QUATRAIN-QUATRE-BRAS noted by S, denoting the vector part by V, and writes (3) aß Saß Vaß, where Saß -TaTẞcose, Vaß TaTßesin 0. Reversing the order we derive in a similar manner (4) ẞa =TBTa (-cos e sin 0), whence by comparison with (3) we obtain the important formulas Saß Spa, Vaß = — νβα, αβ + βα 2Saß(a +ẞ)2 = a2+ 2Saß82. Hamilton then proceeds to define β = = the quotient of two vectors -, where ẞ and a are β a α unit vectors, to be such that. a = B. Observe, however, that (cos + e sine) a = cos 0+e sine) (B cos y sin 0) = ẞcos20+(y+eß) sin @cos + ey sin2 = ẞ, since eßy and ey = B. It folβ lows that we may write (5) α = cos e sin 0 = - Ba (see 4) as an operator that partakes of the nature of e = — -k, the difference being that e, a unit vector, turns a through a right angle, whereas - Ba = cos 0 + € sin turns a through the angle 0. Hamilton calls cos + sin @ the versor of the operation, and also employs U as its notaβ -= a tion. If a and B are any vectors, then (6) Τρ Τα -= a ᎢᏰ (cos + e sin @), where is an absolute numΤα ber and therefore also a tensor. Hence the product or quotient of two vectors is a four obtained of the general field of mathematics. Every quaternion formula is equivalent to some proposition in spherical trigonometry or that of the plane; in its scalor equations we descend to principles of Cartesian geometry, while the theory of quadric surfaces and quadric functions of position, so numerous in mathematical physics, is simplified to the solution of linear vector equations, reducible in turn to the theory of a single function. Somewhat later, but quite independently, Grassmann published his Ausdehnungslehre (Leipsic, 1844), in which he develops a system even more general than Hamilton's, and the same may be said of the very valuable article on the general question by Prof. Pierce of Harvard on Linear Associated Algebra,' published for private circulation only. Aside from the physical and geometrical applications made by Hamilton, those of Tait are the most remarkable, and it is worthy of note that the leaders in mathematical physics like Tait, Clifford and Clark-Maxwell employ frequently the concepts of quaternions. The future development of the calculus has and still is a matter of considerable interest, many mathematicians believing it to be the greatest achievement of the 19th century. The French and German writers hold it at present in slight esteem, and to-day there is probably a greater interest in the subject in America than even in England. The original works are 'The Lectures' and "The Elements' (London, 1866), which are very voluminous. The German translation by Glass (Leipsic, 1882) is preferable because of the translator's notes. More available are Tait's 'An Elementary Treatise on Quaternions' (Oxford, 1871) and the much simpler work of Kelland and Tait, 'Introduction to Quaternions' (London, 1882), and Hardy, 'Elements of Quaternions (Boston, 1887). J. BRACE CHITTENDEN, PH.D., Prof. of Mathematics, Brooklyn Polytechnic Institute. Quat'rain, a poetical composition of four verses rhyming alternately, a usual form for epigrams, epitaphs, proverbs, etc.; stanzas of four verses in a longer piece of poetry are also called quatrains, if each stanza expresses by itself a complete idea. Quatre-Bras, kätr-brä ("four branches"), Belgium, in the province of Brabant; at the intersection of the main roads from Brussels and Charleroi and from Namur to Nivelles; about 20 miles south-southeast of Brussels. It is famous for the battle fought here, 16 June 1815, between the English under Wellington and the French under Ney. It was Napoleon's plan at the opening of the campaign of 1815 to fight the armies of the allies singly, as he felt himself unable to cope with their combined forces. At the head of an army of 129,000 men, supported by 350 guns, he made a rapid march into Belgium, crossing the Sambre almost before the allies were aware of his movements. His army was divided into three corps: the right wing, (cos + sin 0), 48,000 strong, being under Grouchy; the centre, QUATRE-VINGT-TREIZE-QUAY severe struggle, retired upon Wavre 16 June. On the same day Ney attacked Wellington at Quatre-Bras. The English, 32,000 strong, repulsed Ney, and maintained their position, though at a loss of over 5,000 men, including the Duke of Brunswick. The English general, hearing of the retrograde movement of the Prussians, fell back on the 17th on Waterloo, where the decisive battle was fought on the following day. Quatre-Vingt-Treize, kätr-văn-trāz (Ninety-Three'), a work of historical fiction by Victor Hugo, published in 1874. The time is, as the title indicates, 1793. The scene is Vendée, during the civil war. In its style it displays much unevenness, but