21. The sine of an arc is the distance of one extremity of the arc from the diameter through the other extremity.

cosine of AM'. These lines are respectively equal to OP and OP'.

23. The tangent of an arc is the perpendicular to the radius at one extremity of the arc, limited by the prolongation of the diameter through the other extremity.

Thus, AT (Fig. 4) is the tangent of the arc AM, and AT""' is the tangent of the arc AM'.

24. The cotangent of an arc is the tangent of its complement, "complement tangent" being contracted into cotangent.

Thus, BT" (Fig. 4) is the cotangent of the arc AM, and BT" is the cotangent of the arc AM'.

The sine, cosine, tangent, and cotangent of an arc, a, are, for convenience, written sin a, cos a, tan a, and cot a.

These functions of an arc may also be considered as functions of the angle which the arc measures.

Thus, (Fig. 4) PM, NM, AT, and BT", are respectively the sine, cosine, tangent, and cotangent of the angle AOM, as well as of the arc AM.

25. The sine of an arc is equal to the sine of its supplement; and, in general, any function of an arc is equal to the corres

ponding function of its supplement. Thus, if A is any arc

or angle,

NOTE.

These relations exist between the numerical values

of the functions; the algebraic signs, which they have in the different quadrants, are not considered.

26. A Natural Sine, Cosine, Tangent, or Cotangent, is the sine, cosine, tangent, or cotangent, of an

radius is 1.

arc whose

A Table of Natural Sines is a table from which the natural sine, cosine, tangent, or cotangent of any arc may be found.

The Table of Natural Sines, beginning at page 63 of the tables, gives the values of the sines and cosines only. If the tangent or cotangent of an arc, A, is desired, it may be found by the relation,

tan A =

sin A
cos A

cot A =

cos A sin A

TABLE OF LOGARITHMIC SINES.

27. A Logarithmic Sine, Cosine, Tangent, or Cotangent is the logarithm of the sine, cosine, tangent, or cotangent of an arc whose radius is 10,000,000,000.

A Table of Logarithmic Sines and Tangents is a table. giving the logarithm of the sine and cosine, tangent and cotangent of any arc or angle.

The logarithm of the tabular radius is 10.

Any logarithmic function of an arc or angle may be found by multiplying the corresponding natural function by 10,000,000,000, and then taking the logarithm of the result; or more simply, by taking the logarithm of the corresponding natural function, and then adding 10 to the result.

28. In the table, beginning at page 18 of the tables, the logarithmic functions are given for every minute from 0° to 90°. In addition, their rates of change for each second, are given in the column headed "D."

For the sine and cosine, there are separate columns of differences, which are written to the right of the respective columns; but for the tangent and cotangent, there is but a single column of differences, which is written between. them.

The angle obtained by taking the degrees from the top of the page, and the minutes from any line on the left hand of the page, is the complement of that obtained by taking the degrees from the bottom of the page, and the minutes from the same line on the right hand of the page. But, by definition, the cosine and the cotangent of an arc are, respectively, the sine and the tangent of the complement of that arc (Arts. 22 and 24); hence, the columns designated sine and tang, at the top of the page, are designated cosine and cotang at the bottom.

29. To find the logarithmic functions of an angle which is expressed in degrees and minutes.

If the angle is less than 45°, look for the degrees at the top of the page, and the minutes in the left hand column; then follow the corresponding horizontal line to the column designated at the top by sine, cosine, tang, or cotang, as the case may be; the number there found is the logarithm required. Thus,

If the angle is greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right hand column; then follow the corresponding horizontal line backwards to the colnmn designated at the bottom by sine, cosine, tang, or cotang, as the case may be; the number there found is the logarithm required. Thus,

30. To find the logarithmic functions of an angle which is expressed in degrees, minutes, and seconds.

Find the logarithm corresponding to the degrees and minutes as before; then multiply the corresponding number taken from the column headed "D" (which is millionths), by the number of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent.

EXAMPLES.

1. Find the logarithmic sine of 40° 26' 26".

The same rule is followed for the figures discarded (in this

case 16), as in Art. 7.

2. Find the logarithmic cosine of 53° 40′ 46′′.

If the angle is greater than 90°, we find the required function

of its supplement (Art. 25).

3. Find the logarithmic tangent of 118° 18′ 25′′.

This is done by reversing the preceding rule: Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithm is not found in the table, then find the next less logarithm, and take from the table the corresponding degrees and minutes, and set them aside. Subtract the logarithm found in the table, from the given logarithm, annex