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43 Degrees. M Sine, Co-line. Tang Co-tang

Secant. Co-sec. M 9.833783 9.864127 9.969656 10.030344 10.135878 10.166217 60 1/9.8339199.8640109.969909 10.030091 10.135990 10.166081 59 29.834054 9.863892 9.970162 10.029838 10.136108 10.165946 58 39.834189 9.863774 9.970416|10.029584 10.136226 10.165811 57 49.834325 9.869656 9.970669 10.029331 10.136344 / 10.165675 56 59.8344609.863538 9.970922 10.029078 10.136462 10.165540 55 69.834595 9.863419 9.971175 10.028825 10.136581 10.165405 54 79.834750 9.863301 | 9.971429 10.028571 | 10.136699 10.165270 53 89.834865 9.863183 9.971682 10.028318 | 10.136817 10.165135 52 99.834999 9.863064 9.971935 10.028065 10.136936 10.165001 51 10 9.835134 9.862946 9.972188 10.027812 10.137054 10.164866 50 119.835269 9.8628279.972441 10.027559 10.137172 10.164731| 49 12 9.835403 9.862709 9.972694 10.027306 | 10.137291 10.164597 | 48 13 9.835538 9.8625909.972948 10.027052 10.137410 10.164462 47 14 9.835672 9.869471 9.973201 10.026799 10.137529 10.164328 46 15 9,835807 9.862353 9.973454 10.026546 10.137647 10.164193 45 16 9.835941 9.862254 9.973707 | 10.026293 10.137766 10.164059| 44 17 9.836075 9.862115 9.973960 10.026040|10.137885 10.163925 43 18 9.836209 9.8619969.974213 10.025787 | 10.138004 10.163791 42 19 9.836343 9.861877 9.974466 10.025534 10.138128 10.163657 41 209.836477 9.861758 9.974719 10.025281 10.138242 10.163523/ 40 219.836611 9.861638 9.974973 10.025027 10.138362 10.163389 39 22 9.8367459.861519 9.975226 10.024774 10.138481 10.163255 38 23 9.8368789,861400 9.975479 10.024521 10.138600 10.163122 37 24 9.837012 9.861280 9.975732 10.024268 10.138720 10.162988 36 25 9.8.37146 9.861161 9.975985 10.024015| 10.138839 10.102854 | 35 26 9.837279 9.8610419.976238 10.023762 10.138959 10.16-2721 34 27 9.837412 9.860922 9.970191 10.023509 10.139078 10.162588 33 28 9.8375469.860802 9.976744 10.023256 10.139198 10.162454 32 29 9.837679 9.8606829.976997 10.023003 10.139318 10.162321 31 30 \9.837812 9.8605629.977250 10,022750 W.139438 10.162188 30 31 9.837945 9.860442 9.97750310.022497 10.139558 10.162055 29 32 9.838078 9.860322 9.977756 10.022244 10.139678 10.161922/ 28 33 9.8382119.8602029.978009 10.021991 10.139798 10.161789/ 27 34 9.838944 9.860082 | 9.97826210.021788 10.139918 10.161656 26 35 9.8384779.859962 9.97851510.021485 10.140038 10.161523 25 36 9.8386109.8598429.978768 10.021232 10.140158 10.161390 24 37 9.838742 9.859721 9.979021 10.020979/10.140279 10.161258 23 389.838875 9.859601 9.979274 10.02072610.140399 10.161125 | 22 39 9.839007 9.859480 9.979527 10.020473 10.140520 10.160993 21 40 9.839140 9.859360 9.979780 10.020220 10.140640 10.160860 20 41 9,839272/9.859239 9.980033 10.019967 10.140761 10160728 19 429.839404 9.859119 9.980286 10.019714 10.140881 10.160596 18 43 9.839536 9.858998 9.980538 10.019462 10.141002 10.160-164) 17 44 9.839668 9.858877 9.980791 ( 10.019209 10.141123 10.160332) 16 45 9.839800 9.858756 9.981044 10.018956 | 10.141244 10.160200 15 46 9.839932 9.858635 9.981297 10.01 8703 10.141365 | 10.160068 14 47 9.840064 9.858514 9.981550 10.018450 10.141486 10.159936 13 48 9.840196 9.858393 9.981803 10.018197 | 10.141607 10.159804 12 49 9.840328 9.858272 9.982056 10.017944 10.141728 10.159672| 11 50 9.840459 9.858157 9.982309 10.017691 10.141849 10.159541 10 51 9.840591 9.858029 9.982562 10.017438 10.141971 10.159409 9 52 9.840722 9.857908 9.982814 10.017186 10.142092 10.159278 8 539.840854 9.857786 9.983067 10.016933 10.142214 10.159146 7 54 9.840985 9.857665 9.983320 10.016680 10.142335 10.159015 6 55 9.841116 9.857543 9.983573 10.016427 10.142457 10.158884 5 56 9.841247 9.857422 9.983826 10.016174 10.142578 10.158753 4 57 9.841378 9.857300 9.984079 10.015921 10.142700 10.158622 3 58 9.841509 9.857178 9.984331 10.015669 10.142822) 10.158491 2 599,841640 9.857050 9.984584 10.015416 10.142944 10.158360 609.841771 8569349.984837 10.015168 10.143066 10.158229 0 M Co-sine. Sine. Co-tang. Tang. Co-sec. Secant. M

