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QUATERNIONS , in See also:mathematics. The word "quaternion " properly means " a set of four." In employing such a word to denote a new mathematical method, See also:Sir W. R. See also: We will See also:tree of Surinam in See also:honour of the See also:negro Quassi or Coissi, who employed the intensely See also:bitter bark of the tree (See also:Quassia amara) as a remedy for See also:fever. The See also:original quassia was officially recognized in the See also:London See also:Pharmacopoeia of 1788. In 1809 it was replaced by the bitter See also:wood or bitter ash of See also:Jamaica, Picraena excelsa, which was found to possess similar properties and could be obtained in pieces of much larger See also:size. Since that date this wood has continued in use in See also:Britain under the name of quassia to the exclusion of the Surinam quassia, which, however, is still employed in See also:France and See also:Germany. Picraena excelsa is a tree 5o to 6o ft. in height, and resembles the common ash in See also:appearance. It has large See also:compound leaves composed of four therefore confine ourselves, so far as his predecessors are concerned, to attempts at interpretation which had geometrical applications in view. One geometrical interpretation of the negative sign of algebra was See also:early seen to be See also:mere reversal of direction along a line. Thus, when an See also:image is formed by a See also:plane See also:mirror, the distance of any point in it from the mirror is simply the negative of that of the corresponding point of the See also:object. Or if See also:motion in one direction along a line be treated as See also:positive, motion in the opposite direction along the same line is negative. In the See also:case of See also:time, measured from the See also:Christian era, this distinction is at once given,by the letters A.D. or B.C., prefixed to the date. And to find the position, in time, of one event relatively to another, we have only to subtract the date of the second (taking See also:account of its sign) from that of the first. Thus' to find the See also:interval between the battles of See also:Marathon (490 B.C.) and See also:Waterloo (A.D. 1815) we have
+1815—(—490) =2305 years.
And it is obvious that the same See also:process applies in all cases in which we See also:deal with quantities which may be regarded as of one directed See also:dimension only, such as distances along a line, rotations about an See also:axis, &c. But it is essential to See also:notice that this is by no means necessarily true of operators. To turn a line through a certain See also:angle in a given plane, a certain operator is required; but when we wish to turn it through an equal negative angle we must not, in See also:general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the See also:reverse operator, unless the angle involved be an See also:odd multiple of a right angle. This is, of course, on the usual See also:assumption that the sign of a product is changed when that of any one of its factors is changed,—which merely means that—i is commutative with all other quantities.
See also: In this sense the imaginary expression a + b -J — r is constructed by measuring a length a along the fundamental line (for real quantities), and from its extremity a line of length b in some direction perpendicular to the fundamental line. But he did not attack the question of the See also:representation of products or quotients of directed lines. The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary. In 1806 (the See also:year of publication of Buee's See also:paper) See also:Jean See also:Robert Argand published a pamphlet2 in which precisely the same ideas are developed, but to a considerably greater extent. For an interpretation is assigned to the product of two directed lines in one plane, when each is expressed as the sum of a real and an imaginary part. This product is interpreted as another directed line, forming the See also:fourth See also:term of a proportion, of which the first i Strictly speaking, this See also:illustration of See also:Tait's is in See also:error by unity because in our See also:calendar there is no year denominated zero. Thus the interval between See also:June the first of i B.C. and June the first of I A.D. is one year, and not two years as the See also:text implies. (A.McA.) 2 Essai sur une maniere de representer See also:les Quantites Imaginaires dans les Constructions Geometriwues. A second edition was published by J. Houel (See also:Paris, 1874). There is added an important Appendix. consisting of the papers from Gergonne's Annales which are referred to in the text above. Almost nothing can, it seems, be learned of Argand's private See also:life, except that in all See also:probability he was See also:born at See also:Geneva in 1768.term is the real (positive) unit-line, and the other two are the factor-lines. Argand's work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by J. F. See also:Francais. This writer stated that he had found the germ of his remarks among the papers of his deceased See also:brother, and that they had come from See also:Legendre, who had himself received them from some one unnamed. This led to a See also:letter from Argand, in which he stated his communications with Legendre, and gave a resume of the contents of his pamphlet. In a further communication to the Annales, Argand pushed on the applications of his theory. He has given by means of it a See also:simple See also:proof of the existence of n roots, and no more, in every rational algebraic equation of the nth See also:order with real coefficients. About 1828 John See also:Warren (1796—1852) in See also:England, and C. V. Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of See also:Cauchy, See also:Gauss and others, the properties of the expression a + b Al — i were developed into the immense and most important subject now called the theory of complex See also:numbers (see NUMBER). From the more purely symbolical view it was developed by See also:Peacock, De See also:Morgan, &c., as See also:double algebra. Argand's method may be put, for reference, in the following See also:form. The directed line whose length is a, and which makes an angle 0 with the real(positive) unit line, is expressed by a (See also:cos 0+i See also:sin 0), where i is regarded as + -1. The sum of two such lines (formed by adding together the real and the imaginary parts of two such expressions) can, of course, be expressed as a third directed line)—the See also:diagonal of the parallelogram of which they are conterminous sides. The product, P, of two such lines is, as we have seen, given by 1:a(cos 0-{-isin0) : : a'(cos0'-{-isin0'):P, or P=aa' (cos (0+0')+isin (0+ 0') Its length is, therefore, the product of the lengths of the factors, and its inclination to the real unit is the sum of those of the factors. If we write the expressions for the two lines in the form A+Bi, A'+B'i, the product is AA'—BB'-l-i(AB'+BA'); and the fact that the length of the product line is the product of those of the factors is seen in the form (See also:A2 +132) (A'2+B'2) _ (AA' — BB')2+ (AB' + BA')2. In the See also:modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors. Argand's attempts to extend his method to space generally were fruitless. The reasons will be obvious later; but we mention them just now because they called forth from F. J. Servois (Gergonne's Annales, 1813) a very remarkable comment, in which was contained the only yet discovered trace of an anticipation of the method of Hamilton. Argand had been led to deny that such an expression as ii could be expressed in the form A+Bi,—although, as is well known, See also:Euler showed that one of its values is a real quantity, the exponential See also:function of—sr/2. Servois says, with reference to the general representation of a directed line in space: " L'analogie semblerait exiger que le trinome fut de la forme p cos a+q cos 13+r cos a, 13, y etant les angles d'une droite avec trois axes rectangulaires; et qu'on eut (p cos a + q cos l3 + r cos 7) (p' cos a + q' cos I + r' cos y) =cos2a+cos213+cos2y=1. Les valeurs de p, q, r, p', q,' r', qui satisferaient a See also:cette See also:condition seraient absurdes; mais seraient-elles imaginaires, reductibles a la forme gEnerale A+B v -1 ? Voila une question d'analyse fort singulibre que je soumets a vos lumieres. La simple proposition que je See also:vous en fais suffit pour vous faire voir que je ne crois point que toute fonction analytique non reelle soit vraiment reductible a la forme A+B s/ -1." As will be seen later, the fundamental i, j, k of quarternions, with their reciprocals, furnish a set of six quantities which satisfy the conditions imposed by Servois. And it is quite certain that they cannot be represented by See also:ordinary imaginaries. Something far more closely analogous to quaternions than anything in Argand's work ought to have been suggested by De 1Vloivre's theorem (1730). Instead of regarding, as Buee and Argand had done, the expression a(cos 0 + i sin 0) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in which 0 is measured, turns it in that plane through the angle 0, and at the same time increases its length in the ratio a : 1. From the new point of view we see at once, as it were, why it is true that (cos 0+ i sin 0)Y. = cos m0+ i sin m0. For this equation merely states that in turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle. To make this process applicable to any plane in space, it is clear that we must have a See also:special value of i for each such plane. In other words, a unit line, See also:drawn in any direction whatever, must have -1 for its square. In such a system there will be no line in space specially distinguished as the real unit line: all will be alike imaginary, or rather alike real. We may See also:state, in passing, that every quaternion can be represented as a (cos 0+ a sin 0),—where a is a real number, 0 a real angle, and 1r a directed unit line whose square is -1. Hamilton took this See also:grand step, but, as we have already said, without any help from the previous work of De Moivre. The course of his investigations is minutely described in the See also:preface to his first great work (Lectures on Quaternions, 1853) on the subject. Hamilton, like most of the many inquirers who endeavoured to give a real interpretation to