A good rule to adopt in such a case is to write the nearest EVEN number. For example,

for the half of .261 81 we write .130 90;

Returning to our example, we find, by taking the number corre

which is the form in which it is to be written in order to apply the rule of characteristics. The corresponding number is 0.816 50.

We have here a case in which, had we neglected considering the surplus 10 as we habitually do, the characteristic of the answer would have been 4 instead of 9 or 1. The easiest way to treat such cases is this:

When we have to divide a logarithm in order to extract a root, instead of increasing the characteristic by 10, increase it by 10 x index of root.

Thus we write

Dividing by 2,

log = 19.823 91 20.
V

log = 9.911 96 - 10,

which is in the usual form.

3. To find the cube root of 1.

This logarithm is in the usual form, and gives

The affix

30,

or

10 x divisor, can be left to be understood in these cases as in others. All that is necessary to attend to is that instead of supposing the characteristic to be one or more units less than 10, as in the usual run of cases, we suppose it to be one or more units less than 10 × divisor.

9. The Arithmetical Complement.

When a logarithm is subtracted from zero, the remainder is called its arithmetical complement.

If L be any logarithm, its arithmetical complement will be — L. Hence if

then

that is,

L = log n,

arith. comp. = − L = log=;

The arithmetical complement of a given logarithm is the logarithm of the reciprocal of the number corresponding to the given logarithm. Notation. The arithmetical complement of a logarithm is written co-log. It is therefore defined by the form

Finding the arithmetical complement. To find the arithmetical complement of log 2 = 0.301 03, we may proceed thus:

0.000 00

log 2, 0.301 03

co-log 2, 9.698 97-10.

We subtract from zero in the usual way; but when we come to the characteristic, we subtract it from 10. This makes the remainder too large by 10, so we write 10 after it, thus getting a quantity which we see to be log 0.5.

-

We may leave the -10 to be understood, as already explained.

The arithmetical complement may be formed by the following rule:

Subtract each figure of the logarithm from 9, except the last significant one, which subtract from 10. The remainders will form the arithmetical complement.

For example, having, as above, the logarithm 0.301 03, we form, mentally, 909; 9-36; 9-09; 9-18; 90=9; 10-37; and so write

9.698 97

as the arithmetical complement.

To form the arithmetical complement of 3.284 00 we have 9 - 3 = 6; 9-27; 98 = 1; 10 4 = 6. The complement is

therefore

.716 00.

The computer should be able to form and write down the arithmetical complement without first writing the tabular logarithm, the subtraction of each figure being performed mentally.

Use of the arithmetical complement. The co-log is used to substitute addition for subtraction in certain cases, on the principle: To add the co-logarithm is the same as to subtract the logarithm.

Example. We may form the logarithm of in this way by addition:

log 3, 0.477 12

co-log 2, 9.698 97

log, 0.176 09

Here there is really no advantage in using the co-log. But there is an advantage in the following example:

2763 419.24

To find the value of P =

We add to the loga

99

rithms of the numerator the co-log of the denominator, thus:

The use of the arithmetical complement is most convenient when

the divisor is a little less than some power of 10.

10. Practical Hints on the Art of Computation.

The student who desires to be really expert in computation should learn to reduce his written work to the lowest limit, and to perform as many of the operations as possible mentally. We have already described the process of taking a logarithm from the table without written computation, and now present some exercises which will facilitate this process.

1. Adding and subtracting from left to right. If one has but two numbers to add it will be found, after practice, more easy and natural to write the sum from the left than from the right. method is as follows:

The

In adding each figure, notice, before writing the sum, whether the sum of the figures following is less or greater than 9, or equal to it.

If the sum is less than 9, write down the sum found, or its last figure without change.

If greater than 9, increase the figure by 1 before writing it down. If equal to 9, the increase should be made or not made according as the first sum following which differs from 9 is greater or less than 9.

If the first sum which differs from 9 exceeds it, not only must we increase the number by 1, but must write zeros under all the places where the 9's occur. If the first sum different from 9 is less than 9, write down the 9's without change.

The following example illustrates the process:

750276 8 3 5 7 8 5 8 8 9 2 8 3 7

8 2 39 1 7 1 6 4 5 0 4 1 1 0 2 5 9 8

1 5 7 4 1 9 4 0 0 0 2 8 9 9 9 9 5 4 3 5 Here and 8 are 15. 52 being less than 9, we write 15 without change. 30 being less than 9, we write 7 without change. 9+2 being greater than 9, we increase the sum 3 +0 by 1 and write down 4. 7+1 being

less than 9, we write the last figure of 9+ 2, or 1, without change. 6+ 7 being greater than 9, we increase 7+1 by 1 and write down 9. Under 6 + 7 we write down 3 or 4. To find which, 8+1=9; 3+6=9; 5+ 4 = 9; 7+5= 12. This first sum which is different from 9 being greater than 9, we write 4 under 6 + 7, and O's in the three following places where the sums are 9. 7+5 12. Since 8+09, we write down 2. Before deciding whether to put 8 or 9 under 80, we add 5+4 = 9;819;8+1=9; 9+0=9; 2+2 = 4. This being less than 9, we write 8 under 8 +0, and 9's in the four following places. Since 5+8=13> 9, we write 5 under 2+2. Since 9+ 3 = 12 > 9, we write 4 under 5+ 8. Since 8+7= 15 > 9, we write 3 under 9+3. Finally, under 8+7 we write 5.

=

This process cannot be advantageously applied when more than two numbers are to be added.

EXERCISES.

Let the student practise adding each consecutive pair of the following lines, which are spaced so that he can place the upper margin of a sheet of paper under the lines he is adding and write the sum upon it.

227289

25

09

131

20

19 17

88195

669027

08

685

8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 4

Subtracting. We subtract each figure of the subtrahend from the corresponding one of the minuend (the latter increased by 10 if necessary), as in arithmetic.

Before writing down the difference, we note whether the following figure of the subtrahend is greater, less, or equal to the corresponding figure of the minuend.

If greater, we diminish the remainder by 1 and write it down.* If less, we write the remainder without change.

If equal, we note whether the subtrahend is greater or less than the minuend in the first following figure in which they differ.

If greater,, we diminish the remainder by 1, as before, and write 9's under the equal figures.

* If the student is accustomed to carrying 1 to the figures of the minuend when he has increased the figure of his subtrahend by 10, he may find it easier to defer each subtraction until he sees whether the remainder is or is not to be diminished by 1, and, in the latter case, to increase the minuend by 1 before subtracting.