32. When the antecedent of a ratio is greater than its consequent, the ratio is called a ratio of the greater inequality. Thus 5:3, 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 I be added to both terms of the ratio 5:3, it be 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, it 3 2 becomes 3+x: 2+x, that is becomes ; these fractions re 3+ t duced to a common denominator as before, become 6+3x and 4+2x b 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 4:6, but 3 18 == and 5 30 4 20 the latter of which being the 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 9+3x 2+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: bone nth of the ratio of a: :b. 41.B. Let a: 1 be a given ratio, then a2: 1, a3 : 1, aa : 1, aa: I, 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, . . n, are called the measures of the ratios of a, a2, a3, aa, a" to 1 respectively, or the logarithms of the quantities a, a2, a3, a*, . . . : . a". 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, &c. be any number of given ratios; these compounded by Art. 40. will be (axbxcxd: bxcxd a xbx cx d bxcxdxe xe, or) α == or a: e, the ratio of the first antecedent e a to the last consequent e. 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 4 5 (4×5:3×2 or) is greater than either or as appears by 3 2' 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' 3 ΟΥ 4 2 , 5 as appears by reducing these fractions as before. 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 15 4 60 5 given ratios, namely, the result will be X = 4' which is the latter of the given ratios; and if from the com 15 5 pounded ratio we decompound the latter given ratiot the 28' 15 7 105 3 result will be X == =) the former given ratio: 28 5 140 4 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 multiply 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 51 9 and 7 8' 3. From the ratio compounded of the ratios 8:7, 3:4, and 5:9, subtract the ratio compounded of the ratios 1:2, 8:3, 7. From a x take 3 a 5 x, and from ax: y2 take y : 2 ax. 8. From the ratio compounded of a: b, x: z, and 5: 4, take the ratio compounded of 5b: x, and 2 a: 3z. 48. If the terms of a ratio be nearly equal, or their difference when compared with either of the terms very small, then if this difference be doubled, the result will express double the given that is, the ratio of the squares of its terms, nearly. ratio; Let the given ratio be a+x: a, the quantity x being very small when compared with a, and consequently still smaller when compared with a+x; then will (a+x)2, or) a2+2 ax+x2: a2 be the ratio of the squares of the terms a+x and a: and because x is small when compared with a, x.x (or x2) is small when compared with 2a.x, and much smaller than a.a; wherefore if on account of the exceeding smallness of x2, compared with the other quantities, it be rejected, then (instead of a2+2 ax+x2 : a2) we shall have a2 +2ax: a2; that is, (by dividing the whole by a) a+2x: a, for the ratio of the squares of a+x: a, which was to be shewn. EXAMPLES.-1. Required the ratio of the square of 19 to the square of 20? α Here a=19, x=1, and = 19 19 a+x 20' therefore by the preceding =- ; that is, the ratio of the square of 19 to the 21 |