This expression will terminate when the sign of the square root has been separated m which the polygon 0; since by definition sin. 90° = unity, the circumference of (66.) It is obvious that the greater the number of sides the more nearly will the circumference of the polygon approach to that of the circle, and by this means we might obtain an approximate value of the ratio (denoted by the symbol ) which the circumference of a circle bears to its diameter. For suppose we find, by performing the operations indicated in the above expression, that the numerical value of the circumference of a polygon of s sides, and of that of another of 2s sides, agree as far as a certain number of decimal places; then it will easily be understood that the value of the circumference of a polygon of any greater number of sides will also coincide with that in which there are s sides, to the same number of decimal places; and consequently the same may be said of the circumference of the circle, to which, by constantly increasing the number of sides of the polygon as above-mentioned, the circumference of the polygon constantly approximates as its limit. It was, in fact, by the laborious method of actually extracting the square roots as indicated in the above expression that the earlier mathematicians calculated the value of T. About the year 1600, Ludolph van Collen or à Ceulen, a Dutch mathematician, by means of the same formula, and certain artifices by which he abridged the numerical computation, determined the value of to be between 3.14159 26535 39793 23846 26433 83279 50238, and By means of certain series which will be given hereafter, this computation can be performed with great comparative facility. SECTION V. Numerical Values of Trigonometrical Functions of Angles whose Magnitude is given numerically.-Construction and Arrangement of Tables of Natural Sines, Cosines, &c. Examples of the Use of these Tables. THE formulæ of the preceding sections express the general relations existing between the quantities involved in them, without reference to their numerical values. In the practical applications, however, of mathematics, the determination of numerical values forms, for the most part, the ultimate object of our investigations; and therefore it becomes essential that we should be able to assign such values to the different trigonometrical functions of an angle whose magnitude is numerically given. To prevent the labour, however, of making these numerical computations for each particular problem, which (particularly in astronomical calculations) would be immense, tables have been formed in which these values are tabulated, for every angle ascending in arithmetical progression from 0 to 90°, the common difference being some small quantity, usually one minute or one second. Before we proceed farther with the solutions of triangles, we shall explain the methods of computing and arranging these tables of natural sines, cosines, &c. Possessed as we now are of such tables, computed with great accuracy, the detail of the methods of calculating them might, at first sight, appear superfluous; they form, however, a useful exercise for the student, and, moreover, it may sometimes become necessary, in investigations of a delicate nature, to obtain the values of the trigonometrical functions to a greater degree of accuracy than that afforded by the tables in common in which they are given to seven or eight places of decimals. We shall now, therefore, proceed to explain these methods. The most important is that by which the values of the sines of successive angles are computed, by means of the values of the sines of those immediately preceding; but there are certain angles for which these values can be computed independently, without ascending to them by these successive steps, and to these we shall first direct our attention. use, (67.) The numerical values contained in the following table result immediately from our definitions of the respective trigonometrical functions ; and it is also shown for what values of the angle these values are positive or negative. The symbol o denotes a quantity whose value is infinitely great. See Algebra, 116, 117. And the perpendicular M P on A C bisects A C in M = 2 sin. 18° cos. 18. Art. (43.) Hence substituting and dividing by cos. 18°, we have, 4 (cos. 18°) - 3 2 sin. 18° = 4 { (1 · (sin. 18°)}-3 14 (sin. 18°) (72.) Thus we have, by independent methods, the sines and cosines of 18°, 30°, 45°, 60°, 72°, and 90°. Also, we have, Art. (44.), if A be less than 45°, By the same formula the sine of 9° may be calculated; and sin. 3° sin. (18° 15°) sin. 18° cos. 15° cos. 18°, sin. 15°. Whence sin. 3° is known; and in a similar manner we obtain the sines of every angle in the series 3°, 69, 9°, 12°, &c.,...90°. The following table exhibits these values: |