the imaginary of common algebra, found that at least two kinds, orders or ranks of quantities were necessary for the purpose. But, instead of dealing with points on a line, and then wandering out at right angles to it, as Buee and Argand had done, he See also:chose to look on algebra as the See also:science of " pure time," l and to investigate the properties of " sets " of time-steps. In its essential nature a set is a linear function of any number of " distinct " units of the same See also:species. Hence the simplest form of a set is a " couple "; and it was to the possible See also:laws of combination of couples that Hamilton first directed his See also:attention. It is obvious that the way in which the two See also:separate time-steps are involved in the couple will determine these laws of combination. But Hamilton's special object required that these laws should be such as to See also:lead to certain assumed results; and he therefore commenced by assuming these, and from the assumption determined how the separate time-steps must be involved in the couple. It we use See also:Roman letters for mere numbers, capitals for instants of time, Greek letters for time-steps, and a See also:parenthesis to denote a couple, the laws assumed by Hamilton as the basis of a system were as follows: (B1, B2) — (Ai, A2) = (B, —A,, B2—A,) = (a, ) ; (a, b) (a, 19) = (aa—b,6, ba+a/3) 2 To show how we give, by such assumptions, a real interpretation to the ordinary algebraic imaginary, take the simple case a=o, b= I, and the second of the above formulae gives (0, 1)(a, $)=(- Q, a). Multiply once more by the number-couple (o, I), and we have (0,1)(0, 1)(a,0)=(0, 1)(— ,a)=(—a,—R)=(—I,0)(a,0)=—(a,0). Thus the number-couple (o, I), when twice applied to a step-couple, simply changes its sign. That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a somewhat extravagant one. Some Eastern potentate, possessed of See also:absolute See also:power, covets the vast possessions of his See also:vizier and of his See also:barber. He determines to rob them both (an operation which may be very satisfactorily expressed by —I); but, being a wag, he chooses his own way of doing it. He degrades his vizier to the See also:office of barber, taking all his goods in the process; and makes the barber his vizier. Next See also:day he repeats the operation. Each of the victims has been restored to his former See also:rank, but the operator —I has been applied to both. Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x+iy+jz, by which he proposed to represent the directed line in space whose projections on the co-ordinate axes are x, y, z. The composition of two such lines by the algebraic I Theory of Conjugate Functions, or Algebraic Couples, with a Preliminary and Elementary See also:Essay on Algebra as the Science of Pure Time, read in 1833 and 1835, and published in Trans. R. I. A. xvii. ii. (1835). 3 Compare these with the See also:long-subsequent ideas of Grassmann.addition of their several projections agreed with the assumption of Buee and Argand for the case of coplanar lines. 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 See also: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 See also: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 See also: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 See also:felt that the new See also: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 See also:discovery is See also:October 16, 1843. Suppose, for simplicity, the factor-lines to be each of unit length. Then x, y, z, x', y', z' See also: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 See also: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 See also: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 See also: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 See also:internal See also:character of its own. Hamilton, having gone thus far, proceeded to evolve these results from a characteristic See also: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 See also:convention (b) that to See also: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 See also:reversing either, and therefore will recover its original direction when both are reversed. But this would obviously be inconsistent with (a). If it be perpendicular to the factor lines, (a) shows that it must have simultaneously every such direction. Hence it must be a mere number. Again, the product of two lines at right angles to one another cannot, even in part, be a number. For the reversal of either factor must, by (b), change its sign. But, if we look at the two factors in their new position by the See also:light of (a), we see that the sign must not change. But there is nothing to prevent its being represented by a directed line if, as further applications of (a) and (b) show we must do, we take it perpendicular to each of the factor lines. Hamilton seems never to have been quite satisfied with the apparent heterogeneity of a quaternion, depending as it does on a numerical and a directed part. He indulged in a great deal of See also: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 See also: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 " See also: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 See also: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 See also: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 See also: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 ' See also:Die Ausdehnungslehre, Leipsic, 1844; 2nd ed., vollstandig and in strenger Form bearbeitet, See also:Berlin. 