surveying, may, by the 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, I 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 the 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. ThusCo-tangent of 33° 20' is 1.52043 of 55° 54′ is .67705 Divide by the diff. = feet. .84338)93.0000(110.27 Ans. 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 PROBLEM XII.-To find the latitude and departure, or northing, &c. 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 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. Dep. 16.65 .16 16.81 Northing 33.36, Easting 16.81. 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 and departure for 6 chains and 22 links on a course N. 38° 27', W. 3.8677826 4.8711930 365th root of $1.05, or amount of $1. for 1 day, 1.00013368072 0.00005805 365th root of $1.06, or amount of $1. for 1 day, 1.00015965359 0.00006933 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, 0.00176577 0.00210882 1296000 6.1126050 57.295780 1.7581226 3437.74677 3.5362739 206264.8 5.3144251 0.000004848-6.6855749 0.000009696 -6.9866049 0.000014544 -5.1626961 0.000290888-4.4637261 0 017453293-2.2418774 0.017452406 -2.2418553 320 25051500 081760 3.2455127 5280 3.7226340 63360 4.8018152 640 2.8061800 1024005.0103000 3097600 6.4910254 27178400 7.4452679 4014489600 9.6036304 7924 3.8989445 2535680 6.4040945 13946240 7.1444572 Circumference of the Equator, in miles, Radius of Earth's orbit, in miles, Sun's horizontal parallax, A TABLE OF LOGARITHMS OF NUMBERS FROM 1 TO 10,000. N. N. 5 7 8 0.903090 33 9 10 0.954243 34 1.041393 36 12 15 16 Log. Log. N. 0.000000 26 1.414973 51 0.301030 1.431364 52 0.477121 28 1.447158 53 0.602060 29 1.462398 54 1.732394 0.698970 30 1.477121 55 1.740363 6 0.778151 31 1.491362 56 1.748188 0.845098 32 1.505150 57 1.755875 82 1.518514 58 1.763428 83 1.531479 59 1.770852 84 1.924279 1.544068 60 1.778151 85 1.929419 1.556303 61 1.785330 86 1.934498 1.079181 37 1.568202 62 1.792392 87 1.939519 13 1.113943 38 1.579784 63 1.799341 88 1.944483 14 1.146128 39 1.591065 64 1.806180 89 1.949390 1.176091 40 1.602060 65 1.812913 90 1.954243 1.204120 41 1.612784 66 1.819544 91 1.959041 1.230449 42 1.623249 67 1.826075 92 1.963788 1.633468 68 1.832509 93 1.968483 1.278754 44 1.643453 69 1.838849 94 1.973128 1.301030 45 1.653213 70 1.845098 95 1.977724 21 1.322219 46 1.662758 71 1.851258 96 22 1.342423 47 1.672098 72 1.857333 97 23 1.361728 48 1.681241 73 1.863323 98 24 1.380211 49 1.690196 74 1.869232 25 1.397940 50 1.698970 75 1.875061 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 N. 0123 4 5 6 7 | 8 | 9 | D. 100000000, 0434 0868 1301| 1734 2166; 2598 3029 3461] 3891 432 101 4321 4751 5181 5609 6038 6466 6894 7321 7748 8174 428 102 8600 9026 9451 9876 .300 .724 1147 1570 1993 2415 424 103 012837 3259 3680 4100 4521 4940 5360 5779 6197 6616 419 104 7033 7451 7868 8284 8700 9116 9532 9947.361 .775 416 105 021189 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 106 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 107 9384 9789 .195 .600 1004 1408 1812 2216 2619 3021 404 108 033424 3826.4227 4628 5029 5430 5830 6230 6629 7028 400 109 7426 7825 8223 8620 9017 9414 9811.207.602.998 396 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4932 393 111 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 112 9218 9606 9993 .380 766 1153 1538 1924 2309 2694 386 113 053078 3463 3846 4230 4613 4996 5378 5760 6142 6524 382 114 6905 7286 7666 8046 8426 8805 9185 9563 9942 .320 379 115 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 372 117 8186 8557 8928 9298 9668 .38 .407 .776 1145 1514 369 118 071882 2250 2617 2985 3352 3718 4085 4451 4816 5182 366 119 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 120 079181 9543 9904 .266 .626.987 1347 1707 2067 2426 360 121 082785 3144 3503 3861 4219 4576 4934 5291 5647 6004 357 122 6360 6716 7071 7426 7781 8136 8493 8845 9198 9552 355 123 9905 .258 .611.963 1315 1667 2018 2370 2721 3071 351 124 093422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 125 6910 7257 7604 7951 8298 8644 8990 9335 9681..26 346 126 100371 0715 1059 1403 1747 2091 2434 2777 3119 3462 343 127 3804 4146 4487 4828 5169 5510 5851 6191 6531 6871 340 128 7210 7549 7888 8227 8565 8903 9241 9579 9916 .253 338 129 110590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 113943 4277 4611 4944 5278 5611 5943 6276 6603 6940 333 131 7271 7603 7934 8265 8595 8926 9256 9586 9915 .245 330 132 120574 0903 1231 1560 1888 2216 2544 2871 3198 3525 328 133 3852 4178 4504 4830 5156 5481 5806 6131 6456 6781 325 134 7105 7429 7753 8076 8399 8722 9045 9368 9690..12 323 135 130334 0655 0977 1298 1619 1939 2260 2580 2900 3219 321 136 3539 3858 4177 4496 4814 5133 5451 5769 6086 6403 318 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 315 138 9879.194 .508 .822 1136 1450 1763 2076 2389 2702 314 139 143015 3327 3639 3951 4263 4574 4885 5196 5507 5818 311 140 146128 6433 6748 7058 7367 7676 7985 8294 8603 8911 309 141 9219 9527 9835 142 .449 .756 1063 1370 1676 1982 307 142 152288 2594 2900 3205 3510 3815 4120 4424 4728 5032 305 143 5336 5640 5943 6246 6549 6852 7154 7457 7759 8061 303 144 8362 8664 8965 9266 9567 9868.168 .469 .769 1068 301 145 161368 1667 1967 22662564 2863 3161 3460 3758 4055 299 146 4353 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 147 7317 7613 790S 8203 8497 8792 9086 9380 9674 9968 295 148 170262 0555 0848] 1141| 1434 1726 2019 2311 2603 2895 293 149 3186 3478 3769 4060| 4351 4641 4932 5222 5512 5802 291 150 176091 6381 6670 6959 7248 7536 7825 8113 8401 8689 289 151 8977 9264 9552 9839 .126 .413 .699 .985 1272 1558 287 152 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 153 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 154 7521 7803 8084 8366 8647 8928 9209 9490 9771..51 281 155 190332 0612 0892 1171 1451 1730 2010 2289 2567 2846 279 156 3125 3403 3681 3959 4237 4514 4792 5069 5346 5623 278 5899 6176 6453 6729 7005 7281 7556 7832 8107 8382 276 8657 8932 9206 9481 9755 .29 .303 .577 .850 1124 274 159 201397 1670 1943 2216 2488 2761 3033 3305 3577 3848 272 137 157 158 |