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given here. But as we are already supplied with accurate trigonometrical tables, the computation of the canon is, to the great body of our students, a subject of speculation, rather than of practical utility. Those who wish to enter into a minute examination of it, will of course consult the treatises in which it is particularly considered.

There are also numerous formulæ of merification, which are used to detect the errors with which any part of the calculation is liable to be affected. For these, see Legendre's and Woodhouse's Trigonometry, Lacroix's Differential Calculus, and particularly Euler's Analysis of Infinites.

Note K. p. 127. The following rules for finding the sine or tangent of a very small arc, and, on the other hand, for finding the arc from its sine or tangent, are taken from Dr. Maskelyne's Introduction to Taylor's Logarithms.

To find the logarithmic sine of a very small arc. From the sum of the constant quantity 4.6855749, and the logarithm of the given arc reduced to seconds and decimals, subtract one third of the arithmetical complement of the log. arithmic cosine.

To find the logarithmic TANGENT of a very small arc. To the sum of the constant quantity 4.6855749, and the logarithm of the given arc reduced to seconds and decimals, add two thirds of the arithmetical complement of the logarithmic cosine.

To find a small arc from its logarithmic sine. To the sum of the constant quantity 5.3144251, and the given logarithmic sine, add one third of the arithmetical complement of the logarithmic cosine. The remainder diminished by 10, will be the logarithm of the number of seconds in the arc.

To find a small arc from its logarithmic TANGENT. From the sum of the constant quantity 5.3144251, and the given logarithmic tangent, subtract two thirds of the arithmetical complement of the logarithmic cosine.' The remainder diminished by 10, will be the logarithm of the number of seconds in the arc.

For the demonstration of these rules, see Woodhouse's Trigonometry, p. 189.

A TABLE OF

NATURAL SINES AND TANGENTS,

TO EVERY TEN MINUTES OF A DEGREE

IF the given angle is less than 45°, look for the title of the column, at the top of the page; and for the degrees and minutes, on the left. But if the angle is between 45o and 90°, look for the title of the column, at the bottom; and for the degrees and minutes, on the right.

D. M. Sine. Tangent. Cotangent. Cosine. D. M. 0° 0' 0.0000000 0.0000000 Infinite. 1.0000000 200 0 10 0029089 0029089313.77371 0.999995 50

0058177 0058178 171.88510 9999831 40

0087265 00S7269114.58865 9999619 30 40 0116353 0116361 85.939791 9999323 20 0° 501 0145139| 01 454511 68.750087 9998912/89° 10

20 30

1° 0 0.0174324 0.0174551/ 57.299962 0.9998177599 0

10 0203609 0203650 49.103881 9997927 50 20 0232090 0232753 42.96 1077 9997292 40

(261859 38.188159 9996573 30 40 0290817 0290970 31.367771 99957701 20 1.0 50 031992: 0320036/ 31.211577 9991SSiSSO 10

30 0281969

2° 0 0.03 18995 0.0319208 28.636253 0.9993905889 0 10 0378065 0378335 26.431600 9992851

50 20

0107131 0 107469 21.511758 9991709 40 30 0136191 0136609 22.903760 9990182

40 0465253 0165757| 21.470101 9989171 120 50 0494308 019 1913 20.205553 99877758 10

30 20

13° 0 0.0523360 0.0521078 19.081137 0.999029587 0

10 0552106 0553251 18.074977 998 1731
20
0581448 0582131 17.169337 99-3082

40 30

0610185 0611626 16.319555 9981318 40 0639517 0610929 15.60 1781 997995:30 20 3° 501 06685-11 0670013 14.921417 9977627156.10

30

1° 0 0.0697365 0.0699205 11.300666 0.99750040

10 0726580 0723505 13.730738 997:35369 20 0755589 0757755 13.196883 9971013 30 0781591 0787017 12.706205 9169173 30

40 0813587 0816293 12.250505 9966919 4° 50'1 0812576 0815583 11.826167 99614108 10

10

5° 0' 0.0871557| 0.0874987) 11.430052 0.98619871050

0900532 090 1206 11.059131 9959370 201 0929199 09335 101 10.711913 30 0958158! 09962890 10.385397 99533963 30

40 0987108 0992257| 10.078031 9951132 20 5° 50' 1016351 1021641 9.7881732 991821751° 10 D. M. Cosine. Cotangent.! Tangent.

Sine. D. M.

H

50

D. M. 1 Sine. Tangent. | Cotangent. Cosine. D. M.
6° 00.10152550.1051012 9.5143615 0.991521984° 0

10 1074210 10501629.2553035 9912136
20

1103126 11098999.0093261 9938969 40 30 1132032 1139336 8.7768874 9935719 30

40 1160929 1168332 8.5555168 9932384 20 6° 50 1189816) 1193329 8.3119555 992806583° 10'

7° 0' 0.12186930.12278168.1443 1610.9925 162 83° 0 10 1217560 125373847.9530224 9921871

50 20 1276116 12369137.7703506 9918201 40 30 1305262 13165257.5957541 9914149 30

40 1331096 13161297.4257061 9910610 7° 50' 1362919 13757577.2687255| 9906687182° 10

20

8° 0 0.13917310.140108517.11536970.9902681 82° 0'

10 1420531 14350816.9692335 9898590 50
20

1419319 14617816.8269 137 9891416 40 30 1478091 14915106.691 1562 9890159 30

40 1506857 1521262 6.5605535 9885817 20 8o 50 1535607 155 1010 6.4348129 9881392 81° 10

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go 00.15643130.15838 01 6.31375150.987688381° 0

10 1593069 1613677 6.1970279 9872291 50 20

1621779 16135376.05 1351 9867615 40 30 1630476 16731265.9757614 9862856 30

10791591 17033143.8708042 9838013 20 90 50 170782 1733292 5.7693654 9853087 80° 10

40

1109 010.17361620.1763270 5.67128180.9818078 80° 0

10 17651211 17932795.5763786 9812985 50 20 17937-16! 182331955.45 15052 9837808 40 30

1822353 18533905.39.33172 9325-19 30 40 18509 19 18531955.30927931 9827206 20 10° 50 1879528/ 19136325.2250617 9821781 79° 10'

11° 0 0.19080900.1913-035.111551010.9S16272797 0

10 1936636 197 10085.0634352 9810680 50 20 1965 166 200 121 1.9-9 (0271 9805005 40 30 1993679 2031523 1.9151570 9799217 30 40

20:22176 20618311.81300 15 9793106 20 11-30 205065) 2095181 1.7728565 9787183789 10 D. M. Cosine. Cotangent. Tangent. Sine. D. M.

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