acteristic multiplied by it will give a negative result; but that which is to be carried from the decimal part will be positive; therefore, their difference will be the characteristic of the product. Ex. 1. Required the square, or second power, of 31.

2. Required the cube, or third power, of .25.

35. Divide the logarithm of the given number by the index of the root; and the quotient will be the logarithm of the required root (Art. 12).

When the characteristic of the logarithm is negative, and does not contain the given divisor without a remainder, we may increase the characteristic by any number that will make it exactly divisible, provided we prefix an equal positive number to the decimal part of the logarithm.

Ex. 1. Required the square, or second root, of 1296.

2. Required the cube, or third root, of .00048.

Log .00048

=

(Log .00048)+3=

4.681241

2.893747 Ans. .078297.

Here, the negative characteristic 4 not being exactly divisible by 3, it is increased by 2 to make it so, and then the 2 borrowed is restored, by regarding 2 as prefixed to the decimal part. 3. Required the fourth root of .434296.

4. What is the tenth root of 2?

Ans. .811794.

Ans. 1.0718.

BOOK II.

PLANE TRIGONOMETRY.

DEFINITIONS AND ELEMENTARY PRINCIPLES.

36. TRIGONOMETRY is the science which treats of methods of computing angles and triangles.

37. PLANE TRIGONOMETRY treats of methods of computing plane angles and triangles.

38. The MAGNITUDE OF ANGLES is represented by numbers expressing how many times they contain a certain angle fixed upon as the unit of angular measure.

For this purpose a right angle is generally divided into 90 equal parts called degrees, each degree into 60 equal parts called minutes, each minute into 60 equal parts called seconds; then an angle is expressed by the number of degrees, minutes, seconds, and decimal parts of a second, which it contains.

39. Degrees, minutes, and seconds, are marked by the symbols °, ', "; thus, to represent 16 degrees, 9 minutes, 23.5 seconds, we write 16° 9' 23.5.

40. Since angles at the centre of a circle are to each other as the arcs of the circumference intercepted between their sides (Geom., Prop. XVII. Bk. III.), these arcs may be regarded as the measures of the angles, and the number of units of arc intercepted on the circumference may be used to express both the arc and the corresponding angle.

41. A degree of arc is ʊ of a circumference; a minute, of a degree; a second, go of a minute; and these arcs subtend angles of a degree, a minute, and a second, respectively, at the centre.

42. For simplifying calculations, the radius employed in measuring angles, being constant, is taken at an assumed value of unity, as the linear unit of measure.

43. Since the value of the constant ratio of the circumference to the diameter of a circle, represented by л, is 3.14159 (Geom., Prop. XV. Sch. 1, Bk. VI.), if the radius of a circle is denoted by r, its circumference is 2 πr, where л= 3.14159. Hence, as r is taken as unity, any number of degrees may be expressed as a multiple or fractional part of л. Thus 360° 2 л, 180° = π, 90° = and 30°

π

2'

π

6°

44. The COMPLEMENT OF AN ANGLE, or arc, is the remainder obtained by subtracting the angle or arc from 90°. Thus the complement of 45° is 45°, and the complement of 31° is 59°.

When an angle, or arc, is greater than 90°, its complement is negative. Thus the complement of 127° is - 37°.

Since the two acute angles of a right-angled triangle are together equal to a right angle, they are complements one of the other.

45. The SUPPLEMENT OF AN ANGLE, or arc, is the remainder obtained by subtracting the angle or arc from 180°. Thus the supplement of 110° is 70°.

When the angle is greater than 180°, its supplement is negative. Thus the supplement of 200° is-20°.

Since the three angles of any triangle are together equal to two right angles, any one of them is a supplement of the sum of the other two.

TRIGONOMETRIC FUNCTIONS.

46. TRIGONOMETRIC FUNCTIONS are the quantities by which angles are subjected to computation.

These we shall consider, in accordance with the best modern authorities, as ratios formed by comparing the sides of a rightangled triangle, and thus capable of comparison one with another by means of their geometrical properties.

These ratios have received the special names of sine, tangent, scant, cosine, cotangent, and cosecant.

There are also sometimes employed the quantities termed versed sine, coversed sine, and suversed sine.

47. The SINE of an angle is the ratio of the opposite side to the hypothenuse.

Thus, in any right-angled triangle, A B C,

if the sides be denoted by p, b, h, we shall

B

48. The TANGENT of an angle is the ratio of the opposite side to the adjacent side.

49. The SECANT of an angle is the ratio of the hypothenuse to the adjacent side.

50. The COSINE, COTANGENT, and COSECANT of an angle are respectively the SINE, TANGENT, and SECANT of its complement. Hence, since the acute angles of a right-angled triangle are complements one of the other (Art. 44), we have, according to the definitions,

that the cosecant, cotangent, and cosine of an angle are respec

tively the reciprocals of the sine, tangent, and secant of the angle.

52. If the cosine of A be subtracted from unity, the remainder is called the versed sine of A; if the sine of A be subtracted from unity, the remainder is called the coversed sine of A; and if the cosine of A be added to unity, the sum is called the suversed sine of A. Hence,

vers A1 cos A, covers A = 1 — sin A,

suvers A=1+cos A.

(6)

53. The values of trigonometric ratios remain the same so long as the angle continues the same.

B'

B

Let BA C be any angle; in A B take any point, B, and draw BC perpendicular to A C; also take any other point, B', and draw B'C' perpendicular to A C. Then, since the triangles ABC, AB' C' are similar, their sides have to one another the same ratio (Geom., Art. 210), and therefore sin A, tan A, A

&c. will have the same values, whether A B C

C' C

or A B'C' be the triangle by the sides of which they are expressed. It is also evident that their values would change with a change of the angle. Hence,

The trigonometric ratios determine the angles, and conversely; that is, any determinate values being given for the one, determinate values can be found for the other.

con

54. The terms sine, tangent, secant, &c., were formerly * sidered to be functions of an arc, and denoted certain trigonometric lines.

D

Α

Cot.

T'

Sec

Thus, let O be the centre of any circle, A A" its diameter, and A Bany arc; draw the radius OA' at right angles to AA", and draw tangents to the circle at the points A and A'; produce OB to meet the first tangent in 7and the second tangent in T"; draw BD perpendicular to O A, and B D' perpendicular to OA'. Then, by the old definitions, the lines of the figure are considered to

Suvers Cos. Vers

DA

"The modern method has now completely superseded the ancient method

in English works." . Todhunter's Trigonometry, p. 49.