Page:The New International Encyclopædia 1st ed. v. 16.djvu/671

* QUADRATURE. 587 QUADRILLE. process of determining the area of a surface. The term conies from the conception that we find a square whose area is equal to that of a given surface. The quadrature of the circle is one of the three great problems of antiquity, the others being the triseetion of an angle (q.v.) and the du]ilication of the cube. (See Cube.) These problems, like that of perpetual motion, have had their devotees in all ages since the advent of geometry and physics. The quadrature of the circle means the determination of the area of a circle of given radius, or the construction by the use of only the straight edge and the compasses of a square whose area is equal to that of the given circle. It was known to the Greek geometers that the area of a circle is half the rectangle whose sides are its radius and circumference respec- tively; so that the determination of the length of the circumference of a circle in terms of the radius, or the evaluation of w, is precisely the same problem as that of the quadrature of the circle. A brief outline of the history of attempts to evaluate the ratio ir is given in the article Circle. The quadrature of curves can often be effected by means of another curve, a so-called 'quadra- trix.' An impor- tant type of this curve is that prob- ably invented by Hippias of Elis (c.tno B.C.), used both for quadrature and triseetion. and called the quadra- trix of Dinostratus. The curve, probably the most ancient of the transcendental ones, may be de- lined as the plane locus of the intersection of a straight line revolving uniformly about a point, and another straight line moving uniformly parallel to a given direction. If in the figure CO = r is the uniformly re- volving radius, and PQ, the line moving parallel to OY, the locus of P, their intersection, or the curve OPR, is the quadratrix. Its rectangular QOADnATHIX.
 * (r-

-a;)tan ^—; r is a mean pro- 2r equation is y: portional between the quadrant OB and the segment CD; and thus the circumference of a circle may be expressed in t«rms of the radius. Ahcnce, if it were possible to construct D geomet- rically, the quadrature of the circle would be effected by elementary geometry, a condition which is always understood when it is said that the quad- rature of the circle cannot be effected. Another important form of the quadratrix is that of Tschirnhausen (1687). This curve may be de- fined as the locus of the point P. lying at the ajime time upon LQ parallel to BO, and upon MP parallel to OA (OAB being a quadrant of radius OA = r), where L moves over the quad- rant and M moves over the radius r uniformly. The equation of the curve is y :=?-sin-^. It has been used for the multisection of angles and the quadrature of curves. Consult: jVIontucla, Histoire des reclierches stir la f] tin lira t tire du cercle (Paris, 1754); New- ton, Tractatus de Quadratura Ctirvarnm (Lon- VoL. XVI.— :i8. don, 170fi); Klein, I'titnuun, I'robloits of Elemen- tart) (Icuinelry (Gottingen, 18!)5; American ed., Boston. 1897); SchcUbach, Vvber mechanische Quadrulur (Berlin, 2d ed., 1884). QUADRATURE. In astronomy, a planet is said to be in quadrature when there is a right angle at the earth between the direction of the planet and the direction of the sun. QUAD'RIEN'NIUM UTILE, u'ti-le (Lat., useful four }-cars). In Scotch law, the four years after majority during which a person is entitled to revoke or set aside any deed made to his prejudice during minority. This protection was also given by the Roman law to minors, to enable them to neutralize any unfair advantage that may have been taken of their inexperience during minority. See Ixfaxt. QUADRILATERAL (from Lat. quadrilate- rtis, four-sided, from <itiaituor, four -f- latvs, side). A polygon (q.v.) of four sides. Among the remarkable properties of the quadrilateral are the following: The lines joining the mid- points of the successive sides form a parallelo- gram; the lines joining the mid-points of the opposite sides bisect each other. The bisectors of the four angles of a quadrilateral form an- other quadrilateral whose opposite angles are supplemental; a <iuadrilateral of the latter kind is iuseriptible in a circle. The sum of the squares on the four sides of a quadrilateral is equal to the sum of the square on the diagonals plus four times the square on the line joining their mid-points. A square can be constructed eqi»al to any polygon, and hence equal to any ' quadrilateral, but the Z area of a ipiadrilateral cannot in general be ex- .  pressed as an algebraic ISOSCELES TBAPEZOID. funCtioU of the SidcS. A irapeztiim is a quadrilateral no two of whose sides are parallel, and a trapezoid is a plane quadrilateral having two of its sides parallel. If the parallel, ll the /- angles at the ex- / tremities of either £ TRAPEZOID. parallel side are equal, the trape- zoid is said to be isosceles. If the other two sides of the trapezoid are parallel, the figure be- comes a parallelogram. See Mensuration; Pakallelogram. QUADRILATERAL. A common designa- tion for the strong military line formed by the four fortresses of JIantua, Peschiera. Verona, and Legnago, which constituted a great bulwark for Austria in maintaining her dominion in Northern Italy in the nineteenth century. See Italy; Fortification. QUADRILLE, kwa-drll' (Fr., square). A dance of French origin, consisting of consecutive dance movements, generally five in number, danced by couples, or sets of coui)les. oposite to and at riglit angles to one another. The name is derived from the fact that the dancers are arranged into squares consisting each of four couples. The dance originated in the French ballets of the eighteenth century and was almost immediately adopted by society. Its modern form <latcs from the beginning of the nineteenth century. The names of the figures are: Le pan-