Quadrature (mathematics)
In mathematics, quadrature is a historical term which means determining area. Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in mathematical analysis.
Mathematicians of ancient Greece, according to the Pythagorean doctrine, understood determination of area of a figure as the process of geometrically constructing a square having the same area (squaring), thus the name quadrature for this process. The Greek geometers were not always successful (see quadrature of the circle), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lunes of Hippocrates and the quadrature of the parabola. By Greek tradition, these constructions had to be performed using only a compass and straightedge.
For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side (the geometric mean of a and b). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b, then the height (BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to the point where it crosses the circle, equals the geometric mean of a and b. A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.
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