I02 



NA TURE 



[Dec. 2, li 



I have not ventured to attempt any remarks upon the 

 wide region of pure mathematics, or even upon the pro- 

 gress of such portions of it as have attracted tlie greatest 

 share of interest among ourselves. I have felt that, as 

 one who has resided and lectured in Cambridge for the 

 past fifteen years, the most appropriate subjects for my 

 address would be those upon which my residence in the 

 University during an eventful period, or my experience as 

 a lecturer, might to some extent qualify me to speak. 

 Still, even when so restricted, I have found it no easy 

 matter to decide upon the subjects to which I was most 

 desirous of drawing your attention to-night. 



I should like to have spoken at length upon the theory 

 of elliptic functions. For fourteen years I have lectured 

 regularly, each year, upon this subject, and no lectures of 

 mine have been of so much interest to me. I be- 

 lieve that the time is rapidly approaching when the 

 elementary portions of the theory will be regarded as 

 necessarily forming part of the common course of reading 

 of all students of mathematics, so that a familiarity with 

 sn's, en's, dn's, and their properties will become as essen- 

 tial as the differential calculus to the mathematical equip- 

 ment of every person who has made mathematics one of 

 his subjects of study. 



Quite apart from its far-reaching influence in all branches 

 of pure mathematics and its widespread applications in 

 mathematical physics, there are special reasons which 

 make the theory of elliptic functions a subject of peculiar 

 interest in a course of mathematical studies, and one to 

 which it is important that the student should be intro- 

 duced as early as possible in his career, whether he be read- 

 ing mathematics for its own sake, or for the sake of its 

 applications, or for its advantages as a mental training. It 

 is the first mathematical "theory" that he meets with in 

 his reading — meaning by a " theory" a body of theorems 

 and properties of functions so related to each other that 

 the student cannot fail to see from the equations them- 

 selves that they form a consistent and remarkable system of 

 facts, worthy of study on their own account, irrespective 

 of any applications of which they may be susceptible. It 

 is true that trigonometry, if regarded as the theory of 

 singly periodic functions, is a theory in this sense, but it is 

 reached by the student at too early a stage for him to be 

 enabled to appreciate the nature and importance of facts 

 that are expressed in tlie mathematical language of for- 

 mulas, and even if it were not so, the manner in which 

 the subject is treated in text-books (the functions being 

 derived from the circle and applied to the solution of 

 triangles, &c., before they are considered analytically) 

 makes it difficult to separate the mathematical theory from 

 its various applications. In analytical geometry, which 

 the student next meets with in his reading, a method of 

 representing curves by equations is explained, and applied 

 to the investigation and proof of properties of conies. In 

 his next suljject, differential calculus, he is introduced to 

 new conceptions and processes of the very highest import- 

 ance and the most fundamental character, and is taught 

 to apply them to the investigation of maxima and minima, 

 tangents and asymptotes to curves, envelopes, &c. Then 

 come the elements of the integral calculus and of differ- 

 ential equations : the former consisting of a few chapters 

 giving methods of integrating various classes of functions, 

 followed by applications to curves and surfaces ; and the 

 latter of rules and methods for treating such equations as 

 admit of finite solution. 



Not one of these subjects, in the form in which they are 

 necessarily presented to students, is an end in itself or 

 exists for itself: they consist of ideas, methods, processes, 

 and rules,which the student is taught to apply and to under- 

 stand ; they contain the conceptions with which he has to 

 make himself as familiar as with the commonest facts of 

 life, the tools which he is to have ever ready to his hand 

 for use. And in the course of acquiring this knowledge 

 he is made acquainted with numerous connected series of 



propositions — such as the properties of conies — besides 

 various important results of more purely analytical in- 

 terest. But all of these developments are presented to 

 him in a form which throws no light upon the manner in 

 which they were originally discovered, and, though the 

 propositions are made to follow one another in clear logi- 

 cal order, the student cannot but be sensible that he is 

 travelling, not along a natural highway, but upon a well- 

 worn road, artificially constructed for his convenience. 

 It is not till he reaches the subject of elliptic functions 

 that he has the opportunity of seeing how, by means of 

 the principles and processes that he has learned, a theory 

 can be developed in which one result leads on of itself to 

 another, in which every system of formulae suggests ideas 

 and inquiries about which the mind is eager to satisfy 

 itself, and opens to the view fresh formulae connected by 

 unsuspected relations with others already obtained, so 

 that he cannot resist the feeling that the subject is taking 

 its own course, and that he is merely a bewildered spectator, 

 delighted with the results which unfold themselves before 

 him. He feels that the formulae are, as it were, developing 

 the subject of themselves, and that his part is passive : it 

 is for him to follow where the formulae point the way, 

 and be amazed by the new wonders to which they lead 

 him. 



It may be that in using this language I am expressing 

 the feelings of a mathematician, rather than those of a 

 student on reading the elements of the subject for the first 

 time ; still I am convinced that the attributes I have 

 just referred to are those which distinguish a genuine 

 mathematical theory from a mere collection of useful 

 principles and facts, and that no one can have studied 

 elliptic functions without realising that mathematics is not 

 only a weapon of research but a real living language — a 

 language that can reveal wonderful and mysterious worlds 

 of truths, of which, without its help, the mind could have 

 gained not the least conception. It seems to me, there- 

 fore, of the highest importance that the student should be 

 introduced to a real mathematical theory at the earliest 

 stage at which his knowledge will permit of his deriving 

 from it the peculiar advantages which I have mentioned. 

 Thus only can he obtain expanded views or a true under- 

 standing of the science he is studying. Higher algebra 

 and theory of numbers afford other conspicuous examples 

 of the perfection that a pure mathematical theory can 

 exhibit, but they do not lie so directly in the line of a 

 general mathematical course of studies. Regarded from 

 this latter point of view, elliptic functions has the addi- 

 tional merit of being a subject whose importance is 

 recognised, on account of its physical applications, even 

 by those to whom the gift of duly appreciating the 

 wonders of pure mathematics seems to have been partially 

 denied. 



I should have liked also to have spoken at some length 

 upon another subject that is constantly in my thoughts : 

 I mean the pressing need of text-books upon the higher 

 branches of mathematics. Of text-books for use in 

 schools we have an abundance, and each month produces 

 a fresh supply ; but it is only occasionally that we have to 

 welcome a work intended for the use of the higher 

 University student or the mathematician. Every one of 

 us must sometimes ha\'e felt the want of an introductory 

 treatise that would give the reader the fundamental pro- 

 positions in some branch of mathematics which exists 

 only in memoirs and papers scattered throughout the 

 wilderness oi Journals and Transactions of Societies. We 

 can scarcely expect to have provided for us, in many high 

 subjects, text-books so admirable and thorough as Dr. 

 Salmons ; still I cannot refrain from expressing the hope 

 that in the future the number of advanced mathematical 

 treatises may not be so infinitesimal compared with the 

 number of memoirs as at present. I could mention several 

 subjects that are almost at a standstill, because advance 

 is impracticable for want of avenues by which new workers 



