Quadrature (geometry)

In

(or squaring the circle). Quadrature problems served as one of the main sources of problems in the development of calculus. They introduce important topics in mathematical analysis.

History

Antiquity

, though not all Greek mathematicians adhered to this dictum.

For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side ${\displaystyle x={\sqrt {ab}}}$ (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.

Problems of quadrature for

became the highest achievement of analysis in antiquity.

• 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 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.

For the proofs of these results, Archimedes used the method of exhaustion attributed to Eudoxus.^[3]

Medieval mathematics

In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola (Opus Geometricum, 1647),^[3]: 491 and Alphonse Antonio de Sarasa, de Saint-Vincent's pupil and commentator, noted the relation of this area to logarithms.^{[3]: 492 [4]}

Integral calculus

.

The

integral calculus

came a universal method for area calculation. In response, the term quadrature has become traditional, and instead the modern phrase finding the area is more commonly used for what is technically the computation of a univariate definite integral.

See also

• Gaussian quadrature
• Hyperbolic angle
• Numerical integration
• Quadratrix
• Tanh-sinh quadrature

Notes

References

• Boyer, C. B. (1989) A History of Mathematics, 2nd ed. rev. by
  ISBN 0-471-54397-7
  ).
• Eves, Howard (1990) An Introduction to the History of Mathematics, Saunders,
  ISBN 0-03-029558-0
  ,
• Christiaan Huygens (1651) Theoremata de Quadratura Hyperboles, Ellipsis et Circuli
• Jean-Etienne Montucla (1873) History of the Quadrature of the Circle, J. Babin translator, William Alexander Myers editor, link from HathiTrust
  .
• Christoph Scriba (1983) "Gregory's Converging Double Sequence: a new look at the controversy between Huygens and Gregory over the 'analytical' quadrature of the circle", Historia Mathematica 10:274–85.