30. Hence, one ratio is greater than another, when the antecedent of the former ratio is a greater multiple, part, or parts of its consequent, than the antecedent of the latter ratio is of its consequent; or, when the fraction constituted by the terms of the first ratio, is greater than that constituted by the terms of the latter. Thus 6: 2 is greater than 8: 4, for 6 contains 2 three times, 6 8 is greater than 4 whereas 8 contains 4 but twice, or 2 31. Having two or more ratios given, to determine which is the greater. RULE. Having expressed the given ratios in the form of fractions, (Art. 27.) reduce these fractions to other equivalent ones having a common denominator, (Vol. I. P. 1. Art. 180.) The latter will express the given ratios having a common consequent, wherefore the numerators will express the relative magnitudes of the ratios respectively. EXAMPLES.-1. Which is the greater ratio, 7: 4, or 8: 5? These ratios expressed in form of fractions, are 24 8 and 5' whence 7x5=35, and 8x4=32, these are the new numerators; also 4×5=20, the common denominator. being the greater, shews that the ratio of 7 : 4, is greater than the ratio of 8:5. 1 2. Which is the greater ratio, that of 8: 11, or that of 23:32 ? 8 23 These ratios expressed like fractions, are and which 32' 11 reduced to other equivalent fractions with a common denominator, become 256 253 and respectively; the former of these being the 352' 352 greater, shews that the ratio 8: 11, is greater than the ratio 23:32. 3. Which is greatest, the ratio of 18: 25, or that of 19:27 ? Ans. the former. 4. Which is the greatest, and which the least, of the ratios 9:10, 37: 41, and 75: 83? 32. When the antecedent of a ratio is greater than its consequent, the ratio is called a ratio of the greater inequality. Thus 53, 11: 7, and 2 : 1, are ratios of the greater inequality. 33. When the antecedent is less than its consequent, the ratio is called a ratio of the lesser inequality. Thus 35,7: 11, and 1: 2, are ratios of the lesser inequality. 34. And when the antecedent is equal to its consequent, the ratio is called a ratio of equality. ·Thus 5:5, 1:1, and a : a, are ratios of equality. 35. A ratio of the greater inequality is diminished by adding a common quantity to both its terms. Thus, if 1 be added to both terms of the ratio 5:3, it be 5 20 comes 6:4; but = and 6 18 the latter of which (being the ratio arising from the addition of 1 to the terms of the given ratio) is the least, and therefore the given ratio is diminished: and in general, if x be added to both terms of the ratio 3 : 2, ; these fractions re it becomes 3+x: 2+x, that is becomes 3 2 3+x 2+x 6+3x duced to a common denominator as before, become and 4+2x When the antecedent is a multiple of its consequent, the ratio is named a multiple ratio; but when the antecedent is an aliquot part of its consequent, the ratio is named a submultiple ratio. If the antecedent contains the Consequent There is a great variety of denominations applied to different ratios by the early writers, which is necessary to be understood by those who read the works either of the ancient mathematicians, or of their commentators, and may be seen in Chambers' and Hutton's Dictionary: at present it is usual to name ratios by the least numbers that will express them. 6+2x 4+2x respectively; and since the latter is evidently the least, it follows that the given ratio is diminished by the addition of x to each of its terms. 36. A ratio of the lesser inequality is increased by the addition of a common quantity to each of its terms. Thus if 1 be added to both terms of the ratio 3:5, it becomes 3 18 4:6, but = and 5 30' greater, shews that the given ratio is increased: in general, let 2:3 have any quantity x added to both its terms, then the ratio 2 2+x becomes 2+x: 3+x, that is becomes ; these reduced to a 3 3+x common denominator, become and of which the 6+2x 6+3x latter being the greater, it shews that the given ratio is increased. 37. Hence, a ratio of the greater inequality is increased by taking from each of its terms a common quantity less than either. but Thus by taking 1 from the terms of 4:3, it becomes 3:2, the given ratio is increased. the latter being the greater, shews that 38. And a ratio of the lesser inequality is diminished by taking from each of its terms a common quantity less than either. Thus by taking 2 from the terms of 3:4, it becomes 1:2, 39. Hence, a ratio of equality is not altered by adding to, or subtracting from, both its terms any common quantity. 