46 Degrees

44 Degrees.
Sine. Co-side.
M

ang
Co-lang

Secant. Co-sec. M 09.841771 9.856934 | 9.984837 | 10.015163 10.143066 10.158229 60 19.841902 9.856912 9.985090 10.014910 10.143188 10.158098 59 99.842033 9.856690 9.985343 10.014657 10.143310 10.157967 58 39.842165 9.856508 | 9.985596 10,014404 10.143432) 10.157837 57 49.842294 9.856446 9.985848 10.014152 10.143554 10.157706) 56 59.842424 9.856323 9.986101 10.01.3899 10.145677 10.157576 55 69.842555 9.856201 9.986354 10.013646 10.143799'10.157445 54 79.842685 9.856078 9.986607 10.013393 | 10.143922 10.157315 53 89.842815 9.855956 9.986860 10.013:40 10.1440 14 10.157185 52 99.842946 9.855833 9.987112 10.012888 10.144167 10.157054 51 10 9.843076 / 9.855711 9.987365 10.012635 10.144289 10.156924 50 119.843206 9.855588 9.987618 10.012382 10.144412 10.156794 49 129.843336 9.855465| 9.987871 10.012129 | 10.144535 10.156664 48 13 9.843466 9.855342 9.988123 10.011877 10.144658 10.156534 47 149.843595 9.855219 9.988376 10.011624 10.144781 | 10.156405 46 15 9.843725 9.86 5096 9.988629 10.01137110.144904 10.156275 45 169.843855 9.854973 9.988882 10.011118 | 10.145027 10.156145 44 179.843984 19.854850 9.989134 10.010866 10.145 150 10.156016 43 189.844114 9.854727 | 9.989387 10.010613 10.145273 10.155886 42 19 9.844243 9.854603 9.989640 10.010360 10.145397 10.1557 57 41 209.844372 9.854480 9.989893 10.010107 10.145520 / 10.155628 40 219.844502 9.854356 9.990145 10.009855 10.145644 10.155498 39 22 9.844631 9.854233 9.990398 10.009602 10.145767 10.155369 38 23 9.844760 9.854109 9.990651 10.009349 10.145891 10.155240 37 249.8448899.853986 9.990903 | 10.009097 10.14601410155111 36 25 9.845018 9.853862/ 9,991156 10.008844 10.146138 10.154982 $5 26 9.845147 9.853738 9.991409 10.008591 10.146262 10.154853 34 27 9,845276 9.853614 9.991662 10.008338 10.146386 10.154724 33 289.845405 9.853490 9.991914 10.008086 10.146510 10.154595 32 29 9.845533 9.853366 9.992167 10.007833 10,146634 10.154467 31 30 9.845662 9.853242 9.992420 10.007580 10.14:6758 10.154338 30 319.845790 9853118 9.992672 10.007328 10.146882 10.154210 29 32 9.845919 9.852994 | 9.992925 10.007075 10.147006 10.154081 28 33 9.846047 9.8528699.993178 10.006822 10,147131 10.153953 27 349.846175 9.852745 9.993430 10.006570 10.147255 10.153825 26 35 9.846304 9.852620 9.993683 10.006317 10.147380 10.153696 25 36 9.846432 9.852496 9.99393610.006064 10.147504 10.153568 24 379.846560 9.852371 | 9.994189 10.005811 10.147629 10.153440 23 38 9.846688 9.852247 | 9.994441 10.005559 10,147753 10.153312 22 399.8468169.8521 22 9.994694 10.605306 10.147878 10,153181 21 40 | 9.846944 9.851997 9.994947 10.005053 10.148003 10.153056 20 41 9.8470719.851872 9.995199 10.004801 10.148128 10.152929 19 42 9.847199 9.851747 9.995452 10.004548 10.148253 10.152801 43 9.847327 | 9.851622 9.995705 10.004295 10.148378 10.152673 17 44 9.847454 9.851497 9.995957 10.004043 10.148503 10.152546 16 45 9.847582 9.851372 9.996210 10.003790/10.148623 10.152418 15 46 9.847709 9.851246 9.996463 10.003537 10.148754 10.152291 14 479.847836 9.851121 9.996715 10.003285 10.148879 10.152164 13 489.847964 9.850996 9.996968 10.003032 10.149004 10,152036 12 49 9.848091 9.850870 9.997221 10.002779 10.149130 10.1519091 11 50 9.848218 9.850745 9.997473 10.002527 10.149255 10.151782 10 51 9.848345 9.850619 9.997726 10.00227 4 10.149381 10.151655 9 52 9.848472 9.850493 9.997979 10.002021 10.149507 10.151528 8 539.848599 9.850368 | 9.998231 10.001769 10.149632 10.151401 7 54 9.8487269.850242 9.998484 10.00151610.149758 10.151274 6 55 9,848852 9.850116 9.998737 10.001263 10.149884 10.151148 5 56 9.848979 9.849990 9.998989 10.001011 10.150010 10.151021 4 579.849106 9.849864 9.999242 10.000758 10.150:36 10.150894 58 9.849232 9.849738 9.999495 10.000505 10.150262 10.150768 599.849359 9.8496119.999747 | 10.000253 10.150389 10.150641 609.849485 9.849485 10.000000 110.000000 10.150515 10.150515 0 Co-sine. Sine. Co-tang. Tang Co-sec. Secant.

45 Degrees.

18

N

M

TABLE III.

Natural Sines.

In this table the natural sines are exhibited to every degree and ininute 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 180° ; 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 Dearest natural sine or eo-sine in the latter.

EXAMPLE I.

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

535090 The difference at the bottom of the column containing the pa

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

to 184 is

Sum is the natural sine required

535274

EXAMPLE II.

Required the natural Co-sine of 71° 40' 25", or 108° 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

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 495964, answering to 29° 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 natural

Co-sine 369805.

The natural Co-sine next greater to that given is 368936, to which answers 68° 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, 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 I.

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 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-side ; the degrees, &c. corresponding to wbich, 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 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 259 42' 20".

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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 80° 46' 59''; therefore its supplement 99° 13' 1" are the degrees, &c. required.

[graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]
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