2l8 



NATURE 



[July 6, 1882 



from the elementary exposition of them given by Prof. 

 Smith in his introduction to the Papers, pp. xl. et seqq. 

 We should like, however, if we might venture to differ 

 from so great an authority, to take exception to his defini- 

 tion of flatness by means of the notion of planeness, and to 

 the introduction of the idea and the word curvature into 

 an elementary exposition of the properties of space. This 

 seems at best an explanation of the less by the more diffi- 

 cult ; and, after all, the use in this case of the word curva- 

 ture is of questionable propriety (although sanctioned by 

 the highest authority), inasmuch as it suggests not only 

 true but also false analogies. It is very well in the hands 

 of a mathematician, to whom it suggests merely that a 

 certain common apparatus of mathematical formulas is 

 applicable to a particular class of manifoldness and to a 

 particular kind of surface ; but to the mathematically 

 untrained or half-trained reader the word suggests the 

 paradox that portions of space on the two sides of a plane 

 in elliptic space both are and are not congruent. Much 

 harm has, we are persuaded, been done by this unfor- 

 tunate usage of words. A similar piece of mystery making 

 has been practised with //-dimensional space ; the language 

 of mathematicians concerning which has been retailed to 

 ordinary simple-minded people as if it had the literal 

 sense they naturally attach to it. 



Clifford's papers on the geometry of hyper-space began 

 with his translation of Riemann's famous Habilitations- 

 schrift on the hypotheses which lie at the basis of geo- 

 metry. He establishes a close connection between the 

 generalised geometry and the generalised algebra in the 

 Preliminary Sketch of Biquaternions, to our mind one of 

 the ablest of his papers. He farther develops the subject 

 in the memoirs " On the Motion of a Solid in Elliptic 

 Space, 1 ' " On the Theory of Screws in a Space of Con- 

 stant Positive Curvature," "On the Free Motion under 

 no forces of a Rigid System in an //-fold Homaloid." 

 The kinematic of elliptic space as given by Clifford, and 

 developed quite recently by Dr. Ball, forms one of the 

 most elegant and attractive of modern geometrical theories. 

 The starting point may be said to be the finding of the 

 analogue in elliptic space to Euclid's parallel. In the 

 modern geometrical sense a parallel {i.e. a line intersect- 

 ing another at an infinite distance) cannot of course 

 exist in elliptic space except as an imaginary line. If, 

 however, we define a parallel as the straight equidis- 

 tant from a given straight line, then through every 

 point in space two parallels (a right and a left parallel 

 as Clifford calls them) can be drawn to a given 

 straight line. This appears at once by drawing at the 

 given point a tangent plane to the equidistant surface of 

 the given straight line, which it will be remembered is, 

 in elliptic space, an anticlastic surface of revolution of the 

 second degree, every zone of which is congruent with 

 e»ery other of the same breadth. This tangent plane 

 meets the surface in two rectilinear generators, which 

 intersect at the given point and have the property of equi- 

 distance from the given line. Parallels in this sense are 

 of course imaginary in hyperbolic space, Euclid's parallel 

 being the transition case for parallels in both senses. It 

 seems a pity that a new word has not been used for this 

 species of parallel. 



It follows at once by synthetic reasoning of the simplest 

 kind (in which we may in fact dispense with the aid of 



biquaternions or analytical aid of any kind) that almost 

 all the properties of Euclidian parallels and parallelo- 

 grams have their counterpart in the theory of Clifford's 

 parallels, due attention being paid to the distinction of 

 right and left. It is shown that a motion of a rigid body 

 is possible in elliptic space such that every point moves in a 

 right parallel, or every point in a left parallel, to a given 

 straight line. A motion of the first kind is called a right 

 vector, a motion of the second kind a left vector. The 

 composition of two right vectors gives a right vector, and 

 two left vectors a left vector ; whereas the composition of 

 a right vector with a left vector gives the most general 

 motion of a rigid body, which Clifford calls a motor. It 

 was to represent the ratio of two such displacements that 

 Clifford invented his Biquaternion. Translation, strictly 

 analogous to that in Euclidian space, i.e. rotation about 

 the line at infinity does not exist in elliptic space. We 

 may, of course, cause a body so to move that every 

 point of it remains equidistant from a given line, and 

 in the same initial plane with that line. Such a displace- 

 ment is the same as a rotation about the polar of the 

 given line, and is hence called by Clifford a Rotor. We 

 have then the fundamental proposition, that every motor 

 can te represented in an infinite number of ways as the sum 

 of two rotors, but uniquely as the sum of two rotors whose 

 axes are polar conjugates. It is the abolition of the line 

 at infinity, whereby duality is made perfect, that gives the 

 peculiar completeness and elegance to the properties of 

 elliptic space, and fit it to be the paradise of geometers, 

 where no proposition needs to wander disconsolate, bereft 

 of its reciprocal. 



To the second great branch of mathematical theory 

 above alluded to, Clifford made exceedingly important 

 contributions in his memoirs on the " Applications of 

 Grassmann's Extensive Algebra," and "On the Classifi- 

 cation of Geometric Algebras." Following, to some 

 extent, in the footsteps of B. Peirce, whose epoch-making 

 memoir has been given to the public at last in the 

 American Journal of Mathematics for the current yean 

 Clifford treats the subject with an incisive vigour all his 

 own. The point of view (indicated by the word geometric) 

 is no doubt limited, just as Peirce's is in another way ; 

 and there may be some doubt, as yet, as to the exact 

 nature of the foundations upon which the reasoning 

 rests. There is a lingering trace of the old sophistry in 

 Peirce's work, here and there, at least so it appears to 

 us ; a reliance still upon ideas a priori, and a reluctance 

 to abandon the restrictions imposed upon algebra by its 

 arithmetical origin. Yet there can be no question as to 

 the great value of the results already obtained and the 

 immense extension of the mathematical horizon thereby 

 effected. Already the attention of mathematical workers 

 has been powerfully drawn to the matter, and there is 

 hope that ere long another great theory equal in im- 

 portance to the Mannigfaltigkeitslehre will drive its roots 

 through the mathematical soil, 



We have dwelt on two of the subjects touched upon in 

 the "Papers," because they seem to us to be of the 

 greatest immediate importance, and to show Clifford at 

 his best as an original mathematician. But it must not 

 be supposed that there is no other food for the mind 

 mathematical in this volume. On the contrary, not one 

 of these papers but is full of delight and edification, even 



