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 complementary 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 column 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. 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 XVII. I°. 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. 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 120° 24', subtract it from 180°, and look for the log, sine of the remainder (namely 69° 36′), which is 9'971870; or log. sine 120° 24′ = 9'971870. 3. Find the log. tangent of 128° 55′ 47′′. Supplement of the given angle = 51° 4′ 13′′. + 56 0'092720 Tab. diff. 431 X 13 1293 431 56,03 or 56 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 XVIII.* If A denote any angle less than 90°, then For sine......(90+ A) take out......cosine 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 re quired to find the angle, we use RULE XIX. 1o. Find in the tables (XXV, Norie) the next lower† log. sine, log. cosine, &c., and note the corresponding degrees and minutes. 20. 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. * 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 1°. 3°. If the given value be that of a log. sine, log tangent, or log. secant, add these seconds to the degress 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. EXAMPLES. 1. Given log. sine = 9°422195 (or ī‘422195): find the angle. 2, 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 ī·873242): find the angle. 86 357 = *240896, a negative mantissa = 1·759104, adding 10 to the index gives 9759104 for the tabular log. tangent; and very nearly corresponding to this is found the angle 29° 52'. Required the Angles (to the nearest second), the Log. Sine of which is : 1. 9'741279 5. 8.600700 9. 9'929638 13. I'559234 2. 9'518317 6. 9'926100 IO. 9'500000 16. 9700000 17. 8.846217 18, 8*462167 9. 8.967391 8. 9'998970 IO. 9'000000 7. 8. 0'022716 9. 0'315400 IO. 0'497691 10. 0.070362 13. 0630000 5. O'121000 8. 11*467931 II. O'900000 12. 14. 0061462 15. 0109761 G 4. 0974476 7. 11000873 Find the Arc to the Log. Cotangent of 7. 8.327691 10. 0000276 I. 9'742691 4. 0060431 13. 8.460000 9. NAVIGATION. DEFINITIONS. NAVIGATION is a general term denoting that science which treats of the course. The place of a ship is determined by either of two methods, which It has been customary to employ the term NAVIGATION in a restricted Navigation and Nautical Astronomy are the two great co-ordinate divisions of the The |