G H K if G be greater than L, H is greater than M; and if equal, equal; and if less, less*. Again, because C is * 5 Def. 5. to D, as E is to F, and H, K are taken equimultiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; and if equal, equal; and if less, less : but if G be greater than L, it has D F been shewn that H is Mgreater than M; and if equal, equal; and if less, less: therefore if G be greater than L, K is greater than N; and if equal, cqual; and if less, less: and G, K are any equimultiples whatever of A, E; and L, N any whatever of B, F: therefore * . 5 Def. 5. as A is to B, so is E to F. Wherefore, ratios that, &c. A N Q. E. D. G K C F PROP. XII. THEOR. the antecedents is to its consequent, so shall all the Let any number of magnitudes A, B, C, D, E, F, HIequimultiples whatever A E Nwhatever, L, M, N: then, because A is to B, as C is to D, and as E to F; and that G, H, K are equimultiples of A, C, E, and L, M, N, equimultiples of B, D, F; if G be greater than L, H is greater than M, and K greater than N; and if equal, equal ; and if less, less *. wherefore if G be greater * 5 Def. 5. than L, then G, H, K together are greater than L, M, N together; and if equal, equal; and if less, less: but G, and G, H, K together are any equimultiples of A, and A, C, E together; because if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole *: for the same reason * 1.5. L, and L, M, N are any equimultiples of B, and B, D, F: therefore as A is to B t, so are A, C, E, together + 5 Def. 5. to B, D, F together. Wherefore, if any number, &c. Q. E. D. PROP. XIII. THEOR. H See N. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth has to the sixth. Let A the first bave the same ratio to B the second which the third has to D the fourth, but the third a greater ratio to D the fourth, than E the fifth has to F the sixth : also the first A shall have to the second B a greater ratio than the fifth E has to the sixth F. Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and some of D and F such, that the multiple of C is greater than the multiple of D, but the multiple of E is not M Agreater than the multi B D * 7 Def. 5. ple of F*: let these be N- L- same multiple of A; and whatever multiple K is of D, + Hyp. take N the same multiple of B: then, because A is to Bt, as C to D, and of A and C, M and G are equimultiples; and of B and D, N and K are equimultiples ; if M be greater than N, G is greater than K; and if equal, • 5 Def. 5. equal; and if less, less * but G is greater+ than K; + Constr. therefore M is greater than N: but H is not + greater + Constr. than Li and M, H are equimultiples of A, E; and N, L equimultiples of B, F; therefore A has a greater * 7 Def. 5. ratio to B, than E has to F* Wherefore, if the first, &c. Q. E. D. Cor. And if the first have a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth ; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth. See N. PROP. XIV. THEOR. third has to the fourth; then, if the first be greater than # 8. 5. fore also has to D. ABCD All Let the first A have the same ratio to the second B which the third C has to the fourth D: if A be greater than C, B shall be greater than D. Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to B*: but, as A is to Bt, so is C to D; there + Hyp. greater ratio than C has to B*: but of two magnitudes, that to which the same * 13. 5. has the greater ratio is the lesser * : therefore D is less 10, 5. than B ; that is, B is greater than D. Secondly, if A be equal to C, B shall be equal to D. For A is to B, as C, that is, A to D: therefore B is equal to D*. * 9.5. Thirdly, if A be less than C, B shall be less than D. For C is greater than A; and because C is to D, as A is to B, therefore D is greater than B, by the first case; that is, B is less than D. Therefore, if the first, &c. Q. E. D. * PROP. XV. THEOR. A Magnitudes have the same ratio to one another which their equimultiples have. Let AB be the same multiple of C, that DE is of F: C shall be to F, as AB to DE. Because AB is the same multiple of C, that DE is of F, there are as many magnitudes in AB equal to C, as there are in DE equal to F: let AB be divided into magnitudes, GH each equal to C, viz. AG, GH, HB; and K DE into magnitudes, each equal to F, viz. IIH LE DK, KL, LE: then the number of the BC EF first AG, GH, HB, is equal to the number of the last DK, KL, LÊ: and because AG, GH, HB are all equal, and that DK, KL, LE, are also equal to one another; therefore * AG is to DK as GH to *7.5. KL, and as HB to LE: but as one of the antecedents to its consequent *, so are all the antecedents together * 12. 5. to all the consequents together; wherefore, as AG is to DK, so is AB to DE: but AG is equal to C, and DK to F; therefore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q. E. D. PROP. XVI. THEOR. If four magnitudes of the same kind be proportionals, they shall also be proportionals when taken alternately. * 11. 5. Let A, B, C, D be four magnitudes of the same kind, which are proportionals, viz. as A to B, so C to D: they shall also be proportionals when taken alternately; that is, A shall be to C, as B to D. Take of A and B any equimultiples whatever E and F: and of C and D take any equimultiples whatever G and H: and because E is the same multiple of A, that F is of B, and that magnitudes bave the same * 15. 5. ratio to one another * which their equimultiples have; + Hyp. therefore A is to B, as E is to F: but as A is to B sot D- -H* 15. 5. so* is G to H: but it was proved that as C is to D, so is E to F; therefore, as E is to F, so* is G to H. But when four magnitudes are * 14. 5. proportionals *, if the first be greater than the third, the second is greater than the fourth ; and if equal, equal ; if less, less; therefore if E be greater than G, F likewise is greater than H; and if equal, equal; if + Constr. less, less : and E, F are anyt equimultiples whatever * 5 Def. 5. of A, B; and G, H any whatever of C, D: therefore * A is to C as B to D. If, then, four magnitudes, &c. -G B * 11.5. PROP. XVII. THEOR. See N. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so let CD be to DF: they shall also be proportionals taken separately, viz. as AE to EB, so shall CF be to FD. Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD take any equimultiples whatever KX, NP: and because GH is the same multiple of AE, that HK is of EB, therefore GH is the same multiple* of AE, that * 1. 5. GK is of AB: but GH is the same multiple of AE, that LM is of CF; therefore GK is the same multiple of AB, that LM is of CF. Again, because LM is the same niultiple of CF, that MN is of FD; therefore LM is the same multiple* of CF, that LN is of CD: * 1. 5. but LM was shewn to be the same multiple of CF, that GK is of AB; therefore GK * X is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of KH AB, CD. Next, because HK is the same NH multiple of EB, that MN is of FD; and HB DM that KX is also the same multiple of EB, EH FH that NP is of FD; therefore HX is the same multiple* of EB, that MP is of FD. G Á • L * 2. 5. And because AB is to BEt, as CD is to + Hyp: DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are equimultiples*; therefore if GK be greater than HX, then * 5 Def. 5. LN is greater than MP; and if equal, equal; and if less, less : but if GH be greater than KX, then, by adding the common part HK to both, GK is greatert + 4 Ax. 1. than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than + 5 Ax. 1. NP: therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, that if GH be equal to KX, LM is equal to NP; and if less, less: but GH, LM are any equimultiples whatever of AE, CFt, and KX, NP are any + Constr. whatever of EB, FD: therefore*, as AE is to EB, so * 5 Def.5. is CF to FD. If then, magnitudes, &c. Q. E. D. PROP. XVIII. THEOR. If magnitudes, taken separately, be proportionals, they See N. shall also be proportionals when taken jointly: that is, |