The logarithmic tangent of an arc less than two degrees is found in a similar manner. Required the logarithmic tangent of 0° 27′ 36′′. Tabular number from page 114, The logarithm of 1656′′ is 4.685584. 3.219060. Logarithmic tangent of 0° 27' 36" is 7.904644. The column headed log. cot. A+log. A" is found by adding the logarithmic cotangent to the logarithm of the arc expressed in seconds. Hence, to find the logarithmic cotangent of an arc less than two degrees, we must subtract from the tabular number the logarithm of the arc in seconds. Required the logarithmic cotangent of 0° 27′ 36′′. Tabular number from page 114, The logarithm of 1656" is 15.314416. 3.219060. Logarithmic cotangent of 0° 27′ 36′′ is 12.095356. The same method will, of course, furnish cosines and cotangents of arcs near 90°. The secants and cosecants are omitted in this table, since they are easily derived from the cosines and sines. The logarithmic secant is found by subtracting the logarithmic cosine from 20; and the logarithmic cosecant is found by subtracting the logarithmic sine from 20. Thus we have found the logarithmic sine of 24° 27' 34" to be 9.617051. Hence the logarithmic cosecant of 24° 27′ 34′′ is 10.382949. The logarithmic cosine of 54° 12′ 40′′ is 9.767008. Hence the logarithmic secant of 54° 12' 40" is 10.232992. To find the Arc corresponding to a given Logarithmic Sine or Tangent If the given number is found exactly in the table, the corresponding degrees and seconds will be found at the top of the page, and the minutes on the left. But when the given number is not found exactly in the table, look for the sine or tangent which is next less than the proposed one, and take out the corresponding degrees, minutes, and seconds. Find, also, the difference between this tabular number and the number proposed, and, corresponding to this difference, at the bottom of the page will be found a certain number of seconds, which is to be added o the arc before found. Required the arc corresponding to the logarithmic sine 9.750000. The next less sine in the table is The arc corresponding to which is 34° 13' 0". 9.749987. The difference between its sine and the one proposed is 13, corresponding to which, at the bottom of the page, we find 4" nearly. Hence the required arc is 34° 13' 4". In the same manner we find the arc corresponding to logarithmic tan gent 10.250000, to be 60° 38′ 57′′. When the arc falls within the first two degrees of the quadrant, the odd seconds may be found by dividing the difference between the tabular number and the one proposed, by the proportional part for 1". We thus find the arc corresponding to logarithmic sine 8.400000, to be 1o 26′ 22′′ nearly. We may employ the same method for the last two degrees of the quadrant when a tangent is given; but near the limits of the quadrant it is better to employ the auxiliary table on page 114. If we subtract the corresponding tabular number on page 114 from the given logarithmic sine, the remainder will be the logarithm of the arc expressed in seconds. Required the arc corresponding to logarithmic sine 7.000000. We see, from page 22, that the arc must be nearly 3'; the correspond ing tabular number on page 114 is 4.685575. The difference is 2.314425; which is the logarithm of 206."265. Hence the required arc is 3' 26."265. In the same manner we find the arc corresponding to logarithmic tangent 8.184608, to be 0° 52′ 35′′. TABLE FOR THE LENGTHS OF CIRCULAR ARCs, p. 135. This table contains the lengths of every single degree up to 60, and at intervals of ten degrees up to 180; also for every minute and second up to 20. The lengths are all expressed in decimal parts of radius. Required the length of an arc of 57° 17' 44."8. Take out from their respective columns the lengths answering to each of these numbers singly, and add them all together thus: 'hat is, the length of an arc of 57° 17' 44."8 is equal to the radius of he circle. TRAVERSE TABLE, pp. 136–141. This table shows the difference of latitude and the departure to four ecimal places for distances from 1 to 10, and for bearings from 0° to )°, at intervals of 15'. If the bearing is less than 45°, the angle will › found on the left margin of one of the pages of the table, and the disnce at the top or bottom of the page; the difference of latitude wil be found in the column headed lat. at the top of the page, and the de parture in the column headed dep. If the bearing is more than 45°, the angle will be found on the right margin, and the difference of latitude will be found in the column marked lat. at the bottom of the page, and the departure in the other column. The latitudes and departures for different distances with the same bearing, are proportional to the distances. Therefore the distances may be reckoned as tens, hundreds, or thousands, if the place of the decimal point in each departure and difference of latitude be changed accordingly. Required the latitude and departure for the distance 32.25, and the bearing 10° 30'. On page 136, opposite to 10° 30', we find the following latitudes and departures, proper attention being paid to the position of the decimal points. TABLE OF MERIDIONAL PARTS, pp. 142–148. This table gives the length of the enlarged meridian on Mercator's Chart to every minute of latitude expressed in geographical miles and tenths of a mile. The degrees of latitude are arranged in order at the top of the page, and the minutes en both the right and left margins. Under the degrees and opposite to the minutes are placed the meridional parts corresponding to any latitude less than 80°. Thus the meridional parts for latitude 12° 23' are 748.9; TABLE OF CORRECTIONS TO MIDDLE LATITUDE, p. 149. This table is used in Navigation for correcting the middle latitude The given middle latitude is to be found either in the first or last verti cal column, opposite to which, and under the given difference of latitude is inserted the proper correction in minutes, to be added to the middle latitude to obtain the latitude in which the meridian distance is accu rately equal to the departure. Thus, if the middle latitude is 41°, and the difference of latitude 14°, the correction will be found to be 251 which, added to the middle latitude, gives the corrected middle latitud 41° 25'. 17 18 19 20 16 1.204120 41 1.612784 66 1.819544 1.826075 69 1.832509 93 1.968483 1.973128 1.653213 70 1.845098 95. 1.977724 N.B. In the following table, the two leading figures in the first column of logarithms are to be prefixed to all the numbers of the same horizontal line in the next nine columns; but when a point (.) occurs, its place is to be supplied by a cipher, and the two leading figures are to be taken from the next lower line. A 5 6 7 89 D. 2166 2598 3029 3461 3891 | 432 6466 6894 7321 7748 8174 | 428 .724 1147 1570 1993 2415 424 | 4940 5360 5779 6197 6616419 9116 9532 9947 .361.775 | 416 | 3252 3664 4075 4486 4896 | 412 7350 7757 8164 8571 8978 408 | 1408 1812 2216 2619 3021 404 5430 5830 6230 6629 7028 400 9414 9811.207.602.998 396 3362 3755 4148 4540 4932 393 7275 7664 8053 8442 8830 389 1153 1538 1924 2309 2694 | 386 4996 5378 5760 6142 6524 382 8805 9185 9563 9942 | .320 | 379 2582 2958 3333 3709 4083 | 376 6326 6699 7071 7443 7815 373 ..38 .407.776 1145 1514 369 | 3718 4085 4451 4816 5182 366 7368 7731 8094 8457 8819 363 .987 1347 1707 2067 2426 360 4576 4934 5291 5647 6004 357 | ៗ .625 4219 || 5 6 8 | 9 D. 87 86 76 130 174 130 173 346 389 86 129 172 344 387 428 43 86 128 171 213 256 298 212 254 297 211 253 295 338 380 126 168 210 125 167 209 208 250 252 294 251 293 334 376 291 336 378 333 374 |