cal purposes, and the others, when required, are readily expressed in terms of them. We proceed to explain a method for computing a table of natural sines and cosines. It was shown, in Book V, that the linear value of the arc 180°, in a circle whose radius is unity, is 3.141592653. This divided by 180 x 60, the number of minutes in 180°, will give the length of one minute of arc, which is .00029088820867. But there can be no sensible difference between the length of the arc 1' and its sine; and, within narrow limits, that sine will increase directly with the arc. Beyond this, the error which would arise from taking the arc for its sine, upon which the above proceeds, would affect the final decimal figures; and we must, therefore, continue the computation of the series by other processes. To find the values of the cosines of arcs, from 1' to 10', we have cos. √1- sin. = = 1 - sin., nearly. That is, when the sines are very small fractions, as is the case for all arcs below 10', we can find the cosine by subtracting one half of the square of the sine from unity. The natural sines of arcs, differing by 1', from 10' up to 1o, may be computed from those of arcs less than 10', by means of equation (11), group B, which is sin. (a + b) = 2sin. a cos. b sin. (a And when a = b, this equation becomes sin. 2a = 2sin.a cos.a. Eq. (30). b); To find the sine of 11', we make a = 6', and b = 5'; sin. 11' = 2sin. 6' cos. 5'- sin. 1'= .00319976913. then sin. 3° 2sin. 2o cos. 1° - sin. 1o, etc., etc., etc. This process may be continued until we have found the sines and cosines of all arcs differing by 1', from 0 to 90°, the values of the cosines being deduced successively from those of the sines by means of the formula, In this calculation, we began by assuming that, for small arcs, the sines and the arcs were sensibly equal. It must be remembered that this is but an approximation; and although the error in the early stages of the process is not sufficient to affect any of the decimal figures which enter the tables, it will finally become so, since it is constantly increased in the operations by which the sines and cosines of the larger arcs are deduced from those of the smaller. When the error has been thus increased until it reaches the order of the last decimal unit of the table, which assigns our limit of error, we must have the means of detecting and correcting it. This consists in calculating the sines and cosines of certain arcs by independent processes, and comparing them with those found by the above method. We have seen, for example, (Prop. 7, B. V), that the chord of The following elegant method of deducing, from the sine of an arc, the sine and cosine of one half the arc, is given, assuming that the student is familiar with the simple algebraic principles upon which it depends. Let us take the natural sine of 18°, which is .3090170, and make x = sine, and y the cosine of 9° = 18° .3090170 (2); Eq. (30). x2 + 2xy + y2 = 1.3090170; Subtracting (4) from (3), 2y = 1.975377 hence, y = cos.9° = .9876885 Now, by making x = the sine of 4° 30', and y = cosine of 4° 30′, and as before we obtain the sine and cosine of 4° 30'; and another operation will give the sine and cosine 2o 15', etc., etc. We may in this manner compute the sines and cosines of all arcs resulting from the division of 18o by 2, and we may make their values accurate to any assigned decimal figure. This has been carried far enough to show how a table of natural sines, etc., could be computed; but in consequence of the tedious numerical operations which the process requires, other methods are resorted to in the actual construction of the table. The Calculus furnishes formule giving the values of the sines and cosines of arcs developed into rapidly converging series, and from these the sines and cosines of all arcs from 0° to 90°, can be determined with great accuracy and with comparatively little labor. In the last two columns on each page of Table II, will be found the values thus computed of the sines and cosines of every degree and minute of a quadrant. * When an arc is less than 45°, the cosine exceeds the sine; and when the arc is between 45° and 90°, the sine exceeds the cosine. Hence, when the arc is 9°, y, its cosine, exceeds x, its sine; and we therefore placed the minus sign before the second member of Eq. (4). |