Antiquity was the first curved figure to have its exact area calculated mathematically.
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. 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. 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. For a quadrature of a
rectangle with the sides
a and
b it is necessary to construct a square with the side 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. proved that the area of a parabolic segment is 4/3 the area of an inscribed triangle. Problems of quadrature for
curvilinear figures are much more difficult. The
quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. Nevertheless, for some figures a quadrature can be performed. The quadratures of the surface of a sphere and a
parabola segment discovered by
Archimedes 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 (specifically, of a triangle whose vertices are the parabola's two intersection points with the secant line and its intersection with a tangent line of the same slope). For the proofs of these results, Archimedes used the
method of exhaustion attributed to
Eudoxus.
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),
Integral calculus John Wallis algebrised this method; he wrote in his
Arithmetica Infinitorum (1656) some series which are equivalent to what is now called the
definite integral, and he calculated their values.
Isaac Barrow and
James Gregory made further progress: quadratures for some
algebraic curves and
spirals.
Christiaan Huygens successfully performed a quadrature of the surface area of some
solids of revolution. The
quadrature of the hyperbola by
Gregoire de Saint-Vincent and
A. A. de Sarasa provided a new
function, the
natural logarithm, of critical importance. With the invention of
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 ==