SECTION III. CONSTRUCTION OF TRIGONOMETRICAL TABLES. Natural sines and cosines for every ten seconds of the quadrant. 58. In order that the replacing of the angles by their Trigonometrical Ratios may be attended with real utility, it is requisite that when the angle is assigned we should know the numerical values of its Trigonometrical Ratios, and con- . versely. The best way of attaining this object is to form Tables, in which the values of the Trigonometrical Ratios are registered side by side with the angles to which they correspond. We must therefore now shew how to calculate the sines, consines, &c., of all angles between zero and 90°, for every 10", that being the interval at which the angles succeed one another in the best tables; and it must not be thought that unnecessary accuracy is here studied, for in the present state of Astronomical Science, an error amounting to a small fraction of a second is often of serious importance. But we must first establish the truth of the following Propositions. 59. The circular measure of an angle between zero and a right angle, is greater than its sine and less than its tangent. Let BAC, B'AC (fig. 17.) be two equal angles, each less than 90°, with circular measure 0. From any point C in AC draw CB, CB' perpendiculars to AB, AB'; join BB' cutting AC in N, and with centre A and radius AB describe a circular arc cutting AC in D, and which will manifestly pass through B'. Then arc BDB' is greater than BB'; .. BD> BN, and BD BN or 0 > sin 8. Also, admitting the principle that the boundary of a convex curvilinear figure entirely contained within another is less than that of the containing figure, 60. As the angle whose circular measure is 9 is continually approaches to unity, and has unity for its ultimate value. For since lies between sin 0 and tan 0, nished, is 1; therefore, à fortiori, the ultimate value of sin Ꮎ 61. If be the circular measure of an angle between zero .. à fortiori, sin 0 >0 This result will be useful to us in estimating the degree of approximation in the next Article. 62. To find the sine of 10". Let e denote the circular measure of an angle containing 10"; then since the circular measure of 180° is π = 3.1415926535, and the ratio of two magnitudes is the same whatever be the unit in which they are expressed, the above value, it is found that these limits coincide in the first twelve places of decimals; therefore to twelve places of decimals, and cos 10" is found by substituting this value in the formula cos 10" = √1 sin3 10′′. Hence if 0 be the circular measure of an angle containing n seconds, then 0 = n sin 1"; for if h be the circular measure of 1", then ◊ = nh; but .. à fortiori, sin 10′′> 0 - (·00005)o, or sin 10">00004 84813 68110 or sin 10"> 00004 84813 68078, also sin 10" < 0 <00004 84813 68110, erefore the value of sin 10", correct to 12 places of decimals, is sin 10" 00004 84813 68. Substituting this value of sin 10" in the formula cos 10" VI- sin' 10", we have h = sin 1" exact to at least 12 places of decimals; therefore with equal exactness, 0 = n sin 1". 63. The sine and cosine of 10" being known, the sines and cosines of all angles between 0 and 90°, from 10′′ to 10′′, may be computed. Making A = n.10", B = 10" in the formula we find sin (4 + B) = 2 sin A cos B – sin (A – B), sin (n + 1) 10′′ = 2 sin n 10′′. cos 10′′ – sin (n − 1) 10′′; now 2 cos 10" differs from 2 by a very small quantity k, putting, therefore, 2 − k for 2 cos 10", and transposing, we get sin (n + 1) 10′′ – sin n 10′′ = sin n 10′′- sin (n−1) 10′′ – k sin n 10"; hence making n = 1, 2, &c. sin 20′′, sin 30", &c. become successively known; and in general the difference of the sines of consecutive angles (n + 1) 10" and n 10" will be obtained by diminishing the difference of the sines of the preceding angles n 10′′ and (n − 1) 10′′, which is already calculated, by k sin n 10"; so that for each new sine, the only laborious operation will be to multiply the last obtained sine by k. It is necessary to take sin 10′′ and cos 10" with a great many more decimal places than we mean eventually to preserve, in order that the accumulated error, in the long series of operations for forming a table of sines, may have no influence on that order of decimals which we wish to have accurate in the last results. 64. Having computed the sines of angles ascending by intervals of 10", from 0° to 60°, the sines of angles ascending by the same interval from 60° to 90° may be found from the formula sin (60o + 4) – sin (60o – 4) = 2 cos 60o sin A = sin ▲, since cos 60° = 4, or sin (60o + 4) = sin A + sin (60° A+ - A), by putting 4 = 10′′, 20′′, &c. successively. • The numerical value of k is 00000 00023 504. 65. The sines of all angles from 0° to 90° being computed, no calculation is requisite for the cosines; for sin 10" is the same thing as cos 89°.59′.50′′, sin 20" the same as cos 89°.59′.40", and so on; so that a complete table of sines is also a complete table of cosines. 66. The tangents and secants are of course known from the sines and cosines; and complete tables of tangents and secants are also complete tables, respectively, of cotangents and cosecants, just as in the case of the sine and cosine. When the tangents of all angles as far as 45° are computed, the tangents of angles between 45° and 90° may be found from the formula (Art. 56.) tan (45o + 4) = tan (45° – A) + 2 tan 24, by putting A = 10", 20", &c. successively. Also, since (Art. 56.) the tables of tangents and cotangents will give, by simple addition, the cosecants and secants of angles which are even multiples of 10". 67. When an angle, besides degrees and minutes, contains a number of seconds not a multiple of 10, its sine, cosine, &c. are not found exactly in the Tables; they may however be deduced from those of the angle nearest to it, as will be shewn in a future Article, on the principle that the increments of the sines, cosines, &c. of angles, are proportional to the increments of the angles; the calculations being precisely similar to those employed in treating of the logarithms of numbers. As this is a point which will be fully illustrated when we come to speak of the logarithmic sines, cosines, &c. of angles, which are of far greater practical importance than the natural sines, cosines, &c. it is unnecessary to say more upon it here. 68. The Tables never go beyond 45°; for angles greater than 45o, the sines, tangents, and secants are determined by the |