Garrett Birkhoffhas had a lifelong connection with Harvard mathematics. He 
was an infant when his father, the famous mathematician G. D. Birkhoff, 
joined the Harvard faculty. He has had a long academic career at Harvard: 

A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointment from 
1936 until his retirement in 1981. His research has ranged widely through algebra, 
lattice theory, hydrodynamics, differential equations, scientific computing, 
and history ofmathematics. Among his many publications are books on lattice 
theory and hydrodynamics, and the pioneering textbook A Survey of Modern 
Algebra, written jointly with S. Mac Lane. He has served as president of SIAM 
and is a member of the National Academy of Sciences. 
Mathematics at Harvard, 1836-1944 

GARRETT BIRKHOFF 

O. OUTLINE 
As my contribution to the history of mathematics in America, I decided 
to write a connected account of mathematical activity at Harvard from 1836 
(Harvard's bicentennial) to the present day. During that time, many mathematicians 
at Harvard have tried to respond constructively to the challenges 
and opportunities confronting them in a rapidly changing world. 

This essay reviews what might be called the indigenous period, lasting 
through World War II, during which most members of the Harvard mathematical 
faculty had also studied there. Indeed, as will be explained in §§ 1-3 
below, mathematical activity at Harvard was dominated by Benjamin Peirce 
and his students in the first half of this period. 

Then, from 1890 until around 1920, while our country was becoming 
a great power economically, basic mathematical research of high quality, 
mostly in traditional areas of analysis and theoretical celestial mechanics, 
was carried on by several faculty members. This is the theme of §§4-7. 
Finally, I will review some mathematical developments at Harvard in the 
quarter-century 1920-44, during which mathematics flourished there (and at 
Princeton) as well as anywhere in the world. 


GARRETT BIRKHOFF 

Whereas §§1-13 of my account are based on reading and hearsay, much of 
§§ 14-20 reviews events since 1928, when I entered Harvard as a freshman, 
and expresses my own first-hand impressions, mellowed by time. Throughout, 
I will pay attention not only to "core" mathematics, but also to "applied" 
mathematics, including mathematical physics, mathematical logic, statistics, 
and computer science. I will also round out the picture by giving occasional 
glimpses into aspects of the contemporary scientific and human environment 
which have influenced "mathematics at Harvard". Profound thanks are due 
to Clark Elliott and I. Bernard Cohen at Harvard, and to Uta Merzbach and 
the other editors of this volume, for many valuable suggestions and criticisms 
of earlier drafts. 

1. BENJAMIN PEIRCE! 
In 1836, mathematics at Harvard was about to undergo a major transition. 
For a century, all Harvard College students had been introduced to the infinitesimal 
calculus and the elements of physics and astronomy by the Hollis 
Professor of Mathematics and Natural Philosophy. Since 1806, the Hollis 
Professor had been John Farrar (1779-1853), who had accepted the job after 
it had been declined by Nathaniel Bowditch (1773-1838), 2 a native of Salem. 

Bowditch's connection with mathematics at Harvard was truly unique. 
Having had to leave school at the age of 10 to help his father as a cooper, 
Bowditch had been almost entirely self-educated. After teaching himself Latin 
and reading Newton's Principia, he sailed on ships as supercargo on four 
round trips to the East Indies. He then published the most widely used book 
on the science of navigation, The New American Practical Navigator, before 
becoming an executive actuary for a series of insurance companies. 

After awarding Bowditch an honorary M.A. in 1802, Harvard offered him 
the Hollis Professorship of Mathematics and Natural Philosophy in 1806. 
Imagine Harvard offering a professorship today to someone who had never 
gone to high school or college! Though greatly honored, Bowditch declined 
because he could not raise his growing family properly on the salary offered 
($ 1200/yr.), and remained an actuarial executive. A prominent member of 
Boston's American Academy of Arts and Sciences, he did however stay active 
in Harvard affairs. 

In American scientific circles, Bowditch became most famous through his 
translation of Laplace's Mecanique Celeste, with copious notes explaining 
many sketchy derivations in the original. He did most of the work on this 
about the same time that Robert Adrain showed that Laplace's value of 1/338 
for the earth's eccentricity (b -a)/a should be 1/316. Bowditch decided 
(correctly) that it is more nearly 1/300.3 

Bowditch's scientific interests were shared by a much younger Salem native, 
Benjamin Peirce (1809-1880). Peirce had become friendly at the Salem 


MATHEMATICS AT HARVARD, 1836-1944 

Private Grammar School with Nathaniel's son, Henry Ingersoll Bowditch, 
who acquainted his father with Benjamin's skill in and love for mathematics, 
and Peirce is reputed to have discussed mathematics and its applications with 
Nathaniel Bowditch from his boyhood. In 1823, Benjamin's father became 
Harvard's librarian;4 in 1826, Bowditch became a member of the Harvard 
Corporation (its governing board). By then, the Peirce and Bowditch sons 
were fellow students in Harvard College, and their families had moved to 
Boston. 

There Peirce's chief mentor was Farrar. For more than 20 years, Farrar had 
been steadily improving the quality of instruction in mathematics, physics, 
and astronomy by making translations of outstanding 18th century French 
textbooks available under the title "Natural Philosophy for the Students at 
Cambridge in New England". Actually, his own undergraduate thesis of 1803 
had contained a calculation of the solar eclipse which would be visible in New 
England in 1814. Although very few Harvard seniors could do this today, it 
was not an unusual feat at that time. 

By 1829, Nathaniel Bowditch had become affluent enough to undertake 
the final editing and publication of Laplace's Mecanique Celeste at his own 
expense, and young Peirce was enthusiastically assisting him in this task. 
Peirce must have found it even more exhilarating to participate in criticizing 
Laplace's masterpiece than to predict a future eclipse! 

In 1831, Peirce was made tutor in mathematics at Harvard College, and 
in 1833 he was appointed University Professor of Mathematics and Natural 
Philosophy. As a result, two of the nine members of the 1836 Harvard College 
faculty bore almost identical titles. In the same year, he married Sara Mills of 
Northampton, whose father Elijah Hunt Mills had been a U.S. senator [DAB]. 
They had five children, of whom two would have an important influence on 
mathematics at Harvard, as we shall see. 

By 1835, still only 26, Peirce had authored seven booklets of Harvard 
course notes, ranging from "plane geometry" to "mechanics and astronom".5 
Moreover Farrar, whose health was failing, had engaged another able recent 
student, Joseph Lovering (1813-92), to share the teaching load as instructor. 
In 1836, Farrar resigned because of poor health, and Lovering succeeded to 
his professorship two years later. For the next 44 years, Benjamin Peirce and 
Joseph Lovering would cooperate as Harvard's senior professors of mathematics, 
astronomy, and physics. 

Nathaniel Bowditch died in 1838, and his place on the Harvard Corporation 
was taken by John A. Lowell (1798-1881), a wealthy textile industry 
financier. By coincidence, in 1836 (Harvard's bicentennial year) his cousin 
John Lowell, Jr. had left $250,000 to endow a series of public lectures, with 
John A. as sole trustee of the Lowell Institute which would pay for them. 
Lowell (A.B. 1815) must have also studied with Farrar. Moreover by another 


GARRETT BIRKHOFF 

coincidence, having entered Harvard at 13, John A. Lowell had lived as a 
freshman in the house of President Kirkland, whose resignation in 1829 after 
a deficit of $4,000 in a budget of $30,000, student unrest, and a slight 
stroke had been due to pressure from Bowditch!6 As a final coincidence, by 
1836 Lowell's tutor Edward Everett had become governor of Massachusetts, 
in which capacity he inaugurated the Lowell Lectures in 1840. 

In 1842, Peirce was named Perkins Professor of Astronomy and Mathematics, 
a newly endowed professorship. By that time, qualified Harvard 
students devoted two years of study to Peirce's book Curves and Functions, 
for which he had prepared notes. Ambitious seniors might progress to Poisson's 
Mecanique Analytique, which he would replace in 1855 with his own 
textbook, A System ofAnalytical Mechanics. This was fittingly dedicated to 
"My master in science, NATHANIEL BOWDITCH, the father of American 
Geometry". 

Meanwhile, Lovering was becoming famous locally as a teacher of physics 
and a scholar. His course on "electricity and magnetism" had advanced well 
beyond Farrar, and over the years, he would give no less than nine series of 
Lowell Lectures [NAS #6, 327-44].7 

In 1842-3, Peirce and Lovering also founded a quarterly journal called 
Cambridge Miscellany of Mathematics, Physics, and Astronomy, but it did 
not attract enough subscribers to continue after four issues. They also taught 
Thomas Hill '43, who was awarded the Scott Medal by the Franklin Institute 
for an astronomical instrument he invented as an undergraduate. He later 
wrote two mathematical textbooks while a clergyman, and became Harvard's 
president from 1862 to 1868 [DAB 20, 547-8]. 

2. PEIRCE REACHES OUT 
During his lifetime Peirce was without question the leading American 
mathematical astronomer. In 1844, perhaps partly stimulated by a brilliant 
comet in 1843, "The Harvard Observatory was founded on its present site ... 
by a public subscription, filled largely by the merchant shipowners of Boston" 
[Mor, p. 292]. For decades, William C. Bond (1789-1859) had been advising 
Harvard about observational astronomy, and he may well have helped Hill 
with his instrument. In any event, Bond finally became the salaried director 
of the new Observatory, where he was succeeded by his son George (Harvard 
'45). Peirce and Lovering both collaborated effectively with William 
Bond in interpreting data.8 Pursuing further the methods he had learned 
from Laplace's Mecanique Celeste, Peirce also analyzed critically Leverrier's 
successful prediction of the new planet Neptune, first observed in 1846. 

Meanwhile, our government was beginning to play an important role in 
promoting science. At about this time, the Secretary of the Navy appointed 


MATHEMATICS AT HARVARD, 1836-1944 

Lieutenant (later Admiral) Charles Henry Davis head of the Nautical Almanac 
Office. Like Peirce, Davis had gone to Harvard and married a daughter 
of Senator Mills. Because of Peirce and the Harvard Observatory, he 
decided to locate the Nautical Almanac Office in Cambridge, where Peirce 
served from 1849 to 1867 as Consulting Astronomer, supervising for several 
years the preparation (in Cambridge) of the American Ephemeris and Nautical 
Almanac, our main government publication on astronomy. This activity 
attracted to Cambridge such outstanding experts in celestial mechanics as Simon 
Newcomb (1835-1909) and G.W. Hill (1838-1914), who later became 
the third and fourth presidents of the American Mathematical Society.9 Others 
attracted there were John M. van Vleck, an early AMS vice-president, 
and John D. Runkle, founder of a short-lived but important Mathematical 
Monthly. 

Similarly, Alexander Dallas Bache [DAB 1,461-2], after getting the Smithsonian 
Institution organized in 1846 with Joseph Henry as its first head 
[EB 20, 698-700], became head of the U.S. Coast Survey. Bache appointed 
Peirce's former student B. A. GouldIO as head of the Coast Survey's longitude 
office and Gould, who was Harvard president Josiah Quincy's son-in-law, decided 
to locate his headquarters at the Harvard Observatory and also use 
Peirce as scientific adviser. Peirce acted in this capacity from 1852 to 1874, 
aided by Lovering's selfless cooperation, succeeding Bache as superintendent 
of the Coast Survey for the last eight of these years. II In both of these roles, 
Peirce showed that he could not only apply mathematics very effectively (see 
[Pei, p. 12]); he was also a creative organizer and persuasive promoter. 

Indeed, from the 1840s on, Peirce was reaching out in many directions. 
Thus, he became president of the newly founded American Association for 
the Advancement of Science in 1853. He was also active locally in Harvard's 
Lawrence Scientific School (LSS) during its early years. 