in its characters, vivid scenes, and tragic catastrophe ranks among the author's best. Quatrefages de Bréau, Jean Louis Armand de, zhon loo-e är-män de kätr-fäzh de brã-ō, French naturalist: b. Berthezène, Gard, France, 10 Feb. 1810; d. Paris, France, 12 Jan. 1892. He was graduated from the University of Strasburg and in 1832 settled in Toulouse where he practised medicine and shortly afterward founded the Journal de Médecine et de Chirurgie de Toulouse. In 1838-40 he occupied the chair of zoology at the University of Toulouse and in the latter year removed to Paris where he made a special study of invertebrates. In 1850 he was appointed professor of natural history at the Lycée Napoléon, Paris, and in 1855 professor of anthropology at the Musée d'Histoire Naturelle, which post he occupied until his death. He was elected to the Academy of Sciences in 1852 and was widely known by his anthropological investigations. He was the first expounder of the doctrine of phlebenterism, and strongly opposed to Darwin's theories. He was author of many important works, among which are: (Souvenirs d'un Naturaliste' (1854); "Histoire Naturelle des Annelés Marins et d'Eau Douce (1865); La Race Prussienne) (1871); L'Espèce Humaine (1877); Introduction à l'Etude des Races Humaines' (1887-9); Théories Transformistes) (1892); etc. was Quatrefoil, kǎt'er-foil, in architecture, an opening or a panel divided by cusps or foliations into four leaves, or more correctly the leafshaped figure formed by the cusps. It is an ornament which has been supposed to represent the four leaves of a cruciform flower, and is common in the tracery of Gothic windows. Bands of small quatrefoils are much used as ornaments in the perpendicular Gothic style, and sometimes in the decorated. Quatremère, kätr-mar, Etienne Marc, French Orientalist: b. Paris, France, 12 July 1782; d. there 18 Sept. 1857. He was educated at the Collège de France, was employed at the Bibliothèque Impériale in 1807 and in 1809 accepted the chair of Greek in the University of Rouen. He was elected to the French Institute in 1815, became professor of Hebrew at the Collège de France in 1819 and in 1838 was made professor of Persian in the School for Living Oriental Languages. He made numerous valuable translations and was author of important geographical and historical works in addition to his work as an Orientalist. He published: Recherches Critiques et Historiques sur la Langue et la Littérature de l'Egypte (1808); 'Histoire des Sultans Mamelouks▸ (1837-45); etc. Quatremère de Quincy, de kǎn-sē, Antoine Chrysostome, French archæologist and politician: b. Paris 28 Oct. 1755; d. there 8 Dec. 1849. He was prominent politically under the republic, consulate, empire, and restoration; was made permanent secretary of the Academy of Beaux-Arts in 1816, and was editor in that department of the Journal des Savants'; elected professor of archæology at the Royal Library in 1818 and serving as deputy in 1820-1. He wrote: Dictionnaire de l'Architecture) (17861825); 'Histoire de la Vie et des Ouvrages de Rafaël' (1824); Canova et ses Ouvrages' (1834); etc. Quatrocento, kwät-rō-chěn'tō, in art, a term applied to the characteristic style of the artists of the 14th century; it was hard, rigid, and peculiar in color as well as in form and pose. It was the intermediate stage of that progressive period of art, which, commencing with Fra Angelico, reached excellence with Leonardo da Vinci. Quay, kwa, Matthew Stanley, American politician: b. Dillsburg, York County, Pa., 30 Sept. 1833. He was graduated from Jefferson the bar in 1854, and elected prothonotary of College in 1850, studied law, was admitted to Beaver County in 1856 and 1859. During the Civil War he was in active service as colonel of the 134th Pennsylvania regiment, being present at the battle of Fredericksburg; he was also asState military agent at Washington, and military sistant commissary general of Pennsylvania, secretary to the governor. In 1865-7 he was a member of the Pennsylvania legislature; in 1872-8, and again in 1879-82, he was State secre1878-9; and State treasurer in 1885. As early as tary; recorder for the city of Philadelphia in 1869 he was secretary of the executive committee of the Republican State committee, and since then has become the undisputed leader of his party in Pennsylvania. In national politics committee since 1885, and in 1888 was its chairhe has been a member of the Republican national man, conducting a successful Presidential campaign. In 1887 he was elected United States Senator, serving continuously till 1899; in that year a deadlock in the legislature prevented his re-election. This was in part owing to the fact that in 1898 on the failure of the