From this work we have the following rule:

Find the tabular logarithmic function next less than the given function, and the corresponding degrees and minutes; divide the difference of these logarithms by the difference for 1", and annex the quotient to the degrees and minutes.

NOTE 1. The method for finding the angle corresponding to a given logarithmic cosine or cotangent is the same, except that we find the next greater tabular logarithmic function, instead of the next less.

NOTE 2. The labor of division may be saved by using the tables of proportional parts.

Ex. 2. Find the angle whose log tan 9.87258.

Page 74,

Given log tangent = 9.87258

log tan 36° 42' = 9.87238. Tab. diff. = 26.

Ex. 3. Find the angle whose log cos = 9.27235.

Page 48,

Given log cosine = 9.27235

log cos 79° 12' = 9.27273. Tab. diff. = 67.

Ex. 1. To find log sin 0° 45' 37".28.

log sin 0° 45' 37" 8.12284. Tab. diff. = 16.

Page 30,

Hence, diff. for 0".28

=

.. log sin 0° 45′ 37′′.28 = 8.12288

NOTE.- The tables of proportional parts may be used, as explained in

Page 28,

...

log cos 89° 22' 35" = 8.03678. Tab. diff. = 19.

P.P. for 0".6 = - 11.4

log cos 89° 22′ 35′′.63 = 8.03666.

Ex. 4. To find log sin 4° 36' 58'.6.

Page 42,

=

.. log tan 5° 14' 46".4 8.96294.
log cos 5° 38' 32".8 = 9.99789.

Ex. 6. To find log cot 85° 45′ 23′′.7.

NOTE.

.. log cot 85° 45′ 23′′.7 8.87038.

When the logarithmic function is given, the angle may be found by reversing the above operations, as in Art. 14.

TABLE III. THE NATURAL* TRIGONOMETRIC FUNCTIONS. 16. This table (pp. 84-87) contains the natural sines, tangents, etc., of angles from 0° to 90°, at intervals of 10', calculated to four places of decimals. If greater accuracy is required it may be obtained by the proportional parts.

* A table which gives the values of the trigonometric sines, cosines, etc., is called a table of natural trigonometric functions.

17. To find the sine, tangent, etc., of a given angle.

If the sine or tangent is required, we look for the degrees in the left-hand column, and the minutes at the top of the page. If the cosine or cotangent is required, we look for the degrees in the right-hand column, and the minutes at the bottom of the page. The use is similar to that of the table of logarithmic functions, as may be seen by the following examples:

18. To find the angle corresponding to a given sine, tangent, cosine, or cotangent.

Ex. 1. Find the angle whose cosine is .4585.