The Lawrence Scientific School. The LSS is best understood as an early 
attempt to promote graduate education in pure and applied science, including 
mathematics. Established when Harvard's president was Edward Everett, 
John A. Lowell's former tutor, and its treasurer was Lowell's then affluent 
business associate Samuel Eliot, the LSS was named for Abbott Lawrence, another 
New England textile magnate who had been persuaded to give $50, 000 
to make its establishment possible in 1847. By then, the cloudy academic 
concepts of "natural philosophy" and "natural history" were becoming articulated 
into more clearly defined "sciences" such as astronomy, geology, physics, 
chemistry, zoology, and botany. Correspondingly, the LSS was broadly conceived 
as a center where college graduates and other qualified aspirants could 
receive advanced instruction in these sciences and engineering. Its early scientifically 
minded graduates included not only several notable "applied" mathematicians 
such as Newcomb and Runkle, but also the classmates Edward 


GARRETT BIRKHOFF 

Pickering and John Trowbridge. John Trowbridge would later join and ultimately 
replace Joseph Lovering as Harvard's chief physicist, while Pickering 
would become director of the Harvard Observatory and an eminent astrophysicist. 
Most important for mathematics at Harvard, Trowbridge would 
have as his "first research student in magnetism" B. O. Peirce. 

The leading "pure" scientists on the LSS faculty were Peirce, the botanist 
Asa Gray, the anatomists Jeffries Wyman, and the German-educated Swiss 
naturalist Louis Agassiz (1807-73), already internationally famous when he 
joined the faculty of the LSS as professor of zoology and geology in 1847. 

The original broad conception of the Lawrence Scientific School was best 
exemplified by Agassiz. Like Benjamin Peirce, Agassiz expected students 
to think for themselves; but unlike Peirce, he was a brilliant lecturer, who 
soon "stole the show" from Gray and Peirce at the LSS. Although he did 
not influence mathematics directly, we shall see that three of his students 
indirectly influenced the mathematical sciences at Harvard: his son Alexander 
(1835-1912); the palaeontologist and geologist Nathaniel Shaler; and the 
eminent psychologist and philosopher William James. 

The Lazzaroni. Like Alexander Bache, Louis Agassiz and Benjamin 
Peirce were members of an influential group of eminent American scientists 
who, calling themselves the "Lazzaroni", tried to promote research. However, 
very few tuition-paying LSS students had research ambitions. The majority 
of them studied civil engineering under Henry Lawrence Eustis ('38), who 
had taught at West Point [Mor, p. 414] before coming to Harvard. Most 
of the rest studied chemistry, until 1863 under Eben Horsford, whose very 
"applied" interests made it appropriate to assign his Rumford Professorship 
to the LSS. 

Peirce's students. Peirce had many outstanding students. Among them 
may be included Thomas Hill, B. A. Gould, John Runkle, and Simon Newcomb. 
Partly through his roles in the Coast Survey, the Harvard Observatory, 
and the Nautical Almanac, he was responsible for making Harvard our 
nation's leading research center in the mathematical sciences in the years 
1845-65. 

Although most of his students "apprehended imperfectly what Professor 
Peirce was saying", he also was "a very inspiring and stimulating teacher" 
for those eager to learn. We know this from the vivid account of his teaching 
style [Pei, pp. 1-4] written by Charles William Eliot (1834-1926), seventy 
years after taking (from 1849 to 1853) the courses in mathematics and 
physics taught by Peirce and Lovering. The same courses were also taken by 
Eliot's classmate, Benjamin's eldest son James Mills Peirce (1834-1906). After 
graduating, these two classmates taught mathematics together from 1854 
to 1858, collaborating in a daring and timely educational reform. At the 
time, Harvard students were examined orally by state-appointed overseers 


MATHEMATICS AT HARVARD, 1836-1944 

whose duty it was to make sure that standards were being maintained. "Offended 
by the dubious expertness and obvious absenteeism of the Overseers, 
... the young tutors [Eliot and J. M. Peirce] obtained permission to substitute 
written examinations, which they graded themselves" [HH, p. 15]. 

After 1858, J. M. Peirce tried his hand at the ministry, while Eliot increasingly 
concentrated his efforts on chemistry (his favorite subject) and universityadministration. 
Josiah Parsons Cooke ('48), the largely self-taught Erving 
Professor of Chemistry, had shared chemicals with Eliot when the latter was 
still an undergraduate [HH, p. 11]. Then in 1858, these two friends successfully 
proposed a new course in chemistry [HH, p. 16], in which students 
performed laboratory exercises "probably for the first time". Eliot demonstrated 
remarkable administrative skill; at one point, he was acting dean of 
the LSS and in charge of the chemistry laboratory. In these roles, he proposed 
a thorough revision in the program for the S.B. in chemistry, based 
on a "firm grounding in chemical and mathematical fundamentals". He then 
served with Dean Eustis of that School and Louis Agassiz on a committee 
appointed to revamp the school's curriculum as a whole [HH, pp. 23-25]. 
However Eliot's zeal for order and discipline antagonized the more informal 
Agassiz, and Eliot's reformist ideas were rejected. After losing out to the 
more research-oriented "Lazzarone" Wolcott Gibbs in the competition for 
the Rumford professorship [HH, pp. 25-27], in spite of J. A. Lowell's support 
of his candidacy, Eliot left Harvard (with his family) for two years of 
study in France and Germany. 

MIT. Academic job opportunities in "applied" science in the Boston area 
were improved by the founding there of the Massachusetts Institute of Technology 
(MIT). Since the mid-1840s, Henry and then his brother William 
Rogers, "one of those accomplished general scientists who matured before 
the age of specialization", had been lobbying with J. A. Lowell and others for 
the benefits of poly technical education, and in 1862 William Rogers became 
its first president. Peirce's student John Runkle joined its new faculty in 1865 
as professor of mathematics and analytical mechanics, and later became the 
second president of MIT. [2 Recognizing Eliot's many skills, Rogers soon also 
invited him to go to MIT, and Eliot accepted [HH, pp. 34-7]. There Eliot 
wrote with F. H. Storer, an LSS graduate and his earlier collaborator and 
hiking companion, a landmark Manual ofInorganic Chemistry. 

3. PEIRCE'S GOLDEN YEARS 
When the Civil War broke out in 1861, J. M. Peirce was a minister in 
Charleston, South Carolina. Benjamin promptly had James made an assistant 
professor of mathematics at Harvard, to help him carry the teaching 
load. Benjamin's brilliant but undisciplined younger brother Charles Sanders 


GARRETT BIRKHOFF 

Peirce (1839-1914), after being tutored at home, had graduated from Harvard 
two years earlier without distinction. Nevertheless, Benjamin secured 
for him a position in the U.S. Coast Survey exempting Charles from military 
service. As we shall see, this benign nepotism proved to be very fruitful. 

A year later, his former student Thomas Hill became Harvard's president, 
having briefly (and reluctantly) been president of Antioch College. At Harvard, 
Hill promoted the "elective system", encouraging students to decide 
between various courses of study. He also initiated series of university lectures 
which, like the LSS, constituted a step toward the provision of graduate 
instruction. Peirce participated in this effort most years during the l860s, 
lecturing on abstruse mathematics with religious fervor. 

From the l850s on, Peirce had largely freed himself from the drudgery of 
teaching algebra, geometry, and trigonometry. Moreover, whereas Lovering's 
courses continued to be required, Peirce's were all optional (electives), and 
were taken by relatively few students. Peirce was frequently "lecturing on 
his favorite subject, Hamilton's new calculus of quatemions" [Pei, p. 6], to 

W. E. Byerly ('71) among others. In 1873 Byerly was persuaded by Peirce 
to write a doctoral thesis on "The Heat of the Sun". In it, he calculated the 
total energy of the sun, under the assumption (common at the time) that 
this was gravitational. 13 The calculation only required using the calculus and 
elementary thermodynamics. Nevertheless, Byerly became Harvard's first 
Ph.D., and a pillar of Harvard's teaching staff until his retirement in 19l2! 
Benjamin Peirce had long been interested in Hamilton's quatemions a + 
bi + cj + dk; moreover Chapter X of his Analytical Mechanics (1855) contained 
-a masterful chapter on 'functional determinants' of n x n matrices. 14 
During the Civil War, Benjamin and Charles became interested in generalizations 
of quatemions to linear associative algebras. After lecturing several 
times to his fellow members of the National Academy of Sciences on this 
subject, Benjamin published his main results in 1870 in a privately printed 
paper. IS This contained the now classic "Peirce decomposition" 

x = exe + ex (1 -e) + (1 -e )xe + (1 -e)x (1 -e) 

with respect to any "idempotent" e satisfying ee = e. 
An 1881 sequel, published posthumously by J. J. Sylvester in the newly 
founded American Journal of Mathematics (4, 97-229), contained numerous 
addenda by Charles. Most important was Appendix III, where Charles 
proved that the only division algebras of finite order over the real field R 
are R itself, the complex field C, and the real quatemions. This very fundamental 
theorem had been proved just three years earlier by the German 
mathematician Frobenius. 16 
Charles Peirce also worked with his father in improving the scientific instrumentation 
of the U.S. "Coast Survey", which by 1880 was surveying the 


MATHEMATICS AT HARVARD, 1836-1944 

entire United States! But most relevant to mathematics at Harvard, and most 
distinguished, was his unpaid role as a logician and philosopher. An active 
member of the Metaphysical Club presided over by Chauncey Wright (,52), 
another of Benjamin's ex-students who made his living as a computer for 
the Nautical Almanac, Charles gave a brilliant talk there proposing a new 
philosophical doctrine of "pragmatism". He also published a series of highly 
original papers on the then new algebra of relations. Although surely "imperfectly 
apprehended" by most of his contemporaries, these contributions 
earned him election in 1877 to the National Academy of Sciences, which had 
been founded 12 years earlier by the Lazzaroni. 

In his last years, increasingly absorbed with quaternions, Benjamin 
Peirce's unique teaching personality influenced other notable students. These 
included Harvard's future president, Abbott Lawrence Lowell, and his brilliant 
brother, the astronomer Percival. They were grandsons of both Abbott 
Lawrence and John A. Lowell! To appreciate the situation, one must realize 
that the two grandfathers of the Lowell brothers were John A. Lowell, the 
trustee of the Lowell Institute, and Abbott Lawrence, for whom Harvard's 
Lawrence Scientific School (see §2) had been named. From 1869 to 1933, 
the presidents of Harvard would be former students of Benjamin Peirce! 

Two others were B. O. Peirce, a distant cousin of Benjamin's who would 
succeed Lovering (and Farrar) as Hollis Professor of Mathematics and Natural 
Philosophy; and Arnold Chace, later chancellor of Brown University. 
Peirce's lectures inspired both A. L. Lowell and Chace to publish papers in 
which "quaternions" (now called vectors) were applied to geometry. Moreover 
both men would describe in [Pei] fifty years later, as would Byerly, how 
Peirce influenced their thinking. Still others were W. E. Story, who went on 
to get a Ph.D. at Leipzig, and W. I. Stringham (see [S-G]). Story became 
president of the 1893 International Mathematical Congress in Chicago, and 
himself supervised 12 doctoral theses including that of Solomon Lefschetz 
[CMA, p. 201]. 

Benjamin Peirce's funeral must have been a very impressive affair. His 
pallbearers fittingly included Harvard's president C. W. Eliot and ex-president 
Thomas Hill; Simon Newcomb, J. J. Sylvester, and Joseph Lovering; his 
famous fellow students and lifelong medical friends Henry Bowditch and 
Oliver Wendell Holmes (the "Autocrat of the Breakfast Table"); and the new 
superintendent of the Coast Survey, C. P. Patterson. 

