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are parallel, or 0=0, the product, which is now the square of any that of Grassmann. But it is to be observed that Grassmann, (unit) line is -1. And when the two factor lines are at night angles I though he virtually accused Cauchy of plagiarism. does not to one another, or @ = /2, 'the product is simply ix' +jy' + ks", the unit line perpendicular to both. Hence, and in this lies the main appear to have preferred any such charge against Hamilton. element of the symmetry and simplicity of the quaternion calculus, He does not allude lo Hamilton in the second edition of his all systems of three mutually rectangular unit lines in space have

work. But in 1877, in the Mathematische Annalen xii., he the same properties as the fundamental system i, j, k. In other words, if the system (considered as rigid) be made to turn about

gave a paper “ On the Place of Quaternions in the Ausdehtill the first factor coincides with i and the second with j, the pro

nungslehre," in which he condemns, as far as he can, the nomenduct will coincide with k. This fundamental system, therefore, clature and methods of Hamilton. becomes unnecessary; and the quaternion method, in every case,

There are many other systems, based on various principles, which takes its 'reference lines solely from the problem to which it is have been given for application to geometry of directed lines, but applied. It has therefore, as it were, a unique internal character those which deal with products of lines are all of such complexity tits own.

as to be practically useless in application. Others, such as the Hamilton, having gone thus far, proceeded to evolve these results

Barycentrische Calcil of Möbius, and the Méthode des equipollences rom a characteristic train of a priori or metaphysical reasoning. of Bellavitis, give elegant modes of treating space problems, so

Let it be supposed that the product of two directed lines is some long as we confine ourselves to projective geometry and matters of thing which has quantity; i.e. it may be halved, or, doubled, for that order; but they are limited in their field, and therefore need instance. Also let us assume (a) space to have the same properties not be discussed here. More general systems, having close analogies in all directions, and make the convention (b) that to change the to quaternions, have been given since Hamilton's discovery was sign of any one factor changes the sign of a product. Then the published. As instances we may take Goodwin's and O'Brien's product of two lines which have the same direction cannot be, even papers in the Cambridge Philosophical Transactions for 1849. (See in part, a directed quantity. For, if the directed part have the same also ALGEBRA: special kinds.) direction as the factors, (b) shows that it will be reversed by reversing either, and therefore will recover its original direction when

Relations to other Branches of Science.-The above narrative both are reversed. But this would obviously be inconsistent shows how close is the connexion between quaternions and the with (a). If it be perpendicular to the factor lines, (a) shows that I ordinary Cartesian space-geometry. Were this all the gain by it must have simultaneously every such direction. Hence it must

their introduction would consist mainly in a clearer insight into be a mere number. Again, the product of two lines at right angles to one another

the mechanism of co-ordinate systems, rectangular or not-a cannot, even in part, be a number. For the reversal of cither factor very important addition to theory, but little advance so far as must, by (b), change its sign. But, if we look at the two factors practical application is concerned. But, as yet, we have not in their new position by the light of (a), we see that the sign must

taken advantage of the perfect symmetry of the method. not change. But there is nothing to prevent its being represented by a directed line is, as further applications of (a) and (b) show we

When that is done, the full value of Hamilton's grand step must do, we take it perpendicular to cach of the factor lines. Hamilton becomes evident, and the gain is quite as extensive from the seems never to have been quite satisfied with the apparent hetero- practical as from the theoretical point of view. Hamilton, in geneity of a quaternion, depending as it does on a numerical and

fact, remarks, “I regard it as an inelegance and imperfection a directed part. He indulged in a great deal of speculation as to

in this calculus, or rather in the state to which it has hitherto the existence of an extra spatial unit, which was to furnish the maison d'être of the numerical part, and render the quaternion

been unfolded, whenever it becomes, or secms to become, homogeneous as well as linear. But for this we must refer to his necessary to have recourse ... to the resources of ordinary own works.

