QUATRE-BRAS 



QUEBEC 



523 



involving; only three numbers ; while a scalar, 

 which might be denned as the quaternion which 

 changes one vector into a parallel one, is still more 

 degenerate, involving only one number viz. itself. 

 There is still one very important representation 

 of a quaternion to consider. This is done most 

 simply aa follows : Let o be the two vectors OA, 

 OB (fig. 5). Resolving 

 ft along and perpen- 

 dicular to a we get /3 = 

 OM + ON ; and hence 



Fig 5. 



But OA.OM, being the 

 product of two parallel 

 vectors, is minus the 

 product of the lengths 

 or tensors. On the 



other hand, the product OA.ON, being the product 

 of two perpendicular vectors, is a vector perpen- 

 dicular to the plane of the paper with tensor equal 

 to twice the area of the tnangle OAB. Thus the 

 quaternion a/3 is equal to the sum of a scalar and a 

 vector; and generally for any quaternion (q) we 

 have the relation 



q = S.q + V.q, 



where S selects the scalar part and V the vector 

 part. The geometrical meanings of S and V 

 operating on oft are easily seen to DC these 



S.o/3 = - TaT/3 cos 0, V.a/3 = iTaT/3 sin 9, 



where is the unit vector perpendicular to a 

 and 8. 



We end with a few illustrations. Thus, if a is 

 a constant vector, and p a variable vector, the 

 equation S.a/> = c, a constant, means that the 

 resolved part of p along the direction of a is con- 

 stant, and that therefore the extremity of p traces 

 out a plane perpendicular to a. The versor that 

 tarns any line through an angle in a given plane 

 has the form cos t> i sin 6, where / is the right 

 versor perpendicular to that plane. Demoivre's 

 theorem (see DEMOlVRE) at once follows if we 

 write i = V - 1. Finally, if ft represents a force 

 acting at the extremity of o, V.a/3 is the vector 

 moment of the force aliout the origin ; and in the 

 almost self-evident equation 



V.a (/3 + 1 ) = V.o/3 + V.a/3i 



we have a completely general demonstration of 

 Varignon's theorem of moments. See MOMENT. 



Hamilton'* Lecture* on Quaternion* (185:5) and his 

 Elrmtnt* of Quaternion* (1866) are still the classical 

 work! on the subject. Tail's Element* of Qiiatemiims 

 (3d ed. 1890) is probably better fitted as a text-book for 

 the student to work through, and contains some original 

 applications of high physical interest. Kelland and Tail's 

 Introduction to Quaternion* (1874) may be recommended 

 to the beginner. Tail's treatise has been translated into 

 French and German. 



Quatre-Bras, a village of Belgium, aliout 10 

 mih'-< SSE. of Waterloo, situated at the intersection 

 of the great roads from Brussels to Charleroi, and 

 from Nivelles to Naiiuir, whence its name ('four 

 arms'). On 16th June 1815, two days before the 

 battle of Waterloo (q.v. ), Quatre-Bras was the 

 scene of a desperate battle between the Kn^lish 

 under Wellington and the French under Ney. 

 The honours of the field remained with the former ; 

 bat the severe defeat of Bliicher the same day at 

 Ligny compelled Wellington to retreat. The loss 

 on the English side was 5200, on the French 4140, 

 amongst the Allies lieing the Duke of Brunswick, 

 the gallant chief of the Black Brunswickers. A 

 monument to his memory, a bronze lion 10 feet 

 high, wan erected in 1890. 



Quatrefages, JEAN Louis ARMAXD UK, a 

 naturalist, was born at Berthezenic ((!:ird) on 



B 



Quatrefoil. 



