E. T. Bell and at Caltech between the Wars Judith R. Goodstein and Donald Babbitt

E. T. (Eric Temple) Bell, num- who was in the act of transforming what had been ber theorist, a modest technical school into one of the coun- novelist (as John Taine), and try’s foremost scientific institutes. It had started a man of strong opinions out in 1891 as Throop University (later, Throop about many things, joined Polytechnic Institute), named for its founder, phi- the faculty of the California lanthropist Amos G. Throop. At the end of World Institute of Technology in War I, Throop underwent a radical transformation, the fall of 1926 as a pro- and by 1921 it had a new name, a handsome en- fessor of mathematics. At dowment, and, under Millikan, a new educational forty-three, “he [had] be- philosophy [Goo 91]. come a very hot commodity Catalog descriptions of Caltech’s program of in mathematics” [Rei 01], advanced study and research in pure mathematics having spent fourteen years in the 1920s were intended to interest “students on the faculty of the Univer- specializing in mathematics…to devote some sity of Washington, along of their attention to the modern applications of with prestigious teaching mathematics” and promised “to provide defi- stints at Harvard and the nitely for such a liaison between pure and applied All photos courtesy of the Archives, California Institute of Technology. of Institute California Archives, the of courtesy photos All University of Chicago. Two Eric Temple Bell (1883–1960), ca. mathematics by the addition of instructors whose years before his arrival in 1951. training and interests have been in both fields [CIT Pasadena he had received 28]”. Indeed, the mathematical physics faculty at the American Mathematical Society’s coveted Caltech at that time was probably as good as any- Bôcher Memorial Prize for outstanding work ap- 1 where in the country. Even if he did not write the pearing in the Society’s Transactions. His swift catalog copy himself, Millikan saw the application election to the National Academy of Sciences was of mathematics to other fields as a major consid- expected. eration from the very beginning of his decision to Bell had been lured to Pasadena by the re- expand mathematics at Caltech. nowned experimental physicist Robert A. Millikan, Before Bell’s arrival the mathematics faculty consisted of William Birchby, Harry Van Buskirk, Judith R. Goodstein is university archivist emeritus at the California Institute of Technology. Her email address is Luther Wear, and , all holdovers [email protected]. from the Throop era. Birchby and Van Buskirk taught elementary courses; they were primarily Donald Babbitt is professor of mathematics emeritus at the University of California, Los Angeles. His email address is mathematics teachers rather than researchers. [email protected]. Although he had earned a Ph.D. in mathemat- ics from Johns Hopkins University, Wear, who 1The theory Bell advanced in his long and fundamental memoir (“Arithmetical Paraphrases”, I and II) provided handled some of the more advanced courses, did many applications to the theory of numbers. (Co-winner no research. Bateman, an English mathematical Solomon Lefschetz’s paper offered essentially a complete physicist, was the only member with sterling aca- topological theory of algebraic surfaces.) demic credentials. Following his graduation from Cambridge University (B.A. in 1903; Smith Prize, DOI: http://dx.doi.org/10.1090/noti1009 1905; and M.A. in 1906), Bateman had studied

686 Notices of the AMS Volume 60, Number 6 at Göttingen and in Paris. He later taught at the at the institute now University of Liverpool and the University of Man- that matrices had chester before immigrating in 1910 to the United come into fashion States. Like Wear, he had a Hopkins doctorate in physics, thanks to (1913). the recent invention Throop Polytechnic Institute hired Bateman of matrix mechanics in 1917 as a professor of aeronautical research as a way to formu- and mathematical physics. By then he had al- late quantum the- ready compiled an impressive record as a math- ory. But Bell came to ematician: some seventy scientific papers on understand even be- topics ranging from geometry to earthquake fore setting foot on waves, a British Association report on the history the campus that Mil- and theory of integral equations, a Cambridge likan’s “conversion” University monograph on Maxwell’s equations, was something of a and a textbook in press on differential equations.2 delusion. As he told To make ends meet, he lectured at the Bureau of Aristotle Michal, a Standards and reviewed papers for the Weather job-hunting post- Caltech mathematics faculty and Bureau. At Throop, Bateman was given a free hand doc in mathematics teaching fellows, 1932. Front row, from to develop the theoretical side of aeronautics, to whom he had met at left: Aristotle Michal, Harry Bateman, suggest experimental work in a small wind tunnel Harvard, “Our first Eric T. Bell, and Harry C. Van Buskirk. under construction on campus, and to offer some job will be to con- Rear row: William Birchby, James H. advanced courses. He taught the first courses in vert Millikan to pure Wayland, Carlton C. Worth, Luther E. aerodynamics and related subjects at the institute, mathematics—he Wear, Robert S. Martin, [unknown], and including propeller theories, aerology, and elastic- thinks he is already J. Lawrence Botsford. ity [Gre-Goo 84].3 Bateman’s forte was not the kind converted, but he hasn’t passed the half of his sal- of “applied mathematics” that the Hungarian-born vation” [Bel 26a]. In a letter to Harvard mathemati- engineer Theodore von Kármán, the first director cian George Birkhoff, he noted that the institute’s of Caltech’s Guggenheim Aeronautical Labora- stars were the theoretical physicist Paul Epstein, tory, later brought to bear so successfully on the an expert on quantum theory; the “inexhaustible” problems of the aircraft industry. The theoretical Harry Bateman, who wore many scientific hats; and physicist Paul Ehrenfest, writing to his wife from Richard Chace Tolman, who was “a mathematician Pasadena in 1924, marveled at Bateman’s uncanny gone wrong on chemistry” [Bel 26b]. mathematical ability but was not persuaded that In short, Bell was well aware of the lesser sta- the mathematician grasped the physics underlying tus of pure mathematics at Caltech, but he had his calculations. Ehrenfest describes a discussion hopes of turning Millikan around. “To my way of he had with Bateman about some of his papers thinking,” he told Michal, “Pasadena has the most in mathematical physics as follows: “After a few interesting possibilities of anyplace in the United questions over a half hour I could hardly see what States” [Bel 26b]. In the same letter, Bell spoke of he did or did not accomplish. He is a very pleas- the institute’s need “to build up the library and, by ant fellow … can calculate wonderfully, i.e., he has prevailing on Millikan to take on a few good young ‘calculational intuition’, but jumps around help- men, to make the place as strong in mathematics lessly in the volcano of his calculations, he has no (pure) as it is in applied.” Millikan, according to physical intuition” [Ehr 24]. Bell, had given him a free hand “to work out my Millikan had apparently given Bell some indi- own problems,” with the understanding that Bell cation that he might build up pure mathematics was to “help with the math. physics,” a task Bell thought more suitable for H. P. Robertson, who had 2Cambridge University also published Bateman’s monu- been his student at the . mental tome Partial Differential Equations of Mathemati- Bell had reasons for optimism. Pure mathemat- cal Physics in 1932, which is still available as a print-on- ics had blossomed into an active and expanding demand paperback. discipline in the United States in the decade fol- 3In addition to various topics in analysis, Bateman did lowing the end of World War I. In 1926 his own research in “classical” areas of physics (electrodynamics, publishing record, which had begun at Washing- relativity, and hydrodynamics) and taught vector analysis, ton in 1915, ran to sixty-eight titles, a substantial potential theory, transcendental functions, hydrodynam- number for a ten-year period (and just a fraction ics, and integral equations—the sorts of things that every of his career total of more than three hundred, young classical theorist of his generation was expected to know. Between 1918 and 1928 most of his publications not counting his fictional output as John Taine). It dealt with electromagnetic theory, which later prompted was a number probably matched only by Leonard MIT mathematical physicist and polymath Edwin Bidwell Dickson, who specialized in algebra and number Wilson to call him “probably the most powerful and con- theory at Chicago. Nevertheless, before hiring sistent mathematician now working in this field” [Wil 27]. Bell, Millikan had polled Dickson, Birkhoff, and

