THE NAPIERIAN SYSTEM.

406. The Napierian system, or that invented by Baron Napier, has the modulus arbitrarily assumed as unity, or

M = 1.

Denoting the logarithms in the Napierian system by loge, we have, from formula (A),

log. (1 +p) = p −22 +?2

(B)

This series may be applied to the calculation of Napierian logarithms, if p is a proper fraction, but unless p be very small, it converges so slowly that a large number of terms are required to obtain the accuracy desirable; and if p be greater than 1, the series becomes divergent, and consequently altogether unsuitable.

It can, however, be converted into another more convenient series. Thus,

Subtracting equation (2) from equation (1), and observ

ing that

—

log. (1 − p) — log. (1 − p) = loge (1 ±1),

log. (1+1) = log. (~ + 1) = log. (z + 1) — log. z.

Substituting these values in equation (3), we have

407. This series, (C), may be employed in computing the Napierian logarithm of any number, when the like logarithm of the preceding number is known.

But, by Art. 396, the logarithm of 1 is 0; therefore, making +1 successively equal to the prime numbers 2, 3, 5, 7, etc., and observing that the logarithm of any composite number is equal to the sum of the logarithms of its prime factors (Art. 399), we obtain the following

Napierian logarithms are sometimes called hyperbolic logarithms, from having been originally derived from the hyperbola. They are also sometimes called natural logarithms, from being those which occur first in the investigation of a method of edealating log withms.

COMMON LOGARITHMS.

408. To compute common logarithms, it is necessary to find the value of M, the modulus of the system, when the base is 10.

Let

10* = n,

(1)

where x is the common logarithm of n.

Then, taking the Napierian logarithm of each member, by Art. 401 we have

1

where

(2)

loge 10

is the factor depending upon the base (Art.

405), and, therefore, the modulus required.

409. The common logarithm of a number may be derived from the Napierian logarithm of that number, by multiplying it by .4342945.

For, by equation (2), Art. 408, the common logarithm of a number is equal to the Napierian logarithm of that number multiplied by the modulus of the common system. Thus, representing a common logarithm by log,

410. A table of common logarithms, exact to any number of decimal places, may also be constructed directly from the series already employed. For, multiplying both members of (C) by the modulus of the common system, we have

log (z+1) - log z

= .868589

( 2+1+3(23+1)2 +5 (22+1)2+(2+1)2+.....) (D)

Make 1 successively equal to 2, 3, etc., and we find

1

log 3 = log 2+.868589 (+3.5 +5.5+...)

= 0.301030

= 0.477121

=0.602060

=1.000000

etc.

411. In the common system of logarithms, if the logarithm of any number be known, we can immediately determine the logarithm of the product or quotient of that number by any power of 10.

For,

and

=

log (m X 10") = log m+ log 10" log m+n,
log m―n.

log (17) = log m log 10" =

10n

That is, if the logarithm of any number be known, we can determine the logarithm of any number which has the same succession of figures, but differs merely by the position of the decimal point (Arts. 394, 395). Thus, if

Here, the negative sign placed over the characteristic indicates that it alone is negative, the decimal part being always positive.

Common logarithms are often called Briggs's logarithms, from the name of their inventor.

412. Let

BASE OF THE NAPIERIAN SYSTEM.

e* = 10,

where e is the Napierian base, and x is the Napierian logarithm of 10. Then, taking the common logarithm of each member, we have

x loge = 1, or loge=

1

loge 10

=.4342945 (Art. 408.)

That is, the modulus of the common system is the common logarithm of the Napierian base. From Arts. 409, 410, it is evident that the base of the Napierian system is between 2 and 3. Calculated to seven places of decimals it is found that

e 2.7182818.

TABLES OF COMMON LOGARITHMS.

413. A TABLE OF LOGARITHMS usually contains all the whole numbers between 1 and a given number, with their logarithms.

The characteristics of the first 100 numbers are inserted, but for all other numbers the decimal part only of the logarithms is given, while the characteristic is left to be supplied by inspection (Arts. 394, 395).

The numbers are in the column headed N, and their logarithms, or the decimal parts of their logarithms, are opposite on the same line. Sometimes there is a column headed D, in which are the mean or average differences of the ten logarithms against which they are placed.