Glaisher on the Inclusion of the Theory of Jacobian Elliptic Functions in Mathematics Curricula

 I quote below excerpts from a Presidential Address delivered by J. W. L. Glaisher to the London Mathematical Society and published in its Proceedings in November 1886. He discusses the theory of Jacobian elliptic functions.

"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 tho 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 have liked to speak 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 believe that the time is rapidly approaching when the elementary portions of the theory will be regarded as necessarily forming a part of the common course of reading of all students of mathematics, so that a familiarity with sn's, cn's, dn's, and their properties, will become as essential as the differential calculus to the mental analytical equipment 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 introduced as early as possible in his career, whether he be reading 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 themselves 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 the mathematical language of formula); 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 conics. In his next subject, differential calculus, he is introduced to new conceptions and processes of the very highest importance 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 differential 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 find methods for treating such equations as admit of finite solution. Not one of these subjects, in tho form in which they arc 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 understand; 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. In the course of acquiring this knowledge, he is made acquainted with numerous connected series of propositions—such as the properties of conics—besides various important results of more purely analytical interest. 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 logical 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 enquiries 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 formula) 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 to 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, therefore, 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 understanding of the science he is studying. Higher algebra and the 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 additional 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." (Emphases added.)