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the 17th letter of the English, Latin and other alphabets of western Europe. As in Latin so in English it is always followed by the vowel u: as in English so probably in ancient Latin qu was equal to cu or kw: q is, therefore, a superflous letter, and has no place in a scientific alphabet, save as standing for some sound differing from that of k: in A. J. Ellis's palæotype alphabet q stands for nasal ng. Some of the ancient Latin grammarians regarded Q as a contracted form of CV (that is cu); but others, and with them modern philologists, recognize in Q a modified form of the koppa of early Greek Q, derived fro mthe Phoenician alphabet. (See ALPHABET). This character Occurs in very ancient Latin inscriptions; but later the down stroke was written aslant, Q, whence the form Q. Q was not employed in Anglo-Saxon writing, cw being used instead: cwen, queen; cwellan, to quell; nor was it used in early German writing, except for words borrowed from Latin; but afterward words of native Germanic origin were spelled with q: quälen, to torment; quer, athwart.

In some systems for the transliteration of the semitic languages Q is used for the Hebrew P, Arabic . The sound Qu in Germanic languages is cognate with what appears to be an unlabialized guttural in the mother tongue of the Indo-Europeans. The labialized guttural Q is often related to pure labial sounds: thus the Latin quattuor, "form,” corresponds to the Oscan petora. In words borrowed from the French, such as coquette, Qu is pronounced as a simple k. Q stands for "Quintus,” Q. E. D. for Lat. quod erat demonstrandum, "which was to be proved"; Q. E. F. for quod erat faciendum, which was to be done"; qr. for "quire" or "quarter"; qt. for "quart"; q.v. for Lat. quod vide, "which see"; q.d. for Lat. quasi dictum, "as if said"; Q. C. for "Queen's Counsel."

QUA (kwä) BIRD. See HERONS.

QUACKENBOS, kwǎk'ĕn-bos, George Payn, American educator: b. New York, 4 Sept. 1826; d. New London, N. H., 24 July 1881. He was graduated at Columbia in 1843 and admitted to the bar. Later he abandoned

law and for many years conducted a large private school in New York. He published 'Advanced Course of Rhetoric and Composition' (1854); School History of the United States' (1857); Natural Philosophy) (1859), and many other once popular textbooks.

QUACKENBOS, John Duncan, American physician, son of G. P. Quackenbos (q.v.): b. New York, 22 April 1848. He was graduated from Columbia in 1866 and from the College of Physicians and Surgeons in 1871, and has since practised in New York. He became ad

junct professor of English at Columbia in 1884, and was professor of rhetoric in Barnard College 1891-93. He is a specialist in mental diseases and has published, among various other works, History of the World'; 'History of Ancient Literature'; 'Tuberculosis'; 'Typhoid Fever'; 'New England Roads and Roadside Attractions; (Hypnotism in Mental and Moral Culture'; 'Hypnotic Therapeutics'; 'Enemies and Evidences of Christianity'; 'Magnhild' (1919), etc.

QUACKENBUSH, kwǎk en bush, Stephen Platt, American naval officer: b. Albany, N. Y., 23 Jan. 1823; d. Washington, D. C., 4 Feb. 1890. He entered the navy as midshipman in 1840, received rank as lieutenant in 1855 and lieutenant-commander in 1862. In the early part of the Civil War he had charge of the Delaware, Unadilla, Pequot, Patapsco and Mingo in the blockading squadron, covered General Burnside's army at Aquia Creek and Roanoke Island, participated in the battles of Elizabeth City and New Berne, N. C., engaged the Confederate batteries at Winton, N. C., and destroyed the town. He later took part in the engagements at Sewell's Point Landing, Wilcox Landing and Malvern Hill and received at the last-mentioned place a wound from which he lost his right leg. In 1863 he captured the Princess Royal loaded with materials for a Confederate ironclad. In 1864 while dredging Charleston Harbor for torpedoes his ship, the Patapsco, was struck by a torpedo and sunk. Later he was in command of the Mingo, protecting Georgetown, S. C. He became rearadmiral in 1884 and was retired in 1885.

QUADI, kwā'dī, an ancient powerful people of southeastern Germany, of the Suevic race. They inhabited the country bounded by Mount Gabreta, the Hercynian forest, the Sarmatian Mountains and the Danube; their neighbors being the Gothini and Osi on the north, the Jazyges Metanasta on the east, the Pannonians on the south and the Marcomanni on the west. With the last-named people they were on terms of alliance. In the reign of Tiberius the Romans erected a kingdom of the Quadi; but in the reign of Marcus Aurelius the Quadi joined the great German confederacy against the empire, and in 174 were on the point of destroying the imperial legions in a great battle when a sudden storm, attributed to the prayers of the Christian soldiers in the emperor's army, enabled the Romans to recover from their confusion and achieve a complete victory. The independence of the Quadi was recognized by Commodus in 180. They disappear from history about the close of the 4th century.

