ART. 10. To find d.x", m and n being positive integers.

But I being invariable, its differential is 0; therefore,

From this article, and Arts. 8 and 10, it is evident that

d.xnx-1dx,

whether n is integral or fractional, positive or negative.

ART. 12. Let the two ascending series, Ax2 + Bæb + Сx2 + Dad + &c., and Mx" + Nx" + Px2 + Qx2 + &c., be always equal; so that whatever value may be assigned to x, we shall still have,

Ax+B+C+Dxd+ &c. = MxTM+Nx" +Рx2+Qxa+&c. ; a=m, b = n, c = p, &c.;

then,

and

A = M, B = N, C = P, &c.

For, if possible, let a be less than m, and divide the equation by ; then

A+Baba+ Cx-a + Dxd--a + &c. =

Mama+Na+ Px + Qx2 + &c.

But as the series are both ascending ones, b, c, d, m, n, p, q, &c. are all greater than a. Hence, if x = 0, all the terms of this equation, except the first, will vanish; hence in that case A = 0, which is evidently absurd. Therefore, a is not less than m; and in a similar way it may be proved that m is not less than a therefore a = m, and the above equation be

Bxb+ Ca + Dad + &c. Then, by the same process of reasoning, we find b = n; and B = N. Hence the proposition is manifest.

ART. 13. An important application of the property just announced may be exhibited in the demonstration of Newton's binomial theorem.

It is evident that in the general development of (1 + x)", the first term must be 1; for when x = 0, (1 + x)" = 1o = 1. We may therefore assume

(1 + x)" = 1 + Аx2 + BÃa + Сx2 + Dx2 + &c.

in which A, B, &c., are unknown, but determinate coefficients; and p, q, r, &c., unknown exponents, integral or fractional, positive or negative. Suppose x a variable quantity. Then, differentiating both sides of this equation, and dividing by dx, we have,

n.(1 + x)1−1 = pÃæ3 ̈1+qВxa ̈ ̈1 + rСxTM-1 + sDx3--1 + &c. Multiplying by 1 + x,

n.(1 + x)" = pАx2-1 + qВx¶-1 + rСxTM---1 + sDx−1 + &c. рАx2 + qВx2 + rСx2 + sDx® + &c.

Then, from first equation, multiplying by n,

n(1 + x)" = n + nАx2 + nВx2 + nСx2 + &c. These equations being identical, we have by transposition, pАx+qBx-1 + rСxTM-1 + sDx2--1 + &c. = n +

Dx

(n − p) Ax2+(n − q) Bxa + (n − r) Cx2 + (n − s) Dã3 + &c. ; and by comparing, first the exponents, and then the coefficients, (Art. 12,) we have

From this we readily obtain the development of (a + b)".

n

1 n

2

x3 + &c.

3

As the equation d.x" nx"-1dx, on which this demonstration is founded, is equally correct whether n is integral or fractional, positive or negative, it is evident that the preceding development of (a + b)" is also correct, whatever may be the value of n.

Of Logarithms.

The calculations which are connected with Trigonometry are much facilitated by the use of logarithms; it will therefore be proper, in a treatise on that science, to explain their nature and use.

ART. 14. If we take a series of numbers in geometrical proportion, beginning with a unit, as 1, a, a2, a3, a1, a3, ao, a1, &c., it is manifest that the product of any two of these terms is a term whose exponent is the sum of the exponents of the factors. Thus,

Hence it appears that a, being assumed equal to any number at pleasure, if we can find such values of m, n, &c., that am a" = A, a" = B, &c., A, B, being given numbers, then calling m the logarithm of A, n the logarithm of B, &c.; the logarithm of AB will be the sum of the logarithms of A and B.

A

The logarithm of

will be the logarithm of A diminished by

B

the logarithm of B.

In other words, the business of multiplying and dividing by given numbers may be effected by the addition and subtraction of their logarithms.

=

As a A, a given number; we readily perceive that, by assuming different values of a, we shall change the value of m; that is, we shall have different numbers to denote the logarithm of a given number A, by varying the value of a. Thus it appears there may be an indefinite variety of systems, according to the various values which may be taken for a. This quantity a is called the radix or base of the system.

ART. 15. To investigate a formula by which the logarithm of any given number may be computed, we may assume ay; y being any given number whatever; then x = logarithm of y: and the object in view is to find a general expression for x in terms of y. If we suppose x to be variable, it is manifest that y = a* must also be variable.

In the first place, if x = 0, then y = ao = 1, whatever value may be assigned to a; it is therefore evident that the logarithm of 1 is 0 in every system.

Now, let

y' = ax+h = a*.ah = yah :

and, to reduce this second member to a more manageable form, put

then, (Art. 13,)

1+b= a;

-1

y'=y(1+b)"=y+y { hb+h.” 7 1b2 + h h = 1 h − 2 b3 + &c.

Therefore,

2

2 3

}