See also:

• QUADRATRIX (from Lat. quadrator, squarer)

QUADRATRIX (from See also:

• LAT

Lat. quadrator, squarer)  , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics, a See also:

• CURVE (Lat. curvus, bent)

curve having ordinates which are a measure of the See also:

• AREA

area (or See also:

• QUADRATURE (from Lat. quadratura, a making square)

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 See also:

• QUADRATRIX (from Lat. quadrator, squarer)

quadratrix of Dinostratus was well known to the See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

ancient See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek geometers, and is mentioned by See also:

• PROCLUS, or PROCULUS (A.D. 410-485)

Proclus, who ascribes the invention of the curve to a contemporary of See also:

• SOCRATES

Socrates, probably Hippias of See also:

• ELIS
• ELIS, or ELEIA
• ELIS, PHILOSOPHICAL SCHOOL OF

Elis . Dinostratus, a Greek geometer and See also:

• DISCIPLE

disciple of See also:

• PLATO

Plato, discussed the curve, and showed how it effected a See also:

• MECHANICAL

mechanical See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of squaring the circle . Pappus, in his Collections, treats of its See also:

• HISTORY

history, and gives two methods by which it can be generated . (I) Let a See also:

• SPIRAL

spiral See also:

• LINE

line be See also:

• DRAWN

drawn on a right circular See also:

• CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

cylinder; a See also:

• SCREW (O.E. scrue, from O. Fr. escroue, mod. ecrou; ultimate origin uncertain; the word, or a similar one, appears in Teutonic languages, cf. Ger. Schraube, Dan. skrue, but Skeat, following Diaz, finds the origin in Lat. scrobs, a ditch, hole, particularl

screw See also:

• SURFACE

surface is then obtained by See also:

• DRAWING

drawing lines from every point of this spiral perpendicular to its See also:

• AXIS (Lat. for " axle ")

axis . The orthogonal See also:

• PROJECTION

projection of a See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section of this surface by a See also:

• PLANE

plane containing one of the perpendiculars and inclined to the axis is the quadratrix . (2) A right cylinder having for its See also:

• BASE

base an Archimedean spiral is intersected by a right circular See also:

• CONE (Gr. Kwvos)

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

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 See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

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

radius AB . The See also:

• LOCUS (Lat. for " place "; in Gr. rinros)

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

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 (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

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 See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

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 See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

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 (Gr. K46os, a cube)

cube (q.v.) and trisecting an See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

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 See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

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 .