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VIII.

QUATERNIONS AND THE AUSDEHNUNGSLEHRE,

[Nature, vol. xliv. pp. 79–82, May 28, 1891.]

The year 1844 is memorable in the annals of mathematics on account of the first appearance on the printed page of Hamilton's Quaternions and Grassmann's Ausdehnungslehre. The former appeared in the July, October, and supplementary numbers of the Philosophical Magazine, after a previous communication to the Royal Irish Academy, November 13, 1843. This communication was indeed announced to the Council of the Academy four weeks earlier, on the very day of Hamilton's discovery of quaternions, as we learn from one of his letters. The author of the Ausdehnungslehre, although not unconscious of the value of his ideas, seems to have been in no haste to place himself on record, and published nothing until he was able to give the world the most characteristic and fundamental part of his system with considerable development in a treatise of more than 300 pages, which appeared in August 1844.

The doctrine of quaternions has won a conspicuous place among the various branches of mathematics, but the nature and scope of the Ausdehnungselehre, and its relation to quaternions, seem to be still the subject of serious misapprehension in quarters where we naturally look for accurate information. Historical justice, and the interests of mathematical science, seem to require that the allusions to the Ausdehnungselehre in the article on "Quaternions" in the last edition of the Encyclopædia Britannica, and in the third edition of Prof. Tait's Treatise on Quaternions, should not be allowed to pass without protest.

It is principally as systems of geometrical algebra that quaternions and the Ausdehnungselehre come into comparison. To appreciate the relations of the two systems, I do not see how we can proceed better than if we ask first what they have in common, then what either system possesses which is peculiar to itself. The relative extent and importance of the three fields, that which is common to the two systems, and those which are peculiar to each, will determine the relative rank of the geometrical algebras. Questions of priority can only relate to the field common to both, and will be much simplified by having the limits of that field clearly drawn.

Geometrical addition in three dimensions is common to the two systems, and seems to have been discovered independently both by Hamilton and Grassmann, as well as by several other persons about the same time. It is not probable that any especial claim for priority with respect to this principle will be urged for either of the two with which we are now concerned.

The functions of two vectors which are represented in quaternions by and are common to both systems as published in 1844, but the quaternion is peculiar to Hamilton's. The linear vector function is common to both systems as. ultimately developed, although mentioned only by Grassmann as early as 1844.

To those already acquainted with quaternions, the first question will naturally be: To what extent are the geometrical methods which are usually called quatemionic peculiar to Hamilton, and to what extent are they common to Grassmann? This is a question which anyone can easily decide for himself. It is only necessary to run one's eye over the equations used by quatemionic writers in the discussion of geometrical or physical subjects, and see how far they necessarily involve the idea of the quaternion, and how far they would be intelligible to one understanding the functions and but having no conception of the quaternion or at least could be made so by trifling changes of notation, as by writing or in places where they would not aflect the value of the expressions. For such a test the examples and illustrations in treatises on quaternions would be manifestly inappropriate, so far as they are chosen to illustrate quaternionic principles, since the object may influence the form of presentation. But we may use any discussion of geometrical or physical subjects, where the writer is free to choose the form most suitable to the subject. I myself have used the chapters and sections in Prof. Tait's Quaternions on the following subjects: Geometry of the straight line and plane, the sphere and cyclic cone, surfaces of the second degree, geometry of curves and surfaces, kinematics, statics and kinetics of a rigid system, special kinetic problems, geometrical and physical optics, electrodynamics, general expressions for the action between linear elements, application of to certain physical analogies, pp. 160–371, except the examples (not worked out) at the close of the chapters.

Such an examination will show that for the most part the methods of representing spatial relations used by quaternionic writers are common to the systems of Hamilton and Grassmann. To an extent comparatively limited, cases will be found in which the quatemionic idea forms an essential element in the signification of the equations.

The question will then arise with respect to the comparatively limited field which is the peculiar property of Hamilton, How important are the advantages to be gained by the use of the quaternion? This question, unlike the preceding, is one into which a personal equation will necessarily enter. Everyone will naturally prefer the methods with which he is most familiar; but I think that it may be safely affirmed that in the majority of cases in this field the advantage derived from the use of the quaternion is either doubtful or very trifling. There remains a residuum of cases in which a substantial advantage is gained by the use of the quatemionic method. Such cases, however, so far as my own observation and experience extend, are very exceptional. If a more extended and careful inquiry should show that they are ten times as numerous as I have found them, they would still be exceptional.