People's Bank where State funds were deposited, he had been accused of being party to a conspiracy for the misappropriation of public funds; the trial occurred in April 1899, and resulted in his acquittal. He was then appointed Senator ad interim by the governor, and elected to the Senate in 1901 for the term expiring in 1905. In the Senate he was one of the strongest opponents of the Panama Canal. His power as a political organizer lies chiefly in his adroit methods of reconciling opposing factions and hostile interests. A striking example of this was his method of meeting the reform movement of 1902 by supporting as candidates for governor of the State and mayor of Philadelphia, men who were endorsed by the reformers and not prominent in the regular Republican organization, thus gaining the support of what might have been a dangerous opposition. Quay, a landing-place along a line of coast or a river bank, or round a harbor. They QUAYLE QUEBEC are usually constructed of stone, but sometimes of wood and iron. See WHARF. Quayle, kwal, William Alfred, American Methodist clergyman and author: b. Missouri 1860. He was graduated from Baker University, Baldwin, Kan., in 1885, was ordained to the Methodist ministry the next year and has since held prominent pastorates in Kansas City and Indianapolis. He was professor of Greek at Baker University 1885-90, and president of that institution 1890-2. He has lectured widely and has published: The Poet's Poet and Other Essays (1897); A Hero and Some Other Folk (1900); The Blessed Life' (1901); 'The Gentleman in Literature' (1902); etc. Quebec, kwē-běk' (Fr. kė-běk), province of Canada, in British North America. It is bounded on the north by Labrador and Hudson Bay, on the east by Labrador and the Gulf of St. Lawrence, on the south by the Bay of Chaleurs, New Brunswick, and the States of Maine, New Hampshire, Vermont and New York, and on the southwest by the River Ottawa and the province of Ontario. Its length is nearly 1,000 miles on a due east and west course, its breadth is 300 miles, and its area 351,873 square miles. The surface of the country is varied and picturesque, embracing several ranges of mountains and lofty hills, diversified by numerous rivers, lakes and forests. The Notre Dame or Green Mountains, a continuation of the Appalachian range, extend along nearly the whole of the south side of the St. Lawrence. The Laurentian Mountains skirt the northern bank of the same river and the Ottawa, but 130 miles west of Montreal cross the Ottawa and curve in the direction of Kingston, whence they run westward to the shores of Lakes Superior and Huron. Quebec abounds in large rivers, bays and lakes. The St. Lawrence, navigable for ships of over 5,000 tons as far as Montreal, flows through the entire length of the province, receiving a short distance above Montreal the waters of the Ottawa, a river 600 miles in length. The St. Maurice, which rises in Lake Oskelaneo and empties into the St. Lawrence at Three Rivers, is over 400 miles in length. he rivers Baliscan, Jacques Cartier and Montmorency, the last named famous for its falls, also enter the St. Lawrence from the north. The Sanguenay, rising in Lake St. John and flowing into the St. Lawrence at Tadousac, is one of the most remarkable bodies of water in the world, varying in depth from 100 to 1,000 feet. The Richelieu, draining Lake Champlain, discharges into the St. Lawrence from the south, and so do the St. Francis, Chaudiere, Chateauguay, Yamaska, Etchemin, Loup, Assumption, Benancour and North. The longest lake in the province is Lake St. John, with an area of 360 square miles. Its principal islands are Anticosti, Bonaventure and the Magdalen Islands, situated in the Gulf of St. Lawrence, 50 miles north of Prince Edward Island. The Laurentian formation, which extends through the central portion of the province, holds thick beds of limestone and iron ore and large deposits of graphite. Farther west the Laurentian formation is succeeded by the Potsdam sandstone. On this rests a dolomitic limestone, and the limestone of the Lower Silurian age, known as the Chazy and Trenton groups and affording the best building stone of the province. The cold in Quebec in winter is generally steady; the thermometer often registers 20 degrees below zero, and the snow, falling to a great depth, lies on the ground from November is generally warm and pleasant, is indicated at until April. The extreme heat of summer, which 90°; Autumn, which lasts six or eight weeks, is the finest season of the year. Vegetation develops rapidly, there is a rich and loamy soil in much of the country, and in the eastern townships of the province farming is the chief industry. The principal crops are