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TABLE II.

Logarithmic Sincs, Tangente, and Secanis.

This table contains the logarithmic, or, as they are sometimes called, the artificial sins, tangents, and secants, to each degree and minute of the quadrant, with their complements or co-sines, co-tangents, and co-secants, to six places of figures besides the index.

To find the Logarithmic Sine, Co-sine, &c. of any Number of Degrees and Minutes.

If the given degrees be under 45, they are to be taken from the top, and the minutes from the left side column, opposite to which in that column with the name of the logarithm at the top, will be found the required logarithm. But if the degrees be more than 45, they will be found at the bottom of the page, and the minutes in the right side column ; likewise the name of the logarithm is to be taken from the bottom of the page.

When the given degrees exceed 90, they are to be subtracted from 180 degrees, and the logarithm of the remainder taken out as before. Or the logarithmic, sine tangent, &c. de grees more than 90, is the logarithmic co-sine, co-tangent, &c. of their excess above 90 degrees.

EXAMPLES.

Required the log. sine of 36 32
co-sine of

61 18
tangent of 54 17
co.tang. of 42 50
secant of 19 27
CO-secant of 70 33

side of
or sine of 71 24
or co-sine of 18 86.

logarithm,

9.774729 9.681443 10.143263

10.032877 * 10.025519 10.025519

108 567

9.976702

To find the Degrees and Minutes nearest corresponding to a given Logarithmic Sine,

Co-sine, &c. Look in the column marked at the top or bottom with the name of the given logarithm, and when the nearest to it is found, the corresponding degrees and minutes will be those required, observing that when the name is at the top of the column, the degrees are to be taken from the top and the minutes from the left side column, but if the name is at the bottom, the corresponding degrees will be there likewise, and the minutes in the right side column.

EXAMPLES.
The degrees and minutes corresponding to the
log. sine 9.265390 are 10° 37'

tangent 9.70156 arc 26° 42'
co sine 9.528461
70 16

secant 10.25413 56 9 The logarithmic sides, &c. taken out to degrees and minutes only are in general sufficiently accurate, but in some of the more rigid astronomical calculations, it is frequently necessary to take them out to the nearest second ; when this is the case they are to be found in the fol. lowing manner :

To find the sine, tangent, &c. of an arch expressed in degrees, minutes and seconds.

RULE. Find the sine, tangent, &c. answering to the given degree and minute, and also that answer. ing to the next greater minute ; multiply the difference between them by the given number of seconds, and divide the product by 60; then the quotient added to the sine, tangent, &c of the given degree and minute, or subtracted from the co-sine, co-tangent, &c. will give the quantity required, nearly.

EXAMPLES. 1. Required the log. sine of 41° 35' 45” 2. Required the log. tangent of 39° 54' 48", sine of 41° 35' log. 9.821977

tangent of 39° 54' log. 9.807163 sine of 41 36 9.822120

tangent of 39 55 9.807465 Diff. 143

Diff. 302 then, as 60'' : 45" :: 143 : 107, which added then, as 60'' : 48" :: 302 : 241* which added to the log. answering to 41° 35' gives 9.822084, to the number corresponding to 39° 54' gives the log sine of 41° 35' 45'

the log. tangent of 59° 54'. 48'' = 9.807404 If the arch be less than three degrees it will be necessary to use the following rule :

To the arithmetical compliment of the given degrees and minutes reduced to seconds, add the logarithm of the given degrees, minutes, and seconds, reduced to seconds, and the log. sine, tangent, &c. of the given degrees, and minutes, the sum, rejecting 10 from the index, will be the log, sine, tangent, &c. of the proposed number of degrees, minutes, and seconds.

The arithmetical compliment of any log. is found as in the common log: ; but when the index is 10 or greater than 10, the left hand figure of it must be rejected.

EXAMPLE Required the arithmetical complement of 9.265390.

For the first figure 9, write 0; for 2, 7; for 6, 3; for 5, 4 ; for 3, 6; for 9,1; and for 0, 0; thus the arithmetical complement is 0.234610.

In the same manner, the arithmetical complement of 9.528461 is 0.471539, the arithmetical complement of 9.701560 is 0.298440, and the arith. complement of f10.254130 is 9.745870.

