COPYRIGHT, 1899, BY LONGMANS, GREEN, AND CO. ALL RIGHTS RESERVED. FIRST EDITION, 1899. REPRINTED, JULY, 1902; APRIL, 1905; FEBRUARY, 1906. ELECTROTYPED BY J. S. CUSHING & CO., NORWOOD, MASS. II. FIVE-PLACE LOGARITHMS OF THE SINE, COSINE, TANGENT, AND COTANGENT FOR EACH MINUTE FROM 0° TO 90° (2) LOGARITHMS OF THE SINE, COSINE, TANGENT, AND CO- 86-90 (3) VALUES OF THE SINE, COSINE, TANGENT, AND COTAN- NOTE. These tables have been arranged primarily for students in elementary trigonometry, and the explanations are intended for beginners in that branch of mathematics. Tabular differences and proportional parts should be calculated, and not copied from tables, by those who use logarithmic and trigonometric tables for the first time. The editor may be allowed to take this opportunity of expressing his belief that the principles and use of common logarithms can be easily explained in the school course in arithmetic, and practical applications given which will be interesting and advantageous to young pupils. EXPLANATION OF THE TABLES. TABLE I. COMMON LOGARITHMS. N.B. The meaning and properties of logarithms are explained in works on algebra. 1. The first page of the table gives the characteristics and mantissas of numbers from 1 up to 100. The remainder of the table gives only mantissas. The characteristics are obtained by the following rule, which is deduced in algebra:* When the number is greater than 1, the characteristic is positive, and is one less than the number of figures 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. The first three figures of a number of four figures are found in the left-hand column marked N; the fourth figure of the number is found in the lines at the top and the foot of the page. The last three figures of the mantissa are found in the same line as the first three figures of the number, and in the same column as the fourth figure of the number. The first two figures of the mantissa are in the column headed 0, and are printed only once. They are found either in the same line as the last three figures, or in the first line above which contains a whole mantissa. If, however, a precedes the last three figures of the mantissa, the first two figures are found in the following line. * *This rule may be easily deduced in arithmetic. 2. To find the logarithm of a number. RULE: Write the characteristic, and then annex the mantissa found by means of the table. (a) A number of four figures. = log 3552 3.55047: log 355.72.55108; log 35.74 = 1.55315; log 36.341.56038; log 536.22.72933; log 5.371 = 0.73006. (b) A number of less than four figures. In this case, annex ciphers, or suppose them to be annexed, and proceed as in case (a). log.213=1.32838; log 47.6 = 1.67761; log .0375 = 2.57403. (c) A number of more than four figures. To find log 47653. The characteristic is 4. The mantissa, as shown in algebra, is the same as the mantissa of log 4765.3. Log 4765.3 lies between log 4765 and log 4766. Hence the mantissa of log 4765.3 is between the mantissas of log 4765 and log 4766. It is assumed that the change in the mantissa is proportional to the change in the number, as the latter increases from 4765 to 4766; that is, mantissa of log 4765.3 = mantissa of log 4765 + .3 × (mantissa of log 4766 — mantissa of log 4765).* mantissa of log 4765=.67806 difference for .3=.3x9= mantissa of log 4766=.67815 mantissa of log 4765=.67806 27 NOTE 1. By general agreement, a number with six or more decimal places is reduced to a number with five in the following way: If a number less than 5 is in the sixth decimal place, then the number in the fifth place is left unchanged; if a number greater than 5 is in the sixth place, or if there is a 5 in the sixth place and it is followed by figures other * It is assumed that when a number varies from one value to another, the change in the mantissa is proportional to the change in the number if the latter change is small in comparison with the number. This is not strictly correct, but is accurate enough for practical purposes. |