4. Find the value of .035063. By Art. 95, log .035063 = log.035063. log.035063 8.5449 — 10 20. - 20 (see Note 3) 3)28.544930 9.5150 10 = log.3274. Note 3. To divide a negative logarithm, add to both parts such a multiple of 10 as will make the negative portion of the characteristic exactly divisible by the divisor, with -10 as the quotient. Thus, to divide 8.5449-10 by 3, add 20 to both parts of the logarithm, giving the result 28.5449-30. Dividing this by 3, the quotient is 9.5150-10. ARITHMETICAL COMPLEMENT. 103. The Arithmetical Complement of the logarithm of a number, or briefly the Cologarithm of the number, is the logarithm of the reciprocal of that number. The following rule is evident from the above: To find the cologarithm of a number, subtract its logarithm from 10-10. Note. The cologarithm may be obtained from the logarithm by subtracting the last significant figure from 10 and each of the others from 9, 10 being written after the result in the case of a positive logarithm. It is evident from the above that the logarithm of a fraction is equal to the logarithm of the numerator plus the cologarithm of the denominator. Or in general, to find the logarithm of a fraction whose terms are composed of factors, Add together the logarithms of the factors of the numerator, and the cologarithms of the factors of the denominator. Note. The value of the above fraction may be found without using cologarithms, by the following formula: log .51384 8.709.0946 = log .51384 - log (8.709 x .0946) = log .51384 (log 8.709 + log .0946). The advantage in the use of cologarithms is that the written work of computation is exhibited in a more compact form. EXAMPLES. 105. Note. A negative quantity can have no common logarithm, as is evident from the definition of Art. 79. If negative quantities occur in computation, they may be treated as if they were positive, and the sign of the result determined irrespective of the logarithmic work. Thus, in Ex. 3, p. 65, the value of 721.3 × (-3.0528) may be obtained by finding the value of 721.3 x 3.0528, and putting a negative sign before the result. See also Ex. 34, p. 66. VII. SOLUTION OF RIGHT TRIANGLES. 106. The six elements of a triangle are its three sides and its three angles. We know by Geometry that, in general, a triangle is completely determined when three of its elements are known, provided one of them is a side. The solution of a triangle is the process of computing the unknown from the given elements. 107. To solve a right triangle, two elements must be given in addition to the right angle, one of which must be a side. The various cases which can occur may all be solved by aid of the following formulæ : CASE I. When the given elements are a side and an angle. The proper formula for computing either of the remaining sides may be found by the following rule: Take that function of the angle which involves the given side and the required side. CASE II. When both the given elements are sides. First calculate one of the angles by aid of either of the formulæ involving the given elements, and then compute the remaining side as in Case I. |