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But, assuming the distributive principle, the product of two lines appeared to give the expression xx' —yy' —zz'+i (yx'+xy') +j (xz'+zx') +ij (yz' +zy') For the square of j, like that of i, was assumed to be negative unity . But the interpretation of ij presented a difficulty—in fact the

main difficulty of the whole investigation—and it is specially interesting to see how Hamilton attacked it . He saw that he could get a hint from the simpler case, already thoroughly discussed, provided the two factor lines were in one plane through the real unit line . This requires merely that y : z :: y' : z' ; oryz'—zy'=o; but then the product should be of the same form as the separate factors . Thus, in this special case, the term in ij ought to vanish . But the numerical factor appears to be yz'+zy', while it is the quantity yz'—zy' which really vanishes . Hence Hamilton was at first inclined to think that ij must be treated as nil . But he soon saw that " a less harsh supposition " would suit the simple case . For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factor-lines are coplanar provided ij = —ji, for it would then assume the form ij(yz' —zy') . He had now the following expression for the product of any two directed lines: xx' —yy' — zz' +i (yx' + xy' ) +j(xz' +zx' ) +ij(yz'—zy') . But his result had to be submitted to another test, the Law of the Norms . As soon as he found, by trial, that this law was satisfied, he took the final step ." This led me," he says, " to conceive that perhaps, instead of seeking to confine ourselves to triplets, . . . we ought to regard these as only imperfect forms of Quaternions, . . . and that thus my old conception of sets might receive a new and useful application." In a very

short time he settled his fundamental assumptions . He had now three distinct space-units, i, j, k; and the following conditions regulated their combination by multiplication: z3=j'=k'=—1, ij = — ji = k, jk = — kj =i, ki=—ik=j.3 And now the product of two quaternions could be at once expressed as a third quaternion, thus (a+ib+jc+kd) (a'+ib'-{-jc'+kd') =A+iB+jC+kD, where A=aa'—bb'—cc'—dd', B =ab'+ba'+cd' —dc', C =ac'+ca'+db' —bd', D =ad'+da'+bc'—cb' . Hamilton at once found that the Law of the Norms holds,—not being aware that Euler had long before decomposed the product of two sums of four squares into this very set of four squares . And now a directed line in space came to be represented as ix+jy+kz, while the product of two lines is the quaternion — (xx' +yy' +zz') +i(yz' —zy') +j(zx' —xz') +k(xy' —yx') . To any one acquainted, even to a slight extent, with the elements of Cartesian geometry of three dimensions, a glance at the extremely suggestive constituents of this expression shows how justly Hamilton was entitled to say: " When the conception . . . had been so far unfolded and fixed in my mind, I felt that the new instrument for applying calculation to geometry, for which I had so long sought, was now, at least in part, attained." The date of this memorable discovery is October 16, 1843 . Suppose, for simplicity, the factor-lines to be each of unit length . Then x, y, z, x', y', z' express their direction-cosines . Also, if 8 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx'+yy'+zz'=cos 0, yz'—zy"=x" sin 0, &c., so that the product of two unit lines is now expressed as —coso+(ix"+jy"+kz") sin B . Thus, when the factors 3 It will be easy to see that, instead of the last three of these, we may write the single one ijk = -1 .are parallel, or B=o, the product, which is now the square of any , that of Grassmann . But it is to be observed that Grassmann, (unit) line is —i . And when the two factor lines are at right angles ~ though he virtually accused Cauchy of

plagiarism, does not to one another, or 0=ir/2, the product is simply ix"+jy''+kz", the unit line perpendicular to both . Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k . In other words, if the system (considered as rigid) be made to turn about till the first factor coincides with i and the second with j, the pro-duct will coincide with k . This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied . It has therefore, as it were, a unique internal character of its own . Hamilton, having gone thus far, proceeded to evolve these results from a characteristic train of a priori or metaphysical reasoning . Let it be supposed that the product of two directed lines is some-thing which has quantity; i.e. it may be halved, or doubled, for instance . Also let us assume (a) space to have the same properties in all directions, and make the convention (b) that to change the sign of any one factor changes the sign of a product . Then the product of two lines which have the same direction cannot be, even in part, a directed quantity . For, if the directed part have the same direction as the factors, (b) shows that it will be reversed by

reversing either, and therefore will recover its original direction when both are reversed .

This is a full and informative treatment which places Grassmann in context, and in particular shows the mathematical treatment of the conditions, relations, and assumptions, though written in algebraic form, to have a non Grassmannian conception. Thus when Hamilton also went on to derive the principles of Quaternions metaphysiclly, he treated the subject as had Geassmann in a philosophical and dialectic approach. The comparison from that viewpoint would show the similarity in their thiking, but would contrast the different apprehension of an expression. For Grassmann every expression was a new form of magnitude. Thus by applyig established rules for magnitudes and by studying and applying the specific laws for a given magnitude calculation could be simplified and insights obtained more readily. Thus Grassmann formed the habit of studying the rules governing and entire field of relations and then applying those rules to specific cases.

