, we get
or a : c :: a2 : b2—that is, 1st : 3rd :: square of 1st : square of 2nd. Now, the ratio of 1st : 3rd is, by Def. x., the duplicate ratio of 1st : 2nd. Hence the duplicate ratio of two magnitudes means the square of their ratio, or, what is the same thing, the ratio of their squares (see Book VI. xx.).
4. If four magnitudes be continual proportionals, the ratio of 1st : 4th is equal to the cube of the ratio of 1st : 2nd. This may be proved exactly like 3. Hence we see that what Euclid calls triplicate ratio of two magnitudes is the ratio of their cubes, or the cube of their ratio.
We also see that there is no necessity to introduce extraneous magnitudes for the purpose of defining duplicate and triplicate ratios, as Euclid does. In fact, the definitions by squares and cubes are more explicit.
xiii. In proportionals, the antecedent terms are called homologous to one another; as also the consequents to one another.
If one proportion be given, from it an indefinite number of other proportions can be inferred, and a great part of the theory of proportion consists in proving the truth of these derived proportions. Geometers make use of certain technical terms to denote the most important of these processes. We shall indicate these terms by including them in parentheses in connexion with the Propositions to which they refer. They are useful as indicating, by one word, the whole enunciation of a theorem.
Every Proposition in the Fifth Book is a Theorem.
PROP. I.—Theorem.
If any number of magnitudes of the same kind (a, b, c, &c.), be equimultiples of as many others (a′, b′, c′, &c.), then the sum of the first magnitudes (a + b + c, &c.) shall be the same multiple of the sum of the second which any magnitude of the first system is of the corresponding magnitude of the second system.