nuity is a(1+r), and a second payment becomes due; hence the whole sum due at the end of the second year is a+a(1+r). At the end of the third year a third payment a becomes due, together with the interest on a+a(1+r); hence the whole sum due at the end of the third year is a+a(1+r)+a(1+r)2, or a{1+(1+r)+(1+r)2}, and so on. Hence the amount due at the end of n years is a{1+(1+r)+(1+r)2+(1+r)3+. +(1+r)n−1}. .... These terms form a geometrical progression in which the ratio is 1+r. Hence, by Art. 332, the sum of the series is 422. To find the present value of an annuity, to continue for a certain number of years, allowing compound interest. The present value of the annuity must be such a sum as, if put out to interest for n years at the rate r, would amount to the same as the amount of the annuity at the end of that period. If P denote the present value of the annuity, then the amount of the annuity will be P(1+r)", which must be equal to Ex. 1. How much will an annuity of 500 dollars amount to in 15 years at four per cent. compound interest? (1+2) =1.7987 (1+r)”—1= .7987, whose log. is 1.9024 the log. of .04 is 2.6021 1.3003 the log. of 500 is 2.6990 The amount is $9983, whose log. is 3.9993. Ex. 2. What is the present value of an annuity of 500 dol lars to continue for 20 years, interest being allowed at the rate of four per cent. per annum? (1+r)n =2.188 (1+r)”—1=1.188, whose log. is 0.0748 the log. of (1+r)" is 0.3400 1.7348 ==12500, whose log. is 4.0969 The present value is $6787, whose log. is 3.8317. Ex. 3. How much will an annuity of 600 dollars amount to in 12 years at three per cent. compound interest? Ex. 4. What is the present value of an annuity of 600 dollars to continue for 12 years at three per cent. compound interest? Ex. 5. In what time will an annuity of 500 dollars amount to 5000 dollars at 4 per cent. compound interest? Increase of Population. Ans. In 8 years. 423. The natural increase of population in a country is sometimes computed in the same way as compound interest. Knowing the population at two different dates, we compute the rate of increase by Art. 419, and from this we may compute the population at any future time on the supposition of a uniform rate of increase. Such computations, however, are not very reliable, for in some countries the population is stationary, and in others it is decreasing. Ex. 1. The number of the inhabitants of the United States in 1790 was 3,930,000, and in 1860 it was 31,445,000. What was the average increase for every ten years? Ans. 34 per cent. Ex. 2. Suppose the rate of increase to remain the same for the next ten years, what would be the number of inhabitants in 1870? Ans. 42,330,000. Ex. 3. At the same rate, in what time would the number in 1860 be doubled? Ans. 23 years. Ex. 4. At the same rate, in what time would the number in 1860 be tripled? To find the Logarithm of any given Number. 424. If m and n denote any two numbers, and x and y their x+y х logarithms, then + will be the logarithm of √mn. For, ac 2 cording to Art. 396, ax+y=mn, and, taking the square root of x+y x+y each member, we have a =√mn. Therefore, is the 2 logarithm of √mn, since it is the exponent of that power of the base which is equal to √mn. Now, in Briggs's system, the logarithm of 10 is 1, of 100 is 2, etc. Hence the logarithm of √10 x 100 is logarithm of 31.6228 is 1.5. So, also, the logarithm of √10 × 31.6228. is 1+2 ; that is, the 2 1+1.5 ; that is, the logarithm of 17.7828 is 1.25, and so on for any number of logarithms. In this manner were the first logarithmic tables computed; but more expeditious methods have since been discovered. It is found more convenient to express the logarithm of a number in the form of a series. 425. Logarithms computed by Series.-The computation of logarithms by series requires the solution of the equation a*=n, in which a is the base of the system, n any number, and x is the logarithm of that number. In order that a and n may be expanded into a series by the binomial theorem, we will convert them into binomials, and assume a=1+b and n=1+m; then we shall have (1+b)x=1+m, where x is the logarithm of 1+m, to the base 1+b, or a. Involving each member to a power denoted by y, we have (1+b)*=(1+m)”. Expanding both members by the binomial theorem, we have 1+xyb+xy(xy−1) ̧2+xy (xy−1)(xy—2) ¿3+, etc.— 2 2.3 1+ym+ y ( y − (y — 1) (y—2), 2 2.3 -m3 +, etc. Canceling unity from both members and dividing by y, we have (xy−1)(xy—2)¿3+, etc.)= 2.3 This equation is true for all values of y; it will therefore be true when y=0. Upon this supposition, the equation becomes We have thus obtained an expression for the logarithm of the number 1+m or n. This expression consists of two factors, viz., the quantity M, which is constant, since it depends simply upon the base of the system; and the quantity within the parenthesis, which depends upon the proposed number. The constant factor M is called the modulus of the system. 426. To determine the Base of Napier's System.-In Napier's system of logarithms the modulus is assumed equal to unity. From this condition the base may be determined. Equation (1), Art. 425, in this case becomes m2 x=m- - •+. +, etc. 4 Reverting this series, Art. 383, Ex. 3, we obtain But, by hypothesis, a*=n=1+m; therefore х a2=1+x+t + 2 2.32.3.4+, etc. If x be taken equal to unity, we have 1 1 1 2 2.3 2.3.4+, etc. By taking nine terms of this series, we find a=2.718282, which is the base of Napier's system. 427. The logarithm of a number in any system is equal to the modulus of that system multiplied by the Naperian logarithm of the number. If we designate Naperian logarithms by Nap. log., and logarithms in any other system by log., then, since the modulus of Napier's system is unity, we have where 1+m may designate any number whatever. 428. To render the Logarithmic Series converging.—The formula of Art. 425, m2 m3 log. (1+m)=M(m-2 + etc.), (1.) can not be employed for the computation of logarithms when m is greater than unity, because the series does not converge. This series may, however, be transformed into a converging series in the following manner: |