the lengths of the sides of triangles are taken from the line of lines, and the degrees and minutes from the lines of sines, tangents, or secants. Thus, in Art. 135, ex. 1, 35 R26 : sin 48°. To find the fourth term of this proportion by the sector, make the lateral distance 35 on the line of lines, a transverse distance from 90 to 90 on the lines of sincs; then the lateral distance 26 on the line of lines, will be the transverse distance from 48 to 48 on the lines of sines. For a more particular account of the construction and uses of the Sector, see Stone's edition of Bion on Mathematical Instruments, Hutton's Dictionary, and Robertson's Treatise on Mathematical Instruments. NOTE H. p. 124. The error in supposing that arcs less than 1 minute are proportional to their sines, cannot affect the first ten places of decimals. Let AB and AB' (Fig. 41.) each equal I minThe tangents of these arcs BT and B'T are equal, as are also the sines BS and B'S. The arc BAB' is greater than BS+B'S, but less than BT+ B'T. Therefore BA is greater than BS, but less than BT: that is, the difference between the sine and the arc is less than the difference between the sine and the tangent. Now the sine of 1 minute is 0.000290888216 And the tangent of 1 minute is 0.000290888204 The difference is. 0.0000000000012 The difference between the sine and the arc of 1 minute is less than this; and the error in of 1', and of 0' 52" 44" 3" 45 arcs, as in Art. 223, is still less. supposing that the sines are proportional to their NOTE I. p. 125. There are various ways in which sines and cosines may be more expeditiously calculated, than by the method which is 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 sinall 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 are 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, sec Woodhouse's Trigonometry, n. 189 5° 00.0871557 0.0874887 11.430052 0.9961947 85° 0' 10 0900532 0904206 11.059431 9959370 20 0929199 0933540 10.711913 30 0958458 0962890 10.385397 40 0987408 0992257 10.078031 50 9956708 40 9953962 30 9951132 20 50 50 1016351 1021641 9.7881732 9948217S4° 10' D. M. Cosine. Cotangent. Tangent. Sine. D. M. |