We have now to inquire what we find in the Ausdehnungslehre in the way of a geometrical algebra, that is wanting in quaternions. In addition to an algebra of vectors, the Ausdehnungslehre affords a system of geometrical algebra in which the point is the fundamental element, and which for convenience I shall call Grassmann's algebra of points. In this algebra we have first the addition of points, or quantities located at points, which may be explained as follows. The equation

in which the capitals denote points, and the small letters scalars (or ordinary algebraic quantities), signifies that
and also that the centre of gravity of the weights etc., at the points etc., is the same as that of the weights etc., at the points etc. (It will be understood that negative weights are allowed as well as positive.) The equation is thus equivalent to four equations of ordinary algebra. In this Grassmann was anticipated by Möbius (Barycentrischer Calcul, 1827). We have next the addition of finite straight lines, or quantities located in straight lines (Liniengrössen). The meaning of the equation will perhaps be understood most readily, if we suppose that each member represents a system of forces acting on a rigid body. The equation then signifies that the two systems are equivalent. An equation of this form is therefore equivalent to six ordinary equations. It will be observed that the Liniengrössen and are not simply vectors; they have not merely length and direction, but they are also located each in a given line, although their position within those lines is immaterial. In Clifford's terminology, is a rotor, a motor. In the language of Prof. Ball's Theory of Screws, represents either a twist or a wrench.

We have next the addition of plane surfaces (Plangrössen). The equation

signifies that the plane passes through the point common to the planes and and that the projection by parallel lines of the triangle on any plane is equal to the sum of the projections of and on the same plane, the areas being taken positively or negatively according to the cyclic order of the projected points. This makes the equation equivalent to four ordinary equations.

Finally, we have the addition of volumes, as in the equation

where there is nothing peculiar, except that each term represents the six-fold volume of the tetrahedron, and is to be taken positively or negatively according to the relative position of the points.

We have also multiplications as follows: The line (Liniengrösse) is regarded as the product of the points and The Plangrösse which represents the double area of the triangle, is regarded as the product of the three points and or as the product of the line and the point or of and or indeed of and The volume which represents six times the tetrahedron, is regarded as the product of the points and or as the product of the point and the Plangrösse or as the product of the lines and etc, etc.

This does not exhaust the wealth of multiplicative relations which Grassmann has found in the very elements of geometry. The following products are called regressive, as distinguished from the progressive, which have been described. The product of the Plangrösen and is a part of the line in which the planes and intersect, which is equal in numerical value to the product of the double areas of the triangles and multiplied by the sine of the angle made by the planes. The product of the Liniengrösse and the Plangrösse is the point of intersection of the line and the plane with a numerical coefficient representing the product of the length of the line and the double area of the triangle multiplied by the sine of the angle made by the line and the plane. The product of three Plangrössen is consequently the point common to the three planes with a certain numerical coefficient. In plane geometry we have a regressive product of two Liniengrössen, which gives the point of intersection of the lines with a certain numerical coefficient.

The fundamental operations relating to the pointy line, and plane are thus translated into analysis by multiplications. The immense flexibility and power of such an analysis will be appreciated by anyone who considers what generalized multiplication in connection with additive relations has done in other fields, as in quaternions, or in the theory of matrices, or in the algebra of logic. For a single example, if we multiply the equation

by ( and being any two points), we have
which will be recognised as expressing an important theorem of statics.

The field in which Grassmann's algebra of points, as distinguished from his algebra of vectors, finds its especial application and utility is nearly coincident with that in which, when we use the methods of ordinary algebra, tetrahedral or anharmonic coordinates are more appropriate than rectilinear. In fact, Grassmann's algebra of points may be regarded as the application of the methods of multiple algebra to the notions connected with tetrahedral coordinates, just as his or Hamilton's algebra of vectors may be regarded as the application of the methods of multiple algebra to the notions connected with rectilinear coordinates. These methods, however, enrich the field to which they are applied with new notiona Thus the notion of the coordinates of a line in space, subsequently introduced by Plücker, was first given in the Ausdehnungslehre of 1844. It should also be observed that the utility of a multiple algebra when it takes the place of an ordinary algebra of four coordinates, is very much greater than when it takes the place of three coordinates, for the same reason that a multiple algebra taking the place of three coordinates is very much more useful than one taking the place of two. Grassmann's algebra of points will always command the admiration of geometers and analysts, and furnishes an instrument of marvellous power to the former, and in its general form, as applicable to space of any number of dimensions, to tiie latter. To the physicist an algebra of points is by no means so indispensable an instrument as an algebra of vectors.

Grassmann's algebra of vectors, which we have described as coincident with a part of Hamilton's system, is not really anything separate from his algebra of points, but constitutes a part of it, the vector arising when one point is subtracted from another. Yet it constitutes a whole, complete in itself, and we may separate it from the larger system to facilitate comparison with the methods of Hamilton.

We have, then, as geometrical algebras published in 1844, an algebra of vectors common to Hamilton and Grassmann, augmented on Hamilton's side by the quaternion, and on Grassmann's by his algebra of points. This statement should be made with the reservation that the addition both of vectors and of points had been given by earlier writers.

In both systems as finally developed we have the linear vector function, the theory of which is identical with that of strains and rotations. In BAmilton's system we have also the linear quaternion function, and in Grassmann's the linear function appHed to the quantities of his algebra of points. This application gives those transformations in which projective properties are preserved, the doctrine of reciprocal figures or principle of duality, etc. (Grassmann's theory of the linear function is, indeed, broader than this, being coextensive with the theory of matrices; but we are here considering only the geometrical side of the theory.)

