APPENDIX. TO find the Seconds in an Angle corresponding to any given Logarithm; and, conversely, to find the Logarithm answering to any Angle containing Seconds. See articles 84 and 138. The seconds of a degree may be found by proportion as follows. Let 9.6699740 be the log. sine of an angle A. The nearest log. sine to this in the common tables is 9-6699420, which answers to 27° 53' to the nearest minute. Now the difference between these two logarithms is 320; and in the tables it appears that the difference of the logarithms corresponding to 1'is 39.8. Hence 39·8: 320 :: 1' : 8' (130 Pl. Tr.). Consequently the angle A = 27° 53′ 8". In this manner we can find the angle to seconds corresponding to any logarithm. In general, let ao b' represent any arc or angle, whose log. lis next less than the given log. L, if it be the log. of a sine, tangent, or secant; or next greater, if it be the log. of a cosine, cotangent, or cosecant; and let d be the difference of the logarithms corresponding to 1"; then d: L:: 1" : c". Hence a b' c" is the arc or angle corresponding to the given log. L. On the contrary, we can find the logarithm answering to any angle containing seconds, as follows. Let the arc a° b' c' be given to find its logarithm, then 1": c": d: D. Therefore 1+D is the log. required, if it be the log. of a sire, tangent, or secant; and 1-D is the log., if it be the log. of a cosine, cotangent, or cosecant.* In small arcs the logarithmic sines and tangents increase so irregularly that the differences for seconds are not in the same ratio as the differences for minutes. * The two following rules were communicated by Joseph Clay, Esq. Now as a small arc decreases, and approximates to a right line, the differences of the logarithms of numbers expressing the different lengths of the decreasing arc approximate to the differences of the logarithms of the sines and tangents (130 Pl. Tr.). Therefore in small arcs containing seconds, the following rules give results much nearer the truth than the preceding rules by proportion. To find the Logarithmic Sine or Tangent of any Arc containing Seconds. small Reduce the degrees and minutes of the given arc to seconds, and find the logarithm of that number of seconds. Reduce the whole arc to seconds, and find the logarithm of that number of seconds. Subtract the first logarithm from the second. Find the logarithmic sine or tangent of the degrees and minutes in the arc proposed, to which add the difference of the logarithms found above. The sum is the logarithmic sine or tangent of the given arc. Required the log. tangent of 4° 25' 37". 4° 25' =15900", whose log. is 4.201397 4° 25' 37" 15937", whose log. is 4.202407 Given the Logarithmic Sine or Tangent of any small Arc, to find the Arc. Find the logarithmic sine or tangent next less than the given logarithm, and subtract it from the given logarithm. Reduce the degrees and minutes answering to the logarithm next less than the given logarithm to seconds, and find the logarithm of that number of seconds. To this last logarithm add the difference of the logarithms found above, and the number answering to the sum of these logarithms is the number of seconds in the arc required. Required the arc answering to the log. sine 8-682553. Log. sine next less 8.681043, whose arc is 2° 45′ = 9900" = The number answering to this log. is 9934.5" 2° 45′ 34•5′′ the arc required. Of the Tangent and Secant of an Arc of 90°. Difficulties have arisen from the supposition that an arc of 90° has a tangent and a secant, both infinite. The following error arises from this supposition. In a right-angled spherical triangle, radius: cos. angle at the base :: tan. hyp.: tan. base (53). Now when the base is 90°, the hypothenuse is also 90° (40). Therefore if these two equal arcs have any tangents, they must be equal; therefore radius=cos. angle at the base, whatever that angle may be. This false conclusion arises from the supposition that an arc of 90° has a tangent. As the arc increases to 90°, the tangent and secant increase without limit, cease to exist when the arc is 90°, and then begin again at a quantity indefinitely great. Hence it follows that, when a tangent or a secant enters into a calculation, and the arc becomes 90°, we can draw no infallible conclusion from the calculation. THE END. |