TABLE. III.

Natural Sines.

In this table the natural sines are exhibited to every degree and minute of the quadrant, and arranged so that the degrees corresponding to the sines are to be taken from the top of the page with their minutes in the left side columns, and the degrees answering to the cosines from the bottom with their minutes in the right side columns.

The natural sine or co-sine of any number of degrees, &c. more than 90, is the same as the natural sine or co-sine of its supplement, found by subtracting them from 1800; or the natural sine or co-sine of an arch greater than 90° is the natural co-sine or sine of its excess above 90°.

To find the natural Sine or Co-sine of a given Number of Degrees, Minutes, and Seconds:

Or, to find the degrees, Minutes, and Seconds, corresponding to a given natural Sine or Co-sine.

These are to be found as directed for the logarithmic sines, &c. except that the differences to 100" are to be taken from the bottom of that column containing the given degrees in the former case, or the nearest natural sine or co-sine in the latter.

EXAMPLE I.

Required the natural Sine of 32° 21' 45", or its Supplement 147° 38' 15". The natural sine of 32o 21' is

The difference at the bottom of the column containing the na tural sine of the given degrees and minutes is 409, this multiplied by 45, pointing off two figures in the product,

is

Sum is the natural sine required

EXAMPLE II.

535090

+184

535274

Required the natural Co-sine of 719 40′ 25′′, or 1080 19′ 35′′.

The natural co-sine of 71° 40' is

314545

The difference 460, multiplied by 25, pointing off two figures, is -115

Remainder is the natural co-sine required

EXAMPLE III.

314430

Required the Degrees, Minutes, and Seconds, answering to the natural Sine 495994.

The natural sine next less to that given is 495964, answering to 290 44'; the difference between this natural sine and the given one is 30, to which two cyphers being added, and that divided by 422, the difference at the bottom of the column, gives the quotient 7" to be annexed to 29° 44'. Hence 29' 44' 7", or its supplement 150° 15' 53", are the degrees, &c. required.

EXAMPLE IV.

Required the degrees, Minutes, and Seconds, answering to the natu ral Co-sine 368805.

The natural Co-sine next greater to that given is 368936, to which answers 68o 21'; the difference between this natural sine and the given one is 131, to which two cyphers being added, and that divided by 451, the difference found at the bottom of the column, gives the quotient 29". Hence 68° 21′ 29′′, or its supplement, 1110 38′ 31′′ are the degrees, &c. required.

To find the natural versed Sine of a given Number of Degrees, Minutes, and Seconds.

If the given arch be less than 90°, find its natural co-sine, which subtract from 1000000, and the remainder will be the natural versed sine required. But if the given arch exceed 90°, find the natural cosine of its supplement, which add to 1000000, and the sum will be the natural versed sine required.

EXAMPLE 1.

Required the natural versed Sine of 20° 39′.

The natural co-sine of 20° 39' is 935752, which subtracted from 1000000, leaves 064248, the natural versed sine of 20° 39'.

EXAMPLE II.

Required the natural versed Sine of 146° 38′ 40.′′

The natural co-sine of 33° 21′ 20′′ (the supplement of 146° 38′ 40′′) is 835274, which added to 1000000, the suni 1885274 is the natural versed sine required.

To find the Degrees, &c. corresponding to a given natural versed Sine.

Take the difference between the given natural versed sine and 1000000, and the remainder will be a natural co-sine; the degrees, &c. corresponding to which, will be those required, if the given natu ral versed sine be less than 1000000, but if otherwise, it will be their supplement.

EXAMPLE I.

Required the Degrees, &c. answering to the natural versed sine 098965. The above subtracted from 1000000, leaves 901035, which taken as a natural co-sine, corresponds to 25° 42′ 20′′.

EXAMPLE II.

Required the Degrees, &c. answering to the natural versed Sine i 160172. Here 1000000 subtracted from the above, leaves 160172, which taken out as a natural co-sine, corresponds to 80° 46′ 59′′; therefore its supplement 999 13' 1" are the degrees, &c. required.

