in geometry, denotes the reducing a figure to a square. Thus, the finding of a square which shall contain just as much surface or area as a circle, an ellipsis, or a triangle, is the quadrature of a circle, an ellipsis, or a triangle. The quadrature, especially amongst the ancient mathematicians, was a great postulation. The quadrature of curvilinear spaces is of difficult investigation; and in this respect extremely little was done by the ancients, except the finding the quadrature of the parabola by Archimedes. In the year 1657, Sir Paul Neil, Lord Brouncker, and Sir Christopher Wren geometrically demonstrated the equality of some curvilinear figures to rectilinear spaces; and soon afterwards the same thing was proved both at home and abroad of other curves, and which were brought under an analytical calculus. The first specimen of this was given to the public in 1688 by Mercator, in a demonstration of Lord Brouncker's quadrature of the hyperbola by Dr Wallis's reduction of a fraction into an infinite series by division. Sir Isaac Newton, however, had before discovered a method of attaining the quantity of all quadruple curves analytically by his fluxionary calculus. It was disputed between Sir Christopher Wren and Mr Huygens which of them had first discovered the quadrature of any determinate cycloidal space. Leibnitz afterwards found that of another space; and in 1669 Bernoulli discovered the quadrature of an infinity of cycloidal spaces, both segments and sectors. (For the history of the quadrature of the circle, see the third supplement to the fourth volume of Montucla's Mathematical Recreations.)

in astronomy, that aspect of the moon when she is ninety degrees distant from the sun; or when she is in a middle point of her orbit, between the points of conjunction and opposition, namely, in the first and third quarters.