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104 Q U A Q U A Perhaps to the student there is no paxt of elementary mathematics so repulsive as is spherical trigonometry. Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with re- gard to axes fixed in the system, is a matter of trouble- some complexity by the usual methods. But every quaternion formula is a proposition in spherical (some- times degrading to plane) trigonometry, and has the full advantage of the symmetry of the method. And one of Hamilton's earliest advances in the study of his system (an advance independently made, only a few months later, by Cayley) was the interpretation of the singular operator ?( )?~S where q is a quaternion. Applied to any directed line, this operator at once turns it, conically, through a definite angle, about a definite axis. Thus rotation is now expressed in symbols at least as simply as it can be exhibited by means of a model. Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms. The physical properties of a heterogeneous body (pro- vided they vary continuously from point to point) are known to depend, in the neighbourhood of any one point of the body, on a quadric function of the coordinates with reference to that point. The same is true of physical quantities such as potential, temperature, &c., throughout small regions in which their variations are continuous; and also, without restriction of dimensions, of moments of inertia, &c. Hence, in addition to its geometrical applications to surfaces of the second order, the theory of quadric functions of position is of fundamental importance in physics. Here the symmetry points at once to the selection of the three principal axes as the directions for i, y, k; and it would appear at first sight as if quaternions could not simplify, though they might improve in ele- gance, the solution of questions of this kind. But it is not so. Even in Hamilton's earlier work it was shown that all such questions were reducible to the solution of linear equations in quaternions; and he proved that this, in turn, depended on the determination of a certain operator, which could be represented for purposes of calculation by a single symbol. The method is essentially the same as that developed, under the name of "matrices" by Cayley in 1858; but it has the peculiar advantage of the simplicity which is the natural consequence of entire freedom from conventional reference lines. Sufficient has already been said to show the close con- nexion between quaternions and the theory of numbers. But one most important connexion with modern physics must be pointed out, as it is probably destined to be of great service in the immediate future. In the theory of surfaces, in hydrbkinetics, heat-conduction, potentials, &c., we constantly meet with what is called Laplace's operator, d? d? d? viz., jT2 + J~2+T~2- We know that this is an invariant; i.e., it is independent of the particular, directions chosen for the rectangular coordinate axes. Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression i jr+Jj~+^J could be, like ix +jy + Jcz, effectively expressed by a single letter. He chose for this purpose V- And we now see that the square of y is the negative of Laplace's operator; while v itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity. Thus, applied to a potential, it gives the direc- tion and magnitude of the force; to a distribution of temperature in a conducting solid, it gives (when mul- tiplied by the conductivity) the (lux of heat, ttc. No better testimony to the value of the quaternion, method could be desired than the constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like Clerk-Maxwell (in his Electricity and J/<n/- netisni). Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely trans- lating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions. Works on the Subject. Of course the great works on this sub- ject are the two immense treatises by Hamilton himself. Of these the second (Elements of Quaternions, London, 1866) was posthumous incomplete in one short part of the original plan only, but that a most important part, the theory and applications of V. These two works, along with Hamilton's other papers on quaternions (in the Dublin Proceedings ami Transactions, the Philo- sophical Magazine, &c. ), are storehouses of information, of which but a small portion has yet been extracted. A German translation of Hamilton's Elements has recently been published by Glan. Other works on the subject, in order of date, are Allegret, Essai sur le Calcul dcs Quaternions (Paris, 1862); Tait, An Elementary Treatise on Quaternions (Oxford 1867, 2d ed. 1873; German translation by v. Scherif, 1880, and French by Plarr, 1882-84); Kelland and Tait, Introduction to Quaternions (London, 1873, 2d ed. 1882); Hoiiel, Elements de la Theorie des Quaternions (Paris, 1874); Unverzagt, Theorie der Quuternionen (Wiesbaden, 1876); Laisant, Introduction a la Methode des Quaternions (Paris, 1881); Graefe, Vorlesungen uber die Theorie der Quaternionen (Leipsic, 1884). An excellent article on the " Principles " of the science, by Dillner, will be found in the Mathernatische Annalen, vol. xi., 1877. And a very valuable article on the general question, Linear Associated Algebra, by the late Prof. Peirce, was unfortunately lithographed for private circulation only. Sylvester and others have recently pub- lished extensive contributions to the subject, including quaternions under the general class matrix, and have developed much farther than Hamilton lived to do the solution of equations in quaternions. Several of the works named above are little more than compilations, and some of the French ones are painfully disfigured by an attempt to introduce an improvement of Hamilton's notation; but the mere fact that so many have already appeared shows the sure progress which the method is now making. (P. G. T.) QUATREMERE, ETIENNE MARC (1782-1857), one of the most learned of modern Orientalists, came of an eminent family of Parisian merchants. His father was a victim of the Revolution, his mother a pious woman devoted to works of charity and venerated after her death almost as a saint. The son retained much of what was best in the old spirit of the Parisian bourgeoisie its industry, sobriety, and independence of character, along with a certain narrowness of view. He was sincerely religious, with strong Jansenist and Gallican tendencies, a touch of rationalism, and a great dislike of modern growths of Catholicism. His whole life was spent alone among his books, and his works always display the most extensive and accurate erudition in which indeed, and not in criticism or original ideas, his strength lay. Employed in 1807 in the manuscript department of the imperial library, he passed to the chair of Greek in Rouen in 1809, entered the academy of inscriptions in 1815, taught Hebrew and Aramaic in the College de France from 1819, and finally in 1827 became professor of Persian in the School of Living Oriental Languages. Quatremere's first work was Recherches sur la langue et la littera- ture de I'Egypte (1808), showing that the language of ancient Egypt must be sought in Coptic. His Mem. sur les Nabateens (1835) has been mentioned under NABAT^EANS, and his translation of MAKIU'ZI'S history of the Mameluke sultans in the article on that author. The valuable notes to the latter book show his erudition at the best. He published also among other works a translation of Rashid al-Din's Hist, des Mongols de la Perse (1836); Mem. geog. et hist, sur VEgypte (1810); the text of Ibn Khaldun's Prolegomena; and a vast number of useful memoirs in the Jour. As. His numerous reviews in the Jour, des Savants should also be men- tioned. Quatremere made great lexicographic collections in Ori- ental languages, fragments of which appear in the notes to his