a10-b-10-a-b; the same result as before. 31. Hence, since an arithmetical complement added makes the result 10 too great, a corresponding allowance must be made in any operation in which arithmetical complements of logarithms are used. MULTIPLICATION BY LOGARITHMS. 32. Add the logarithms of the factors; and the sum will be the logarithm of their product (Art. 9). The term sum, here used, is to be understood in its algebraic signification. Therefore, since the characteristic alone of a logarithm is negative (Art. 15), whatever there is to be carried from the decimal part, in the operation, must either be added to a positive characteristic, or subtracted from one that is negative. Also, when the characteristics of the logarithms are not either all positive or all negative, the difference between their sums must be taken, and the sign of the greater prefixed. Ex. 1. Multiply 120, 101, and .015 together. Here, the 2 cancels a positive 2, so we have but 2 to set down. 2. Multiply 3.26 by .0085. 3. Multiply 6651 by 108. DIVISION BY LOGARITHMS. Ans. .02771. Ans. 718308. 33. Subtract the logarithm of the divisor from the logarithm of the dividend, and the difference will be the logarithm of their quotient (Art. 10). Or, Add the arithmetical complement of the logarithm of the divisor to the logarithm of the dividend, and the sum, less 10, will be the logarithm of the quotient (Art. 31). The term difference, here used, is to be understood in its algebraic signification. Therefore, the sign of the characteristic of the divisor must be changed; and then, if the characteristics of the divisor and dividend have the same sign, their sum must be taken, but when of different signs, their difference, with the sign of the greater, for the characteristic of the logarithm of the quotient. Also, if 1 is carried from the decimal part, it must be regarded as positive, and must be united with the characteristic of the divisor before it is changed. Here, in the first operation, 1 carried from the decimal part to the 2 changes it to 1, which being taken from 2, leaves 3 to set down; and, in the second operation, 10 is taken from the sum of the characteristics (Art. 31). 6. Find the 4th term of the proportion, 720: 196 :: 155.5. Ans. 42.33. INVOLUTION BY LOGARITHMS. 34. Multiply the logarithm of the given number by the exponent of the power to which the number is to be raised; and the product will be the logarithm of the required power (Art. 11). Since the exponent of any power is positive, a negative char 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 = 4.681241 (Log .00048)÷3= 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 to of a circumference; a minute, of a degree; a second, 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 2πr, where = 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° 2л, 180° and 30° = = 90° π = 2' П 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, secant, cosine, cotangent, and cosecant. There are also sometimes employed the quantities termed versed sine, coversed sine, and suversed sine. |