centre of the arc to the extremity of the cotangent: thus, OT' is the cosecant of AM, or of AM", and OT"" is the cosecant of AM', or of AM". The term co, in combination, is equivalent to complement of; thus, the cosine of an arc is the same as the sine of the complement of that arc, the cotangent is the same as the tangent of the complement, and so on. The eight trigonometrical functions above defined are also called circular functions. RULES FOR DETERMINING THE ALGEBRAIC SIGNS OF CIRCULAR FUNCTIONS. 58. All distances estimated upwards are regarded as pos itive; consequently, all distances estimated downwards must are positive, and N'M', BT", &c., are negative. All distances estimated from the centre in a direction to towards the extremity of the arc are regarded as positive; consequently, all distances estimated in a direction from the second extremity of the arc must be considered negative. Thus, OT, regarded as the secant of AM, is estimated in a direction towards M, and is positive; but OT, re garded as the secant of AM", is estimated in a direction from M", and is negative. These conventional rules, enable us at once to give the proper sign to any function of an arc in any quadrant. 59. In accordance with the above rules, and the definiions of the circular functions, we have the following princi les: The sine is positive in the first and second quadrants, and negative in the third and fourth. The cosine is positive in the first and fourth quadrants, and negative in the second and third. The versed-sine and the co-versed-sine are always positive. The tangent and cotangent are positive in the first and third quadrants, and negative in the second and fourth. The secant is positive in the first and fourth quadrants, and negative in the second and third. The cosecant is positive in the first and second quadrants, and negative in the third and fourth. LIMITING VALUES OF THE CIRCULAR FUNCTIONS. 60. The limiting values of the circular functions are those values which they have at the beginning and end of the different quadrants. Their numerical values are discovered by following them as the arc increases from 0° around to 360°, and so on around through 450°, 540°, &c. The signs of these values are determined by the principle, that the sign of a varying magnitude up to the limit, is the sign at the limit. For illustration, let us examine the limiting values of the sine and tangent. If we suppose the arc to be 0, the sine will be 0; as the arc increases, the sine increases until the arc becomes equal to 90°, when the sine becomes equal to +1, which is its greatest possible value; as the arc increases from 90°, the sine goes on diminishing until the arc becomes equal to 180°, when the sine becomes equal to +0; as the arc increases from 180°, the sine becomes negative, and goes on increasing numerically, but decreasing algebraically, until the arc becomes equal to 270°, when the sine becomes equal to -1, which is its least algebraical value; as the arc increases from 270°, the sine goes on decreasing numerically, but increasing algebraically, until the arc becomes 360°, when the sine becomes equal to 0. It is - 0, for this value of the arc, in accordance with the principle of limits. + လ The tangent is 0 when the arc is 0, and increases till the arc becomes 90°, when the tangent is; in passing through 90°, the tangent changes from to 12, and as the arc increases the tangent decreases, numerically, but increases algebraically, till the arc becomes equal to 180°, when the tangent becomes equal to 270°, the tangent is again positive, equal to +; from 270° to 360°, the - 0; from 180° to and at 270° it becomes tangent is again negative, and at 360° it becomes equal to 0. - If we still suppose the arc to increase after reaching 360°, the functions will again go through the same changes, that is, the functions of an arc are the same as the functions that are increased by 360°, 720° &c. By discussing the limiting values of all the circular func tions we are enabled to form the following table: 61. Let AM represent any arc denoted by a. Draw the lines as represented in the figure. Then we shall have, by definition OM = OA = 1; PM = ON = sin a ; OT' cosec a. The symbols sin2a, cos2a, &c., denote the square of the sine of a, the square of the cosine of a, &c. From Formula (1) we have, by transposition, |