10th February 1810, studied medicine at Strasburg, 

 and in 1838 was appointed professor of Zoology at 

 Toulouse. But this post ne soon resigned and 

 went to Paris, to study further for himself. In 

 1850 he was elected professor of Natural History in 

 the Lw.ee Napoleon, and in 1855 of Anatomy and 

 Ethnology at the Natural History Museum in 

 Paris. He devoted his attention principally to 

 anthropology and the lower animals, especially 

 annelids. His chief works are L'Espece Hmnaine 

 (1877; 8th ed. 1886; Eng. trans. 1879) ; Sourcnirs 

 d'un Naturaliste ( 1854 ; Eng. trans. 1857 ) ; Unite 

 de I'Espece Surname (1861); Crania Ethnica 

 ( 1875-82) ; La Race Prussienne ( 1879 ; Eng. trans. 

 1872) ; Les Pygmees ( 1887) ; Histoire Naturelle des 

 Anneles (2 vols. 1866); Darwin et ses Precttrs- 

 eurs Francois (1892); and Theories Transformistes 

 ( 1892). He died 13th January 1892. See ANTHRO- 

 POLOGY. 



Quatrefoil, an opening in tracery, a panel, 

 &c., divided by cusps or feather- 

 ings into four leaves. This form 

 is much used as an ornament in 

 Gothic architecture. 



Quatremere, ETIENNE 

 MARC, a French orientalist, was 

 born in Paris, 12th July 1782, and 

 from his earliest childhood to his 

 latest years was immersed in 

 study ; he lived more after the 

 fashion of a mediaeval recluse than a modern 

 scholar. Employed in 1807 in the manuscript 

 department of the Imperial Library, he was pro- 

 moted in 1809 to the Greek chair in the College of 

 Rouen, and in 1819 to the chair of Ancient Orien- 

 tal Languages in the College de France, and in 1827 

 be liecame professor of Persian in the School for 

 Modern Oriental Languages. He died 18th Sep- 

 teml>er 1857. Although a man of vast and accurate 

 knowledge, he had little critical insight or origin- 

 ality. His principal works are Beclierches sur la 

 Langue et la Litttratitre de I'f'gypte ( 1808 ), proving 

 that the language of ancient Egypt is to be sought 

 for in modern Coptic ; Memoires Geographiques et 

 Historiques sur l'gyj>te (1810); M (moire sur les 

 Nabateens ( 1835 ) ; flistoire des Sultans Mameloucks 

 (1837), from the Arabic of Makrizi ; Histoire des 

 Mongols de la Perse (1836), from the Persian of 

 Rashid ed-Din ; an edition of the Arabic text of 

 the Prolegomena of Ibn-Khaldun ; and a multitude 

 of articles scattered through the pages of the 

 Juunittl Asiatique and the Journal des Savants. 

 Besides this, he gathered materials for Arabic, 

 Coptic, Syriac, Turkish, Persian, and Armenian 

 dictionaries. 



<tliatt.ro Cento (Hal., 'four hundred, 'a con- 

 traction for one thousand four hundred ; cf. ClNQUE 

 CENTO), in Italian a term for the 15th century, its 

 literature and art ; the early Renaissance. Out- 

 standing Quattrocentisti in art are Ponatello, Delia 

 Robbia, Bninellesco, Masaccio, Ghirlandajo, Lippo 

 Lippi, and Mantegna. 



Quebec* a province of the Dominion of Canada, 

 lies to the east of Ontario, and l>etween that pro- 

 vince and New Brunswick. Deducting the surface 

 of its inland waters, including the River and Gulf 

 of St Lawrence, the area of Quebec, including 

 recent additions, is 347,350 sq. in. The surface 

 of the country is varied and grand, consisting of 

 extensive rivers and lakes, large stretches of agri- 

 cultural land, and immense forests. Two ranges 

 of mountains run through the province from south- 

 west to north-east, that on the south side of the St 

 Lawrence being called the Notre Dame or Green 

 Mountains, stretching from Quebec to Gaspe, 

 while on the north side of the river is the Lauren- 

 tian Range (see CANADA). The chief river in the 



