5. What are the ratios 3009)2: 3010), and 10000]2 : 10005]2? 49. Hence it appears, that in a ratio of the greater inequality, the above proposed ratio of the squares is somewhat too small; but in a ratio of the less inequality, it is too great. 50. Hence also, because the ratio of the square root of a+ 2x: a is a+x : a nearly, it follows that if the difference of two quantities be small with respect to either of them, the ratio of their square roots is obtained very nearly by halving the said difference. EXAMPLES.-1. Given the ratio 120: 122, required the ratio 121=the ratio of 120+: 122, nearly. 2. Given the ratio 10014: 10013, to find the ratio of their square roots? Ans. 20027: 20026. 4. Given 9990: 9996 and 10000 10000.5, to find the ratios of their square roots respectively? 51. By similar reasoning it may be shewn, that the ratio of the cubes, of the fourth powers, of the nth powers, is obtained by taking 3, 4, n times the difference respectively, provided 3, 4, or n times the difference is small with respect to either of the terms. And likewise, that the ratio of the 3rd, 4th, or nth roots are obtained nearly by taking,, part of the difference respectively. 52. When the terms of a ratio are large numbers, and prime to each other, a ratio may be found in smaller numbers nearly equivalent to the former, by means of what are called continued fractions c. a Thus, let the given ratio be expressed by, and let b contain a, c times, with a remainder d; let a contain d, e times, with a remainder f; again, let d contain f, g times, with a remainder h, and so on; then by multiplying each divisor by its quotient, and adding the remainder to the pro a) b (c d) a (e f) d (g h) f (k 1) h (m n) l (p f; this value substituted for a in the preceding equation, we substituting this value for d in the preceding equation, we shall The method of finding the approximate value of a ratio in small numbers, has been treated of by Dr. Wallis, in his Treatise of Algebra, c 10, 11. and in a tract at the end of Horrox's Works; by Huygens, in Descript. Autom. Planet. Op. Reliq. p. 174, t. 1; by Mr. Cotes in his Harmonia Mensurarum, and by several others. substituting this value for f in the preceding equation, we shall e+ et. h g+ therefore by substituting as before, = (c+ α Now in this continued fraction, if one term only (viz. c or taken, it will be an approximation to the ratio b - in small num α ce+1 bers: if two terms, viz. c+ —— (= { ±1) be taken, it will be a e 1 b α nearer approximation than the former, to the ratio ; but neces sarily expressed by a greater number of figures: if three terms be g b nearer approximation to the ratio — expressed by still more figures; a EXAMPLES.-1. Required a series of ratios in smaller numbers, continually approximating to the ratio of 12345 to 67891! 12345) 67891 (5 6166) 12345 (2 13) 6166 (474 52 96 91 56 52 4) 13 (3 12 1 Here b=67891, a=12345, c=5, d=6166, e=2, f=13, g=474, h=4, k=3, l=1. an approximation to the given ratio, in the least whole numbers possible. ce+1 5×2+1 11 Secondly, e a nearer approximation. Thirdly, =( cge+c+g 5×474x2+5+474 5219 a ge+1 2. Required approximate values for the ratio 753171: 3101000 in more convenient numbers? OPERATION. 753171) 3101000 (4 3012684 88316) 753171 (8 46643) 88316 (1 41673) 46643 (1 4970) 41673 (S 1913 &c. Here a=753171, b=3101000, c=4, d=S8316, e=8, f= 46643, g=1, h=41673, k=1, l=4970, m=8, n=1913. Therefore—=—, the first approximation. ce+1 4x8+1 33 e =) the second approximation. 3. The ratio of the diameter of a circle to its circumference is nearly as 1000000000 to 3141592653; required approximating values of this ratio in smaller numbers? |