To find the degrees, minutes, and seconds, answering to a given logarithmic sine, tangent,&86.

RULE.

Find the degrees and minutes answering to the next less logarithmic sine, tangent, &c. which subtract from that given; multiply the remainder by 60, and divide the product by the difference between the next less and next greater logarithms, and the quotient will be the seconds to be annexed to the degrees and minutes before found; or, which is the same, as the differ. ence between the next less and the next greater logarithms is to the difference between the next less and the given logarithms, so is 60 seconds to a number of seconds to be annexed to the number of degrees and minutes, answering to the less logarithm found.

EXAMPLES,

1. Find the degrees, minutes, and seconds (less than 90) answering to the log. sine 9.828846 next less log. 42° 23' 9.828716

given log. 9.828840 greater 42 24 9.828855 next less 9.828716

[blocks in formation]

As 139 : 130 :: 60" : 56'' which annexed to 42° 23' gives 42° 23' 56' answering to the log. sine 9.828846, subtracting 42° 23' 56' from 180° there remains 137° 36' 4" the log. sine of which is also 9.828846. 2. Required the degrees, minutes, and seconds answering to the log tangent 9.975120.

next less log. 43° 21' 9.974973 given log. 9.975120
greater 43 22 9.975226 next legs 9.974973

[blocks in formation]

As 243 : 147 :: 60% : 32"' which annexed to 43° 21' gives 43° 21' 32" the degrees, minutes and seconds required.

If the given logarithm is that of the sine or tangent of a small arch--then, to the arithme. tical complement of the next less logarithm in the tables, add the given logarithm and the loga. rithm of the degrees and minutes, in seconds, answering to the next less logarithm, the sum, rejecting radius, will be the logarithm of the number of seconds in the required arch.

This difference is to be added when the log. of the degrees and minutes next below is less than the log. next above the given sine, tangent, &c.; but when greater it is to be subtractedo

t In this last example the index is 10, therefore I reject the left hand figure, and the repaining figures, v.254130, are to be subtracted from 10.000000 according to the Rule.

59 58 57 56 55 54 53 52 51 50 49 48

46 45

22

Sine 0 Degree. M O'' 10" 20" 30" 401 0

5.685575 5.986605 6.162696 6.287635 6.463720 6.530673 6.588665 6.639817 6.685575 2 6.764756 6.799518 6.831703 6.861666 6.889695 3 6.940847 6.964328 6.986605 7.007794 7.027997 4 7.065786 7.083515 7.100548 7.116938 7.132733 5 7.162696 7.176936 7.190725 7.204089 7.217054 6 7.241877 7.253776 7.265358 7.276639 7.287635

7.308824 7.319043 7.329027 7.338787 7.348332 8 7.366816 7.375770 7.884544 7.393145 7-401578 9 7.417968 7.425937 7.433762 | 7.441449 7.449002 10 7.463725 7.470904 7.477966 7.484915 7.491754 11 7.505118 7.511649 7.518083 7.524423 7.590672 12 7.542906 7.548897 7.554806 | 7.560635 7.566387 13 7.577668 7.583201 7.5886647.594059 7.599388 14 7.609853 7.614993 7.620072 7.625093 7.630056

7.639816 7.644015 7.649361 7.654056 7.658701 16 7.667 841 7.672345 7.676799 7.681208 7.685573 17 7.694173 7.698410 7.702606 7.706762 7.710879 18 7.718997 7.722999 7.726965 7.730896 7.734791 19 7.742477 7.746270 7.750031 7.753758 7.757454 20 7.764754 7.768358 7.771932 7.775477 7.778994 21 7.785943 7.789376 7.792782 7.796162 7.799515

7.806146 7.809423 7.812677 7.815905 7.819111 23 7.825451 7.828586 7.831700 7.834791 7.837 860 24 7.843934 7.846939 7.849984 7.852888 7.855833 25 7.861662 7.864548 7.867414 7.870262 7.873092 26 7.878695 7.881470 7.884228 7.886968 7.889690 27 7.895085 7.897758 7.900414 7.903054 7.905678 28 7.910879 7.913457 7.916019 7.918566 7.921098 29 7.926119 7.928608 7.931082 7.933543 7.935989 SO 7.940842 7.943248 7.9456417.948020 7.950387 31 7.955082 7.957410 7.959727 7.962031 7.964322 32 7.968870 7.971126 7.973370 7.975603 7.977824 33 7.980233 7.984421 7.986598 7.988764 7.990919