4. ELIOT TAKES HOLD17 
When Thomas Hill resigned from the presidency of Harvard in 1868, the 
Corporation (with J. A. Lowell in the lead) recommended that Eliot be his 
successor, and the overseers were persuaded to accept their nomination in 
1869. After becoming president, Eliot immediately tried (unsuccessfully) to 


GARRETT BIRKHOFF 

merge MIT with the LSS [Mor, p. 418].18 At the same time, he tried to build 
up the series of university lectures, inaugurated by his predecessor, into a 
viable graduate school. It is interesting that for the course in philosophy 
his choice of lecturers included Ralph Waldo Emerson and Charles Sanders 
Peirce; in 1870-71 thirty-five courses of university lectures were offered; but 
the scheme "failed hopelessly" [Mor, p. 453]. 

To remedy the situation, Eliot then created a graduate department, with 
his classmate James Mills Peirce as secretary of its guiding academic council. 
This was authorized to give M.A. and Ph.D. degrees, such as those given 
to Byerly and Trowbridge. At the same time, Eliot transferred his former 
rival Wolcott Gibbs from being dean of the LSS to the physics department, 
and chemistry from the LSS to Harvard College, where his former mentor 
Josiah Parsons Cooke was in charge. A year earlier, Eliot had proposed an 
elementary course in chemistry, to be taught partly in the laboratory. This 
gradually became immensely popular, partly because of its emphasis on the 
chemistry of such familiar phenomena as photography [Mor, p. 260]. 

By 1886, all members of the LSS scientific faculty had transferred to Harvard 
College. Moreover undergraduates wanting to study engineering had no 
incentive for enrolling in the LSS rather than Harvard College. As a result 
of this, and competition from MIT and other institutions, there was a steady 
decline in the LSS enrollment, until only 14 students enrolled in l886! Its 
four-year programme in "mathematics, physics, and astronomy", inherited 
from the days of Benjamin Peirce, had had no takers at all for many years, 
and was wisely replaced in 1888 by a programme in electrical engineering. 

Meanwhile, the new graduate department itself was struggling. After 1876, 
the Johns Hopkins University attracted many of the best graduate students. 
Two of them would later be prominent members of the Harvard faculty: 
Edwin Hall in physics and Josiah Royce in philosophy. During its entire 
lifetime (1872-90), the graduate department awarded only five doctorates in 
mathematics, including those of Byerly and F. N. Cole (the latter earned in 
Germany, see §5). 

To remedy the situation, Eliot made a second administrative reorganization 
in 1890. From it, the graduate department emerged as the graduate 
school; the LSS engineering faculty joined the Harvard College faculty in a 
new Faculty of Arts and Sciences. J. M. Peirce's title was changed from secretary 
of the graduate department to dean of the Graduate School of Arts 
and Sciences, and the economist Charles F. Dunbar ('51) was made dean of 
the Faculty of Arts and Sciences. 

Additional programmes of study were introduced into the LSS, and 
Nathaniel Shaler made its new dean. This was a brilliant choice: the enrollment 
in the LSS increased to over 500 by 1900, and Shaler's own Geology 
4 became one of Harvard College's most popular courses. Shaler was 


MATHEMATICS AT HARVARD, 1836-1944 

also allowed to assume management of the mining companies of his friend 
and neighbor, the aging inventor and mining tycoon Gordon McKay [HH, 
pp. 213-15], Shaler persuaded McKay in 1903 to bequeath his fortune to 
the school, where it supports the bulk of Harvard's program in "applied" 
mathematics to this day! 

Mention should also be made of the appointment in 1902 of the selfeducated, 
British-born scientist Arthur E. Kennelly (1851-1939). Joint discoverer 
of the upper altitude "Kennelly-Heaviside layer" which reflects radio 
waves, Kennelly had been Edison's assistant for 13 years and president of 
the American Institute of Electric Engineers. In his thoughtful biography of 
Kennelly [NAS 22: 83-119], Vannevar Bush describes how Kennelly's career 
spanned the entire development of electrical engineering to 1939. Bush and 
my father [GDB III, 734-8] both emphasize that Kennelly revolutionized 
the mathematical theory of alternating current (a.c.) circuits by utilizing the 
complex exponential function. Curiously, this major application is still rarely 
explained in mathematics courses in our country, at Harvard or elsewhere! 

For further information about the changes I have outlined, and other interpretations 
of the conflicting philosophies of scientific education which motivated 
them, I refer you to [HH] and especially [Love]. The latter document 
was written by James Lee Love, who taught mathematics under the auspices 
of the Lawrence Scientific School from 1890 to 1906, when the LSS was 
renamed the Graduate School of Applied Science. Officially affiliated with 
Harvard until 1911, Love returned to Burlington, North Carolina, in 1918 to 
become president of the Gastonia Cotton Manufacturing Company. Reorganized 
as Burlington Mills, this became one of our largest textile companies. 
During these years, Love donated $50,000 to the William Byerly Book Fund. 

5. A DECADE OF TRANSITION 
When Benjamin Peirce died, his son James had been ably assisting him 
in teaching Harvard undergraduates for more than 20 years. Byerly had 
joined them in 1876. At the time, B. o. Peirce was still studying physics and 
mathematics with John Trowbridge and Benjamin Peirce, but he became an 
instructor in mathematics in 1881, assistant professor of mathematics and 
physics in 1884, and Hollis Professor of Mathematics and Natural Philosophy 
(succeeding Lovering) in 1888. In the decade following Benjamin Peirce's 
death, the triumvirate consisting of J. M. Peirce (1833-1906), W. E. Byerly 
(1850-1934), and B. O. Peirce (1855-1913) would be Harvard's principal 
mathematics teachers. 

As a mathematician, J. M. Peirce has been aptly described as an "understudy" 
to his more creative father [Mor, p. 249]. However, "to no one, 
excepting always President Eliot, [was] the Graduate School so indebted" 
for "the promotion of graduate instruction" [Mor, p. 455]. Moreover his 


GARRETT BIRKHOFF 

teaching, unlike that of his father, seems to have been popular and easily 
comprehended. In the 1880s, he and Byerly began giving in alternate years 
Harvard's first higher geometry course (Mathematics 3) with the title "Modern 
methods in geometry -determinants". Otherwise, his advanced teaching 
covered mainly topics of algebra and geometry in which Benjamin and C. S. 
Peirce had done research, such as "quaternions", "linear associative algebra", 
and "the algebra of logic". 

While James Peirce was administering graduate degrees at Harvard as secretary 
of the academic council, Byerly was cooperating most effectively in 
making mathematics courses better understood by undergraduates. His Differential 
Calculus (1879), his Integral Calculus (1881), and his revised and 
abridged edition of Chauvenet's Geometry (1887), presumably the text for 
Math. 3, were widely adopted in other American colleges and universitiesY 

In 1883-4, Byerly and B. O. Peirce introduced a truly innovative course in 
mathematical physics (or "applied mathematics") which has been taught at 
Harvard in suitably modified form ever since. Half of this course (taught by 
Byerly) dealt with the expansion of "arbitrary functions" in Fourier Series and 
Spherical Harmonics, this last being the title of a book he wrote in 1893. The 
other half treated potential theory, and Peirce wrote for it a book, Newtonian 
Potential Function, published in three editions (1884, 1893, 1902). Like 
Byerly's other books, they were among the most influential and advanced 
American texts of their time. 

B. O. Peirce was an able and scholarly, if traditional, mathematical physicist. 
A brilliant undergraduate physics major, his "masterly" later physical 
research was mostly empirical. Although it was highly respected for its thoroughness, 
and Peirce became president of the American Physical Society in 
1913, it lay in "the unexciting fields of magnetism and the thermal conduction 
of non-metallic substances". His main mathematical legacy consisted in 
his text for Mathematics 10, and his Table ofIntegrals... , originally written 
as a supplement to Byerly's Integral Calculus. This was still being used at 
Harvard when I was an undergraduate, but such tables may soon be superseded 
by packages of carefully written, debugged, and documented computer 
programs like Macsyma. 
In short, Harvard's three professors of mathematics regarded their profession 
as that of teaching reasonably advanced mathematics in an understandable 
way. Their success in this can be judged by the quality of their 
students, who included M. W. Haskell, Arthur Gordon Webster, who became 
president of the American Physical Society in 1903, Frank N. Cole, W. F. 
Osgood, and Maxime Bacher. In 1888, when the AMS was founded, two of 
them had inherited the titles of Benjamin Peirce and Lovering; only Byerly 
had the simple title "professor of mathematics".21 


MATHEMATICS AT HARVARD, 1836-1944 

c. S. Peirce. When his father died, C. S. Peirce (1839-1914) was at 
the zenith of his professional career. From 1879 to 1884, he was a lecturer 
at Johns Hopkins as well as a well-paid and highly respected employee of 
the Coast Survey (cf. [SMA, pp. 13-20]). While there, he discovered the 
fundamental connection between Boolean algebra and what are today called 
"partially ordered sets" (cf. American J. Math. 3 (1880), 15-57), thus foreshadowing 
the "Dualgruppen" of Dedekind ("Verbande" or lattices in today's 
terminology). Unfortunately, in describing this connection, he erroneously 
claimed that the distributive law a(b V c) = ab V ac necessarily relates least 
upper bounds x V y and greatest lower bounds xy. 
Indeed, the 1880s were a disastrous decade for C. S. Peirce. His lectures 
at the Johns Hopkins Graduate School were not popular; his personality 
was eccentric; and his appointment there was not renewed after Sylvester 
returned to England. He also lost his job with the Coast Survey soon after 
1890. Although he continued to influence philosophy at Harvard (see §8), he 
never again held a job with any kind of tenure. An early member of the New 
York Mathematical Society, his brilliant turns of speech continued to enliven 
its meetings [CMA, pp. 15-16], but he was not taken seriously. 

6. OSGOOD AND B6cHER22 
By 1888, when the American Mathematical Society (AMS) was founded (in 
New York), a new era in mathematics at Harvard was dawning. Frank Nelson 
Cole (Harvard '82) had returned three years earlier after "two years under 
Klein at Leipzig" [Arc, p. 100]. "Aglow with enthusiasm, he gave courses in 
modern higher algebra, and in the theory of functions of a complex variable, 
geometrically treated, as in Klein's famous course of lectures at Leipzig." 
His "truly inspiring" lectures were attended by two undergraduates, W. F. 
Osgood (1864-1943) and Maxime Bacher (1867-1918), "as well as by nearly 
all members of the department," including Professors J. M. Peirce, B. O. 
Peirce, and W. E. Byerly. 

After graduating, Osgood and Bacher followed Cole's example and went to 
Germany to study with Felix Klein, who had by then moved to Gottingen.23 
After earning Ph.D. degrees (Osgood in 1890, Bacher with especial distinction 
in 1891), both men joined the expanding Harvard staff as instructors for three 
years. Inspired by the example of Gottingen under Klein, they spearheaded 
a revolution in mathematics at Harvard, where they continued to serve as 
assistant professors for another decade before becoming full professors (Osgood 
in 1903, Bacher in 1904). All this took place in the heyday of the Eliot 
regime, under the benevolent but mathematically nominal leadership of the 
two Peirces and Byerly. 

