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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 180o; or the natural sine or co-sine of an arch greater than 900 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 sides, &c. ex. cept 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 320 21' is

535090 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,

+ 184 is

Sum is the natural sine required

535274

LXAMPLE II.

Required the natural Co-sine of 7 1.0 40' 25", or 108° 19' 35''. The natural co-sine of 710 40' is

314545 The difference 460, multiplied by 25, pointing off two figures, is -115 Remainder is the natural co-sine required

314430

EXAMPLE III.

Required the Degrees, Minutes, and Seconds, answering to the patu

ral Sine 495994. The natural sine next tess 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 t' to be annexed to 290 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 680 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 680 21' 39'', or its supplement, 111° 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 patural co-sine of 20° 39' is 935752, which subtracted from 1000000, leaves 064248, the natural versed sine of 200 39'.

EXAMPLE II.

Required the natural versed Sine of 1460 38' 40.** The natural co-sine of 33° 21'20" (the supplement of 146° 38' 40') is 835274, which added to 1000000, the sum 1835274 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 natural versed sine be less than 1000000, but if otherwise, it will be their supplement.

EXAMPLE I.

Required the Değrees, &c.answering to the natural versed şine 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 1160172.

Here 1000000 subtracted from the above, leaves 160172, which taken out as a natural co-sine, corresponds to 800 46' 59'; therefore its supplement 999 13' 1" are the degrees, &c. required.

M

M

1

0° 1° 2°
30 40 5° 6° 70

go 9° ooooooo|017452034899052336069756'0871561104528 131869139173 15643+ 60 I looo2910177430351901052626070047087446 104818. 122158,139461 156722 59 21000 582 0180340354811052417 070337 087735 1051071224471139749 15700958 31000873018325035772053207070627,088025 105396 1227351140037 157246, 57 41001164 018616036062/053498f070917/088315105686 123024140325'157584'564 51001454 0189071036353053788071207 088605 105975123313/140613 15787155 6001745019197 036644054079 071497 088894 106 264 123601140901,15815854 7 002036019488036934054364 0717880891841106553|123890141189,158445153 81002327 019779|037225054660072078/089474 106843 124179141477;158732152

910026180200700375160549502072368089763 107132 124467 141765.159020151 101002909020361037806055241 0726580900531074211 24756 142053159307 50 ulo032001020652038097055531072948090343 107710 125045142341|159594149 1210034910209420383880558220732381090633 107999 125333 142629115988145 130037820212330386781056112073528090922 108289125622 142917116016847 141004072 021524038969 056402 073818 091212108578125910143205 160455 46 15004363 02181510392601056693 074108 091502108867 126199 143493160743145 16004654 02210603955005698307439910917911091561264881437801161030 441 17004945 022397 0398411057274 0746891092081109445 126776144068161317 43 181005236 022687 0401321057;640749791092371109734) 127065144356161604 42 191005527022978040422057854 075269092660 11002312735344644 161894141 201005818 0232690407131258145 0755590929501103131127642 144932162178140 21006109 023560 041004/058435075849 093239110602 127930145220 16246539 221006399023851041294058726 076139 093529110891128219 145507 16275238 23100669002414110415851059016076429 093819 1111801 1285071457951163039 37 24/006981024432 04:876059306076719094108 111469 128796146083/163326136 251007272024723 042166059597077009 094398111758 129084,146371116361335 26007563 02501404245710598871077299 094687 11 2047129373146659163900 34 27 007854025305042748060177 077589 094977 112336 129661146946 164187133 -81008145025595 0430381060468 077879095267 112625129949147234 16447432 29008436 025886 043329060758078169 095556112914130238147522164761131 30008727 026177043619 061049 078459095846113203 130526 147809165048/30 311009017 026468043910061339 078749 0961351113492130815148097 165334 29 321009308026759044201 0616290790391096425 113781131103 148385165621 28 33 009599 027049 044491 061920 0793291096714114070131391 14867216590827 34 009890 027340 0447820622101079619097004114359-31680 148960 166195 26 35010181027631045072062500079909097293 1146.18131968 149248 166482 25 361010472027922045363 0627911080199 097583|114937 132256 149535 16676924 371010763028212045654 063081 080489 0978721152261132545 149823 16705623 38 011054028503045944 063371 080779 098162115515132833, 150111|167342 224 39011344 028794046235063661 081069 09845101 5804133121.150398167629 21 40 011635 0290850465250639521081359 098741116093 133410 15068616791620 41011926 029375 0468160642421081649 099030116382 133698150473 168203/19 420122:7c29666047106064532081939099320176674133986 15126116848918 43012508 029957047 397 064823 082228 099609 116960134274151548168776 17 44012799030248047688 065113082518 099899117249 134563151836 169063 16 450130901030539 047978065403 0828081001881175371348511152123169350115 460133801030829 048269 065693083098 100477 117826 1351391152411 169636 14 4710 r3671031120048559065984 083388 100767118115 135427 152698 16992313 4810139621031414048850066274 083678101056 48404 135716152986170209 12 49014253 031702 0491400665641083968 101346118693-36004 153-73 1704961 50014544 081992049431 066854084258 101635118982 1 36292153561|150783 10 s1014835 032283049721 067145 084547 101924\11927011365801538481171069 9 521015126032574050012 067435084837 102214 19559 136868154136,171356 8 53015416032864050302 0677251085127 10250311984813715611544231-10437 54 015707 03315510505931068015085417102993 120137 137445154710171929 55015998033446 050883068306 085707 103082 120426137733 154998072216 5 56016289 03373710511741068596085997 103371_1207141380211552851152;0214 571016580034027 051464;0688860862861036611210091383091155572172789 3. 581016871034318051755069176 0365761429501212921385971155860173075 59017162 034609 05 20451069466108686610423912158511388851 156147 173362 60017452 034899 05233606975608-156 104528121869 139173156434 1736481 0 89° 88° 87° 800 8) 840

