pressed in y, and hence depends for its value on y alone. But A being independent of y, its value must depend on the base of the system; and hence, The expression for the logarithm of any number is composed of two factors, one dependent on the number, and the other on the base of the system in which the logarithm is taken. The factor which depends on the base, is called the modulus of the system of logarithms. 267. If we take the logarithm of 1 + y in a new system, and denote it by l. (1 + y), we shall have 1.(1 + y) = A' (-+-+-&c.) (5), 2 3 4 5 in which A' is the modulus of the new system. If we suppose y to have the same value as in equation (4), we shall have 1. (1 + y) : 1. (1 + y) : : A' : A; for, since the series in the second members are the same, they may be omitted. Therefore, The logarithms of the same number, taken in two different systems, are to each other as the moduli of those systems. 268. Having shown that the modulus and base of a system of logarithms are mutually dependent on each other, it follows, that if a value be assigned to one of them, the corresponding value of the other must be determined from it. If then, we make the modulus A' = 1, the base of the system will assume a fixed value. The system of logarithms resulting from such a modulus, and such a base, is called the Naperian System. This was the first system known, and was invented by Baron Napier, a Scotch mathematician. With this modification, the proportion above becomes and 1.(1 + y): 1.(1 + y) : : 1 : A, Hence we see that, The Naperian logarithm of any number, multiplied by the modulus of another system, will give the logarithm of the same number in that system. The modulus of the Naperian System being unity, it is found most convenient to compare all other systems with the Naperian; and hence, the modulus of any system may be defined to be, The number by which it is necessary to multiply the Naperian logarithm in order to obtain the logarithm of the same number in the other system. 269. Again, A x l.(1 + y) = 1.(1 + y) gives 1.(1 + y) = 1.(1+y) A That is, the logarithm of any number divided by the modulus of its system, is equal to the Naperian logarithm of the same number. 270. If we take the Naperian logarithm and make y = 1, equation (5) becomes a series which does not converge rapidly, and in which it would be necessary to take a great number of terms to obtain a near approximation. In general, this series will not serve for determining the logarithms of entire numbers, since for every number greater than 2 we should obtain a series in which the terms would go on increasing continually. The following are the principal transformations for converting the above series into converging series, for the purpose of obtaining the logarithms of entire numbers, which are the only logarithms placed in the tables. First Transformation. 1 Taking the Naperian logarithm in equation (5), making y=-, This series becomes more converging as z increases; besides, the first member of the equation expresses the difference between the logarithms of two consecutive numbers. Making z = 1, 2, 3, 4, 5, &c., in succession, we have The first series will give the logarithm of 2; the second series will give the logarithm of 3 by means of the logarithm of 2; the third, the logarithm of 4, in functions of the logarithm of 3... &c. The degree of approximation can be estimated, since the series are composed of terms alternately positive and negative (Art. 241). Second Transformation. A much more converging series is obtained in the following Subtracting the second series from the first, observing that 1.(1 + x) - - 1. (1 (1-x)= r. (1+ 1.(1+) = 2 (2 x3 25 -), we obtain +에++이+...... +3+ This series will not converge very rapidly unless x is a very 1 + x small fraction, in which case, will be greater than unity, 1-x but will differ very little from it. 1 + x 1-x Make 1 = 1 +, z being an entire number. We have 2 (1 + x) 2 = (1 - x) (z + 1); whence, x 1 Hence, the preceding series becomes l'.(z + 1) – l.z = 2 )-1.z = 2 ( 1 12z+1 1 1 or +...). + + 3 (22 + 1)35 (2z + 1)5 This series gives the difference between the logarithms of two consecutive numbers, and converges more rapidly than series (6) Making successively, z = 1, 2, 3, 4, 5..., we find 1.2=2(+ + + ...), 1.3-1.2=2(1+3+55 + + ...), 1 1.4-1.3=2(+373 +++...), 1 Let z=100; there will result P.101=1.100+2(201 + 1 1 +...); whence we see, that knowing the logarithm of 100, the first term of the series is sufficient for obtaining that of 101 to seven places of decimals. There are formulas more converging than the above, from which we may obtain a series of logarithms in functions of others already known, but the preceding are sufficient to give an idea of the facility with which tables may be constructed. We may now suppose the Naperian logarithms of all numbers to be known. The Naperian logarithm of 10 may be deduced from the first and fourth of the above equations, by simply adding the logarithm of 2 to that of 5 (Art. 258). This number has been calculated with great exactness, and is 2.302585093. 271. We have already observed that the base of the common system of logarithms is 10 (Art. 257). We will now find its modulus. We have, l. (1 + y) : 1. (1 + y) :: 1 : A (Art. 267). If now, we multiply the Naperian logarithms before found, by this modulus, we shall obtain a table of common logarithms (Art. 268). All that now remains to be done is to find the base of the Naperian system. If we designate that base by e, we shall have (Art. 267), l'.el.e:: 1 : 0.434294482. But l'e = 1 (Art. 263): hence, hence, 1.:l.e::1: 0.434294482, 1.e 0.434294482. But as we have already explained the method of calculating the common tables, we may use them to find the number whose logarithm is 0.434294482, which we shall find to be 2.718281828: hence e = 2.718281828. We see from the last equation but one that, the modulus of the common system is equal to the common logarithm of the Naperian base. Of Interpolation. 272. A table of logarithms is a tabulated series of numbers, showing the value of x in the equation a = N, corresponding to all the integral values of N, between 1 and some higher number which marks the limit of the table. It has already been remarked that in the system in common use, the value of the base a, is 10. And generally, any mathematical table consists of a series of values of some letter in an algebraic expression, corresponding to equi-distant values of the function on which it depends. The principle of interpolation, which is of great value in practical science, has for its object to find from the tabulated numbers |