angle of elevation of the top of the tower was 33° 45'; required the height of the tower. Ans. 83.998 ft.

6. Wanting to know the horizontal distance between two inaccessible objects E and W, the

following measurements were made:

АВ =
BAW = 40° 16'

From

equal to 200 yards; from the former of these points, A could be seen, and from the latter, B; and at each of the points C and D, a staff was set up. a dis tance CF was measured, not in the direction DC, equal to 200 yards, and from D, a distance DE, equal to 200 yards, and the following angles taken:

Required the distances AP, BP, and CP.

B

This problem is used in locating the position of buoys in maritime surveying, as follows. Three points A, B, and C, on shore are known in position. The surveyor stationed at a buoy P, measures the angles APC and BPC. The distances AP, BP, and CP, are then found as follows:

Suppose the circumference of a circle to be described through the points A, B, and P. Draw CP, cutting the circumference in D, and draw the lines DB and DA.

The angles CPB and DAB, being inscribed in the same segment, are equal (B. III., P. XVIII., C. 1); for a like reason, the angles CPA and DBA are equal hence, in the triangle ADB, we know two angles and one side; we may, therefore, find the side DB. In the triangle ACB, we know the three sides, and we may compute the angle B. Subtracting from this the angle DBA, we have the angle DBC. Now, in the triangle DBC, we have two sides and their included angle, and we can find the angle DCB. Finally, in the triangle CPB, we have two angles and one side, from which data we can find CP and BP. In like manner, we can find AP.

ANALYTICAL TRIGONOMETRY.

47. ANALYTICAL TRIGONOMETRY is that branch of Mathematics which treats of the general properties and relations of trigonometrical functions.

DEFINITIONS AND GENERAL PRINCIPLES.

48. Let ABCD represent a circle whose radius is 1, and suppose its circumference to be divided into four equal parts, by the diameters AC and BD, drawn perpendicular to each other. The horizontal diameter AC, is called the initial diameter; the vertical diameter BD, is called

A

the secondary diameter; the point A, from which arcs are. usually reckoned, is called the origin of arcs, and the point B, 90° distant, is called the secondary origin. Arcs estimated from A, around towards B, that is, in a direction contrary to that of the motion of the hands of a watch, are considered positive; consequently, those reckoned in a con trary direction must be regarded as negative.

The arc AB, is called the first quadrant; the arc BC, the second quadrant; the arc CD, the third quadrant; and the arc DA, the fourth quadrant.

The point at which

an arc terminates, is called its extremity, and an arc is said to be in that quadrant in which its extremity is situated. Thus, the arc AM is in the first quadrant, the arc AM' in the second, the arc AM" in the third, and the arc AM"" in the fourth.

49. The complement of an arc has been defined to be the difference between that arc and 90° (Art. 23); geometrically considered, the

M

M

A

M

complement of an arc is the arc included between the extremity of the arc and the secondary origin. Thus, MB is the complement of AM; M'B, the complement of AM'; M"B, the complement of AM", and so on. When the arc is greater than a quadrant, the complement is negative, according to the conventional principle agreed upon (Art. 48).

The supplement of an arc has been defined to be the difference between that arc and 180° (Art. 24); geometrically considered, it is the arc included between the extremity of the arc and the left hand extremity of the initial diameter. Thus, MC is the supplement of AM, and M"C the supplement of AM". The supplement is negative, when the arc is greater than two quadrants.

51. The cosine of an arc is the distance from the secondary diameter to the extremity of the are thus, NM is the cosine of AM, and NM' is the cosine of AM'.

The cosine may be measured on the initial diameter: thus, OP is equal to the cosine of AM, and OP' to the cosine of AM'.

52. The versed-sine of an arc is the distance from the sine to the origin of arcs: thus, PA is the versed-sine of AM, and P'A is the versed-sine of AM'.

53. The co-versed-sine of an arc is the distance from the cosine to the secondary origin: thus, NB is the coversed-sine of AM, and N"B is the co-versed-sine of AM".

54. The tangent of an arc is that part of a perpendicular to the initial diameter, at the origin of arcs, in cluded between the origin and the prolongation of the diam eter through the extremity of the arc thus, AT is the tangent of AM, or of AM", and AT" is the tangent of AM', or of AM""'.

55. The cotangent of an arc is that part of a perpendicular to the secondary diameter, at the secondary origin, included between the secondary origin and the prolongation of the diameter through the extremity of the arc thus, BT is the cotangent of AM, or of AM", and BT" is the cotangent of AM', or of AM"".

56.

The secant of an arc is the distance from the cerv tre of the arc to the extremity of the tangent: thus, OT is the secant of AM, or of AM", and OT" is the se cant of AM', or of AM""'.

57. The cosecant of an arc is the distance from the