by G. A. Bliss
Graduate work at Chicago
In the graduate school at the University of Chicago my work at first was mostly in mathematical astronomy under. It was fascinating to me, and I owe to Professor Moulton a very great debt for his inspiring teaching, and especially for his encouragement in the writing of my first published paper. It was entitled “The motion of a heavenly body in a resisting medium” and had an influence on myself quite out of proportion to the value of the paper itself. At the end of my first graduate year I applied for a fellowship in astronomy which was not granted, though I understood that Professor Moulton recommended it. My failure to secure this appointment was a great disappointment to me, in view of my somewhat difficult financial circumstances, but the effect on me was undoubtedly good because it caused me to take stock of my own interests and plans for the future. After some careful consideration I concluded that it was really the mathematics which had the great attraction for me, and I decided to try for a Ph.D. in that subject rather than in astronomy. I have never regretted the decision, though it seems clear to me that one could find great satisfaction in any domain of applied mathematics. I must have decided to make mathematics my principal subject before the end of my first graduate year, because I received the master’s degree in mathematics, apparently in the summer quarter of 1898.
The three leading men in mathematics at the University of Chicago at that time (1898) were E. H. Moore, , and . They supplemented each other beautifully. Moore was the fiery enthusiast, keenly interested in the popular mathematical research movements of his day, one after another, at that time especially in group and field theory. Bolza was a product of the meticulous German school of analysis led by . He was an able, aggressive, and widely read research scholar. Maschke was a geometer, much more easy-going than the other two, but brilliant in his research, and the best lecturer on geometry to whom I have ever listened. Under the leadership of these three men Chicago was unsurpassed at that time in America as an institution for the study of higher mathematics. With such representatives of the best mathematical scholarship near at hand one did not need to go abroad to study. Like many other members of our present American mathematical community I am greatly indebted to Moore and Maschke and Bolza for the example of their distinguished research and scholarship and their continuously friendly and encouraging interest.
My last two years of graduate study passed rapidly, full of the inspiration and hard work well known to every one who has been a candidate for the doctor’s degree. Bolza had a beautifully written record, made by himself, of the now famous course in the calculus of variations which Weierstrass had given in 1879, and let me make a copy of it for my own use. It is a course which contains most important results, quite new in 1879, and which has had an unusually wide influence, though in 1898 it was relatively little known. A record of it over Weierstrass’ own name was first published in 1927 in volume VII of his Mathematische Werke. The possession of this masterpiece, and Bolza’s fascinating lectures, gave me a dominant interest in the calculus of variations, though I was exposed to many other mathematical theories and thoroughly enjoyed them. I wrote my doctor’s thesis under Bolza’s direction on “The geodesic lines on the anchor ring.” It was a study of the forms of these curves on the anchor ring and a classification of them according to their possession or non-possession of conjugate points. I have found the paper a useful one and from my correspondence concerning it at various times believe that others have had the same experience.
Early teaching experience
My first teaching position was in 1900 at the University of Minnesota. I enjoyed the students and the teaching from the start.Vorlesungen über Variationsrechnung had just appeared. It was the only treatise on the subject which contained an account of the ideas of Weierstrass, and was not any too complete in that respect either. I determined to make myself as far as possible an expert in the theory of the calculus of variations, and began by reading Kneser. It was the kind of a program which I still recommend strongly to my young graduate students. If one expects to do mathematical research he must first be a master of the literature of some field, or some portion of a field. When he has become such a master he will find plenty of research questions pressing upon him for answer. Many students never reach this stage, either because they are early distracted by other interests, or because they are too anxious to do research and try to do it without having the fundamental scholarship which justifies it. In my case I found the next two years of absorbing scientific interest. I read Kneser and much other literature of the calculus of variations, and wrote my first completely independent research paper “The second variation when one end-point is variable.” It contained an analysis of the movement of a focal point of a curve transversal to an extremal w!en the curvature of the transversal curve varies. I believe that the first paper a student writes after he has received his Ph.D., or at any time as a result entirely of his own interest and enthusiasm, is the most important one for him. In my case the paper just mentioned above was the one which first gave me some confidence that I might in time become a productive scholar.