1862. See also the collected works of See also:Mobius, and those of See also:Clifford, for a general explanation of Grassmann's method.appear to have preferred any such See also: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 See also: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 See also: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 See also: 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 See also:trigonometry. Also, every-thing See also:relating to change of systems of axes, as for instance in the See also: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 See also:matter of troublesome complexity by the usual methods. But every quaternion See also: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 See also:Arthur See also: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 See also:model. Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms. The See also:physical properties of a heterogeneous See also: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 See also: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 See also: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 See also: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 See also: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 See also:symbol. The method is essentially the same as that developed, under the name of " matrices," by Cayley in 1858; but it has the See also: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, See also:heat-See also:conduction, potentials, &c., we constantly meet with what is called " See also:Laplace's operator," viz. d2 22 ye + dz2. We know that this is an invariant; i.e. it is See also: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 See also:rate of most rapid change of that quantity. Thus, applied to a potential, it gives the direction and magnitude of the force; to a See also:distribution of temperature in a conducting solid, it gives (when multiplied by the conductivity) the See also:flux of heat, &c. No better testimony to the value of the quaternion method could be desired than the See also:constant use made of its notation by mathematicians like Clifford (in his Kinematic) and by physicists like Clerk-See also:Maxwell (in his See also:Electricity and See also: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 See also:language of quaternions. (P. G. T.) Supplementary Considerations.—There are three fairly well-marked stages of development in quaternions as a geometrical method. (I) See also: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 See also:Professor P. G. Tait, who was Hamilton's See also: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. Tait himself may be regarded as the See also:chief contributor to this See also:stage. (3) Geometrical applications, different in kind from, though more or less allied to, those in connexion with which the method was originated. These last include (a) C. J. Joly's projective geometrical applications starting from the interpretation of the quaternion as a point-symbol;' these applications may be said to require no addition to the quaternion algebra; (b) W. K. Clifford's biquaternions and G. Combebiac's tri-quaternions, which require the addition of quasi-scalars, independent of one another and of true scalars, and analogous to true scalars. As an algebraic 1 It appears from. Joly~'s and Macfarlane's references that J. B. See also:Shaw, in See also:America, independently of Joly, has interpreted the quaternion as a point-symbol.method quaternions have from the beginning received much attention from mathematicians. An See also:attempt has recently been made under the name of multenions to systematize this algebra. We select for description stage (3) above, as the most characteristic development of quaternions in See also:recent years. For (3) (a) we are constrained to refer the reader to Joly's own See also:Manual of Quaternions (1905). The impulse of W. K. Clifford in his paper of 1873 (" Preliminary See also:Sketch of Bi-Quaternions," Mathematical Papers, p. 181) seems to have come from Sir R. S. See also:Ball's paper on the Theory of Screws, published in 1872. Clifford makes use of a quasi-scalar w, commutative with quaternions, and such that if p, q, &c., are quaternions, when p+wq= p'+wq', then necessarily p=p', q=q'. He considers two cases, viz. See also:w2=1 suitable for non-Euclidean space, and w2=o suitable for Euclidean space; we confine ourselves to the second, and will See also:call the indicated bi-quaternion p+wq an octonion. In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indefinitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to p, and, like Hamilton's vector, wp is not localized; wp is therefore called a vector, though it differs from Hamilton's vector in that the product of any two such vectors wp and wo- is zero because w2=o . p+wo where p, o- are rotors (i.e. p is a rotor and coo- a vector), is