CHAPTER XVI
THE LEADING PROPOSITIONS OF BOOK III
In the American textbooks Book III is usually assigned to proportion. It is therefore necessary at the beginning of this discussion to consider what is meant by ratio and proportion, and to compare the ancient and the modern theories. The subject is treated by Euclid in his Book V, and an anonymous commentator has told us that it "is the discovery of Eudoxus, the teacher of Plato." Now proportion had been known long before the time of Eudoxus (408-355 B.C.), but it was numerical proportion, and as such it had been studied by the Pythagoreans. They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry. It was because of this that Eudoxus worked out a theory of geometric proportion that was independent of number as an expression of ratio.
The following four definitions from Euclid are the basal ones of the ancient theory:
A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.
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.