Post-graduate work abroad
After two years at Minnesota the future there did not look very bright for me, though the years themselves had been most interesting and profitable ones. My father had died in 1900, but I had saved some money out of my very moderate salary, and my mother offered to supplement this out of her own slender resources in order that I might have the benefit of a year of study abroad. It was something which she never should have done, and which can never be repaid except by helping to provide another such opportunity for another such student. I spent all of my year in Göttingen except for brief visits to Berlin, Italy, Paris, and England. The leaders in Göttingen at that time were, , and , a notable group. Among the younger men were , , Abraham, , and , certainly inspiring comrades. As a visitor with his Ph.D. already behind him I was invited to attend the Mathematische Gesellschaft which met every other week. Klein was the dominant figure in the Gesellschaft, a marvel of wide acquaintance with mathematics and mathematicians, and ready with many interesting comments on their books and personalities. After the meeting was over Hilbert and Minkowski would lead a party of a dozen or so to a neighboring cafe where the evening would be spent in lively mathematical discussions.
My year at Göttingen was a most profitable one. I listened to lecture courses by Hilbert, Minkowski, and Klein, and took an active part as a student in the seminar of Hilbert and Minkowski. Hilbert was lecturing on Mechanik der Kontinua, Minkowski on Minimalflächen, and Klein on his Encyklopädie. My reports in the seminar were on some problems in conformal representation in which Hilbert was interested. None of these was closely associated with my chief interest, the calculus of variations, but I found in the excellent library of the mathematical reading room most interesting records of courses in the calculus of variations by Weierstrass, Hilbert, Sommerfeldt, and Zermelo. These alone would have justified my trip to Europe. The profit becomes much greater if one adds to them the inspiration of distinguished professors, the benefit of acquaintance with the group of younger men many of whom afterward became famous, and the great pleasure of lifelong friendship withand , then young American candidates for the doctor’s degree at Göttingen. A paper which emerged from my studies of that year was a discussion of necessary and sufficient conditions for a minimum for a problem of the calculus of variations in the plane with both end-points variable.
In their seminar Hilbert and Minkowski first assigned reports to students in pairs. My colleague was an assistant in a neighboring department of the University who had a quite remarkable acquaintance with mathematical books. I was much discouraged at my own ignorance after our first conference, but I soon found that he had very moderate command of mathematical techniques. I have thought sometimes that our relatively informal exercises and reports in our graduate courses in America tend to develop control of technique earlier than the more formal lecture courses of the German universities. At any rate after our first report I was called upon to make all of the rest of them. It was rather rough going because of my poor German. One day Hilbert interrupted me suddenly with the remark “Ach Herr Bliss, dies Mal haben sie sich gierrt,” whereupon both he and Minkowski rushed to the board, as was their custom, and began writing and talking excitedly with no regard for each other, Hilbert with his face very close to the board because he was so nearsighted, all to the great delight of the students. Presently Hilbert turned to me, as much pleased as I was to find that there had been no mistake, and said that I must have meant “eine eingeschnittene Ellipse.” An ellipse must be cut in from the ends of the major axes to the foci in order to be represented conformally on a rectangle by the method I was trying to explain. The word “eingeschnitten” was not in my vocabulary at that time, but I venture to say that it is now permanently a part of it.
Klein was a dominant figure in German mathematics, and indeed in the German educational world in general. As a younger man he did beautiful research in many domains of pure mathematics. When he grew older he turned almost entirely to applied mathematics, in which his research has seemed to me much less significant, and became rather critical, so it seemed to me, of research in pure mathematics. When I was in Göttingen he looked really ill, and though he lived many years after that I think that he had always to be very careful of his health. One day I was asked to give a paper at the Gesellschaft and presented the paper mentioned above. When I had finished Klein rather took my breath away by asking “But, Herr Bliss, who would be interested in this?” That was for the moment a poser to me, but I tried to explain my belief that one who had been especially a student of the calculus of variations might find something of interest in the results of the paper. It seems that I mixed the words “man” and “sie” so that my remark had rather the meaning that “If you knew anything about the calculus of variations you would be interested.” I still remember the breathless moment which followed my remark, incomprehensible to me at that time but comprehensible enough several years later when Max Abraham retold the incident to some of us with, I hope, much exaggeration. Certainly I could never intentionally be disrespectful to one who has wielded the uniquely distinguished influence of Klein. As for my paper I must say in self defense that I have had many inquiries for the reprints which unfortunately were lost somewhere between Germany and the United States. As I look at the paper again from time to time it seems to me a rather good one, the first relatively complete treatment of a problem of the calculus of variations with two variable end-points.
