practically used. We shall not enter here upon the method of forming the tables themselves. The following is a specimen of the way in which the logarithms of numbers are usually tabulated: Thus, if the number consist of four figures only, we have simply to copy out the figures in the column headed 0, prefix a decimal point, and the proper characteristic. Ex. Log 7991 = 3.9026011, log 7.995 = 9028185. When we speak of a number consisting of four figures only, we include such numbers as 003654, 07682, &c., the number of zeros immediately following the decimal points not being counted. Thus, log 07997 = 2.9029271 log 007992 3.9026555. = When the number contains five figures, as, for instance, 79936, we look along the line containing the first four figures -viz., 7993—of the number until the eye rests upon the column headed 6, the fifth figure. We then take the first three figures of the column headed 0, and affix the four figures of the column headed 6 in the horizontal line of the first four figures of the number. Thus, log 79936 = 4.9027424 log 079927 2.9026935. = It will be seen from the portion of the logarithmic table above extracted, that when the first three figures of the logarithm-viz., 902—have been once printed, they are not repeated, but must be understood to belong to every four figures in each column, until they are superseded by higher figures, as 903. When, however, this change is intended to be made at any place not at the commencement of a horizontal row, the first of the four figures corresponding to the change is usually printed either in different type, or, as above, with a bar over it. Thus we have above 0031, indicating that from this point we must prefix 903 instead of 902. Thus, log 79.986 = 1.9030140, 26. To find the logarithm of a number not contained in the tables. Ex. Find the logarithm of 799-1635. Since the mantissa of the number 79916.35 is the same as the mantissa of the given number, and that the first five figures are contained in the tables, we may proceed as follows— (1.) Take out from the tables the mantissa corresponding to the number 79916. This is 9026337. (2.) Take out the mantissa of the next higher number in the tables-viz., 79917. This is 9026392. (3.) Find the difference between these mantissæ. This is called the tabular difference, being the difference of the mantissæ for a difference of unity in the numbers. We find (4.) Then assuming that small differences in numbers are proportional to the differences of the corresponding logarithms, we find the difference for 35: 35 x 0000055 retaining only 7 figures. This is often called d. = = ⚫0000019, (5.) Now adding this value of d to the mantissa for the number 79916, we get the mantissa corresponding to the number 79916.35. (6.) Lastly, prefix to this mantissa the proper characteristic. The whole operation may stand thus M. of log 79916 = 9026337 (1). * Thus log 79916.35 = log (100 x 799·1635) = 2 + log 799-1635. In the next article we shall show how the required difference may be obtained by inspection from the tables. 27. Proportional parts. We saw in the example just worked that the tab. diff. (omitting the useless ciphers) is 55, and if we examine the table in Art. 25, we shall find the difference between the mantissæ of any two consecutive numbers there to be 54 or 55-generally 54. The number 54 is therefore placed in a separate column at the right of the table, and headed D. The student will understand that the tab. diff. changes from time to time, and is not always 54 or 55. Now assuming as in (4.) of the last article, we have— We find therefore the numbers 5, 11, 16, 22, 27, 32, 38, 43, 49 placed in a horizontal row at the bottom marked P, in the columns respectively headed 1, 2, 3, 4, &c. Hence, if we require the difference for (say) 7, we take out the number 38 from the horizontal row marked P, instead of being at the trouble to find it by actual computation. The following example will illustrate how we proceed when we require the difference for a decimal containing more than one decimal figure. No explanation is needed. = -9027843 :. M. of log 79943726 Hence, log 7994-37263-9027843. 28. Having given the logarithm of a number to find the number. After the explanations of Art. 26, the method of working the following examples will be easily understood :— Ex. 1. Find the number whose logarithm is I-9030173. Taking from the tables the mantissæ next above and below, we have Hence, subtracting (1) from (z)— 33 = d, the difference between the logarithms of the re quired number and the next lower. Now = 33 61, the difference between the next lower number and the required number. Hence ·9030173 = M. of log 79986-61; ... I-9030173 = log ·7998661; ... ·7998661 is the number required. Ex. 2.* Find the value of (1·023)3 × We have (-00123)+ (1.32756)* log N = 3 log 1023 + log 00123-4 log 1.32756. Now, 3 log 1.023 = 3 × 0098756 log 00123 = + (3-0899051) = (4+1-0899051) = ⚫0276268 = I-2724763 * The logarithms used in this example are taken from the tables. = .. 2.5078867 log ·03220229. Hence 03220229 is the number required. Trigonometrical Tables. 29. We use trigonometrical tables much in the same way as we do tables of ordinary logarithms of numbers. Tables have been formed of natural sines, cosines, &c., and also of logarithmic sines, cosines, &c. It is with the latter only we shall now deal, though many of our remarks apply equally to the former. As the values of the natural sines and cosines of all angles between 0° and 90° are (Art. 11) less than unity, it follows (Art. 21) that their logarithms are negative. To avoid, however, printing them in a negative form, and for other reasons, it is usual to add 10 to their real value, and hence in using them we must allow for this. The same thing is also done in the case of logarithmic tangents, cotangents, secants, and cosecants. We generally express the true logarithmie sine by log sin, and the tabular logarithmic sine by L sin. It must be remembered in using the tables that, although (Art. 11) the sine, secant, and tangent of an angle increase as |