assigned to b; we must, therefore, consider the observations which follow, as applying only to cases, in which a differs essentially from unity.

In order to express more clearly, that a has a constant value, and that the two other quantities b and c are indeterminate, I shall represent them by the letters x and y; we then have the equation a*=y, in which each value of y answers to one value of x, so that either of these quantities may be determined by means of the other.

241. This fact, that all numbers may be produced by means of the powers of one, is very interesting, not only when considered in relation to algebra, but also on account of the facility with which it enables us to abridge numerical calculations. Indeed, if we take another number y', and designate by the corresponding value of x, we shall have a = y', and, consequently, if we multiply y by y', we have

if we divide the same, the one by the other, we find

lastly, if we take the mth power of y, and the nth root, we have

It follows from the first two results, that knowing the exponents and belonging to the numbers y and y', we may, by x x' taking their sum, find the exponent which answers to the product yy', and by taking their difference, that which answers to the quotient 2. From the last two equations it is evident, that

y

the exponent belonging to the mth power of y may be obtained. by simple multiplication, and that which answers to the nth root, by simple division.

Hence it is obvious, that by means of a table, in which against the several numbers y, are placed the corresponding values of x, y being given, we may find x, and the reverse; and the multiplication of any two numbers is reduced to simple addition, because, instead of employing these numbers in the operation, we may add the corresponding values of x, and then seeking in the table

the number, to which this sum answers, we obtain the product required. The quotient of the proposed numbers, may be found, in the same table, opposite the difference between the corresponding values of x, and, therefore, division is performed by means of subtraction.

These two examples will be sufficient to enable us to form an idea of the utility of tables of the kind here described, which have been applied to many other purposes since the time of Napier, by whom they were invented. The values of x are termed logarithms, and, consequently, logarithms are the exponents of the powers, to which a constant number must be raised, in order that all possible numbers may be successively deduced from it.

The constant number is called the base of the table or system of logarithms.

I shall, in future, represent the logarithm of y by ly; we have then x = ly, and since y = a*, we are furnished with the equation y = aly.

242. As the properties of logarithms are independent of any particular value of the number a, or of their base, we may form an infinite variety of different tables by giving to this number all possible values, except unity. Taking, for example, a = 10, we have y = (10), and we discover at once, that the numbers

1, 10, 100, 1000, 10000, 100000, &c., which are all powers of 10, have for logarithms, the numbers 0, 1, 2, 3, 4, 5, &c.

The properties mentioned in the preceding article may be verified in this series; thus if we add together the logarithms of 10 and 1000, which are 1 and 3, we perceive, that their sum, 4, is found directly under 10000, which is the product of the proposed numbers.

:

243. The logarithms of the intermediate numbers, between 1 and 10, 10 and 100, 100 and 1000, &c. can be found only by approximation. To obtain, for example, the logarithm of 2, we must resolve the equation (10) = 2, by the method given in art. 221, finding first the entire number approaching nearest to the value of x. It is obvious at once, that x is between 0 and 1, since (10) = 1, (10)1 = 10; we make therefore x = 1, the

0

equation then becomes (10) = 2, or 10 = 2; now z is found

between 3 and 4; we suppose, therefore, z=3+1, and hence

As the value of z' is between 3 and 4, we make

and after a few trials we discover that z" is between 9 and 10. The operation may be continued further; but as I have exhibited this process merely to show the possibility of finding the logarithms of all numbers, I shall confine myself to the supposition of z" = 9; we have then, going back through the several steps, 23, 2 = 23, x = 3/23/3.

93

8

This value of x, reduced to decimals, is exact to the fourth figure, as it gives

x = 0,30107.

By calculations carried to a greater degree of exactness, it is found, that

x = 0,3010300,

the decimal figures being extended to seven places.

Regarding this value of x as an exponent, we must conceive the number 10 to be raised to the power denoted by the number 3010300, and the root of the result to be taken for the degree denoted by 10000000; we thus arrive at a number approaching very nearly to 2; that is (10)

3010300 10000000

2, very nearly; the

3010299 10000000

first member is a little greater than 2; but (10) is smaller.*

* The method explained in this article becomes impracticable, when the numbers, the logarithms of which are required, are large; another method however, which may be very useful, is given by Long, an English geometer, in the Philosophical Transactions for the year 1724, No. 339.

244. By multiplying the logarithm of 2, successively, by 2, 3, 4, &c., we obtain logarithms of the numbers, 4, 8, 16, &c., which are the 24, 3d, 4th, &c. powers of 2.

By adding to the logarithm of 2 the logarithms of 10, 100, 1000, &c. we obtain those of 20, 200, 2000, &c., it is evident, therefore, that if we have the logarithms of the former numbers, we may find the logarithms of all numbers composed of them, which latter can be only powers or products of the former. The number 210, for example, being equal to

=

As the process for determining x in the equation (10) y is very laborous, we may, reversing the order, furnish ourselves with the several expressions for x, then forming a table of the values of y corresponding to those of x, we shall afterwards, as will be perceived, be able, in any particular case, to determine x by means of y.

We take first for z the values comprehended between 0,1 and 0,9; we have then only to determine the value of y, which answers to x = 0,1, or (10), because the several other values of y, namely,

(10), (10)TM, &c.

are the 2a, 3d, &c. powers of the first.

By extracting the square root, we discover at once, that

then taking the fifth root of this result, we have

(10)T = 1,258925412. By a similar process, we deduce from

the value of

(10) = 1,258925412,

5

√ (10)1ʊ — (10) = (10) Too = 1,122018454 ;

then taking the fifth root, we have

and raising the result to the 2a, 3d,..... 9th powers, we obtain the values of y, corresponding to those of x comprehended between 0,01 and 0,09.

It will be readily seen, that by this method, we may also find the

245. Logarithms, which are always expressed by decimals,
are composed of two parts, namely, the units placed on the left
of the comma, and the decimal figures found on the right. The

values of y for those of x between 0,001 and 0,009, between 0,0001
and 0,0009; thus we shall be furnished with the following table.

By means of this table, we may find the logarithm of any number
whatever, by dividing it by 10 a sufficient number of times. To
obtain, for example, the logarithm of 2549, we first divide this num-