Later teaching positions
In the autumn of 1903 a position at Minnesota would still have been open to me, but Professor E. H. Moore, head of the mathematical department of the University of Chicago, wrote to me that I could have an associateship there for one year if I was interested. It seemed to me an excellent opportunity to spend another year in an inspiring mathematical environment, and I accepted his suggestion with much alacrity. My anticipation proved to be well justified. My sojourn abroad had not weakened, but had strengthened, my respect for the Chicago department of mathematics, and for my professors there, and I enjoyed and profited much from my associations with them as a member of the faculty during the year 1903–4.
In the autumn of 1904 I went as assistant professor to the University of Missouri wherewas head of the department of mathematics. It was my first experience with life in a small town and I enjoyed my surroundings immensely. Hedrick was ideally congenial as a leader. While I was there he organized an association of Missouri teachers of mathematics. The first meeting was held at the University in Columbia, Missouri, and we all helped as much as we could to make it a success. Hedrick had, however, a knack for making things go irrespective of our assistance. I had hardly become a part of the University circle in Columbia when in the spring of 1905 I had an invitation to go to Princeton. It was with great regret that I left, carrying with me very permanent memories of a most happy year and of my friendly associations with Hedrick. In one of my classes , now professor at Cornell University, was a student. He was extraordinarily precocious and able.
In the autumn of 1905 the then new preceptorial system at Princeton was greatly extended. About fifty new “preceptors” were added to the faculty. I was one of them. Some of us did not like the word “preceptor” very much, because it suggested tutor or assistant and had no established meaning in the educational world. We were assured that we were the equals of assistant professors elsewhere and that we might think of ourselves as such. I always did so and have always listed myself as such to a world which has now pretty much forgotten the preceptorial system. Princeton is an exceedingly friendly but somewhat conservative place, and I have doubts if the fifty preceptors really became an integral part of the community during the three years that I was there. Those of us in mathematics had a most interesting time, however, with our science and our students.
R. L. Moore were among the leaders and my friendship for them and admiration of their scholarship have remained with me permanently. During one of my years at Princeton was a visiting professor. He was about our own age but very unapproachable. We tried to give him a good hearing in his lectures on Dynamical Theory of Gases, but anything like free discussion proved to be impossible. I learned more of him later by reading his book on electro-magnetic theory, but have never been enthusiastic about his writing as viewed from the standpoint of pure as contrasted to applied mathematics.was Dean and also head of the department of mathematics during my sojourn in Princeton. He was much beloved in his community, not aggressive in mathematical research, but most sympathetic and encouraging to the younger men in his department. We regarded him with great affection. Frequently on Saturday nights the younger men in the department met in the rooms of one or another of them for refreshments and mathematical discussions. The discussions were the liveliest and in many way the most interesting ones in which I have ever taken part. , , and
Teaching at Chicago
About the time I was in Princeton I spent most of my summers teaching in universities other than Princeton. In 1907 a summer term at Wisconsin, where I renewed my associations with Mason and his family, gave me much pleasure. He was at Yale while I was at Princeton and I saw him there from time to time also. But most of my summer teaching was in the Department of Mathematics at Chicago. In the early part of 1908 Maschke died, and during the summer of that year, while teaching at Chicago, I was offered an associate professorship there. Maschke’s loss to his Department was a very great one, in fact irreparable, and I was sure that they would have preferred an older man to replace him. But the challenge to me was one which I could not refuse and I was anxious to accept. Professor Fine at Princeton was most generous in his willingness to facilitate my move, and after teaching some weeks at Princeton in the autumn of 1908, while he made readjustments in his staff and schedule, I joined the Department at Chicago and have remained there ever since.
The next two years were rather strenuous ones for me. I undertook to give some of the advanced courses in geometry which Maschke had always given before, metric differential geometry, analytic projective geometry, higher plane curves. It was most interesting work and I have always appreciated the wider experience in mathematics which this teaching gave me though it took much time. In 1909and I were the lecturers at the Princeton Colloquium of the American Mathematical Society. My part of the colloquium was on fundamental existence theorems for implicit functions and ordinary and partial differential equations in the theory of functions of real variables. I have always hoped to find time to pursue much further the theory of singular points of real transformations of the plane which I developed in those lectures.
In 1910 Bolza withdrew from the department of mathematics at Chicago and went back to Germany to live in Freiburg in Baden. His mother there was quite old and was not expected to live long, though as a matter of fact she did live for many years. Bolza’s departure was another great loss to Chicago., then of Illinois, had made a name for himself in projective differential geometry, and we were fortunate in securing him in the autumn of 1910 as a member of our mathematical staff. His arrival and Bolza’s departure brought my teaching back again into my chosen field of analysis.
