negative characteristics. In order to avoid the necessity of entering negative numbers, 10 is added to every logarithm before it is registered in the tables of logarithmic sines. Thus, on referring to the Table of natural sines (Table XXVI, Norie), we find that sine 160275637. If we calculate the logarithm of 0 275637, we find that its value is '440338; if to this 10 is added, we find that Log. sine 16° 9'440338. In trigonometrical operations this is convenient, but principally because the extraction of roots very seldom occurs. The same thing is done, for the sake of uniformity, with logarithmic tangents, though only those of angles under 45° would, as just stated, have negative indices. may It be observed here that the uniform addition of 10 to the index gives the logarithm of 10000 million times the natural number. Thus, 9'599327 is the log. of 3979486000, and this latter number is the natural sine corresponding to a radius of 10000 millions, instead of a radius of unity. The table of logarithmic sines, cosines, tangents, cotangents, secants, and cosecants, contain all ares from 1' of a degree through all magnitudes up to a quadrant or 90°, the log. of radius as just stated being 10. At the top of the page is placed the number of degrees, and in the left hand column each minute of the degree, opposite to which are arranged the numerical values of the log. sine, cosine, &c., of the corresponding angle in those columns, at the top of which those terms are placed. The headings of the columns run along the top, thus, as far as 44°. The degrees from 45° to 90° are placed at the bottom of the page, and the minutes of the degree arranged in a right hand column, so that the angles read off on the right hand side are complementry to those read off at the points exactly opposite on the left hand side, the values of the sines, cosines, tangents, &c., being found in the columns at the bottom of which those terms are found. This arrangement is rendered practicable by the circumstance of the sines, tangents, &c., of angles being respectively equal to the cosines, cotangents, &c., of the complements of the same angle. Besides the columns headed "sine, tangent," &c., are three smaller columns headed "Diff." They contain the difference between the values of the sines, cosines, &c., of consecutive logarithms corresponding to a change of 100" in the arc; and it should be kept in mind that the same difference is common to the sine and cosecant, to the tangent and cotangent, and to the secant and cosine. They are inserted for the convenience of finding the values of the sines and cosines, &c., of angles which are expressed in degrees, minutes, and seconds.* In the use of these Tables, as in that of the natural sines, two questions present themselves:-First, having given the angle in degrees, minutes, and seconds, required the log. sine, log. cosine, &c. Second, having given the log. sine, log. cosine, &c., required the value of the angle in degrees, minutes, and seconds. RULE XXVI. If the angle whose logarithmic sine, tangent, &c., it is required to find, be given in degrees and minutes, look for the degrees, if the angle be less than 45° at the top of the page, and for the minutes in the left hand column: if the angle be greater than 45°, look for the degrees at the bottom of the page, and for the minutes in the right hand column: the logarithm of the proposed function of the angle will be found opposite the numbers in its proper column. If the value of the angle be given in degrees, minutes, and seconds, we proceed by RULE XXVII. 1°. Find from the Table the sine, cosine, &c., which corresponds to the degrees and minutes. 2°. Multiply the tabular difference by the seconds, and divide by 100. When an angle of degrees, minutes, and seconds, and of less magnitude than 3°, occurs in calculation, neither the logarithmic sine nor the logarithmic tangent will be found very accurately from the ordinary Tables. In some books, as Hutton's "Mathematical Tables," a special Table is given, containing the logarithmic sines and tangents to every second in the first two degrees of the quadrant. By that Table we should find the correct log. tang. of 1° 25' 45, to be 2.3970503, whereas, by using the tab. diff. for 1° 25' and 1° 26', in the ordinary Table, we should get the less accurate result, 2.3970448, because for such small angles, the successive tabular differences for one minute, show too rapidly a wide departure from equality. Norie gives the log. sin, and log. tang. to every ten seconds of the first two degrees of the quadrant, and Raper gives the log. sines to every second up to 1' 30', and to every ten seconds up to 4° 30'. When an angle of degrees, minutes, and seconds, and within less than 3° of 90' occurs in calculation, we cannot, for the reason just stated, obtain very accurately from the ordinary Tables either the logarithmic or the natural tangent. Thus the true log. tang. of 88 4' 15 is 16029497; but by the ordinary Tables we should get for the last three figures 552. 3. If the required quantity be a sine, tangent, or secant, add the result to the last figures obtained in 1°; if it be a cosine, cotangent, or cosecant, subtract. The result will be the required sine, cosine, &c. When the angle is greater than 90°, subtract it from 180°, and look for the remainder, which is called its supplement, in the Tables. Thus, to find the log. sine of 110° 24', subtract it from 180', and look for the log. sine of the remainder (namely 69° 36'), which is 9971870; or log. sine 110° 24' =9'971870. 3. Find the log. tangent of 128° 55′ 47′′. Supplement of the given angle = 51° 4′ 13′′. Tab. diff. 13" gives + 56 0.092720 431 X 13 1293 431 56,03 But a readier way will, in general, be to diminish the given angle by 90° and to look out the remainder according to the following rule: RULE XXVIII.* If A denote any angle less than 90°, then For sine......(90 + A) take out......cosine A ,, tangent....(90 + A) secant (90 + A) cosine....(90 + A) cosecant ..(90 + A) ,, cotangent.. (90 + A) · cotangent A cosecant A sine A secant A tangent A Thus, to find the log, cosine of 110°, or log. cosine (90+20), take out the log. sine 20°, which is 9'534052. To find the log. secant of 160° 12', take out the cosecant 70° 12' which is 10:026465. Required the log. sine, tangent, secant, cosine, cotangent, and cosecant corresponding to the following are: If the value of the log. sine, log. cosine, &c., be given, and it is required to find the angle, we use 1o. RULE XXIX. Find in the Tables (XXV, Norie) the next lower† log. sine, log. cosine, &c., and note the corresponding degrees and minutes. 2°. Subtract this from the given log. sine, log. cosine, &c., multiply the difference by 100, i.e., annex two cyphers, divide by the tabular difference, and consider the result as seconds. 3°. If the given value be that of a log. sine, log. tangent, or log. secant, add these seconds to the degrees and minutes found in 1°.; if it be that of a log. cosine, log. tangent, or log. cosecant, subtract. The result will be the required angle. *This rule may easily be remembered by observing that to the sine, tangent, and secant, co is prefixed, while from the cosine, cosecant, and cotangent the co is dropped, and in each case the excess above 90° of the angle is used. + If the given log. be a cosine, cosecant, or cotangent, we may seek out the next greater to the given log.: then proceed by 2° to find the seconds, which add to the degrees and minutes as found by 1o. EXAMPLES. I. 2. Given log. sine 9'422195 (or 1'422195): find the angle. Given log. sine 9'422195 Tab. log. sine next less 9:421857 = log. sine 15° 19' Tab. diff. for 100′′ = 768)·33800(44′′ additional seconds 3072 3080 Therefore 9'422195 log. sine of 15° 19′ 44′′. Given log. cosine = 9'873242 (or 7·873242): find the angle. Tab. log. cosine next less 9.873223 = log. cosine 41° 41′ Required the Angles (to the nearest second), the Log. Sine of which |