It must not, however, be unnoticed, that the want of precision in the determination of the angle is partly owing to the construction of the Logarithmic and Trigonometrical Tables. The Tables referred to, and in common use*, are computed to seven places of figures; but, if we had Tables † computed to a greater number of places, to double the number, for instance, then the logarithmic sines of all angles between 89° 56′ 18", and 89° 57′ 9", would not be expressed, as they are in Tables now in use, by the same figures. In such circumstances, we should obtain conclusions very little remote from the truth; but then, such Tables would be extremely incommodious for use, and would, in all common cases, give results to a degree of accuracy quite superfluous and useless. Moreover, such Tables, even in the extreme cases which we have mentioned, are not essentially necessary : since their use can be superseded, by abandoning the first method of solution, and recurring either to the 2d, 3d, or 4th method. When the angle (A) sought then is nearly = 90°, the first method must not be used, but one of the latter methods, in which either the sine, cosine or tangent of half the angle is determined; and, in such an extreme case, it is a matter of indifference whether, instead of the first method, we substitute the 2d, or 3d, or 4th. But, in other cases, it is not a matter of indifference: for since, as it has been shewn, the variation or the increment of the sine is as the cosine, and of the cosine as the sine, these two variations are equal at 45°, but beyond 45°, up to 90°, that of the sine is less, and that of the cosine greater; * Shirwin's 8vo. Hutton's 8vo. Taylor's 4to. † In Vlacq's Tables, published at Gouda, 1633, we have Arcs. 89° 56′ 10′′. 20 40 Log. Sines. ..9.9999997300 ....97530 .... ...97958 50.. 89 57 0 10...... ...98154 ....98346 98525 and, the contrary happens between 45° and 0; consequently we have this Rule: If the angle sought be < 90°, use the second method; if > 90°, use the third method. The 4th method may be used, and commodiously, for all values of the angles sought from O up to angles nearly = 180°: when, however, the angle (A) is nearly = 180°, tan. , which is A 2 nearly tan. 90°, is very large, and its variations, (which are as the square of the secant *) are also very large and irregular. If, therefore, we use Sherwin's Tables, which are computed for every minute only of the quadrant, the logarithms corresponding to the seconds, taken out by proportional parts, will not be exact: for, in working by proportional parts, it is supposed, if the difference between the logarithmic tangents of two arcs differing by 60 seconds bed, that the difference between the logarithmic tangents of the first arc, and of another arc, that differs from it n 60 only by n seconds is d: now, this is not true for arcs nearly equal to 90°; and an example will most simply shew it: by Sherwin's Tables, by expanding and neglect-}=(tan. 1"+tan. A. tan. 1") (1+tan. A tan. 1") ing terms involving tan.2 1" = tan. 1" (1 + tan.2 A) nearly; since tan. 1" is an assigned quantity, tan. (A+1") - tan. A x 1 + tan. A sec. A. log. tan. 89° 30′ = 12.0591416 log. tan. 89 29 = 12.0449004....[2] log. diff. corresponding to 60" = 142412 diff. corresponding to 30"= 71206 ........ [4] ... by Rule, log. tan. 89° 29′ 30′′ ([2] + [4]) = 12.0520210 whereas true log. tan. 89° 29' 30', by Taylor's Log. = 12.0519626 Again, log. tan. 89° 50′ = 12.5362727 = 413930 .. by the Rule log. tan. 89° 49' 6" = 12.4990190 whereas the true log. tan. 89° 49' 6", by Taylor's Logms.=12.498845 In these instances, the log. tangent, determined by the proportional parts, is too large, which it plainly must be; for, the logarithmic increment of the tangent increasing as the arc does, that is, the increment during the last 30" being greater than the increment during the first 30", if we take half the whole increment for the increment due to the first 30", or onetenth of the whole increment, for the increment due to the first 6", we plainly take quantities too large. The same reason would, it is true, hold against calculating logarithmic tangents of any arcs by proportional parts, if the values of logarithmic tangents were exactly put down in Tables; but (we speak of the Tables in ordinary use) the values are expressed by seven places only of figures; and, as far as seven places, the irregularities in the successive differences of the logarithmic tangents of arcs that are of some mean value, between 0 and 90°, do not appear; thus, by Sherwin's Tables, log. tan. 44° 30′ = 9.9924197 log. diff. corresponding to 60" = 2527 ... by the Rule log. tan. 44° 29′ 30′′ = 9.99229335 and the true log. tan. by Taylor's Tables = 9.9922934. It appears then, from the assigned reason, and by the instances given, that an angle nearly 90o cannot exactly be found from its logarithmic tangent. The determination of the angle by means of proportional parts will be wrong in seconds by Sherwin's Tables; and will be wrong in the parts of seconds by Taylor's Tables. From the whole of what has been said then, it appears that in computing the values of angles, two inconveniences may occur, either when the successive logarithmic numbers are too nearly alike, as in the case of sines of angles nearly 90°, or too widely different, as in the case of the tangents of angles nearly equal to 90°. It is the business of the Analyst to provide formulæ, by which these inconveniences may be remedied or avoided; and hence have arisen the different methods for attaining, apparently, the same end. Before we entirely relinquish this digression, we wish to observe, that, although the log. sine or log. tangent of the angle A may be determined exactly either by the first or the fourth method, yet, if it should be very small, its value cannot, with sufficient exactness, be determined by the Tables in common use. For, very small angles cannot be exactly found from their logarithmic sines and tangents; not exactly in seconds, by Sherwin's Tables, nor exactly, in parts of seconds, by Taylor's Tables; and therefore, as great exactness may be required, and is commonly required, in those cases, in which a very small angle is to be determined, the Tables are not to be used. They are to be superseded by a peculiar computation, of which, without demonstration, Dr. Maskelyne has given the rule in his Introduction to Taylor's Logarithms, p. 17 and 22. This rule and similar rules will be stated and demonstrated in a subsequent part of this Work, when the analytical series for the sine and tangent of an arc are deduced. To the several cases of the solution of oblique triangles, examples have been given, but, merely arithmetical examples; it may proper therefore, to subjoin a feigned case of practice and observation, in order to shew, more plainly, the use and application of the formule of solution. An observer at A wishes to determine his distance from two inaccessible objects B, C, and also the distance BC, of the same objects. |