called a motor, and has the geometrical significance of Ball's wrench upon, or twist about, a See also:screw. Clifford considers an octonion p+wq as the quotient of two See also:motors p+wo, p'+wa'. This is the basis of a method parallel throughout to the quaternion method; in the See also:specification of rotors and motors it is independent of the origin which for these purposes the quaternion method, pure and simple, requires. Combebiac is not content with getting rid of the origin in these limited circumstances. The fundamental geometrical conceptions are the point, line and plane. Lines and complexes thereof are sufficiently treated as rotors and motors, but points and planes cannot be so treated. He glances at Grassmann's methods, but is repelled because he is seeking a unifying principle, and he finds that Grassmann offers him not one but many principles. He arrives at the tri-quaternion as the suitable fundamental concept. We believe that this tri-quaternion solution of the very interesting problem proposed by Combebiac is the best one. But the first thing that strikes one is that it seems unduly complicated. A point and a plane See also:fix a line or axis; viz. that of the perpendicular from point to plane, and therefore a calculus of points and planes is ipso facto a calculus of lines also. To fix a weighted point and a weighted plane in Euclidean space we require 8 scalars, and not the 12 scalars of a tri-quaternion. We should expect some species of biquaternion to suffice. And this is the case. Let r,, co be two quasi-scalars such that n2=i, wn=w, rw=w2=o. Then the biquaternion 174+wr suffices. The plane is of vector magnitude zVq, its equation is zSpq=Sr, and its expression is the bi-quaternion 17Vq+wSr; the point is of scalar magnitude Sq, and its position vector is 0, where 1Vi3q=Vr (or what is the same, [Vr+q.Vr. q–']/Sq), and its expression is nSq+wVr. (See also:Note that the 2 here occurring is only required to ensure See also:harmony with tri-quaternions of which our See also:present biquaternions, as also octonions, are particular cases.) The point whose position vector is Vrq–1 is on the axis and may be called the centre of the bi-quaternion; it is the centre of a See also:sphere of See also:radius Srg–1 with reference to which the point and plane are in the proper quaternion sense polar reciprocals, that is, the position vector of the point relative to the centre is Srq 1. Vq/Sq, and that of the See also:foot of perpendicular from centre on plane is Srq'. Sq/Vq, the product being the (radius)2, that is (Srq^')2. The axis of the member xQ+x'Q' of the second-order complex Q, Q' (where Q=nq+wr, Q'=nq'+wr' and x, x' are scalars) is parallel to a fixed plane and intersects a fixed transversal, viz. the line parallel to q'q-1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed See also:ellipse which intersects the transversal; the axis is on a fixed ruled See also:surface I See also:Sidney, See also:Spenser and See also:Daniel, are really quatorzains. They to which the plane of the ellipse is a tangent plane, the ellipse being the See also:section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear parallel to the transversal. In the third-order complex the centre See also:locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q-1 is meaningless. Putting r —t7 = E we get Combebiac's tri-quaternion under the form Q=%p-l-rlq+wr. This has a reciprocal Q-1= p-1=nq 1 —wp-1rq 1, and a conjugate KQ (such that K[QQ'] _ KQ'KQ, K[KQ]=Q) given by KQ= EKq+i Kp+wKr; the product QQ' of Q and Q' is Epp'+'ggq'+w(pr'+rq'); the quasi-vector 2(1—K)Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different sense from the See also:rest of this See also:article) z(1+K)Q is Combebiac's scalar (Sp+Sq)+Combebiac's plane. Combebiac does not use K; and in place of E, 77 he uses µ=r7—E, so that µ2=1,wµ= —µw =w, w2=o. Combebiac's tri-quaternion may be regarded from many simplifying points of view. Thus, in place of his general tri-quaternion we might deal with products of an odd number of point-plane-scalars (of form µq+wr) which are themselves point-plane-scalars; and products of an even number which are octonions; the quotient of two point-plane-scalars would be an octonion, of two octonions an octonion, of an octonion by a point-plane-scalar or the inverse a point-plane-scalar. Again a unit point µ may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (b) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar. If .Q=Upder74+cwr and we put Q=(r+zwt)(Ep+nq)X (r+wt) 1 we find that the quaternion t must be 2f(r)/f(q—p), where f(r)=rq—Kpr. The point p=Vt may be called the centre of Q and the length St may be called the radius. If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius. Thus Q-1, Q, Q2, Q3, . . . have a common centre and common radius. Q and KQ have a common centre and equal and opposite radii; that is, the t of KQ is the negative conjugate of that of Q. When Sze=o, (r+zwu) ( ) (1+1wu)_1 is an operator which shifts (without further change) the tri-quaternion operand an amount given by u in direction and distance. Additional information and CommentsThere are no comments yet for this article.
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