7.995198 7.997322 7.999435 8.001538 8.003631 55 8.007787

8.009850 8.011903 8.013947 8.015981 36 8.020021 8.022027 8.024023 8.026011 8.027989 37 8.031919 8,033871 8.095814 8.037749 8.039675

8.043501 8.045401 8.047294 8.049178 8.051054 39 8.054781 8.056633 8.058477 8.060314 8.062142 40 8.065770 8.067582 8.069380 8.071171 8.072955

8.076500 8.078261 8.080016 8.081764 8.083504 42 8.086965 8.088684 8.190398 8.092104 8.093804 43 8.097183 8.098863 8.100537 8.102204 8.103864 448.107167 8.108809 8.110444 8.112074 8.11 8697 45 8.116926 8.118332 8.120131 8.121725 8.123313 46

8.126471 8.128042 8.129606 8.131166 8.132720 47 8.135810 8.137348 8.138879 8.140406 8.141927 48 8.144953 8.146458 8.147959 8.149453 8.150943

8.155382 8.156852 8.158316 8.159776 50 8.162681 8.164126 8.165566 8.167002 8.168433 51 8.17 1280 8.172697 8.174109 8.175517 8.176920 52 8.179713 8.181 102 8.182488 8.183868 8.185245 5.3 8.187985 8.189548 8.190707 8.192062 8.193413 54 8.196102 8.197440 8.198774 8200104 8.201430 55

8.204070 8.205384 8.206694 8.208000 8.209302 56 8.211895 8.21 3185 8.214472 8.215755 8,217034 57

8.219581 8.220849 8.222113 8.223374 8.224631
58 8.927133 8.228380 8.229622 8.230861 8.232096
59
8.234557 8.235782 8.237003 8.238221

8.239436
60"
50%

40" 30" 20"

45 42 41 40 39 38 37 36 S5 S4 33 32 SI 30

50%
6.384545
6.726967
6.916024
7.047303
7.147973
7.229643
7.298358
7.357672
7.409850
7.456426
7.498487
7.536832
7.572065
7.604652
7.634963
7.663297
7.689894
7.714957
7.738651
7.761119
7.782482
7.802843
7.822292
7.840907
7.858757
7.875902
7.892396
7.908287
7.923616
7.938422
7.952741
7.966602
7.98003
7.993064
8.005714
8,018005
8.029959
8.041592
8,052992
8.063963
8,074731
8.085238
8.095497
8.105519
8.115315
8 124895
8.134268
8.1434 13
8.152128
8.161 231
8.169859
8 178319
8.186617
8.194760
8.202752
8.210601
8.218369
8.225884
8.233328
8.240647

10"

29

28

27 26 25 21 23 22 21 20

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19 18 17 16 15

13 12 11 10

49 8.153903

9 8 7 6 5

S 2 1 0

M

Co-sine 89 Degrees.

0

20"

38

Tangent 0 Degree. M 10"

30 40% 50" 0

5.685575 5.986605 6.162696 6.287635 6.384545 59 6.469726 6.530673 6.588665 6.689817 6.685575 6.726968 58 2 6.764756 6.799518 6.831703 6.861666 6.889695 6.910024 57 3

6.940847 6.964329 6.986605 7.0077947.027998 7.047303 50 4.7.0657867.0835157.100548 7.116939 7.132738 7.147973 55 5 7.162696 7.176937 7.190725 7.204089 7,217054 7.229643 6 7.241878 7.253777 7.265359 7.276640 7.287635 7.298359 53 7 7.308825 7.319048 7.329028 7.898788 7.348938

7.357675

52 8 7.366817 7.375772 7.384546 7.399146 7.401579 7.409852

51 9 7.417970 7.425939 7.438764 7.441451 7.4490047.456128 50 10 7.463727, 7.470906 7.977968 7.484917 7.491756 7.498490 11 7.505120 7.511651 7.518085 7.5244267.530675

