Elements of Algebra

142 ELEMENTS SECT. III. the terms of which are called quadrangular numbers, or squares; since they represent the squares of the natural numbers, as we have already seen; and this denomination is the more suitable from this circumstance, that we can always form a square with the number of points which those terms indicate, thus: $1, \quad 4, \quad 9, \quad 16, \quad 25, \quad \ldots$ 431. We see here, that the side of any square contains precisely the number of points which the square root indicates. Thus, for example, the side of the square $16$ consists of $4$ points; that of the square $25$ consists of $5$ points; and, in general, if the side be $n$, that is, if the number of the terms of the progression, $1, 3, 5, 7, \ldots$ which we have taken, be expressed by $n$, the square, or the quadrangular number, will be equal to the sum of those terms; that is to $n^2$, as we have already seen, Article 422; but it is unnecessary to extend our consideration of square numbers any farther, having already treated of them at length. 432. If now we call the difference $3$, and take the sums in the same manner as before, we obtain numbers which are called pentagons, or pentagonal numbers, though they cannot be so well represented by points *. * It is not, however, that we are unable to represent, by points, polygons of any number of sides; but the rule which I am going to explain for this purpose, seems to have escaped all the writers on algebra whom I have consulted. I begin with drawing a small polygon that has the number of sides required; this number remains constant for one and the same series of polygonal numbers, and it is equal to $2$ plus the difference of the arithmetical progression from which the series is produced. I then choose one of its angles, in order to draw from the angular point all the diagonals of this polygon, which, with the two sides containing the angle that has been taken, are to be indefinitely produced; after that, I take these two sides, and the diagonals of the first polygon on the indefinite lines, each as often as I choose; and draw, from the corresponding points marked by the compass, lines parallel to the sides of the first polygon; and divide them into as many equal parts, or by as many points as there are actually in the diagonals and the two sides produced. This rule is general, from the triangle up to the polygon of an infinite number of sides: and the division