The most conspicuous feature of the revolution resulting from the appointments 
of Bacher and Osgood was a sudden increase in research activity. By 


GARRETT BIRKHOFF 

1900, Osgood had published 21 papers (six in German), while Bacher had 
published 30 in addition to a book On the series expansions ofpotential theory, 
and a survey article on "Boundary value problems of ordinary differential 
equations" for Klein's burgeoning Enzyklopiidie der Mathematischen 
Wissenschaflen, both in German. Moreover Bacher and Yale's James Pierpont 
had given the first AMS Colloquium Lectures in 1896, to an audience of 
13, while Osgood and A. G. Webster (a Lawrence Scientific School alumnus) 
had given the second, in 1898.24 

Similar revolutions had taken place in the 1890s at other leading American 
universities. Most important of these was at the newly founded University of 
Chicago, where the chairman of its mathematics department, E. H. Moore, 
was inspiring a series of Ph.D. candidates [LAM, §3]. Under the leadership 
of H. B. Fine, who had been stimulated by Sylvester's student G. B. Halsted, 
Princeton would blossom somewhat later. Meanwhile, Cole had become a 
professor at Columbia, secretary of the AMS, and editor of its Bulletin (cf. 
[Arc, Ch. VD. The Cole prize in algebra is named for him. 

Thus it was most appropriate for Osgood, Bacher, and Pierpont to cooperate 
with E. H. Moore (1862-1932) of Chicago in making the promotion 
of mathematical research the central concern of the AMS. Feeling "the great 
need of a journal in which original investigations might be published" [Arc, 

p. 56], these men succeeded in establishing the Transactions A mer. Math. Soc. 
[Arc, Ch. V]. From 1900 on, this new periodical supplemented the American 
Journal ofMathematics, complete control over which Simon Newcomb was 
unwilling to relinquish. The Annals of Mathematics was meanwhile being 
published at Harvard from 1899 to 1911, with Bacher as chief editor. Primarily 
designed for "graduate students who are not yet in a position to read 
the more technical journals", this also "contained some articles ... suitable 
for undergraduates." 
Harvard continued to educate many mathematically talented students during 
the years 1890-1905, including most notably J. L. Coolidge, E. V. Huntington, 
and E. B. Wilson, all for four years; and for shorter periods E.R. 
Hedrick ('97-'99), Oswald Veblen ('99-'00), and G. D. Birkhoff ('03-'05) . 
At the same time, there was a great improvement in the quality and quantity 
of advanced courses designed "primarily for graduate students", but taken 
also by a few outstanding undergraduates. By 1905, the tradition of Benjamin 
Peirce had finally been supplanted by new courses stressing new concepts, 
mostly imported from Germany and Paris; in 1906 J. M. Peirce died. 

By that time, Harvard's graduate enrollment had increased mightily. From 
28 students in 1872, when Eliot had appointed J. M. Peirce secretary of his 
new "graduate department", it had grown to 250 when Peirce resigned as dean 
of Harvard's "graduate school", to become dean of the entire Faculty of Arts 
and Sciences. A key transition had occurred in 1890, when the graduate "department" 
was renamed a "school", and the Harvard catalog first divided all 


MATHEMATICS AT HARVARD, 1836-1944 

courses into three tiers: "primarily for undergraduates", "for undergraduates 
and graduates", and "primarily for graduates", as it still does. 

The Peirces and Byerly had explained to their students many of the methods 
of Fourier, Poisson, Dirichlet, Hamilton, and Thomson and Tait's Principles 
of Natural Philosophy (1867). However, they had largely ignored the 
advances in rigor due to Cauchy, Riemann, and Weierstass. For example, Byerly's 
Integral Calculus of 1881 still defined a definite integral vaguely as "the 
limit of a sum of infinitesimals", although Cauchy-Moigno's Ler;ons de Calcui 
Integral had already defined integrals as limits of sums '£f(xi)!!J.xi, and 
sketched a proof of the fundamental theorem of the calculus in 1844, while 
in 1883 volume 2 of Jordan's Cours d'Analyse would even define uniform 
continuity. 

The key graduate course (Mathematics 13) on functions of a complex variable 
became modernized gradually. Under J. M. Peirce, it had been a modest 
course based on Briot and Bouquet's Fonctions Elliptiques. In 1891-92, Osgood 
followed this with a more specialized course on elliptic functions as 
such, and the next year with another treating abelian integrals, while Bacher 
gave a course on "functions defined by differential equations", in the spirit 
of Poincare. Then, from 1893 to 1899, Bacher developed Mathematics 13 
into the basic full course on complex analysis that it would remain for the 
next half-century, introducing students to many ideas of Cauchy, Riemann, 
and Weierstrass. Then, beginning in 1895, he and Osgood supplemented 
Mathematics 13 with a half-course on "infinite series and products" (Mathematics 
12) which treated uniform convergence. By 1896, Osgood had written 
a pamphlet Introduction to Infinite Series covering its contents. 

In his moving account of "The life and services of Maxime Bacher" (Bull. 
Amer. Math. Soc. 25 (1919), 337-50) Osgood has described Bacher's lucid 
lecture style, and how much Bacher contributed to his own masterly treatise 
Funktionentheorie (1907), which became the standard advanced text on the 
subject on both sides of the Atlantic. (Weaker souls, whose mathematical 
sophistication or German was not up to this level, could settle for GoursatHedrick.) 
Osgood's other authoritative articles on complex function theory, 
written for the Enzyklopadie der Mathematischen Wissenschaften and as Colloquium 
Lectures,24 established him as America's leading figure in classical 
complex analysis. 

On a more elementary level, Osgood wrote several widely used textbooks 
beginning with an Introduction to Infinite Series (1897). Ten years later, 
his Differential and Integral Calculus appeared, with acknowledgement of its 
debt to Professors B. O. Peirce and Byerly. There one finds stated, for the 
first time in a Harvard textbook, a (partial) "fundamental theorem of the 
calculus". These were followed by his Plane and Solid Analytic Geometry 
with W. C. Graustein (1921), his Introduction to the Calculus (1922), and his 


GARRETT BIRKHOFF 

Advanced Calculus (1925), the last three of which were standard fare for Harvard 
undergraduates until around 1940. Osgood also served for many years 
on national and international commissions for the teaching of mathematics. 

Less systematic than Osgood, Bacher was more inspiring as a lecturer and 
thesis adviser. As an analyst, his main work concerned expansions in SturmLiouville 
series (including Fourier series) associated with the partial differential 
equations of mathematical physics (after "separating variables"). His Introduction 
to the Study ofIntegral Equations (1909, 1914) and his Le<;ons sur 
les Methodes de Sturm ... (1913-14) were influential pioneer monographs. 
Like Bacher's papers which preceded them, they established clearly and rigorously 
by classical methods24a precise interpretations of many basic formulas 
concerned with potential theory and orthogonal expansions (Mathematics 
lOa and Mathematics lOb). 

Several of Bacher's Ph.D. students had very distinguished careers, most 
notable among them being G. C. Evans, who in the 1930s would pilot the 
mathematics department of the University of California at Berkeley to the 
level of preeminence that it has maintained ever since. Others were D. 

R. Curtiss (Northwestern University), Tomlinson Fort (Georgia Tech), and 
L. R. Ford (Rice Institute). 
By 1900, the presence of Osgood, Bacher, Byerly, and B. O. Peirce had 
made Harvard very strong in analysis. Moreover this strength was increased 
in 1898 by the addition to its faculty of Charles Leonard Bouton (18601922), 
who had just written a Ph.D. thesis with Sophus Lie.25 However, it 
was clear that advanced instruction in other areas of mathematics, mostly 
given before 1900 by J. M. Peirce, needed to be rejuvenated by new ideas. 
The first major step in building up a balanced curriculum was taken by 
Bacher. In the 1890s, he had given with Byerly in alternate years Harvard's 
first higher geometry course (Mathematics 3) with the title "Modern methods 
in geometry -determinants". Then, in 1902-3, he inaugurated a new version 
of Mathematics 3, entitled "Modern geometry and modern algebra", with 
a very different outline leading up to "the fundamental conceptions in the 
theory of invariants." The algebraic component of this course matured into 
Bacher's book, Introduction to Higher Algebra (1907), in which §26 on "sets, 
systems, and groups" expresses modern algebraic ideas. This book would 
introduce a generation of American students to linear algebra, polynomial 
algebra, and the theory of elementary divisors. But to build higher courses 
on this foundation, without losing strength in analysis, would require new 
faculty members. 

7. COOLIDGE AND HUNTINGTON 
Harvard's course offerings in higher geometry were revitalized in the first 
decades of this century by the addition to its faculty of Julian Lowell Coolidge 


MATHEMATICS AT HARVARD, 1836-1944 

(1873-1954). After graduating from Harvard (summa cum laude) and Balliol 
College in Oxford, Coolidge taught for three years at the Groton School before 
returning to Harvard. At Groton, he began a lifelong friendship with Franklin 
Roosevelt, which illustrates his concern with the human side of education (see 
§ 15). Indeed, somewhat like his great-great-grandfather Thomas Jefferson, 
our "most mathematical president", Coolidge was unusually many-sided.26 

From 1900 on, Coolidge gave in rotation a series of lively and informative 
graduate courses on such topics as the geometry of position, non-Euclidean 
geometry, algebraic plane curves, and line geometry. After he had spent two 
years (1902-4) in Europe and written a Ph.D. thesis under the guidance of 
Eduard Study and Corrado Segre, these courses became more authoritative. 
In time, the contents of four of them would be published as books on NonEuclidean 
Geometry (1909), The Circle and the Sphere (1916), The Geometry 
ofthe Complex Domain (1924), and Algebraic Plane Curves (1931). 

In 1909-10, Coolidge also initiated a half-course on probability (Mathematics 
9), whose (;ontents were expanded into his readable and timely Introduction 
to Mathematical Probability (1925), soon translated into German 
(Teubner, 1927). Coolidge's informal and lively expository style is well illustrated 
by his 1909 paper on "The Gambler's Ruin".27 This concludes by 
reminding the reader of "the disagreeable effect on most of humanity of anything 
which refers, even in the slightest degree, to mathematical reasoning 
or calculation." The preceding books were all published by the Clarendon 
Press in Oxford, as would be his later historical books (see §19). These later 
books reflect an interest that began showing itself in the 1920s, when he wrote 
thoughtful accounts of the history of mathematics at Harvard such as [JLC] 
and [Mor, Ch. XV] which have helped me greatly in preparing this paper. 

A vivid lecturer himself, Coolidge always viewed research and scholarly 
publication as the last of four major responsibilities of a university faculty 
member. In his words [JLC, p. 355], these responsibilities were: 

1. To inject the elements of mathematical knowledge into a large number 
of frequently ill informed pupils, the numbers running up to 500 each year. 
Mathematical knowledge for these people has come to mean more and more 
the calculus. 
2. To provide a large body of instruction in the standard topics for a 
College degree in mathematics. In practice this is the one of the four which 
it is hardest to maintain. 
3. To prepare a number of really advanced students to take the doctor's 
degree, and become university teachers and productive scholars. The number 
of these men slowly increased [at Harvard] from one in two or three years, 
to three or four a year. 
4. To contribute fruitfully to mathematical science by individual research. 

GARRETT BIRKHOFF 

Coolidge's sprightly wit and his leadership as an educator led to his election 
as president of the Mathematical Association (MAA) of America in the mid1920s, 
during which he also headed a successful fund drive of the American 
Mathematical Society [Arc, pp. 30-32]. 

An important Harvard contemporary of Coolidge was Edward Vermilye 
Huntington (1874-1952). After completing graduate studies on thefoundations 
of mathematics in Germany, he began a long career of down-to-earth 
teaching, at first under the auspices of the Lawrence Scientific School. Concurrently, 
he quickly established a national reputation for clear thinking by 
definitive research papers on postulate systems for groups, fields, and Boolean 
algebra. These are classics, as is his lucid monograph on The Continuum and 
Other Types ofSerial Order (Harvard University Press 1906; 2d ed., 1917). 