In his earliest writings on quaternions, Hamilton does not discuss the linear function. In his Lectures on Quaternions (1853), he treats of the inversion of the linear vector function, as also of the linear quaternion function, and shows how to find the latent roots of the vector function, with the corresponding axes for the case of real and unequal roots. He also gives a remarkable equation, the symbolic cubic, which the functional symbol must satisfy. This equation is a particular case of that which is given in Prof. Cayley's classical Memoir on the Theory of Matrices (1858), and which is called by Prof. Sylvester the Hamilton- Cayley equation. In his Elements of Quaternions (1866), Hamilton extends the symbolic equation to the quaternion function.

In Grassmann, although the linear function is mentioned in the first Ausdehnungslehre, we do not find so full a discussion of the subject until the second Ausdehnungslehre (1862), where he discusses the latent roots and axes, or what corresponds to axes in the general theory, the whole discussion relating to matrices of any order. The more difficult cases are included, as that of a strain in which all the roots are real, but there is only one axis or unchanged direction. On the formal side he shows how a linear function may be represented by a quotient or sum of quotients, and by a sum of products, Lückenausdruck.

More important, perhaps, than the question when this or that theorem was first published is the question where we first find those notions and notations which give the key to the algebra of linear functions, or the algebra of matrices, as it is now generally called. In vol. xxxi, p. 35, of Nature, Prof. Sylvester speaks of Cayley's "ever-memorable" Memoir on Matrices as constituting "a second birth of Algebra, its avatar in a new and glorified form," and refers to a passage in his Lectures on Universal Algebra, from which, I think, we are justified in inferring that this characterization of the memoir is largely due to the fact that it is there shown how matrices may be treated as extensive quantities, capable of addition as well as of multiplication. This idea, however, is older than the memoir of 1858. The Lückenausdruck, by which the matrix is expressed as a sum of a kind of products (lückenhaltig, or open), is described in a note at the end of the first Ausdehnungslehre. There we have the matrix given not only as a sum, but as a sum of products, introducing a multiplicative relation entirely different from the ordinary multiplication of matrices, and hardly less fruitful, but not lying nearly so near the surface as the relations to which Prof. Sylvester refers. The key to the theory of matrices is certainly given in the first Ausdehnungslehre, and if we call the birth of matricular analysis the second birth of algebra, we can give no later date to this event than the memorable year of 1844.

The immediate occasion of this communication is the following passage in the preface to the third edition of Prof. Tait's Quaternions:

"Hamilton not only published his theory complete, the year before the first (and extremely imperfect) sketch of the Ausdehnungslehre appeared; but had given ten years before, in his protracted study of Sets, the very processes of external and internal multiplication (corresponding to the Vector and Scalar parts of a product of two vectors) which have been put forward as specially the property of Grassmann."

For additional information we are referred to art "Quaternions," Encyc. Brit., where we read respecting the first Ausdehnungslehre:

"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. . . . 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."

I shall leave the reader to judge of the accuracy of the general terms used in these passages in comparing the first Ausdehnungslehre with Hamilton's system as published in 1843 or 1844. The specific statements respecting Hamilton and Grassmann require an answer.

It must be Hamilton's Theory of Conjugate Functions or Algebraic Couples (read to the Royal Irish Academy, 1833 and 1835, and published in vol. xvii of the Transactions) to which reference is made in the statements concerning his "protracted study of Sets" and "mode of multiplying couples." But I cannot find anything like Grassmann's external or internal multiplication in this memoir, which is concerned, as the title pretty clearly indicates, with the theory of the complex quantities of ordinary algebra.

It is difficult to understand the statements respecting the Ausdehnungslehre, which seem to imply that Grassmann's two kinds of multiplication were subject to some kind of limitation to a plane. The external product is not limited in the first Ausdehnungslehre even to three dimensions. The internal, which is a comparatively simple matter, is mentioned in the first Ausdehnungslehre only in the preface, where it is defined, and placed beside the external product as relating to directed lines. There is not the least suggestion of any difference in the products in respect to the generality of their application to vectors.

The misunderstanding seems to have arisen from the following sentence in Grassmann's preface: "And in general, in the consideration of angles in space, difficulties present themselves, for the complete (allseitig) solution of which I have not yet had sufficient leisure." It is not surprising that Grassmann should have required more time for the development of some parts of his system, when we consider that Hamilton, on his discovery of quaternions, estimated the time which he should wish to devote to them at ten or fifteen years (see his letter to Prof. Tait in the North British Review for September 1866), and actually took several years to prepare for the press as many pages as Grassmann had printed in 1844. But any speculation as to the questions which Grassmann may have had principally in mind in the sentence quoted, and the particular nature of the difficulties which he found in them, however interesting from other points of view, seems a very precarious foundation for a comparison of the systems of Hamilton and Qrassmann as published in the years 1843–44. Such a comparison should be based on the positive evidence of doctrines and methods actually published.

Such a comparison I have endeavoured to make, or rather to indicate the basis on which it may be made, so far as systems of geometrical algebra are concerned. As a contribution to analysis in general, I suppose that there is no question that Grassmann's system is of indefinitely greater extension, having no limitation to any particular number of dimensions.