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: +.83471) × 2 = 2 +1.66942 = 1.66942. 4. Division: -27412+4(37.27412-40) + 4 = 9.31853 — 10. As in Ex. 3 care is taken that, finally, the number after the log. arithm 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 6.] EXERCISES. 7 M+N, M-N. An expression is said to be adapted to logarithmic computation when it is expressed by means of factors only. Thus, 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. in the subject. The study begins with an explanation of certain ratios which are used in it continually, and most of the numbers that appear in the solution of its problems are incommensurable. If one quantity is half as great as another quantity in magnitude, it is said that the ratio of the first quantity to the second is as one to two, or one-half. This ratio is sometimes indicated thus, 1:2; but more usually it is written in the fractional form, In this example the magnitude of the second quantity is twice that of the first, and the ratio of the second quantity to the first is 2:1, or, adopting the more usual style,, i.e. 2. The ratio of two quantities is simply the number which expresses the magnitude of the one when compared with the magnitude of the other. This ratio is obtained by finding how many times the one quantity contains the other, or by finding what fraction the one is of the other. It follows that a ratio is merely a pure number, and that it can be obtained only by comparing quantities of the same kind. Thus the ratio of the length 3 feet to the length 2 inches is 36, i.e. 18; the ratio of the weight 2 pounds to the weight 3 pounds is. But one cannot speak of the ratio of 3 weeks to 10 yards, for there is no sense in the questions: How many times does 3 weeks contain 10 yards? What fraction of 10 yards is 3 weeks? When it is said that a line is ten inches long, this statement means that a line one inch long has been chosen for the unit of length, and that the first line contains ten of these units. Thus the number used in telling the length of a line is the ratio of the length of this line to the length of another line which has been chosen for the unit of length. The measure of any quantity, such as a length, a weight, a time, an angle, etc., is the number of times the quantity contains or, the fraction that the quantity is of a certain quantity of the same kind which has been adopted as the unit of measurement. In other words, the measure of a quantity is the ratio of the quantity to the unit of measurement. For example, if half an inch is the unit of length, then the measure of a line 8 inches long is 16; if a foot is the unit of length, then the measure of the same line is ; if a second is the unit of time, then the measure of an hour is 3600; if an hour is the unit of time, then the measure of a second is 300. |