n+1 cos y+x-xx cos y+x x cos y+nzxx cos y + (n+1)z 1—2cx + x2 = cos y +z+xx cos y +2z+x2 × cos y + zz+x3× cos y +42 n-1 + ** × cos y + 52 &c. + x ·x cos y + nz. (7). Take x = 1, and the last expression will become cos y + zcos y + cos y + nz cos y + (n + 1)% cos y + z-cos y+cos y +nz→ cos y + (n + 1)z cos y +2+cos y +2x+cos y+32+cos y+42+ cos y+5z &c. + cos y + não Now, by Lemma 4, cos y—cos y+2=2 sin× sin y+2=; and cos y+nz—cos y +(n+1)z=2 sin2 × sin y +(2n+1) Also, by Lemma 5, 1 — cos z =2(sin)', and therefore cos y +2. cos y + cos y + nz · - cos y + (n + 1)z +cos y + zz + cos y + 4z &c. + cos y + zz. But, by Lemma 3, sin y + (2n + 1) − sin y+ = 2 sin n =cos y+2+cos y +az+cos y + zz + cos y + 42&c,+cos y + nz. By dividing the expression for the sum of the sines, obtained in Art. 4, by the expression for the sum of the cosines just. deduced, we shall have the following remarkable formula or theorem ; If the following concise investigation, and the subsequent remarks, shall be so fortunate as to induce a more correct estimate of the utility of these valuable theorems, than seems at present to prevail, even among respectable mathematicians, the wish of the writer will be amply satisfied. represent any equation whatever, finite or infinite; and suppose that, on making x R+%, the equation transforms to n.n-1. n-1 n-1.n-2.n-3 1.2.3 1.2 • 1 db PR +..= =)D'OR 3dR and so of the rest; agreeably to TAYLOR's theorem. [3] da 2dR 3 Now, if R be so nearly equal to the correct value of x, that the difference z bears but a small ratio to R, the numerical value of A, it assumed to be az, will generally agree, to a certain number of figures, with its actual value in equation 2. To that extent, therefore we shall have which is the symbol of NEWTON's and RAPHSON's methods, as well as of the second case of BARLOW's method, first published in No. 12 of this Repository, and since that time in the 8vo edition of Bonnycastle's Algebra. The effect of equation 4 may very commodiously be augmented by substituting it in the result of the more correct equation az + bz', which thus produces which symbolize the other case of BARLOW's method, or that which he directs us to employ when R 1. The more terms of the equation that are deficient, the more nearly will the correction thus obtained agree with that deduced from HALLEY's method, with which this is accurately coincident in the case of pure powers. In other cases since it neglects the same quantities cz' &c. as Halley's, but on the other hand adopts a quantity, n-3 2.n-2 N4 + PR + 2 2 3.n-3 N-5 N-1.1 + 3 which the other does not, it will sometimes be the more accurate method. At the same time, it will be in general much the more convenient, in regard of the smaller numbers which it employs. Adding R, this formula gives (R+ z =) x= 2aR + (n + 1) A 2aR + (n-1) A X R ...... [7] But, if az + bz' be treated as a quadratic, and we which is the favorite irrational method of HALLEY. To improve its efficacy, call the value thus found r, and, taking in two additional terms of equation 2, we shall have This amendment, which usually quintuples the original number of true places, is due to the same eminent mathematician, who also remarks, that, by successively substituting in these formulae the corrected values of and r, the advantage may be carried to any extent, without finding new values for a, b, c, &c. Tho' it is perhaps a digression from the general train of this investigation, yet it will make the syllabus of theorems more complete if we observe here, that when ox is simply x" N, the root of a pure power. From eq. 7, or by adding R to the preceding expression, and reducing, the well-known theorem of HUTTON, for the same purpose. From eq. 8, after the proper reductions, expressed. Each of these theorems admits of various enunciations, by combining any of the symbols, N, R, A, (=N + R*). But, to return and conclude, the accurate value of z is and, if the approximate value r be here introduced in the place of z, it gives z= $ (R + r) — OR which exhibits the method of Double Position. ..[14] Concerning these epitomes, of all the approximating rules which are adopted for general use, the following observations naturally occur : 1. Whenever Newton's method fails, in consequence of an ambiguous or inadequate assumption for R, Halley's rational method must also fail; since it is only a corollary to the former. The same may be said of more complicated theorems, such as those in Simpson's Tracts and Algebra, which proceed on the supposition that— is a distinct approximation to the value of z. 2. The coefficients arising from substituting R+z for the unknown quantity a in any proposed equation, are respectively equal to the quantities arising from substituting R for Ꮖ. either in Newton's limiting equations, or in the differential coefficients (so called) which are derived from the given expression in x by taking the successive fluxions, making at each step = 1, and dividing by the exponent of the step. For equation 3 shews these three views to be identical. In practice, the first mode of obtaining these coefficients will be naturally adopted in finite equations, when R, a good conjectural value of x, is known; the second may be necessary, when R is ambiguous; the third, in case of surds and transcendental expressions. 3. Hence it follows that, excepting of course, the rules for the roots of pure powers, none of the methods above investigated are restricted to equations of any particular class; but their operation applies to transcendental and irrational quantities, as well as to such as are finite. It does not however appear, that any one of the inventors was aware of the universal influence of his principles; and, |