So 



NATURE 



[May .28, 1 89 1 



The functions of two vectors which are represented in qua- 

 ternions by Saj8 and Vo;8 are common to both systems as pub- 

 lished in 1844, but the quaternion is peculiar to Hamilton's. 

 The linear vector function is common to both systems as ulti- 

 mately developed, although mentioned only by Grassmann as 

 early as 1844. 



To those already acquainted with quaternions, the first ques- 

 tion will naturally be : To what extent are the geometrical 

 methods which are usually called quaternionic 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 

 quaternionic 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 So;8 and Vo)3, but having no con- 

 ception of the quaternion afi, or at least could be made so by 

 trifling changes of notation, as by writing S or V in places 

 where they would not affect the value of the expressions. For 

 such a test the examples and illustrations in treatises on qua- 

 ternions 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 dis- 

 cussion 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 "Qua- 

 ternions " 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 v 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 quaternionic idea forms an essential element in the significa- 

 tion 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 precedinjj, 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 quaternionic 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 " Ausdehnungs- 

 lehre " in the way of a geometrical algebra, that is wanting in 

 quaternions. In addition to an algebra of vectors, the " Aus- 

 dehnungslehre " affords a system of geometrical algebra in which 

 the point is the fundamental element, and which for conve- 

 nience 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 



aK-\-b^ + cQ + &c. = fE -f /F -i- &c., 

 in which the capitals 'denote points, and the small letters scalars 

 (or ordinary algebraic quantities), signifies that 



« + i^-l-c-t-&c. = e + f + &c., 

 and also that the centre of gravity of the weights «, b, c, &c., at 

 the points A, B, C, &c., is the same as that of the weights e, f, 

 &c., at the points E, F, &c. (It will be understood that nega- 

 tive weights are allowed as well as positive.) The equation is 

 thus equivalent to four equations of ordinary algebra. In this 

 Grassmann was anticipated by Mbbius ( " Bary centrischer Calcul," 

 1827). 



We have next the addition of finite straight lines, or quantities 

 located in straight lines {Liiiietigrossen). The meaning of the 

 equation 



AB + CD -f &c. = EF -f Gil -t- &c. 



NO. I 126, VOL. 



44] 



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 ta 

 six ordinary equations. It will be observed that the Linien- 

 grossen AB and CD 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 imma- 

 terial. In Clifford's terminology, AB is a rotor, AB -(- CD a 

 motor. In the language of Piof. Ball's "Theory of Screws," 

 AB 4- CD represents either a twist or a zurctu/i. 



We have next the addition of plane surfaces {Plangrossen). 

 The equation 



ABC + DEF -t- GHI = JKL 



signifies that the plane JKL passes through the point common 

 to the planes ABC, DEF, and GHI, and that the projection 

 by parallel lines of the triangle JKL on any plane is equal to the 

 sum of the projections of ABC, DEF, and GHI 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 



ABCD + EFGH = IJKL, 

 where there is nothing peculiar, except that each term repre- 

 sents 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 (Linien- 

 gtdss,e) AB is regarded as the product of the points A and B. 

 The Plangrbsse ABC, which represents ihe double area of the 

 triangle, is regarded as the product of the three points A, B, 

 and C, or as the product of the line AB and the point C, or of 

 BC and A, or indeed of BA and C. The volume ABCD, which 

 represents six times the tetrahedron, is regarded as the product 

 of the points A, B, C, and D, or as the product of the point A 

 and the Plangrossc BCD, or as the product of the lines AB and 

 BC, &c., &c. 



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 Plangrossen ABC and DEF is a part of the line in which 

 the planes ABC and DEF intersect, which is equal in numerical 

 value to the product of the double areas of the triangles ABC 

 and DEF multiplied by the sine of the angle made by the 

 planes. The product of the Liniengrdsse AB and the Plan- 

 grosse CDE 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 multi- 

 plied by the sine of the angle made by the line and the piane. 

 The product of three Plangrossen is consequently the point 

 common to the three planes with a certain numerical coefficient. 

 In plane geometry we have a regressive product of two Linien- 

 grosse, which gives the point of intersection of the lines with a 

 certain numerical coefficient. 



The fundamental operations relating to the point, 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 multipli- 

 cation 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 



AB -f- CD + &c. - EF A- GH -f &c. 



by PQ (P and Q being any two points), we have 



ABPQ -j- CDPQ -f &c. = EFPQ -f GHPQ -f &c., 

 which will be recognized as expressing an important theorem of 

 statics. 



The field in which Grassmann's algebra of points, as distin- 

 guished from his algebra of vectors, finds hs especial application 

 and utility, is nearly coincident with that in which, when we 

 use the methods of ordinary algebra, tetrahedral or anharmonic 

 co-ordinates are more appropriate than rectilinear. In fact, 

 Grassmann's algebra of points may be regarded as the applica- 

 tion of the methods of multiple algebra to the notions connected 

 with tetrahedral co-ordinates, just as his or Hamiltons 

 algebra of vectors may be regarded as the application ol 



