Editing Numerical integration (section)

== History ==
{{main|Quadrature (geometry)}}
The term "numerical integration" first appears in 1915 in the publication ''A Course in Interpolation and Numeric Integration for the Mathematical Laboratory'' by [[David Gibb (mathematician)|David Gibb]].<ref>{{cite web|url=http://jeff560.tripod.com/q.html|title=Earliest Known Uses of Some of the Words of Mathematics (Q)|website=jeff560.tripod.com|access-date=31 March 2018}}</ref>

"Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of [[mathematical analysis]]. [[Greek mathematics|Mathematicians of Ancient Greece]], according to the [[Pythagoreanism|Pythagorean]] doctrine, understood calculation of [[area]] as the process of constructing geometrically a [[square (geometry)|square]] having the same area (''squaring''). That is why the process was named "quadrature". For example, a [[quadrature of the circle]], [[Lune of Hippocrates]], [[The Quadrature of the Parabola]]. This construction must be performed only by means of [[Compass and straightedge constructions|compass and straightedge]].

The ancient Babylonians used the [[trapezoidal rule]] to integrate the motion of [[Jupiter (planet)|Jupiter]] along the [[ecliptic]].<ref>{{cite journal|author1=Mathieu Ossendrijver|title=Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph|journal=Science|date=Jan 29, 2016|doi=10.1126/science.aad8085|volume=351|issue=6272|pages=482–484|pmid=26823423|bibcode=2016Sci...351..482O|s2cid=206644971 }}</ref>

[[File:Geometric mean.svg|thumb|left|220px|Antique method to find the [[Geometric mean]] ]]
For a quadrature of a rectangle with the sides ''a'' and ''b'' it is necessary to construct a square with the side <math>x =\sqrt {ab}</math> (the [[Geometric mean]] of ''a'' and ''b''). For this purpose it is possible to use the following fact: if we draw the circle with the sum of ''a'' and ''b'' as the diameter, then the height BH (from a point of their connection to crossing with a circle) equals their geometric mean. The similar geometrical construction solves a problem of a quadrature for a parallelogram and a triangle.

[[File:Parabola and inscribed triangle.svg|thumb|200px|{{center|The area of a segment of a parabola}}]]
Problems of quadrature for curvilinear figures are much more difficult. The [[quadrature of the circle]] with compass and straightedge had been proved in the 19th century to be impossible. Nevertheless, for some figures (for example the [[Lune of Hippocrates]]) a quadrature can be performed. The quadratures of a sphere surface and a [[The Quadrature of the Parabola|parabola segment]] done by [[Archimedes]] became the highest achievement of the antique analysis.
* The area of the surface of a sphere is equal to quadruple the area of a [[great circle]] of this sphere.
* The area of a segment of the [[parabola]] cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment.
For the proof of the results Archimedes used the [[Method of exhaustion]] of [[Eudoxus of Cnidus|Eudoxus]].

In medieval Europe the quadrature meant calculation of area by any method. More often the [[Method of indivisibles]] was used; it was less rigorous, but more simple and 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), and [[Alphonse Antonio de Sarasa]], de Saint-Vincent's pupil and commentator, noted the relation of this area to [[logarithm]]s.

[[John Wallis]] algebrised this method: he wrote in his ''Arithmetica Infinitorum'' (1656) series that we now call the [[definite integral]], and he calculated their values. [[Isaac Barrow]] and [[James Gregory (mathematician)|James Gregory]] made further progress: quadratures for some [[algebraic curves]] and [[spiral]]s. [[Christiaan Huygens]] successfully performed a quadrature of some [[Solid of revolution|Solids of revolution]].

The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new [[Function (mathematics)|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 "''computation of a univariate definite integral''" is more common.