June/July 2013 Notices of the AMS 687 Millikan wanted to hire Bell, added Dickson, “You could hardly…get a better man,” and he cautioned that Chicago also had its eye on Bell. Nor would the Seattle mathematician likely leave the Pacific Northwest without an annual salary of $5,000 or better, a figure the famously penny-pinching Mil- likan stored away for future reference. As Dickson noted, Bell did not lack for academic suitors. Flush with sharing the Bôcher Prize, he spent the summer of 1925 lecturing on his own material at the University of Chicago and the fall semester at Harvard. He also had offers of profes- sorships from the University of Michigan, Bryn Mawr, and in his pocket, while Chicago began to woo him. Millikan, in discussing A gathering of H. P. Robertson’s friends on the the need for expansion of Caltech’s mathematics steps of the Athenaeum, 1936. Standing, from faculty before his Executive Council in early 1926, left: Virginia Thomas, Tracy Thomas, Howard concluded that everything short of kidnapping Percy Robertson, Ethel Bateman, Pipo von Kármán, should be done to acquire Bell. Bell, in turn, used Angus Taylor, Patsy Taylor, Toby Bell, Eric Temple Columbia’s tantalizing offer of $7,500 to secure Bell, Mary Bowen, and Aristotle Michal. Seated, a quick response from Millikan. Within a month, from left: Hazel Mewborn and Luddye Michal . he and Millikan had reached an agreement for an annual salary of $6,000. Princeton’s Oswald Veblen—research-minded mathematicians who had put the University of Mathematics on the West Coast Chicago, Harvard, and Princeton on the map as outstanding centers in mathematics. More impor- Bell accepted Millikan’s invitation to come to tant from Millikan’s point of view, all three were Caltech because the institute had already acquired members in good standing of the National Acad- a certain cachet in scientific circles and, perhaps emy of Sciences. As Millikan wrote to Veblen in late just as important, because it was situated on the December 1924, “From the standpoint of physics West Coast. A graduate of in and mathematical physics we are fairly competent mathematics in 1904, Bell had earned a master’s here at the Institute to form judgments in which degree in mathematics from the University of we have some confidence, but from the standpoint Washington in 1908. He took his Ph.D. (1912) at of mathematics I feel keenly my own incompe- Columbia University but signed up to teach in the tence” [Rei 93]. From the start, Millikan had Pacific Northwest even before he had turned in his insisted on inviting only scientists of National dissertation. He liked to say that the West Coast, Academy caliber to join him in Pasadena, which while underdeveloped, had the potential to equal helps explain why the recently anointed Nobel anything the East Coast establishment had to offer. Prize winner asked his fellow academicians what Bell chose Caltech over other institutions in part to Bell’s chances were of election to the National prove his point. He railed against the stuffy tradi- Academy. tions at schools such as Harvard; indeed, he would Veblen volunteered only that Dickson thought later warn at least one younger colleague that “the highly of Bell’s work. Birkhoff, winner of the first Eastern places” were “not the whole cheese,” add- Bôcher Prize, gave Bell high marks for his specialty ing that even Washington was “better than some (“[H]e is great in his field, theory of numbers”) and potty Eastern college” [Bel 26c]. In the West, one his publishing record (“very prolific”) but ques- could at least breathe fresh air. tioned whether Bell’s long list of papers told the Bell’s championing of the Far West mirrors whole story (“[W]hen it comes to papers outside the regional factionalism that accompanied the his specialty, his work is not always of high order”) growth of mathematics in America. His instincts [Bir 25]. Dickson, however, lavished praise on Bell, proved sound: during the 1920s and 1930s math- describing him as “an A1 mathematician of very ematics on the West Coast began coming into its exceptional ability in research of high order on own. In 1924 E. R. Hedrick, the first president of fundamental subjects…I have long been strongly the Mathematical Association of America (an or- impressed by his unusual originality and his suc- ganization itself born of regional factionalism), left cess in research of fundamental character” [Dic the University of Missouri to become chairman 25]. He predicted that Bell would be elected to of the mathematics department at UCLA, then the academy before Bateman was, which proved a brand new school in Southern California. An correct: Bell would be elected in 1927; Bateman’s able administrator, Hedrick worked considerable election followed in 1930, two years after he be- improvements in mathematics during his tenure. came a fellow of the Royal Society of London. If The thirties saw the arrival at UCLA of refugee

688 Notices of the AMS Volume 60, Number 6 Max Zorn, followed by Angus Ellis Taylor and Above all, there was Bell’s tumultuous love/hate Tracy Thomas, who was Veblen’s best differen- relationship with H. P. (Bob) Robertson. Berkeley tial geometry student at Princeton. The rise of mathematician Abraham Haskel Taub, a longtime mathematics at UC Berkeley is an even greater friend and colleague of Robertson’s, analyzed it success story. In 1934 Griffith C. Evans, who had this way: “Bell and Bob were both strong people. built up mathematics at Rice University in Texas, Bell taught Bob, learned from him, fought with him took over the chair of Berkeley’s department. With in fun and sometimes not in fun, and both cared Evans at the helm, it became a department rivaling deeply for each other” [Tau 62]. those of Harvard, Chicago, and Princeton. Evans added a number of mathematicians of the first “A Kid Named Robertson” rank, including the Polish-American mathematical Our story begins back in 1922, when Bell, who was statistician Jerzy Neyman. Similarly, Stanford’s offering a course in mechanics at the University mathematics department took off in the 1930s of Washington, encountered “a kid named H. P. with the arrival of Hungarian refugees Gábor Szegö Robertson” in his class who breezed through the and George Pólya. assignments. In a letter to , a for- Unlike most American institutions of higher mer student who later became a distinguished stat- learning, Caltech lacked traditional departments istician, Bell marveled: “He was just 19 last month, and department chairs. Bell became a member and he goes through the most difficult problems of the Physics, Mathematics, and Electrical Engi- and theory like a shot. Even complicated set-ups neering Division, which reported to Millikan. Bell in problems by Lagrange’s equations don’t bother would be responsible for the graduate work in him in the least…Robertson is a prize” [Bel 22]. mathematics, and the mathematicians looked to Bell took Robertson under his wing—much to him to speak up for the field and take the lead the consternation of Washington’s conservative in dealing with Millikan. Bell’s plans to build up mathematics in Pasadena included both Robertson mathematicians, who believed that research did (who would shortly cap off a two-year fellowship not go hand in hand with good teaching—and in Germany with a fellowship year at Princeton) got him interested in the theory of relativity. Bell and Michal, a specialist in Fredholm and Volt- taught a relativity course in the university’s math- erra integral transformations and their Fréchet- ematics department, and Robertson took it in his Gâteaux-Volterra generalized differentials. “I shall senior year. Bell also persuaded Robertson to stay first try for Robertson (Millikan wants him), and another year at Washington to continue studying then for you,” he told Michal. “You and he will not mathematics, electricity, and relativity, using, for conflict; he is primarily an applied mathemati- the last, Hermann Weyl’s Raum-Zeit-Materie as his cian” [Bel 26b]. Bell planned to “ditch” (his word) on textbook. “If I were 15 years younger,” Bell con- Robertson the task of teaching the applied side of fided to Hotelling in 1922, “I would go into relativ- mathematics. “I have so many things I really want ity; as it is, I hate to scrap the detailed knowledge to do that I can’t take time to be expert in math. of the theory of numbers which has taken so long physics,” he wrote to Robertson, just before leav- to acquire” [Bel 22]. ing Seattle. “For instance, I have recently opened Robertson, however, followed the path of rela- up a whole new field in ‘General Arith.’ where there tivity and never looked back. With a bachelor’s are hundreds of things to be done, and where the degree in 1922 and a master’s in 1923 from Wash- job can be best executed by one man working with ington under his belt, he enrolled at Caltech for ad- a lot of able students” [Bel 26c]. Michal would sign vanced graduate work in the fall of 1923 on Bell’s on in 1929, but it took almost two more decades recommendation. There were at the time at least to snare Robertson. three scientists at the institute who were interested The story of mathematics at Caltech in the in relativity: Tolman, especially as it applied to interwar years is marked by a tangle of person- cosmology; Bateman, in the more technical math- alities and rivalries among Bell’s small group of ematical problems; and Epstein, Caltech’s sole mathematicians. Relations between Michal and theoretical physicist. At one of Caltech’s weekly Bell would eventually degenerate into name-calling physics research conferences later that year, Rob- 4 and shouting. Michal also seemed to be getting ertson gave a talk on relativity in which he credited the most graduate students, which upset Bell. Bell with sparking his interest in the subject. Mil- likan and Epstein, who were in the audience, con- 4One of Michal’s main goals in the thirties was to general- gratulated Robertson afterward on a fine talk; Rob- ize the analysis and geometry on finite-dimensional mani- ertson dodged the compliment by heaping praise folds to abstract infinite-dimensional manifolds, although on Bell. Millikan filed the name away. “Thanks for he does not seem to have proved any new deep theorems. Other mathematicians at the time, including Bell, viewed it tooting my horn,” Bell later wrote to Robertson. “I as abstraction for its own sake, with no sense of problem. know what they want and they’ve got him—Bate- Subsequently, however, these topics became important man” [Bel 24]. Still, Bell appeared ready to pack and active fields in pure and applied mathematics. his bags and head south. As he told Robertson

