An Easy Introduction to the Mathematics

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, 9:7, and 20: 21.

4. From a : b decompound x: y. Ans. ay: bx.

=7:24, the

5. From 11: 12 decompound 12:11. Ans. 121: 144. 6. From 3: 4 take 3: 4. Ans. 1: 1.

7. From a x take 3 a 5 x, and from ar: 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: 3 z.

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 ratio; that is, the ratio of the squares of its terms, nearly.

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 +2 ax: 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?

Here a=19, x=1, and

article,

= ; that is, the ratio of the square of 19 to the a+2x 21

109

3. Let the ratio 1092: 1112 be required? Ans.

113

501

4. Required the ratio 10012: 10002?

Ans.

500

5. What are the ratios 3009): 3010), and 100002: 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 120+: 122?

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.

Thus, let the given ratio be expressed by

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

and let b contain

a) b (c

d) a (e

f) d (g

h) f (k

Th(m

n) L (p

Hence the given fraction __ = ac+d=) c+a, but a=de+

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.