until the last significant figure is reached, which must be The arithmetical complement is denoted by taken from 10. the symbol (a. c.). represent any two logarithms whatever, Since we may add 10 to, and a - b their difference. and subtract it from, ab, have, without altering its value, we a − b = a + (10 − b) — 10. . (10.) But, 10-6 is, by definition, the arithmetical complement of b: hence, Equation (10) shows that the difference between two logarithms is equal to the first, plus the arithmetical complement of the second, minus 10. Hence, to divide one number by another by means of the arithmetical complement, we have the following RULE. Find the logarithm of the dividend, and the arithmetical complement of the logarithm of the divisor, add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required. The operation of subtracting subtracting 10 is always performed mentally. RAISING ΤΟ POWERS BY MEANS OF LOGARITHMS. 18. From the principle proved in Art. 7, we have the following RULE. Find the logarithm of the number, and multiply it by the exponent of the power; then find the number corresponding to the resulting logarithm, and it will be the power required. 19. From the principle proved in Art. 8, we have the following RULE. Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required. EXAMPLES. 1. Find the cube root of 4096. The logarithm of 4096 is 3.612360, and one-third of this is 1.204120. The corresponding number is 16, which is the root sought. When the characteristic is negative and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa. 2. Find the 4th root of .00000081. The logarithm of .00000081 is 7.908485, which is equal to 8 +1.908485, and one-fourth of this is 2.477121. The number corresponding to this logarithm is .03: hence, .03 is the root required. PLANE TRIGONOMETRY. 20 PLANE TRIGONOMETRY is that branch of Mathematics which treats of the solution of plane triangles. In every plane triangle there are six parts: three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts, is called the solution of the triangle. 21. A plane angle is measured by the arc of a circle included between its sides, the centre of the circle being at the vertex, and its radius being equal to 1. Thus, if the vertex A be taken as a centre, and the radius AB be will measure the angle A (B. III., P. XVII., S.). A B Let ABCD represent a circle whose radius is equal to 1, and AC, BD, two diameters perpendicular to each other. These diameters divide the circumference into four equal parts, called quadrants; and because each of the angles at the centre is a right angle, it follows that a right angle is measured by a quad D A rant. An acute angle is measured by an arc less than a quadrant, and an obtuse angle, by an arc greater than a quadrant. 22. In Geometry, the unit of angular measure is a right angle; so in Trigonometry, the primary unit is a quadrant, which is the measure of a right angle. For convenience, the quadrant is divided into 90 equal parts, each of which is called a degree; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are denoted by the symbols °, ', "!. Thus, the expression 7° 22' 33", is read, 7 degrees, 22 minutes, and 33 seconds. Fractional parts of a second are expressed decimally. A quadrant contains 324,000 seconds, and an arc of 7° 22′ 33′′ contains 26553 seconds; hence, the angle measured by the latter arc, is the 2553th part of a right angle. In like manner, any angle may be expressed in terms of a right angle. 23. The complement of an arc is the difference between that arc and 90°. The complement of an angle is the difference between that angle and a right angle. Thus, EB is the complement of AE, and FB is the complement of AF. In like manner, EOB is the complement of AOE, and FOB is the complement of AOF In a right-angled triangle, the F acute angles are complements of each other. B E 24. The supplement of an arc is the difference between |