algebra, for the solution of equations in quaternions." This Hamilton was not the only worker at the theory of sets. The refers to the use of the x, y, co-ordinates,-associated, of course, year after the first publication of the quaternion method, there with i, j, k. But when, instead of the highly artificial expression appeared a work of great originality, by Grassmann,' in which ixtiy+kz, to denote a finite directed line, we employ a single results closcly analogous to some of those of Hamilton were letter, a (Hamilton uses the Greek alphabet for this purpose), given. In particular, two species of multiplication (“inner" and find that we are permitted to deal with it exactly as we and “outer") of directed lines in one plane were given. The should have dealt with the more complex expression, the results of these two kinds of multiplication correspond respec- immense gain is at least in part obvious. Any quaternion may tively to the numerical and the directed parts of Hamilton's now be expressed in numerous simple forms. Thus we may quaternion product. But Grassmann distinctly states in his regard it as the sum of a number and a line, ata, or as the preface that he had not had leisure to extend his method to product, By, or the quotient, de', of two directed lines, &c., angles in space. Hamilton and Grassmann, while their earlier while, in many cases, we may represent it, so far as it is required, work had much in common, had very different objects in view. | by a single letter such as 9, 1, &c. Hamilton had geometrical application as his main object; when Perhaps to the student there is no part of elementary mathehe realized the quaternion system, he felt that his object was matics so repulsive as is spherical trigonometry. Also, everygained, and thenceforth confined himself to the development thing relating to change of systems of axes, as for instance in of his method. Grassmann's object seems to have been, all the kinematics of a rigid system, where we have constantly to along, of a much more ambitious character, viz. to discover, if

consider one set of rotations with regard to axes fixed in space, possible, a system or systems in which every conceivable mode and another set with regard to axes fixed in the system, is a of dealing with sets should be included. That he made very matter of troublesome complexity by the usual methods. But great advances towards the attainment of this object all will every quaternion formula is a proposition in spherical (sometimes allow; that his method, even as completed in 1862, fully degrading to plane) trigonometry, and has the full advantage of attains it is not so certain. But his claims, however great they the symmetry of the method. And one of Hamilton's earliest may be, can in no way conflict with those of Hamilton, whose advances in the study of his system (an advance independently

ultiplying couples (in which the inner "and outer " | made, only a few months later, by Arthur Cayley) was the multiplication are essentially involved) was produced in 1833, interpretation of the singular operator al Dol, where q is a and whose quaternion system was completed and published quaternion. Applied to any directed line, this operator at once before Grassmann had elaborated for press even the rudimentary turns it, conically, through a definite angle, about a definite portions of his own system, in which the veritable difficulty of axis. Thus rotation is now expressed in symbols at least as the whole subject, the application to angles in space, had not simply as it can be exhibited by means of a model. Had even been attacked. Grassmann made in 1854 a somewhat quaternions effected nothing more than this, they would still savage onslaught on Cauchy and De St Venant, the former of have inaugurated one of the most necessary, and apparently whom had invented, while the latter had exemplified in applica- | impracticable, of reforms. tion, the system of “ clefs algébriques,which is almost precisely The physical properties of a heterogeneous hody (provided

they vary continuously from point to point) are known to depend, Die Ausdehnungslehre, Leipsic, 1844; 2nd ed., vollständig und

in the neighbourhood of any one point of the body, on a quadric in strenger Form bearbeitet, Berlin, 1862. See also the collected works of Möbius, and those of Clifford, for a general explanation of

function of the co-ordinates with reference to that point. The Grassmann's method.