7.536835

48 12

7.5429097.548900 1.7.5548087.560638 7.566390. | 7.572068 47 13 7.577671 7.583204 7.588667 7.594062 7.599391 7.604655

46 14 7.6098577.614996 7.620076 | 7.625097 7.630060 | 7.634968 45 15 7.639820 7.644619 7.649366 7.654061 7.658706 7.663301 44 16 7.667849 7.672350 7.676804 7.681213 | 7.685578 | 7.689900 43 17 7.694179 7.698416 7.702612 7.706768 7.710885 7.71 4962 18

7.719003 7.723005 7.726972 7.730902 | 7.734797 7.738658 41 19 7.742484 7.746277 7.750037 7.753765 7.757462 7.761127 40 20 7.764761 7.768365 7.771940 7.775485 7.779002 7,782490 39 21 7.785951 7.789384 7.792790 | 7.796170 7.799524 7.802852 22

7.806155 7.8094327.812686 7.815915 7.819120 7.822302 37 23 7.825460 7.828596 7.831710 7.834801 7.837870 7.840918 36 24 7.843944 7.846950 7.849935 7.852900 | 7.855844 | 7.858769 35 25 7.861674 7.864560 7.867426 7.870274 7.873104 7.875915 26 7.878708 7,881483 .7.884240 7.886981 7.889704 7.892410 33 27

7.895099 7.897771 7.900428 7.903068 7.905692 7.908301 32 28 7.910894 7.913471 7.916034 7.918581 7.921113 7.923631 31 29 7.926134 7.928628 7.931098 7.933559 7.936006 7.938439 SU SO 7.940858 7.943265 7.945657 7.948037 7.950404 7.952758 29 31 7.955100 7.957428 7.959745 7.962049 7.964341 7.966621

28 32 7.968889 7.971145 7.973389 7.976622 7.977844 | 7.980054 27 33 7:982253 7.984441 7.986618 7.988785 7.990940 7.993085 | 26 S4 7.995219 | 7.997343 7.999456 8.001560 8.003653 8.005786 25 35 8.007809 8.009872 8.011926 8.013970 8.016004 8.018029 36 8.020044 8.022051 8.024047 8.026035 8.028014 8.029984 28 37 8.031945 8.033897

8.035840 8,037775 8.039701 8.041618 22 38 8.043527 8.045428 8.047321 8.049205 8.051081 8.052949 21 39 8.054809 8.056661 8.058506 8.060342 8.062171 8.065992 40 8.065806 8.067612 8.069410 8.071201 8.072985 8.074761 19 41 8.076531 8.078293 8.080047 8,081795 8.083536 8.085270 18 42 8.086997 8.088717 8,090430 8.092137 8.093837 8.095530 17 43 8.097217 8,098897 8.100571 8.102239 8.103899 8.105554 16 44 8.107202 8.108845 8.110481 8.112110 8.113734 8.115352 15 45 8.116963 8.118569 8.120169 8.121763 8.123351 8.124938 14 46 8.126510 8,128081 8,129646 8.131206 8.132760 8.134308 13 47 8.135851 8.137389 8.138921 8.140447 8.141969 8.143185 12 48 8.144996 8.146501 8.148001 8.149497 8.150987 8.152472 49 8.153952 8.155426 8.156896 8.158361 8.159821 8.161276 10 50 8.162727 8.164172 8.165613 8.1670498.168480 8.169906 9 51 8.171528 8.172745 8.174158 .8.175566 8.176969 8.178368 8 52 8.179763 8.181152 8.182538 8.188919 8.185296 8.186668 53 8.188036 8.189400 8.1907 60 8.192115

8.193466 8.194813 54 8.196156 8.197494 8.198829 8.2001 59 8.201485 8.202808 55 8.204126 8.205440 8.200750 8.208057 8.209359 8.210658

8.211953 8.213243 8 214530 8.215814 8.217093 8.218369 57 8.219641 8.220909 8.2221748.2234348.224692 8.225945 58 8.227195 8.228442 8.229685 8.230924 8-232160 8.283392 59 8.234621 8.235846 8.237068 8.238286 8.239501 8.240713 60" 50"

40"
30"
20"

M
Co.taogenc 89 Degrecs,

56

10"

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