Co-fine of 3A (= co-fine of 2A xy-co-fine of A= . Co-fine of 4A (= co-fine of 3A xy-co-fine of 2A= 34 — 3y2 y2 — y* — 4y2+2; = 2 whence, univerfally, the co-fine of the multiple-arch A will be truly represented by ny 12 n + 2 x2-6x=7x ; &c. which feries, as well as Χ 3 4 X that for the fine, is to be continued till the indices of y become nothing, or negative. But, if you wonld have the fine expreffed in terms of x only, then, because the fquare of the fine + the fquare of the co-fine is always equal to the fquare of the radius, and therefore, in this cafe, x+y=1, it is manifeft that the fines of all the odd multiples of the given arch A, wherein only the even powers of y enter, may be exhibited in terms of only, without furd quantities fo that 44x2 being fubftituted for its equaly, in the fines of the aforementioned arches, we fhall have. : ft. Sine of 3A = 3x 2d. Sine of 5A=5x- 20x3+ 16x5; 3d. Sine of 7A=7x56x3+ 112x3—64x7; 4th. Sine of 9A9x—120x3+432x3—576x2+256x9; &c. And, generally, if the multiple-arch be denoted by nA, then the fine thereof will be truly represented by n 122 I nx-- X. 2.3 I 2 n2 9 n 25 5.6 2.3 3.4 2.3 4.5 9 From From this feries the fine of the fub-multiple of any arch, where the number of parts is odd, may also be found, fuppofing (s) the fine of the whole arch to be given: for let x be the required fine of the fub-multiple, and ʼn the number of equal parts into which the whole arch is divided; then, by what has been already fhewn, × ×3 &c. =s: from the solution of which equa 4.5 tion the value of x will be known. Hence alfo, we have an equation for finding the fide of a regular polygon infcribed in a circle: for feeing the fine of any arch is equal to half the chord of double that arch, let v and be wrote above for x and s respectively, and n n2. I 23 then our equation will become ทบ X 8. 3 4.5 32 23 n n2. 2.3 4 &c. w, or nv n2 = 2 x 2 &c. = w; ex 2.3 4.5 16 preffing the relation of chords, whofe correfponding arches are in the ratio of 1 to n. But, when the greater of the two arches becomes equal to the whole periphery, its chord (w) will be nothing, and then the equation, by dividing the whole by nv, will be reduced to 4.5 n'. 9 n2-25 5.6 64 &c.o; where n is the num ber of fides, and v the fide of the polygon. From the foregoing feries, that given by Sir Isaac Newton, in Phil. Tranf. mentioned in p. 242 of this Treatife, may also be eafily derived. For, if the arch A and its fine x be taken indefinitely small, they will be to one another in the ratio of equality, indefinitely near, by what what has been proved at p. 246; in which cafe, the general expreffion, by writing A instead of x, will become n2 - 25 × A2 &c. I 2.3 4.5 6.7 Therefore, if n be now supposed indefinitely great, fo that the multiple-arch nA may be equal to any given arch z, the fquares of the odd numbers, 1, 3, 5, &c. in the factors n' · 1, n2—9, n2— 25, &c. may be rejected as nothing, or inconfiderable, in refpect of n2; and then the foregoing feries will become nA n3A3 n'A\ &c. wherein, if for nA, its equal 2.3.4.5.6.7 z, be substituted, we fhall then have z — 2.3.4.5.6.7 &c. which is the fine 2.3.4.5 Moreover the aforegoing general expreffions may be applied, with advantage, in the folution of cubic, and certain other higher equations, included in this form, viz. z”—az”—2 + n -3 2n 3n n-4 n-5 xn−5 x whence, as it is proved above, that the former part of the equation (and therefore its equal) reprefents the cofine of n times the arch whose co-fine is y, we have the following rule. Find, from the tables, the arch whose natural co-fine is (or its log. co-fine = log. žƒ• log.f-log.) the radius R being unity; take the nth part of that arch, and find its a co-fine, which multiply by 2√2, and the product will be n the true value of z, in the propofed equation z"-az"-2 Thus, let it be required to find the value of z, in the cubic equation z3 432% = 1728; then, we fhall haven 3, a=432, and f= 1728; confequently M = 864 f a 2 1442 n thereto -) = ,5, and the arch correfponding 60°; whence the co-fine of (20°) thereof will be found,9396926; and this, multiplied by 24 (= 24 2) gives 22,55262 for one value of z. n But befides this, the equation has two other roots, both of which may be found after the very fame manner: for, fince 0,5 is not only the co-fine of 60°, but also of 60° + 360°, and 60o +2 × 360°, let the co-fine of (140°) of the former of these arches be now taken; which is ,7660444, and must be expreffed with a negative fign, because the arch corresponding is greater than one right angle, and lefs than three. Then, the value thus thus found being, in like manner, multiplied by 24 ( = 2√2), we shall thence get ( = n 18,38506 for an other of the roots: whence the third, or remaining root will also be known; for, feeing the equation wants the fecond term, the pofitive and negative roots do here mutually destroy each other; and therefore the remaining root must be 4,16756, the difference of the two former, witth a negative fign. From a given circle ABCH it is proposed to cut off a fegment. ABC, fuch, that à right line DE drawn from the middle of the chord, AC, to make a given angle therewith, Shall divide the arch BC of the femi-fegment into two equal parts BE and EC. F B E Let the chord BC be drawn, and upon the diameter HDB let fall the perpendicular EF: put the radius OB of the circle = r, and the tangent of the given angle CDE (anfwering to that radius)t, and let OF =%; then will EFVrrez, and BC (= 2EF)=2√rr— zz, and confequently BD (= BC2 4x2-422 -) = A D C H BH which taking BF=r-z, we have DF = But, by trigonometry, EF: DF:: rad. : tang. DEF, 1 |