June/July 2013 Notices of the AMS 689 in closing, “I’d sell my and publish often. By this time, the chemist Linus left foot to get into a Pauling had begun to create a stir at the institute, job in California.” and Bell warned Robertson that there was “one and Before Bell managed only one way to beat him. You know what it is as to plant both feet in well as I do” [Bel 27a]. the Southland, Rob- There is no evidence to suggest that Robertson ertson had obtained cared one way or the other about who was top a Caltech Ph.D., with a dog on campus, but judging by this admonition it major in mathematical might have mattered to Bell, who in any case seems physics and a minor to have enjoyed lecturing Robertson on the need in mathematics. He to write up his work and submit it for publica- learned quantum tion. “Millikan and the lot are watching you to see mechanics and physi- whether you can do it…for your own sake, if not cal hydrodynam- God’s, make good, and do it in a hurry” [Bel 28]. ics from visiting In the spring of 1928 Robertson asked Millikan scientists Ehrenfest if he could remain at Princeton for a second year. and Vilhelm Bjerknes; The German mathematician Hermann Weyl was statistical mechanics, set to arrive, and it seemed too good an oppor- electricity and magne- tunity to pass up, especially as Weyl’s book on tism, and advanced dy- relativity theory had played an influential role in namics from Epstein; Robertson’s undergraduate education. The request and various other sub- was granted, and soon Robertson embarked on an jects under Bateman’s English translation of Weyl’s celebrated book on guidance, including group theory and quantum mechanics [Wey 31]. List of contributors to E. T. Bell’s vector analysis, poten- Late that fall, Caltech officials authorized Bell to retirement gift in 1953. More than tial theory, transcen- open negotiations with Aristotle Michal, who had thirty friends and colleagues signed dental functions, and moved on to a teaching position at Ohio State the flyleaf of the 1670 edition of integral equations. University. Michal had emigrated from his native Diophantus’ Arithmetica, including He completed his dis- Smyrna to the United States in time to go through Jake Zeitlin, the legendary Los sertation in 1925 on high school, which he followed with undergradu- Angeles rare book dealer who a topic in relativity, ate work at Clark University and graduate work at brokered the sale. “On dynamical space- Rice, culminating in a Ph.D. under Griffith Evans. times which contain a Michal played hard to get, but in March of 1929— conformal Euclidean four months and a flurry of letters later, mainly 3-space”, under Bateman’s direction and then over how much he would be paid—he sent Bell a crossed the Atlantic for a year’s study in Germany telegram accepting a salary of $4,500 and an as- as a National Research Council Fellow in math- sociate professorship effective September 1930. ematics. The fellowship was renewed for a second In the meantime, Robertson was expected back in year with the promise of a third either at Harvard Pasadena to begin teaching in the fall. Events then or Princeton. Robertson spoke of permanently took an unexpected turn. locating in Pasadena from 1928 on, “but that is as yet unsettled,” he wrote in spring 1927 to a family The Contretemps with Robertson friend [Abe 27]. “I have had suggestions from other On March 20, a day after receiving Michal’s tele- universities looking for an assistant professor [of] gram, Bell wrote an uncharacteristically icy letter mathematics or mathematical physics, and it may to Robertson in Princeton—a letter totally out of be to our advantage to go to one of these in spite keeping with the usually informal, sometimes of our preference for Pasadena.” By this time, as bawdy, tone of the correspondence between the he later recalled in a brief autobiographical sketch, two friends—in which he intimated that Robertson “I found myself very much more interested in might be better off teaching somewhere other than the physical application than in the mathematics at Caltech. “This is rather an official letter, writ- itself” [Rob 51a]. ten at the request of [Millikan’s assistant, Earnest] In June 1927 Millikan wrote to Robertson, who Watson and Dr. Millikan,” Bell began by way of was still in Germany and expecting to continue his explanation. “They want to know whether you are N.R.C. Fellowship at Princeton, and offered him a definitely planning to return here next year… As faculty position, effective September 1928, with you know, the rule here is no advancement in ei- the status of assistant professor of mathematics ther rank or salary unless a man is productive. So if on leave until that time. Robertson seemed de- you see anything that you consider more attractive, lighted by the invitation and quickly sent back his acceptance letter. From then on, Bell began hound- please let us know as soon as possible. We must ing his former student to publish, publish quickly, know definitely by April 1st.” While “I personally hope you will accept, and fight it out on the lines

