RATIO. 



321 



E F G II 



Itatio. For ai, by hypothesis, A is exact- 



m ~~Y~ mm/ ly as great, wlu-n compared to B, as 

 C is when compared to 1), it is evi- 

 dent that the il<ml>le, triple, or any 

 multiple of A, will be exactly as 

 jreat when compared to B, as the 

 double, triple, or the same multiple 

 of C, when compared to 1) ; and, 

 therefore, E is exactly as great when 

 compared to B, as G is when com- 

 pared to D. From this it follows, 

 that E is exactly as great when compared to the dou- 

 ble, triple, or any multiple of B, as G is when com- 

 pared to the double, triple, or the same multiple of 1). 

 Consequently E is exactly as great when compared to 

 F, as G is when compared to H ; or E has the same 

 ratio to F that G has to H. 



VII. If the first of four magnitudes has to the se- 

 cond, the same ratio that the third has to the fourth, 

 and if any like aliquot parts whatever be taken of the 

 first and third, and any like aliquot parts whatever of 

 the second and fourth ; the part of the first will have 

 the same ratio to the part of the second, that the part 

 of the third has to the part of the fourth. 



Let A the first have to B the second, the same ratio 

 that C the third has to D the fourth ; 

 and let E, G be any aliquot parts 

 whatever of A and C, and F, H any 

 whatever of B, D ; and then E will 

 have the same ratio to F that G has 

 toH. 



For A being exactly as great when 

 compared to B, as C is when com- 

 pared to D, it is evident that the 

 half, third, or any aliquot part of A, 

 will be exactly as great when com- 

 pared to B, as the half, third, or the 

 same aliquot part of C is when com- 

 pared to D. But E,G are alike ali- 

 quot parts of A,C, and, therefore, E 

 has the same ratio to B that G has 

 to D. Again, as E is exactly as 

 great when compared to B, as G is when compared to 

 I)* it is evident that E must be exactly as great when 

 compared to the half, third, or any aliquot part of B, as 

 G is when compared to the half, third, or the same ali- 

 quot part of D. But F,H are like aliquot parts of B,D, 

 and, therefore, E is exactly as great when compared to 

 Fi as G is when compared to H ; and E has the same 

 ratio to F, that G has to H. 



REMARK. The preceding articles contain the at- 

 tempt to attain the first of the three objects already men- 

 tioned. The sixth article is the 4th Proposition in the 

 5th Book of Euclid, and by the same mode of reason- 

 ing as is employed in that article, the 7th, 8th, 9th, 

 10th, llth, 13th, 14th and 15th Propositions in the 5th 

 Book may be demonstrated, as also Simson's Proposi- 

 tions A, B, C, D. We now proceed to the second ob- 

 ject, or the demonstration of the 5th Definition. 



VIII. If the first of four magnitudes has the same 

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

 if any equimultiples whatever be taken of the first and 

 third, and also any whatever of the second and fourth ; 

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

 the second, the multiple of the third will be equal to 

 the multiple of the fourth ; if greater, greater ; if less, 

 less. For, by article 6', the multiples will be propor, 



VOL. XVII. PART I. 



A B C D 



E F G H 



tionali, and, therefore} the assertion it true, by arti- 

 cle 6. 



The same things being allowed as above, it evident- 

 ly follows, that if the multiple of the third be greater 

 than the multiple of the fourth, the multiple of the first 

 will be greater than the multiple of the second ; and if 

 the multiple of the third be less than the multiple of the 

 fourth, the multiple of the first will be lean than the 

 multiple of the second. 



IX. If the first of four magnitudes has the same ra- 

 tio to the second that the third has to a magnitude lets 

 than the fourth, then it is possible to take certain equi- 

 multiples of the first and third, and certain equimulti- 

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

 of the first shall be greater than the multiple of the se- 

 cond, but the multiple of the third not greater 

 than the multiple of the fourth. 



Let A; B, C, DE be four magnitudes, and let 

 A have the same ratio to B that C has to FE, a 



1 



N P M L K 



magnitude less than DE ; then it 

 is possible to take certain equi- 

 multiples of A, C, and certain 

 equimultiples of B, DE, such, ABC 

 that the multiple of A shall be 

 greater than the multiple of B, 

 but the multiple of C not great- 

 er than the multiple of DE. 



Of DF, FE, take such equi- 

 multiples GH, HI, that each of 

 them may be greater than C. 

 Then of C take H the double, L 

 the triple, &c. until a multiple of 

 C be obtained greater than HI. Let M be the multiple 

 of C, which first becomes greater than HI, and L the 

 multiple of C, which is next less than M, and then HI 

 is not less than L. But, by the construction, GH is 

 greater than C ; and as M is equal to L and C toge- 

 ther, M is greater than HI, but not greater than GI. 

 Let N be the same multiple of A that AI is of C, and 

 P the same multiple of B that HI is of FE ; and then, 

 as A, B, C, FE are proportionals, and as M is greater 

 th,an HI, N is greater than P, by article S. Again, as 

 GH, HI, are equimultiples of DF, FE, by the first 

 proposition in the 5th book of Euclid, GI is the same 

 multiple of DE that HI is of FE, or that P is of B. 

 Consequently, certain equimultiples, N, M, have been 

 taken of A the first and C the third ; and certain equi- 

 multiples, P and GI, of B the second and DE the 

 fourth ; such, that N is greater than P, but AI is not 

 greater than GI. 



X. If the first of four magnitudes has the same ratio 

 to the second, that the third has to a magnitude great- 

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

 of the fourth. 



Let A, B, C, DE be four magnitudes, and let A the 

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

 third has to FE a magnitude greater than DE ; then 

 it is possible to take certain equi- 

 multiples of A and C, and certain 

 equimultiples of B and DE ; such, 

 that the multiple of A shall be 

 less than the multiple of B ; but A B C 



2 s 



