that arc and 180°.

The supplement of an angle is the dif

ference between that angle and two right angles.

Thus, EC is the supplement of AE, and FC the supplement of AF. In like manner, EOC is the supple

ment of AOE, and FOC the supplement of AOF.

In any plane triangle, either angle is the supplement of the sum of the other two.

25. Instead of employing the arcs themselves, we usually employ certain functions of the arcs, as explained below. A function of a quantity is something which depends upon that quantity for its value.

The following functions are the only ones needed for solving triangles :

26. The sine of an arc is the distance of one extremity of the arc from the diameter, through the other extremity. Thus, PM is the sine of

AM, and P'M' is the sine of
AM'.

If AM is equal to M'C, AM and AM' will be supplements of each other; and because MM' is parallel to AC, PM will be equal to P'M'

T"

B

T

M'

IN

MT

P'

A

M

(B. I., P. XXIII.): hence, the sine of an arc is equal to the sine of its supplement.

D

27. The cosine of an arc is the sine of the complement of the arc.

Thus, NM is the cosine of AM, and NM' is the cosine of AM'. These lines are respectively equal to OP and OP'.

It is evident, from the equal triangles of the figure, that the cosine of an arc is equal to the cosine of its supple

ment.

28. 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

angles AOT and 40T"", have a common base 40, and the angles at the base equal; consequently, the remaining parts are respectively equal: hence, 'AT is equal to AT"". But AT is the tangent of AM, and AT" is the tangent of AM': hence, the tangent of an arc is equal to the tangent of its supplement.

It is to be observed that no account is taken of the algebraic signs of the cosines and tangents, the numerical values alone being referred to.

29. The cotangent of an arc is the tangent of its complement.

Thus, BT' is the cotangent of the arc AM, and BT" is the cotangent of the are AM'.

The sine, cosine, tangent, and are, for convenience, written sin a,

cotangent of an arc, α,
cos a, tan
and cot a.

These functions of an arc have been defined on the supposition that the radius of the arc is equal to 1; in this case, they may also be considered as functions of the angle which the arc measures.

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

O as

30. It is often convenient to use some other radius than 1 ; in such case, the functions of the arc, to the radius 1, may be reduced to corresponding functions, to the radius R. Let AOM represent any angle, AM an arc described from 0 a centre with the radius 1, PM its sine; A'M' an arc described from 0 as a centre, with any raradius R, and P'M' its sine. Then, because OPM and OP'M' are similar triangles, we shall have,

M'

M

PA PA'

OM : PM :: OM' : P'M', or, 1: PM :: R: P'M' ;

whence,

PM

P'M'
>
R

and, P'M'

PM × R;

and similarly for each of the other functions.

That is, any function of an arc whose radius is 1, is equal to the corresponding function of an arc whose radius is R, divided by that radius. Also, any function of an arc whose radius is R, is equal to the corresponding func tion of an are whose radius is 1, multiplied by the ra

dius R.

By making these changes in any formula, the formula will be rendered homogeneous.

TABLE OF NATURAL SINES.

31.

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 by means of which the natural sine, cosine, tangent, or cotangent of any are, may be found.

Such a table might be used for all the purposes of trigonometrical computation, but it is found more convenient to employ a table of logarithmic sines, as explained in the next article.

TABLE OF LOGARITHMIC SINES.

32. A LOGARITHMIC SINE, COSINE, TANGENT, or COTANGENT is the logarithm of the sine, cosine, tangent, or cotangent of an are whose radius is 10,000,000,000.

A TABLE OF LOGARITHMIC SINES is a table from which the logarithmic sine, cosine, tangent, or cotangent of any are may be found.

The logarithm of the tabular radius is 10.

Any logarithmic function of an arc may be found by multiplying the corresponding natural function by 10,000,000,000 (Art. 30), 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 (Art. 5).

33. In the table appended, the logarithmic functions are given for every minute from 0° up to 90°. In addition, their rates of change for each second, are given in the column headed "D."

The method of computing the numbers in the column. headed "D," will be understood from a single example. The

logarithmic sines of 27° 34', and of 27° 35', arc, respectively, 9.665375 and 9.665617. The difference between their mantissas is 242; this, divided by 60, the number of seconds in one minute, gives 4.03, which is the change in the mantissa for 1", between the limits 27° 34′ and 27° 35'.

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 logarithm of the tangent increases, just as fast as that of the cotangent decreases, and the reverse, their sum being always equal to 20. The reason of this is, that the product of the tangent and cotangent is always equal to the square of the radius; hence, the sum of their logarithms must always be equal to twice the logarithm of the radius, or 20.

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 are (Arts. 26 and 28): hence, the columns designated sine and tang, at the top of the page, are designated cosine and cotang at the bottom.

USE OF THE TABLE.

To find the logarithmic functions of an arc which is ex pressed in degrees and minutes.

34. If the arc is less than 45°, 100k for the degrees at the top of the page, and for the minutes in the left hand column; then follow the corresponding horizontal line till you