3. Required the logarithmic sine of 159° 20' This will be the same with that of its supplement 200 40'. For convenience the supplements of arcs in the 1st quadrant are placed on the right of the page in the tables at top, and on the left at bottom. By looking therefore for 159° at top on the right, and immediately under for 20' in the column м, on the range with this in the column entitled sine at top I find 9.54769, the same that would have been found for 20° 40'. 59. CASE II. If the given number be composed of degrees, minutes, and seconds, find the logarithm of the degrees and minutes as above, and then to know how much this should be increased for the given number of seconds, in case of the sine or tangent, or diminished in case of the cosine or cotangent, observe that the number in the column marked Diff. is the increase of the logarithm for the number of seconds in the column headed м, against which it stands, and will be the quantity to add to the logarithmic sine or tangent before found, or to subtract from the logarithmic cosine or cotangent. The number in the column Diff. is calculated by subtracting one of two consecutive logarithms in the table, which differ by 1', from the other,* and dividing the remainder by 60, the number of seconds in a minute; the quotient is the difference of logarithms corresponding to a difference of 1'' in the numbers to which they belong, or is the increase of the logarithm for 1" increase of arc. This quotient, multiplied by any number of seconds, will give the increase of the logarithm for that number of seconds. This calculation depends upon the principle mentioned at Art. 51, that the differences of logarithms are proportional to the differences of their corresponding numbers. See also Art. 23 App. I. 60. The logarithmic sines and cosecants, cosines and secants, tangents and cotangents, have each pair but one column of differences between them, the reason of which will appear from the following demonstration. The secant and cosecant may be easily computed from the cosines and sines. Thus (Art. 33): hence, = sec. = R COS log. sec 20―log. cos for log. of the quotient difference of logs. (Art. 49); and log. of the * This diff. is given in the first five pages of the table so that the seconds must be calculated as here described when the given arc is 40 or less, and any number of minutes. square of a number equal to twice the log. of the number (Art. 56), and RULE. To obtain log. cosec, subtract log. sine from 20. From the above it appears that as the log. cos decreases with the increase of arc the log. sec increases by the same amount, and as the log. sin increases the log. cosec decreases by the same amount. Again, by Art. 37, we have tan X cot R2 applying logarithms to this equation, since the log. of a product = the sum of the logs. of the factors, and the log. of a power = log. of the number raised to the power multiplied by the index of the power, we have log. tan log. cot = 2 log R = 20 log. R being 10. Therefore having two arcs a and b, since log. tan + log. cot in both is 20 we have log. tan a+ log. cot a = log. tan blog. cot b, or transposing, that is, the difference of the logarithmic tangents of two arcs is equal to the difference of their logarithmic cotangents. EXAMPLES. 1. Required the logarithmic sine of 40° 26' 28". I find the log. sine of 40° 26' to be 9.81195; in the column M, at the left, I find the given number of seconds 28, and on the same horizontal line in the column of diff. I find 7, which added to the log. before found The logarithmic tangent of any given number of degrees, minutes, and seconds, is found in a similar manner from the column entitled tangent. * A similar arrangement for finding the difference corresponding to any number of seconds will be found in the table of natural sines and cosines, for the former on the left and for the latter on the right of the page. 2. Required the logarithmic cosine of 8° 40′ 40′′. I find the cosine of 8° 40′ to be 9.99501; the tabular difference in the adjoining column against the seconds 40" is 1; subtracting* this result from 9.99501, the remainder is 9.99500, the logarithmic cosine sought. The difference for the seconds may be calculated as follows. 3. Required the log. sine of 32° 10′ 23′′. 61. In the tables of Callet are found the logarithmic sines, tangents, cosines, and cotangents for every 10" in the quadrant; and the columns of differences contain the differences of the consecutive logarithms, or the increase of the logarithm for 10" at that part of the quadrant. To take out therefore a logarithmic sine, &c., from these tables, take out for the degrees, minutes, and tens of seconds, and take out also the number from the column of diff.; cut off one figure on the right of the latter, which is equivalent to dividing by 10, and multiply it by the number of seconds by which the given arc differs from an exact number of tens of seconds. The product will be the number by which to increase or diminish the logarithm already taken out, according as the trigonometrical line to which it corresponds is an increasing or decreasing function of the arc. EXAMPLES OF THE APPLICATION OF THE TABLES OF CALLET. 1. To find the log. tan of 49° 12′ 25′′•8 Of 49° 12' 20" log. tan is 10.0639854+ Tab. diff. 425 × 5.8 246.5 10.0640100 log. required. * It will be recollected that as the arc increases in the first quadrant the cosine diminishes. + The characteristic in the column of tangents and cotangents when 10, is printed 0. and when 11, is printed 1., in the tables of Callet. 2. To find the log. cot of 101° 25′ 43′′ = Of 78° 34' 10" log. cot is Diff. for 10" or 1084X7 log. cot of 78° 34' 17" 758.8 9.3056846 log. required. In the tables of Callet are to be found the logarithms of trigonometrical lines of arcs given in grades, &c., of the centesimal division of the circle. The following is an example of the use of the table. To find the log. sine of 928 75 84`` of 928 75' the log. sin is 9.9971776 The first 4 figures of the log. 9.997 are taken from the top or bottom of the column, the former when the logarithm is above a black horizontal line drawn across the column, the latter when below. As the diff. 78 is the diff. of two consecutive logarithms corresponding to 100 two figures must be pointed off to the right after multiplying by the 84." PROBLEM. 62. To find the degrees, minutes, and seconds answering to any given logarithmic sine, cosine, tangent, or cotangent. The method is, of course, exactly the reverse of that just given. Look for the given logarithm in the proper column, which you will know from its title, either at the top or bottom, and if you find it exactly, the degrees will be found at the top of the page, and the minutes on the same horizontal line with your logarithm, in the first column at the left, if the title of the column be at top, but the degrees will be found at the bottom of the page, and the minutes in the column at the right, if the title of the column which contains your logarithm be at the bottom. If the given logarithm cannot be found, take the next less logarithm contained in the tables, subtract it from the given, and seek the remainder in the column marked diff.; the number on the same horizontal line in the column м is seconds, which add to the degrees and minutes belonging to the logarithm found in the tables, if your given logarithm be that of a sine or tangent, but which subtract from the degrees and minutes, if a cosine or cotangent. Or more accurately, to find the seconds multiply the remainder above mentioned, by 60, and divide the product by the difference between two consecutive logarithms in the table.* EXAMPLES. 1. Required the number of degrees, minutes, and seconds, of which the logarithmic sine is 9.88005. I find the next less logarithm in the column marked sine at bottom, to be 9.87996, which subtracted from the given logarithm, leaves 9; this found in the column diff. adjoining, against it in the column м is 50, which is seconds. Taking the degrees from the bottom of the page, and the minutes from the column at the right, and in the same horizontal line with the logarithm 9.87996, I have 49° 20' 50" for the number required. Or more accurately, since the diff. 9 corresponds to any number of seconds from 48" to 52" calculate as follows: 2. Required the number of degrees, minutes, and seconds, of which the log. cotangent is 10.00869. I find the next less logarithm in the table to be 10.00859, that of 440 26', which subtracted from the given logarithm, leaves 10, corresponding to which in the column м is either 23" or 24"; more accurately 10 25)600(24" 50 100 100 and the required number is 44° 26'-24′′ or 44° 25′ 36′′. * This last rule is on the principle that the difference of the logarithmic functions is proportional to the difference of their arguments, the difference of the arguments in this case being 60". |