He indulged in a great deal of speculation as to the existence of an extra-spatial ,unit, which was to furnish the raison d'etre of the numerical part, and render the quaternion homogeneous as well as linear . But for this we must refer to his own works . Hamilton was not the only worker at the theory of sets . The year after the first publication of the quaternion method, there appeared a work of great originality, by Grassmann,' in which results closely analogous to some of those of Hamilton were given .

In particular, two species of multiplication (" inner " and "

outer ") of directed lines in one plane were given . The results of these two kinds of multiplication correspond respectively to the numerical and the directed parts of Hamilton's quaternion product . But Grassmann distinctly states in his preface that he had not had leisure to extend his method to angles in space . Hamilton and Grassmann, while their earlier work had much in common, had very different objects in view . Hamilton had geometrical application as his main object; when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his method . Grassmann's object seems to have been, all along, of a much more ambitious character, viz. to discover, if possible, a system or systems in which every conceivable mode of dealing with sets should be included . That he made very great advances towards the attainment of this object all will allow; that his method, even as completed in 1862, fully attains it is not so certain . But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked . Grassmann made in 1854 • a somewhat savage onslaught on Cauchy and De St Venant, the former of whom had invented, while the latter had exemplified in application, the system of " clefs algebriques," which is almost precisely ' Die Ausdehnungslehre, Leipsic, 1844; 2nd ed., vollstandig and in strenger Form bearbeitet, Berlin . 1862 . See also the collected works of Mobius, and those of Clifford, for a general explanation of Grassmann's method.appear to have preferred any such charge against Hamilton . He does not allude to Hamilton in the second edition of his work .But in 1877, in the Mathematische Annalen, xii., he gave a paper " On the

Place of Quaternions in the Ausdehnungslehre," in which he condemns, as far as he can, the nomenclature and methods of Hamilton . There are many other systems, based on various principles, which have been given for application to geometry of directed lines, but those which deal with products of lines are all of such complexity as to be practically useless in application . Others, such as the Barycentrische Calciil of Mobius, and the Methode des equipollences of Bellavitis, give elegant modes of treating space problems, so long as we confine ourselves to projective geometry and matters of that order; but they are limited in their field, and therefore need not be discussed here . More general systems, having close analogies to quaternions, have been given since Hamilton's discovery was published . As instances we may take Goodwin's and O'Brien's papers in the Cambridge Philosophical Transactions for 1849 . (See also ALGEBRA: special kinds.) Relations to other Branches of Science.—The above narrative shows how close is the connexion between quaternions and the ordinary Cartesian space-geometry . Were this all, the gain by their introduction would consist mainly in a clearer insight into the mechanism of co-ordinate systems, rectangular or not—a very important addition to theory, but little advance so far as practical application is concerned . But, as yet, we have not taken advantage of the perfect symmetry of the method . When that is done, the full value of Hamilton's grand step becomes evident, and the gain is quite as extensive from the practical as from the theoretical point of view . Hamilton, in fact, remarks,2 " I regard it as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded, whenever it becomes, or seems to become, necessary to have recourse . . . to the resources of ordinary algebra, for the solution of equations in quaternions." This refers to the use of the x, y, z co-ordinates,–associated, of course, with i, j, k . But when, instead of the highly artificial expression ix-}-jy+kz, to denote a finite directed line, we employ a single letter, a (Hamilton uses the Greek alphabet for this purpose), and find that we are permitted to deal with it exactly as we should have dealt with the more complex expression, the immense gain is at least in part obvious .Any quaternion may now be expressed in numerous simple forms . Thus we may regard it as the sum of a number and a line, a-ba, or as the product, (3y, or the quotient, 3e-', of two directed lines, &c., while, in many cases, we may represent it, so far as it is required, by a single letter such as q, r, &c . Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical

trigonometry . Also, every-thing 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 regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods . But every quaternion formula is a proposition in spherical (sometimes 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 Arthur Cayley) was the interpretation of the singular operator q( )q-', 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 (provided 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 co-ordinates with reference to that point . The 2 Lectures on Quaternions, § 513 . 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, j, k; and it would appear at first sight as if quaternions could not simplify, though they might improve in elegance, 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 connexion between quaternions and the theory of numbers . But one most important connexion with modern physics must be pointed out . In the theory of surfaces, in hydrokinetics, heat-conduction, potentials, &c., we constantly meet with what is called " Laplace's operator," viz. d2 22 ye + dz2 . We know that this is an invariant; i.e. it is independent of the particular directions chosen for the rectangular co-ordinate axes . Here, then, is a case specially adapted to the isotropy of the quaternion system; and Hamilton easily saw that the expression idx +j –+k- dcould be, like ix+jy+kz, effectively expressed by a single letter . He chose for this purpose V . And we now see that the square of V 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 direction and magnitude of the force; to a

distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the flux of heat, &c . 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 Magnetism) . 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 translating the unwieldy Cartesian expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions . (P . G . T.) Supplementary Considerations.—There are three fairly well-marked stages of development in quaternions as a geometrical method . (I) Generation of the concept through imaginaries and development into a method applicable to Euclidean geometry . This was the work of Hamilton himself, and the above account (contributed to the 9th ed. of the Ency . Brit. by Professor P . G . Tait, who was Hamilton's pupil and after him the leading exponent of the subject) is a brief resume of this first, and by far the most important and most difficult, of the three stages . (2) Physical applications .