AB, BC, CD, &c. be taken, and if PB be taken = p, p being the coefficient of the second term of the quadratic equation Z2 - p Z + 1 = 0, of which the roots are, a, β; and it has been already proved, that if, (putting AB then 2.cos. 2 (AB), or 2 cos. (AC)=x+ 2 2 for A), 1 22 2. cos. 3 (AB), or 2. cos. AD = x2+, &c. 2 2 1 3 hence, PB being 1 x +=, X 1 ,ora, or, a + β, PC = a2 + β2, PD = a3 + β3, &c. = &c. Vieta, p. 295, Opera Mathematica, Leyden, 1646, expresses the values of these chords, not by the sums of the powers of the roots, but by expressions equivalent to such sums: thus, he puts for PB, 1 N, or N; (N) he represents by 1 Q, (N) by 1 C, (N)5 by 1 QC, &c. then but, N2 - 2, N3 - 3N, &c. express the sums, of the squares, of the cubes, &c. of the roots of an equation & - Nx+1; for, the formula for the sum of the mth powers is Vieta, therefore, is not to be entirely excluded from the honour due to the invention of the preceding theorem. Vieta calculated by chords; and, his formulæ, which we have just given, are, in fact, the same as the expressions for cos. 2 A, cos. 3 A, &c. given in pages 42 and 44. Vieta also has, p. 297, given another form, exhibiting the relations between the chords of AB, AC, AD, &c. He puts the chord of AB = 1 and the relation of the chord of AC to the chord of AB, N, consequently, N= chord AC * See Simpson's Essays, p. 106. 3 ) AB 1 = 2 sin. 3 A, and the chord AE= AB is put = 1, the = A, (2 cos. A)3 - 2 x 2. cos. A = 2. sin. 4 A, &c. which are the same, in fact, as [s"] [s"] given in page 47. We may also employ the above mode of expressing the cosines of multiple arcs, in deducing de Moivre's formula, which is *(cos. A + - 1. sin. A) = cos. mA + - 1. sin. mA, for since 1 1 cos. A=(x+), sin. A=(-22+2 2 1 * Lagrange, p. 116 Calcul des Fonctions, says, that this form is as remarkable for its simplicity and elegance, as it is for its generality and utility: and M. Laplace, in the Leçons des Ecoles Normales, considers the invention of this formula to be of equal importance with that of the Binomial Theorem. If we expand these expressions, and then add them, we shall From the above mode of representing the cosines of multiple arcs we may also deduce, and concisely, the formule of Cotes, page 113, &c.* Theor. Log. Praf. in Harmonia Mensurarum, and of De Moivre, Misc. Analyt. p. 16, &c. thus, * The Theorem of Cotes was not announced to the public by its Author, but by the Editor of his Works, Dr. Robert Smith, who informs us, page 113, Preface, that after various conjectures and trials, he extracted it and its meaning from the deceased Author's loose papers " Revocavi tandem ab interitu Theorema Pulcherrimum." M. Lagrange conjectures, and with probability, that Cotes arrived at his Theorem by the way of Vieta's Theorems. See page 54. In the expression, 2.cos.m 1 then xm + xm and x + == 2 cos. θ, or, xam-2.cos. 0 × x+1=0, Now, from x +2.cos. θ m θ , m therefore, if x were deduced from the first expression in terms of cos, , or, in other words, if a were the root of the equa θ x + 1 = 0, that same value of x, or root a, substituted in the second expression, xTM + 1 xm = 2 cos. 0, would make it a true equation, or a would be a root of the equation Hence, by the doctrine of equation, x - a, is a divisor, both of x2 since θ 2.cos.x+1, and of x2m - 2 cos.0.xm+1; and similarly, one root, x is a divisor both of x2 - 2 cos. 2 cos. 0.2m + 1; and consequently, (x-a) (x-1), or, 2.cos.x + 1 is a divisor of Now, by Table, p. 16, and by the preceding reasoning, it appears that the arcs H |