The problem is simply this: _Having given the diameter of a circle, to determine, by a geometrical construction, in which only straight lines and circles shall be made use of, the side of a square, equal in area to the circle_. As I have said, the problem is solved, if, by a construction of the kind described, we can determine the length of the circumference; because then the rectangle under half this length and the radius is equal in area to the circle, and it is a simple problem to describe a square equal to a given rectangle. To illustrate the kind of construction required, I give an approximate solution which is remarkably simple, and, so far as I am aware, not generally known. Describe a square about the given circle, touching it at the ends of two diameters, AOB, COB, at right angles to each other, and join CA; let COAE be one of the quarters of the circumscribing square, and from E draw EG, cutting off from AO a fourth part AG of its length, and from AC the portion AH. Then three sides of the circumscribing square together with AH are very nearly equal to the circumference of the circle. The difference is so small, that in a circle two feet in diameter, it would be less than the two-hundredth part of an inch. If this construction were exact, the great problem would have been solved. One point, however, must be noted; the circle is of all curved lines the easiest to draw by mechanical means. But there are others which can be so drawn. And if such curves as these be admitted as available, the problem of the quadrature of the circle can be readily solved. There is a curve, for instance, invented by Dinostratus, which can readily be described mechanically, and has been called the quadratrix of Dinostratus, because it has the property of thus solving the problem we are dealing with. As such curves can be described with quite as much accuracy as the circle—for, be it remembered, an absolutely perfect circle has never yet been drawn—we see that it is only the limitations which geometers have themselves invented that give this problem its difficulty. Its solution has, as I have said, no value; and no mathematician would ever think of wasting a moment over the problem—for this reason, simply, that it has long since been demonstrated to be insoluble by simple geometrical methods. So that, when a man says he has squared the circle (and many will say so, if one will only give them a hearing), he shows that either he wholly misunderstands the nature of the problem, or that his ignorance of mathematics has led him to mistake a faulty for a true solution. (From _Chambers’s Journal_, January 16, 1869.) _A NEW THEORY OF ACHILLES’ SHIELD._