After the like manner we demonstrate, 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 same 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 sixth and if the first be less than the third, the fourth will be less than the fixth. Which was to be demonstrated. PRO P. XXI. THEOR. If there be three magnitudes, and other three, which taken two and two have the same ratio, but in a cross order, and if the first be greater than the third : then will the fourth be greater than the Sixth: if the first be equal to the third, the fourth will be equal to the sixth : and if the first be less 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, so is E to F, and as B is to c, so is D to E. And a be greater than c: I say D is greater than F; 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. 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, so is E to F, and inversely as c is to B, so is E to D:A BC DEF Therefore e will have a greater ratio to F than it has to D. But when the same magnitude has ratios to two others, that which has the greater ratio [by 19. 5.) is the leller magnitude: Therefore F is less than D : and so D will be greater than F. In like manner we demonftrate if a be equal to c; n will be equal to F; and if A be less than c; D will be less than F, IF 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 firft be greater than the third ; Enen will the fourth be greater than the sixth : 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 same ratio ; the first will bave to the last of the first magnitudes, the same ratio which the first bas to the last of the obers. Note, It is common to express this in tbe words, by equality. Let there be any number of magnitudes A, B, C, and others D, E, F as many, which taken two and two in order are in the fame ratio, as A is to e, so is D to E, and as B is to c, so is E to F: I fay, by equality, they shall be in the same ratio, as A is to c, so 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. GK MABC DE FHLN 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, so is H to L. Also by the same reason K will be to M, as L is to N. And since 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 same 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, so will [by 5. def. 5.) o 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 same ratio ; they will be also [by equality) in the fame ratio. Which was to be demonstrated. PRO P. XXIII. THEOR. If there be three magnitudes, and three others, which taken two and two in a cross order have the same ratio ; the first hall have to the last of the first magnitudes the same ratio wbich the first has to the last of the others?. Note, This is usually referr'd to in these words, by equality of cross proportion. And by 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 same ratio; viz. as A is to B, so 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.! the parts of magnitudes have the same ratio, as their equimultiples; it will be as A to B, so is G to H. the same reason as e is to F, so is M to N: But as a is to B, so is E to F. Therefore as G is to H, so [by 11.5.] will m be to N. And becaufe it is as B to c, fo is D to E, and H, K are taken equimultiples of B, D, and GHLABC DEFKMN I, M 1, M any others of c, E; [by 15. 5.) it shall be as his to I, so 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 same 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 ton: and if G be less than L; K will also be less 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.] o be to F. Wherefore if there be three magnitudes, and other three, which taken two and two in a cross order, are in the saine ratio, the first shall have to the last of the firfe magnitudes, the same ratio which the first has to the last of the others. * Euclid proposes only three magnitudes here; but they may be any number whatsoever, as well as they may in prop. 22. a geometrician seldom having occasion for any more. The fourteenth and twenty-second propofitions of the seventh book answer to these propofitions in numbers, PRO P. XXIV. THEOR. If the first of fix magnitudes has the same ratio te the second, as the third bas to the fourth, and the fifth has the same ratio to the second as the fixth bas to the fourth : then all the first and fifth magnitudes have the same ratio to the second, as the third has to the fourth. For let the first of fix magnitudes A B have the same ratio to the second c, as the third D e has to the fourth F: and let the fifth B G have the same ratio to the second c, that the fixth E H, has to the fourth F:I say the first and fifth magnitudes taken together, will have the famę ratio to the second c, as D H the third and fixth magnitudes put together has to the fourth F. For because as B G is to c, so 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, so is D E to F, and as c is to to BG, so is F to E H; by equality [by H 22. 5.) A B will be to B G, as D E is to EH. But since when magnitudes divi- G ded are proportional, [by 18. 5.) they will be also proportional when they are compounded : it will be therefore as E AG is to c B, so is DH to HE. But B as G B is to c, so is DH to F: Therefore by equality [by 22. 5.) it will be às A G is to c, so is D H to ]. If therefore the first of fix magnitudes has the same ratio to the second, as the third has to the fourth, and the fifth has the same ratio to the second as the fixth AC DF has to the fourth : then shall the first and fifth magnitudes conjointly taken have the same ratio to the second, as the third and fixth magnitudes conjointly taken has to the fourth. Which was to be demonftrated. PRO P. XXV. THEOR. If four magnitudes be proportional, the greatest and least 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, so is E to F, and let A B be the greatest of them, and F the least: I say A B and F aré greater than c D and E. For make A G equal to E, and c H B equal to F. Then because A B is to c D, as E is to D to f; it shall be as A B to CD, so is A G to CH. And because one magni H Η tude A B is to another C D, as the part À G 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 i d of the other, as one magnitude A B is to the other c D. But [by jup- À O EF position] |