* Lectures on Quaternions, $ 513. XXII 12

same is true of physical quantities such as potential, temperature, method quaternions have from tke beginning received much &c., throughout small regions in which their variations are attention from mathematicians. An attempt has recently been continuous; and also, without restriction of dimensions, of made under the name of multenions to systematize this algebra. moments of inertia, &c. Hence, in addition to its geometrical We select for description stage (3) above, as the most charapplications to surfaces of the second order, the theory of quadric acteristic development of quaternions in recent years. For functions of position is of fundamental importance in physics. (3) (a) we are constrained to refer the reader to Joly's own Here the symmetry points at once to the selection of the three Manual of Qualernions (1905). principal axes as the directions for i, j, k; and it would appear The impulse of W. K. Clifford in his paper of 1873 (" Preat first sight as if quaternions could not simplify, though they liminary Sketch of Bi-Quaternions," Mathematical Papers, might improve in elegance, the solution of questions of this p. 181) seems to have come from Sir R. S. Ball's paper on the kind. But it is not so. Even in Hamilton's earlier work it Theory of Screws, published in 1872. Clifford makes use of a was shown that all such questions were reducible to the solution quasi-scalar w, commutative with quaternions, and such that if of linear equations in quaternions; and he proved that this, in P,9, &c., are quaternions, when ptwerp' two', then necessarily turn, depended on the determination of a certain operator, p=Ć, q=(. He considers two cases, viz. W=I suitable which could be represented for purposes of calculation by a for non-Euclidean space, and wi=o suitable for Euclidean single symbol. The method is essentially the same as that space; we confine ourselves to the second, and will call the developed, under the name of " matrices,” by Cayley in 1858; | indicated bi-quaternion ptwq an oclonion. In octonions the but it has the peculiar advantage of the simplicity which is analogue of Hamilton's vector is localized to the extent of being the natural consequence of entire freedom from conventional confined to an indefinitely long axis parallel to itself, and is reference lines.

called a rotor; if p is a rotor then wp is parallel and equal to e, Sufficient has already been said to show the close connexion and, like Hamilton's vector, wp is not localized; wp is therefore between quaternions and the theory of numbers. But one called a vector, though it differs from Hamilton's vector in that most important connexion with modern physics must be pointed the product of any two such vectors wp and wo is zero because out. In the theory of surfaces, in hydrokinetics, heat-con- | wi=0.ptwo where p, o are rotors (i.e. p is a rotor and wo a duction, potentials, &c., we constantly meet with what is called vector), is called a motor, and has the geometrical significance of

Ball's wrench upon, or twist about, a screw. Clifford considers Laplace's operator," viz. Tuttit Joz. We know that this

an octonion p+wq as the quotient of two motorsptwo,p' two'. is an invariant; i.e. it is independent of the particular directions This is the basis of a method parallel throughout to the chosen for the rectangular co-ordinate axes. Here, then, is a

quaternion method; in the specification of rotors and motors case specially adapted to the isotropy of the quaternion system; it is independent of the origin which for these purposes the

quaternion method, pure and simple, requires. and Hamilton easily saw that the expression in tiž+ki, Combebiac is not content with getting rid of the origin in could be, like ix+jy+kz, effectively expressed by a single

| these limited circumstances. The fundamental geometrical letter. He chose for this purpose V. And we now see that the

conceptions are the point, line and plane. Lines and comsquare of V is the negative of Laplace's operator; while V itself,

plexes thereof are sufficiently treated as rotors and motors, when applied to any numerical quantity conceived as having a

but points and planes cannot be so treated. He glances at definite value at each point of space, gives the direction and the

Grassmann's methods, but is repelled because he is seeking rate of most rapid change of that quantity. Thus, applied to

a unifying principle, and he finds that Grassmann offers him

not one but many principles. He arrives at the tri-quaternion a potential, it gives the direction and magnitude of the force; |

as the suitable fundamental concept. to a distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the flux of heat, &c.

We believe that this tri-quaternion solution of the very

interesting problem proposed by Combebiac is the best one. No better testimony to the value of the quaternion method could be desired than the constant use made of its notation by mathe

But the first thing that strikes one is that it seems unduly maticians like Clifford (in his Kinematic) and by physicists like Clerk complicated. A point and a plane fix a line or axis, viz. Maxwell (in his Electricity and Magnetism). "Neither of these men that of the perpendicular from point to plane, and therefore professed to employ the calculus itself, but they recognized fully

a calculus of points and planes is ipso facio a calculus of lines the extraordinary clearness of insight which is gained even by merely translating the unwieldy Cartesian expressions met with

also. To fix a weighted point and a weighted plane in in hydrokinetics and in electrodynamics into the pregnant language | Euclidean space we require 8 scalars, and not the 12 scalars of quaternions.

