according to what has been above shown, 17 has to 19 a greater ratio than 13 has to 15. So that the general terms upon which a greater, equal, or less, ratio exists are as follows: Isaid to have to B a less ratio than C has to D. The student should understand all up to this proposition perfectly before proceeding further, in order fully to comprehend the following propositions of this book. We therefore strongly recommend the learner to commence again, and read up to this slowly, and carefully reason at each step, as he proceeds, particularly guarding against the mischievous system of depending wholly on the memory. By following these instructions, he will find that the parts which usually present considerable difficulties will present no difficulties whatever, in prosecuting the study of this important book. PROP. IX. THEO. Magnitudes which have the same ratio to the same magnitudes are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. .. Magnitudes which have the same ratio, &c. :0 = A B This may be shown otherwise, as follows:- Let A B = A C, then BC, for, as the fraction A the fraction and the numerator of one equal to the C' numerator of the other, therefore the denominator of these fractions are equal, that is, BC. Again, if B A C : A, B = C. B must C. B For, as A PROP. X. THEO. That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two; and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the less of the two. which is absurd according to the hypothesis. is not or = O, and |