690 Notices of the AMS Volume 60, Number 6 proposed,” Bell wrote in closing, he emphasized would end up like the again that there was “absolutely no chance here diffident Bateman, for a man who is not productive” [Bel 29a]. On whose priority in cer- April 1 , Robertson sent Millikan a telegram saying tain matters concern- that “under present circumstances” he felt obliged ing special relativity, to request release from his appointment. Marietta Bell believed, went un- Fay, Robertson’s daughter, recalls that “both my appreciated [Bel 24]. parents were deeply hurt by Bell’s letter. They However, Robertson’s never forgave him” [Rei 93]. intransigence suggests A few days later, in a letter to Bell and Millikan, that he simply did not Robertson explained that he had gotten the im- like being told what to pression that “the Institute is not satisfied with do by other mathema- my work during the two years which have elapsed ticians. since the appointment” [Rob 29a]. On April 20, After he turned Bell replied, wondered if they were still on speak- Caltech down, Rob- ing terms, and said that he thought Robertson ertson was offered a had “made the mistake of your life in turning position in Princeton's down what was offered” at Caltech. As a “final mathematics depart- parting piece of advice,” he warned Robertson to ment. He refused that “get busy, and show the world that you are doing offer and requested something,” all the while insisting that he took instead a position “personally the full responsibility” for the events like the one he had that caused Robertson to resign [Bel 29b]. Robert- just spurned from son responded that there were no hard feelings Caltech, which would Title page of Bell’s Arithmetica. The but that Bell’s apparent change of attitude toward have allowed him to 1670 edition includes the claim made him in the space of one year, and what seemed to teach physics as well by Fermat that he had proved what Robertson to be the institute’s view of him as an as mathematics. The became known as Fermat’s Last inadequate “producer”, had left him no alternative university countered Theorem but lacked space in the but to resign, as “the only self-respecting thing to by offering him an as- margin to show his work. Bell’s long- do” [Rob 29b]. sistant professorship time research interest in Diophantine Robertson’s mother visited Bell and his wife, of mathematical phys- equations set the stage for his own Toby, in Pasadena the following year and added ics with a seat in both book about Fermat’s Last Theorem, her own postscript to this bizarre tale. It seemed departments, a situa- The Last Problem, published to her, as she wrote to her son and daughter-in-law, tion he accepted but posthumously in 1961. that the Bells’ admiration for her son was mixed would later vow never with jealousy, stemming from the esteem in which to repeat [Rob 51a]. With the exception of a one- both Richard Tolman and Princeton’s mathemati- year sabbatical leave in 1935–1936 (which he spent cians held him [A M Rob 30]. in Pasadena “at the scene of some of my former Just how “unproductive” had Robertson actually indiscretions” [Rob 35a]), Princeton remained his been? According to his bibliography, he published academic home until his permanent return to eleven papers between 1924 and 1929 on topics Caltech in 1947. During his tenure at Princeton, ranging from differential geometry and quantum Robertson rose to prominence as one of the piv- theory to relativistic cosmology, including an ar- otal figures in the United States in the working ticle in 1929 in Physical Review on the uncertainty out of the general theory of relativity. So when he principle, which remains to this day among his returned to Caltech, he did so on his own terms. most cited papers [Rob 29c]. It was not a great As he later recalled, “I had had enough of two sets number, to be sure, compared with the twenty-nine of department meetings and of the intellectual papers Bell had whipped out in the first six years arrogance of mathematicians, and elected to go of his own career. Bell put a premium on showman- into the Physics Department, although I had been ship, visibility, and a high profile, which is probably offered my choice” [Rob 51a]. why he kept telling Robertson to publish his work Bell never quit hounding his protégé to publish in bits and pieces and to do it quickly rather than more often. He wrote to Oswald Veblen in the win- wait until he had pulled things together. The Amer- ter of 1931, “If you happen to think of it, would ican mathematics community, Bell argued, likely you mind jogging Robertson up to get out some of wouldn’t understand it either way. Bell continually his stuff on mathematical physics? As it is, other admonished Robertson to put himself forward in people are running away with bits of it under his every way possible—to line up offers of positions nose” [Bel 31a]. Even Robertson’s election to the at other institutions, for example, which would National Academy in 1951 provided Bell with am- impress Millikan, who “eats up this sort of thing” munition. “This is something many years overdue, [Bel 28]. He seems to have feared that Robertson mainly your own fault, for straddling the fence

June/July 2013 Notices of the AMS 691 between two sciences and not letting your fellows know on which side your balls hung,” he wrote [Bel 51]. Robertson, no stranger to such repartee, replied in kind, “Thank you for your obscene congratulations…You attribute the delay in me sitting on the fence with one orchid hanging over on either side; I say they were only waiting for them to shrivel up and drop off before I would be ripe for the honor” [Rob 51b].

Bringing Up the Next Generation Bell was living in a boarding house in San Francisco at Morgan Ward, Bell’s first graduate student, the time of the 1906 earthquake and buried these two entered Caltech in 1924 (where he was one of only math books in the backyard. forty-eight graduate students altogether) after earning his bachelor’s degree at UC Berkeley. He became Caltech’s first Ph.D. in mathematics, receiving his degree summa cum laude in 1928 with a dissertation on the foundations of general arithmetic;5 this was followed by his appointment as a research fellow. Like physics and chemistry at Caltech, mathematics tapped its own students from the 1920s on. In 1929 Ward joined the fac- ulty as an assistant professor of mathematics and, aside from a year at Princeton in 1934–1935, remained at Caltech until his death in 1963. His research interests ranged from recurring series, Diophantine equations, abstract arithmetic and lattice theory6 to functional equations and numeri- cal analysis. Like Bell, he had a deep interest in the theory of numbers and a “great contempt for those who proliferate easy empty generalizations of the Several days later, Bell rescued his copy of Théorie des great classic ideas of mathematics,” according to Nombres, Édouard Lucas’s 1891 classic text in number Derrick H. Lehmer, who got to know Ward while theory, from the scorched earth. spending the 1930–1931 academic year at Caltech on an N.R.C. Fellowship in mathematics [Leh 93]. The onset of the Great Depression in 1929 may have dampened Bell’s spirits a bit. In response to a 1931 letter from Veblen hinting that he would welcome an invitation to visit Pasadena while Al- bert Einstein was in residence, Bell wrote: I fear it is out of the question. The financial stringency has hit us hard. The mathematicians never did have any funds available to pay outside lecturers. The one time when we did pay a lecturer, namely Harald Bohr,

5This is how Ward viewed his thesis: “We have now reached a logical stopping place. In Part I, we have sub- jected the concept of a class and a binary operation to a careful scrutiny, and shown that for the purposes of general arithmetic, where all classes are denumerable, we can assume the class ordered without any loss of gen- Beneath his name and the names of two cities, erality, or replaced by the numbers one, two, three, …We Seattle and San Francisco, Bell wrote the following have thus ‘arithmetized’ general arithmetic in Kronecker’s inscription: “This book went through the San Francisco sense.” http://resolver.caltech.edu/caltechETD: Fire April 16, 1906.” Although Bell discarded most of etd-03042005-135853. his correspondence in later years, he kept these two 6Eighteen papers on these topics appeared in the Annals charred books, which are now housed in the Caltech of Mathematics, then (and now) the leading American archives. journal of mathematics.

