+ 10 of tbis. bears a greater Proportion, + is the lefser Magnitude. Therefore F is less than D; and so D shall be greater than F. After the same manner we demonstrate, if A be equal to C, D will also be equal to F; and if A be less than C, D will be also less than F. lf, therefore, there are three Magnitudes, and others equal to them in Number, which, taken two and two, are in the same Proportion, and the Proportion be perturbate ; if the first Magnitude be greater ihan the third, then the fourth will be greater than the sixth ; but if the first be equal to the third, then is the fourth equal to the sixth; if less, lefs; which was to be demonstrated. PROPOSITION XXII. THEOREM. equal to them in Number, which, taken two and shall be in the same Proportion by Equality. L! ET there be any Number of Magnitudes, A, B, and C; and others D, E, and F, equal to them in Number, which, taken two and two, are in the fane Proportion; that is, as A is to B, so is D to E; and as B is to C, lo is E to F. I say, they are also proportional by Equality; viz. as. A is to C, fo is D to F. For let G and H be any Equimultiples of A and D; and K and L any Equimultiples of B and E; and likewife M and N, any Equimultiples of C and F. Then because A A B C DEF is to B, as D is to E; and G and H are Equimultiples of A and D; GKMHLN and K and L Equimultiples of B and * 4 of tbis. E, it shall be, *as G is to K, lo is H to L. For the fame Reason, also, it Order, Order, are in the fame Proportion; therefore if G ex. ceeds M, * H will exceed N; if G be equal to M, * 20 of bike then I shall be equal to N; and is G be less than M, H Mall be less than N. But G and I are Equimultiples of A and D; and M and N any other Equi: multiples of C and F. Whence as A is to C, so thall D+ be to F. Therefore, if there be any Number of Def. s. of Magnitudes, and others equal to them in Number, which, tbis. taken two and two, are in the same Proportion ; then they fall be in the same Proportion by Equality, which was to be demonstrated, PROPOSITION XXIII. PROBLEM. them in Number, which, taken two and two, are A, B, and C; and others equal For, let G, H, and L, be Equi- GHKL MN Then, because G and Hare Equi- * 15 of bisa But A is to B, as E to F. Therefore, I as G is to H, fo is M to'N. Again, because B is to C as D is 11s of rbis. to £; and H and L are Equimultiples of B and D; as likewise K and Many Equimultiples of C and E it Ihall be as H is 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, and K, and others, L, M, and N, equal to them in Number, which, taken two and two, are in the fame Propor tion, and their Analogy is perturbate; then if G ex* 21 of this.ceeds K, also L * will exceed N; and is G be equal to K, then I will be equal to N; and if G be less than K, L will likewise be less than N. But G and L are Equimultiples of A and D; and K and N Equimultiples of C and F. Therefore, as A is to C, lo Thall D be to F. Wherefore, if there be three Magnia tudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion; and if their Analogy be perturbate, then shall they be also in the same Proportion by Equality; which was to be demonstrated. PROPOSITION XXIV. PROBLEM. the second, es the third to ibe fourth; and if H For, because B G is to C, as E H is to F; it thall be (inverfly.), as C is to B G, fo is F to E H. Then, fince AB is to C, as D E is to F; and as C is to BG, fo is F to EH; it shall be, * by Equality, as * 22 of this. A B is to GB, fo is D E to E H. And because Mag. nitudes, being divided, are proportional, they shall also bet proportional when compounded. Therefore, aš + 18 of ebis. AG is to BG, fo is D H to HE: But as G B is I to 1 Hyp. C, lo allo is HE to F. Wherefore, by Equality*, it fall be, as A G is to C, fo is D H to F. Therefore, if the first Magnitude has the same Proportion to the sea cond, 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 fifih, have the fame Proportion to the second, as the third compounded with the fixth has to the fourth ; which was to be demonstrated. PROPOSITION XXV. PROBLEM. and the least of them, will be greater ihan the other two. LET four Magnitudes, A B, CD, E, F, be pro portional, whereof A B is to CD, as E is to F; let A B be the greatest of them, and F the least. I lay, A B and F, are B greater than CD and E. For, let A G be equal to E, and CH to F. Then, because A B is to C D as E is to F; and since A G G+ and CH are each equal to E and D F; it shall be as A B is to D C, so is A G to CH. And because the Whole AB is to the Whole CD, as the Part taken away A G is to the Part taken away CH; it shall also be *, as the Residue G B to the Residue HD, so is the Whole AB to the Whole CD. But A B is A CEF greater than CD; therefore, also, H + 29 of Ibis. GB fhall be greater than HD. And since AG is equal to E, and C H to F; A G 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 G B is the greater, if A G and F are added to GB; and CH and E to HD; then A B and F will necessarily be greater than CD and E. Wherefore, if four Magnitudes be preportional; the greatest and the least of them will be greater than the other two; which was to be demonftrated. The End of the Fifth Book. EUCLID'S |