wheat, barley, oats, rye, peas, buckwheat, potatoes, clover, hay and corn. The value of horses, cattle, sheep and swine in the province in 1902 was $2,413,416. The total yield and value of the fisheries of the province in the same year was $2,174,459, salmon and cod making up more than half of this total. The number of vessels engaged in the fisheries in 1901 was 786, having a tonnage of 25,605, and manned by 6,214 men. The value of the imports entered for home consumption by the province in 1902 was $73,634,001, and the value of the goods exported from the province in the same year was $91,057,201. There are 147 banks (including branches) in the province, and a clearing-house at Montreal with annual transactions in 1902 of $1,089,976,730. There are also 157 post-office savings banks in the province. There were in 1902 3,445 miles of steam railway and 171 miles of electric railway in operation in the province. The value of Quebec's mineral production in 1901 was $3,761,639. The receipts of the province in 1902 were $4,515,170 and its expenditures $4,490,677, of which $418,536 was used for charities and $455,185 for public instruction. There are published in Quebec 15 daily newspapers and 180 periodicals of other descriptions, or one for every 8,355 of population. Educational matters in the province of Quebec are under the control of a superintendent of public instruction assisted by a council consisting of 35 members, and divided into committees for the management of Roman Catholic and Protestant schools, respectively. Compulsory school age is from 5 to 16 years. The schools are maintained partly by local taxation and partly by government grants and are individually controlled by local boards. There are in the province 5,298 elementary schools, 568 model schools, 166 academies, II normal schools, 19 classical colleges, 4 universities, 4 institutions for blind and deaf-mutes, and 8 schools of arts and manufactures. Chief among the higher institutions of learning are Laval University at Quebec, and McGill University and the Presbyterian College at Montreal. There are also in the province 5 industrial and the same number of reformatory schools. The system of government established in Canada under the Union Act of 1867 is a federal union having a general or central government controlling matters essential to the general development, the permanency and the union of the whole Dominion, and a number of provincial organizations, each governed by a lieutenantgovernor, nominated and removable by the government of the Dominion, and advised by a council responsible to the people's representatives, and with a legislature composed in Quebec of two houses-a council appointed by the Crown, and an elective assembly. The Parliament of Canada consists of a Senate and House QUEBEC of Commons. The Senate, as at present constituted, has 81 members, 24 of whom are from Quebec. The House of Commons consists at present of 213 members, the basis of representation fixed under the provisions of the Act of Confederation, being that the province of Quebec is always to have 65 representatives, and each of the other provinces such a number as will give the same proportion of representatives to its population as the number 65 bears to the population of Quebec as ascertained by a decennial census. In the province of Quebec the qualifications for the electoral franchise are ownership or occupancy of real property, position as teachers or clergymen after five months' domicile in electoral district; income or personal property of specified amount-real, or real and personal, valued at $300, $200 and $100 (fishermen). Income, rentiers $100; others $300. Absentees in the United States may vote if they have returned with their families and have resided in an electoral district one month before election. Voting in elections is by ballot. No property qualification is demanded from a member of the Commons, nor is he limited to a residence in the district from which he is elected. The municipal organization of the province comprises townships, being rural districts of an area of eight or ten square miles; villages with a population of over 750; towns with a population of over 2,000, and cities established from the growth of towns when their population exceeds 15,000. The townships and villages are administered by a reeve and councillors; the towns by a mayor and councillors, and the cities by a mayor and aldermen. The judiciary of the province consists of superior, district and county courts, whose judges are appointed by the governor-general in council, and from which appeal can be had to the supreme court of Canada presided over by a chief justice and five associate judges. Final appeal, except in criminal