692 Notices of the AMS Volume 60, Number 6 was provided for by a crumb dropped gaps, particularly in combinatorial topology and from the physicists’ banquet…I was to point-set topology, subjects offered at Princeton, have got a new man this year, but the Texas, Virginia, and Michigan. “There really was money wasn’t forthcoming. In the past not a mathematics department in an administra- they have usually paid railway fare to tive sense,” Taylor remembered. “I don’t think essential meetings; this year that also is there was much planning of curricula. There was cut out, so I shall have to pay my way to very little guidance of graduate students” [Tay 81]. New Orleans. However, this depression Be that as it may, early in 1936 Veblen asked can’t last forever. [Bel 31b] Robertson, then at Caltech on sabbatical, for his opinion of Angus Taylor. Robertson replied: In fact, the Depression dragged on, and no new faculty appointments were made in mathematics I do know Taylor fairly well, and I do until the early 1940s. think he is really very good…But the fly While at Princeton, Ward confided to Robertson in the ointment is this, as you probably that he was not entirely happy with his situation know by this time: Taylor has just yes- at Caltech. In a letter to H. S. Vandiver at the terday accepted an instructorship here University of Texas, Robertson said that Ward at the Institute, and has withdrawn his “would welcome an opportunity for a change; this application for an N[ational] R[esearch] is, of course, most confidential” [Rob 35b], and on F[ellowship]. I am sorry he has done the same day he wrote to J. Robert Oppenheimer this just now, for I think he would profit at Berkeley: “The purpose of the present note is much more at this point by contact with to find out whether and to what extent you have other schools of analysis—particularly the ear of Evans or other influential people in the the Johnny [von Neumann]-operator- Mathematics Department. Because if you have, school! But he seems quite sure that they might be interested to know…that Morgan he is doing the best thing by staying on Ward has always had a soft spot in his heart for here—and teaching brats on a twelve- Berkeley…I have gotten to know Ward much bet- hour basis. [Rob 36] ter this [year] and have reason to believe he would by no means resent an opportunity for a change” Veblen refused to accept Taylor’s decision and [Rob 35c]. Ward in fact received a promotion and managed to persuade him to accept the fellowship a raise at Caltech soon after his return, prompt- from the National Research Council and come to ing Robertson to remark, “I suppose that makes Princeton as a postdoc, where he worked with Salo- things look different there, doesn’t it? I must say mon Bochner. Taylor’s fellowship was renewed for it increases my respect for your Chief [Millikan]” 1938–1939, and then, unexpectedly, he received an [Rob 35d]. offer of appointment at UCLA for that academic Angus Taylor, Aristotle Michal’s first student year. Keen to return to California, he tried to find and one of his best, was another promising star. out whether Caltech planned to ask him back in Taylor and the other graduate students in math- 1939. “So there I was, in April or May of 1938, a bit ematics at Caltech in the 1930s were pushed to the up in the air about my future,” he later recalled. research frontier as quickly as possible. Because he “Then unexpectedly, I received word from E. T. already knew the theory of functions of a complex Bell urging me to accept the UCLA offer. That told variable, Taylor skipped Bateman’s course, which me what I hadn’t known for sure about Caltech’s leaned heavily on Whittaker & Watson’s A Course interest in me. After discussing the matter with of Modern Analysis. (He later described Bateman people at Princeton, and against their advice, I as “a very gentle and nice man…in a little rut all resigned the second year of the fellowship and by himself” [Tay 81].) Instead, under Michal’s took the job at UCLA. I’ve never regretted it” [Tay guidance, Taylor became proficient in Lebesgue 81]. Morgan Ward, who was hoping Taylor would measure and integration, the theory of abstract return to Pasadena, blamed Bell. Taylor, he wrote spaces and functional analysis, and Riemannian Robertson, would have done “his fair share of the and non-Riemannian geometry. “Michal did some work, something which A. D. Michal and E. T. Bell lecturing, but made the students do quite a bit of avoid successfully.” Bell, he added, “seems down it themselves,” Taylor recalls [Tay 81]. He studied [on] him, but I cannot find out exactly why—he abstract algebra under Bell, who, he says, “didn’t seems to have made the wrong remark to him lecture. He had all the students tell the class what some time ago, and Bell has treasured it” [War 38]. was in the books, so the students did all the lectur- In 1939 Ward received an offer from Johns Hop- ing; Bell commented and criticized.” In a memoir kins and, perhaps with Taylor’s fate in mind, asked for the Mathematical Association of America, Millikan “what my immediate prospects might Taylor characterized Bell as “a stimulating person, be” [War 39]. He also took the opportunity to an- given to expressing strong opinions [but] I don’t nounce that his student Robert Dilworth, who had think he spent much time preparing what he was done his graduate work on a new algebraic theory going to say in class” [Tay 84]. The curriculum had of lattices, had just been awarded a prestigious

June/July 2013 Notices of the AMS 693 Sterling Fellowship at Yale in case Millikan was to associate professor. In the last year of the war, thinking of hiring an assistant professor in the he served as an analyst with the 8th Air Force in future. While at Yale, Dilworth continued this Great Britain. In 1951 Dilworth would become a research, which is well documented in his book full professor (and a major figure in lattice theory), (with Peter Crawley) Algebraic Theory of Lattices, and while continuing his research in algebra he published in 1973.7 Meanwhile, Bell told Millikan also worked in other fields, including probability that if Hopkins wanted Ward, Caltech “should do theory and statistics. the utmost we can to make it worth his while to Bell was disappointed by the turn of events. stay” in Pasadena [Bel 39]. Ward was promoted to If Millikan wanted a mathematician more tuned professor the following year. to applied mathematics, then Robertson was the The first substantial revamping of Caltech’s man. Others on the campus, including Ward and mathematics faculty in more than a decade came von Kármán, who had turned to Robertson for help during the war. In 1942 a new assistant professor on a turbulence problem during his Pasadena sab- in mathematics was needed to replace the retir- batical, also plugged for Robertson. But Millikan, as ing Van Buskirk. Ward’s former student Robert Bell wrote to Robertson, was unwilling to “spend Dilworth and Michal’s former student Angus Tay- any real money,” and Bell “frankly” saw “no chance lor were the candidates. The screening took place until, if ever, Pa [Millikan] retires and a more en- “in house”. Millikan turned to several physicists lightened financial policy is adopted” [Bell 42]. at the institute for advice. They assured him that In 1945 Robert Millikan retired as Caltech’s Dilworth’s research was potentially of greater head, and the following year physicist Lee A. use to physicists, although the spectroscopist DuBridge became the institute’s president. Sub- William Houston questioned the criteria Millikan stantial salary increases went into effect shortly appeared to be using in deciding between the two after DuBridge took office. Bell’s original band of mathematicians (“I was rather surprised to have mathematicians also passed into history: Harry it mentioned [by you] that he [Dilworth] seemed Bateman, the AMS’s Gibbs lecturer in 1943, died to be more inclined toward the applications of three years later en route to New York to receive mathematics than many other mathematicians” an award from the Institute of Aeronautical Sci- [Hou 42]). In a letter to one of us [JRG] many years ences.8 Aristotle Michal died of heart disease in later, Taylor wrote that he knew “that Bell and 1953. That year, E. T. Bell retired from Caltech, Michal were at odds over getting me to go back twenty-seven years after coming to Pasadena. The to Caltech from Princeton…Millikan made it quite transformation of the mathematics faculty, which clear, it seemed to me, that he was only inter- Millikan had largely regarded as a service depart- ested in what Dilworth or I might be useful for in ment, into a first-class research group began under teaching the physics graduate and undergraduate the inspired leadership of H. Frederic Bohnenblust, students, and not at all in our scholarly potential who arrived as a full professor of mathematics in in research” [Tay 91]. But he had no idea that he 1946. Applied mathematics as a distinct discipline had been a candidate for a tenure-track position at came to Caltech in the mid-1960s. To avoid friction the school along with Dilworth several years later. with the pure mathematicians, it was organized “I would not have left UCLA for Caltech then [in as a research school within Caltech’s engineering 1942], in any event” [Tay 81], he later reported to division. John Greenberg, a historian of mathematics from Pasadena. Bell’s Legacy Millikan, in his letter to Michal explaining his Whereas Bell was a prolific writer of novels, poetry, decision to appoint Dilworth instead of Michal’s and science fiction under his pseudonym of John former student, assured him that “there has never Taine, he achieved true rock star recognition, or been any question about the excellence of Taylor’s something approaching it, among mathemati- teaching work,” while advancing an argument that cians and nonmathematicians alike for Men of Michal was in no position to challenge: “Our trust- Mathematics [Bel 37], published in 1937. Still ees thought it undesirable to try to run the risk of popular today, discusses the creating unpleasant feeling between UCLA and the personalities and mathematics of a host of great California Institute by trying to pull Taylor away mathematicians, including Niels Henrik Abel, from them…and so decided to try to get Dilworth , David Hilbert, and Bernhard here for next fall. There is of course never anything permanent about a first appointment” [Mil 42]. 8The Bateman Manuscript Project, based in part on notes Dilworth returned to Caltech as an assistant pro- he left, started up several years later under the direction fessor in 1943 and two years later was promoted of Arthur Erdélyi and aided by three research associates, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. 7“He is also well known for ‘Dilworth’s Theorem’ in com- Tricomi, culminating in three volumes of Higher Tran- binatorics, which arose in work on posets,” notes Alfred scendental Functions, supplemented by two volumes of Hales, a student of Dilworth’s [Hal 2012]. Tables of Integral Transforms.