In 1913 I was made a full professor at the University of Chicago.as the leader, with Dickson, Moore, and of course many others, were engaged about that time in strengthening The American Mathematical Monthly and in the organization of the Mathematical Association of America. In the summer of 1918, after the United States had been drawn into the war, we undertook in our Department to give courses in navigation to men who had been accepted and were waiting in Chicago to enter the naval school there. We had a number of sections. Mine had about one hundred students in it. To my surprise when the course was over the men in it presented me with a very fine watch. I have never felt that acceptance of it was appropriate, since they were the ones who were making the sacrifices and preparing themselves for service. But I must say that I have valued their gift much, especially as a reminder of their seriously attentive and successful interest.
Aberdeen proving ground
In late summer of 1918 my old friend Oswald Veblen began writing me from Aberdeen Proving Ground saying that he needed mathematicians. He was major in charge of the Range Firing Section there, with the duty of supervising the preparation of range tables. Our work at Chicago with the prospective naval students was by that time well organized, and it seemed to me the most effective work which I could then do. But I had confidence in Veblen’s judgment and after repeated letters decided to join him at Aberdeen as what they called a “scientific expert.” Veblen was right, of course, about the need of mathematics in ballistics. The old Siacci theory broke down early in the war because it contained an approximation not justifiable for gun elevations greater than twenty degrees. F. R. Moulton, Veblen, and their associates had devised effective methods of successive approximation for solving the differential equations of trajectories in all cases, but the theory and application of the so-called “differential corrections” of trajectories was still very imperfect. These are corrections made to the normal range to account for the effects of wind, variations from normal in the density of the air and the weights of the projectile and powder charge, and in the case of long ranges for the rotation of the earth. This was my particular field of study, and I was astonished to find that the most advanced mathematics which I knew was needed and effective. The wind correction, for example, is a function of the wind curve when wind velocity is plotted as a function of altitude. It is what the mathematicians sometimes call a functional or a function of a line. The calculation of the differentials of such functionals is a principal problem of my special field, the calculus of variations. I found that the methods of that calculus were effective, and that in many cases with suitable adjustments they reduced by three-fourths the time necessary for the computation of range corrections. The saving for anti-aircraft trajectories was not so great, but the method is applicable there also, and I have left a detailed description of it for that case in a blue print in the archives at the Proving Ground. These results and savings were important for the Range Firing Section because of the large number of computers engaged in the construction of range tables and the constant flow of data into the Section concerning new experimental firings and new types of guns. I am happy to say that there are many applications other than in ballistics of the method which I devised. Some of them I have pursued to my own satisfaction.
In 1921 I was elected president of the American Mathematical Society. It is a position of great honor, and also of great responsibility. The Society was having financial difficulties, a condition which seems to be normally associated with healthy progress. E. R. Hedrick suggested a campaign to increase membership. I must confess that I doubted the possibility of success sufficient to be profitable, and that I undertook the labor of the campaign with great reluctance because of the time it would take from my scientific research interests. Once started, however, both Hedrick and I devoted ourselves to it heartily. The results well justified our efforts, especially as this first campaign of the Society encouraged later and more successful ones which have developed into a continuous membership campaign under the guidance of our skillful secretary,.
At Chicago, as at other universities, the years following the war were exceedingly crowded ones. There seemed no limit to the number of students who were seeking college educations and higher degrees. Our situation was aggravated by the serious illness and consequent retirement of Wilczynski from active work in 1923. Fortunately we were able to secure, his very able successor in the field of projective differential geometry, to carry on his work. Our climax was reached in 1927. Advanced graduate courses in one summer quarter sometimes attracted over eighty students, and it seemed almost impossible to take care of the large number of candidates seeking higher degrees in mathematics. Fortunately the head of our Department, E. H. Moore, was conservative about increasing our permanently appointed personnel, for after 1927 there was at first a gradual reduction in the demands upon us, and then a rapid decrease during the depression years. We were fortunate in not having expanded our Department unduly, and in being able to retain our regularly appointed staff intact without salary reductions.
In 1927 Professor Moore had passed the normal retiring age and was anxious to lay aside his administrative responsibilities as head of the Department. The Trustees appointed me chairman, at first without public announcement, and Mr. Moore was retained as nominal head.
