nearly in the middle between 3 and 4, but rather nearer the latter than the former, let 3.5 and 3.6 be taken for the two assumed numbers.

Then log. 3.5.5440680, which, being multiplied by 3.5, gives 1.904238=first result;

And log. 3.6.5563025, which, being multiplied by 3.6, gives 2.002689 for the second result.

1.098451

: .1 :: .002689: 00273

For the first correction; which, taken from 3.6 leaves x=3.59727, nearly.

And as this value is found, by trial, to be rather too small, let 3.59727 and 3.59728 be taken as the two assumed numbers.

Then log. 3.59727.5559731, which being multiplied by 3.59727, gives 1.9999854- first result.

And log. 3.59728=.5559743, which, being multiplied by 3.51728, gives 1.9999953-second result.

.0000099 : .00001 :: .0000047 00000474747 For the second correction; which, added to 3.59728, gives x 3.59728474747, extremely near the truth.

2. Given x-2000, to find an approximate value of x. Ans. x=4.82782263

3. Given (6x)=96, to find the approximate value of x. Ans. x 1.8826432 4. Given x=123456789, to find the value of x.

Ans. 86400268

5. Given xî—x—(2x—x≈)*, to find the value of x

Ans. x 1.747933

OF THE

BINOMIAL THEOREM.

THE binomial theorem is a general algebraical expression, or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series; the form of which, as it was first proposed by Newton, being as follows:

Where p is the first term of the binomial, the second

m

term divided by the first, the index of the power, or

n

root, and A, B, C, &c. the terms immediately preceding those in which they are first found, including their signs + or -.

Which theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for r, a, m, and.n, in either of the above formulæ, and then finding the result according to the rule (1).

(1) This celebrated theorem, which is of the most extensive

EXAMPLES.

1. It is required to convert (a2+x) into an infinite

se in algebra, and various other branches of analysis, may be otherwise expressed as follows:

It may here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative,

Where the law of formation of the several terms of

the series is sufficiently evident.

the first of these expressions may be exhibited in the more simple form

1.
2 3 4

Where the last term is called the general term of the series, because if 1. 2, 3, 4, &c. be substituted successively for n, it will give all the rest.