integral part of the logarithm is called the index or charac teristic. The two great advantages of the common system, as will now be shown, are: (1) The characteristic of a logarithm can be written on mere inspection; (2) The position of the decimal point in a number affects the characteristic alone, the mantissa being always the same for the same sequence of figures. log.11, log .012, log .0013, log .0001-4, etc. (2) From (1) and (2) comes the following rule for finding the characteristic: When the number is greater than 1, the characteristic is positive and is one less than the number of digits to the left of the decimal point; when the number is less than 1, the characteristic is negative, and is one more than the number of zeros between the decimal point and the first significant figure. When a change is made in the position of the decimal point in a number, the value of the number is changed by some integral power of 10. Its logarithm is then changed by a whole number only, and, consequently, its mantissa is not affected. For example, 25.382538 x 10-2, 25380002538 × 103; and hence, log 25.38 log 2538-2, log 2538000=log 2538 +3. = Accordingly, it is necessary to put only the mantissas of sequences of integers in the tables. 5. Negative characteristics. In common logarithms the mantissa is always kept positive. Thus, for example, log 25380 4.40449; log .002538 = log log 2538 - log 1000000 = 3.40449 — 6 = 3.40449. (Never put - 2.59551.) 2538 = 1000000 This logarithm is usually written 3.40449, in order to show that the minus sign affects the characteristic alone. In order to avoid the use of negative characteristics, 10 is often added to the logarithm and 10 placed after it. Thus 3.40449 is written 7.40449 - 10. The second form is more convenient for purposes of calculation. Special care is necessary in dealing with logarithms because of the fact that the mantissa is always positive, while the characteristic may be either positive or negative. Some typical examples involving negative characteristics are given below. A result like (1) is always put in the form (2), in which the number placed after the logarithm is -10. Ex. 3 may also be worked thus: (-1+.83471) x 2-2 +1.669421.66942. × : 4. Division. 3.27412 ÷ 4 =(37.27412 – 40) ÷ 4 = 9.31853 – 10. As in Ex. 3 care is taken that, finally, the number after the logarithm be- 10. 5. 2.34175 ÷ 5 = (48.34175 – 50) ÷ 5 = 9.66835 – 10. The method of finding the logarithms in the tables when the numbers are given, and the way to find the numbers when the logarithms are given, are usually explained in connection with the tables of logarithms. 6. Exercises in logarithmic computation. On looking at the laws of logarithms, (3)–(6), Art. 3, it is apparent that logarithms cannot assist in the operations of addition and subtraction. Logarithms are of no service in computing expressions of the forms M+N, M-N. An expression is said to be adapted to logarithmic computation when it is expressed by means of factors only. Thus, 1 an. br c1028 is not. is adapted to logarithmic computation, but a+b-2 a2+19 7a-5b Let R = √.005. Then log R = log (.005): =log .005= 3.69897 = 17.69897-20 8.84948-10 = 2.84948. ... R.07071. 3. Find 742 × .0769. Let R be the value. Then log R = log √742 × .0769 = log (742 × .0769) = } log (742 × .0769) = } [log 742 + log .0769]. Let R be the value. Then log R = † (log 456 + log 372 — log 350 – log 249). 9. Find the value of √63.42 × 74.95, √6.35 × 10.87, √14.21 × 17.29. 63.9 x 72.11 31.21 x 41.7 10. Find the value of 7.81 X 6.95' 11.39 × 15.71' CHAPTER II. TRIGONOMETRIC RATIOS OF ACUTE ANGLES. 7. The name Trigonometry is derived from two Greek words which taken together mean 'I measure a triangle.'* At the present time the measurement of triangles is merely one of several branches included in the subject of trigonometry. The more elementary part of trigonometry is concerned with the calculation of straight and circular lines, angles, and areas belonging to figures on planes and spheres. It consists of two sections, viz. Plane Trigonometry and Spherical Trigonometry. Elementary trigonometry has many useful applications, for instance, in the measurement of areas, heights, and distances. An acquaintance with its simpler results is very helpful, and sometimes indispensable, in even a brief study of such sciences as astronomy, physics, and the various branches of engineering. Some modern branches of trigonometry require a knowledge of advanced algebra. Their results are used in the more advanced departments of mathematics and in other sciences. This work considers only the simpler portions of trigonometry, and shows some of its applications. The truths of elementary trigonometry are founded upon geometry, and are obtained and extended by the help of arithmetic and algebra. A knowledge of the principal facts of plane geometry, and the ability to perform the simpler processes of algebra, are necessary on beginning the study of plane trigonometry. Instruments for measuring lines and angles, and accuracy in computation are required in making its practical applications. 8. Ratio. Measure. On entering upon the study of trigonometry it is very necessary to have clear ideas concerning the terms ratio and incommensurable numbers as explained in arithmetic and algebra, for these terms play a highly important part * See historical sketch, p. 165. |