694 Notices of the AMS Volume 60, Number 6 Riemann, concentrating on those born from the historian. Ann Hibner Koblitz, who has written eighteenth century on. Bell followed his account of about Sonia Kovalevskaya, thinks “[Bell] might well men (and the occasional woman) in mathematics become known to future generations of mathema- with The Development of Mathematics, issued by ticians and historians as the legend maker of the McGraw-Hill in 1940, with a revised edition five . It is to him that mathema- years later [Bel 45]. This was a sweeping account ticians are largely indebted for distorted impres- of the history of mathematics, starting with its sions of their predecessors” [Joc-Efr]. Koblitz’s tart beginnings in ancient Babylonia and Egypt and remarks are echoed by Roger Cooke, who deems charting its progress to 1945. Bell’s treatment of Kovalevskaya an “infuriatingly Readers of The Development of Mathematics patronizing, innuendo-laden mistreatment” [Joc- will require some mathematical sophistication to Efr]. Nevertheless, E. T. Bell was undoubtedly one fully appreciate it. In his review in Isis, I. Bernard of America’s leading mathematicians in number Cohen, the dean of American historians of sci- theory and in the first half of the ence at that time, wrote, “Within one restriction, twentieth century. (For another nontechnical dis- the present book is excellent; that restriction cussion of Bell’s mathematical work, see Lincoln consists in the fact that it really begins on p. 99, K. Durst’s interesting Appendix to [Rei 01].) He with ch. 7: ‘The Beginning of Modern Mathematics, was awarded the Bôcher Prize for his 1921 papers 1637–1687.’ In the first 98 pages, there are many in the Transactions of the American Mathematical statements that one will take exception to” [Coh Society on arithmetical paraphrases, a notion due 41]. Cohen’s objections included Bell’s breezy to Bell himself [Bel 21]. Its applica- description of al-Khwarizmi’s algebraic methods tion is to derive a huge number of identities for (“a psychiatrist might say it was the death instinct various arithmetic functions. An example of the having its way”) and “the grand manner” in which principle of arithmetical paraphrase is the follow- Bell demolished Plato’s detractors (“Of all changes ing: Suppose there are integers n , a , b …n , a , that mathematical thought has suffered in the 1 1 1 k k b such that n sin(a x + b y) +…+ n sin(a x + b y) past 2,300 years, the profoundest is the twentieth- k 1 1 1 k k k = 0 as function of integers x, y. Then, for any odd century conviction, apparently final, that Plato’s function f of two integral variables, we have: n f (a , conception of mathematics was and is fantastic 1 1 b ) +…+ n f (a , b ) = 0. If f is an odd arithmetical nonsense of no possible value to anyone”). Like 1 k k k Cohen, most reviewers gave the book high praise function, then this would be an identity for this while also calling attention to Bell’s unorthodox arithmetical function. style. D. R. Curtiss, writing in National Mathematics In 1926 Bell was selected to give the distin- Magazine, noted, “After a few drier pages there is guished AMS Colloquium Lectures that led to his always a pungent remark on human frailties, a bit 1927 Colloquium publication Algebraic Arithmetic of grim humor, sometimes an aside of a half page [Bel 27b]. This volume expanded on his notion of or more on what dictators are doing to mathemat- arithmetical paraphrases and Euler algebra in a ics, and what philosophers or theologians would very general and abstract way. Leonard Dickson, do if they could” [Cur 41]. Rudolph Langer, who in his review in the Bulletin of the American Math- reviewed the book for Science, probably spoke for ematical Society, writes: many when he declared, “The presentation of the This book of marked originality is of whole is admirable. It is flowing and graceful and vital interest to advanced students in often characterized by a genuine and delightful various branches of mathematics, in- humour” [Lan 41]. cluding the theory of numbers, abstract Bell also had his detractors, some of whom algebra, elliptic and theta functions, accused him of being flippant. Reviewers of Men Bernoullian numbers and functions of Mathematics complained of his inclination to and the foundation of mathematics… sacrifice historical accuracy for a more colorful A leading feature of the book seems story. The most blatant example is his exaggerated to the reviewer to be its success in a account of the life of Évariste Galois, who died at twenty following a celebrated duel. Did Galois re- systematic attempt to find a unified ally create group theory in its entirety during the theory for each of various important last night before the duel? Not according to Tony problems in the theory of numbers,

Rothman, who in Genius and Biographers: The 9 Fictionalization of Evariste Galois” [Rot 82] accuses Bell gave Tom M. Apostol, the number theorist who took Bell’s place at the institute in 1950, his own personal copy Bell of inventing history. Based on his research on of Algebraic Arithmetic, signed and dated 12 December Galois, Rothman concludes that Bell “consciously 1927. Apostol recalls discussing the book with his student or unconsciously saw his opportunity to create a Basil Gordon “and both of us agreed that it was impen- legend … Unfortunately, if this was Bell’s intent, etrable” [Apo 2012]. Gian-Carlo Rota told he succeeded.” Today’s historians of mathematics that he doubted “if anyone has read it all the way through” are even less generous in their praise of Bell as a [Rei 93]. Apostol is of the same opinion.

