May 28, 1891] 



NATURE 



81 



the methods of multiple algebra to the notions connected 

 with rectilinear co-ordinates. These methods, however, enrich 

 the field to which they are applied with new notijns. Thus the 

 notion of the co-ordinates of a line in space, subsequently intro- 

 duced by Plucker, 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 

 co-ordinates, is very much greater than when it takes the place 

 of three co-ordinates, for the same reason that a multiple algebra 

 taking the place of three co-ordinates 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 the 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 any- 

 thing 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, aug- 

 mented on Hamilton's side by the quaternion, and on Grass- 

 mann'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 Hamilton's system we have also the 

 linear quaternion function, and in Grassmann's the linear 

 function applied to the quantities of his algebra of points. This 

 application given those transformations in which projective 

 properties are preserved, the doctrine of reciprocal figures or 

 prmciple of duality, &c. (Grassmann's theory of the linear 

 function is, indeed, broader than this, being co-extensive 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. lu 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 lor 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. Cay ley's classical "Memoir on the 

 Theory of Matrices " (1858), and which is called by Prof. Syl- 

 vester 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 s-o full a dis- 

 cussion 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 srain 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, 

 Liickenausdruck. 



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 this journal, Prof. 

 Sylvester speaks of Cayley's "ever-memorable" " Memoir on 

 Matrices " as constituting " a secjnd birth of Algebra, its avatar 

 in a new and glorified form," and refers to a passage in his 

 " Lectures on Universal Algebra" from which, 1 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 Liickenausdruck, by which the matrix is expressed 

 as a sum of a kind of _^rpducts {liickenhallig, 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 follow- 

 ing 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. "Quater- 

 nions," "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 slates 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 "Ausdeh- 

 nungslehre " 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 183s, 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 limita- 

 tion 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 con- 

 sideration 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 lime which he should wish to devote 

 to them at ten or fifteen years (see his letter to Prof. Tait in 

 the North British Revie-M 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 Grassmann as published in the 

 years 1843-44. Such a comparison should be based on the 

 positive evidence of doctrines and methods actually published. 



NO. II 26, VOL. 44] 