From 1907-8 on, he gave biennially a course (Mathematics 27) on "Fundamental 
Concepts of Mathematics", cross-listed by the philosophy deptartment 
(see the end of §8), which introduced students to abstract mathematics. He 
also became coauthor in 1911 (with Dickson, Veblen, Bliss, and others) of 
the thought-provoking survey Fundamental Concepts ofModern Mathematics 

(J. W. Young, ed.); 2d ed. 1916. This survey still introduced mathematics 
concentrators to 20th century axiomatic mathematics when I began teaching, 
25 years later. It is interesting to compare this book with Bacher's address on 
"The Fundamental Conceptions and Methods of Mathematics" (Bull. Amer. 
Math. Soc. 11 (1904), 11-35), and with §26 of his Introduction to Higher 
Algebra. 
In the 1 920s, Huntington broadened his interests. Four years after making 
"mathematics and statistics" the subject of his retiring presidential address 
to the MAA (A mer. Math. Monthly 26 (1919), 421-35), he began teaching 
statistics in Harvard's Faculty of Arts and Sciences. Offered initially in 1923 
as a replacement to a course on interpolation and approximation given earlier 
(primarily for actuaries) by Bacher and L. R. Ford, it was given biennially 
from 1928 on as a companion to the course on probability for which Coolidge 
wrote his book. 

Finally, as a related sideline, he invented in 1921 a method of proportions 
for calculating how many representatives in the U.S. Congress each state is 
entitled to, on the basis of its population.28 This method successfully avoids 
the "Alabama paradox" and the "population paradox" that had flawed the 
methods previously in use. Adopted by Congress in 1943, it has been used 
successfully by our government ever since. 

8. PASSING ON THE TORCH 
As I tried to explain in §5, the mathematics courses above freshman level 
offered at Harvard in the l870s and l880s could be classified into two main 


MATHEMATICS AT HARVARD, 1836-1944 21 

groups: (i) courses on the calculus and its applications in the tradition of Benjamin 
Peirce's texts (including his Analytical Mechanics), designed to make 
books on classical mathematical physics (Poisson, Fourier, Maxwell) readable, 
and (ii) courses on topics in algebra and geometry related to the later 
research of Benjamin and C. S. Peirce. Broadly speaking, Byerly and B. O. 
Peirce revitalized the courses in the first group with their new Mathematics 
10, while J. M. Peirce made comprehensible those of the second. It was primarily 
J. M. Peirce's courses that Coolidge and Huntington replaced, giving 
them new content and new emphases. 

The first major change in the mathematics courses at Harvard initiated 
by Bacher and Osgood concerned Mathematics 13 and its new sequels, and 
these changes bear a clear imprint of the ideas of Riemann, Weierstrass, and 
Felix Klein, who had "passed the torch" to his enthusiastic young American 
students. We have already discussed this change in §6. 

The emphasis on "the theory of invariants" in Bacher's revitalized Mathematics 
3 and his Introduction to Higher Algebra (cf. §6) also reflects Felix 
Klein's influence, while the emphasis on "elementary divisors" clearly stems 
from Weierstrass. It is much harder to trace the evolution of ideas about the 
foundations of mathematics. In § 11 of his article in the Ann. of Math. 6 
(1905), 151-89, Huntington clearly anticipated the modern concepts of relational 
structure and algebraic structure, as defined by Bourbaki, far more 
clearly than Bacher had in his 1904 article on "The Fundamental Conceptions 
and Methods of Mathematics", and probably influenced §26 of Bacher's 
Introduction to Higher Algebra. However, it would be hard to establish clearly 
the influence of this pioneer work. Indeed, although supremely important for 
human culture, the evolution of basic ideas is nearly impossible to trace reliably, 
because each new recipient of an idea tends to modify it before "passing 
it on". 

c. S. Peirce, conclusion. This principle is illustrated by the evolution of 
two major ideas of C. S. Peirce: his philosophical concept of "pragmatism", 
and his ideas about the algebra of logic. Both of these ideas were transmitted 
at Harvard primarily through members of its philosophy department, as we 
shall see. 
The idea of pragmatism was apparently first suggested in a brilliant philosophicallecture 
given by C. S. Peirce at Chauncey Wright's Metaphysical Club 
in the 1870s. In this lecture, Peirce claimed that the human mind created 
ideas in order to consider the effects of pursuing different courses of action. 
This lecture deeply impressed William James (1842-1910), whose 1895 Principles 
of Psychology was a major landmark in that subject [EB 12, 1863-5]. 
During our Civil War, James had studied anatomy at the Lawrence Scientific 
School and Harvard Medical School, inspired by Jeffries Wyman and Louis 
Agassiz. After spending the years 1872-76 as an instructor in physiology at 


GARRETT BIRKHOFF 

Harvard College, and twenty more years in preparing his famous book, James 
turned to philosophy and religion. 

In 1906, James finally applied Peirce's idea to a broad range of philosophical 
problems in his Lowell Lectures on "Pragmatism... ", published in 
book form. In turn, James' lectures and writings on psychology and "pragmatism" 
strongly influenced John Dewey (1859-1952), whose philosophy dominated 
the teaching of elementary mathematics in our country during the 
first half of this century [EB 7, 346-7]. It is significant that the last three 
chapters of Bertrand Russell's History of Western Philosophy are devoted to 
William James, John Dewey, and the "philosophy of logical analysis" underlying 
mathematics, as Russell saw it. 

Peirce's concern with logic overlapped that of Huntington with postulate 
theory. Actually, C. S. Peirce was a visiting lecturer in philosophy at Harvard 
and a Lowell lecturer on logic in Boston in 1903, and Huntington's article 
on the "algebra of logic" in the Trans. A mer. Math. Soc. 5 (1904), 288-309, 
contains a deferential reference to Peirce's 1880 article on the same subject, 
and a letter from Peirce which totally misrepresents the facts, and shows how 
far he had slipped since 1881. The facts are as follows. 

Never analyzed critically at Harvard, Peirce's pioneer papers on the algebra 
of relations and his 1881 article basing Boolean algebra on the concept 
of partial order inspired the German logician Ernst Schroder. First in his 
Operationskreis des Logikkalkuls, and then in his three volume Algebra der 
Logik (1890-95), Schroder made a systematic study of Peirce's papers. In 
turn, these books stimulated Richard Dedekind to investigate the concept of 
a "Dualgruppe" (lattice; see §16), in two pioneer papers which were ignored 
at the time. 

Although Huntington did impart to Harvard students many of the other 
fundamental concepts of Dedekind, Cantor, Peano and Hilbert, transmitting 
them in his course Mathematics 27 and to readers of the books cited in §7, 
he paid little attention, if any, to this work of SchrOder and Dedekind. 

Indeed, it was primarily through Josiah Royce that the ideas of C. S. Peirce 
had any influence at Harvard. Royce, whose interests were many-sided, made 
logic the central theme of his courses. In turn, he influenced H. M. Sheffer 

(A.B. '05) and C. I. Lewis (A.B. '06), two distinguished logicians who wrote 
Ph.D. theses with Royce and later became members of the Harvard philosophy 
department (see §12). 
Royce also influenced Norbert Wiener, who wrote a Ph.D. thesis comparing 
Schroder's algebra of relations with that of Whitehead and Russell at Harvard 
in 1913, and later became one of America's most famous mathematicians. 
Indeed, an examination of the first 332 pages of Wiener's Collected Works 


MATHEMATICS AT HARVARD, 1836-1944 

(MIT Press, 1976) shows that until 1920 he felt primarily affiliated with 
Harvard's philosophy department. 

9. FROM ELIOT TO LOWELL 
As the preceding discussion indicates, great advances were made at Harvard 
in mathematical teaching and research during Eliot's tenure as president 
(1869-1909). However, besides many ambitious mathematical courses, Harvard 
also offered in 1900 a number of very popular 'gut' courses. After 
30 years of President Eliot's unstructured "free elective" system, it became 
possible to get an A.B. from Harvard in three years with relatively little effort. 
Moreover, whereas athletic excellence was greatly admired by students, 
scholastic excellence was not. Someone who worked hard at his studies might 
be called a "greasy grind", and a social cleavage had developed between "the 
men who studied and those who played".29 

Abbott Lawrence Lowell, who himself became the world's leading authority 
on British government without attending graduate school,3o had in 1887 
drawn attention "to the importance of making the undergraduate work out 
... a rational system of choosing his electives ... [with] the benefit of the 
experience of the faculty" [Low, p. 11]. Fifteen years later, he spearheaded 
in 1901-2 a faculty committee whose purpose was to reinstate intellectual 
achievement as the main objective of undergraduate education ([Yeo, Ch. 
V], [Mor, xlv-xlvi]). After six more years of continuing faculty discussions 
in which Osgood and Bacher were both active [Yeo, pp. 77-78], and many 
votes, Eliot appointed in 1908 a committee selected by Lowell "to consider 
how the tests for rank and scholarly distinction in Harvard College can be 
made a more generally recognized measure of intellectual power" [Yeo, p. 
80]. In 1909 Lowell succeeded Eliot as president at the age of 52. 

In his inaugural address [Mor, pp. lxxix-lxxxviii], Lowell outlined his 
plan of concentration and distribution, stating that a college graduate should 
"know a little of everything and something well" [Low, p. 40]. Having in 
mind the examples of Oxford and Cambridge Universities, he also proposed 
creating residential halls (at first for freshmen) to foster social integration. I 
shall discuss the fruition of these and other educational reforms of Lowell's 
in §12 below. His ideas have been expressed very clearly by himself and by 
Henry Yeomans,31 his colleague in the government department and frequent 
companion in later life. For the moment, I shall describe only some major 
changes in undergraduate mathematics at Harvard which he encouraged, that 
took place during the years 1906-29. 

Calculus instruction. During its lifetime (1847-1906), the Lawrence Scientific 
School had shared in the teaching of elementary mathematics at Harvard. 
In 1910, during its transition into a graduate school of engineering 


GARRETT BIRKHOFF 

(completed in 1919), this responsibility was turned over to the mathematics 
department, doubling the latter's elementary teaching load. At the time, 
"nine-tenths of all living [Harvard] graduates who took an interest in mathematics 
at college got their inspiration from Mathematics C," which then 
covered only analytic geometry through the conic sections. 

This seemed deplorable to Lowell, who knew that the calculus, its extensions 
to differential equations, differential geometry, and function theory, and 
its applications to celestial mechanics, physics, and engineering, had dominated 
the development of mathematics ever since 1675. Aware of this domination, 
he sometimes identified the phonetic alphabet, the Hindu-Arabic 
decimal notation for numbers, symbolic algebra, and the calculus, as the 
four most impressive inventions of the human mind. 

Lowell soon persuaded the faculty to require each undergraduate to take 
for "distribution" at least one course in mathematics or philosophy, presumably 
to develop power in abstract thinking. Through the visiting committee 
of the Harvard mathematics department (see below), he also encouraged devoting 
substantial time in Mathematics C to the calculus. Within a decade, 
"half of the Freshman course was devoted to the subject [of the calculus], and 
in 1922 the Faculty of Arts and Sciences, through the President's deciding 
vote, passed a motion that no mathematics course where the calculus was not 
taught would be counted for distribution" [Mor, p. 255]. This change was 
followed by steadily increasing emphasis (at Harvard) on the calculus and 
its applications, until "In 1925-26, 327 young men, just out of secondary 
school, were receiving a half-year of instruction in the differential calculus" 
[Mor, p. 255]. 

Visiting Committees. Since 1890, the activities of each Harvard department 
have been reviewed by a benevolent visiting committee, which reports 
triennially to the board of overseers. Beginning in 1906, Lowell's brother-inlaw 
William Lowell Putnam played a leading role on the visiting committee 
of the mathematics department, and in 1912, Lowell invited George Emlen 
Roosevelt, a first cousin of Franklin Delano Roosevelt, to join it as well.32 
Both men had been outstanding mathematics students, and their 1913 report 
with George Leverett and Philip Stockton contained "the important suggestion 
that the bulk of freshmen be taught in small sections" [Mor, p. 254]. 

