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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 verification, 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 logarithmic 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.

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 45° and 90°, look for the title of the column, at the bottom; and for the degrees and minutes, on the right.

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20 0/0.0348995 0.0349208 28.636253 0.9993908 88° 0'

[blocks in formation]

40 0987408 0992257 10.078031 50 50 1016351 1021641 9.7881732

D. M. Cosine. Cotangent. Tangent.

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D. M.

Sine.

Tangent. Cotangent. Cosine.

D. M.

10

20

6° 0'0.1045285 0.1054042 9.5143645 0.9945219 84° 0' 1074210 1080462 9.2553035 9942136 1103126 1109899 9.0098261

50

9938969

40

30 1132032

1139356 8.7768874

9935719

30

1168832 8.5555468

40 1160929 6° 50' 1189816 1198329 8.3449558

7° 0' 0.1218693 0.1227846 8.1443464 0.9925462 83° 0'

9932384

20

[blocks in formation]

8° 0/0.1391731 0.1404085 7.1153697 0.9902681 82° 0

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90 0/0.1564345 0.1583844 6.3137515 0.9876883 81° 0'

[blocks in formation]

10° 00.1736482 0.1763270 5.6712818 0.9848078 80° 0'

9862856

30

9858013 20

[blocks in formation]

40 10° 50 1879528 1913632 5.2256647 9821781 79° 10'

110 00.1908090 0.1943803 5.1445540 0.9816272 79° 0'

10

20

1936636 1974008 5.0658352 9810680 1965166 2004248 4.9894027 30 1993679 2034523 4.9151570 40 2022176 2064834 4.8430045 11° 50 2050655 2095181 4.7728568 D. M. Cosine. Cotangent. Tangent.

[blocks in formation]
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