June/July 2013 Notices of the AMS 695 including its interrelations with algebra recent times. During the Second World War, artifi- and analysis. [Dic 30] cial intelligence came of age, alongside the inven- tion of the digital computer. The study of networks However, it seems that most mathematicians for simulating the central nervous system helped who were interested in understanding what Alge- to motivate graph theory and, in general, helped braic Arithmetic was all about had a difficult time to stimulate a revival of combinatorial theory after figuring it out, especially with regard to Bell’s dis- the war. Bell’s work on exponential and iterated cussion of the “umbral calculus”. Dickson may well exponential numbers, exponential polynomials, have been one of the very few people (or perhaps and the symbolic or “umbral” calculus was much the only one) who understood the mathematical cited in the literature on combinatorial analysis content of the book.9 of the fifties and sixties. Chief among the math- Some of Bell’s major contributions were: ematicians who referred to his work were Leonard Bell series [Bel 15]. These series are important arithmetic series in number theory and were so Carlitz, a one-time NRC Fellow at Caltech (in 1930), named by T. M. Apostol in [Apo 76]. and John Riordan of the Bell Laboratories, a one- Paraphrases [Bel 21], for which he won the time collaborator of Claude Shannon, the founder Bôcher Prize. of artificial intelligence in the U.S. Bell contrib- Euler Algebra [Bel 23]. The Euler algebra, roughly uted, among other things, to the development of speaking, is the formal algebra generated by the formal power series and generating functions as Cauchy algebra C of power series in one variable tools for obtaining recurrence relations, which are and the Dirichlet algebra D of Dirichlet series. It important for treating combinatorial problems in has been an important tool in the theory of gen- a unified way. The solution of recurrence relations erating functions. derived from some kind of counting problem is Algebraic Arithmetic [Bel 27b]. In this book, Bell perhaps the most common, basic, and elementary puts his theory of paraphrases and Euler algebra application of generating functions in combinato- into an abstract setting. Included is a heuristic and rial theory [Tan 75]. impenetrable discussion of the classical umbral Bell numbers and Bell polynomials often figure calculus of Blissard. The remainder of the book as an important feature of combinatorial and is also very difficult reading. The umbral calculus probabilistic problems. (For an example of the was not sorted out and made rigorous until the influence of Bell’s use of generating functions in 1970s by Gian-Carlo Rota, his students, and others. a combinatorial setting, see [Rio 57], [Rio 58].) It See, for example, [R, K, & O 73]. struck two mathematicians as odd that it took Bell polynomials [Bel 34]. These were called ex- so long for the followups to what they deemed ponential polynomials by Bell. Although we do not Bell’s “classic paper” on exponential polynomi- know when they were first called Bell polynomials, als to appear [G & H 62]. Rota acknowledged the they were so called by John Riordan in his classic widespread role that Bell’s exponential numbers 1958 book on combinatorics [Rio 58]. played in a great many problems of enumeration Bell numbers [Bel 38]. These were called iterated and of probability [Rot 64]. Thus so did the Bell exponential integers by Bell and are important polynomials manifest themselves as an important in the theory of partitions (among other areas of feature of many combinatorial and statistical combinatorics) [Rot 64]. They were so named in problems [Rio 58]. 1948 by Becker and Riordan in [B & R 48]. Rota later tried to do what Bell failed to do in 1940, namely, put the “umbral calculus” or “sym- Appendix bolic calculus” on rigorous foundations. Rota’s Bell’s Contributions to Combinatorics purpose was to provide even greater unity to the by John Greenberg10 results of combinatorial analysis. In saying that writing the paper was like “assembling a dinosaur Authors’ note: Bell’s most significant mathemati- from a few charred bones in the desert,” the au- cal contributions were in the field we now call thor meant to pay homage to his predecessors, combinatorics. The following description of Bell’s such as Carlitz and Riordan, not to belittle them contributions to combinatorics is the work of the [R, K, & O 73]. While Bell himself did not provide late John Greenberg, who received his Ph.D. in ironclad foundations for his uses of formal power history in 1979 from the University of Wisconsin series and generating functions, Bell, unlike some under the direction of Daniel Siegel. This essay was of the authors who later cited him, also realized to be part of his unfinished history of the Caltech the difficulty, and his concerns gave rise to more mathematics department. than one “Bell Problem” [Gou 74]. E. T. Bell would have been surprised to see where Though Bell took note himself of “the interest his real influence upon mathematics lay, at least in in combinatorial analysis and elsewhere” of some of the objects of his attention [Bel 38], it’s hard to 10John Greenberg was a postdoc for one of us (JRG) in the imagine that he wouldn’t have been surprised at Caltech archives for three years (1981–1984). the fact that this was the area where his greatest

696 Notices of the AMS Volume 60, Number 6 influence lay. As a historian of mathematics, Bell source of information about Bell and mathemat- ridiculed the late eighteenth-century German ics at Caltech between the wars can be found in combinatorial school, labeling it a “ridiculous the H. P. Robertson papers, housed in the Caltech interlude” [Man 75]. Bell ridiculed the German archives, which Reid consulted. However, materi- formal manipulation of binomial and multinomial als were given in installments, beginning in 1971 coefficients and formal expansion of powers of (and processing was begun by then archivist Carol infinite series. It seems like a strange edict for Finerman) and ending in 1998. Following the final someone as well versed in the combinatorial art as donation in 1998, the entire collection was inte- Bell himself! “Bell [was an] inveterate combinato- grated, rearranged, and described by Dr. Erwin. rialist and number theorist, [a] person well versed We have made extensive use of these supplemental in operations with series” [Gou 74]. There is some materials. irony that the revival of combinatorial analysis was The following abbreviations are used: AM, Aris- brought about in large part by the advent of the totle Michal papers; RAM, Robert Andrews Millikan computer and the wholesale creation of branches papers; HPR, Howard Percy Robertson papers, all of mathematics to minister to computer science, in the Institute Archives, California Institute of for if mathematics was a science that stood on its Technology; GB, George Birkhoff papers, Harvard own, as Bell seemed to think, his own work with University Archives; OV, Oswald Veblen papers, the greatest impact was that which was pressed . into the service of other sciences. If Bell abhorred the applied mathematician, he nevertheless helped [Abe 27] H. P. Robertson, letter to W. H. Abel, 5 March 1927 (HPR, box 1.1). to open up new areas of applied mathematics in [A M Rob 30] Anna McLeod Robertson, letter to H. P. spite of himself. Millikan, his old nemesis, would Robertson, 22 October 1930 (HPR, box 27.1). have been tickled to death! For all the clairvoy- [Apo 76] Tom M. Apostol, Introduction to Analytic Num- ance, wit, and sagacity that ushered forth from ber Theory (New York: Springer Science & Business the pen of Bell the science fiction writer, Bell the Media, 1976). mathematician never could have foreseen or an- [Apo 2012] ______, email of 19 August 2012 to J. R. ticipated the uses to which his work would be put Goodstein and D. G. Babbitt. after his death. [B & R 48] H. W. Becker and John Riordan, The arith- metic of Bell and Sterling numbers, Amer. J. Math. 70: Acknowledgments 385–394 (1948). [Bel 15] E. T. Bell, An arithmetical theory of certain We thank Alfred W. Hales and Tom M. Apostol numerical functions, University of Washington Publi- for comments on the manuscript; Sara Lippin- cations, Mathematical and Physical Sciences, 1: 1–44 cott for editorial support; Shelley Erwin, Loma (1915). Karklins, and Elisa Piccio in the California Institute [Bel 21] ______, Arithmetical paraphrases. I, II, Trans. of Technology Archives in Pasadena for assistance Amer. Math. Soc. 22: 1–30, 198-219 (1921). in locating documents, letters, and photographs; [Bel 22] ______, letter to H. Hotelling, 3 April 1922 Jim Staub, in graphic resources, for photographing (HPR, box 1.12). [Bel 23] ______, Euler algebra, Trans. Amer. Math. Soc., Bell’s books; Dana Roth for bibliographic research; 25: 135–154 (1923). and the archivists at the manuscript and special [Bel 24] ______, letter of 8 January 1924 to H. P. Rob- collections division at the University of Washing- ertson (HPR, box 1.12). ton, the Library of Congress (the Oswald Veblen [Bel 24] ______, letter of 8 January 1924 to H. P. Rob- papers), and the Harvard University Archives (the ertson (HPR, box 1.12). George Birkhoff papers) for providing manuscript [Bel 26a] ______, letter of 13 June 1926 to Aristotle material from their collections. Michal (AM, box 1.15). [Bel 26b] ______, letter to A. Michal, 21 March 1926 References (AM, box 1.15). [Bel 26c] ______, letter to A. Michal, 13 June 1926 (AM, Although he wrote at length about the lives of box 1.15). other mathematicians, E. T. Bell glossed over [Bel 26d] ______, letter of 15 May 1926 to A. Michal many details of his own biography. Bell also sys- (AM, box 1.15). tematically destroyed much of his professional [Bel 26e] ______, letter of 15 June 1926 to H. P. Robert- correspondence before he retired. H. P. Robertson, son (HPR, box 1.12). his friend and colleague, declined to write Bell’s [Bel 27a] ______, letter of 20 October 1927 to H. P. biographical memoir for the National Academy. Robertson (HPR, box 1.12). Constance Reid’s The Search for E. T. Bell, Also [Bel 27b] ______, Algebraic Arithmetic, AMS Colloquium Known as John Taine, published in 1993, provided Publication, 7, 1927. [Bel 28] ______, letter of 22 February 1928 to H. P. the first authoritative portrait of the Scottish-born Robertson (HPR, box 1.12). mathematician who concealed much about his [Bel 29a] ______, letter of 20 March 1929 to H. P. Rob- family and his childhood in San Jose, CA. Reid’s ertson (HPR, box 1.12). book, which reads like a detective story, is the [Bel 29b] ______, letter of 20 April 1929 to H. P. Rob- starting point for anyone studying Bell. A rich ertson (HPR, box 1.12).