(P. G. T.) of a tri-quaternion. We should expect some species of biSupplementary Consideralions.--There are three fairly well- quaternion to suffice. And this is the case. Let m, w be two. marked stages of development in quaternions as a geometrical quasi-scalars such that ma=n, wn=w, nw=w:=o. Then the bimethod. (1) Generation of the concept through imaginaries quaternion ng twr suffices. The plane is of vector magniand development into a method applicable to Euclideantude V9, its equation is Spe=Sr, and its expression is the geometry. This was the work of Hamilton himself, and the bi-quaternion Va+wSr; the point is of scalar magnitude above account (contributed to the oth ed. of the Ency. Brit. by S9, and its position vector is B, where VBq=Vr (or what is Professor P. G. Tait, who was Hamilton's pupil and after him the same, B=(Vr+q. Vr.q-]/Sg), and its expression is n Sa+wVr. the leading exponent of the subject) is a brief résumé of this (Note that the here occurring is only required to ensure first, and by far the most important and most difficult, of the harmony with tri-quaternions of which our present bithree stages. (2) Physical applications. Tait himself may be quaternions, as also octonions, are particular cases.) The regarded as the chief contributor to this stage. (3) Geometrical point whose position vector is Vrois on the axis and may applications, different in kind from, though more or less allied

be called the centre of the bi-quaternion; it is the centre of a to those in connexion with which the method was originated. sphere of radius Srol with reference to which the point and These last include (a) C. J. Joly's projective geometrical applica- plane are in the proper quaternion sense polar reciprocals, tions starting from the interpretation of the quaternion as a that is, the position vector of the point relative to the centre point-symbol:1 these applications may be said to require no is Sra! Vq/Sq, and that of the foot of perpendicular from addition to the quaternion algebra;'(6) W. K. Clifford's bi- centre on plane is Srai. Sq/V9, the product being the (radius), quaternions and G. Combebiac's tri-quaternions, which require that is (Srq')? The axis of the member XQ+x'Q of the the addition of quasi-scalars, independent of one another and of second-order complex Q, Q' (where Q=ngtwr, Q=no' twr true scalars, and analogous to true scalars. As an algebraic and x, x' are scalars) is parallel to a fixed plane and intersects " It appears from Joly's and Macfarlane's references that J. B.

a fixed transversal, viz. the line parallel to d god which Shaw, in America, independently of Joly, bas interpreted the

| intersects the axes of Q and Q'; the plane of the member quaternion as a point-symbol.

contains a fixed line; the centre is on a fixed ellipse which Wiersects the transversal; the axis is on a fixed ruled surface | Sidney, Spenser and Daniel, are really quatorzains. They to which the plane of the ellipse is a tangent plane, the ellipse consist of three quatrains of alternate rhyme, not repeated in being the section of the ruled surface by the plane; the ruled the successive quatrains, and the whole closes with a couplet. surface is a cylindroid deformed by a simple shear parallel A more perfect example of the form could hardly be found to the transversal. In the third-order complex the centre than the following, published by Michael Drayton in 1602: locus becomes a finite closed quartic surface, with three (one

Dear, why should you commend me to my rest, always real) intersecting nodal axes, every plane section of

When now the night doth summon all to sleep? which is a trinodal quartic. The chief defect of the geometrical

Methinks this time becometh lovers best;

Night was ordained together friends to keep. properties of these bi-quaternions is that the ordinary algebraic

How happy are all other living things scalar finds no place among them, and in consequence Q- is

Which though the day conjoin by several flight, meaningless.