M

0°

1°

2o

50

6o

9° M

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M

M

16

190

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Diff. to

100" 477 475 473

| 47 | 469 467 465

462 460 457

M 20°

26°

7°

21° 22° 238 -4° 8° 29° M 0342020 358358 374607|390731|406737|422618 438371 453990 469472 484810|60| 1342293 358640 374876 390999407002422882 438633 454250 469728 485064 59 2342567 358911375146 391267 407268 423145 438894 454509 469985 48531858 3 342840 359183 3754 6 391534 407534423409 439155 454768 470242 485573 57 4 343113 359454 375685|391802 407799 423673 439417 455027 470499 48582756 5343387 359725 375955 392070 408065|423936 439678 455286 470755 4860855 6343660 359997 376224 392337 408330 424199 439939 455545 471012 486335 54 7243933 360268 376494 392605 408596 424463 440200 455804 471268 486590 53 344206 360540 376763 392872 408861 4247 26 440462 456063 471525 486844 52 9 344479 360811 377033 393140 409127 424990 440723 456322 471782 487098 51 10 344752 361082 377302 393407 409392 425253 440984 456580 472038 48735250 11345025 361353 377571 393675|409658 425516|441245 456839 472294 487606|49| 12345298 361625 377841 393942 409923 425779441506 457098 472551 487860 48 13:345571 361896 378110 394209 410188 426042 441767 457357 472807488114 47 14 345844 362167 378 379 394477 410454426306 442028 457615473063 488 67 46 15346117 362438 378649 394744410719 426569 442289 457874 473320 488621 45 16 346390 362709 378918 395011410984 42683-442550 458133 473576 488875 44 17 346663 362980 379187 395278 411249 427095 442810458391 473832 489129 43 18346936 363251 379456 395546411514 427358 443071 458650 474088 489382 42 19 347208 363522 379725 395813 411779 427621 443332 458908 474344 489636 41 20 34748 363793 379994 39608041 2045 427884 443593 459166 474600 489890 401 21347754 364054 380263|396347 41 23:0428147 443853 459425 474856 490143 39 22 348027 364335 380532 346614412575428410 444114 459683 475112 490397 38 23 348299 364606 380801 396881 412840 428672 444375 459942 475368 490650 37 24 348572 364877 381070 397148 413104428935 444635 460200 475624 490904 36 25348845 365148 381339 397415 413369 429198 444896 460458 475880 491157 35 26 349117 365418 381608 397682 413634 429461 445156 460716 476136 491411 34 27 349390 365689 381877 397949 413899 429723 445417 460974 476392491664 33 28 349662 365960 382146 398215 414164 429986 445677 461232476647 491917 32 29 349935 366231 382415 398482 414429 430249 445937 461491 476903 49217031 30 350207 366501 382683 398749 414693 430511 446198 461749 477159 492424 30 31 330480 366772|382952|399016414958430774 446458 462007 477414 492677|29| 32350752 367042 383221 399283 415223 431036 446718 462265 477670 492930 28 33 351025 367313 383490|399549415487 431299 446979 462523 477925 493183 27 34 351297 367584 383758 399816 415752431561 447239 462780 478181 493436 26. 39351569 367854 384027 400082416016431823 447499 463038478436 493689|25| 36 351842 368125 384295 400349 416281 432086 447759 463296 478692 493942 24 37 352114 368395 384564 400616416545 432348 448019 463554 478947 494195 23 38352386 368665 384832 400882416810 432610 448279 463812 479203 494448 22 39 352658 368936 385101 401149 417074 432873 448539 464069 479458 494700|21| 40 352931 369206 385369 401415 417338 433135 448799|464327 479713 494953 20 41 353203 369476 385638 401681 417603433397 449059 464684 479968 495206 19 42 353475 369747 385906 401948 417867 433659 449319 464842 480223 495459 18 43 353747 370017 386174 402214 418131 433921 449579 465100 480479 495711 17 44354019 370287 386443 402480 418396 434183 449839 465357 480734 495964 16 45 354291 370557 386711 402747 418660 434445 450098 465615 480989 49621715| 46 354563 370828 386979 403013 418924 434707 450358 465872 481244 496469|14| 47 354835 371098 387 247 403279 419188 434969 450618 466129 48149949672213| 48 355107 371368 387516 403545 419452 435231 450878 466387 481754 496974 12 49 355379 371638 387784 4038114:9716 435493 451137 466644 482009 497226 11 50 355651 371908 388052 404078 419980 435755 451397 466901 482263 497479 10 51 355923 372178 388320 404344420244 436017451656 467158 482518,497731 52 356194 372448 388588 404610420508436278 451916 467416 482773 497983 8 53356466 372718 388856 404876420772436540 452175 467673 483028 498236 7 54 356738 372988 389124 405142421036 436802 452435 467930 483282 498488|| 6| 55 357010 373258 389392 405408 421300 437063 452694 468187 483537 498740 56 357281 3735281389660 405673 421563 437325 452953 468444 483792 498992 57 357553 3737971389928 405939 421827 437587453213 468701 484046.499244 58 357825 374067 390196 406205 422091 437848 453472 468958 484301 499496 2 59 358096 374337 390463 406471422355 438110 453731 469215 484555 499748 1 60 358368 374607 390731 406737 422618 438371 453990 469472 484810 500000 0 M 699 68" 67° 66° 659 64 63° 62° 619.

60° M