ber, which being taken two and two in each Order, * 20 of tbis. are in the same Proportion. If G exceeds M, *H will exceed N; if G be equal to M, then H shall be equal to N; and if G be less than M, H shall be lefs than N. But G, H, are Equimultiples of A, D, and M, N, any other Equimultiples of C and F. Whence Def . 5. as A is to C, so shall + D be to F. Therefore, if there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the same Proportion, then they shall be in the same Proportion by Equality; which was to be demonstrated. of ibis. PROPOSITION XXIII. THE O'R EM them in Number, which, taken two and two, are Proportion by Equality. nitudes A, B, C, and o- For let G, H, L, be Equimultiples of A, B, D, and K, M, N, any Equimultiples of C, E, F. Then because G, H, are Equimultiples of A and B, and fince Parts have the same Proportion as their like Multiples when taken corre* 15 of töis. fpondently, it shall be * as A is to B, fo is G to H; and by the fame Reason, as E is to F, so is M to N. † 11 of rbis. But A is to B as E is to F. Therefore, t as G is to H, fo is M to N. Again, because B is to C, as D is 2 is to E, and H, L are Equimultiples of B and D, as likewise K, M any Equimultiples of C, E; it Ihall be as H to K, so is L to M. But it has been also proved, that as G is to H, so is M to N. Therefore, because three Magnitudes, G, H, K, and others, L, M, N, equal to them, in Number, which taken two and two are in the same Proportion, and their Analogy is perturbate; then if G exceeds K, also L * will exceed * 21 of tbis, N; and if G be equal to K, then L will be equal to - N; and if G be less than K, L will likewise be less than N. But G, L are Equimultiples of A, D; and K, N Equimultiples of C, F. Therefore, as A is to C, so shall D be to F. Wherefore, if there be three Magnitudes , and others equal to them in Number, which taken two and two are in the fame Proportion; and if their Analogy be perturbate, then all they be also in the same Proportion by Equality; which was to be demonstrated. PROPOSITION XXIV. THEOREM. the second, as the third to the fourth; and if the G H ET the first Magnitude AB E the second C, as the third DE has B to the fourth F. Let also the fifth - BG have the fame Proportion to the second C as the fixth EH has 1 F For because BG is to C, as EH is to F, it shall be (inversely) as C is to BG, fo is F to EH. Then fince AB is to C; as DE is to F, and as C is to BG, fo is F to 22 of this. EH; it shall be * by Equality as A B is to BG, fo is DE to EH. And because Magnitudes, being divided, † 18 of this are proportional, they shall also bet proportional when compounded. Therefore, as AG is to GB, fo is DH Hyp to HE: But as GB is I to C, so also is HE to F. Wherefore, by Equality *, it shall be as AG is to C, fo is DH to F. Therefore, if the first Magnitude has the fame Proportion to the fécond, as the third to the fourth; and if the fifth has the same Proportion to the second, as the sixth has to the fourth; then shall the first , compounded with the fifth, have the same Proportion to the second, as the third compounded with the sixth has to the fourth; which was to be demonstrated. PROPOSITION XXV. THE ORE M. If four Magnitudes be proportional, the greatest and the least of them, will be greater than the other two. be Le propor* tional, whereof AB is to CD, as E is to F; let AB be the greatest of them, and F the least. I say AB, and B F, are greater than CD and E. For let AG be equal to E, G H Part taken away CH; it shall * 19 of this. also be * as the Residue GB to the Residue HD; fo is the A C E F and and CH to F, AG and F will be equal to CH and E. But if equal things are added to unequal things, the wholes shall be unequal. Therefore GB, HD being unequal, for GB is the greater: If A G, and F, are added to GB, and CH, and E, to HD; AB and F will necessarily be greater than CD and E. Wherefore, if four Magnitudes be proportional, the greatest, and the least of them,' will be greater than the other two; which was to be demonstrated. The End of the FIFTH Book, L EUCLI D's 148 E U C L I D's ELEMENTS BOOK VI. DÉFINITION S. 'S 1. IMILAR Right-lined Figures, are fuck as have each of their several Angles equal to one another, and the Sides about the equal Angles proportional to each other. II. Figures are said to be reciprocal, when the An tecedent and Consequent Terms of the Ratio's are in each Figure. III. A Right Line is said to be cut into mean and extreme Proportion, when the whole is to the greater Segment, as the greater Segment is to the leser. IV. The Altitude of any Figure, is a perpendicular Line drawn from the Top, or Vertex to the Base. when the quantities of the Ratio's being multi- PRO |