We have given the foregoing definitions in the order of Euclid, as given by Simson, Lardner, and others;2[ 2]Except that viii. is put before vii., because it relates, as v. and vi., to the equality of ratios, whereas vii. is a test of their inequality. but it is evidently an inverted order; for vi. viii. are definitions of proportion, and v. is only a test of proportion, and is not a definition but a theorem, and one which, instead of being taken for granted, requires proof. The following explanations will give the student clear conceptions of their meaning:—
1. If we take two ratios, such as 6 : 9 and 10 : 15, which are each equal to the same thing (in this example each is equal to
), they are equal to one another (I. Axiom i.). Then we may write it thus—
This would be the most intelligible way, but it is not the usual one, which is as follows:—6 : 9 :: 10 : 15. In this form it is called a proportion. Hence a proportion consists of two ratios which are asserted by it to be equal. Its four terms consist of two antecedents and two consequents. The 1st and 3rd terms are the antecedents, and the 2nd and 4th the consequents. Also the first and last terms are called the extremes, and the two middle terms the means.
2. Since a proportion consists of two equal ratios, and each ratio can be written as a fraction, whenever we have a proportion such as
we can write it in the form of two equal fractions. Thus:
