + D:D. 

 .'. in any proportion, &c. Q. E. D. 



COR. If the proportionals are homogeneous, then, by 

 alternation (Prop. XI.), 

 A + B :C + D::A:C;andA + B:C + D::B:D. 



PROPOSITION XIV. THEOREM. 



In any proportion, the difference between an antece- 

 dent and its consequent is to either term, as the ditier- 

 enco between the other antecedent and consequent is to 

 the like term. 



Let A : B : : C : D be any proportion ; then taking 

 any antecedent and consequent, as A, B, suppose first 

 that B>A, and consequently (Prop. IV.) that D>C. 



Take any equimultiples of B, D, and the same of 

 B A, D C ; and arrange the terms as usual 



A : B : : C : D 



mB mD 



: B A 



mB mA 



C 



D-C 



,C 



Then (Del. VI.) A cannot be contained oftener in iB 

 than C is contained in nil) ; but A is contained in m A 

 just as often as C is contained in iC, namely, m tinns, 

 without remainder. Therefore, A cannot be contained in 

 mB mA oftener than C is contained in mD i/iC. 



In like manner may it be shown that C cannot be 

 contained in mD C ofteuer than A is contained in 

 mB mA; .'. (Def. VI.) 



A : B A : : C : D C ; .'. (Prop. XIII.) 

 B : B A : : D : D C 



And inverting these two proportions (Prop. VII.), 



B A : A : : D C : C, and B A : B : : D C : D 

 which proves the theorem when A > B. 



Next let A>B, and consequently (Prop. IV.) C>D. 



By inversion, B : A : : D : C : .'.as proved al>ovc, 

 A B : B : : D : D, and A B : A : : C D : C, 

 which proves the theorem when A > B ; .'. A v> B : A 

 : : C c D : C ; and A v> B : B : : C c/> D : C .'. in any 

 proportion, <fcc. Q.E. D. 



PROPOSITION XV. THEOREM. 



If there be three magnitudes and other three, such 

 that, whichever set be taken, the first in that set is to 

 the second, as the second in the other set is to tho third ; 

 then if the first in one set be greater than the third, thu 

 lirst in the other set also will be greater than tho third. 



Let the two sets of magnitudes be 



A, B, C 1 , . ( A : B : : E : F 



D, E, F j 8uch that ( D : E : : B ; C 



If A>C then must D>F ; and if D>F, then A>C. 

 Let A > C : and take mA, mC, such equimultiples of 

 A, C that iA may contain B of truer than mC contains B 

 (Ax. 4) ; take also inE, the same multiple of E. Then 

 since by the second proportion mE contains D only at 

 as mC contains B, .'. mA contains B oftener than 

 mE contains D. 



But wE contains F as often as mA contains B (Prop. 

 VII., Cor.), therefore mE contains F oftener than mE 

 contains D, /. D > F. So that if A > C, then must 

 D > F. 



And in like manner may it be demonstrated that if 

 D>F then must A>C ; .'. ij there be three maijnitudet, 

 ic. Q. E. D. 



PROPOSITION XVI. THEOREM. 



If there be three magnitudes and other three such 

 that whichever set be taken, the first in that set is to the 

 second, as the second in the other set is to the third ; 

 tli.-ii the first in the one set will be to the third, as the 

 first in the other set is to the third. 



