1.4 X 10 (because sixth figure is required) = 14, corresponding to 5 for the sixth figure. Hence, x = 164.615.

TABLE II. Constants and Their Logarithms. (Page 20.)

The loga

11. No description of this table is necessary. rithms are given to seven places, instead of five, in case a greater degree of accuracy should be required. If only the first five places are used, the fifth figure must be increased by 1, if the sixth figure is 5, or more.

TABLE III. Logarithmic Sines, Cosines, Tangents and
Cotangents. (Pages 21–66.)

12. The logarithms of the trigonometric functions are used in computation much more frequently than the functions themselves, which are called natural functions. For this reason this table is given more prominence than that of the natural functions. The table gives the logarithms of the functions for each minute from 0° to 90°. The functions of angles not expressed evenly in minutes can be found by interpolation, as explained below.

Since sec and csc are the reciprocals of cos and sin respectively, their logs can always be found by taking the cologs of the latter.

The sin and cos of all angles and the tan of angles less than 45° are less than unity; hence, their logarithms have negative characteristics. For this reason the characteristics of all these logarithms are increased by 10 in the tables.

13. To Find the Logarithmic Function of an Angle Less than 90°.

Enter the table with the given number of degrees, which will be found at the top of the page, if it is 44° or less, but at the bottom of the page, if it is greater than 44°. The function required is read at the top or bottom of the page, according as the number of degrees is at the top or bottom, and the required logarithm is taken from the corresponding column. The minutes are read in the left hand column of the page, if the degrees are read at the top, but in the extreme right hand column of the body of the table if the degrees are read at the bottom.

EXERCISES.

1. Find log sin 24° 38'. 24° is at the top of page 46, and the log sin column for 24° is the first column of logarithms on the page. Running down the page until we come to 38', we find log sin 24° 38' 9.61994.

=

2. Find log tan 57° 16'. 57° is at the bottom of page 54. Running up the page in the column marked at the bottom log tan, until we come to the line with 16' on the right, we find log tan 57° 16'

0.19192.

Verify the following statements:

log sin 39° 16′ = 9.80136,
log tan 63° 24′ = 0.30037,
log cos 41° 31' = 9.87434,
log cot 26° 12′ = 0.30798,
log cos 31°

= 9.93307,

log cos 8° 19' -
9.99541,
log cot 54° 9' = 9.85887,
log tan 82° 56′ = 0.90670,
log cot 7° = 0.91086,
log sin 19° 12′ = 9.51702.

14. Interpolating for Seconds.

Find the logarithmic functions for the degrees and minutes as before; then apply a correction for the seconds, as explained below. This correction must be added if the function is sin or tan, and subtracted if the function is cos or cot.

Find log sin 16° 28′ 35′′.

log sin 16° 28'

9.45249, and the tabular difference is 43; that is, the log sin increases by 43, while the angle increases by 1'. Hence, the proportional increase for 1" is 48, and for 35′′ it is #8 X 35 25.08..., the nearest integer to which is the required correction. Hence,

301

log sin 16° 28′ 35′′

= 9.45249.00025 9.45274.

The auxiliary table of proportional parts for tabular difference 43 will give the same result. The column to the left of the vertical line in these auxiliary tables gives the number of seconds, arranged in the order 6, 7, 8, 9, 10, 20, 30, 40, 50. If the correction for 1, 2, 3, 4, or 5 seconds is required it is obtained by taking one-tenth of that for 10, 20, 30, 40, or 50 respectively. The work can be arranged concisely as follows, but it is desirable in actual practice to compute the correction mentally and to write only the complete logarithm:

On pages 22 to 27 of the table, on account of the large number of differences which occur, owing to the rapid variation of the logarithms, different arrangements of the tables of Prop. Pts. are made. If the logarithm required falls on pages 25 to 27, and it happens that the tabular difference is one for which a table of proportional parts is given, the procedure is the same as above; otherwise as follows: Find log tan 3° 51′ 26′′

= 188.

log tan 3° 51' = 8.82799, tab. diff. This tabular difference is not given, so we use the auxiliary tables for 185 and 3 (because 185 + 3

=

188) instead.

Hence, the total correction to be added is 82 and log tan 3°

In a case of this kind it is, perhaps, just as easy to compute the correction without using the auxiliary tables.

On pages 22 to 24 the Prop. Pt. is given for one second for each tabular difference for log sin, log tan, and log cot. Log cos varies so slowly in this part of the table that no auxiliary tables are necessary.

and log sin 1° 48′ 53′′ = 8.49708+ .00354 = 8.50062

On account of the very rapid variation in the log sin and log tan at the beginning of the table, the theory that the variation of the log is proportional to that of the angle, leads to results which are sometimes appreciably in error. For this reason, when great precision is required, Table IV., pp. 67, 68, should be used in finding the log sin and log tan of angles less than 4°. An explanation of this table is given below, § 19. Verify the following statements:

log tan 5° 38′ 5′′
log sin 1° 12′ 38′′
log cos 26° 28′ 37′′
log cot 9° 1' 43'' = 0.79889,
log sin 45° 43′ 28′′ 9.85491.

=

15. To Find the Logarithmic Function of an Angle > 90°.

According to the theorems demonstrated in Elements of Trigonometry §§ 28-31, and the rules on page 40, summarizing the results, the functions of any angle can be found if those of all angles less than 90° are known. These results are given here in the form of the following rules:

I. To find the function of an angle between 90° and 180° subtract the angle from 180° and look for the same function of the difference, or subtract 90° from the angle and look for the co-function of the difference.

II. To find a function of an angle between 180° and 270° subtract the angle from 270° and look for the co-function of the differ

ence, or subtract 180° from the angle and look for the same function of the difference.

III. To find a function of an angle between 270° and 360° subtract the angle from 360° and look for the same function of the difference, or subtract 270° from the angle and look for the co-function of the difference.

The second alternative in each of these rules is better if the angle has minutes and seconds, for there is less danger of making a mistake in taking the difference.

EXERCISES.

1. Find log cos 117° 19′ 35′′.

By rule I. log cos 117° 19′ 35′′= log (—sin 27° 19′ 35′′).

NOTE. In taking the logarithm of a negative quantity we proceed as if the quantity were positive. To the logarithm when found, we prefix the symbol (—) or annex the symbol n. Neither of these signs affect the operations to which the logarithm may be subjected, but are used merely to remind the computer that the corresponding numbers are negative.

=

9.66187,

log sin 27° 19′ 35′′ -
log cos 117° 19′ 35′′ = (—) 9.66187.

2. Find log tan 242° 20′ 17′′.
By rule II. log tan 242° 20′ 17′′/ =
Verify the following statements:

log tan 62° 20′ 17′′ 0.28054.

=

16. To Find an Angle Given one of its Logarithmic Functions.

A further glance at the general constitution of the table is first necessary. Upon each page of the table are four columns of logarithms, the first and fourth are logarithmic sines and cosines, the second and third are logarithmic tangents and cotangents. The logarithms increase, going toward the back of the table in the first and second columns, and then passing into the fourth and third columns respectively, they increase, going toward the front of the table. Remembering this, the place of any given logarithm in the table can be found readily.