820 81°

Natural Co-sines. Diff. to 485

480

2

M

8,

M

485

484 | 484 | 483 | 483 | 482 | 481

100

478

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35 183665 200793 217859 234859 251788/268640 2854103020931318684 335178 23 36183951 201078 218143 235142 252069 268920285688 302370 318939 335452 24 37 184237/201363218427 235425252351 269200 285967/302647 319235 335726 23 38184523 201648218711235708 252632 269480 286246 30292413195111336000 22 39184809201933|218995 2359901252914 269760286525 303202 319786 336274 21 40185095 202218 219279 236273 253195 270040 286803 3034791320062 336 547 20 411185381|202502 219562 236556 253477 270320 287082 3037561320337 33682119 42 185667 2027872198461236838 253758 2706002873613040331320613 33709518 43 185952 203072 220130237121254039 270880 287639/304310 320889 337369,17 144 186238/203357220414 237403 25432112711602879181304587 321164 337643 16 59 186524/203642220697 237686 254602271440 288196 304864 325439/33791715 46186810203927220981237968 254883/271720288475,305141 321715338190 14 47 187096 204211 221265238251 255165 272000/288753 305418 321990 338404 13 48187381 204496 221548 238533 255446 272280 289032305695 322266 33873812

49 187667204781221832238816255727272560 289310 305972 32254133901211 50:87953 205065 222116 239098 256008 272840 289589306249 32281633928510 5111882381205350222399 239381 256289 273120 289867306526 323092 3395599 52 188524205635222683239663256571 273400 290145 306803 323367 3398328 53 188810205920 222967|239946 256852 27 3679290424 307080 323642 3401067 54189095206204 223250 240228257133 273959 2907023073573239173403806 5511893811206489 223534 240510 257414 274239 290981 307633 324193340653 51 56189667 206773223817240793 2576951274519/291259307910 324468 3409274 57189952/207058/224101|241075 257976274798291537 3081871324743 3412003 58 1902381207343 224384 241357258257 275078 291815 308464 325018 341473 591905231207627 224668241640 258538 275358) 29209413087401325293 341747 60190809 207912 2249512419222;8819275637292372 309017 3255681342020 79° 789 77" 76° 75° 74" 730 720 71"

Natural Co-sines. Diff, to 100" 477

473

2

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M м

700

29°

M

42

20° 21° 22° 23 140

26° 27 0342020 358368 374607 390731 4061737 422618438371 453990 469472 484810260 1342293 358640 374876390999 407002 422882 438633 454250 469728 48506459 2342567 358911375146391267 407268 423145/438894454509 464985 485318 58 3342840 359183|375416391934 407534423404 439155 454768 470242 485573 57! 4/343113 359454 375685391802 407799 +23673439417 455027 470499 485827 56 5 3433873597253759551392070 4080651423936 439678455286 470755 486081 55 6343660 359997 3762241392337 408330424199 439939 455545471012 486335154 7 343933360268 376494 392605 408596424463440200 455804 471268 486590 53 83442063605401376763392872 4088614247 26 440462 4560631471525 486844) 52

