terms of the geometric series. The numbers, it will be observed, are the powers of the base, and the logarithms are the indices of those powers.

Further to illustrate the use of logarithms, we give the following table:—

1. Required the quotient of 32768 divided by 2048. The indices or logarithms of these numbers are, respectively, 15 and 11. The difference of these logarithms is 4, which is the logarithm of 16, the quotient required. Hence the difference of the logarithms of two numbers is the logarithm of their quotient.

2. Required the third power of 32. The logarithm of 32 is 5. Multiply this by 3, the index of the power to which 32 is to be raised, and the product, 15, is the index of 32768, the required power. Hence, to involve a number to a given power, we multiply its logarithm by the index of the power to which it is to be raised.

3. Required the fourth root of 4096. The index of this is 12. Divide this index by 4, the degree of the root to be extracted, and the quotient will be 3, which is the logarithm of 8, the root required. Hence, to extract the root of a number, we divide its logarithm by the number expressing the degree of the root to be extracted, and the quotient is the logarithm of the root required.

3. The table in Art. 2 contains only the integral powers of 2, that being sufficient for the purpose of illustration; but a complete table contains all the numbers of the natural series, within the limits of the table, together with the indices, or logarithms. The logarithms in such a table will, in most instances, be fractions. Thus, the logarithms corresponding to any of the numbers between 4 and 8 would be 2 and some fraction;

of any number between 8 and 16, the logarithm would be 3 and a fraction; and so on.

This, in general, can only

4. Calculation of Logarithms. Since all numbers are considered as the power of some one base, we will have, if a be the base, and n the number, a = n. The determination of the logarithm will then consist in solving the above equation so as to find x. be done by approximation. The details to which it would lead are entirely foreign to the present work. Those who desire to become acquainted with the subject may consult the author's "Treatise on Algebra."

5. Bases. Theoretically, it is of no importance what number is assumed as the base of the system; but practical convenience suggests that 10, the base of our system of notation, should also be the base of the system of logarithms. By the use of this base, it becomes unnecessary to insert in the table of logarithms their integral portions. For, as will be seen hereafter, the figures in the decimal portion of the logarithm depend on the figures in the number, while the integral portion of the logarithm depends solely on the position of the decimal point in the number.

6. Assuming, then, 10 for a base, we have the following series:

Numbers,

1, 10, 100, 1000, 10000, 100000, 1000000;

Logarithms, 0 1 2

3

4

5

6.

The logarithm of any number between 1 and 10 will be wholly decimal; between 10 and 100, it will be 1 and a decimal; and so on.

If the powers of 10 be continued downwards, we have

The logarithm of any number between .1 and 1 is therefore-1 a decimal, of a number between .01 and .1 it is -2+ a decimal, &c.

7. Indices of Logarithms. The integral portion of every logarithm is called the index, the decimal portion being sometimes called the mantissa. From the above series, it is manifest that, if the number is greater than 1, the index is positive, and one less than the number of integral figures. Thus, 246.75 coming between 100 and 1000, its logarithm will be 2 and a decimal. If the number is less than 1, the index will be negative. For example, the logarithm of .0024675, which comes between .001 and .01, will be -3+ a decimal.

8. Mantissæ. The mantissæ of logarithms to the base 10 depend solely on the figures of the number, without any regard to the position of the decimal point.

Let the logarithm of 31.416 be 1.497151: then, since 314.16 is 10 times 31.416, its logarithm will be 1.497151 + 1 = 2.497151. Similarly, the logarithm of 31416, which is 1000 times 31.416, will be 1.497151 +34.497151.

Again, .031416 = 31.416 ÷ 1000: its logarithm is therefore 1.497151-3=-2.497151, in which the sign is understood to belong solely to the index 2, and not to the mantissa. Since, then, the index can be supplied by attention to the position of the decimal point, the mantissæ alone are inserted in the body of a table of logarithms. The annexed table will illustrate the above more fully:

9. Table of Logarithms. A table of logarithms consists of the series of natural numbers, with their logarithms, or, rather, the mantisse of their logarithms, so arranged that

one can be readily determined from the other. In the table of logarithms appended to this treatise, the mantissæ of the logarithms of all numbers, from 1 to 9999 inclusive, are given. On the first page are found the numbers from 1 to 99, with their logarithms in full. The remaining pages contain only the mantissæ of the logarithms. The first column, headed N, contains the numbers, from 100 to 999; and the second, headed 0, the mantissæ of their logarithms. Thus, the logarithm of the number 897 is 2.952792; the index being 2, because there are three integral figures in the number.

The remaining columns contain the last four figures of the mantissæ of the logarithms of numbers of four figures, the first three of which are found in the first column, and the fourth, at the head. Thus, if the number were 8976, the last four figures 3083 of the mantissa of its logarithm would be found in the column headed 6; the first two, 95, found in the second column, being common to them all. The logarithm of 8976 is, therefore, 3.953083.

10. To denote the point in which the second figure changes, when such change does not take place in the first logarithmic column, the first of the four figures from the change to the end of the line is printed as an index figure; thus, on page 25 of the tables, we have the lines

89

N.

0

456

1 2 3 4 8965 9060 9155 9250 9346 9441 9536 9631 457 9916 011 °106°201 °296 °391 °486 581 °676 0771 458 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718

5

6

7

9726❘ 9821

In such cases the first two figures are found in the next line. The logarithm of 4575 is, therefore, 3.660391.

11. To find the Logarithm of a number from the tables. If the number consists of one or two figures only, its logarithm is found on the first page of the table. If the two figures are both integers, the index is given also; but, if the one or both figures be decimal, the decimal part only

of the logarithm should be taken out. Thus, the logarithm of 8 is 0.903090; of 59 is 1.770852.

If the number be wholly or part a decimal, the index must be changed in accordance with the principles laid down in Art. 7. Thus, the index must be one less than the number of figures in the integral part of the natural number. But when the natural number is wholly a decimal the index is negative, and must be one more than the number of ciphers between the first significant figure and the decimal point. Thus, the logarithm of

.8 is -1.903090; of .059 is -2.770852.

If the number consists of three figures, look for it in the remaining pages of the table, in the column headed N. Opposite to it, in the first column, will be found the decimal portion of the logarithm; the first two figures of the logarithm, being common to all the columns, are printed but once, to save room. Thus, the logarithm of

272 is 2.434569; of 529 is 2.723456;

the index being placed in accordance with the above rule.

If the number consists of four figures, the first three must be found as before; and the fourth, at the top of the table. The last four figures of the logarithm are found opposite to the first three figures of the number, and under the fourth; the first two figures of the logarithm being found in the first logarithmic column. Thus, if the number were 445.8, look for 445 in the column headed N, and opposite thereto, in the column headed 8, the figures 9140 are found; these affixed to 64, found in the first column, give 649140 for the decimal portion of the logarithm; and, as there are three integral figures, the index is 2. Hence, the complete logarithm is 2.649140.

If there are more than four figures in the number, find the logarithm of the first four figures as before. Take the difference between this logarithm and the next greater in the table; multiply this difference by the remaining figures in the number, and from the product separate as many figures from the right hand as are contained in the mul