1. To find the square root of 365.

Log. 365-2.562293,

Divide by 2, the index of the root. Root 19.10496, its log. 1.2811461.

2. To find the 10th root of 2.

Log. 20.301030

Divided by 10 = 0.030103

Whose number, 1.0717, is the root.

3. To find the square root of 0.093.

Log. 0.093 2.968483

=

Divided by 2, gives 1.484241 Whose number, Q.304959, is the root.

4. To find the cube root of 0.00048.

Log. 0.00048 = 4.681241

Divided by 3, gives 2.893747

Whose number, 0.0782973, is the root.

CHAPTER II.

PLANE TRIGONOMETRY.

24. PLANE TRIGONOMETRY is that branch of Mathematics which treats of the methods of finding, by calculation, the unknown sides and angles of a plane triangle, from those sides and angles that are given.

25. For the purposes of trigonometrical calculations, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds.

26. As the circumference of a circle may be regarded as a proper measure of angles, having their vertices at the centre, the four right angles, which can be formed about the same point, are measured by 360 degrees; two right angles, by 180 degrees; one right angle, by 90 degrees; and an angle less than a right angle, by an arc less than 90 degrees.

27. Degrees, minutes, and seconds are usually designated by the respective characters, Thus, 16° 12′ 15′′ is read,

16 degrees, 12 minutes, and 15 seconds.

28. The complement of an angle is what remains after subtracting the angle from 90°. The sum of an angle and its complement, is equal to 90°.

29. The supplement of an angle is what remains after subtracting the angle from 180°. The sum of an angle and its supplement, is equal to 180°.

30. The sine of an angle is the perpendicular let fall from one extremity of the arc which measures it, on the diameter passing through the other extremity. Thus, BD (Pl. I. Fig. 1)

31. The cosine of an angle, or arc, is the part of the diameter intercepted between the foot of the sine and centre. OD is the cosine of AB.

32. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. AC is the tangent of the arc AB.

33. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. OC is the secant of the arc AB.

34. The three lines BD, AC, and OC, depend for their values on the arc AB and the radius OB, and are always determined by them. They are thus designated: BD=sin. AB, or sin. AOC, AC=tung. AB, or tang. AOC, OC=sec. AB, or sec. AOC.

35. If ABE be equal to a quadrant, or 90°, then EB is the complement of AB (28). Let the lines ST and FIB be drawn perpendicular to OE. Now, IB is the sine, ET the tangent, and OT the secant of the arc EB. Instead of writing sine, tangent, and secant, of this complemental arc EB, the lines are usually designated by means of the arc AB. Thus, BI=OD, is called the cosine of AB, or cosine of AOC, written cos. AB; ET is called the cotangent of AB, or cotangent of AOC, written cot. AB; and OT, the cosecant of AB, or cosecant of AOC, written cosec. AB. In general, if A be any arc or angle, we have the cos. A=sin. (90°—A), cot. A=tang. (90°-A), and cosec. A=sec. (90°—A).

36. The triangles OBD, and OCA, being similar, and also similar to the triangles OBI, OTE, the relations which exist between the lines that depend for their values on the arc AB, are readily ascertained.

=

37. The radius, sine, and cosine of any arc, forming a rightangled triangle, if we denote the radius by R, we have R2 sin.+cos.2. The radius, secant, and tangent, forming a rightangled triangle, the sec.2 R2+tang."; also, cosec.2=R2+ cot.2. The similar triangles ODB, OAC, give, for any arc or

=

Cos. A: sin. A :: R: tang. A=R

Cos. A R :: R: sec. A=

And the similar triangles, OIB, OET, give,

sin. A

cos. A'

R2

cos. A

Sin. A cos. A :: R: cot. A

cos. A

R

sin. A'

R2

sin. A'

Sin. A : R :: R : cosec. A=

Also, OAC and OTE being similiar,

OA : AC :: ET: OE, or, R2=tang. A cot. A.

We see, from these relations, that the cosine of an arc can be found, when radius and the sine are given; and if radius, the sine, and cosine be known, that the secant, cosecant, tangent, and cotangent are determined from them.

38. We see, in examining the figure, that if the point B coincides with the point A, or the arc AB=0, the sin. AB becomes =0, the tang. AB becomes =0, the sec. AB=R, the cos. AB =R; the cot. AB, and cosec. AB, become infinite, since OC and ET are then parallel. When the arc AB=45°, sin. A= cos. A, tang. A=cot. A, sec. A=cosec. A. When the arc AB becomes AE, or equal to 90°, then the sin. A, or sin. 90°= R; cos. A, or cos. 90=0; tang. A and sec. A become infinite, as the lines AC and OC do not intersect.

39. If the arc ABEF, greater than 90, be considered, FH is its sine (30); OH, its cosine (31); AQ, its tangent (32); and OQ, its secant (33). But FH is the sine of the arc GF, the supplement of ABF (29), and OH is its cosine (31); hence, the sine of an arc is equal to the sine of its supplement; and the cosine of an angle, to the cosine of its supplement.

Furthermore, AQ is the tangent of the arc AEF (32), and OQ is its secant (33); GL is the tangent, and OL the secant, of the supplemental arc GF. But, as AQ is equal to GL, and OQ to OL, it follows, that the tangent of an arc or angle is equal to its supplement; and the secant of an arc, to the secant of its supplement.*

* These relations are between the values of the trigonometrical lines; the

40. If, in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent be calculated for every minute or second of the quadrant, and arranged in a table, such table is called a table of sines and tangents. If the radius of the circle be 1, the table is called a table of natural sines. A table of natural sines is, therefore, a table which shows the values of the sines, cosines, tangents, and cotangents, to seconds or minutes, of all the arcs of a quadrant. The corresponding values of the secants and cosecants are usually omitted, being readily found from those of the cosine and sine (37). 41. If the values of the sine, cosine, tangent, and secant be known for arcs or angles less than 90°, they are also known for arcs or angles which are greater. is greater than 90, its supplement is values of these lines are the same for an angle and its supplement (39).

For, if an arc or angle less than 90°, and the

42. We have not considered the sines, cosines, &c. of arcs greater that 180; for, as the sum of the three angles of a plane triangle is equal to 180°, it follows, that no larger arc can enter into the calculations of the sides and angles of plane triangles.

THEOREM.

43. The sides of a plane triangle are proportional to the sines of their opposite angles.

Let ABC (Pl. I. Fig. 2) be a triangle; then,

CB: CA :: sin. A: sin. B.

For, with A as a centre, and AD, equal to the less side BC, as a radius, describe the arc DI; and with B as a centre, and BC as a radius, describe the arc CL. Now, ED is the sine of the angle A (30), and CF is the sine of the angle B, to the same radius AD or BC. But, by similar triangles, AD DE AC: CF; AD, being equal to BC, we have, BC sin. A :: AC: sin. B, or

BC

:

AC:: sin. A : sin. B.

By comparing the sides AB, AC, in a similar manner, we should obtain,