QUATEBNIONS. 599 QUATREMERE. I shows that ;i =_—/.-, 7.7 = — j,i7^ = —y, whence The severe defeat of liliiclicr the same day %^~2_^i rj"'. 3t Ligny rendered Wellington's liard-won victorV I— -7 Z '■' almost valueless; and the British eommander re- . fci— — ifc _ ;. tired next morning through Jeniappes to Water- These relations, together with the consequent loo in order to keep up his enmniunication with equation i;/.: = — ^1 and the relation already men- the Prussian army. The Duke of Brunswick, tioned, that i" — y —k' — —, form the basis commanding the German troops, was killed in of the quaternion theory. They show at once this battle. See Waterloo. that in this theory multiplication is not com- QTJATREFAGES DE BREAU, ka'tr'-fazh' de bra'cV, Jean Loiis Abmamd de (1810-92). A French naturalist, born at Berthezfene (Gard). He was educated at Strassburg. where he ob- tained a doctorate in medicine in 18.S2, began practice at Toulouse, and established there the Journal de. medecine et de chintrgw dc Toulouse. From 1838 to 1840 he was professor of zoiilog^- in the faculty of sciences of the University of Toulouse, and from 1842 was at Paris, where he designed plates for the R^gne animal illus- trc, wrote for the Revue des Deux Mondes, in 1850 was appointed professor of natural his- tory at the Lycee Napoleon, Paris, and in 185.5 professor of anthropology at the JIusee d'llistoire Xaturelle. He was elected to the Academy of Sciences in 1852. He was a note- worthy teacher, and became particularly known for his anthropological investigations and his studies of the invertebrates, especially the annelids. The doctrine of phlebenterism. accord- ing to which the intestinal raniitications of a certain division of gastropods, known as Phle- benterata, have a respiratory funetion. was first expounded by him. He published an extensive list of works, including: Soureuirs d'un nalura- liste (1854) ; Histoire naturellc des aiinelrs marins (1865) ; La race prussienne (1871), which involved him in a scientific controvcr.sy with Virchow; Homines fossiles et hommes sauvages (1884); and Introduelion a I'etudc des races lunnaines (2 vols., 1887-89), considered his most important publication. For a complete list of his works, consult ilalloizel, Godefroy. Liste chro- nologique des tracaux de M. Arniand de Quntre- fages de Breau (extrait du bull(>tiii de la SociiT-tS Naturellc d'Autun) (Autun, ISIJ.'J). QTTATBEFOIL (OP. qiiatrefeuille, from quatre, from Lat. quattuor, four + feuiUe, from Lat. folium, leaf). An heraldic bearing meant to represent a flower with four leaves. See Her- aldry. QXJATEEMERE, ka'tr'mfir', Etienne JUkc (1782-1857). A learned French Orientalist, bom in Paris. He studied at the College de France under the celebrated Arabist Silvestre tie Sacj', and in 1807 was employetl in the manuscript department of the Biblioth&que luipi'riale. In 1800 he became professor of tireek in the College of Rouen, and in 1815 was elected a member of the French Institute. In 1810 he was called to the chair of Hebrew in the College de France, and in 1S38 was made jirofessor of Persian in the school for modern Oriental languages. He died at Paris. Quatremfre"s earliest works were devoted to Egv'ptian subjects. In his Recherches critiques et historiques sur la laiigue et la lifti- rature de I'Egypte (1808) he demonstrated that Coptic is the true representative of ancient Egyptian, but he later declined to accept the discoveries of Cliampollion. and would never ad- mit that the Egyptian hieroglypbics could be read phonetically. His geographical and his- torical works are of very great value, especially mutative. To illustrate the application of quaternions, let a = j-i i-H) -- sic, and /3 = a;' i + »/';' -+- z' k. Then o^ = —(xx'+ i/g' + ez') + {yz' — zy')i + {zx' — xz')j + {xy' — yx')k, and /3a = — (xx' + yy' -f zz') — (yz' — zy')i {zx' — xz')j — {xy' — yx')k. Hence Sa/3 =S/3a (i.e. the -ralars are equal), VajS = — V^a (i.e. the ver- -iiis are opposites), and a/3 -|- /So =2 S o/S. X ( iw (a + ^)- = a^ + a/3 +^a + /32 = a^ + 25a/3 + ^-, which is the ordinary trigonometric formula for c- — (J- — 2ab ■ cos C +6=. Also V (a-f /3)(a — /3)= Va- — Va/3 + V/3o — V/S^ = — Va/3 + V/3a, because Va2=— Vo2 and hence is zero; and this equals = 2Vap. Taking the tensors of both sides of the equation V(a + /S) = — 2Vo/3, we have the theo- rem : The parallelogram whose sides are paral- lel and equal to the diagonals of a given paral- lelogram has twice the area of the latter. Fur- thermore 5(a-(-/3)(a — /3) = a^ — /3£, and vanishes onl_v when a^ = P'-, or ra = t/3 ; whence the diagonals of a ])arallelogram are perpendicular t-o each other when and oul.v when the sides are all equal. The chief application of quat- Diions, however, is in physical problems, and I Ml- these reference must be made to works u])on the subject. It is evident that the complex mimljer admits of still further generalization, to the form a = o,j, -f oJ, + 03%+ (i„-rf'„-i + <ij„. This theory has been developed by Weier- strass ( Gijttinger Naclirichten, 1884-86) , Schwarz, Dedekind, Holder, and others. The leading works on quaternions and the related Ausdehnungskhre of Grassinann (q.v.) are the following: Grassmann, Ausdehnungs- Ichre (1844); Hamilton, Lectures on Quater- nions (Dublin, 1853) ; Elements of Quaternions (London, 180G; 2d ed. 1809); Taft, Elementary Treatise on Quaternions (Oxford, 1867): 2d ed. 1873) ; Hoiiel, Thcorie des quaternions (Paris, 1874) ; Laisant, Methode des quaternions (Paris, 1881); ilcAulay, Vtility of Quaternions in Physics (London, 1893). QXTATEAIIT, kwot'ran (Fr. quatrain, from (imitrc. from Lat. quattuor, four). A name given (originally by the French) to a little poem of four verses (lines) rhyming alternately, or even sometimes to four verses of a longer poem, such as a sonnet, if they contain a complete idea within themselves. Epigrams, epitaphs, pro- verbs, etc., arc often expressed in quatrains. QUATRE-BEAS, katr' bra. A village in the Province of Brabant. Belgium, about 19 miles southeast of Brussels (Map: Belgium, C 4). It is situated at the intersection of the great roads from Brussels to Charleroi, and from Nivelles to Namur, whence its name. On June 10, 1815, two days before the battle of Waterloo, Quatre-Bras was thr scene of a desperate and sanguinary battle between the British and their German allies under Wellington and the French under Ney, in which the former were victorious. I
