3x2

compared with 1,

a2

is very small compared with 3

203

3x

; a

fortiori is very small compared with 1 or with a3

3x

a

Hence 1± 1, or a 3x : a, is a near approximation

α

to (a ±x)3: a3, if a be small when compared with a.

Similarly it may be shewn that a ± 4x: a; a ±5x: a; &c. are approximations respectively to (a ±x)1: a1; (a±x)3 : a3; &c.

Again, since √a±æ : √a = √1

: 1,

a

The utility of the rules here proved will be sufficiently manifest from the following Examples, when it is observed by what a troublesome process the several proposed ratios would be found without the rules.

Ex. 1. (1·5241)1 : (1·524)1 = 1·5240 + 4 × 0·0001 : 1·524 nearly = 1·5244 : 1·524 nearly.

APPENDIX II.

EQUATIONS.

In the infinite variety of equations which ingenious persons may put together, it is not to be supposed that any general Rules can be laid down for every operation necessary to their solution.

There are, however, peculiar artifices of more frequent occurrence than others, which shall be exhibited in the following examples.

Ex. 4. Va+x+ax = b; required .

3

3

(Va + x + √ a − x)3 = 2a + 3 √√√ a2 − x2. (√ a + x + √ a − x),

3

and va+x+a-x a by the supposition is equal to b;

.. cubing both sides of the proposed equation,