This new plan allowed an increasing number of able graduate students 
in mathematics to be self-supporting by teaching elementary courses (based 
on Osgood's texts). For example, during the years 1927-40, S. S. Cairns, 

G. A. Hedlund, G. Baley Price, C. B. Morrey, T. F. Cope, J. S. Frame, D. C. 
Lewis, Sumner Myers, J. H. Curtiss, Walter Leighton, Arthur Sard, John W. 
Calkin, Ralph Boas, Herbert Robbins, R. F. Clippinger, Lynn Loomis, Philip 
Whitman, and Maurice Heins served in this role. At the same time, a few 
outstanding new Ph.Do's were invited to participate in Harvard's research environment 
by becoming Benjamin Peirce instructors. Among these, one may 

MATHEMATICS AT HARVARD, 1836-1944 

mention John Gergen, W. Seidel, Magnus Hestenes, Saunders Mac Lane, Holbrook 
MacNeille, Everett Pitcher, Israel Halperin, John Green, Leon Alaoglu, 
and W. J. Pettis in the decade preceding World War II. 

Besides giving benign and wise advice, the visiting committees of the mathematics 
department established and financed for many decades a departmental 
library, where for at least seventy years the bulk of reading in advanced 
mathematics has taken place. Among the many grateful users of this library 
should be recorded George Yale Sosnow. More than 60 years after studying 
mathematics in it around 1920, he left $300,000 in his will to endow its 
expansion and permanent maintenance. 

10. GEORGE DAVID BIRKHOFF 
A major influence on mathematics at Harvard from 1912 until his death 
was my father, George David Birkhoff (1884-1944). His personality and 
mathematical work have been masterfully analyzed by Marston Morse in 
[GDB, vol. I, xxiii-lvii], reprinted from Bull. Amer. Math. Soc. 52 (1946), 
357-91. Moreover I have already sketched some more personal aspects of 
his career in [LAM, §7 and §§14-15]. Therefore, I will concentrate here on 
his roles at Harvard. 

When my father entered Harvard as a junior in 1903, he had already been 
thinking creatively about geometry and number theory for nearly a decade. 
According to his friend, H. S. Vandiver [Van, p. 272] "he rediscovered the 
lunes of Hippocrates when he was ten years old". In this connection, I still 
recall him showing my sister and me how to draw them with a compass (see 
Fig. 1) when I was about nine, joining the tips of these lunes with a regular 
hexagon, and mentioning that with ingenuity, one could construct regular 
pentagons by analogous methods. By age 15, he had solved the problem 
(proposed in the Amer. Math. Monthly) of proving that any triangle with 
two equal angle bisectors is isosceles. 

Before entering Harvard, he had proved (with Vandiver) that every integer 
bn 26 16 

an -(n > 2) except 63 = -has a prime divisor p which does not 
divide ak -b k for any proper divisor k of n. He had also reduced the question 
of the existence of solutions of xm yn + ym zn + zm xn = °(m, n not both even) 
to the Fermat problem of finding nontrivial solutions of ul + VI + WI = 0, 
where t = m2 -mn+n2. Indeed, he had already begun his career as a research 
mathematician when he entered the University of Chicago in 1902. There 
he soon began a lifelong friendship with Oswald Veblen, a graduate student 
who had received an A.B. from Harvard (his second) two years earlier.33 

I have outlined in [LAM, §7] some high points of my father's career during 
the final "formative years" in Cambridge, Chicago, Madison, and Princeton 
that preceded his return to Cambridge. He himself has described with feeling, 
in [GDB, vol. III, pp. 274-5], his intellectual debt to E. H. Moore, Bolza, 


GARRETT BIRKHOFF 

FIGURE 1 

and Bacher, thanking Bacher "for his suggestions, for his remarkable critical 
insight, and his unfailing interest in the often crude mathematical ideas which 
I presented". It was presumably under the stimulus of Bacher (and perhaps 
Osgood) that he wrote his first substantial paper (Trans. Amer. Math. Soc. 
7 (1906), 107-36), entitled "General mean value and remainder theorems". 
The questions raised and partially answered in this are still the subject of 
active research.34 Moreover his 1907 Ph.D. thesis, on expansion theorems 
generalizing Sturm-Liouville series, was also stimulated by Bacher's ideas 
about such expansions, at least as much as by those of his thesis adviser, 

E. 
H. Moore, about integral equations. 
Return to Harvard. As Veblen has written [GDB, p. xvii], my father's 
return in 1912 as a faculty member to Harvard, "the most stable academic 
environment then available in this country," marked "the end of the formative 
period of his career". He had just become internationally famous for his 
proof of Poincare's last geometric theorem. Moreover Bacher had devoted 
much of his invited address that summer at the International Mathematical 


MATHEMATICS AT HARVARD, 1836-1944 

Congress in Cambridge, England, to explaining the importance and depth of 
my father's work on boundary value problems for ordinary differential equations. 
Equally remarkable, my father had been chosen to review for the Bull. 
Amer. Math. Soc. (17, pp. 14-28) the "New Haven Colloquium Lectures" 
given by his official thesis supervisor, E. H. Moore, and Moore's distinguished 
Chicago colleagues E. J. Wilczynski and Max Mason. 

It is therefore not surprising that, in his first year as a Harvard assistant 
professor, he and Osgood led a seminar in analysis for research students, 
or that he remained one of the two leaders of this seminar until 1921. By 
that time, it "centered around those branches of analysis which are related 
to mathematical physics". This statement reflected interest in the theory of 
relativity (see §11). It may seem more surprising that the reports of the 
visiting committee of 1912 and 1913 took no note of this unique addition 
to Harvard's faculty, until one remembers that their main concern was with 
the mathematical education of typical undergraduates! 

1912 as a milestone. By coincidence, 1912 also bisects the time interval 
from 1836 to 1988, and so is a half-way mark in this narrative. It can also 
be viewed as a milestone marking the transition from primary emphasis on 
mathematical education at Harvard to primary emphasis on research. Since 
Byerly retired and B. O. Peirce died in 1913, it also marks the end of Benjamin 
Peirce's influence on mathematics at Harvard. Finally, since I was one 
year old at the time, it serves as a convenient reminder that all the changes 
that I will recall took place during two human life spans. 

During the next two decades, G. D. Birkhoff would supervise the Ph.D. 
theses of a remarkable series of graduate students. These included Joseph 
Slepian (inventor of the magnetron), Marston Morse, H. J. Ettlinger, J. L. 
Walsh, R. E. Langer, Carl Garabedian (father of Paul), D. V. Widder, H. W. 
Brinkmann, Bernard Koopman, Marshall Stone, C. B. Morrey, D. C. Lewis, 

G. Baley Price, and Hassler Whitney. Four of them (Morse, Walsh, Stone, 
and Morrey) would become AMS presidents. 
In retrospect, my father's role in bringing topology to Harvard (as Veblen 
did to Princeton), at a time just after L. E. J. Brouwer had proved some 
of its most basic theorems rigorously, seems to me especially remarkable. 
So does his early introduction to Harvard of functional analysis, through 
his 1922 paper with O. D. Kellogg on "Invariant points in function space", 
his probable influence on Stone and Koopman, and his "pointwise ergodic 
theorem" of 1931. But deepest was probably his creative research on the 
dynamical systems of celestial mechanics. It was to present this research that 
he was made AMS colloquium lecturer in 1920, and to honor it that he was 
awarded the first Bacher prize in 1922. 

It is interesting to consider my father's related work on celestial mechanics 
as a continuation of the tradition of Bowditch and Benjamin Peirce, which 


George David Birkhoff 


MATHEMATICS AT HARVARD, 1836-1944 

was carried on by Hill and Newcomb, and after them by E. W. Brown at Yale. 
Brown became a president of the AMS, and my father was happy to teach his 
course on celestial mechanics one year in the early 1920s, and to coauthor 
with him, Henry Norris Russell, and A. o. Lorchner a Natural Research 
Council Bulletin (#4) on "Celestial Mechanics". This document provides 
a very readable account of the status of the theory from an astronomical 
standpoint as of 1922, including the impact of Henri Poincare's Methodes 
nouvelles de la Mecanique celeste. 35 

Although my father's lectures were not always perfectly organized or models 
of clarity, his contagious enthusiasm for new mathematical ideas stimulated 
students at all levels to enjoy thinking mathematically. He also enjoyed 
considering all kinds of situations and phenomena from a mathematical 
standpoint, an aspect of his scientific personality that I shall take up 
next. 

11. MATHEMATICAL PHYSICS 
Among research mathematicians, my father will be longest remembered 
for his contributions to the theory of dynamical systems (including his ergodic 
theorem), and his work on linear ordinary differential and difference 
equations. These were admirably reviewed by Marston Morse in [GDB, I, 
pp. xv-xlix; Bull. A mer. Math. Soc. 52, 357-83], and it would make little 
sense for me to discuss them further here. At Harvard, however, there were 
very few who could appreciate these deep researches, and so from 1920 on, 
my father's ideas about mathematical physics and the philosophy of science 
aroused much more interest. These were also the themes of his invited addresses 
at plenary sessions of the International Mathematical Congresses of 
1928 and 1936, and of most of his public lectures. Accordingly, I shall concentrate 
below on these aspects of his work (cf. Parts V and VI of Morse's 
review). 

Relativity. Of all my father's "outside" interests, the most durable concerned 
Einstein's special and general theories of relativity. Unfortunately, 
it is also this interest that has been least reliably analyzed. Thus Morse's 
review suggests that it began in 1922, whereas in fact his 1911 review of 
Poincare's Gottingen lectures concludes with a discussion of "the new mechanics" 
of Einstein's special theory of relativity (cf. [GDB, III, pp. 193-4] 
and Bull. A mer. Math. Soc. 17, pp. 193-4). Moreover, he had touched on 
these theories and discussed "The significance of dynamics for general scientific 
thought" at length in his 1920 colloquium lectures,36 before initiating 
in 1921-22 an "intermediate level" course on "space, time, and relativity" 
(Mathematics 16) having second-year calculus as its only prerequisite. He 
promptly wrote (with the cooperation of Rudolph Langer) a text for this 
course, entitled Relativity and Modern Physics (Harvard University Press, 


GARRETT BIRKHOFF 

1923, 1927). In 1922, he also gave a series of public Lowell lectures on 
relativity. Two years later, he gave a similar series at U.C.L.A. (then called 
"the Southern Branch of the University of California"), and edited them into 
a book entitled The Origin, Nature, and Influence ofRelativity (Macmillan, 
1925). It was not until 1927 that he finally published in book form his deep 
AMS colloquium lectures, in a book Dynamical Systems, which omitted many 
of these topics which he had presented orally seven years earlier. 

Bridgman, Kemble, van Vleck. My father's interest in relativity and the 
philosophy of science was shared by his friend and contemporary Percy W. 
Bridgman ( 1882-1961). (Bridgman's notes of 1903-4 on B. O. Peirce's Mathematics 
10 are still in the Harvard archives, and it seems likely that my father 
attended the same lectures.) Bridgman would get the Nobel prize 25 years 
later for his ingenious experiments on the "physics of high pressure", his own 
research specialty, but in the 1920s he amused himself by writing the classic 
book on Dimensional Analysis (1922, 1931), by giving a half-course on "electron 
theory and relativity", and writing a thought-provoking book on The 
Logic ofModern Physics (1927). The central philosophical idea of this book, 
that concepts should be examined operationally, in terms of how they relate 
to actual experiments, is reminiscent of the pragmatism of William James 
and C. S. Peirce. 

