The Here, 1 taken from I, gives 2 for a result. subtraction, as in this case, is always to be performed in the algebraic sense. 3. Divide 37.149 by 523.76. Ans. 0.0709274. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of 17. The ARITHMETICAL COMPLEMENT of a logarithm is the result obtained by subtracting it from 10. is the arithmetical complement of 1.869544. Thus, 8.130456 The arithmetical complement of a logarithm may be written out by commencing at the left hand and subtracting each figure from 9; until the last significant figure is reached, which must be The arithmetical complement is denoted by taken from 10. Let a and represent any two logarithms whatever, Since we may add 10 to, without altering its value, we and α b, their difference. and subtract it from, ab, But, 106 is, by definition, the arithmetical complement of b: hence, Equation (10) shows that the difference be tween two logarithms is equal to the first, plus the arith metical complement of the second, minus 10. Hence, to divide one number by another by means the arithmetical complement, we have the following of RULE. Find the logarithm of the dividend, and the arithmetical complement of the logarithm of the divisor, add them toge ther, and diminish the sum by 10; the number correspond ing to the resulting logarithm will be the quotient required. 3. Multiply 358884 by 5672, and divide the product Applying logarithms, the logarithm of the 4th term, is equal to the sum of the logarithms of the 2d and 3d terms, minus the logarithm of the 1st: Or, the arithmetical complement of the 1st term, plus the logarithm of the 2d term, plus the logarithm of the 3d term, minus 10, is equal to the logarithm of the 4th term. The operation of subtracting 10, is performed mentally. RAISING OF POWERS BY MEANS OF LOGARITHMS. 18. From Article 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 EXTRACTING ROOTS BY MEANS OF LOGARITHMS. 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 Mathematica 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 equal to 1, be the intercepted arc BC will measure the angle A (B. III., P. XVII., S.). 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 B D |