A, B, C, &c. 1, A', B', C', &c. and 1, A', B', C', &c. denote the multiples corresponding to the arcs n.a, n + 1.a, and n-1.a; then A+1=A', B+ A'=B', C+B'C', &c. Whence the values of A, B, C, &c. are determined, either by the method of finite differences, adopting the appropriate notation, or from the theory of functions. Thus in the first table, AA=1, and A=n-2; AB=A'=n-3, and B n-3.n-4 ; AC = B'= 2 Wherefore in general (1.) Sin na=2n-1.cn-1s-n-2.2-3cm-3s+ 1 (2.) Cos na 2-1.cnn.gn-3.cn-2+ The third and fourth tables are evidently formed by multiplying constantly by 2cos 2a or 2-4s2, and subtracting the term preceding; or the multiplication by 4s2 produces the second differences of the successive quantities. Hence in the former, AAA=4n", ΔΔΒ=4A", &c.; wherefore AA=n+1.n+1, and A= 2.3 ; 3.4 and B-n.n-1.n+1.n-3.n+3 But in the fourth table, 2.3.4.5 AAA+4, AAB+4A", AAC=4B"; and consequently AA= 2n+2, and A=2; AB=2(2n+2.n+2)=n.n+In+2 3 and n2 (4.) Cos na=1--82+ n2 n24 n2 n2-4 n2-16 s+, &c. In the fifth and sixth tables, the coefficients are evidently the same as those of the power of a binomial, only proceeding from both extremes to the middle terms. Hence, according as n is odd or even, (5.) 2n-1 sin an sin nan.sin(n-2)an." 1 N-I 2 sin(n-4)a 2n-1 sin an = cos nan.cos(n-2)an. cos(n-4)a (6.) 2n-1 cos an=cos na+n.cos(n-2)a+n..cos(n-4a)+ In these three expressions, half the last term, which corresponds to the middle in the expansion of the binomial, is to be taken, when n is an even number. . It will be satisfactory likewise to subjoin an investigation of the sine of the multiple arc, as derived from the Theory of Functions. It appears from inspecting the successive formation of the sines of the multiple arcs, 1. that the odd powers only of s occur; 2. that the coefficient of the first term is only n, and the other coefficients are its functions of third, fifth, &c. orders; and 3. that since, in the case when n=1, the rest of the coefficients evidently vanish, those coefficients in general, as affected by opposite signs, must in each term produce a mutual balance. /// ///// Let therefore sin na=n.s+n.s3+n.s5 &c.; where s denotes the sine of the arc a, and n, n, n, &c. the successive odd orders of the functions of n. It is evident, from (Prop. 3. cor. 2. Trig.) that, by substitution ((n+1)+(n-1))+((n+1)+(n-1))+((n+1)+(n-1))55 /// ///// + &c. =2√(1-s2) sin na=(2-s2-4s, &c.) (ns+ns3+ns, &c.) =2ns+(2n-n)s3+(2n-n-in)s5, &c. Now, equating corresponding terms, and rejecting the powers ofs, we obtain these general results : will //// ///// /// 2n'=2n'; (n+1)+(n+1)=2n-n; (n+1)+(n-1)=2n-n-n. It remains hence to discover the several orders of the functions of n. 1. The equation 2n'=2n' contains a mere identical proposition; but other considerations indicate that n must always denote the first term, or that the first function of n is n itself. 2. The equation (n+1)+(n-1)=2n-n fixes the conditions of the third function of n, which, from the nature of the relation, is obviously imperfect, and wants the second term. Put therefore, n""=an+ẞn; and, by substitution, 2an3+6an+ 2βη=2an3+2βη-n. Equating now the corresponding terms, and 6a=-1, or a=-; but a+3=0, and therefore β=+ I 이 3. Again, in the third equation, (n+1)+(n-1)=2n-n-n, substitute n=an5+ βn3+yn, and the conditions of the fifth order of the function of n will be determined by this compound expression: 2an5 + (20a + 2β)n3+ (10+6+2y)n = 2an+ (23+)n2+(2--)n. Equate the corresponding terms, and 20x + 28=23+, or a= I I In like manner, 104+6β+2y=2y--, and s=一一一元= -10 2.3.4.5 36 .. From the expression for the sine of a multiple arc, may be deduced the series for the sine of any arc, in terms of the arc A itself, and conversely. Let na=A, and therefore a=; if n be supposed indefinitely great, then a must be indefinitely small, and consequently in a ratio of equality to s. Whence, But n being indefinitely great, the composite fractions n2-9 -&c. , &c. are each in effect equal to unit, which forms their extreme limit. Consequently, assuming that modification, sinA=A_A A5 3 2.3+ 2.3.4.5, &c. Again, putting a= A and s=S, suppose n to be indefinitely small, and sin na=na=nA; whence, by substitution, But, if n vanish from all the terms, the series will pass into By a similar investigation, the series for the cosine of an arc CosA-1 + +. &c. 1.2 2.3.4 2.3.4.5.6 These series' are very commodious for the calculation of sines, since they converge with sufficient rapidity when the arc is not a large portion of the quadrant. Though the method explained in the text is on the whole much simpler, yet as the errors of computation are thereby unavoidably accumulated, it would be proper at intervals to calculate certain of the sines by an independent process. The series' now given furnish also various modes for the rectification of the circle. Thus, assuming an arc equal to the 1 1 radius, its sine is, 1 + &c. .841471, and its 2.3 2.3.4.5 cosine is, 1 1 1 2+2.3.4 -&c. 440302. But that arc evidently approaches to 60°, of which the sine is ✓ =,866025, and the cosine .500000. Wherefore (Pr. 1. Trig.) the sine of the difference of these two arcs is .866025×.540302-.841471 × .500000=.04718, and consequently, by the series, that interval itself is .0472. Hence the length of the arc of 60° is 1.0472, and the circumference of a circle which has unit for its diameter is 3 x 1.0472=3.1416; an approximation extremely convenient. 5. The Fifth Proposition may be otherwise demonstrated from the corollaries at p. 363. Let AB and BC, or BC', be two arcs, of which AB is the greater; make AD, or AD', equal to BC, and apply the respective tangents. Because OAE is a right-angled triangle, and OG', OF, are drawn, making equal angles with OA and OE, it follows, that OA2-AE.AG': OA2:: EG': AF, and consequently R2tan AB. tan BC: R2:: tanAB + tanBC: tan(AB + BC). Again, since OG and OF' make equal angles with OA and OE, it is 1 |