In 1916, Bridgman had supervised a doctoral thesis on "Infra-red absorption 
spectra" by Edwin C. Kemble which (as was required by the physics 
department at that time) included a report on experiments made to confirm 
its theoretical conclusions. Five years later, Kemble supervised the thesis of 
John H. van Vleck (1899-1980), grandson of Benjamin Peirce's student John 

M. van Vleck and son of the twelfth AMS president E. B. van Vleck. This 
thesis, entitled "A critical study of possible models of the Helium atom", is 
a case study of the unsatisfactory state of quantum mechanics at that time. 
Quantum mechanics. However, in 1926, Schrodinger's equations finally 
provided satisfactory mathematical foundations for nonrelativistic mechanics, 
shifting the main focus of mathematical physics from relativity to atomic 
physics. In that same year, my father began trying to correlate his relativistic 
concept of an elastic "perfect fluid", having a "disturbance velocity equal to 
that oflight at all densities" [GDB, II, 737-63 and 876-86], with the spectrum 
of monatomic hydrogen, usually derived from Schrodinger's non-relativistic 
wave equation. Although this work was awarded an AAAS prize in 1927, of 
greater permanent value was probably his later use of the theory of asymptotic 
series to reinterpret the WKB-approximations of quantum mechanics, 
which yield classical particle mechanics in the limiting case of very short 
wave length (ibid., pp. 837-56). Related ideas about quantum mechanics 
also constituted the theme of his address at the 1936 International Congress 
in Oslo [GDB II, 857-75]. 


MATHEMATICS AT HARVARD, 1836-1944 

In the meantime, Kemble had taught me most of what I know about quantum 
mechanics. Far more important, he had just about completed his 1937 
book, The Fundamental Principles of Quantum Mechanics. The preface of 
this book mentions his "distress" at "the tendency to gloss over the numerous 
mathematical uncertainties and pitfalls which abound in the subject", and his 
own "consistent emphasis on the operational point of view". 

Like Kemble, van Vleck (Harvard Ph.D., 1922) made non-relativistic quantum 
mechanics his main analytical tool; but unlike Kemble, he attached little 
importance to its mathematical rigor. Instead, he applied it so effectively to 
models of magnetism that he was awarded a Nobel prize around 1970. As a 
junior fellow in 1934, I audited his half-course (Mathematics 39) on "group 
theory and quantum mechanics", and was startled by his use of the convenient 
assumption that every matrix is similar to a diagonal matrix. * The 
courses (Mathematics 40) on the "differential equations of wave mechanics" 
given in alternate years through 1940 by my father, must have had a very 
different flavor. 

12. PHILOSOPHY; MATHEMATICAL LOGIC 
From his philosophical analysis of the concepts of space and time, my father 
also gradually developed radical ideas about how high school geometry 
should be taught. His public lectures on relativity had included (in Chapter 
II, on "the nature of space and time") a system of eight postulates for 
plane geometry, of which the first two concern measurement. They assert 
that length and angle are measurable quantities (magnitudes, or real numbers), 
measurable by "ruler and protractor". Whereas Euclid had devoted his 
axioms to properties of such "quantities", my father saw no good reason why 
high school students should not use them freely. 

A decade later, he proposed a reduced system of four postulates for plane 
geometry, including besides these measurement postulates only two: the existence 
of a unique straight line through any two points, and the proportionality 
of the lengths of the sides of any two triangles ABC and A'B' C' having equal 
corresponding interior angles. His presentation to the National Council of 
Teachers of Mathematics two years earlier had included a fifth postulate: that 
"All straight angles have the same measure, 1800 ."37 This presentation was 
coauthored by Ralph Beatley of Harvard's Graduate School of Education, 
and their ideas expanded into an innovative textbook on Basic Geometry 
(Scott, Foresman, 1940, 1941). 

Less innovative analogous texts on high-school physics and chemistry, 
coauthored by N. Henry Black of Harvard's Education School with Harvey 
*Of course, every finite group of complex matrices is similar to a group of unitary matrices. 


GARRETT BIRKHOFF 

Davis and James Conant,38 respectively, had been widely adopted. However, 
perhaps because it came out just before World War II, the book by 

G. D. Birkhoff and Beatley never achieved comparable success. 
Expanded from 4 (or 5) postulates to 23, and from 293 pages to 578, 
G. D. Birkhoff's idea of allowing high-school students to assume that real 
numbers express measurements of distance and angles was developed by 
E. E. Moise and F. L. Downs, Jr. into a commercially successful text Geometry 
(Addison-Wesley, 1964). 
Aesthetic Measure. According to Veblen, my father "was already speculating 
on the possibility of a mathematical theory of music, and indeed of 
art in general, when he was in Princeton" (in 1909-12). At the core of his 
speculations was the formula 

(1) 
where the Oi are pleasing, suitably weighted elements of order, the Cj suitably 
weighted elements of complexity, intended to express the effort required to 
"take in" the given art object, and M is the resulting aesthetic measure (or 
"value"). Attempts were made to quantify (1) by David Prall at Harvard and 
others, through psychological measurements; those interested in aesthetics 
should read G. D. Birkhoff's book Aesthetic Measure (Harvard University 
Press, 1933). See also his papers reprinted in [GDB, III, pp. 288-307, 32034, 
382-536, and 755-838], the first of which constitutes his invited address 
at the 1928 International Congress in Bologna. 

Of my father's last five papers (##199-203 in [GDB, vol. iii, p. 897]), one 
is concerned with quaternions and refers to Benjamin and C. S. Peirce; a 
second with axioms for one-dimensional "geometries"; and a third with generalizing 
Boolean algebra. His enthusiasm for analyzing basic mathematical 
structures and recognizing their interrelations never flagged. 

Like the relativistic theory of gravitation in flat space-time which was his 
dominant interest in the last years of his life (see §20), these speculative contributions 
are less highly appreciated by most professional mathematicians 
today than his technical work on dynamical systems. However, they made 
him more interesting to the undergraduates in his classes, his tutees, and his 
colleagues on the Harvard faculty. In particular, they contributed substantially 
to his popularity as dean of the faculty, and to the high esteem in which 
he was held by President Lowell and the Putnam family.39 They must have 
also influenced his election as president of the American Association for the 
Advancement of Science. 

A. N. Whitehead. My father's ventures into mathematical physics, the 
foundations ofgeometry, and mathematical aesthetics were comparable to the 
ventures into relativity, the foundations of mathematics, and mathematical 
logic of A. N. Whitehead, who joined Harvard's philosophy department in 

MATHEMATICS AT HARVARD, 1836-1944 

1924. The Whiteheads lived two floors above my parents at 984 Memorial 
Drive, and were very congenial with them. 

The situation had changed greatly since 1910, when Josiah Royce was the 
only Harvard philosopher who found technical mathematics interesting, and 
(perhaps because of William James) Harvard's courses in psychology were 
given under the auspices of the philosophy department. In the 1920s and 
1930s, not only Whitehead, but also C. I. Lewis (author of the Survey ofSymbolic 
Logic) and H. M. Sheffer of the philosophy department (cf. §8) were 
important mathematical logicians. Moreover Huntington's course Mathematics 
27 on "Fundamental concepts ... " (cf. §7) was cross-listed for credit in 
philosophy, and there was even a joint field of concentration in mathematics 
and philosophy. 

In the 1920s, mathematical logic was a bridge connecting mathematics and 
philosophy, making the former seem more human and the latter more substantial. 
Whitehead and Russell's monumental Principia Mathematica was 
considered in the English-speaking world to have revolutionized the foundations 
of mathematics, reducing its principles to rules governing the mechanical 
manipulation of symbols. In particular, its claim to have made axioms 
"either unnecessary or demonstrable" was widely accepted by both mathematicians 
and philosophers.4o 

In the following decade, G6del and Turing would revolutionize ideas about 
the role and significance of mathematical logic; the Association for Symbolic 
Logic would be formed; and the subject would gradually become detached 
from the rest of mathematics, concentrating more and more on its own internal 
problems. However, the addition of W.V. Quine to the Harvard philosophical 
faculty, and the presence in Cambridge of Alfred Tarski for several 
years, continued to stimulate fruitful interchanges of ideas until long after 
World War II. 

13. POSTWAR RECRUITMENT 
The retirement of Byerly in 1913 and the death of B. O. Peirce in 1914, together 
with the deaths ofB6cher and G. M. Green, and the departure of Dunham 
Jackson after six years as secretary in 1919,41 created a serious void in 
Harvard mathematics. This void was filled slowly, at first (in 1920) by Oliver 

D. Kellogg (1878-1932), and William C. Graustein (1897-1942), who had 
earned Ph.D.'s in Germany before the war with Hilbert and Study, respectively_ 
Then came Joseph L Walsh (1895-1973) in 1921, and (after H. W. 
Brinkmann in 1925) H. Marston Morse (1892-1977) in 1926. Both Walsh's 
and Morse's Ph.D. theses had been supervised py my father.42 Like Osgood, 
B6cher, Coolidge, Huntington, and Dunham Jackson, Graustein (A.B. 1910) 
and Walsh (S.B. 1916) had been Harvard undergraduates. 

GARRETT BIRKHOFF 

Kellogg immediately modernized and infused new life into Mathematics 
lOa ("potential theory"), took on the teaching of Mathematics 4 (mechanics), 
and joined my father in running the seminar in analysis. The 1921-22 department 
pamphlet announced that in that seminar, "the topics assigned will 
centre about those branches of analysis which are related to mathematical 
physics". This statement was repeated for two more years, during the first 
of which Kellogg and Einar Hille (as B.P. Instructor) directed the seminar, 
a fact which confirms my impression, described in §12, that in those years it 
was relativity theory and not "dynamical systems" that seemed most exciting 
at Harvard. 

Graustein was an extremely clear lecturer and writer. His and Osgood's 
Analytic Geometry, and his texts for Mathematics 3 (Introduction to Higher 
Geometry, 1930) and Mathematics 22 (Differential Geometry, 1935), were 
models of careful exposition. Combined with Coolidge's lively lectures and 
more informal texts on special topics, they made geometry second only to 
analysis in popularity at Harvard during the years 1920-36. 

Walsh concentrated on analysis. In 1924-5, he expanded Osgood's halfcourse 
Mathematics 12 on infinite series, which had remained static for 30 
years, into a full course on "functions of a real variable" which included the 
Lebesgue integral. He also soon invented "Walsh functions",43 and became 
an authority on the approximation of complex and harmonic functions. His 
interest in this area may have been stimulated by Dunham Jackson, who had 
done distinguished work in approximation theory fifteen years earlier (see 
Trans. AMS 12). Most striking was Walsh's result that, in any bounded 
simply connected domain with boundary C, every harmonic function is the 
limit of a sequence of harmonic polynomials which converges uniformly on 
any closed set interior to C (Bull. A mer. Math. Soc. 35 (1929), 499-544). 

Brinkmann came from Stanford, where H. F. Blichfeldt had interested 
him in group representations. A year's post-doctoral stay in G6ttingen with 
Emmy Noether had not converted him to the axiomatic approach. A brilliant 
and versatile lecturer, his graduate courses were mostly on algebra and number 
theory, in which he interested J. S. Frame and Joel Brenner, see [Bre]. 
However, he also gave a course on "mathematical methods of the quantum 
theory" with Marshall Stone in 1929-30. 

