Elements of Algebra

230 ELEMENTS SECT. IV. CHAP VII. *Of the Extraction of the Roots of Polygonal Numbers.* 656. We have shewn, in a preceding chapter *, how polygonal numbers are to be found; and what we then called a side, is also called a root. If, therefore, we represent the root by $x$, we shall find the following expressions for all polygonal numbers: the IIIgon, or triangle, is $\frac{x^2+x}{2}$, the IVgon, or square, - $x^2$, the Vgon - - - - $\frac{3x^2-x}{2}$, the VIgon - - - - $2x^2-x$, the VIIgon - - - - $\frac{5x^2-3x}{2}$, the VIIIgon - - - - $3x^2-2x$, the IXgon - - - - $\frac{7x^2-5x}{2}$, the Xgon - - - - $4x^2-3x$, the Ngon - - - - $\frac{(n-2)x^2-(n-4)x}{2}$. 657. We have already shewn, that it is easy, by means of these formulæ, to find, for any given root, any polygonal number required: but when it is required reciprocally to find the side, or the root of a polygon, the number of whose sides is known, the operation is more difficult, and always requires the solution of a quadratic equation; on which account the subject deserves, in this place, to be separately considered. In doing this we shall proceed regularly, beginning with the triangular numbers, and passing from them to those of a greater number of angles. 658. Let therefore 91 be the given triangular number, the side or root of which is required. If we make this root $= x$, we must have $\frac{x^2+x}{2} = 91$; or $x^2 + x = 182$, and $x^2 = -x + 182$; consequently, \* Chap. 5, Sect. III. *p 144*