EXAMPLES FOR PRACTICE.

1. Given x3+10x2+5x=2600, to find a near approxi

16x3+40x2-30x+1=0, to find a near

3. Given x+2x+3x+4x2+Ex-54321, to find the

value of x.

4. Given the value of x.

Ans. x 11.00673

Ans. 1.284724

Ans. 8.414455

Ans. 4.510661

Ans. 7.123883

(7x3+4x2)+✔(20x210x)=28, to find

5. Given (144x3--(x2+20)2)+√(196x2 —(x2+24) 2)=114, to find the value of x.

OF EXPONENTIAL EQUATIONS.

An exponential quantity is that which is to be raised to some unknown power, or which has a variable quantity for its index; as

And an exponential equation is that which is formed between any expression of this kind and some other quantity, whose value is known; as

a=b, xx=α, &c.

Where it is to be observed, that the first of these equations, when converted into logarithms, is the same as

x log. a=b, or x=

log. b.

; and the second equation xx=

log. a.

a is the same as x log. x log. a.

In the latter of which cases, the value of the unknown quantity x may be determined, to any degree of exactness by the method of double position, as follows:

RULE.

Find, by trial, as in the rule before laid down, two numbers as near the number sought as possible, and substitute them in the given equation

x log, x=log, a,

instead of the unknown quantity, noting the results obtained from each.

Then, as the difference of these results is to the difference of the two assumed numbers, so is the difference between the true result, given in the question, and either of the former, to the correction of the number belonging to the result used; which correction being added to that number, when it is too little, or subtracted from it, when it is too great, will give the root required, nearly.

And, if the number thus determined, and the nearest of the two former, or any other that appears to be nearer, be taken as the assumed roots and the operation be repeated as before, a new value of the unknown quantity will be obtained still more correct than the first; and so on, proceeding in this manner, as far as may be thought necessary.

EXAMPLES.

1. Given x=100 to find an approximate value of x. Here, by the above formula, we have

x log x=log. 100=2.

And since x is readily found by a few trials, to be nearly in the middle between 3 and 4, but rather nearer the latter than the former, let 3.5 and 36 be taken for the two assumed numbers

Then log. 3.5.5440680, which, being multiplied by 3.5, gives 1.901238 first result;

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

.098451 : .1

2.002689
2.

:: .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. 3.59728=0.555974243134677 to 15 places The log. 3.59727=0.555973035847267 to 15 places which logarithms, multiplied by their respective numbers, give the following products:

Now since only 6 additional figures are to be obtained, we may omit the three last figures in these errors; and state thus as difference of errors 9902681: difference of sup. 1: error 4974656: the correction 502354, which united to 3.59728 gives us the true value of x= 3.59728502354.*

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

5. Given xx. -X=

I

Ans. 8.6400268

= (2x — x2)2, 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:

* The correct answer to this question has been first given by Doctor Adrain” in his edition of Hutton's Mathematics, who plainly proves that Hutton's an swer, which is the same as Bonny castle's, is incorrect; See Hutton's Mathe maties, Vol. 1. p. 263. N. Y. Edition.

Ea.

Where P is the first term

term divided by the first,

2n

m -3n

m

root, and A, B, C, &c. the

n

4n

M- -2n
·BQ+· ・ca+

-DQ, &c.

3n

of the binomial, & the second

the index of the power, or

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 P, Q, un, and n, in either of the above formulæ, and then finding the result according to the rule.*

*This celebrated theorem, which is of the most extensive use in algebra, and various other branches of analysis, may be otherwise expressed as follows:

n`atx n 2n a+x' n 2n

(3] &c. 3n `a+x'

It may here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative, the first of these expressions may be exhibited in a more simple form (a+x)mam.

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.

EXAMPLES.

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

Where the law of formation of the several terms of the

series is sufficiently evident.

2. It is required to convert

b)**, into an infinite series.

1

(a+b), or its equal («+