gree, and one of the second. For, in this case, we can only decompose the ten imaginary roots into five periods of two terms each, or into two periods of five terms each; and, in the first instance, it is obvious, that in order to get the sum of all the p's, as p+p′+p" +p"", &c., the sum of the product of every two, of every three, &c., the equation must necessarily rise to the fifth degree. And if, according to the other division, the roots were resolved into two periods of five terms each, though the values of these periods would be found by a quadratic, yet this would be of no use, as we should thus have only the value of the sum of five of the roots, whereas it appears, from cor. 2, art. 117, that it is only by knowing the sum of two roots the solution of the equation can be determined rationally. It is necessary, therefore, in all cases, to manage the subdivisions so, that the final equation may be a quadratic; that is, so that the number of terms in each period, in the last instance, shall not exceed two; which may always be done, because n being a prime number, n-1 is always even; and thus, when n-1 is any power of 2 (as we can then at every step divide each period into two others), it follows, that the solution of such an equation may always be effected by means of quadratic equations only: and, consequently, a polygon of such a number of sides may be inscribed geometrically in a circle. Now, 5, 17, 257, 65537, are prime numbers of this form, and therefore each of these admits of a geometrical construction. We know also, from other principles, that if any two polygons of an unequal number of sides, prime to each other, can be in scribed geometrically in a circle, that the polygon, the number of sides of which is equal to the product of these two, can also be inscribed geometrically. For let a and b represent the number of sides of two polygons, each inscribable in a circle, a and b being prime to each other; then it is to be demonstrated, that the polygon, the number of whose sides is equal to ab, is also inscribable. Now the angles at the centre of these polygons, will be and if it can be shown that the difference of any multiples of the two first, can be made equal to the third, the truth of what is advanced will be evident. 360°n Let, then, and 2 a 360°m , represent any multiples of the angles of the two first, then the dif 360°(nb — ma) = 360°, or nb-ma= 1 ; and, since a and b are prime to each other, such values of m and n may be found, <a and b, as will answer this condition; and, consequently, the third polygon, of which the number of sides is ab, may be inscribed by means of the two first. Also all polygons, of which the number of sides is any power of 2, may be inscribed by continual bisections: and again, all those whose number of sides is equal to the product of any inscribable polygon, into any power of 2. And hence we have 507 Table containing the least Values of p and q in the Equa- 2 p 13865 13 23 THE FOLLOWING WORKS OF JOHN BONNYCASTLE, ESQ. PROFESSOR OF MATHEMATICS AT THE ROYAL MILITARY ACADEMY, WOOLWICH, 1. AN INTRODUCTION TO ARITHMETIC; or, a complete Exercise Book, for the Use both of Teachers and Students being the first Part of a general Course of Mathematics, with Notes, containing the Reason of every Rule, demonstrated from the most simple and evident Principles: together with some of the most useful Properties of Numbers, and such other Particulars as are calculated to elucidate the more abstruse and interesting Parts of the Science. Price 8s. boards. 2. THE SCHOLAR'S GUIDE TO ARITHMETIC; or, a complete Exercise Book, for the Use of Schools; with Notes, containing the Reason of every Rule, demonstrated from the most simple and evident Principles: together with some of the most useful Properties of Numbers, and general Theorems for the more extensive Use of the Science. Ninth edition, 1808. 12mo. 2s. 6d. bound. 3. 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