The quiet evening yet together brings, Putting 1-n=& we get Combebiac's tri-quaternion under

And cach returns unto his love at night,

O thou that art so courteous unto all, the form Q=&p+ng+wr. This has a reciprocal Q-1= $p-1=ng

Why should'st thou, Night, abuse me only thus, -wp-q", and a conjugate KQ (such that K(QQI=

That every creature to his kind dost call, KQKQ, K[KQ]=Q) given by KQ=&Kq+nRo+wKr; the

And yet 'tis thou dost only sever us? product Qe of and Q is '+ngd' twipr' tro); the

Well could I wish it would be ever day,

If, when night comes, you bid me go away. quasi-vector (1-K)Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different

Donne, and afterwards Milton, fought against the facility sense from the rest of this article) 1(1+K)Q is Combebiac's

and incorrectness of this form of metre and adopted the Italian scalar (Sp+Sq) + Combebiac's planc. Combebiac does not use

form of sonnet. During the 19th century, most poets of K; and in place of g, n he uses urn-Ě, so that u?= 1,wur - uw

distinction prided themselves on following the strict Petrarchan =w, wa=o. Combebiac's tri-quaternion may be regarded from

model of the sonnet, and particularly in avoiding the final many simplifying points of view. Thus, in place of his general

couplet. In his most mature period, however, Keats returned tri-quaternion we might deal with products of an odd number

to the quatorzain, perhaps in emulation with Shakespeare; of point-plane-scalars (of form ug+wr) which are themselves

and some of his examples, such as “ When I have fears," point-plane-scalars; and products of an even number which

“Standing aloof in giant ignorance,” and “Bright Star," are are octonions; the quotient of two point-plane-scalars would

the most beautiful in modern literature. The “Fancy in be an octonion, of two octonions an octonion, of an octonion

Nubibus," written by S. T. Coleridge in 1819, also deserves by a point-plane-scalar or the inverse a point-plane-scalar.

notice as a quatorzain of peculiar beauty. Again a unit point k may be regarded as by multiplication

QUATRAIN, sometimes spelt Quarlain (from Fr. quatre, changing (a) from

The octonion to point-plane-scalar. (b) from four), a piece of verse complete in four rhymed lines. point-plane-scalar to octonion, (c) from plane-scalar to linear

length or measure of the verse is immaterial, but they must be element, (d) from linear element to plane-scalar.

bound together by a rhyme-arrangement. This form has If Q=&p+ng+wr and we put Q=(1+ fwt) (Ep+ng) X

always been popular for use in the composition of epigrams, (1+ wh- we find that the quaternion i must be aj(r)]f(9-p),

on account of its brevity and neatness, and may be considered where (n)=rq-K pr. The point p=Vt may be called the

as a modification of the Greek or Latin epigram at its concisest. centre of Q and the length St may be called the radius. If

QUATREFAGES DE BRÉAU, JEAN LOUIS ARMAND DE Q and Q are commutative, that is, if QQ=Q'Q, then Q and

810-1892), French naturalist, was born at Berthezène, near I have the same centre and the same radius. Thus 0,

Vallerangue (Gard), on the roth of February 1810, the son of a Q, Q?, Q,... have a common centre and common radius.

Protestant farmer. He studied medicine at Strassburg, where Q and KQ have a common centre and equal and opposite

he took the double degree of M.D. and D.Sc., one of his theses radii; that is, the i of KQ is the negative conjugate of that

being a Théorie d'un coup de canon (November 1829); next of Q. When Su=o, (1+lwu) ( ) (1+iwu)? is an operator

year he published a book, Sur les aérolithes, and in 1832 a which shifts (without further change) the tri-quaternion

treatise on L'Extraversion de la vessie. Removing to Toulouse, operand an amount given by u in direction and distance.