cases, lies by leave to the judicial committee of the privy council of England. Police magistrates and justices appointed by the provincial government also have their place in the administration of justice. The crown-lands of the province, of which about 6,625,000 acres have been surveyed, are controlled by its commissioner of colonization and public works. They are sold at from 20 to 60 cents an acre, and the purchaser is required to take possession of the land within six months of the date of the sale, and to occupy it during two years. The population of the province in 1901 was 1,648,898, of whom 477,687 dwelt in towns of more than 4,000. The number of families in the same year was 307,304. Roman Catholics comprise the greater part of the population, in 1901 numbering 1,429,260. The numerous churches and convents of the province attest at every turn the wealth and power of the Church, and the tourist finds the whole land practically parceled out among its saints, as far as concerns the nomenclature of the settlements and villages. Quebec, too, has her Notre Dame de Lourdes, to whose shrine the faithful flock by thousands. Twenty miles east of the city of Quebec, on the banks of the St. Lawrence, is the Church of Ste. Anne de Beaupré, or, as the Saint is more particularly known, La Bonne Ste. Anne, who for two centuries has won fame in Canada for miraculous cures. This historic place rests on a little plateau under the shelter of a lofty mountain of the Laurentides, and consists of a straggling street of wooden houses, with steep roofs and projecting eaves. Here one will see on the fête of Ste. Anne, and at other fixed times, a mass of people from every part of Canada as well as from the United States. A handsome gray stone church attests the faith of the thousands who in past years have offered their supplications at the shrine of La Bonne Ste. Anne. Piles of crutches are deposited in every available corner as so many votive offerings from the countless cripples who have been cured or relieved. The relic through which all the cures are said to be effected consists of a part of the finger bone of Ste. Anne, which was sent in 1668 by the Chapter of Carcassone to Monseigneur de Laval. The situation of many of the villages of Quebec is exceedingly picturesque when they nestle in some nook by the side of river or bay, or overlook from some hill a noble panorama of land and water. The spire of the stone church rises generally from the midst of the houses, always making the most conspicuous object in the surrounding landscape. The houses are for the most part built of wood. The roofs are often curved, with projecting eaves, which afford a sort of veranda, under which the family sit in summer evenings. Some of the more pretentious structures, especially the inns, have balconies running across the upper story. Many of the barns and outhouses have thatched roofs, which are never seen in any other part of Canada. The interiors are very plainly furnished, in many cases with chairs and tables of native manufacture. A high iron stove is the most important feature of every dwelling in a country where the cold of winter is so extreme. Whitewash is freely used inside and outside, and there is on the whole an air of cleanliness and comfort in the humblest cottage. No class of the population of Canada is more orderly or less disposed to crime. Early marriages have always been encouraged by the priests, and large famlies are the rule in the villages. The temperate habits of the people make them valuable employees in mills and manufactories of all kinds, and until a recent period there was a steady exodus from the province to the manufacturing towns of New England. A large proportion of the men employed in the lumbering industry of Canada is also drawn from the province of Quebec. In commercial and financial enterprises, however, the French Canadians cannot compete with their fellow citizens of British origin, who practically control the great commercial undertakings and banking institutions of Quebec, especially in Montreal. Nor, as a rule, can they compare with the English population as agriculturists. The French system of sub-dividing farms among the members of a family has tended to cut up the land unprofitably, and it is a curious sight to see the number of extremely narrow lots throughout the province. The French population also has less enterprise, and less disposition to adopt new machines and improved agricultural implements than the people of the other provinces of Canada. In times past there was no native literature, and little general culture except in small select circles at Quebec and Montreal; but during the past half century, with the increase of wealth, |