After the like manner we demonftrate, that if A be equal to C, D will be equal to F: and if A be less than C, D will be less than F. If therefore there be three magnitudes, and other three, which taken two and two are in the fame ratio, and if the first be greater than the third, then will the fourth be greater than the fixth: If the first be equal to the third, the fourth will be equal to the fixth and if the firft be lefs than the third, the fourth will be less than the fixth. Which was to be demonftrated. PROP. XXI. THE O R. If there be three magnitudes, and other three, which taken two and two have the fame ratio, but in a cross order, and if the first be greater than the third: then will the fourth be greater than the fixth: if the first be equal to the third, the fourth will be equal to the fixth: and if the first be lefs than the third, the fourth will be less than the fixth. Let there be three magnitudes A, B, C, and other three D, E, F which taken two and two are in the same ratio, but in a cross order; viz. as A is to B, fo is E to F, and as B is to C, fo is D to E. And A be greater than c: I fay D is greater than F; if A be equal to C, D will be equal to F: and if A be lefs than C, D will be lefs than F. For because A is greater than C, and B is another magnitude; [by 8. 5] the ratio of A to B will be greater than the ratio of c to B. But as A is to B, fo is E to F, and inverfely as c is to B, fo is E to DA BC DEF Therefore E will have a greater ratio to F than it has to D. But when the fame magnitude has ratios to two others, that which has the greater ratio [by 19. 5.] is the leffer magnitude: Therefore F is less than D and fo D will be greater than F. In like manner we demonstrate if A be equal to C; D will be equal to F ; and if A be less than c; D will be less than F, If Book V. If therefore there be three magnitudes, and others equal to them in number, which taken two and two are in the fame ratio; and their proportion be perturbate, and if by equality, the first be greater than the third; then will the fourth be greater than the fixth: If the first be equal to the third, the fourth will be equal to the fixth and if the first be less than the third, the fourth will be less than the fixth. Which was to be demonstrated. PRO P. XXII. THEOR. If there be any number of magnitudes and others as many, which taken two and two in order are in the fame ratio; the first will have to the last of the first magnitudes, the fame ratio which the firft has to the last of the others. Note, It is common to express this in the words, by equality. Let there be any number and others D, E, F as many, GK MABC DEFHLN of magnitudes A, B, C, which taken two and two in order are in the fame ratio, as A is to E, fo is D to E, and as B is to c, fo is E to F: I fay, by equality, they fhall be in the fame ratio, as A is to c, fo is D to F. For take G, H, equimultiples of A, D, and K, L any other equimultiples of B, E; and M, N other equimultiples of C, F. Then because A is to B, as D is to E, and G, H, are taken equimultiples of A, D, and K, L any others of B, E; [by 4. 5.] it will be as G is to K, fo is H to L. Alfo by the fame reafon K will be to M, as L is to N. And fince there are three magnitudes G, K, M, and others H, L, N equal to them in number, which taken two and two are in the fame ratio: Therefore by equality, [by 20. 5.] if G exceeds M; H will exceed N; if G be equal to M, H will be equal to N: If G be less than M, H will be less than N. And G, H are equimultiples of A, D, and M, N any others of c, F: Therefore as A is to c, fo will [by 5. def. 5.] D be to F. Wherefore if there be any number of magnitudes, and others as many, which taken two and two in order are in the fame ratio; they will be alfo [by equality] in the fame ratio. Which was to be demonftrated. PROP. XXIII. THEOR. If there be three magnitudes, and three others, which taken two and two in a cross order have the fame ratio; the first fhall have to the last of the first magnitudes the fame ratio which the first has to the laft of the others. Note, This is ufually referr'd to in these words, by equality of crofs proportion. Let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a cross order, are in the fame ratio; viz. as A is to B, fo is E to F; and as B is to c, fo is D to E: I fay as A is to C, fo is D to F. For take G, H, K equimultiples of A, B, D, and L, M, N any other equimultiples of C, E, F. Then because G, H are equimultiples of A, B, and [by 15.5.1 the parts of magnitudes have the fame ratio, as their equimultiples; it will be as A to B, so is G to H. And by the fame reafon as E is to F, fo is M to N: But as A is to B, fo is Therefore as Gis to H, fo [by 11.5.] will м be to N. And because it is as B to c, fo is D to E, and H, K are taken equi E to F. multiples of B, D, and G HLAB C DEFKMN Book V 1, м any others of C, E; [by 15. 5.] it shall be as His to L, fo is K to M. But it has been proved that G is to H, as M is to N. Wherefore because there are three magnitudes G, H, L, and others K, M, N equal to them in number, which taken two and two are in the fame ratio; and their proportion is perturbate, then by equality, [by 21. 5.] if G exceeds D; K will exceed N: if G be equal to L; K will be equal to N: and if G be less than L; K will also be lefs than N. But G, K are equimultiples of A, D, and L, N equimultiples of c, F: Therefore as A is to c, fo will [by 5. def. 5.] D be to F. Wherefore if there be three magnitudes, and other three, which taken two and two in a cross order, are in the fame ratio, the firft fhall have to the laft of the firft magnitudes, the fame ratio which the firft has to the last of the others. a Euclid propofes only three magnitudes here; but they may be any number whatfoever, as well as they may in prop. 22. a geometrician feldom having occafion for any more. The fourteenth and twenty-fecond propofitions of the seventh book answer to these propositions in numbers. PROP. XXIV. THEOR. If the first of fix magnitudes has the fame ratio ta the fecond, as the third has to the fourth, and the fifth has the fame ratio to the fecond as the fixth bas to the fourth: then shall the first and fifth. magnitudes have the fame ratio to the fecond, as the third has to the fourth. For let the first of fix magnitudes A B have the fame ratio to the second c, as the third D E has to the fourth F: and let the fifth BG have the fame ratio to the second c, that the fixth E H, has to the fourth F: I fay the first and fifth magnitudes taken together, will have the fame ratio to the fecond c, as D H the third and fixth magnitudes put together has to the fourth F. For because as BG is to c, fo is E H to F; it will be inversely [by cor. 4. 5.] as c is to B G, fo is F to E H. And because as A B is to C, fo is DE to F, and as c is to to B G, fo is F to E H; by equality [by But B If therefore the firft of fix magnitudes has the fame ratio to the second, as the third has to the fourth, and the fifth has H E the fame ratio to the fecond as the fixth ACDF has to the fourth: then shall the first and fifth magnitudes conjointly taken have the fame ratio to the fecond, as the third and fixth magnitudes con jointly taken has to the fourth. Which was to be demonftrated. PROP. XXV. THEOR. If four magnitudes be proportional, the greatest and leaft of them taken together will be greater than the two others taken together b. Let the four magnitudes AB, C D, E, F be proportional, and let it be as A B is to C D, fo is E to F, and let A B be the greatest of them, and F the leaft: I fay A B and F are greater than C D and E. For make AG equal to E, and C H equal to F. Then because A B is to C D, as E is to F, and AG is equal to E, and c H equal G to F; it fhall be as A B to C D, fo is AG to CH. And because one magnitude A B is to another C D, as the part AG of the one is to the part CH of the other; [by 19. 5.] the remaining part GB of the one will be to the remaining part HD of the other, as one magnitude B D H A B is to the other CD. But [by up- ACE F pofition] |