322 



RATIO. 



N P M I 





Ratio, the multiple of C not less than 

 the multiple of DE. 



For of ED, DF, let IG, GH, be 

 taken, such equimultiples, that 

 each of them may be greater than 

 C ; and, as in the last article, let 

 M be taken, such a multiple of C, 

 that it may be greater than IG, 

 but less than IH. By Prop. 1. in 

 the 5th book of Euclid, IH, IG, are equimultiples of 

 FE, DE, and therefore let P be taken, the same multi- 

 ple of B that either of them is of its part ; and let N 

 be the same multiple of A that M is of C. Then, as A 

 B, C, FE, are proportionals, and as M is less than IH, 

 N is less than P, by article 8. Consequently N, the 

 multiple of A the first, is less than P, the multiple 

 of B the second ; but M, the multiple of C the third, 

 is not less than IG, the multiple of DE the fourth. 



XI. If any equimultiples whatever be taken of the 

 first and third of four magnitudes, and any equimulti- 

 ples whatever of the second and fourth ; and if, when 

 the multiple of the first is less than that of the second, 

 the multiple of the third is also less than that of the 

 fourth ; or, if when the multiple of the first is equal to 

 that of the second, the multiple of the third is also 

 equal to that of the fourth ; or, if when the multiple 

 of the first is greater than that of the second, the mul- 

 tiple of the third is also greater than that of the fourth ; 

 then, the first of the four magnitudes will have the 

 same ratio to the second, that the third has to the 

 fourth. 



For, if the first have not the same ratio to the se- 

 cond, that the third has to the fourth, it will have to 

 the second, the same ratio that the third has to a mag- 

 nitude, either greater or less than the fourth. But if 

 the first have the same ratio to the second, that the 

 third has to a magnitude greater than the fourth, then, 

 by article 10, certain equimultiples can be taken of the 

 first and third, and certain equimultiples of the second 

 and fourth, such, that the multiple of the first shall be 

 less than the multiple of the second, but live multiple 

 of the third not less than the multiple of the fourth ; 

 and this would be contrary to the first of the above 

 suppositions. 



Again, if the first has the same ratio to the second, 

 that the third has to a magnitude less than the fourth, 

 then, by article 9> certain equimultiples can be taken 

 of the first and third, and certain equimultiples of the 

 second and fourth ; such, that the multiple of the first 

 shall be greater than the multiple of the second, but 

 the multiple of the third not greater than the multiple 

 of the fourth ; and this would be contrary to the last of 

 the three suppositions. 



Lastly, if the multiple of the first be equal to the 

 multiple of the second, and the multiple of the third to 

 the multiple of the fourth, then the multiple of the first 

 will have the same ratio to that of the second, that the 

 multiple of the third has to that of the fourth ; and 

 consequently, by article 7, the first will have the same 

 ratio to the second, that the third has to the fourth. 



Remark. The fifth definition of the 5th book of 

 Euclid, having been considered as a proposition, and 

 established as such by demonstration, the doctrine of 

 ratio and proportion may be extended as in that book. 

 The same extension, however, may be effected by 

 means of the first seven of the preceding articles, as a 

 foundation connected with this evident truth, that two 

 magnitudes of the same kind must have the same ratio 



to one another, as the numbers which measure them, 

 or express their relative values. Whatever is proved " 

 as to the proportionality of the numbers, must be appli- 

 cable to the magnitudes to which they are strictly 

 analogous. 



XII. If four numbers be proportionals, the product 

 of the first and fourth is equal to the product of the se- 

 cond and third. Thus if N, P, M, Q, be four num- 

 bers, and if it be N : P : : M : Q, then N x Q = P X 

 M. For dividing the first and third of the proportion. 

 als by N, and the second and fourth by P, we have ac- 



cording to article 7, 1 : 1 : : ^ : , and therefore, by 



article 5, -^=p> and N x Q = P X M. 



XIII. If there be four numbers, such that the pro- 

 duct of the first and fourth is equal to the product of 

 the second and third, the first has the same ratio to 

 the second, that the third has to the fourth. Thus, if 

 N, P, M, Q, be four numbers, and if N x Q = P x M, 

 then N : P : : M : Q. For, let R be a fourth propor- 

 tional number to N, P, M ; and then, by the last arti- 

 cle, N x R = P X M. But, by hypothesis, P x M = N 

 X Q ; and therefore NX Q = N x R. Consequently, 

 N . P : : M : Q. 



In the following articles, let the small letters, a, o, c 

 &c. denote the numbers which express the relative va- 

 lues of the magnitudes A, B, C, &c., and then the 

 subsequent explanation applies to them all. The large 

 letters are used in the data and assertions, the de- 

 monstrations are effected by the small letters, and the 

 large are put instead of the small in the conclusion, 

 thereby intimating that the assertion has been proved. 



XIV. If four magnitudes of the same kind be pro- 

 portionals, they will also be proportionals when taken 

 alternately. Thus, if it be A : B : : C : D, then A C 

 : : B ; D. 



For it being a : b : : c : d, by article 12, a x d b 

 X c; and therefore, by article 13, a : c : : b : d, that is 

 A : C : : B : D. 



XV. Ratios that are equal to the same ratio, are 

 equal to one another. That is, if it be A : B : : C : D 

 and C : D : : E : F, then A : B : : E : F. For it bel 

 ing a : b : : c : d, and c : d : : e :f, by article 12, x d 



, , ., f axd a c 



= b x c, and therefore c, and - -. For the 

 o b d 



c e a e 



same reasons, -= 7,, and consequently, ~= -, and 



" J J 



a Xf b X c, and by article 13, a : b : : e : f. Hence 

 A: B:: E: F. 



XVI. If any number of magnitudes be proportion- 

 als, any one of the antecedents has the same ratio toils 

 consequent, that all the antecedents, taken together, 

 have to all the consequents taken together. That is, if 

 it be A : B : : C :D, and C : D : : E : F, then A : B : : A-f. 

 C + E : B + D -f. F. For it being a : b : : c : d, and 

 c:d::e:f, by the last article, a:b::e:f. By article 

 1 2, therefore, we have axd h^c, and a xf= bx*, 

 and consequently a x d -f nxf= b x c -f b xe. To 

 these equals add axb, and then a x 6 -fa xd + Xf=z 



Ratio 



Consequently,^ article 13, a: b : : a-j-c-f- e: b-4-d-l-f- 

 that is, A : B ; : A + C -f- E : B + D -f F. 



XVII. If of four magnitudes the first and second to- 

 gether- ha/ve the same ratio to the second, that the third 

 and fourth together have to the fourth; the first will have 



