NATURE 



217 



THURSDAY, JULY 6, 1882 



CLIFFORD'S MATHEMATICAL PAPERS 

 Mathematical Papers. By William Kingdon Clifford. 

 Edited by Robert Tucker, with an Introduction by H. J. 

 Stephen Smith. (London: Macmillan and Co., 1882.) 

 Mathematical Fragments j being Facsimiles of his Un- 

 finished Papers Relating to the Theory of Graphs. By 

 the late \V. K. Clifford. (London : Macmillan and 

 Co., 1881.) 



ONLY those who wander much through the aridities 

 of modern English mathematical text-books, whose 

 duty compels them daily to read such literature, and who 

 know 



" The mi^pent tyme, the service vaine, 

 Whilk to considder is ane pane," 



can understand the pleasure of reviewing a book like 

 Clifford's Papers. Here there is no occasion to yawn 

 over page after page of commonplace, to mark with 

 wonder the hundredth iteration of an ill-founded infer- 

 ence, to trace with languid amusement the method and 

 arrangement of our ancestors, nay, the hereditary dots 

 and dashes decrepid in the fourth generation. On the 

 contrary, the novelty and variety alike of subject and of 

 treatment is almost confusing, every page shadows forth 

 some new idea, every line is informed with the personality 

 and with the genius of its author. 



Clifford was one of the bright spirits, all too few in 

 number, who, in a generation, whose educational system 

 is devoted to the encouragement of mediocrity and the 

 cultivation of sciolism, saved the English school from the 

 reproach of inability to follow their leaders. He was one of 

 the select few who sat at the feet of Ca>ley and Sylvester, 

 and shared their genius. 'When we compare him with the 

 former of his great masters, he appears at first to want 

 the steadfast purpose and rugged strength of our mathe- 

 matical giant. The extreme, almost boyish vivacity of 

 his style, and the refined elegance and studied variety of 

 his methods give an impression of this kind which a 

 nearer acquaintance with his work speedily dispels. 

 Apart from his great originality, this elegance, popularity 

 in the best sense, of style gave Clifford a specially im- 

 portant place among the leaders of the English School of 

 Mathematicians, a place which there seems to be none 

 left to fill. It was by his assistance that many were led 

 to scale the almost inaccessible heights on which stand 

 the structures of modern mathematics. 



In some respects the exuberant philosophy of his 

 popular works, especially his lectures, in which the more 

 striking conclusions of modern mathematical science 

 were presented to the uninitiated, must have harmed his 

 reputation for solidity of thought. We are also inclined 

 to doubt whether some of the enthusiastic non-mathe- 

 matical souls that thought they had assimilated his 

 teaching, really after all rose to the conception of 

 Riemann's finite space of uniform positive curvature, in 

 which the problem is solved of 



" Einer dem's zu Herzen ging 

 Dass ihm sein Zopf so hinten hing, 

 Der wollt' es anders haben." 



Such a flight is given only to the sons of Genius, and to 

 Vol. xxvi. — No. 662 



those who have in the first place painfully exercised their 

 pinions in less ambitious journeys. Still these lectures of 

 Clifford did good service in drawing the attention of the 

 rising generation to the revolution that is taking-place in 

 the very elements of exact science. If every physical 

 discovery of permanent or passing importance is to have 

 its day in the drawing-room and the lecture-hall, why 

 should the trumpet of mathematical progress not be 

 blown occasionally in the streets of Gath and Ascalon ? 

 If too many be called in this way, some few may still be 

 chosen. To these few the volume of Mathematical Papers 

 will furnish the best help available in the English language 

 to enable them to follow their calling. To our mind the 

 popular lectures are cut too much after the passing fashion 

 of the present day ; and we should be surprised if the 

 majority of those best qualified to judge of Clifford's work 

 did not agree with us that it will be on the present volume 

 that his future fame will rest. In our poor judgment there 

 is ample foil dation. 



It would scarcely be proper here to criticise the papers 

 in detail, with the view of pointing out the exact amount 

 of originality in each. Besides, even if the reviewer felt 

 more confident of his judgment in such a matter, the task 

 were a needless one, for it has been done already, in the 

 admirable introduction, by an authority to whom every 

 English mathematician will at once bow. 



The Papers have a somewhat fragmentary aspect. This 

 is due in part to the immense range of Clifford's mathe- 

 matical sympathy, which led him to write on a great 

 variety of subjects ; but mainly to the fact that many of 

 the papers are actually unfinished, some of the most im- 

 portant being indeed mere sketches. Clifford seems to 

 have cared, comparatively speaking, but little for the 

 mere mathematical Artj his interest was reserved mainly 

 for methods and principles. Accordingly we find him 

 much occupied with new and far-reaching theories ; and 

 many of the memoirs in this volume are merely the out- 

 lines of vast schemes of work, which life and leisure were 

 denied him to accomplish. 



Besides advances in the Theory of Algebraic Integrals, 

 the development of Projective Geometry, and the enor- 

 mous extension of analysis that is included under the title 

 of Higher Algebra, two great generalisations have marked 

 the progress of modern mathematical science. The first 

 of these is the extension of the axioms of geometry, 

 which originated with Gauss, Bolyai, and Lobatschewsky, 

 and was perfected by Riemann, and the theory of an 

 w-fold manifoldness (Mannigfaltigkeitslehre) of which tri- 

 dimensional geometry in this extended sense is only a 

 particular case, Euclidian geometry a more particular 

 case still. The second consists in a somewhat similar 

 extension of the Axioms, or more strictly speaking, of the 

 Laws of Operation, of Algebra, begun independently by 

 Hamilton and Grassmann, and resulting in the first 

 instance in the Quaternions of the one and the Ausdeh- 

 nungslehre of the other. Both these generalisations 

 have been progressive, and both appear to be pregnant 

 with mighty results for the future. Clifford seized upon 

 them with the instinct of genius. They pervade and 

 colour the whole of his work, and the student who wishes 

 clearly to understand the tendency of much that he has 

 done must begin by attaining some mastery over these 

 fundamental novelties. Great assistance will be obtained 



