Celebratio Mathematica

In the gradu­ate school at the Uni­versity of Chica­go my work at first was mostly in math­em­at­ic­al as­tro­nomy un­der F. R. Moulton. 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, O. Bolza, and H. Masch­ke. 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 Wei­er­strass. 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.

My first teach­ing po­s­i­tion was in 1900 at the Uni­versity of Min­nesota. I en­joyed the stu­dents and the teach­ing from the start. Kneser’s 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.

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 Klein, Hil­bert, and Minkowski, a not­able group. Among the young­er men were Zer­melo, E. Schmidt, Ab­ra­ham, Fe­jer, and Carathéodory, 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 with Max Ma­son and O. D. Kel­logg, 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.

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 where E. R. Hedrick was 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 W. A. Hur­witz, 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.

H. B. Fine 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. Eis­en­hart, Veblen, J. W. Young and 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 Jeans 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.

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 1909 Kas­ner and 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. Wil­czyn­ski, 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. Slaught 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.

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, R. G. D. Richard­son.

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 E. P. Lane, 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.

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, T. S. Fiske, and E. W. Brown, 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, Birk­hoff 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.

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.