ELEMENTS OF EUCLID. BOOK V. See N See N. See N. DEFINITIONS. 1. A less magnitude is called an aliquot part or submultiple of a greater, when the less measures the greater. 2. A greater magnitude is called a multiple of a less, when the greater is measured by the less. 3. Ratio is a mutual relation of two magnitudes of the same kind, with respect to quantity. 4. Magnitudes are said to have a ratio to each other, if they be such that the less can be multiplied so as to exceed the greater. 5. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when any submultiple whatsoever of the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth. 6. Magnitudes which have the same ratio are called proportionals. 7. If a submultiple of the first be contained in the second oftener than an equi-submultiple of the third is contained in the fourth; the first is said to have a less ratio to the second than the third has to the fourth: and e contra, the third is said to have a greater ratio to the fourth than the first has to the second. 8. Proportion is the similitude of ratios. 9. Proportion consists of three terms at the least. 10. When three magnitudes are proportional (A to B as B to C) the first is said to have the third (A to C) a duplicate ratio of that which it has to the second (that is of the ratio A to B.) 11. When four magnitudes are in continued proportion (A to B as B to C, and B to C as C to D) the first is said to have to the fourth (A to D) a triplicate ratio of that which it has to the second (that is of the ratio A to B). 12. If there be any number of magnitudes of the same See N. kind (A, D, C and F) the first is said to have to the last (A to F) a ratio compounded of the ratios which the first has to the second, the second to the third, the third to the fourth (A to D, D to C, and C to F) and so on to the last. 13. In proportionals the antecedents are said to be homologous to one another, as also the consequents to one another. Geometers make use of the following terms to express certain modes of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals. 14. By permutation or alternation, when we infer that if there be four magnitudes of the same kind proportional, the first is to the third as the second to the fourth, as is proved in Prop. 33. B. 5. 15. By inversion, when we infer that if there be four magnitudes proportional, the second is to the first as the fourth to the third. Prop. 20. 16. By composition, when we infer that if there be four magnitudes proportional, the sum of the first and second is to the second, as the sum of the third and fourth to the fourth. Cor. Prop. 21. 17. By division, when we infer that if there be four magnitudes proportional, the difference between the first and second is to the second as the difference between the third and fourth to the fourth. Cor. 2. Prop. 25. 18. By conversion, when we infer that if there be four magnitudes proportional, the first is to the sum or difference of the first and second, as the third to the See N. sum or difference of the third and fourth. Prop. 21 and 25 and Cor. 1. Prop. 25. 19. Ex æquali, or ex æquo, when we infer that if there be any number of magnitudes more than two, and as many others, which taken two by two of each series are in the same ratio, the first is to the last in the first series, as the first to the last in the second series. Of this there are two species. 20. Ex æquo ordinate, when the first magnitude is to the second in the first series as the first to the second in the second series, and the second to the third in the first series as the second to the third in the second series, and so on, and we infer as in the preceding definition that the first is to the last in the one series as the first to the last in the other. Prop. 34. 21. Ex æquo perturbate, when the first magnitude is to the second in the first series, as the penultimate to the last in the second series, and the second to the third in the first series as the antepenultimate to the penultimate in the secoud series, and so on; and we infer as in def. 19. that the first is to the last in the one series as the first to the last in the other. Prop. 38. Axioms. 1. Equi-multiples of the same or of equal magnitudes are equal: 2. Those magnitudes of which the same or equal magnitudes are equi-multiples or equi-submultiples, are equal, 3. A multiple or submultiple of a greater magnitude is greater than the equi-multiple or equi-submultiple of a less. 4. That magnitude, of which a multiple or submultiple is greater than the equi-multiple or equi-submultiple of another, is greater than that other magnitude. PROP. I. THEOR. If there be given two equal magnitudes (BC and DE), Plate 5. as often as any third magnitude (A) is contained in one Fig. 1. of them, so often it is contained in the other. First, let one of the given magnitudes BC be a multiple of A, and A is not oftener contained in one of them than in the other. See N. For, if it be possible, let A be oftener contained in (1) Ar. 1. DE than in BC, and as often as A is contained in BC, B. 5. so often take it away from DE, and let a certain part (3). 1. (2) Hypoth. E remain, BC and Dm are therefore equi-multiples of B. 1. the same A, and therefore BC is equal to Dm (1), but BC is equal to DE (2), therefore Dm is equal to DE (3), which is absurd. Now, if it be possible, let A be oftener contained in BC than in DE, take away A as often as it is possible from DE and a part mE shall remain less than A, take it away as often from BC, and since it is contained in BC oftener than in DE, a part nC shall remain greater than A or equal to it, and therefore greater than mE, (4) Constr. (3) Ax. 1. but Bn and Dm are equi-multiples of A (4), and there- B. 5. fore equal (5), also BC and DE are equal (6), therefore (6) Hypoth nC is equal to mE (7), but it was proved to be greater (7) 4r. 3. than it, which is absurd. 2. Let neither of the given magnitudes be a multiple of A, and A is not oftener contained in one than in the other. B. 1. For, if it be possible, let A be oftener contained in BC than in DE, take it away from DE as often as it is possible, and mE shall remain less than A; take often from BC and nC shall remain greater away A as than A and therefore greater than mE; but Bn and Dm are equi-multiples of A and therefore equal (8), also (8) Ar. 1. BC and DE are equal (9), therefore nC is equal to B. 5. mE (10), but it has been proved greater than it, which 10.3. (9) Hypoth. is absurd. In no case therefore is A oftener contained B. 1. in one of the given magnitudes, BC or DE, than in the other. (10) Ax. Fig. 2. Cor. 1. Hence it is evident, that if either of the given magnitudes be a multiple of A, the other is also an equi-multiple of A. Cor. 2. If two magnitudes be equal, as often as one of them is contained in any third, so other contained in the same third. often is the first equal Cor. 3. If there be four magnitudes, the to the second, the third equal to the fourth, as often as the first is contained in the third so often the second is contained in the fourth. PROP. II. THEOR. If there be given two magnitudes (BC and DE), and any submultiple whatsoever of a third magnitude (A) be always contained as often in the one as in the other, those given magnitudes (BČ and DE) are equal. For, if it be possible, let one of them BC be greater than the other, and let its excess be nC, take a submultiple a of A less than nC, a is oftener contained in BC than in Bn, but Bn is equal to DE, therefore a is con(1) Prop. 1. tained in Bn as often as in DE (1), and therefore is contained in BC oftener than in DE, but it is contained as (2) Hypoth. often (2), which is absurd, therefore neither of the given quantities is greater than the other, and therefore they are equal. B. 5. Fig. 5. Cor. If there be four magnitudes, of which the first is equal to the second, and any submultiple of the first be always contained in the third as often as an equi-submultiple of the second is contained in the fourth, the third is equal to the fourth. PROP. II. THEOR. If there be four magnitudes (A, B, CD and EF) of which the first is greater than the second, and the third is equal to the fourth, the first is not contained in the third oftener than the second in the fourth. First, let B be a submultiple of EF, and, if it be possible, let A be contained in CD oftener than B in EF; |