For, if not, let AB be to BE as CD to DG: Then (5. 17) since these four magnitudes are proportionals, they are also proportionals, when taken separately, that is, AE is to EB as CG to GD: But (Hyp.) AE is to EB as CF to FD; therefore CF is to FD as CG to GD: Therefore (5. 14) if CF be greater than CG, FD will be greater than GD, or if less, less: But, since CF and FD together make up the same whole as CG and GD, therefore, if CF be greater than CG, FD will be less than GD, or if less, greater-which is absurd: therefore AB is to BE as CD to DF. Wherefore, If four magnitudes &c. Q.E.D. C EF G B D PROP. XIX. THEOR. If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, then the remainder shall be to the remainder as the whole to the whole. Let the whole AB, be to the whole CD, as AE, a magnitude taken from AB, is to CF, a magnitude taken from CD then the remainder EB shall be to the remainder FD as the whole AB to the whole CD. For, because AB is to CD as AE to CF, therefore, alternately (5. 16), AB is to AE as CD to CF : But, if magnitudes taken jointly be proportionals, they are also proportionals when taken separately (5. 17); therefore, EB is to AE as FD to CF, and, alternately, EB is to FD as AE to CF, that is, as AB to CD (Hyp.). F B D Wherefore, If the whole &c. Q. E.D. COR. If the whole be to the whole as a magnitude taken from the first is to a magnitude taken from the other, then also the remainder shall be to the remainder as the magnitude taken from the first is to that taken from the other; that is, if AB be to CD as EB to FD, then AE shall be to CF as EB to FD. PROP. E. THEOR. If four magnitudes be proportionals, they shall also be proportionals by conversion; that is, the first shall be to its excess above the second as the third to its excess above the fourth. Let AB be to BE as CD to DF: then BA shall be to AE as DC to CF. A For, because AB is to BE as CD to DF, therefore, by division (5. 17), AE is to EB as CF to FD, and, by inversion (5. B), BE is to EA as DF to FC: Therefore, by composition (5. 18), BA is to AE as DC is to CF. Wherefore, If four magnitudes &c. Q. E.D. E с F B D PROP. XX. THEOR. If there be three magnitudes, and other three, which, taken two and two, have the same ratio, then, if the first be greater than the third, the fourth shall be greater than the sixth, and if equal, equal, and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, which, taken two and two, have the same ratio, that is, let A be to B as D to E, and let B be to C as E to F: then, if A be greater than C, D shall be greater than F, and if equal, equal, and if less, less. First, let A be greater than C; D shall be greater than F: For, because A is greater than C, and B is any other magnitude, therefore (5. 8) A has to B a greater ratio than C has to B: But A is to B as D is to E; therefore D has to E a greater ratio than C to B: And because B is to C as E to F, by inversion, C is to B as F is to E: And D was shewn to have to E a greater ratio than C to B; therefore D has to E a greater ratio than F to E, and therefore (5. 10) D is greater than F. Next, let A be equal to C: D shall be equal to F: For, because A is equal to C, and B is any other magnitude, therefore A is to B as C is to B (5. 7): But A is to B as D to Lastly, let A be less than C; D shall be less than F: For C is greater than A; and, as before, C is to B as F to E, and, in like manner, B is to A as E to D; therefore, by the first case, F is greater than D, that is, D is less than F. Wherefore, If there be three &c. ABC DEF Q E.D. AB DEF BC DEF A PROP. XXI. THEOR. If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order, then if the first magnitude be greater than the third, the fourth shall be greater than the sixth, and if equal, equal, and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio, taken two and two, but in a cross order, that is, let A be to B as E is to F, and let B be to C as D is to E: then if A be greater than C, D shall be greater than F, and if equal, equal, and if less, less. First, let A be greater than C; D shall be greater than F: For, because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B: But A is to B as E to F; therefore E has to F a greater ratio than C to B: And because B is to C as D to E, therefore, by inversion, C is to B as E to D: And E was shewn to have to F a greater ratio than C to B; ABO therefore E has to F a greater ratio than E to D; and therefore F is less than D, that is, D is greater than F. DEF Next, let A be equal to C; D shall be equal to F: For, because A is equal to C, and B is any other magnitude, therefore A is to B as C is to B: But A is to B as E to F, and C is to B as E to D; therefore E is to F as E to D, and therefore D is equal to F. Lastly, let A be less than C; D shall be less than F: For C is greater than A; and, as before, C is to B as E to D, and, in like manner, B is to A as F to E; therefore, by the first case, since C is greater than A, F is greater than D, that is, D is less than F. Wherefore, If there be three &c. Q. E. D. BC A B C DEF PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio, then the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N.B. This is usually cited by the words "ex æquali." First, let there be three magnitudes, A, B, C, and other three, D, E, F, which, taken two and two, have R the same ratio, that is, let A be to B as D to E, and B to C as E to F: then A shall be to C as D to F. Take of A, D, any equimultiples whatever G, H, and of B, E, any whatever K, L, and of C, F any whatever M, N: Then, because A is to B as D to E, and that G, H are equimultiples of A, D, and K, L equimultiples of B, E, therefore GK M G is to K as H to L (5.4): And, for A B C DEF same ratio, therefore (5. 20) if G be greater than M, His greater than N, and if equal, equal, and if less, less: But G, H are any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F; therefore A is to C as D to F. Next, let there be four magnitudes, A, B, C, D, and other four E, F, G, H, which, taken two and two, have the same ratio, that is, let A be to B as E to F, B to C as F to G, and C to D as G to H: then A shall be to D as E to H. A. B. C. D. For, because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio, therefore, by the first case, A is to C as E to G: But C is to D as G is to H; therefore also, by the first case, A is to D as E to H: And so we may proceed, whatever be the number of magnitudes. Wherefore, If there be any number &c. Q.E.D. PROP. XXIII. THEOR. If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same |