In medieval Europe, quadrature meant the calculation of area by any method. The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment.įor the proofs of these results, Archimedes used the method of exhaustion attributed to Eudoxus.The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere.The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity. Nevertheless, for some figures a quadrature can be performed. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. Problems of quadrature for curvilinear figures are much more difficult. A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.Īrchimedes proved that the area of a parabolic segment is 4/3 the area of an inscribed triangle. 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. By a certain Greek tradition, these constructions had to be performed using only a compass and straightedge, though not all Greek mathematicians adhered to this dictum.Īntique method to find the geometric meanįor a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b (the geometric mean of a and b). The Greek geometers were not always successful (see squaring the circle), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lune of Hippocrates and the parabola. Greek mathematicians understood the determination of an area of a figure as the process of geometrically constructing a square having the same area ( squaring), thus the name quadrature for this process. History Antiquity The lune of Hippocrates was the first curved figure to have its exact area calculated mathematically. They introduce important topics in mathematical analysis. Quadrature problems served as one of the main sources of problems in the development of calculus. A classical example is the quadrature of the circle (or squaring the circle). In mathematics, particularly in geometry, quadrature (also called squaring) is a historical process of drawing a square with the same area as a given plane figure or computing the numerical value of that area. For other uses, see Quadrature (disambiguation).
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