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 √2 by √3? And when we multiply a ratio by √5, what is the meaning of this operation? If we say that √2 : √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 √3 contained in √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 √5? We certainly take it more than 2 times and less than 3 times, but what meaning can we assign to √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.
Educationally, however, we are forced to proceed as we do. Just as Dedekind's theory of numbers is a simple one for college students, so is the ancient theory of proportion; but as the former is not suited to pupils in the high school, so the latter must be relegated to the college classes. And in this we merely harmonize educational progress with world progress, for the numerical theory of proportion long preceded the theory of Eudoxus.
The ancients made much of such terms as duplicate, triplicate, alternate, and inverse ratio, and also such as composition, separation, and conversion of ratio. These entered into such propositions as, "If four magnitudes are proportional, they will also be proportional alternately." In later works they appear in the form of "proportion by composition," "by division," and "by composition and division." None of these is to-day of much importance, since modern symbolism has greatly simplified the ancient expressions, and in particular the proposition concerning "composition and division" is no longer a basal theorem in geometry. Indeed, if our course of study were properly arranged, we might well relegate the whole theory of proportion to algebra, allowing this to precede the work in geometry.
We shall now consider a few of the principal propositions of Book III.
Theorem. If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally.
In addition to the usual proof it is instructive to consider in class the cases in which the parallel is drawn through the two sides produced, either below the base or above the vertex, and also in which the parallel is drawn through the vertex.
Theorem. The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.