QUADRATURE (Lat. quadrat ura, from quadrare, to square, from quadra. quadrus. square, from quatt nor, four). In mathematics, the process of determining the area of a surface. The term comes from the conception that we find a square whose area is equal to that of a given surface. The quadrature of the circle is one of the three great problems of antiquity. the others being the trisection of an angle (q.v.) and the duplication of the tube. (See C UDE. ) These problems, like that of perpetual motion, have had their devotees in all ages since the advent of geometry and physics. The quadrature of the circle means the determination of the area of a circle of given radius, or the construction by the use of only the straight edge and the compasses of a square whose area is equal to that of the given circle. It was known to the Creek geometers that the area of a circle is half the rectangle whose sides are its radius and circumference respec tively; so that the determination of the length of the circumference of a circle in terms of the radius, or the evaluation of 7r, is precisely the same problem as that of the quadrature of the circle. A brief outline of the history of attempts to evaluate the ratio is given in the article CIRCLE.
The quadrature of curves can often be efl'eeted by means of another curve, a so-called `quadra trix.' An impor tant type of this curve is that prob ably invented by Hippies of Elk (c.40)) me.), used both for quadrature caul trisection, and called the quadra trix of Dinostretus. The curve. probably the most ancient of the transcendental ones, may be de fined as the plane locus of the intersection of a straight line revolving uniformly about a point, and another straight line moving uniformly parallel to a given direction.
If in the figure CI) = r is the uniformly re volving radius, and PO, the line moving parallel to OY, the locus of P. their intersection, or the curve nPR, is the quadratrix. Its rectangular equation is y=(r—x)tan ; r is a mean pro portional between the quadrant OB and the segment CD; and thus the circumference of a circle may be expressed in terms of the radius. Whence, if it were possible to construct D geomet ricelly, the quadrature of the circle would be effected by elementary geometry, a condition which is always understood when it is said that the quad rature of the circle cannot be effected. Another important form of the quadratrix is that of Tschirnhausen (1687). This curve may lie de fined as the locus of the point P, lying at the same time upon LO parallel to 130. and upon :\1P parallel to OA (OAB being a quadrant of radius OA = r). where L moves over the quad rant and moves over the radius r uniformly.
The equation of the curve is y = rsin It has been used for the multisection of angles and the quadrature of curves.
Consult: Ilistoire des reelterches sur la quadrature du eercle (Paris, 1754): New ton, Traetatus de Quadrature eurrarum (Lon don, 1706) ; Klein, Famous Problems of Elemen tary (icomutry (ltZttingen, 1895; American ed., Boston, 1897) ; Schellbach, Veber meehanische Quadrutur (Berlin, 2d ed., Is54).