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 (1703). See Grätz, ''Gesch. der Juden, especially in vol. v. (1806), with the additions and corrections of Harkavy in the Hebrew translation; and Fürst, Gesch. des Karäerthums (1865); S. Pinsker, Liqquṭe Qadmoniyyot'': articles by A. Harkavy and by S. Poznanski in the Jewish Quarterly Review (e.g. x. 238–276, and vols. xviii.–xx.). See also Jewish Encyclopedia, s.v. “Anan,” “Karaites,” &c.

 QARO (or ), JOSEPH BEN EPHRAIM (1488–1575), codifier of Jewish law, whose code is still authoritative with the mass of Jews, was born in 1488. As a child he shared in the expulsion from Spain (1492), and like most prominent Jews of the period was forced to migrate from place to place. In 1535 he settled in Safed, Palestine, where he spent the rest of his life. Safed was then the headquarters of Jewish mysticism. Qaro was himself a mystic, for the tribulations of the time turned many men’s minds towards Messianic hopes; nor was he by any means the only great Jewish legalist who was also a mystic. Mysticism in such minds did not take the form of a revolt against authority, but was rather the spiritual flower of pietism than an expression of antinomianism. It is, however, as a legalist that Qaro is best known. In learning and critical power he was second only to Maimonides in the realm of Jewish law. He was the author of two great works, the second of which, though inferior as an intellectual feat, has surpassed the first in popularity. This was inevitable, for the earlier and greater book was designed exclusively for specialists. It was in the form of a commentary (entitled Beth Yoseph) on the Turim (see ). In this commentary Qaro shows an astounding mastery over the Talmud and the legalistic literature of the middle ages. He felt called upon to systematize the laws and customs of Judaism in face of the disintegration caused by the Spanish expulsion. But the Beth Yoseph is by no means systematic.

Qaro’s real aim was effected by his second work, the Shulḥan ’Arukh (“Table Prepared”). Finished in 1555, this code was published in four parts in 1565. The work was not accepted without protest and criticism, but after the lapse of a century, and in consequence of certain revisions and amplifications, it became the almost unquestioned authority of the whole Jewish world. Its influence was to some extent evil. It “put Judaism into a strait-jacket.” Independence of judgment was inhibited, and the code stood in the way of progressive adaptation of Jewish life to the life of Europe. It included trivialities by the side of great principles, and retained elements from the past which deserved to fall into oblivion. But its good effects far outweighed the bad. It was a bond of union, a bar to latitudinarianism, an accessible guide to ritual, ethics and law. Above all, it gave a new lease of life to the great theory which identified life with religion. It sanctified the home, it dignified common pursuits. When, however, the era of reform dawned in the 19th century, the new Judaism found itself impelled to assume an attitude of hostility to Qaro’s code.

See Graetz, Geschichte der Juden, vol. ix. (English trans. vol. iv.); Ginzberg, in Jewish Encyclopedia, arts. “Caro” and “Codification"; Schechter, Studies in Judaism, second series, pp. 202 seq.

 QUACK, one who pretends to knowledge of which he is ignorant, a charlatan, particularly a medical impostor. The word is a shortened form of “quacksalver” (Du. kwaksalver), in which form it is common in the 17th century, “salver” meaning “healer,” while “quack” (Du. kwakken) is merely an application of the onomatopoeic word applied to the sounds made by a duck, i.e. gabble or gibberish. In English law, to call a medical practitioner a “quack” is actionable per se without proof of special damage (Allen v. Eaton (1630), 1 Roll. Abs. 54). The often-quoted legal definition of a “quack” is “a boastful pretender to medical skill,” but a “quack” may have great skill, and it is the claim to cure by remedies which he knows have no efficacy which makes him a “quack” (see Dakhyl v. Labouchere, The Times, 29th of July 1904, and 5th and 9th of November 1907).  QUADRATRIX (from Lat. quadrator, squarer), in mathematics, a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of 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 ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus, in his Collections, treats of its history, and gives two methods by which it can be generated. (1) Let a spiral line be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix. (2) A right cylinder having for its base an Archimedean spiral is intersected by a right circular 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. ABC is a quadrant in which the line AB and the arc AC are divided into the same number of equal parts. 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 radius AB. The locus of these intersections is the quadratrix. A mechanical construction is as follows: Let AMP be a semicircle with centre O (fig. 2). Let PQ be the 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 O in the same time as the vector M 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 $x⁄2a$, 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 asymptotes are x = ±2na, n being an integer. The intercept on the axis of y is 2a/; 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 (q.v.) and trisecting an 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 x/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), in astronomy, that aspect of a heavenly body in which it makes a right angle with the direction of the sun; applied especially to the apparent position of a planet, or of the moon at first and last quarters. In mathematics, quadrature is the determination of a square equal to the area of a curve or other figure.  QUADRIGA, the ancient four-horsed chariot (Lat. quadrigae, contracted from quadrijugae), which was regarded as one of the seven sacred features in Rome. It was chiefly used, as the triumphal car of generals or emperors. The earliest example mentioned is that which was modelled in terra-cotta and raised on the pediment of the temple of Jupiter Capitolinus. In later time it formed the chief decorative feature which crowned the triumphal arches, and there are numerous representations of it on coins.  QUADRILATERAL, in geometry, a figure enclosed by four straight lines. It is also a military term applied to a combination of four fortresses mutually supporting one another. The fortresses of Namur, Liége, Maastricht, and Louvain, and also those of Silistria, Rustchuk, Shumla, and Varna, were so called. But the most famous quadrilateral was that of the four fortified towns of north Italy—Mantua, Peschiera, Verona, and Legnago, 