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TABLE. III.

Natural Sines.

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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, lo 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 chat column containing the given degrees in the former case, or the nearest natural sine or co-sine in the latter.

1

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 når

tural sine of the given degrees and minutes is 409, this
multiplied by 45, pointing off two figures in the product,

of 184

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Sum is the natural sine required

535274

EXAMPLE II.

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Required the natural Co-sine of 710 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 natu

ral Sine 495994. The natural sine next less to that given is 4:95964, 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 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 68° 21'; the difference between this natural sine and the giver 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 680.21'29', 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 909, 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 I.

Required the natural versed Sine of 200 39'. The natural 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 1460 38' 40"} is 835274, which added to 1000000, the sum 1855274 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 800 46' 59'; therefore its supplement 999 13' 1" are the degrees, &c. required.

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0000000017452034899 052336069756 087156 104528 121869 139173 156434 60 1000291 017743 035190052626 070047 087446 104818 122158 139461 156-21:59 2000582018034 035481 052917 070337 087735 105107 122447 139749 1570091581 3000873 018325035772053207 070627 088025 105396 122735 140037 157296 57 4001164 018616 036062053498070917 088315105686| 123024140325 157584|56| 5001454 018907 036353 053788 071207 088605 105975123313140613 157871155 6001745019197036644 054079 071497 088894 106264123601140901 15815854 7002036 019488 036934 054369 071788 089184 106553 123890 141189' 158445 53 8002327 019779 037225034660 072078 089474 106843124179141477; 158732 52 9002618 020070037516 054950072368 089763 107132 124467 141765159020|51| 10002909 020361 037806 055241 072658 090053 107421 124756 142053159307 50 11003200020652038097055531 072948 090343 107710 125045142341159594 49 12003491 020942038388 055822 073238 090633 107999125333 142629 159881 48 13003782021233 038678 056112 073528 090922 108289 125622 142917160168 47 14004072 021524038969 056402 073818 091212108578125910 143205 160455 46 15004363 021815039260056693 074108 091502108867 126190 143493160743 45 16004654 022106 039550056983074399 091791 109136 126488143780 161030 17004945 022397 039841057274 074689 092081 109445 126776 14406816131743 18005236 022687 040132 057564 074979 092371 109734 127065144356 161604 42 19005527 022978 040422 057854 075269 09266010023 127353144644 16189141 20005818 023269 040713058145 075559 092950110313 127642 144932162178 40 21006109/023560 041004 058435 075849 093239110602 127930 145220 16246539 22 006399 023851041294058726 076139 093529 110891 128219 145507 16275238 23006690 024141 041585 059016076429 093819 111180128507 145795 163039 37 24 006981 024432 041876 059306 0767 19 094108 111469 128796 146083 163326 36 25007272 024723 042166 059597 077009 094398 111758 129084 146371163613|39| 26 007563 025014042457 05y887 077299 094687 112047 129373 146659 16390034 27007854 025305 042748 060177 077589094977112336129661146946 164187|33| 28 008145 025595 043038 060468 077879095267 112625 129949 14723416447432 29008436 025886 043329 060758 078169 095556 112914 130238147522 164761,31 30008727 026177 043619 061049 078459 09584613203130526 147809 16504830 31009017 026468 043910061339 078749 096135113492 130815148097 165334 29 32009308 026759 044201 061629 079039 096425 113781 131103 148385165621 28 33009599 027049 044491 061920 079329 096714 114070131391 148672165908 27 34009890 027340 044782 062210 079619 097004 114359 131680 148960166195 261 35010181 027631045072 062500 079909 097 293 114648131968 149248166482|25| 36 010472 027922 045363 062791 080199 097583 114937 132256 14953516676924 37010763 028212045654 063081 080489 097872 115226 132545 149823 167056 23 38 011054 028503 045944 063371 080779 098162 115515 132833:15011116734222 39 011344 028794 046235 063661 081069 098451 115804 133121 150398 16762921 40011635 029085 046525 063952 081359 098741 116093 133410 150686 167916 20 41011926 029375 046816 064242 081649|09903011638213369815097316820319 42 012217 C29666 047106064532081939 099320 116671 133986 151261 168489 18 43012508 029957 047397 064823 082228 099609 116960134274 151548 168776 17 44 012799 030248 047688 065113 082518099899 127249 134563 151836 169063 16 45 013090030539 047978 065403 082808 100188 117537 13485115212316935015Į 46 013380 030829 048269 065693.083098 100477 117826 135139 152411169636 14 47013671 031120048559065984083388100767 118115135427 1526981649233 48013962 031411 048850 066274 083678 101056118404135716152986170209 121 49 014253 031702049140066564083968 101346 18693 136004153-73170496 11 50014544031992 049431 066854 084258 201635118482136292153561 17078310 51014835032283 049721 067145 084547 101924 19270136580153848 171069 52015126 032574050012067435 084837102214119559 136868 154136 171356 53015416 032864 050302 067725 085127 102503119848 137156 154423 1716437) 54015707033155050593 068015085417 102793120137 137445154710 171929 55015998 033446 050883 068306.085707 103082120426137733154998 172216 5 56 016289 033737051174 068596 085997 103371120714138021155285 172502 4 57016580034027 051464 068886 086286 103661121003138309 155572 172789 3 58016871034318051755 069176'036576 103950121292138597 155860 173075 59 017162 034609|032045 069466 086866 10.4239 121581 138885156147 173362 60017452 034899 052336 069756 08-156104528 121869 139173156434 173648 0 Μ 89° 88° 87° 36° 859 84 81° Natural Co-sines.

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Natural Co-sines

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