Example. Having failed from the Lizard, in lat. 49° 55′ N, on a courfe 49° 59' 10" fouth-wefterly 3429 378 miles : required what longitude and latitude the fhip is found in. Log. of 3429 378 the diftance failed Log. cofine of 49° 59′ 10′′ the course Log. of 2205', or 36° 45' diff. of the latitudes. 3'5352153 9:80819:3 • 3*3434086 Now fubtracting 36° 45′ from 49° 55', the remainder 13° 10' N. is the latitude the fhip is found in. By which latitude, now known, the difference of log. tangents will be found 3372 605, and the further procefs in nothing differing from the fecond rule, whereby the difference of longitude will be found 53° 00'. Thus the dead reckoning by the log. line, and daily account of a fhip's way, are duly kept, and the trouble very little more than by plain failing. These are all the cafes that occur in practice; the reft, that are moftly fpeculative, are either easily reducible to thefe, or else not to be performed by logarithms, and therefore come not at prefent under our cognizance. But it is to be noted, that both the complements of the latitudes are to be estimated from the fame pole of the world; which may be from either; and therefore if one latitude be N, and the other s, to have their complements, you must add 90° to one of them, and fubtract the other from 90, and then the operation will be the fame as in the preceding cafes. Example. Given St. Jago, one of the Cape de Verd iflands, in the latitude of 14° 55′ N; and the ifland St. Helena, in latitude 15° 45′ s, and their difference of longitude 30° 12' E; to find the course and distance. Jago 52° 28′ . 1. tan. +Co lat. {St. Helena 37 71. 1. tan. Log. tang. of the courfe 44° 11' 53" 98 101144965 1. 1812 3.2581582 9-8790845 conft. log 10'1015104 2354 120 its co. log. 6.6281714 Log. of 2567.875 diftance of St. Helena from St. Jago Or if it be thought eafier, when one latitude is N, and the other s, you may add 90° to each of them, the fum of the log. tangents of their halves (abating twice the radius) will be the fame as the difference of the log. tangents of the former. For an example take the fame latitudes as in the preceding. {14° 56'=104° 56' its half{ Then 90°+14° 56' 104° 56′ 15 45 105 45 52 5210 The fum (abating twice the radius) equal to the former 1. tan. 101144965 1. tan. distance 10'1209155 2354°120 Alfo when both latitudes are of the fame name, that is both N or both s, you may add 90° to each of them, the difference of the log. tangents of half thefe fums, will be the fame as of the log. tangents of half the complements of those latitudes. OF THE TABLE FOR THE LENGHTS OF CIRCULAR ARCS. ΤΗ HIS is table 12, and conftitutes page 340. It contains the lengths of every fingle degree up to 180, and of every minute, fecond, and third, each up to 60. The form of it is obvious, the length of each degree, minute, fecond, or third, immediately following it on the fame line in the next column. And the two following examples will fhew the ufe of the table, OF THE TABLE FOR COMPAIRING THE COMMON AND THI HYPERBOLIC LOGARITMS. HIS is table 13, and is the upper part of page 341. It contains the hyperbolic logs. anfwering to the first 100 common logs. and is very useful for fpeedily changing the one into the other. Ex. 1. To find the hyp. log, anfwering to the common log. O'9542425. Beginning at the left hand, and dividing the given number into periods of two figures each, including the index, take out the hyp. log. to each period, omitting 2 figures at the 2d period, four at the 3d, and 6 at the 4th: then add them all together, thus: Ex. 2 To find the common log. anfwering to the hyp. log. 2.1972246. Subtract continually each next lefs tabular hyp, log. from the given num ber, and from the remainders; and the feveral common logarithms anfwering to thefe tabular hyp. logs. joined together, will be the com. log. reqd, thus: 09 hyp. log. given 21972246 2.0723226 1248980 1243396 25 9'9542425 58 2.1972246 anfw,. The remaining pages contain the fmall table of the names and degrees, &c. in the points of the compafs, which needs no iliuftration; and a copious lift of fuch errors, with their corrections, as have been discovered in the principal books of logarithms; among which are many that have been detected by myfelf, both in the Avignon edition of Gardiner, and in Gardiner's own quarto edition, which renders this lift more compleat than any former one, and it will be found very ufeful in correcting those books of tables which are already in the poffeffion of the public. As to all the editions of Sherwin's tables in octavo, the errors in them, amounting to many thousands, are far too numerous to be printed in this work. (2) Numb. 1 to 100, and LOGARITHMS N. 100 L. 00 their Log. with Indices. N. Log. N. Log. N. Log. 10.0000000 511.7075702|| 100 0000000||1501760913||200 3010300 20.3010300 521.7160033 1010043214 151 1789769 201 3031961 30.4771213 53 1.7242759 102 0086002 1521818436 202 3053514 40.6020600 54 1.7323938 103 0128372 153 1846914 203 3074960 50.6989700 55 1.7403627 1040170333 154 1875 207 204 3096302 60.7781513 56 1.7481880 105 0211893 1551903317 205 3117539 70.8450980 57 1.7558749 106 0253059 1561931246 206 3138672 80.9030900 58 1.7634280 107 0293838 157 1958997 207 3159703 90.9542425 59 1.7708520 108 0334238 158 1986571 208 3180633 60 1.7781513109 0374265 10 1.0000000 159 2013971 209 3201463| 131.1139434 631.7993405 112 0492180 162 2095 150 212 3263359 214 3304138 165 2174839 215 3324385 141.1461280 64 1.8061800 1130530784163 2121876 2133283796 151.1760913 65 1.8129134 1140569049 164 2148438 161.2041200 66 1.8195439 115 0606978 171.2304489 67 1.8260748 1160644580 1 166 2201081 216 3344538 18 1.2552725 68 1.8325089 1170681859 167 2227165 217 3364597 19 1.2787536 69 1.8388491 118 0718820 168 2253093 218 3384565 201.3010300 70 1.8450980 119 0755470 169 2278867 2193404441 211.3222193 71 1.8512583 120 0791812 1702304489 220 3424227 221.3424227 721.8573325 121 0827854 171 2329961| |221 3443923 231.3617278 731.8633229 122 0863598 172 2355284 222 3463530 241.3802112 741.8692317 123 0899051 173 2380461 223 3483049 251.3979400 75 1.8750613 124 0934217 1742405492 224 3502480 261.4149733 761.8808136 125 0969100 175 2430380 2253521825 271.4313638 77 1.8864907 126 1003705 176 2455127 226 3541084 28 1.4471580 781.8920946 127 1038037 177 2479733 29 1.4623980 79 1.8976271 128 1072100 178 2504200 30 1.4771213 80 1.9030900129 1105897 179 2528530 311.4913617 811.9084850 130 1139434 321.5051500 82 1.9138139 131 1172713 33 1.5185139 83 1.9190781 1321205739 341.5314789 84 1.9242793 133 1238516 351-5440680 851.9294189 134 1271048 36 1.5563025 861.9344985 135 1303338 37 1.5682017 87 1.9395193 136 1335389 381.5797836 881.9444827 137 1367206 39 1.5910646 891.9493900 138 1398791 401.60206c0 90 1.9542425 139 1430148 411.6127839 911.9590414 140 1461280 421.6232493 92 1.9637878 141 1492191 431.6334685 93 1.9684829 142 1522883 227 3560259 228 3579348 229 3598355 180 2552725 230 3617278 181 2576786 231 3636120 182 2600714 232 3654880 183 2624511 233 3673559 |