he practised medicine for a short time, and contributed various

memoirs to the local Journal de médecine and to the Annales BIBLIOGRAPHY.-In 1904 Alexander Macfarlane published aldes sciences naturelles (824-26). But being unable to conBibliography of Quaternions and allied systems of Mathematics for the International Association for promoting the study of Quaternions tinue his researches in the provinces, he resigned the chair of and allied systems of Mathematics (Dublin University Press); zoology to which he had been appointed, and in 1839 settled the pamphlet contains 86 pages. In 1899 and 1901 Sir W. R.

in Paris, where he found in H. Milne-Edwards a patron and Hamilton's classical Elements of Quaternions of 1866 was republished under C. J. Joly's editorship, in two volumes (London). Joly adds

dela a friend. Elected professor of natural history at the Lycée valuable notes and thirteen important appendices. In 1890 the Napoléon in 1850, he became a member of the Academy of 3rd edition of P. G. Tait's Elementary Treatise on Quaternions Sciences in 1852, and in 1855 was called to the chair of anthroappeared (Cambridge). In 1905 C. J. Joly published his Manual y and ethnography at the Musée d'histoire naturelle. of Quaternions (London); the valuable contents of this are doubled

Other distinctions followed rapidly, and continued to the end by copious so-called examples; every earnest student should take these as part of the main treatise. The above three treatises may of his otherwise uneventiul career, the more important being be regarded as the great storehouses; the handling of the subject honorary member of the Royal Society of London (June 1879), is very different in the three. The following should also be

member of the Institute and of the Académie de médecine, and mentioned: A. McAulay, Octonions, a development of Clifford's Bi-quaternions (Cambridge, 1898); G. Combebiac, Calcul des

commander of the Legion of Honour (1881). He died in triquaternions (Paris, 1902); Don Francisco Pérez de Muñoz, Paris on the 12th of January 1892. He was an accurate Iniroduccion al estudio del cálculo de Cuaterniones y otras Algebras observer and unwearied collector of zoological materials, gifted especiales (Madrid, 1905); A. McAulay, Algebra after Hamilton, or

or with remarkable descriptive power, and possessed of a clear, Multonions (Edinburgh, 1908).

(A. MCA.)

vigorous style, but somewhat deficient in deep philosophic QUATORZAIN (from Fr. quatorze, fourteen), the term used | insight. Hence his serious studies on the anatomical characters in English literature, as opposed to “sonnet," for a poem in of the lower and higher organisms, man included, will retain their fourteen rhymed iambic lines closing (as a sonnet strictly value, while many of his theories and generalizations, especially never does) with a couplet. The distinction was long neglected, in the department of ethnology, are already forgotten. because the English poets of the 16th century had failed to The work of de Quatrelages ranged over the whole field of zoology apprehend the true form of the sonnet, and called Petrarch's from the annelids and other low organisms to the anthropoids and and other Italian poets' sonnets quatorzains, and their own

man. Of his numerous essays in scientific periodicals, the more

important were: Considérations sur les caractères zoologiques des incorrect quatorzains sonnets. Almost all the so-called sonnets

rongeurs (1840): “De l'organisation des animaux sans vertèbres of the Elizabethan cycles, including those of Shakespeare, I des Côtes de la Manche" (Ann. Sc. Nat., 1844); " Recherches sur le système nerveux, l'embryogénie, les organs des sens, et la takes the form of Fr. quai, older ccy or caye, cf. Spanish cayo, circulation des annélides" (ibid., 1844-50); "Sur les affinités et

a bar, barrier or reef. The earlier form in English is “ kay," les analogies des lombrics et des sangsues" (Ibid.); "Sur l'histoire naturelle des tarets" (Ibid., 1848-49). Then there is

and it was so pronounced. “Key” was also earlier prothe vast series issued under the general title of " Etudes sur les nounced "kay,” and the change in pronunciation in the one types inférieurs de l'embranchement des annclés," and the results was followed also in the other. In spelling also the word was of several scientific expeditions to the Atlantic and Mediterranean

lated to "key," in the sense of a reel, or, especially, of coastlands, Italy and Sicily, forming a series of articles in the Revue des deux mondes, or embodied in the Souvenirs d'un natural

