QUADRATURE OF TUE CIRCLE. This is one of the grand problems of antiquity, which, unsolved and probably unsolvable, continue to occupy even in the present day the minds of many curious speculators. The trisection of an angle, the duplication of the cube, and the perpetual motion have found, in every age of the world since geometry and physics were thought of, their hosts of patient devotees. The physical question involved in the perpetual motion (q.v.) is treated of under that head; and we shall now take the opportunity of noticing the mathematical questions involved in the other problems -above mentioned; but more especially that of the quadrature of the circle, in which the -difficulty is of a different nature from that involved in the other two geometrical ones. A few words about them, however, will help as an introduction to the subject.
According to the postulates of ordinary geometry, all constructions must be made by the help of the circle and straight line. Straight lines intersect each other in but one point; and a straight line and circle, or two circles, intersect in two points only. From analytical point of view we may express these facts by saying that the determination -of the intersection of two straight lines involves an equation of the first degree only; while that of the intersection of a straight line and a circle, or of two circles, is reducible to an equation of the second degree. But' the trisection of an angle, or the duplication of the cube, requires for its accomplishment the solution of an equation of the third degree; or, geometrically, requires the intersections of a straight line and a curve of the third degke, or of two conics, etc., all of which are excluded by the postulates of the science. If it were allowed that a parabola or ellipse could be described with a given focus and directrix, as it is allowed that a circle can be described with a given radius about a given -center, the trisection of an angle and the duplication of the cube would be at once brought under the category of questions resolvable by pure geometry; so that the difti •culty in these cases is one of mere restriction of the postulates of what is to be called
.geometry.
It is very different in the case of the quadrature of the circle, which (the reader of the preceding article will see at once) means the determination of the area of a circle of .given radius—literally, the assigning of the side of a square whose area shall be equal to that of the given circle.
The common herd of "squarers of the circle," which grows more numerous every .day, and which includes many men of undoubted sanity, and even of the very highest business talents, rarely have any idea of the nature of the problem they attempt to solve. It will, therefore, be our best course to show first of all what has been done toward the .solution of the problem; we shall then venture a few remarks as to what may yet be done, and in what direction philosophic " squarers of the circle " must look for real advance.
In the first place, then, we observe that mechanical processes are utterly inadmissible. A fair approximation may, no doubt, be got by measuring the diameter of a circular disk of uniform material, and comparing the weight of the disk with that of a square portion of the same material of given side. But it is almost impossible to execute any measurement to more than six places of significant figures; hence, as will soon be shown, this process is at best but a rude approximation. The same is to be said of such obvious processes as wrapping a string round a cylindrical post of known diameter, and compar ing its length with the diameter of the cylinder; only a rude approximation to the ratio of the circumference of a circle to its diameter can thus be obtained.
Before entering on the history of the problem, it must be remarked that the Greek geometers knew that the area of a circle is half the rectangle under its radius and cir cumference (see CIRCLE), so that the determination of the length of the circumference of a circle of given radius is precisely the same problem as that of the quadraturc of the circle.