a schol. 1. pl. trig. If the log. of a mixed number is wanted, find the log. of the whole number, and the fraction, as above, whose log. add to the log. of the whole number, or reduce the mixed number to an improper fraction, and find its log. as above. If the log. of a whole number, and decimal fraction is wanted, find the log. as if all were whole numbers; but prefix the exponents only for the whole numbers. If, at any time, the hyperbolic, or Neper's log. is wanted, the modulus betwixt which and Brigg's log is the decimal .434294481,03, &c. Suppose the hyperbolic log. 10 is wanted, divide Brigg's log. of 10=1,000ooco by the above modulus, gives 2,302585092994; the fame of any other. If the hyperbolic log. is given, being multiplied by the modulus, gives Brigg's log. OF SINES, TANGENTS, &c. THE radius of the circle being supposed divided into any rnumber of parts, the fines of such a number of these parts as the arches, of which they are the fines, are of a quadrant ; and, as the tangent of 45o is equal radius, the parts of the tangent above 45° will be proportionally greater than radius: The same of secants, as they are always greater than radius; but the fines being given, the tangents and fecants can be found from the proportion given page 150; the versed sine may be found thus, because the cofine of any arch + the versed fine, is equal to rad. Therefore, from rad. fubtract the cofine, gives the versed fine required. The above are natural fines, tangents, &c. The log. fines, tangents, &c. are only the log. of the natural sines, tangents, &c. fo that from the log. tables of numbers, find the log. answering the number of the log. fine, tangent, secants, verfed fines, gives the log. fine, tangent, &c. of the arch required. Ex. if the log. fine of 50° is required, its natural fine is .7660,444, of fuch parts as the rad. is 1.000,000, the log. of which fine is 9.8842540; and as many places as the natural fine wants of 1. so many units will the log. fine want of index 9; and as many places as the natural tangent or secant exceeds units, so many units will the log, tangent, or secant, exceed index 10; but the log. fine, tangent, &c. may be found without the log. tables, by an infinite series, the rad. of the log. being 10.0000000; the fame of any other fine. If the tangent of any arch, as of 50° is wanted, then, because the tangent of any arch is a fourth proportional to the cof. fine, and rad. *, therefore adding the log fine to rad. and, fubtracting its cosine, gives the tangent. of that arch. ۱ Ex. Log S. of 50° + rad. 19.8842540-log. of 40o =9.8080675 = 10.0761865 = tang. 50°; and, because the secant is a third proportional to the cofine and rad. therefore, from twice rad. fubtract the cosine of 50o, gives 10.1919325, the secant of 50°. If the versed sine is wanted, suppose of 50°, fubtract its cofine from rad. gives 3572.124, its natural fine; its log. fine, found, as before directed, is 9.5529265 : The fame thing of any other. But, because the versed sine of an arch above 90° is frequently wanted, the excess of the arch above rad. + rad. = the versed fine above 90°, Suppose the versed sine of 130° is wanted, its excess above rad. is 40°, the natural fine of which is 6427.876 + R. =16.27.876, its log. is 10.2155814; or, without the natural fines, thus, the log. of 2 = 30103000 + twice the log. fine of half the arch, = 20.2155814 - rad. = 10.2155814 = log. versed sine of 130°. In the table of fines, tangents, &c. begin, as usual, in all tables of the like nature with one minute, increasing to 45°, the degrees below 45 are placed on the left side of the page; the minutes answering to them increase downwards; those above 45° are placed on the right fide of the page; the minutes answering to them increase from the bottom of the page upwards, as usual. To find the nat. or log. fine, tangent, &c. of fecond and third Minutes, for which a Table is calculated, for every second ; and each 6 to 5 places, at the end of Table of fines, tangents, &c. TO find which, find the degrees and minutes in the table of fines, tangents, &c. and take the difference betwixt that found and next greater; then, as is to that d fference, so is second and third minutes, required in the denon ination of a minute, which is done in decimals, in the table, at the end of the tables of fines, &c. to a fourth proportional; which add to the fine or tangent found in the table of fines, &c. Ex. If the nat. fine of 58°, 19', 22", 36", is wanted, find the fine of 580.19', the difference betwixt which and 58°.20', the next greater is 1,528; which, X the decimal of 22", 36", as found in the table, gives 576.056+8509, 639 = 8510,215, the decimal .056 being rejected. If the log. fine of 58° 19', 22", 36", is wanted, find log. fine 58° 19', the difference betwixt which and 58° 20' is 779 × 3.7, found, as before, is 203.683 + log. fine of 58°19′ = 9.9299112 = 9.9299405, the decimal 683 being rejected, as before: The fame of any other