Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. The process of making something square.
 n. Mathematics The process of constructing a square equal in area to a given surface.
 n. Astronomy A configuration in which the position of one celestial body is 90° from another celestial body, as measured from a third.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. the process of making something square; squaring
 n. a situation in which three celestial bodies form a rightangled triangle, the observer being located at the right angle
 n. the condition in which the phase angle between two alternating quantities is 90°
 n. A painting painted on a wooden panel
from the GNU version of the Collaborative International Dictionary of English
 n. The act of squaring; the finding of a square having the same area as some given curvilinear figure; ; the operation of finding an expression for the area of a figure bounded wholly or in part by a curved line, as by a curve, two ordinates, and the axis of abscissas.
 n. A quadrate; a square.
 n. The integral used in obtaining the area bounded by a curve; hence, the definite integral of the product of any function of one variable into the differential of that variable.
 n. The position of one heavenly body in respect to another when distant from it 90°, or a quarter of a circle, as the moon when at an equal distance from the points of conjunction and opposition.
from The Century Dictionary and Cyclopedia
 n. In geometry, the act of squaring an area; the finding of a square or several squares equal in area to a given surface.
 n. A quadrate; a square space.
 n. The relative position of two planets, or of a planet and the sun, when the difference of their longitudes is 90°.
 n. But when armillæ were employed to observe the moon in other situations … a second inequality was discovered, which was connected, not with the anomalistical, but with the synodical revolution of the moon, disappearing in conjunctions and oppositions, and coming to its greatest amount in quadratures. What was most perplexing about this second inequality was that it did not return in every quadrature, but, though in some it amounted to 2° 39′ , in other quadratures it totally disappeared.
 n. A side of a square.
 n. In electricity, phase difference of 90°, or one quarter period.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. the construction of a square having the same area as some other figure
Etymologies
Examples

(A quadrature is the inverse problem, that of determining the fluents when the fluxions are given.)

More curious than his quadrature is his name; what are we to make of it?

The error by Archimedes is ultimately comparable to the use of that fallacy of the notion of quadrature which was already implicit in the Aristotelean presumption expressed by

Further more, La quadrature du net has investigated the petition … Many of the names are fake, a lot of these “artists” dont exist, are members of the french IFPI, or their names have been used without their knowledge … Oh yes nice petition indeed …

“Further more, La quadrature du net has investigated the petitionâ ¦ Many of the names are fake, a lot of these âartistsâ dont exist, are members of the french IFPI, or their names have been used without their knowledgeâ ¦ Oh yes nice petition indeedâ ¦”

“Further more, La quadrature du net has investigated the petitionâ¦ Many of the names are fake, a lot of these âartistsâ dont exist, are members of the french IFPI, or their names have been used without their knowledgeâ¦ Oh yes nice petition indeedâ¦”

Further more, La quadrature du net has investigated the petition… Many of the names are fake, a lot of these “artists” dont exist, are members of the french IFPI, or their names have been used without their knowledge… Oh yes nice petition indeed…

La quadrature notes that the names appearing in the petition are not artistic names but real names of artists.

Before Newton, quadrature or integration had rested ultimately “on some process through which elemental triangles or rectangles were added together”, that is, on the method of indivisibles.

Isaac Barrow [17] (1630 “ 77) was one of the first mathematicians to grasp the reciprocal relation between the problem of quadrature and that of finding tangents to curves ” in modern parlance, between integration and differentiation.
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the_word_is_what commented on the word quadrature
This would be an excellent word to use in slang/vernacular. Being the process of making "square." In the slang sense you could tell your friends they excel at quadrature (meaning they are square in the slang sense). Very excellent.
March 6, 2010