The secants and co-secants are not often employed in trigonometrical calculations, and are therefore omitted in the annexed table. They may easily be found, if desired, by the following rule. THE LOGARITHMIC SECANT OF AN ANGLE IS FOUND BY SUBTRACTING THE LOGARITHMIC CO-SINE OF THE SAME ANGLE FROM 20.000000 : and THE LOGARITHMIC CO-SECANT IS FOUND BY SUBTRACTING THE LOGARITHMIC SINE FROM 20.000000. Thus, find the secant of 31° 20. 20.000000 Co-sin, 31° 20'= 9.931537 Rem. = sec. 31° 20'=10.068463 Find the co-sec. of 47° 38'. 20.000000 Sin. 47° 38= 9.868555 Rem. =co-sec. 47° 38'=10.131445 When there are seconds, the column headed D is employed. The numbers in this column show how much difference must be allowed in the log, for every second. This difference is carried to two places of decimals. Hence the reason of the following rule. To find a sine or tangent to degrees, minutes, and seconds. Find for degrees and minutes as before ; then multiply the opposite number in the column D, by the seconds, cut off two figures from the right of the product, and add what is left to the log. before found. For co-sines and co-tangents, this product must be subtracted, instead of being added. TO FIND THE DEGREES, OR DEGREES AND MINUTES, ANSWERING TO ANY GIVEN LOGARITHMIC SINE, CO-SINE, TANGENT, OR CO-TANGENT. Under or over the proper name in the table, seek the given log. or the one nearest to it : the degrees will be found at the same extremity of the page as the name, and the minutes at the left or right, according as the name is at the top or bottom. To find an angle to seconds, subtract from the given log. the next less in the table, annex two ciphers to the remainder, and divide it, thus augmented, by the tabular difference. The quotient is seconds, to be added to the degrees and minutes of the tabular log. in case of sines and tangents, but to be subtracted, in case of co-sires and co-tangents. EXPLANATION OF THE TRAVERSE TABLE, OR TABLE OF DIF FERENCE OF LATITUDE AND DEPARTURE. This is calculated for degrees and quarters of degrees, and for any distance up to 100 rods, chains, &c. ; by which the northings and southings, eastings and westings made in a survey may be found. Note. Northings and southings are called difference of latitude, or simply latitude ; eastings and westings are called departure, meridian distance, or longitude. To find the latitude and deparlure, or northing, &c. for any course and distance. 450 at the bottom of the page, and look for the distance in the right or left hand column; against the distance, and directly under or over the course, stand the northing, &c. in whole numbers and decimals. If the course be less than 45°, the northing or southing will be greater than the easting or westing; but if more than 45°, the easting or westing will be the greatest. When the distance exceeds 100, take any two or more numbers, which, added together, will equal the distance, and find the latitude and departure for each of these numbers; add the several latitudes together, and the sum will be the whole latitude ; and so for the departure. And when the distance is in chains and links, or whole numbers and decimals, find the latitude, &c. for the chains or whole numbers, and then for the links or decimals, remembering to remove the decimal point in the table further to the left, according to the given decimal. 1. Required the latitude and departure for 45 rods, on a course N. 15° 15' W. Under 15° 15' and against 45 is 43.42 for the northing, and 11.84 for the westing. 2. Required the latitude and departure for 120 rods, on a course S. 58° 30' E. Take one third of 120, which is 40; against this number, over 58° 30', is 20.90 for the latitude, and 34.11 for the departure. These multiplied by 3 give 62.70 for the southing, and 102.33 for the easting. 3. Required the latitude and departure for 37.36 rods, or 37 chains and 36 links, on a course N. 26° 45' E. For 37. Lat. 33.04 Dep. 16.65 .16 Northing 33.36 Easting 16.81 Note.-When the minutes are not 15, 30, or 45, the northings, &c. must be calculated by natural sines, or by trigonometry. When the latitude and departure are themselves given, the course and distance may be found in the table, thus: look along the columns of latitude and departure, until the given numbers are found opposite each other. The course will then be directly under or over them, and the distance in the right or left hand column. If the numbers cannot be exactly found, take those which come nearest. If either or both are too great to be found in the table, divide both by any number, (10, 100, 1,000, &c. is most convenient,) and use the quotients. The course found thus will be the one required, but the distance must be multiplied by the number used before as a divisor. EXPLANATION OF THE TABLE OF NATURAL SINES. Natural Sines are decimals bearing the same proportion to unity or 1, that the sine of the corresponding number of degrees and minutes bears to radius, or sine of 90°. That is, 1 is assumed as the nat. sine of 90°, and the table calculated accordingly. TO FIND THE NATURAL SINE OF ANY NUMBER OF DEGREES AND MINUTES. If the degrees be less than 45, look for them at the top of the columns, and for the minutes at the left hand ; but if more than 45, look for them at the bottom, and for the minutes at the right hand ; under or over the degrees and against the minutes, will be the natural sine rcquired. The reverse of this will give the degrees and minutes corresponding to any natural sine. TO CALCULATE THE NORTHING OR SOUTHING, &r, FOR ANY COURSE AND DISTANCE, BY NATURAL SINES. Find the nat. sine and co-sine of the course, and into each of these multiply the distance ; the products will be the latitude and departure required. Required the latitude and departure for 6 chains and 22 links on a course N. 38° 27' W. Nat. sine of 38° 27', 0.62183 Nat. co-sine 0.78315 6.22 6.22 124366 124366 373098 156630 156630 469890 3.8677826 Anower. Northing 4.87 4.8711930 Westing 3.87 A TABLE OF LOGARITHMS OF NUMBERS FROM 1 to 10,000. N. 3 4 5 6 7 8 9 10 li 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Log. 0.000000 0.301030 0.477121 0.602060 0.698970 0.778151 0.845098 0.903090 0.954243 1.000000 1.041393 1.079181 1.113943 1.146128 1.176091 1.204120 1.230449 1.255273 1.278754 1.301030 1.322219 1.342423 1.361728 1.380211 1.397940 N. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Log 1.414973 1.431364 1.447158 1.462398 1.477121 1.491362 1.505150 1.518514 1.531479 1.544068 1.556303 1.568202 1.579784 1.591065 1.602060 1.612784 1.623249 1.633468 1.643453 1.653213 1.662758 1.672098 1.681241 1.690196 1.698970 N. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 174 75 Log. 1.707570 1.716003 1.724276 1.732394 1.740363 1.748188 1.755875 1.763428 1.770852 1.778151 1.785330 1.792392 1.799341 1.806180 1,812913 1.819544 1.826075 1.832509 1.838849 1.845098 1.851258 1.857333 1.863323 1.869232 1.875061 N. Log. 76 1.880814 177 1.886491 78 1.892095 79 1.897627 80 1.903090 81 1.908485 82 1.913814 83 1.919078 84 1.924279 85 1.929419 86 1.934498 87 1.939519 88 1.944483 89 1.949390 90 1.954243 91 1.959041 92 1.963788 93 1.968483 94 1.973128 95 1.977724 96 1.982271 97 1.986772 98 1.991226 99 1.995635 100 2.0000no 42 43 44 45 46 47 48 49 50 N.B. In the following table, in the last nine columns of each page, where the first or leading figures change from 9's to O's, points or dots are introduced instead of the O's through the rest of the line, to catch the eye, and to indicate that from thencc the annexed first two figures of the Logarithm in the second column stand in the next lower line. |