In 1927 the University established a number of special professorships, one of which is the Eliakim Hastings Moore Distinguished Service Professorship. L. E. Dickson is the first incumbent. He was one of Moore’s earliest students, and his career in mathematical research has been remarkable. I can think of no more appropriate appointment. One of the purposes of our Department has been to provide Dickson with a most favorable environment for the pursuit of his teaching and research interests, and I believe that we have succeeded. In 1933 I was appointed to a similar professorship bearing the name of Martin A. Ryerson. The appointment was a surprise to me and gave me one of the greatest of the pleasurable thrills of my life.
Editorial and committee work
In the preceding paragraphs I have not mentioned my editorial work. I was associate editor of the Annals of Mathematics from 1906 to 1908, and of the Transactions of the American Mathematical Society from 1909 to 1916. E. H. Moore,, and , the first chief editors of the Transactions, actually taught many members of our then youthful research community to write in good mathematical style. Others of us who assisted in those earlier days attempted to do the same. I spent days at a time trying to master unfamiliar fields and preparing sometimes unwelcome advice to authors. The result was that I developed a very great distaste for editorial work. The papers which especially irked me were theses from other universities than my own which I thought should have been more carefully edited at home. In 1916 a chief editorship of the Transactions was offered to me and I declined it. My decision was probably a mistake, since many of our best mathematicians have unselfishly devoted much time to the Transactions, and since the work is undoubtedly exceedingly important for our mathematical community. I did not escape permanently, however, for I was doing the equivalent of editorial criticism on the National Research Fellowship Board from 1924 to 1936, and have been an editor of the Carus Mathematical Monographs from 1924 on, and chairman of the editorial committee of the University of Chicago Science Series from 1929.
The Fellowship Board was a fine group of men, consisting, when I first became a member, of Simon Flexner, Chairman, K. T. Compton and Mendenhall and Millikan in physics, Johnston and Keyes and Kohler in chemistry,and Bliss and Veblen in mathematics. Johnston and Mendenhall later withdrew and were replaced by others. I often wondered how some of these men found the time for the conscientious work which they did on the Board. Our choices were criticized severely in particular cases, sometimes justly, sometimes not. At one time the three groups examined independently their list of past appointees, marking those who had distinguished themselves as Fellows A, those who would again be appointed B, and the failures C. When they compared notes the number of C’s was in each field about ten percent of the total in that field. This seems to me an excellent record, and I wish that each of our critics might have an opportunity to try to equal or surpass it. But the effort to classify people according to their ability, to say that one is better than another, is distasteful to me, and in editorial work it seems to me a much greater pleasure to try to find the strong points of a paper than to criticize its deficiencies. In general I feel sure that our mathematical community will be more effective and much happier if we cultivate whenever possible an appreciative and cooperative point of view rather than a critically comparative one.
Philosophy of graduate work
When Mason was president of the University of Chicago our Department was under great pressure. We had too many students. He repeatedly urged us to restrict enrollment by setting higher standards for admission. I have always felt that this is a mistaken policy. Experience has shown that it is impossible in many cases to foretell an entering student’s future. Some of the inexperienced ones develop real power, and some of the brilliant ones fade away. Many who are not qualified to take a higher degree get a real thrill and a new impetus for their teaching work out of the advanced courses. I therefore favor keeping graduate lecture courses open to all who have a reasonable justification for their desire to listen in. The time when distinctions should be made is when the student applies for candidacy for a higher degree, after he has given his instructors some indications of what he can be expected to do. The candidates for higher degrees are the ones who take the time, not the listeners in lecture courses.
At the present time there are strong influences at work to cheapen higher degrees in mathematics, as also in other subjects. I hope that our standards can be preserved. In our Department we have been unwilling that a candidate for the master’s degree should divide his requirement between mathematics and some other subject, because if he does so he can present for his degree only the most elementary graduate material in either field. We have insisted on a thesis containing if possible some new result of an elementary sort. The purpose of this is to give the student his first introduction to methods of research and to train him in mathematical writing. I advise students to take master’s degrees on the way to their doctorates because of the great value to them of this preliminary training. We keep a record of the performance of a student in attaining his master’s degree and use it as one of the aids to guide us in determining whether or not he should be recommended for candidacy for the doctor’s degree. Thus we can avoid the necessity of preliminary examinations for the doctor’s degree. I am much opposed to such preliminary examinations because they take much time and tend to place the emphasis on courses rather than research for both students and faculty. The sooner a candidate for the doctor’s degree can start his research the better it will be for everybody.