the low range of reefs or islcts on the coasts of Spanish America, iste (2 vols., 1854). These were followed in quick succession by e.g. on the coast of Florida, the chain of islets known as Florida the Physiologie com parće, metamorphoses de l'homme et des animaux Keys. (1862): Les Polynésiens et leurs migrations (1866); Histoire QUEBEC. a province of the Dominion of Canada. bounded S. naturelle des annclés marins et de l'eau douce (2 vols., 1866); La Rochelle el ses environs (1866); Rapport sur les progrès de l'anthro

y New Brunswick and the United States, W. by Ontario, N. by pologic (1867); Ch. Darwin et ses précurseurs français (1870), a the district of Ungava, and E. by the gulf of St Lawrence and study of evolution in which the writer takes somewhat the same the strip of castern Labrador which belongs to Newfoundland. attitude as A. R. Wallace, combating the Darwinian doctrine in 1 Ungava be considered as added to the province of Quebec. its application to man; La Race prussienne (1871): Crania Ethnica, jointly with Dr Hamy (2 vols., with too plates, 1875-82), a classical

Hudson Strait is the northern boundary. The province includes work based on French and foreign anthropological data, analogous | the island of Anticosti, the Bird Islands and the Magdalen to the Crania Britannica of Thurnam and Davis, and to S, G. Mor- Islands, in the gulf of St Lawrence. The western boundary, ton's Crania Americana and Crania Aegyptiaca; L'Espèce humaine

separating Quebec from Ontario, extends through Point au (1877); Nouvelles Etudes sur la distribution géographique des négritos (1882); lommes fossiles et hommes santages (1884);

Baudet on the river St Lawrence to Point Fortune on the and Histoire générale des races hronaines (2 vols., 1886-89), the first | Ottawa river, from which place the boundary follows the volume being introductory, while the second attempts a complete Ottawa to Lake Temiscaming. From the north end of this classification of mankind.

latter lake it runs due north to Hudson Bay. The province of QUATREFOIL, in Gothic architecture, the piercing of tracery Quebec thus cxtends from Blanc Sablon, a fishing harbour at the in a window or balustrade with small semicircular openings western end of the Strait of Belle Isle (which separates Canada known as “ foils "; the intersection of these foils is termed the from Newfoundland) in 59° 7' W. to Lake Temiscaming in cusp

79° 40' W., a distance of about 1350 miles. The area of the QUATREMÈRE, ÉTIENNE MARC (1782-1857), French province is 351,873 sq. m. The general direction of the province Orientalist, the son of a Parisian merchant, was born in Paris is north-east and south-west, following the course of its chief on the 12th of July 1782. Employed in 1807 in the manuscript physical feature, the river St Lawrence. Speaking generally, department of the imperial library, he passed 10 the chair of it may be said that the province of Quebec comprises the hydro. Greck in Rouen in 1809, entered the Academy of Inscriptions in graphical basin of the river St Lawrence as far west as the 1815, taught Hebrew and Aramaic in the Collège de France intersection of the parallel of 45° N. with the latter. The St from 1819, and finally in 1827 became professor of Persian in Lawrence flows near the southern edge of its basin, only some the School of Living Oriental Languages.

50,000 sq. m. of the area of the province lying south of the river. Quatremère's, first work was Recherches . . . sur la langue et la The province of Quebec falls into three main physiographical lillérature de l'Egypte (1808), showing that the language of ancient

| divisions, viz.: (1) the Laurentian Highlands, (2) the Valley Egypt must be sought in Coptic. His translation of Makrizi's Arabic history of the Mameluke sultans (2 vols., 1837-41) show's his

