of forming them, by adding to the logarithm of 2.7341, respectively, the logarithms of 10, 100, 1000, &c. computed to a base = 3.

This is not the sole principal inconvenience that would arise from using a system of logarithms with a base not equal to 10. We might indeed, as it has been explained, by slight Arithmetical operations, directly find the logarithms of numbers from Tables of no greater extent than those which are in use; but, the reverse operation of finding the number from the logarithm, could not at all conveniently or briefly be performed: for, the logarithm proposed might be nearly equal to, a logarithm which the Tables did not contain. These considerations will, perhaps, be sufficient to shew the very great improvement that necessarily ensued on Briggs's alteration of the logarithmic base. The real value of that alteration does not seem to have been duly appreciated by writers on this subject.

For the description and use of Tables, in which the computed logarithms of numbers are recorded, the Reader is referred to the volumes of the Tables themselves: and, as nothing seems wanting to the plainness and precision of the rules therein delivered, it would be a needless accumulation of matter to insert them here. The principle however of the construction of certain small Tables for proportional parts, that are nearest the margin of every page, requires explanation. The use of these Tables is to find the logarithms of numbers consisting of more than five places. See Sherwin, p. 6, Hutton, p. 128, first edition.

Let the number composed of the first five figures or digits of the number N be n; therefore, the number next to n, or which differs from n by 1, is n + 1; let x be the digit, which placed after the digits composing n, shall make it N, then N=10n + x, and

and from this formula the small Tables of the proportional parts may be computed: for instance,

Let N = 678323, then n = 67832, and n + 1 = 67833,

=

(or taking the nearest whole number) 19: and by putting for x, 1, 2, 3, &c. we may form the small Table which is in the page containing the number 6783, &c. thus:

See Sherwin's Tables, page 6, and at number 6783, and also Hutton's, page 128.

The above proof establishes the truth of the precept for finding the logarithms of numbers consisting of more than 5 places

of figures: the other precept* which directs us to find the number corresponding to a logarithm not found exactly in Tables, may be thus proved.

Let L be the proposed logarithm, N the number: I the tabular logarithm next less; ' the tabular logarithm next greater; n, n', their corresponding numbers.

Let x be the difference of N and n, or let N = n + x; then log. N log. (n + x)

=

log. n(1+2)=log. n + log. (1 + 2)

log. n' = log. (n + 1) = log. n (1 + 1) = log. n + log. (1+1)

L

n+ x = n +

;; from which expression, the precept (Sher

win, p. 8. Hutton, p. 130.) and the small Table are derived for instance, let L.4414728, then (see the Tables),

being the two figures given according to the Rule and Table, 8 corresponding to 125.6, or 126 the nearest integer, and 4 to 62.8, or 63 the nearest integer.

* Hutton, p. 130, first edition: Sherwin, p. 8, fifth edition.

FF

It may now be worth the while to illustrate, by a few more instances, the uses of logarithms; and this will be done chiefly with a view of relieving the Student from any embarrassment which the negative index or characteristic (see p. 221.) as it is called, may occasion.

In the common system of logarithms in which the base is 10, 1 is the logarithm of 10, and 0 the logarithm of 1: consequently, every number than can be assigned between 10 and 1 must have for its logarithm a proper fraction, or (since a fraction may always be decimally expressed) a decimal fraction. The logarithms, therefore, of 2, 3, 4.56, 6.9345, &c. must be such decimals as .3010300, .47712125, .6589648, .8410152, &c. which, as it has been already argued (see pp. 207, &c.) are not to be called artificial numbers, but are computed numbers such as make good the equations,

The Logarithmic Tables contain, in fact, the logarithms only of those numbers which are contained between 1 and 10; and, from these registered logarithms, those of other numbers, less than 1, and greater than 10, are to be derived by means of the properties of logarithms. Thus, in the following extract from Sherwin's Tables:

the logarithms, with a decimal point prefixed to their first figure, are respectively the true or real logarithms of

log. 12.55=log. (1.255 × 10)=log. 1.255 +log. 10=log. 1.255+1,

and since similarly,

log. 125.5 = log. 1.255 = log. 1.255 + 2, &c.

we find by these properties of logarithms, the logarithms of 12.55, 125.5, 1255, &c. to be expressed by

1.0986437,

2.0986437,

3.0986437:

and similarly, we may, from the logarithms of 1.256, 1.257, &c. immediately assign, by prefixing the proper indices or characteristics, the logarithms of 12.56, 125.6, 12560, &c. 12.57, &c. 1257000, &c.

But if .0986437 be, as it is, the real logarithm of 1.255, the

These negative quantities, then, are the real logarithms of the above decimal numbers, that is, the equations,

are, within certain limits of exactness, true equations.

Now although, by means of the registered logarithms, the logarithms of decimal numbers may always be assigned by the preceding method, yet they are not immediately assigned: there intervenes, as an operation, the subtraction of the logarithm taken out of the Tables, either from 1, or 2, or 3, &c. In order to get rid