9/344479 360811 377033|3931401409127 424990 440723 456322471782 487098 S. 10344752 361082 377302 393407 409392'425253440984 456580 472038/48735250 11345025361353377571 393675499658;425516441245/456839 472294 487606 49 12 345298 361625377841393942 409923425779 441506 4570981472551 487860148 13 345571361896 378150394209 410188426042 441767 457357472807 4881

114 471 14345844 362167 378 379 394477 410454426306 442028 457615 473063 498;67 46 15346117 362438 378649 394744410719426569442289 457874473320 488621 45 1613463901362709378918 39501410984,426832 442550458133 47357648887544 17,346663 362980 379187395278 411249427095 4428 10 458391 473832 48912943 181346936 363251379456 395546 411514427358 443071 453650 474088 489382 19 347208363522 379725 395813411779427621443332 458908474344 489636147 20 34748363793 379994 196080 412045427884 443593459166 474600 489890

40 21 347754 36406413802631396347 412310428147+43853/459425 474956490143 39 22 348027 364335 380532 3466 4 4125754284101444114 45968347511 2 490397138! 23348299 364606380801 396881412840428672 4 14 375 459942 4753681490650 37 241348572364877381070 397148413104 428935 4446351460200475624/490904 36 25 348845 365148/381339 397415 413369429198 444896 460458475880491157135 26 349117365418381608 397682 413634429.461 445156460716476136 49141134 27 349390 365084 381877 397949 41389942972; 445417 460974 476392 491664 33 28 349662 365960382146 348215414164429986445677 46123247664?1491917:32 29 3499353662311382415 398482 414429430249 445937 461491 476903 492170131 30 350207 306501 382683 398749 414693 43051446198 461749 477159 49242430 311350480 366772 382952399016 4149584307744-6458 462007|477414 442677 29 32350752 36704238322113992831415223431036 446718462263 477670 49293028 333510253673131383490 399549415487 431299 446979462523147792549318327 34 351297367584383758399816 415752431561 447239462780 478181 493436 26 35351569|367854 384027400082 416016 431823447499 463038 478436493689 25 3635:842 368125384295 400349 41628 432086 447759 463296478692/493942 24 37 352114 368395 384564 4006:6416545 432348448019 463554 478947 49419523 38 352386 3686651384832400882416810 432610448279 463812 474203 494448) 22 39352658368936385101401149 417074 432873 448539464069 479458494700121 40352931 369206 385369 401415 417338 +33135448799 454327147971349495320 411353203 369476385638 401681417603 433397449054 464,84_479468 495206 19 42/353475369747 385906 401948 417867 433659 449314 464842 480223 495459 18 43 353747 3700171386174 402214 4181311433921 449579465100 480479 495711117 44 354019370287 386443 402480 418396 4341834498 39 495357 480734 49596416 45354291 370557 386711 402747 418660 434445450098 46561548-989496217.15 46354563 370828 386979 403013418924 4347071450358/465872 48124449646914

47 354835/371098 387247 403279419188434969 450618 4661291481499 496722113 48355107 371368387516 403545 419452 435231 450878 466387:48 1754,496974 12 49355379 371638387784 403811 414716 435493 451137 466644 482004 49722611 5035565113719081388052 40407814199801435755451397 466901,482263 497479 10 15113559231372178 388320404344/420 244 436017451656 467158.4825184977319 52 356194 372448 388588404610 420508 436278f451916 467416482773 497983 8 53 356466 372718388856 404876 420772 436540452175 467673 483028 4982367 54/356738 372988/389124 405142 421036 436802 452435 467930483282 498488 61 55357010373258389392 405408 421300 437063452694468187 483537 4987405 56 357281 373528 389660405673 421563 437325/452953, 468444 483792 498992 41 573575533737971389928 405939 421827 437587 453213468701;484046 499244 3. 58 357825 374067390196 406205 422091 437848 453472,468958,484301 499496 2 591358096 374337 390463 406471 422355 438110453731.469205484555 499748 1 60358368 3746071390731|406737 422618438371453990 469472,484810 500000. O 699 68 67° 660 65° 649 03° 60 1612 60°

Natural Cosines Diff.to 451 445 441 434 430 426

422

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448

438

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