of the St Lawrence, and (3) the Notre Dame Mountains and erudition at the best. He published among other works Mémoires | the rolling country lying to the south-east of this range. sur les Nabatiens (1835); a translation of Rashid al-Din's Hist. (1) The Laurentian Highlands are sometimes referred to as the des Mongols de la Perse (1836); Mén. giog. et hist. sur l'Égypte “ Laurentian Mountains," as they appear to constitute a mountain (1810); the text of Ibn Khaldun's Prolegomena; and a vast range when viewed from the gull or the river St Lawrence. This number of useful memoirs in the Journal asiatique. His numerous portion of the province, however, is really a plateau having an reviews in the Journal des savunts should also be mentioned. clevation of 1000 to 2000 ft. above sea level, but this plateau Quatremère made great lexicographic collections in Oriental north of latitude 55° falls away to lower levels toward Hudson Bay languages, fragments of which appear in the notes to his various and Hudsor. Strait. Along the extreme castern border of these works. His MS. material for Syriac has been utilized in Payne Laurentian Highlands on the coast of Labrador, however, the Smith's Thesaurus; of the slips he collected for a projected Arabic, country rises to much grcater altitudes, forming an extremely Persian and Turkish lexicon some account is given in the preface rugged district which attains in places an elevation of 6000 ft. to Dozy, Supp. aux dictt. arabes. They are now in the Munich above sca-level. This plateau forms what is known as the Laurentian library.

peneplain and is hummocky in character, the surface, however, A biographical notice by M. Barthélemy Sainte-Hilaire is prefixed ! being but slightly accentuated and the sky line seen from the to Quatremère's Mélanges d'histoire et de philologie orientale (1861). higher points in the area being nearly level. It is densely wooded

and everywhere abounds in lakes, great and small, lying either in QUAY, MATTHEW STANLEY (1833-1904), American poli- basins etched in the rock suríace by glacial action or else bounded tical“ boss," was born in Dillsburg, York county, Pennsyl by the irregularly distributed drift which more or less completely vania on the 20th of September 1822. He graduated at covers the surface of the underlying rocks. From these lakes

issuc very numerous streams tributary to the larger rivers. These Jefferson College (now Washington and Jefferson College) in

lakes and rivers form so continuous a series of waterways that 1850 and was admitted to the bar in 1854. He served in a traveller who knows their courses, and the portages connecting various capacities in the Civil War, and in 1865-1867 was a them, can traverse this immense tract of country in any direction member of the state House of Representatives, becoming

by canoe. These strcams also, cascading down from the elevated

surface of the plateau to sea-level, afford immense water power, secretary of the commonwealth in 1873-1878 and again in

which is used to an increasing cxtcnt as the methods of long-distance 1879-1882, recorder of Philadelphia in 1878-1879, and state electrical transmission of power become more and more perfect. treasurer in 1886–1887. He was chairman of the Republican These waters are, moreover, clear and pure, and the country is one national executive campaign comunittee in 1888. and was a l in which malaria and similar diseases are unknown. Some of the member of the United States Senate in 1887-1899 and again

rivers draining the Laurentian country run in very deep, high

walled valleys or fjords cut in the solid rock; a number of which, in 1901-1904. For nearly twenty years he dominated the

comparable in character although perhaps not in depth to those government of Pennsylvania, and also played a very prominent of Norway and Greenland, pass outward from the central portion part in national affairs. In 1800 he was brought to trial on a l of the peneplain north, cast and south. As an example of such

fjords in the province of Quebec, those occupied by the waters of charge of misappropriating state funds, and, although he was

the Hamilton, Mingan and Saguenay rivers may be cited as well acquitted, the feeling among the reform element in his own

as that, now partially silted up, which is occupied by Lake Temisparty was so bitter against him that the legislature was dead caming and the Mattawa river. The walls of solid gneiss between locked and his re-election was postponed for two years. He which the Saguenay flows are in places from 1500 to 1800 ft. iul died on the 28th of May 1904.

height, while the waters of the river in places reach a depth of

1400 ft. QUAY, a wharf or landing-place for the loading and unloading this Laurentian country in the province of Quebec and its of water-borne cargo. The word, now pronounced like “key," | continuation into the adjacent province contain the chief timber

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