fine, tangent, secant, whether above or below 45°, or of any versed fine, whether above or below 90°. If If an arch is given, to find the degrees, minutes, seconds, and thirds answering to that arch. Find in the table of fines, &c. the nearest, and take the difference betwixt that found and arch given; divide that difference by the difference betwixt that found and next greater, gives a quotient; which, in the table at the end of the fines, &c. shows the second and third minutes answering to it. Ex. Let the arch given be a fine of 9.9876543, the nearest less is 9.9876488, the fine of 76°.24', the difference is 55; which divide by the difference betwixt the sine of 76°, 24, and 76° 25' = 306, gives 17973, which look for in the table of feconds and thirds, gives 10", 45" Observe the same of any other arch, whether tangent, secant, or versed fine. For NUMBERS increasing in their Natural Order, from an Unite to 10,000, With a Table of Artificial SINES, TANGENTS, and SECANTS, the Radius 10,0000000; and to every Degree and Minute of the Quadrant, - 100 0000000 04441 08677 13009 17337 21661 25980 | 30295 34605 38912 ΙΟΙ 0043214 47512 51805 56094 60380 64660 68937 73210 77478 81742 102 0086002 90257 9450998756 03000 07239 11474 15704 19931 24154 103 0128372 32587 36797 41003 45205 49403 53598 57788 61974 66155 104 0170333 74507 78677 82843 87005 91163 9531799467 03613 07755 105 0211893 16027 20157 24284 28406 32525 36639 40750 44857 48960 106 0253059 57154 61245 65333 69416 73496 77572 81644 85713 89777 107 0293838 97895 01948 05997 10043 14085 18123 22157 26188 30214 108 0334238 38257 42273 46285 50293 54297 58298 109 0374265 78248 82226 86202 90173 94141 98106 110 0413927 17873 21816 25755 29691 33623 37551 62295 66289 70279 02066 06023 09977 41476 45398 49315 III 0453230 57141 61048 64952 68852 72749 76642 80532 84418 88301 112 0492180 96056 99929 03798 07663 11525 15384 19239 23091 26939 113 0530784 34626 38464 42299 46131 49959 53783 57605 61423 65237 114 0569049 72856 76661 80462 84260 88055 91846 95634 99419 03200 115 0606978 10753 14525 18293 22058 25820 29578 33334 37086 40834 1160644580 48322 52061 55797 59530 63259 66986 70709 74428 78145 1170681859 85569 89276 92980 96681 00379 04073 07765 11453 15138 118 0718820 22499 26175 29847 33517 37184 40847 44507 48164 51819 66404 70043 73679 77312 80942 84568 88192 119 0755470 59118 62763 120 0791812 95430 69045 0265606265 09870 13473 17073 20669 24263 121 0827854 31441 35026 38608 42187 45763 49336 52906 56473 60037 122086359867157 70712 74265 77814 81361 84905 88446 91984 95519 123 0899051 02581 06107 09631 13152 16670 20185 23697 27206 30713 124 0934217 37718 41216 44711 48204 51694 55180 58665 62146 65624 125 0969100 72573 76043 79511 82975 86437 89896 93353 96806 00257 126 1003705 07151 10594 14034 17471 20905 24337 27766 31193 34616 127 1038037 41456 44871 48284 51694 55102 58507 61909 65309 68705 128 1072100 75491 78880 82267 85650 89031 92410 95785 99159 02529, 129 1105897 09262 12625 15985 19343 22698 26050 29400 32747 36092 130 1139434 42773 46110 49444 52776 56105 59432 62756 66077 69396 131 1172713 76027 79338 82647 85954 89258 92559 95858 99154 02448 132 1205739 09028 12315 15598 18880 22159 25435 28709 31981 35250 133 1238516 1238516 41781 45042 48301 51558 54813 58065 61314 64561 67806, 134 1271048 74288 77525 80760 83993 87223 90451 93676 96899 00119 135 1303338 06553 09767 12978 16187 19393 22597 25798 28998 32195 136 1335389 38581 41771 44959 48144 51327 54507 57685 60861 64034 137 1367206 70375 73541 76705 79867 83027 86184 89339 92492 95643 138 1398791 01937 05080 08222 11361 14498 17632 20765 23895 27022 139 1430148 33271 36392 39511 42628 45742 48854 51964 5507258177 140 1461280 64381 67480 70577 73671 76763 79853 82941 86027 | 89110 141 1492191 95270 98347 01422 04494 07564 10633 13699 16762 | 19824 142 1522883 25941 28996 32049 35100 38149 41195 44240 47282 50322 143 1553360 56396 59430 62462 65492 68519 71544 74568 77589 80608 144 1583625 86640 89653 92663 95672 98678 01683 04685 07686 10684 145 1613680 16674 19666 22656 25644 28630 31614 34596 37575 40553 146 1643529 46502 49474 52443 55411 58376 61340 64301 67261 70218 147 1673173 76127 79078 82027 84975 87920 90864 93805 96744 99682 148 1702617 05551 08482 11412 14339 17265 20188 23110 26029 28947 149 1731863 34776 37688 40598 43506 46412 49316 52218 55118 58016 150 1760913 63807 66699 69590 72478 75365 78250 81133 84013 86892 151 1789769 92645 95518 98389 01259 04126 06992 09856 12718 15578 152 1818436 21292 24147 26999 29850 32698 35545 38390 41234 44075 153 1846914 49752 52588 55422 582 52588 55422 58254 61084 63912 66739 69563 72386 154 1875207 78026 80844 83659 86473 89285 92095 94903 97710 00514 155 1903317 06118 08917 11715 14510 17304 20096 22886 25675 28461 156 1931246 34029 36810 39590 42367 45143 47918 50690 53461 56229 157 1958997 61762 64525 67287 70047 72806 75562 78317 81070 83821 158 1986571 89319 92065 94809 97552 00293 03032 05769 08505 11239 |