differentials of those quantities. If, therefore, we put z=the arc am, om will be represented by dz; and if AT=t, pm=x, AP=V, and CT=s, then Tt will be represented by dt, no by dx, vp(=nm) by dv, and st by ds; hence by the preceding art. rdx rdx I. dz= II. dz III. dz= IV. dz= (270) If these formulæ be expanded, and the integrals of each term be taken, we shall obtain the common series for the arc, in terms of its sine, tangent, &c. I. Thus dz =rdx 1 x2 3x1 3.5 x + 5 + 2p3 2.4 75' 2.4.6 r7 x2 dx 3x dx + 3.5 x dx 3.5.7 x dx + + 2.4.6 76 & 2.4.6.8 78 23 3x5 3.5.7 x9 2.4.6.8.9 89 25 + &c. and by reverting the series t=z + + t3 t5 + 372 5r 23 225 + 72 t7 77.6' &c. 1727 + 315,6 In a similar manner the rest may be found. r2x2-cos z; rad-cos z=vers z; 2.3.4.5.6.7.8.9 7.89 The same series will apply to the chord of any arc by substituting the diameter d, or 2r instead of r.* (271) If 90°-A be substituted for A, then we shall obtain sin3 (90°-A) 3 sin3 (90°-A), &c.; + 90°-A sin (90° — A) + 2.3 r2 5 2.4.5 4 3.5 cos7 A &c.; or 2.3 r2 2.4.5 4 2.4.6.7 6 + (77 — cos? A) &c. A2 COS Ar- + &c. 2.3.4.5.6 752.3.4.5.6.7 77 the 24th part of the cube of the length of the arc nearly. See Note, Chap. IV. (154.). † Emerson's Trigonometry, 2d edit. page 32. &c. (273) A = 90° &c. sec Art + + A2 5A+ 61A6 277A8 50521A1o 2r 24r 7205 8064r7 362880079 + 3.5rs + + (275) There are various methods of constructing a table of sines, but the following, though not the least laborious, is the most simple. These sines are found thus. From the square of the radius subtract the square of the sine, the square root of the difference is the cosine. The radius diminished by the cosine leaves the versed sine; and ✓vers2+sin2=chord, the half of which is the sine of the next arc, &c. From the sine of 7° 30′, find the sine of 3° 45', and so on continually, till the sines are as the arcs, which will be found at the end of the twelfth division from 30°: that is, at the end of the twelfth division, the arc will be 52" 44"" 3"" 45""", and K its sine 0002556634; but at the end of the 11th division the arc is l′ 45′′ 28′′"' 7'''' 30''''', and its sine 0005113269; therefore the sine of the 12th division is just half the sine of the eleventh, in the same manner as the twelfth arc is half the eleventh. Now in indefinitely small arcs, the arcs will be to each other as their corresponding sines, hence 52′′ 44′′" 3"""' 45''''' : 1':: ·0002556634 : 0002908882 the sine of one minute. The cosine of 1'=the square root of the radius 1, diminished by the square of the sine of l', viz. cosine l'=9999999577. And it is shown (261) that 2 cos l'. sin l'—sin 0'0005817764-sin 2', or cos 89° 58' 2 cos 1'. sin 2′ — sin l'='0008726645=sin 3′, or cos 89° 57′ 2 cos l'. sin 3′- sin 2′-·0011635526=sin 4', or cos 89° 56′ 2 cos l'. sin 4' sin 3'0014544406=sin 5', &c. &c. This method may likewise be extended to the cosines by means of the formulæ in art. (262). 2 cos l'. cos l'. 2 cos 1'. cos 2' 2 cos l'. cos 3' 2 cos l'. cos 4' cos 0′9999998308=cos 2′, or sin 89° 58′ cos l'=9999996192=cos 3', or sin 89° 57' cos 2′9999993231=cos 4', or sin 89° 56′ cos 3'9999989423=cos 5', or sin 89° 55′ Proceed thus to find the sine and cosine of every minute of the arc as far as 30°. sin A. COS B (276) Let A and B be any two arcs, then (245). rad sin (A+B) + sin (A-B); let A=30°, then sin a= rad, and cos B = sin (30° +B) + sin (30°—B); hence sin (30°+B)=COS B-sin (30°—B). If, therefore, B=1', 2', 3', 4', &c. successively, we shall have sin 30° 1' cos l' sin 29° 59' sin 30° 2′-cos 2′-sin 29° 58′ sin 30° 3′ =cos 3′-sin 29° 57′, &c. And in this manner find all the sines, and thence all the cosines, as far as 45°. By these means all the sines and cosines from 0° to 90° will be obtained; for sin (45° +A)=cos (45°-A), and cos (45°+A)=sin (45°-A). (277) The sines and cosines being constructed, we have by (209) cos: sin:: rad: tan; hence the tangents are found. tan: rad :: rad: cot; hence the cotangents are found. cos: rad: : rad: sec; hence the secants are found. sin rad :: rad: cosec; hence the cosecants are found. : The versed sines are found by subtracting the cosine from radius for an arc less than a quadrant, and adding the cosine to the radius for an arc greater than a quadrant. (278) ARTIFICIAL, or LOGARITHMIC SINES, &c. are only the logarithms of the natural sines, &c. The natural sines are generally calculated to the radius 1, and being in all cases, except when the arc is 90°, less than radius or 1, they must of course be decimals. Now the logarithm of a whole number and the logarithm of a decimal is the same, only the index or whole number prefixed to the former is affirmative, and that to the latter negative. Hence if logarithmic sines, &c. were immediately formed from natural sines which are calculated to the radius 1, their indices would all be negative. To avoid this, the logarithmic radius instead of being taken 1, as in the natural sines, is generally considered as ten thousand millions. (279) To find the logarithmic sine of 1' to 7 places of figures, without the index. rad=1: rad=10,000,000,000 :: 0002908882 (the natural sine of l', the radius being 1): 2908882 (the natural sine of l', the radius being 10,000,000,000.) The logarithm of 2908882 =64637261 the logarithmic sine of 1'. The indices to the logarithmic sines from 0' to 4', will be 6; from 4' to 35' the indices will be 7; from 35' to 5° 45′ the indices will be 8; from thence upwards to 89° they will be 9; and at 90° the index will be 10 the logarithm of the radius 10,000,000,000. = (280) Hence if we take the natural sine of any arc from a table of natural sines (where the radius is unity), and multiply it by 10,000,000,000; the logarithm of the product will give the logarithmic sine of that arc to as many places of figures as the natural sines are carried to. It It may be proper to inform the learner, that this method will not be exactly true for the first five degrees; because the natural sines in the tables are not carried to a sufficient number of places. (281) In logarithms the operation of multiplication is performed by addition, and division by subtraction. The logarithmic sines being constructed, the tangents, &c. are formed from art. (209) thus: If you double the logarithmic sine of half an arc, and subtract 9.6989700 from the product, the remainder will be the logarithmic versed sine of that arc. |