QUADRATURE OF THE CIRCLE. Tho speculative part of this question might be passed over with a slight description of the means of finding a square equal to a given circle, or of expressing a circle by means of the square on its radius, if it were not that it is connected with one of those 'propensities, the love of the marvellous, which, carried to an undue extent, tend more than others to throw the mind off its balance, and destroy the comfort of the individual. When it is considered that there are still persons who spend their time, means, and energies in the attempt to overcome a difficulty of which they do not even know the character, it is worth while to enter a little more at length upon this celebrated question of the quadraturo of the circle than its mathematical importance would seem to require. We may add that its historical importance is very great.
It is a proposition not very difficult of proof, that if a right-angled triangle have the radius of a circle for its base, and a line equal to the circumference for its altitude, the triangle is equisreal with the circle. Hence the quadrature is reduced to the finding a line equal in length to the circumference, either geometrically or arithmetically ; or to finding an answer to one or other of the following questions : Given a, theiliameter of a circle in units of a given kind, required a number or fraction w, such that a multiplied by ar may be the number of those same units in the circumference. It is easily shown that this number w must be the same for all cireles.
Given the diameter of a circle, required geometrically a method of drawing a straight line equal in length to the circumference.
Those who first proposed these questions found their progress arrested by the insufficiency of their arithmetic and the limitations of their geometry. Tho former question has lung been' settled, and it has been shown that the ratio of the circumference to the diameter is INCOMMENSURABLE. The latter question cannot be called finally settled, since there is no proof in which all agree that the geometrical quadrature is impossible, though there are considerations which render it in the highest degree unlikely, and there are also asserted proofs of the impossibility which some admit, and which make even those who do not absolutely admit them think their conclusion all but proved.
But the mistake of those who produce pretended quadratnrcs often lies in this, that they do not know what is meant by the word geometrical. They imagine that anything is geometrical which deals in notions about space, and deduces that which is not obvious from that which is. But geometry, in the technical sense, is that which results from the use of Euclid's postulates VI ion), which permit nothing but tho junction of two points by a straight line, the indefinite production of that joining line, and the description of a circle with a given centre, and the line juiuing that centre with another given point as a radius. These limitations make the whole difficulty ; otherwise nothing would be more easy than to determine a circle by the QUAIMATRIX, if that were allowed to be drawn, or to suppose a circle to roll on a straight line till the point which first touched the straight line touches it again, in which case the lino rolled over is the length of the circum ference. When, therefore, any one imagines, as is often the case, that he has found a method of squaring the circle, it generally happens that ho only announces the not very new nor surprising fact, that a diffi. culty which exists under certain circumstances may be no difficulty at all under others. But in like manner as no one would be held likely to answer the question " Required the way of building a house without the use of iron,' who should first demand a common hammer and nails, so the greater number of persons who attempt to square the circle intuit not be supposed to meet the geometrical difficulty by assuming powers of which geometry expressly require's the use to ho abandoned, until it can be shown to be given in allowing the simple postulates above mentioned.