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. 241. In order to deduce a logarithmic series sufficiently con verging to be of use in computing the Naperian logarithms of numbers, let us take the logarithmic series and make M'=1. Designating, as before, the Naperian logarithm by 1, we shall have, 1(1 + y) = y − 22 + 22 − 1 + K y2 ყვ 5 &c. (1). If now, we write in equation (1), -y for y, it becomes, Subtracting equation (2) from (1), member from member, 1 (1 + y)z = (1 −y) (z+1), whence, y=2z+1° Substituting these values in equatior. (4), and observing that 1(z + 1) − lz = 2(2z+1 + 3(2z+1)3+ 1 1 5(2+1)s+ &c.)(5), or, by transposition, 1 1 l(z +1) = &z + 2(2z+1 + 3(2z+1)2 + +&c.) (6), 3(2z+1)5(2z+1)5 Let us make use of formula (6) to explain the method of computing a table of Naperian logarithms. It may be remarked, that it is only necessary to compute from the formula the logarithms of prime numbers; those of other numbers may be. found by taking the sum of the logarithms of their factors. The logarithm of 1 is 0. If now we make z= 1, we can find the logarithm of 2; and by means of this, if we make 2= 2, we can find the logarithm of 3, and so on, as exhibited below. 1 1 27 = 1.791759 + 2 (13+ 3. (18)3 + 5. (13)** 18=14+12 19 = 2 × 13 11075+12 .&c. In like manner, we may compute the Naperian logarithms of all numbers. Other formulas may be deduced, which are more rapidly converging than the one above given, but this serves to show the facility with which logarithms may be compated. 241*. We have already observed, that the base of the common system of logarithms is 10. We will now find its modulus. We have, 7(1+ y): log (1+ y) : : 1 : M (Art. 238). If we make y = 9, we shall have, 10: log 10 :: 1: M. But the 7102.302585093, and log 101 (Art. 228); hence, M = 1 2.302585093 common system. = 0.434294482 = the modulus of the If now, we multiply the Naperian logarithms before found, by this modulus, we shall obtain a table of common logarithms (Art. 238). 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. 237), le: loge: : 1 0.434294482. But le 1 (Art. 235): hence, = 1: loge: : 1 : 0.434294482; hence, loge = 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 loga rithm of the Naperian base. Of Interpolation. 242. When the law of a series is given, and several terms taken at equal distances are known, we may, by means of the formula, T = a + nd + "(n − 1) d ̧ + n(n − 1) (n − 2) 1 2 1 2 3 already deduced, (Art. 209), introduce other terms between them, which terms shall conform to the law of the series This operation is called interpolation. In most cases, the law of the series is not given, but only numerical values of certain terms of the series, taken at fixed intervals; in this case we can only approximate to the law of the series, or to the value of any intermediate term, by the aid of formula (1). To illustrate the use of formula (1) in interpolating a terin in a tabulated series of numbers, let us suppose that we have the logarithms of 12, 13, 14, 15, and that it is required to find the logarithm of 12. Forming the orders of differences from the logarithms of 12, 13, 14 and 15 respectively, and taking the first terms of each, we find + 0.000355, d1 = 0.034762, d=0.002577, dy = 0.000355. If we consider log 12 as the first term, we have also Making these several substitutions in the formula, and ne glecting the terms after the fourth, since they are inappreciable. or, by substituting for d, da, &c., their values, and for a its Had it been required to find the logarithm of 12.39, we should have made n = .39, and the process would have been the same as above. In like manner we may interpolate terms between the tabulated terms of any mathematical table. INTEREST. 243. The solution of all problems relating to interest, may be greatly simplified by employing algebraic formulas. In treating of this subject, we shall employ the following notation: Let p denote the amount bearing interest, called the principal; the part of $1, which expresses its interest for one year, called the rate per cent.; the time, in years, that p draws interest; the interest of p dollars for t years; p+ the interest which accrues in the time t. This sum is called the amount. Simple Interest. To find the interest of a sum p for t years, at the rate r, and the amount then due. Sincer denotes the part of a dollar which expresses its interest for a single year, the interest of p dollars for the same |