June/July 2013 Notices of the AMS 697 [Bel 31a] ______, letter of 27 January 1931 to O. Veblen [Man 75] Kenneth R. Manning, The emergence of the (OV). Weierstrassian approach to complex analysis, Arch. [Bel 31b] ______, letter of 5 November 1931 to Hist. Exact Sci. 14: 297-383 (1975). O. Veblen (OV). [Mil 42] Robert A. Millikan, letter of 6 July 1942 to [Bel 34] ______, Exponential polynomials, Ann. Math. A. D. Michal (RAM, box 25.1). 35(2): 258–277 (1934). [R, K, & O 73] G-C. Rota, D. Kahaner, and A. Odlyzko, [Bel 37] ______, Men of Mathematics (New York: Simon On the foundations of combinatorial theory. VIII. & Schuster, 1937). Finite operator calculus, J. Math. Anal. Appl. 42: 684- [Bel 38] ______, The iterated exponential integers, Ann. 760 (1973). [Rei 93] Letter of R. A. Millikan to O. Veblen, December Math. 39(2): 539–557 (1938). 1924, quoted in Constance Reid, The Search for [Bel 39] ______, letter of 18 April 1939 to R. A. Millikan E. T. Bell, Also Known as John Taine (Washington, D.C. (RAM, box 24.22). Mathematical Association of America, 1993). [Bel 42] ______, letter of 29 August 1942 to H. P. Rob- [Rei 01] Constance Reid, The alternative life of E. T. Bell, ertson (HPR, box 1.15). Amer. Math. Monthly 108 (5): 393-402 (2001). [Bel 45] ______,The Development of Mathematics (New [Rio 57] John Riordan, The numbers of labeled colored York: McGraw-Hill, 2d edition, 1945). and chromatic Trees, Acta Mathematica 97: 211-25 [Bel 51] ______, letter of 27 April 1951 to H. P. Robert- (1957). son (HPR, box 1.15). [Rio 58] ______, Introduction to Combinatorial Analysis [Bir 25] ______, letter of 5 January 1925 to R. A. Millikan (New York: Wiley, 1958). (RAM, box 25.3). [Rob 29a] H. P. Robertson, letter of 6 April 1929 to [Bir 26] ______, letter to G. Birkhoff, 25 October 1926 R. A. Millikan (HPR, box 4.19). (GB). [Rob 29b] ______, letter of 11 May 1929 to E. T. Bell [CIT 28] Bull. Calif. Inst. Technol., 37 (121): 91–92 (1928). (HPR, box 1.12). [Coh 41] I. Bernard Cohen, Review of The Development [Rob 29c] ______, The uncertainty principle, Phys. Rev. of Mathematics, Isis 33(2): 291–293 (1941). 34: 163-4 (1929). [Cur 41] D. R. Curtiss, Review of The Development of [Rob 35a] letter to Earnest Watson, 22 April 1935 (HPR, Mathematics, Nat. Math. Magazine 15(8): 435–438 box 6.14). (1941). [Rob 35b] ______, letter of 6 February 1935 to H. S. [Dic 25] L. E. Dickson, letter of 1 January 1925 to R. A. Vandiver (HPR, box 6.13). Millikan (RAM, box 25.3). [Rob 35c] ______, letter of 6 February 1935 to J. R. Op- [Dic 30] L. E. Dickson, Review of Algebraic Arithmetic, penheimer (HPR, box 6.13). [Rob 35d] ______, letter ca. July 1935 to M. Ward (HPR, Bull. Amer. Math. Soc. 36: 455–459 (1930). box 6.13). [Dur 01] Appendix. Some of E. T. Bell’s mathematics, in [Rob 36] ______, letter of 28 March 1936 to O. Veblen [Rei 01], 400–402. (HPR, box 1.13). [Ehr 24] Paul Ehrenfest, letter of 11 January 1924 to [Rob 51a] ______, Further autobiographical facts, re- T. Ehrenfest, quoted in C. Truesdell, An Idiot’s Fugi- quested by the National Academy of Sciences, undated tive Essays on Science: Methods, Criticism, Training, (HPR, box 27.2). Circumstances (New York: Springer-Verlag, 1984), [Rob 51b] ______, letter of 18 May 1951 to E. T. Bell 403–438. (HPR, box 1.15). [Goo 91] Judith R. Goodstein, Millikan’s School: A His- [Rot 64] Gian-Carlo Rota, The number of partitions of a tory of the California Institute of Technology (New set, Amer. Math. Monthly 71: 498-504 (1964). York: W. W. Norton, 1991). [Tan 75] Steven M. Tanny, Generating functions and [Gou 74] H. W. Gould, Coefficient identities for powers generalized alternating subsets, Amer. Math. Monthly of Taylor and Dirichlet series, Amer. Math. Monthly 75: 55-65 (1964). 81: 3–14 (1974). [Tau 62] Abraham Haskel Taub, H. P. Robertson, 1903- [Gre-Goo 84] J. L. Greenberg and J. R. Goodstein, 1961, SIAM Journal 10: 737-801 (1962). Theodore von Kármán and applied mathematics in [Tay 81] Angus Ellis Taylor, letter of 2 November 1981 America, Science 222: 1300–1304 (1983). to J. Greenberg. [G & H 62] H. W. Gould and A. T. Harper, Operational [Tay 84] ______, A life in mathematics remembered, formulas connected with two generalizations of Amer. Math. Monthly 91 (10): 605-618 (1984). Hermite Polynomials, Duke Math. J. 29: 51–63 (1962). [Tay 91] ______, letter of 16 November 1991 to J. R. [Hal 2012] Alfred W. Hales, email of 26 August 2012 Goodstein. [War 38] Morgan Ward, letter of 20 May 1938 to H. P. to D. Babbitt. Robertson (HPR, box 6.13). [Hou 42] William V. Houston, letter of 13 July 1942 to [War 39] ______, letter of 18 April 1939 to R. A. Millikan R. A. Millikan (RAM, box 25.1). (RAM, box 24.22). [Joc-Efr] John J. O’Connor and Edmund Robertson, [Wey 31] Hermann Weyl, The Theory of Groups and MacTutor History of Mathematics, http://www- Quantum Mechanics, translated by H. P. Robertson history.mcs.st-and.ac.uk/Biographies/Bell. (London: Methuen, 1931). html. [Wil 27] Edwin Bidwell Wilson, Some recent specu- [Lan 41] Rudolph Langer, Review of The Development of lations on the nature of light, Science 65: 265-271 Mathematics, Science, New Series 93: 281-283 (1941). (1927). [Leh 93] Derrick H. Lehmer, The mathematical work of Morgan Ward, Mathematics of Computation 61 (203): 307-311 (1993).

698 Notices of the AMS Volume 60, Number 6