Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Let magnitudes which have the same ratio be called proportional.[76] Of these, the first is so loose in statement as often to have been thought to be an interpolation of some later writer. It was probably, however, put into the original for the sake of completeness, to have some kind of statement concerning ratio as a preliminary to the important definition of quantities in the same ratio. Like the definition of "straight line," it was not intended to be taken seriously as a mathematical statement. The second definition is intended to exclude zero and infinite magnitudes, and to show that incommensurable magnitudes are included. The third definition is the essential one of the ancient theory. It defines what is meant by saying that magnitudes are in the same ratio; in other words, it defines a proportion. Into the merits of the definition it is not proposed to enter, for the reason that it is no longer met in teaching in America, and is practically abandoned even where the rest of Euclid's work is in use. It should be said, however, that it is scientifically correct, that it covers the case of incommensurable magnitudes as well as that of commensurable ones, and that it is the Greek forerunner of the modern theories of irrational numbers. As compared with the above treatment, the one now given in textbooks is unscientific. We define ratio as "the quotient of the numerical measures of two quantities of the same kind," and proportion as "an equality of ratios." But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when we multiply a ratio by [sqrt]5, what is the meaning of this operation? If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we assign to "quotient"? If it is the number that shows how many times one number is contained in another, how many _times_ is [sqrt]3 contained in [sqrt]2? If to multiply is to take a number a certain number of times, how many times do we take it when we multiply by [sqrt]5? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to [sqrt]5 times? It will thus be seen that our treatment of proportion assumes that we already know the theory of irrationals and can apply it to geometric magnitudes, while the ancient treatment is independent of this theory.