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 a' the corresponding value of x, we shall have a*'y', and consequently, if we multiply y by y', we have

y y' = a* × ax' = ax+x';

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 two first results, that knowing the exponents x and x' belonging to the numbers y and y', we may, by taking their sum, find the exponent, which answers to the product y y', and by taking their difference, that which answers to the quotient. From the two last equations it is evident, that the exy ponent 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 he 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)1, 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

3, 4,

5,

&c.

0, 1, 2, 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)o 1, (10)1= 10; we make therefore x =

=

1

--

?

the

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

z

1

between 3 and 4; we suppose therefore =3+ and hence

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

we have then.

whence we obtain

1

128

(4)3⁄4” = 2 (4)3 = 1; }, or (}}})~"=;;

and after a few trials we discover that " 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 "9; we have then, going back through the several steps,. 2=28, 2=23, x=23.

93 289

28

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) 1000000 =2, very nearly; the first member is a little greater than 2; but (10) 10000000 smaller.*

8010300

3010299

is

* 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. 559.

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 2d, sd, 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

its logarithm is equal to

2 x 3 x 5 x 7,

12 +13+15+17,

and since 5= 120, we have

15-110-12.

=

As the process for determining x in the equation (10) y is very laborious, 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 the values comprehended between 0,1 and 0,9; we have then only to determine the value of y, which answers to 0,1, or (10)10, because the several other values of y, namely

x=

(10), (10), &c.

are the 24, 34, &c. powers of the first.

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

(10) or (10) 10 — 3,162277660;

then taking the fifth root of this result, we have

√(10) to = (10) 2% = (10)Tổʊ = 1,122018454;

then taking the fifth root, we have

(10) 10 = 1,023292992;

and raising the result to the 24, 3d

.... 9th powers, we obtain the values of y, corresponding to those of a 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 number by