QUADRAGESIMA, the period of the 40 days' fast preceding Easter: the season has a like name in Greek, Tesserakosté, fortieth. The first Sunday of Lent is sometimes called Quadragesima, but in the Roman calendar it is Dominica prima Quadragestima, first Sunday of Lent and the Sundays following are denominated, respectively, the second, third and fourth of Quadragesima, Passion Sunday (Dominica Passionis), and Palm Sunday (Dominica Palmarum). In the Book of Common Prayer these six Sundays are called the first, second, third, fourth and fifth of Lent and "Sunday next before Easter»; but this Sunday is also styled in England and the United States "Passion Sunday," and its week "Passion Week"; its familiar name among Catholics is "Holy Week."

QUADRANGLE, a square or four-sided court or space surrounded by buildings, as often seen in the buildings of a college. Also in geometry, a figure having four angles, and consequently four sides.

QUADRANT. See SURVEYING.

QUADRATE BONE, the squarish bone developed in reptiles and birds, by means of which the lower jaw is articulated or joined to the skull. The lower jaw of these forms is thus not articulated directly or of itself to the skull, as in mammals; and in reptiles and birds each half of the lower jaw is composed of a number of distinct pieces. In mammals, on the contrary, the lower jaw consists simply of two halves united together in front. The os quadratum, or quadrate bone, which thus forms a characteristic structure of birds and reptiles, is generally regarded as corresponding in mammals to one of the little bones or auditory ossicles of the internal ear, named the malleus. See EAR.

QUADRATURE OF THE CIRCLE. The problem involved in the quadrature of the circle requires the determination of the length of a straight line such that the square constructed thereon shall have an area equal to that of a given circle. It can be shown in a variety of ways that, if r is the radius of the circle, the area will be equal to 2 where can only be obtained approximately in terms of a finite number of fractions. On the other hand, it has become a matter of general information that the quadrature of the circle is impossible, and this is true only when the ancient Greek problem is understood, which involves a serious limitation; that is, the quadrature must be effected by means of a geometrical construction in which the mathematician is limited to the use of but two instruments, the straight edge and a pair of compasses. The problem is not solved, therefore, if any other instrument or any equivalent analytic method is employed. For 4,000 years innumerable attempts have been made to discover this construction, all destined to fail, as it was demonstrated by Lindemann in 1882 to be impossible. In reaching this conclusion we are confronted by the fundamental question: What geometrical constructions are, and what are not, possible, when restricted to the use of these instruments? In analysis, operations correspond to constructions. The operation, a × b, involves taking b units a times, which, with an assumed unit of length, is equivalent to laying off a line having a X b units of length, which is accomplished by using the method of propor

tions. In a similar manner, the rational operations of addition, subtraction, multiplication and division find a geometrical solution involving the straight edge alone.

Irrational operations are divided into algebraic and transcendental. Any operation that involves the extraction of a square root only presents the simplest case of an algebraic irrationality, and any construction involving the determination of the points of intersection of two circles, or a circle and straight line, leads to an equation of the second, of the fourth or of some higher degree, whose solution involves the extraction of square roots and rational operations only. Conversely, the necessary and sufficient condition that unknown quantities can be constructed with the straight edge and compasses is that the unknown quantities can be expressed explicitly in terms of the known by an analytic expression involving only a finite number of rational operations and square roots. In other words a Euclidean geometrical solution is impossible when no corresponding algebraic equation exists. When a number like V Z is the root of an algebraic equation with integral coefficients, for example, x2-2=0, and still can not be expressed exactly in terms of a finite series of numbers it is an algebraic irrational number. When the number, like e, the natural base in the theory of logarithms, or ", the ratio of the circumference to the diameter, is not the root of any algebraic equation, with integral_coefficients, it is a transcendental number. Lindemann provided that is a transcendental number and, hence, since it is not the root of any algebraic equation, it cannot be constructed to an assumed unit by the extraction of square roots, that is, by using the straight edge and compasses.

The possiblity of a geometrical solution of a problem in general depends upon a theorem in the theory of numbers to the effect that the degree of the irreducible equation satisfied by an expression composed of square roots only is always a power of 2; whence, if an irreducible equation is not of degree 2n, it cannot be solved by square roots.

Next to the squaring of the circle, the most famous problems of antiquity are the Delium problem of the duplication of the cube and the trisection of an arbitrary angle. Granting the preceding general theorem, these are easily shown to be impossible when restricted to the straight edge and compass.

The duplication of the cube requires the determination of the edge of a cube x, such that its cube shall be twice the volume of a given unit. That is 2. This equation is irreducible, since otherwise VZ would be rational. Moreover, the equation is a cubic and its degree is not of the form 2n. Hence the solution is in general impossible,

The problem of the trisection of an arbitrary angle corresponds to the solution of x=cos +V-1 sin and it follows that in general this is impossible by Euclidean methods.

History. The quadrature of the circle is attempted in the Rhind Papyrus (2000 B.C.), the oldest known mathematical work, in which Ahmes, an Egyptian priest, lays down the empirical rule: "Cut off one ninth of the diameter;