RULE. Divide the logarithm of the number by the index of the root required. Ex. 1. Required the cube root of 482.38. The logarithm of 482.38 is 2.683389. Dividing by 3, we have 0.894463, which corresponds to 7.842, which is therefore the root required. Ex. 2. Required the 100th root of 365. Ans., 1.0608. When the characteristic of the logarithm is negative, and is not divisible by the given divisor, we may increase the characteristic by any number which will make it exactly divisible, provided we prefix an equal positive number to the decimal part of the logarithm. Ex. 3. Required the seventh root of 0.005846. The logarithm of 0.005846 is 3.766859, which may be written 7+4.766859. Dividing by 7, we have 1.680980, which is the logarithm of .4797, which is, therefore, the root required. This result may be verified by multiplying 1.680980 by 7, the result will be found to be 3.766860. Ex. 4. Required the fifth root of 0.08452. PROPORTION BY LOGARITHMS. (16.) The fourth term of a proportion is found by multiplying together the second and third terms, and dividing by the first. Hence, to find the fourth term of a proportion by loga rithms, Add the logarithms of the second and third terms, and from their sum subtract the logarithm of the first term. Ex. 1. Find a fourth proportional to 72.34, 2.519, and 357.48 Ans., 12.448. (17.) When one logarithm is to be subtracted from another, it may be more convenient to convert the subtraction into an addition, which may be done by first subtracting the given log. arithm from 10, adding the difference to the other logarithm, and afterward rejecting the 10. The difference between a given logarithm and 10 is called its complement; and this is easily taken from the table by beginning at the left hand, subtracting each figure from 9, except the last significant figure on the right, which must be subtracted from 10. For To subtract one logarithm from another is the same as to add its complement, and then reject 10 from the result. a-b is equivalent to 10-b+a−10. To work a proportion, then, by logarithms, we must the logarithms of the second and third terms. The characteristic must afterward be diminished by 10. 489 is 2.689309 38750 is 4.588272 The fourth term is 2765, whose logarithm is 3.441700. One advantage of using the complement of the first term in working a proportion by logarithms is, that it enables us to exhibit the operation in a more compact form. Ex. 3. Find a fourth proportional to 73.84, 658.3, and 4872., Ans. Ex. 4. Find a fourth proportional to 5:745, 781.2, and 54.27. BOOK II. PLANE TRIGONOMETRY. (18.) TRIGONOMETRY is the science which teaches how to determine the several parts of a triangle from having certain parts given. Plane Trigonometry treats of plane triangles; Spherical Trigonometry treats of spherical triangles. (19.) The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 minutes, and each minute into 60 seconds. Degrees, minutes, and seconds are designated by the characters, ',". Thus 23° 14′ 35′′ is read 23 degrees, 14 minutes, and 35 seconds. Since an angle at the center of a circle is measured by the arc intercepted by its sides, a right angle is measured by 90°, two right angles by 180°, and four right angles are measured by 360°. D Cotan. L K Cosine Sec. I Sine Tangent (20.) The complement of an arc is what remains after subtracting the arc from 90°. Thus the arc DF is the complement of AF. The complement of 25° 15' is 64° 45'. In general, if we represent any arc by A, its complement is 90° — A. Hence, if an arc is greater than 90°, its complement must be negative. Thus, the complement of 100° 15' is B E Cos. Vers. G H A -10° 15'. Since the two acute angles of a right-angled triangle are to- M gether equal to a right angle, each of them must be the complement of the other. (21.) The supplement of an arc is what remains after subtracting the arc from 180°. Thus the arc BDF is the supplement of the arc AF. The supplement of 25° 15′ is 154° 45'. In general, if we represent any arc by A, its supplement is 180°-A. Hence, if an arc is greater than 180°, its supplement must be negative. Thus the supplement of 200° is —20o. Since in every triangle the sum of the three angles is 180°, either angle is the supplement of the sum of the other two. (22.) The sine of an arc is the perpendicular let fall from one extremity of the arc on the radius passing through the other extremity. Thus FG is the sine of the arc AF, or of the angle ACF. Every sine is half the chord of double the arc. Thus the sine FG is the half of FH, which is the chord of the arc FAH, double of FA. The chord which subtends the sixth part of the circumference, or the chord of 60°, is equal to the radius (Geom., Prop. IV., B. VI.); hence the sine of 30° is equal to half of the radius. (23.) The versed sine of an arc is that part of the diameter intercepted between the sine and the arc. Thus GA is the versed sine of the arc AF. (24). The tangent of an arc is the line which touches it at one extremity, and is terminated by a line drawn from the center through the other extremity. Thus AI is the tangent of the arc AF, or of the angle ACF. (25.) The secant of an arc is the line drawn from the center of the circle through one extremity of the arc, and is lim ited by the tangent drawn through the other extremity. Thus CI is the secant of the arc AF, or of the angle ACF. (26.) The cosine of an arc is the sine of the complement of that arc. Thus the arc DF, being the complement of AF, FK is the sine of the arc DF, or the cosine of the arc AF. The cotangent of an arc is the tangent of the complement of that arc. Thus DL is the tangent of the arc DF, or the cotangent of the arc AF. The cosecant of an arc is the secant of the complement of that arc. Thus CL is the secant of the arc DF, or the cose cant of the arc AF. In general, if we represent any angle by A, COS. cot. A=sine (90°—A). A=tang. (90°—A). cosec. A sec. (90°-A). Since, in a right-angled triangle, either of the acute angles is the complement of the other, the sine, tangert, and secant of one of these angles is the cosine, cotangent, and cosecan of the other. D Cotan. L Cosine ·Sec. I (27.) The sine, tangent, and secant of an arc are equal to the sine, tangent, and secant of its supplement. Thus FG is the sine of the arc AF, or of its supplement, BDF. Also, AI, the tangent of the arc AF, is equal to BM, the tangent of the arc BDF. And CI, the secant of the arc AF, is equal to CM, the secant of the arc BDF. B K E Sine Tangent Cos. Vers. A G H (28.) The relations of the sine, cosine, &c., to each other, may be derived from the proportions of the sides of similar triangles. Thus the triangles CGF, CAI, CDL, being similar, we have, 1. CG: GF:: CA: AI; that is, representing the arc by A, and the radius of the circle by R, cos. A: sin. A :: R: tang. A. R sin. A cos. A Whence tang. A= 2. CG:: CF: CA: CI; that is, cos. A : R :: R: sec. A. 3 GF: CG:: CD: DL; that is, sin. A : cos. A ::R: cot. A. 4 GF: CF:: CD: CL; that is, sin. A : R::R: cosec. A. 5. AI: AC:: CD: DL; that is, tang. A: R:: R: cot. A. The preceding values of tangent and cotangent, secant and cosecant will be frequently referred to hereafter, and should be carefully committed to memory. |