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. A TABLE OF NATURAL SINÈS 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. 2° 0'0.0348995 0.0349208 28.636253 0.9993908 88° 0' 3° 00.0523360 0.0524078 19.081137 0.998629587° 0 10 0552406 0553251 18.074977 9984731 50 20 0581448 0582434 17.169337 9983082 40 30 0610485 0611626 16.349S55 40 0639517 0640829 15.604784 3° 50' 0668544 0670043 14.924417 9981348 30 20 9977627 86° 10' 4° 0' 0.0697565 0.0699268 14.300666 0.9975641 86° 0′ 10 0726580 0728505 13.726738 9973569 50 20 0755589 0757755 13.196883 30 0784591 0787017 12.706205 40 0813587 0816293 12.250505 4° 50' 0842576 0845583 11.826167 9971413 40 9969173 30 9966849 20 9964440 85° 10' 50 40 5° 00.0871557 0.0874887 11.430052 0.9961947 85° 0' 10 0900532 0904206 11.059431 9959370 20 0929499 0933540 10.711913 9956708 30 0958458! 0962890 10.385397 9953962 30 40 0987408 0992257 10.078031 9951132 20 5° 50' 1016351 1021641 9.7881732 994821784° 10' D. M. Cosine. Cotangent. Tangent. Sine. D. M. D. M. Sine. Tangent. Cotangent. Cosine. D. M. 6° 00.1045285 0.10510429.5143645 0.994521984° 0' 10 1074210 1080462 9.2553035 9942136 50 10 7° 0 0.1218693 0.1227846|8.1443464 0.9925462 83° 0 1247560 1257384 7.9530224 9921874 50 8° 0 0.1391731 0.14040857.1153697 0.9902681 82° 0' 9° 0' 0.1564345 0.15838446.3137515 0.9876883 81° 0' 10 1593069 1613677 6.1970279 9872291 50 11° 0 0.1908090 0.19438035.1445540 0.9816272 79° 0' |