It seems to me that there is widespread misunderstanding of the significance of doctor’s degrees in mathematics. The comment is often made that the purpose of such a degree is to train students for research in mathematics, and that the success of the degree is doubtful because most of those who attain it do not afterward do mathematical research. My own feeling about our higher degrees is quite different. The real purpose of graduate work in mathematics, or any other subject, is to train the student to recognize what men call the truth, and to give him what is usually his first experience in searching out the truth in some special field and recording his impressions. Such a training is invaluable for teaching, or business, or whatever activity may claim the student’s future interest.
Not long ago a committee of the Mathematical Association of America published a report in The American Mathematical Monthly advocating the inauguration of a higher degree for teachers not emphasizing training in research. I should hesitate to believe that research experience is not as valuable for teaching as for other types of activity. Of the ten members of the committee nine have doctor’s degrees in mathematics, and five are doctors of our own Department. Some have been active in mathematical research, and others have devoted themselves to other equally important duties in their respective environments. I would venture the opinion that the training for the doctor’s degrees held by these committee members has in each case been an important factor in establishing them in the positions of usefulness and influence which they now hold. The suggestion of a higher degree for prospective college teachers, comparable with the doctor’s degree but not requiring a research thesis, is a very old one, as old as my experience in university work. I think that some one who has confidence in such degrees should try to establish them. But they should not be called doctor’s degrees, since this title has long implied research.
In a recent report R. G. D. Richardson has tabulated the research output of doctors of philosophy in mathematics in the United States, and has given comparative statistics for various American universities. His figures show among other things that many of our doctors do no research after they receive the doctorate; that a considerable number continue to publish in limited amount; and that a relatively small number continue to be active in mathematical research. The report itself is purely statistical, but it has been the basis for criticisms of the training of doctors in mathematics in this country which seem to me unjustified. The fact that many doctors do no mathematical research after receiving their degrees is quite in accord with the purpose of such degrees as described in a preceding paragraph above, provided that the doctors apply in whatever may be their chosen activity the spirit of the investigator. It is not easy to measure success in this respect, but it is clear that our doctors as a group have been given relatively very great and very important educational responsibilities in the universities of the country. Those of us who have devoted our lives to the intimate study of mathematics are of course greatly interested in the relatively small number of doctors who turn out to be continuously interested and successful in research. The percentage of doctors in this group will be small in mathematics, as in any other subject. To insure adequate renewals in the ranks of such scholars it is necessary that our population of graduate students of mathematics shall always be as large as is reasonable in view of the demand for trained mathematicians in our educational and other institutions. Furthermore for the continuation of the activity and interest of this relatively small group of enthusiasts the presence of the much larger number of mathematicians who do some research themselves and who have high appreciation of it in others is essential. No great mathematical school of the past, with its heroes to whom we often ascribe too exclusively its achievements, ever flourished without its public of well-trained and appreciative listeners. Thus it seems to me that doctors of all the types described in Richardson’s statistics are essential to the well being of our American mathematical school, and that criticisms of our doctorate based upon the figures given in his paper are not well founded.
I would make one final comment on Richardson’s report relative to the doctors of our own Department. Their number is larger than that of any other mathematical Department, but the average of their annual publication output is lower there than in a number of other cases. The average of their total number of published pages per year is also the largest. If one is thinking only of mathematical research produced it is evident that magnitude of production can be attained either by a considerable number of workers each producing a limited amount, or by a smaller number of scholars each of whom is very active. At Chicago we have turned out men of both types and though our total productivity is large our average per doctor is relatively smaller. The discussion of these questions impresses one, however, with the futility of attempts at measurement of the relative effectiveness of different departments. What is needed in every case is a group of men devoted to mathematics and ambitious to make the most effective use of their own personalities and interests in research and teaching. In every environment it is important to stimulate the strong graduate student by the challenge of his research thesis, but it is still more important to impress upon him the significance of a first independently conceived research paper following the doctorate. The preparation and publication of such a paper is the first conclusive evidence of true interest in scholarship.
It is clear from the preceding paragraphs that different individuals and different groups have different interests and different ideas about the training of students, and that different methods in research and teaching may achieve comparable successes. For those who have demonstrated their own effectiveness the most important things which a university can offer are freedom and encouraging recognition. There is nothing more discouraging in university work than to be continually told that methods and ideals of the past have for the most part been seriously wrong, even if one knows from his own experience that in his own department of learning the charge is not well justified. Unfortunately the psychology of our most enthusiastic students of the theory of education at the present time seems to be that of the pessimist and critic, as one may verify by an examination of the addresses on almost any recent program of speakers on educational theory.