Booth's multiplication algorithm

2013-09-18T22:46:34Z

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'''Quadrature''' is a historical mathematical term which means determining area. Quadrature problems have served as one of the main sources of problems for [[mathematical analysis]].

== History ==
[[File:Mitjana geomètrica amb teorema de l'altura.PNG|thumb|left|220px|<center>Antique method to find the [[geometric mean]]</center>]]

[[Greek mathematics|Mathematicians of ancient Greece]], according to the [[Pythagoreanism|Pythagorean]] doctrine, understood determination of [[area]] of a figure as the process of geometrically constructing a [[square (geometry)|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 [[lune of Hippocrates|lunes of Hippocrates]] and [[The Quadrature of the Parabola|the quadrature of the parabola]]. By Greek tradition, these constructions had to be performed using only a [[Compass and straightedge constructions|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 <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 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.

[[File:Parabola and inscribed triangle.svg|thumb|200px|<center>The area of a segment of a parabola is 4/3 that of the area of a certain inscribed triangle.</center>]]
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 a [[lune of Hippocrates]]) a quadrature can be performed. The quadratures of the surface of a sphere and a [[The Quadrature of the Parabola|parabola segment]] discovered by [[Archimedes]] became the highest achievement of analysis in antiquity.
:* The area of the surface of a sphere is equal to quadruple 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 proof of these results Archimedes used the [[method of exhaustion]]<ref>{{harvnb|Katz|1998|loc=pg. 113}}</ref> of [[Eudoxus of Cnidus|Eudoxus]].

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),<ref>{{harvnb|Katz|1998|loc=pg. 491}}</ref> and [[Alphonse Antonio de Sarasa]], de Saint-Vincent's pupil and commentator, noted the relation of this area to [[logarithm]]s.<ref>{{harvnb|Katz|1998|loc=pg. 492}}</ref>

[[John Wallis]] algebrised this method; he wrote in his ''Arithmetica Infinitorum'' (1656) some series which are equivalent to what 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 the surface area 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 (some would say archaic), 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]]
* [[Quadrance]]
* [[Quadratrix]]
* [[Tanh-sinh quadrature]]

==Notes==
{{reflist}}

== References ==

* Boyer, C. B., ''A History of Mathematics'', 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
* Eves, Howard, ''An Introduction to the History of Mathematics'', Saunders, 1990, ISBN 0-03-029558-0,
* {{citation|last=Katz|first=Victor J.|title=A History of Mathematics: An Introduction|edition= 2nd|publisher=Addison Wesley Longman|year=1998|isbn=0321016181}}
[[Category:History of mathematics]]
[[Category:History of geometry]]
[[Category:Mathematical terminology]]

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Booth's multiplication algorithm