to it, or less; then the first of the magnitudes has to the second the same ratio that the third has to the fourth. COR.-Conversely, if the first of four magnitudes have to the second the same ratio that the third has to the fourth, and if any equimultiples whatsoever be taken of the first and third, and any whatsoever of the second and fourth; then according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third shall be greater than the multiple of the fourth, equal to it, or less. 6. Magnitudes are said to be proportionals when the first has the same ratio to the second that the third has to the fourth; and the third to the fourth the same ratio which the fifth has to the sixth; and so on, whatever be their number. When four magnitudes, A, B, C, D, are proportionals, it is usual to say that A is to B as C to D, and to write them thusA: B::C:D, or thus, A: B = C: D. 7. In proportionals, the antecedent terms of the ratios are called homologous to one another; so also are the consequents. 8. When four magnitudes are proportional, they constitute a proportion. The first and last terms of the proportion are called the extremes; the second and third, the means. 9. When of the equimultiples of four magnitudes, taken as in the fifth definition, the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first has to the second a greater ratio than the third magnitude has to the fourth; and the third has to the fourth a less ratio than the first has to the second. COR.-Conversely, if the first of four magnitudes have to the second a greater ratio than the third has to the fourth, two numbers m and n may be found, such that, while m times the first magnitude is greater than n times the second, m times the third shall not be greater than n times the fourth. 10. When there is any number of magnitudes greater than two, of which the first has to the second the same ratio that the second has to the third, and the second to the third the same ratio which the third has to the fourth, and so on, the magnitudes are said to be continual proportionals, or in continued proportion. 11. When three magnitudes are in continued proportion, the second is said to be a mean proportional between the other two. Three magnitudes in continued proportion are sometimes said to be in geometrical progression, and the mean proportional is then called a geometric mean between the other two. 12. When there is any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on to the last magnitude. Thus : If A, B, C, D be four magnitudes of the same kind, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D. This is expressed A : D = A: B C: D }} 13. A ratio which is compounded of two equal ratios is said to be duplicate of either of these ratios. COR.-If the three magnitudes A, B, and C are continual proportionals, the ratio of A to C is duplicate of that of A to B, or of B to C. For, by the last definition, the ratio of A to C is compounded of the ratios of A to B, and of B to C; but the ratio of A to B the ratio of B to C, because A, B, C are continual proportionals; therefore the ratio of A to C, by this definition, is duplicate of the ratio of A to B, or of B to C. = 14. A ratio which is compounded of three equal ratios is said to be triplicate of any one of these ratios. COR.-If four magnitudes A, B, C, D be continual proportionals, the ratio of A to D is triplicate of the ratio of A to B, or of B to C, or of C to D. For the ratio of A to D is compounded of the three ratios of A to B, B to C, C to D; and these three ratios are equal to one another, because A, B, C, D are continual proportionals; therefore the ratio of A to D is triplicate of the ratio of A to B, or of B to C, or of C to D. The following technical words may be used to signify certain ways of changing either the order or the magnitude of the terms of a proportion, so that they continue still to be proportionals: 15. By alternation, when the first is to the third, as the second is to the fourth. (V. 16.) 16. By inversion, when the second is to the first, as the fourth is to the third. (V. A.) 17. By addition, when the sum of the first and the second is to the second, as the sum of the third and the fourth is to the fourth. (V. 18.) 18. By subtraction, when the difference of the first and the second is to the second, as the difference of the third and the fourth is to the fourth. (V. 17.) 19. By equality, when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred-that the first is to the last of the first rank of magnitudes, as the first is to the last of the others. Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two: 20. By direct equality, when the first magnitude is to the second of the first rank, as the first to the second of Book V.] DEFINITIONS, AXIOMS, PROPOSITION 1. 265 the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in a direct order. (V. 22.) 21. By transverse equality, when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the last but three to the last but two of the second rank; and so on in a transverse order. (V. 23.) AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude. PROPOSITION 1. THEOREM. If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its submultiple, the same multiple is the sum of all the first of the sum of all the rest. Let any number of magnitudes A, B, and C be equimultiples of as many others D, E, and F, each of each: it is required to prove that A + B + C is the same multiple of D + E + F that A is of D. Let A contain D, B contain E, and C contain F, each any number of times, as, for instance, three times; = = F+F+F; D+E+F taken three times. I. Ax. 2 B, and C were each any other equimultiple A+B+C would be the same multiple COR.-Hence, if m be any number, mD + mE + mF =m(D + E + F). If to a multiple of a magnitude by any number, a multiple of the same magnitude by any number be added, the sum will be the same multiple of that magnitude that the sum of the two numbers is of unity. .. A + B contains C as often as there are units in m + n. COR. 1.-If there be any number of multiples whatsoever, as A=mE, B=nE, C=pE, then A+B+C= (m+n+p)E. COR. 2.-Since A + B + C = (m + n + p)E, = mE, B = nE, and C = pE, and since A |