several logarithms are added or subtracted to form a quotient, we find the results of the following table:

From this table we see that if we form the continued product of eight factors, by adding their logarithms the average error of the sum of the logarithms will be more than a unit in the last place.

As an example of the accumulation of errors, let us form the product 11.13.

We have from the table

log 11 = 1.041 39

"131.113 94

log product, 2.155 33

We see that this is less than the given logarithm of the product 143 by a unit of the fifth order. But if we use seven decimals we have log 11, 1.041 392 7

66 13, 1.113 943 4

2.155 336 1

Comparing this with the computation to five places, we see the source of the error.

If the numbers with which we enter the tables are affected by errors, these errors will of course increase the possible errors of the logarithms.

In determining to what degree of accuracy to carry our results, we have the following practical rule :

It is never worth while to carry our decimals beyond the limit of precision given by the tables, which limit may be a considerable fraction of the unit in the last figure of the tables.

Let us have a logarithm to five places of decimals, 1.929 49, of which we require the corresponding number. Entering the table, we

perceive that the corresponding number is between 85.01 and 85.02. If this logarithm is the result of adding a number of logarithms, each of which may be in error in the way pointed out, we may suppose it probably affected by an error of half a unit in the last figure and possibly by an error of a whole unit or more. That is, its true value may be anywhere between 92.948 and 92.950.

The number corresponding to the former value is 85.012, and that corresponding to the latter 85.016. Since the numbers may fall anywhere between these limits, we assign to it a mean value of 85.014, which value, however, may be in error by two units in the last place. It is not, therefore, worth while to carry the interpolation further and to write more than five digits.

Next suppose the logarithm to be 2.021 70. Entering the table, we find in the same way that the number probably lies between the limits 105.121 and 105.126. There is therefore an uncertainty of five units in the sixth place, or half a unit in the fifth place. If the greatest precision is desired, we should write 105.124. But our last figure being doubtful by two or three units, the question might arise whether it were worth while to write it at all. As a general rule, if the sixth figure is required to be exact, we must use a six- or sevenplace table of logarithms.

Still, near the beginning of the table, the probable error will be diminished by writing the sixth figure.

Now knowing that at the beginning of the table a difference of one unit in the number makes a change ten times as great in the logarithm as at the end of the table, we reach the conclusions:

In taking out a number in the first part of the table, it can never be worth while to write more than six significant figures, and very little is added to the precision by writing more than five.

In the latter part of the table it is never worth while to write more than five significant figures.

Sometimes no greater accuracy is required than can be gained by using four-figure logarithms. There is then no need of writing the last figure. If, however the printed logarithm is used without change, the fourth figure must be increased by unity whenever the fifth figure exceeds 5. When the fifth figure is exactly 5, the increase should or should not be made according as the 5 is too small or too great. To show how the case should be decided, a stroke is printed above the 5 when it is too great. In these cases the fourth figure should be used as it stands, but, when there is no stroke, it should be increased by unity.

12. Applications of Logarithms to the Computation of Annuities and Accumulations of Funds at Compound Interest.

One of the most useful applications of logarithms is to fiscal calculations, in which the value of moneys accumulating for long periods at compound interest is required.

Compound interest is gained by any fund on which the interest is collected at stated intervals and put out at interest.

As an example, suppose that $10 000 is put out at 6 per cent interest, and the interest collected semi-annually and put out at the same rate. The principal will then grow as follows:

Although in business practice interest is commonly payable semiannually, it is in calculations of this kind commonly supposed to be collected and re-invested only at the end of each year. This makes the computation more simple, and gives results nearer to those obtained in practice, because a company cannot generally invest its income immediately. If it had to wait three months to invest each semi-annual instalment of interest collected, the general result would be about the same as if it collected interest only once a year and invested it immediately.

If r be the rate per cent per annum, the annual rate of increase Let us put

will be

100°

p, the annual rate of increase = 100'

p, the amount at interest at the beginning of the time, or the principal;

a, the amount at the end of one or more years.

Then, at the beginning of first year, principal...

....

Interest during the year.

Ρ

pp

Amount at end of year....

p (1+p)

Interest on this amount during second year...... pp (1+p) Amount at end of second year, (1 + p) p (1 + p) = p (1 + p)2 Continuing the process, we see that at the end of n years the amount will be

a = p (1 + p)n.

(1) To compute by logarithms, let us take the logarithms of both members. We then have

log a log p + n log (1+ p).

=

(2)

Example. Find the amount of $1250 for 30 years at 6 per cent per annum.

1. Find the amount of $100 for 100 years at 5 per cent compound interest.

2. A man bequeathed the sum of $500 to accumulate at 4 per cent interest for 80 years after his death. After that time the annual interest was to be applied to the support of a student in Harvard College. What would be the income from the scholarship?

3. If the sum of one cent had been put out at 3 per cent per annum at the Christian era, and accumulated until the year 1800, what would then have been the amount, and the annual interest on this amount?

It is only requisite to give three significant figures, followed by the necessary number of zeros.

4. Solve by logarithms the problem of the horseshoeing, in which a man agrees to pay 1 cent for the first nail, 2 for the second, and so on, doubling the amount for every nail for 32 nails in all.

NOTE. It is only necessary to compute the amount for the 32d nail, because it is easy to see that the amount paid for each nail is 1 cent more than for all the preceding ones.

5. A man lays aside $1000 as a marriage-portion for his new-born daughter, and invests it so as to accumulate at 8 per cent compound interest. The daughter is married at the age of 25. What does the portion amount to?

6. A man of 30 pays $2000 in full for a $5000 policy of insurance on his life. Dying at the age of 80, his heirs receive $7000, policy and dividends. If the money was worth 4 per cent to him, how much have the heirs gained or lost by the investment?

7. What would have been the answer to the previous question, had the man died at the age of 40, and the amount paid been $6000?

Other applications of the formula. By means of the equations (1) and (2) we may obtain any one of the four quantities a, p, p, and n when the other three are given.

CASE I. Given the principal, rate of interest, and time, to find the amount.

This case is that just solved.

CASE II. Given the amount, time, and rate per cent, to find the principal.

Solution. Equation (1) gives

log plog an log (1+ p),

by which the computation may be made.

CASE III. Given the principal, amount, and time, to find the rate. Solution. Equation (2) gives

Example. A man wants a principal of $600 to amount to $1000