fucceed. When m and 2 are found, the initial terms of the feries, and the progref fion of the exponents are both determined.

8. By the fame method of affuming a feries with indeterminate coefficients, even the more common operations may be facilitated; evolution, for inftance, may be converted into involution, and division into multiplication.

Let it be required to extract the fquare root of a2-x2; affume √a2x2=A+Bx2 +Cx++Dx+, &c. then fquaring both fides, and tranfpofing,

A+2ABx2+2 ACx4+2AD≈ +, &c. + x2+ B2x2+ 2BCx+, &c.

Hence A2-a2=0; 2AB+1=0; 2AC+

9. As an example of the application of the fame method to the finding of a quo

tient, let there be given the fraction

I

I

a+bx+dx, and let a+bx+dx2 = A + Bx +

x2+Dx2+, &c.

Then multiplying by a+bx+dx3, and tranfpofing all the terms to one fide of the equation, we have,

[aA+aВx+aCx2+aDx3+, &c.

Abx+Bbx2+Cbx3+, &c.

Ad+Bdx3+, &c.

Hence aA-1 =0, aВ+Ab=0, aC+Bb

+ Ad = o, aD + Cb + Bd=0; that is,

+Ad

the law of continuation is evident, the coefficient of each term being derived in the fame manner, from thofe of the two preceding terms.

In cafes where both divifion and evolution are to be performed at the fame time,

APPENDIX.

No. V:

Of Logarithms.

LL affirmative numbers may be con

A fidered as powers of any one given

affirmative number. The powers of 2, for inftance, may fucceffively become equal, either exactly, or within less than any affigned difference, to all numbers whatsoever, from o upwards. Thofe powers which have integers for their exponents, viz. 2°, 23, 22, 23, 2a, &c. give the geometrical progreffion 1, 2, 4, 8, 16, 32, &c. and the intermediate numbers are expreffed, at least nearly, by powers of 2 having fractional

In like manner, might the powers of 10

be taken to express all numbers thus,

A

In the fame manner, any other number, even a fraction, might be taken, in place of 2 or 10, in the preceding examples, and fuch exponents might be found as would give its powers equal to all numbers, from o upwards.

There are, therefore, no limitations with refpect to the magnitude of the number, of which the powers are to reprefent all other numbers, except that it must neither be equal to unity, nor negative. If it be =1, then all its powers will alfo be 1; and if it be negative, there will be numbers to which none of its powers can poffibly be equal.

=

2. If, therefore, y denote any number whatever, and a a given number, ʼn may n be found fuch that a"y; and n, that is, the

exponent