26o 



NATURE 



[July i6. 1903 



of algebraic numbers, are instances of subjects the inception 

 of which we owe to the Germany of the nineteenth century. 



While, as we have already remarked, the English have 

 shown a considerable interest in some branches of research, 

 it is often said, and I think with truth, that our record in 

 the history of modern mathematics is not worthy of our 

 place among the nations. It is, at any rate, a fact that a 

 considerable number of men spend the greater part of their 

 student life in the special study of mathematics, and after 

 a successful college career are appointed to teaching posts 

 which leave them a fair amount of leisure for the pursuit 

 of their chosen subject, and that, nevertheless, their life 

 is barren of contributions to learning. This state of things, 

 which we must admit to be much more general in this 

 country than on the Continent, is, perhaps, the gravest 

 feature in the situation at present, and it becomes deeply 

 interesting to attempt to trace its course. 



The explanation which I personally favour places the 

 origin of the evil back in student days, and in our methods 

 of instruction. The most casual reader of text-books 

 cannot fail to be struck by the fact that English text-books 

 treat their subjects in much greater detail than is customary 

 on the Continent ; innumerable side-issues are raised, trifles 

 are elaborated, and examples are multiplied a hundredfold. 

 Moreover, topics which have now become comparatively 

 unimportant, or even positively obsolete, are always re- 

 tained, and each text-book differs from its predecessor only 

 in a further increase of prolixity. 



The result is that even the best men cannot, in a student 

 course of many years, wade through this mass of material 

 to the frontier of existing knowledge, and the unfortunate 

 student finds his college career over and his teaching life 

 begun before he has gone anything like far enough to begin 

 independent research. 



I can scarcely conceive a greater benefit to the study of 

 mathematics in this country than a series of short text- 

 books holding closely to the main lines, casting away the 

 rubbish and the trifles, and carrying a student to the 

 furthest boundary of learning in a three years' university 

 course. 



Although the evil relates chiefly to college text-books, it 

 would not be difficult to mention branches of higher learn- 

 ing the progress of which has been arrested for a long 

 period simply by the publication of unreadable accounts of 

 them. 



In order that our research may be the worthy centre of 

 a life-work, it is needful to have not merely the equipment 

 of a full knowledge of the past, but also a clear and well- 

 defined idea as to which are to be considered the chief and 

 which the minor objects of investigation. For the next 

 worse thing to doing no research at all is to spend one's 

 time on matters that are of very little consequence. 



This point is all the more important because there is 

 every indication that we are now at a critical point in the 

 history of mathematics, and that the twentieth century will 

 see progress in somewhat different directions from those 

 which characterised the last half of the nineteenth. 



Let me recall the fact that, from the time of Newton to 

 the death of Cauchy in 1857, the main progress of mathe- 

 matics was in the realm of analysis — the science which is 

 based on Newton's infinitesimal calculus, and which was 

 enriched by all the greatest masters, Euler, Lagrange, 

 Laplace, D'Alembert, the Bernouillis, Taylor, Legendre, 

 Fourier, Gauss, Abel, Jacobi, and Cauchy. 



The latter half of the nineteenth century saw, however, 

 a notable change. As in the hands of these giants even the 

 inexhaustible mine of analysis seemed to be worked out, 

 new subjects came into prominence, such as invariants, the 

 theory of groups, the Mengenlehre, analysis situs, 

 quaternions, and non-Euclidean geometry ; the theory of 

 functions developed itself on lines quite foreign to the older 

 analysts, and the demand for rigorous proofs led many 

 even of those who remained in the domain of analysis, as 

 Du Bois Reymond and Pringsheim, to devote themselves 

 rather to a careful investigation of the foundations than 

 to an extension of the superstructure. Now, however, we 

 seem to be on the threshold of a change. The branches 

 of mathematics the introduction of which we owe to the last 

 generations of German mathematicians are already be- 

 ginning to show signs of exhaustion — bv which I mean 

 that further work in such a subject as the invariant-theorem 

 along the present lines does not promise to yield any great 

 NO. 1759. VOL. 68] 



increase of mathematical power ; the process of under- 

 pinning the edifice has now been, to a great extent, accom- 

 plished, and the work of upbuilding can be recommenced, 

 while the interest of the theory of functions has largely 

 passed over into topics of a distinctly analytical character, 

 such as the theory of automorphic functions, the theory of 

 expansions convergent within a given region, and the theory 

 of summable series. 



All the indications seem to point to the conclusion that 

 pure mathematics is in the process of its natural evolution 

 returning to the old path, and that a new phase of advance 

 in the analysis of differential equations and functions is 

 about to come upon us. 



But though the same, it will be changed ; the work of 

 the last fifty years has given rise to ideas and methods the 

 application of which must necessarily extend the older 

 subjects in altogether new directions, and perhaps lead, to 

 an era worthy to be compared with that of Euler and 

 Lagrange. 



UNIVERSITY AND EDUCATIONAL 

 INTELLIGENCE. 

 The Joule studentship, founded " to assist research, 

 especially among younger men, in those branches of 

 physical science more immediately connected with Joule's 

 work," will shortly become vacant through the termination 

 of the tenure of Dr. Ulrich Behn, who was nominated by 

 the K. Akademie der Wissenschaften of Berlin in 1901. 

 On this occasion the nomination of a student rests with 

 the president and council of the Royal Society, who will 

 make their selection in October next. The studentship is 

 of the value of 100/. in all. Information may be obtained 

 from the assistant secretary of the Royal Society. 



We regret to see the announcement of the death of Sir 

 Joshua Fitch at the age of seventy-nine. The country has 

 thus lost one of its foremost authorities on educational theory 

 and practice. Sir Joshua Fitch was for thirty-one years con- 

 nected with the Education Department, and the wide and 

 varied experience which he acquired give exceptional weight 

 to his views on educational subjects, expressed in many 

 articles, books and addresses. Since his retirement from 

 official life in 1894, he has taken an active part in the 

 formation of sound public opinion upon educational 

 questions. He recognised that the important point to 

 bring before the people was " that education ought to be 

 a national concern, that it should not be left entirely to 

 local, or private, or irresponsible initiative." This principle 

 must be accepted before any substantial provision will be 

 made for educational progress. Sir Joshua took an active 

 part in the reorganisation of the University of London 

 as a teaching university, and throughout his career 

 identified himself with movements which had for their 

 object the coordination and development of the educational 

 forces of the country. 



Of the Education Vote of 11,249,806?. agreed to by Com- 

 mittee of the House of Commons last Thursday, only half 

 a million belongs to secondary education. In the course 

 of a speech made in introducing the vote. Sir William 

 Anson expressed the fear that the traditional educational 

 work was being destroyed, and was not being replaced 

 with anything of a really substantial character. He was 

 especially alarmed at the condition of the smaller grammar 

 schools. " In these schools much attention is now being 

 given to science, with results that are not altogether satis- 

 factory. The classical languages are almost disregarded, 

 and history and geography are neglected." Mr. Balfour 

 spoke to much the same effect in the speech at the Allied 

 Colonial Universities dinner which appears in another part of 

 this issue. The suggestion is that science is not such a good 

 educational instrument as the study of dead languages. It 

 does not need much consideration to see that these con- 

 clusions are unsound. For centuries our grammar schools 

 have been training grounds for teachers of Greek and 

 Latin, and it would be strange if efficient methods had not 

 been evolved. Every encouragement has been given to the 

 humanities both in school and university, and the masters 

 who have controlled the curriculum or guided the studies 

 have been, with rare exceptions, men distinguished for 



