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. Dem.—Let m denote the multiple which the magnitudes of the first system are of those of the second system. Then we have a = ma′ (hyp.), b = mb′, c = mc′. &c., &c. Hence, by addition, PROP. II.—Theorem. If two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a′, b′) are of another (c′), then the sum of the two first is the same multiple of their submultiple which the sum of their corresponding magnitudes is of their submultiple. Dem.—Let m and n be the multiples which a and b are of c. Then we have Therefore Hence a + b is the same multiple of c that a′ + b′ is of c′. This Proposition is evidently true for any number of multiples. PROP. III.—Theorem. If two magnitudes (a, b) be equimultiples of two others (a′, b′); then any equimultiples of the first magnitudes (a, b) will be also equimultiples of the second magnitudes (a′, b′). Dem.—Let m denote the multiples which a, b are of a′, b′; then we have Hence, multiplying each equation by n, we get Hence, na, nb are equimultiples of a′, b′. PROP. IV.—Theorem. If four magnitudes be proportional, and if any equimultiples of the first and third be taken, and any other equimultiples of the second and fourth; then the multiple of the first : the multiple of the second :: the multiple of the third : the multiple of the fourth.