40. If the terms of one ratio be multiplied by the terms of another respectively, namely antecedent by antecedent, and consequent by consequent, the products will constitute a new ratio, which is said to be compounded of the two former; this composition is sometimes called addition of ratios. Thus, if the ratio 3: 4 be compounded with the ratio 2:3, the resulting ratio (3×2:4×3, or) 6: 12 is the ratio compounded of the two given ratios 3: 4 and 2:3, or the sum of the ratios 3:4 and 2; 3. : 41. If the ratio a: b be compounded with itself, the resulting ratio a2 b2 is the ratio of the squares of a and b, and is said to be double the ratio a: b, and the ratio a: b is said to be half the ratio a2: b2; in like manner the ratio a3: b3 is said to be triple the ratio a: b, and a: b one third the ratio a3 : b3 ; also a" ; ba is said to be n times the ratio of a: b, and a: bì one nth of the ratio of a b. : n, are called the meaa" to 1 respectively, or a". 41.B. Let a: 1 be a given ratio, then a2: 1, a3 : 1, aa : 1, a": 1, are twice, thrice, four times, n times the given ratio, where n shews what multiple or part of the ratio a": 1 the given ratio a: 1 is; hence the indices 1, 2, 3, 4, sures of the ratios of a, a2, a3, a*, the logarithms of the quantities a, a2, a3, a1, . . . 42. If there be several ratios, so that the consequent of the first ratio be the antecedent of the second; the consequent of the second, the antecedent of the third; the consequent of the third, the antecedent of the fourth, &c. then will the ratio compounded of all these ratios, be that of the first antecedent to the last consequent. For let a b, b: c, c: d, d: e, &e. be any number of given ratios; these compounded by Art. 40. will be (axbx cxd: bxcxd a xbx cxd bxcxdxe xe, or) α e a to the last consequent e. or a e, the ratio of the first antecedent 43. Hence, in any series of quantities of the same kind, the first will have to the last, the ratio compounded of the ratios of the first to the second, of the second to the third, of the third to the fourth, &c. to the last quantity. 44. If two ratios of the greater inequality be compounded together, each ratio is increased. Thus, let 4:3 be compounded with 5:2, the resulting ratio 20 6 4 5 (4x5:3×2 or) is greater than either or as appears by 3 ? reducing these fractions to a common denominator. Art. 31. 45. If two ratios of the lesser inequality be compounded together, each ratio is diminished. Thus, let 3:4 be compounded with 2:5, the resulting ratio 6 (3×2:4×5 or) is less than either of the given ratios 20' 2 5' as appears by reducing these fractions as before. 3 ΟΥ 4 46. If a ratio of the greater inequality be compounded with a ratio of the less, the former will be diminished, and the latter increased. Thus, let 4:3 be compounded with 2:5, the resulting ratio 47. From the composition of ratios, the method of their decomposition evidently follows; for since ratios may be represented like fractions, and the sum of two ratios is found by multiplying these fractions representing them together, it is plain that in order to take one ratio from another, we have only to divide the fraction representing the former by that representing the latter. Hence, if the ratio of (3: 4 or) 5 3 4 be compounded with the ratio of (5 : 7 or) we obtain the ratio of (15: 28 or) 15 28 7' ; now if from this ratio we decompound the former of the 3 4 15 4 60 5 given ratios, namely the result will be ( X =) which is the latter of the given ratios; and if from the com 15 pounded ratio we decompound the latter given ratio 23' 15 7 result will be X 28 5 5 the 7 140)=the former given ratio: whence to subtract one ratio from another, this is the rule. RULE. Let the ratios be represented like fractions. (Art. 27.) Invert the terms of the ratio to be subtracted, and then multi*ply the correspondent terms of both fractions together; the product reduced to its lowest terms will exhibit the remaining ratio, or that which being compounded with the ratio subtracted, will give the ratio from which it was subtracted. EXAMPLES.—1. From 5:7, let 9 : 8 be subtracted. These ratios represented like fractions, are 5 9 and 7 |