of the first Ratio are not of the same Kind with the Terms of the latter. Therefore instead of that, it may not be improper to add this Demonstration following : If four Magnitudes are proportional, they will be so conversely : For let A B be to B E, as CD a DF. And then dividing it is, as AE is to BE, so is CF to DF: And this inversely is, as BE is to AE, so is DF to CF; which by compounding becomes, as AB is to AE, fo is CD to CF ; which by the 17th Definition is converse Ratio ; By S. Cunn. PROPOSITION XX. THE OR EM. them in Number, which being taken two and two ET A, B, C, be three Magnitudes, in Number, taken two and two in each Order, are in the same Proportion, viz. let A be to B, as D is to E, and B to C, as E to F; and let the first Magnitude A be greater than the third C. I say the fourth D is also greater than the sixth F. And if Á B C A be equal to C, D is equal to F. But if A be less than C, D is less than F. For because A is greater than C, and B is any other Magnitude; and fince a greater Magnitude hath * a greater Proportion *8 of this. to the fame Magnitude than a leffer hath, A will have a greater Proportion to B, than C to B. But as A is to B, so is D to D E F E; and inversly, as C is to B, so is F to E. Therefore allo D will have a greater Proportion to E, than F has to E. But of Magnitudes having Proportion to the same Magnitude, that which has the greater Proportion * 10 of tbis. Proportion is * the greater Magnitude. Therefor is greater than F. In the same manner we dem strate, if A be equal to C, then D will be also e to F, and if A be less than C, then D will be than F. Therefore, if there be three Magnitudes, others equal to them in Number: which being taken and two in each Order, are in the same Ratio. If firft Magnitude be greater than the third, then the fou will be greater than the sixth : But if the first becq to the third, then the fourth will be equal to the fixt and if the first be less than the third, the fourth will less than the fixth; which was to be demonstrated. THEOREM. them in number, which taken two and two, ar ET three Magnitudes, A, B, C, be proportional; and others D, E, F, equal to them in Number, Let their Analogy likewise be perturbate, viz. as A is to B, fo is E to F; and as B is to C, fo is D to E; if the first Magnitude A be greater than the third C. I say, the fourth D is also greater than the sixth F. And if A be equal to C, then D is equal to F; but if A be less than C, then is less than F. A в с For fince A is greater thanC, and B is * 8 of ibis. some other Magnitude, A will have * greater Proportion to B, than C has to B. which the fame Magnitude has a greater D E F + 10 of ibis. Proportion, is f the lefser Magnitude. Therefore а Therefore F is less than D; and so D shall be greater THEORE M. equal to them in Number, which taken two and , C, and others D, E, F, equal to them in Number, which taken two and two, are in the same Proportion, that is, as A is to B, fo is D to E, and as B is to C, I say, they are also proportional by Equality, viz. as A is to C, lo is so is E to F. D to F. For let G, H, be Equimultiples of A,D; and K, L, any Equimultiples of B, E, and likewise M, N, any Equi- A B C D E F multiples of C, É. Then because A is to B, as D is to E, GKM HLN and G, H, are Equimultiples of A, D, and K, L, Equimultiples of B, E; it Thall be * as G is to K, fo is H to L. For the same Reason also it will be, as K is to M, fo is L to N. And since there are three Magnitudes, G, K, M, and others H, L, N, equal to them in Num 4 of tbisa |