help of natural sines and tangents, be solved exactly in the same way, and with the same facility as he would solve a simple question in the Rule of Three. Natural sines are merely decimals, bearing the same proportion to unity, or 1, that the corresponding number of degrees and minutes bears to radius, or 90°. Natural tangents bear the same proportion to unity or 1, that the corresponding number of degrees and minutes bears to 45°, because it is a well known principle, that the sine of 90° and the tangent of 45° are each equal to radius. That is, 1 is assumed as the natural sine of 90° in the table of natural sines, and as the tangent of 45° in the table of tangents, and every other number in each of these tables is calculated accordingly. GENERAL RULE. 1. State the question in every case, as already taught: 2. Multiply the second and third terms together, and divide their product by the first. The manner of taking natural sines and tangents from the tables, is the same as for logarithmic sines and tangents; only that there is in the tables, no column of differences as in the latter, for the more readily finding the odd seconds, when required. But these may be found by making a proportion for the aliquot parts. There are some problems to which natural tangents afford a much more simple and ready solution, than any process by logarithms. The following one, in heights and distances, will illustrate this. EXAMPLE. The altitude of an inaccessible object taken at an unknown distance from its base, is 55° 54'; and when taken again at the distance of 93 feet from the place of the first observation in a direct line with it, the altitude is 33° 20′: Required the height of the object. RULE. Divide the difference of the natural co-tangents of the angles of elevation, by the distance between the stations. Co-tangent of 33° 20' is 1.52043 Thus of 55° 54' is .67705 NOTE. This is the shortest solution possible, and perfectly easy. Again: Given the latitude and departure, in transverse sailing or surveying, to find the course. RULE. Divide the departure by the latitude, the quotient will be the natural tangent of the course: or divide the latitude by the departure, and the quotient will be the co-tangent of the course. Universally, If in any right angled triangle, the perpendicular be divided by the base, the quotient will be the tangent of the angle at the base; and if the base be divided by the perpendicular, the quotient will be the tangent of the angle at the vertex of the perpendicular. OF THE TRAVERSE TABLE, OR TABLE 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, PROBLEM XII.-To find the latitude and departure, or northing, fe. for any course and distance. If the course be less than 45°, look for it at the top, but if more than 45°, 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 and deci mals, 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. NOTE. When the minutes are not 15, 30, or 45, the northings, &c. may be had by proportion, or they may be calculated by natural sines, or by trigonometry. PROBLEM XIII.-To calculate the Northing or Southing, &c. 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 atd departure for 6 chains and 22 links on à course N. 380 27', W. Nat sine of 38° 27', 0.62183 Nat co-sine, 0.78315 6.22 6.22 0.00005805 0.00006933 0.00176577 0.00210882 6.1126050 1.7581226 3.5362739 5.3144251 365th root of $1.05, or amount of $1. for 1 day, 1.00013368072 365th root of $1.06, or amount of $1. for 1 day, 1.00015965359 12th root of $1.05, or amount of $1. for 1 mo. 1.00407412 12th root of $1.06, or amount of $1. for 1 mo. 1.0048675505 360 degrees expressed in seconds, Arc, equal to radius, in degrees, in minutes, Length of an arc of 1"-sine of 1′′ 1296000 57.295780 3437.74677 206264.8 0.000004848 -6.6855749 0.000009696 -6.9866049 of 2" sine of 2" Radius of Earth's orbit, in miles, Sun's horizontal parallax, 95273869 7.9789738 A TABLE OF LOGARITHMS OF NUMBERS FROM 1 TO 10,000. N. Log. 10 N. 0.000000 26 1.414973 51 1.707570 0.301030 27 1.431364 0.477121 28 1.447158 0.602060 29 1.462398 54 0.698970 30 1.477121 55 0.778151 31 1.491362 56 0.845098 32 1.505150 57 8 0.903090 33 1.518514 58 9 0.954243 34 1.531479 59 1.000000 35 1.544068 60 11 1.041393 36 1.556303 61 12 1.079181 37 1.568202 62 13 1.113943 38 1.579784 63 14 1.146128 39 1.591065 64 15 1.176091 40 1.602060 65 16 1.204120 41 1.612784 66 17 1.230449 42 1.623249 67 Log. N. Log. 12882818 76 1.880814 52 1.716003 Log. 1.944483 1.949390. 1.954243 68 1.832509 93 1.968483 1.838849 94 1.973128 1.845098 95 1.977724 1.662758 1.322219 46 1.851258 96 1.982271 22 1.342423 47 1.672098 72 1.857333 97 1.986772 23 1.361728 1.681241 73 1.863323 24 1.380211 49 1.690196 74 1.869232 25 1.397940 50 1.698970 75 1.875061 98 1.991226 99 1.995635 100 2.000000 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 0'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 thence the annexed first two figures of the Logarithm in the second column stand in the next lower line. |