See also:mathematics, a
See also:curve having ordinates which are a measure of the
See also:area (or quadrature) of another curve . The two most famous curvesof this class are those of Dinostratus and E . W . Tschirnhausen, which are both related to the circle . The quadratrix of Dinostratus was well known to the
See also:ancient Greek geometers, and is mentioned by
See also:Proclus, who ascribes the invention of the curve to a contemporary of
See also:Socrates, probably Hippias of Elis . Dinostratus, a Greek geometer and
See also:disciple of
See also:Plato, discussed the curve, and showed how it effected a
See also:mechanical solution of squaring the circle . Pappus, in his Collections, treats of its
See also:history, and gives two methods by which it can be generated . (I) Let a
See also:line be
See also:drawn on a right circular cylinder; a
See also:surface is then obtained by
See also:drawing lines from every point of this spiral perpendicular to its
See also:axis . The orthogonal
See also:projection of a section of this surface by a
See also:plane containing one of the perpendiculars and inclined to the axis is the quadratrix . (2) A right cylinder having for its
See also:base an Archimedean spiral is intersected by a right circular
See also:cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis . From every point of the curve of intersection, perpendiculars are drawn to the axis . Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix .
Another construction is shown in fig . 1 .
See also:ABC is a quadrant in which the line AB and the arc AC are divided into the same number of equal parts. c Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to BC and through the corresponding points on the
See also:radius AB . The locus of these intersections is the quadratrix . A mechanical construction is as follows: Let AMP be a semicircle with centre 0 (fig . 2) . Let PQ be the
See also:ordinate of the point P on the circle, and let M be another point on the circle so related to P that the ordinate PQ moves from A to 0 in the same
See also:time as the vector OM describes a quadrant . Then the locus of the intersection of PQ and OM is the quadratrix of Dinostratus . The cartesian equation to the curve is y = x cot 2a, which shows that the curve is symmetrical about the axis of y, and that it consists of a central portion flanked by infinite branches (fig . 2) . The asym- ptotes are x= 2na, n being an integer . The intercept on the axis of y is 2a/a; therefore, if it were possible to accurately construct the curve, the quadrature of the circle would be effected .
The curve also permits the solution of the problems of duplicating a
See also:cube (q.v.) and trisecting an
See also:angle . The quadratrix of Tschirnhausen is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before . The mutual intersections of the lines drawn from the points of division of the arc parallel to AB, and the lines drawn parallel to BC through the points of division of AB, are points on the quadratrix (fig . 3) . The cartesian equation is y = a cos ,rx/2a . The curve is periodic, and cuts the axis of x at the points x= = (2n–1)a, n being an integer; the maximum values of y are =a . Its properties are similar to those of the quadratrix of Dinostratus .
QUADRATURE (from Lat. quadratura, a making square)
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