Morse applied variational and topological ideas related to those of my 
father (and of Poincare before him). Just before he came to Harvard, he 
had derived the celebrated derived Morse inequalities (Trans. A mer. Math. 
Soc. 27 (1925), 345-96). The main fruit of his Harvard years was his 1934 
Colloquium volume, Calculus of Variations in the Large. The foreword of 


MATHEMATICS AT HARVARD, 1836-1944 

this volume describes admirably its connection with earlier ideas and results 
of Poincare, Bacher, my father, and my father's Ph.D. student Ettlinger.44 

14. My UNDERGRADUATE MATHEMATICS COURSES 
Most of my own undergraduate courses in mathematics were taught by 
these relatively new members of the Harvard staff, and by two other thesis 
students of my father: H. W. Brinkmann and Hassler Whitney, who joined 
the Harvard mathematics staff in the later 1920s (Whitney, Simon Newcomb's 
grandson, as a graduate student). It may be of interest to record my 
own youthful impressions of their teaching and writing styles.45 

In this connection, I should repeat that whereas my description of mathematical 
developments at Harvard before 1928 has been based largely on 
reading, hearsay, and reflection, from then on it will be based primarily on 
my own impressions during fifteen years of slowly increasing maturity. 

Shortly after joining my parents in Paris in the summer of 1928, my father 
ordered me to "learn the calculus" from a second-hand French text which he 
picked up in a bookstall along the Seine. Later that summer, after explaining 
to me Fermat's "method of infinite descent", he challenged me to prove that 
there were no (least) positive integers satisfying X4 +y4 = Z4. After making 
substantial progress, I lost heart, and felt ashamed when he showed me how 
to complete the proof in two or three more steps. 

The next fall, I was fortunate in being taught second-year calculus as a 
freshman by Morse and Whitney. Their lectures made the theory of the 
calculus interesting and intuitively clear; especially fascinating to me was their 
construction of a twice-differentiable function U(x, y) for which Uyx =f. Uxy. 
The daily exercises from Osgood's text gave the needed manipulative skill in 
problem solving. Likewise, the clarity of Osgood and Graustein made it easy 
and pleasant to learn from their Analytic Geometry, in "tutorial" reading (see 
§15), not only the reduction of conics to canonical form, but also the theory 
of determinants. 

I learned the essentials of analytic mechanics (Mathematics 4) from Kellogg 
concurrently. In his lectures Kellogg explained how to reduce systems 
of forces to canonical form, and derived the conservation laws for systems 
of particles acting on each other by equal and opposite "internal" forces. His 
presentation of Newton's solution of the two-body problem opened my eyes 
to the beauty and logic of celestial mechanics, and reinforced my interest 
in the calculus and the elementary theory of differential equations. In an 
unsolicited course paper, I also tried my hand at applying conservation laws 
to deduce the effect of spin on the bouncing of a tennis ball (I had played 
tennis with Kellogg, for several years a next door neighbor), as a function 
of its coefficient of (Coulomb) friction and its "coefficient of restitution". I 


GARRETT BIRKHOFF 

was also delighted to learn the mathematical explanation of the "center of 
percussion" of a baseball bat. 

That spring my father gave me a short informal lecture on the crucial 
difference between pointwise and uniform convergence of a sequence of functions, 
and then challenged me to prove that any uniform limit of a sequence 
of continuous functions is continuous. After I wrote out a proof (in two or 
three hours), he seemed satisfied. In any event, he encouraged me to take as 
a sophomore the graduate course on functions of a complex variable (Mathematics 
13) from Walsh, omitting Mathematics 12. This was only 15 months 
after I had begun learning the calculus. 

This was surely my most inspiring course. Walsh had a dramatic way 
of presenting delicate proofs, lowering his voice more and more as he approached 
the key point, which he would make in a whisper. Each week we 
were assigned theorems to prove as homework. As I was to learn decades 
later, our correctors were J. S. Frame, who became a distinguished mathematician, 
and Harry Blackmun, now a justice on the U. S. Supreme Court. 
They did their job most ably, conscientiously checking my homemade proofs, 
which often differed from those of the rest of the class. What a privilege it 
was! 

Concurrently, I took advanced calculus (Mathematics 5) from Brinkmann, 
who drilled a large class on triple integration, the beta and gamma functions 
and many other topics. He made concise and elegant formula derivations 
into an art form, leaving little room for student initiative. Osgood's Advanced 
Calculus supplemented Brinkmann's lectures admirably, by including 
an explanation of how to express the antiderivative JR(x, yiQ(x))dx of any 
rational function of x and the square root of a quadratic function Q(x) in 
elementary terms, good introductions to the wave and Laplace equations, etc. 
Through Brinkmann's lectures and Osgood's book, I acquired a deep respect 
for the power of the calculus, which I have always enjoyed trying to transmit 
to students. 

My junior year, I took half-courses on the calculus of variations (Mathematics 
15) from Morse, on differential geometry (Mathematics 22a) from 
Graustein, and on ordinary differential equations (Mathematics 32) from 
my father. Morse's imaginative presentation again made me conscious of 
many subtleties, especially the sufficient conditions required to prove (from 
considerations of 'fields of extremals') that solutions of the Euler-Lagrange 
equations are actually maxima or minima. Graustein, on the other hand, 
explained details of proofs so carefully that there was little left for students 
to think about by themselves. I preferred my father's lecture style, which 
included a digression on the three 'crucial effects' of the general theory of 
relativity, and a challenge to classify qualitatively the solutions of the autonomous 
DE x = F(x). (He did not suggest using the Poincare phase 
plane.) 


MATHEMATICS AT HARVARD, 1836-1944 

He mentioned in class the fact (first proved by Picard) that one cannot 
reduce to quadratures the solution of 

(14.1) U" + p(x)U' + q(x)U = O. 
Not knowing anything then about solvable groups or Lie groups, I was skeptical 
and wasted many hours in trying to find a formula for solving (14.1) by 
quadratures. 

Finally, in my senior year, I took a half-course on potential theory (Mathematics 
lOa) with Kellogg. There I found the concept of a harmonic function 
and Green's theorems exciting, but was bored by elaborate formulas for expanding 
functions in Legendre polynomials. I also took one on "analysis 
situs" (combinatorial topology) with Morse, which was an unmitigated joy, 
however, especially because of its classification of bounded surfaces, proofs 
of the topological invariance of Betti numbers, etc. The reduction of rectangular 
matrices of Os and Is to canonical form under row equivalence was 
another stimulating experience. 

15. HARVARD UNDERGRADUATE EDUCATION: 1928-42 
My own undergraduate career was strongly influenced by the philosophy 
of education developed by President Lowell. Lowell was an "elitist", who 
believed that excellence was fostered by competition, and best developed 
through a combination of drill, periodic written examinations, oral discussions 
with experts, and the writing of original essays of variable length. He 
also believed that breadth should be balanced by depth, and that originality 
was a precious gift which could not be taught. His primary educational 
aim was to foster intellectual and human development through his system of 
concentration and distribution of courses, tutorial discussions, "general examinations", 
and senior honors theses. I believe that he wanted Harvard to 
train public-spirited leaders with clear vision, who could think hard, straight, 
and deep. 

During eleven academic years, 1927-38, I slept in a dormitory, ate most 
meals with students in dining halls (from 1936 to 1938 as a tutor), usually 
participated in athletics during the afternoon, and studied in the evening. 
From 1929 on, my primary aim was to achieve excellence as a mathematician, 
and I think the Harvard educational environment of those years was ideal 
for that purpose also. After 1931, I continued to think about mathematics 
during summers, if somewhat less systematically. 

I think I have already said enough about individual mathematics courses at 
Harvard. All my teachers impressed me as trying hard to communicate to a 
mix of students a mature view of the subject they were teaching and (equally 
important) as being themselves deeply interested in it. Of even greater value 
was the encouragement I got in tutorial to do guided reading and 'creative' 


GARRETT BIRKHOFF 

thinking about mathematics and a few of its applications. These efforts were 
tactfully monitored by leading mathematicians, who were surely conscious of 
my limitations and slowly decreasing immaturity, and communicated their 
evaluations to my father. 

My first tutorial assignment was to learn about (real) linear algebra and 
solid analytic geometry as a freshman by reading the book by Osgood and 
Graustein. In Walsh's Mathematics 13, I spent the spring reading period of 
my sophomore year on a much more advanced topic: figuring out how to 
reconstruct any doubly periodic function without essential singularities from 
the array of its poles. In a junior course by G. W. Pierce on "Electric oscillations 
and electric waves", I wrote a term paper on the refraction and reflection 
of electromagnetic waves by a plane interface separating two media having 
different dielectric constants and magnetic permeabilities, and presented my 
results as one of the speakers at a physics seminar the following fall. I surely 
learned more from giving my talk than the audience did from hearing it! 

From the middle of that year on, my main tutorial efforts were devoted 
to planning and writing a senior honors thesis, for which endeavor approximately 
one-fourth of my time was officially left free. My tutorial reading for 
this began with Hausdorff's Mengenlehre (first ed.) and de la Vallee-Poussin's 
beautiful Cours d'Analyse Infinitesimale, from which I learned the foundations 
of set-theoretic topology and the theory of the Lebesgue integral, respectively. 
In retrospect, I can see that this reading and my father's oral 
examination on "uniform convergence" essentially covered the content of 
Mathematics 12 on "functions of a real variable" (cf. [Tex, p. 16]). My 
acquaintance with general topology was broadened by reading Frechet's Thesis 
(1906), which introduced me to function spaces, and his book Les Espaces 
Abstraits. It was also deepened by reading the fundamental papers of 
Urysohn, Alexandroff, Niemytski, and Tychonoff (Math. Ann., vols. 92-95). 
I was fascinated by Caratheodory's paper "on the linear measure of sets" and 
Hausdorff's fractional-dimensional measure, so brilliantly applied to fractals 
by Benoit Mandelbrot in recent decades. This reading was guided and monitored 
by Marston Morse; like all faculty members, his official duties included 
talking with each of his 'tutees' for about an hour every two weeks. 

By that time, Lowell's ambition of establishing "houses" at Harvard similar 
to the Colleges of Oxford and Cambridge had also come to fruition, 
and I became a member of Lowell House, of which Coolidge was the dedicated 
"master". Brinkmann was a resident tutor in mathematics, and J. S. 
Frame a resident graduate student. My mathematical tutorials with Morse 
were supplemented by occasional casual chats on a variety of subjects with 
these and other friendly tutors, as well as (naturally) with my father. The 
Coolidges tried to set a tone of good manners by entertaining suitably clad 
undergraduates in their tastefully furnished residence. 


MATHEMATICS AT HARVARD, 1836-1944 

Hours of study in Lowell House were relieved by lighter moments. One 
of these involved a humorous letter from President Roosevelt to Coolidge, 
which ended "... do you remember your first day's class at Groton? You 
stood up at the blackboard -announced to the class that a straight line is 
the shortest distance between two points -and then tried to draw one. All 
I can say is that I, too, have never been able to draw a straight line. I am 
sure you shared my joy when Einstein proved there ain't no such thing as a 
straight line!" 

As a senior in Lowell House, I wrote a rambling 80 page thesis centering 
around what would today be called multisets (but which I called "counted 
point-sets"), such as might arise from a parametrically defined rectifiable 
curve x(s) allowed to recross itself any number of times. Not taking the hint 
from the fact that pencilled comments by the official thesis reader ended on 
page 41, I submitted all 80 pages for publication in the AMS Transactions, 
and was shocked when a kindly letter from J. D. Tamarkin explained why it 
could not be accepted!46 

In the comfortable and well-stocked Lowell House library, I became acquainted 
with the difficulty of defining "probability" rigorously. But above 
all, in the one room departmental library funded by the visiting committee, 
I discovered Miller, Blichfeldt and Dickson's book on finite groups, and 
soon became fascinated by the problem of determining all groups of given 
finite order. There I also saw Klein's Enzyklopiidie der Mathematischen 
Wissenschaften with its awe-inspiring multivolume review of mathematics 
as a whole. After finishing my honors thesis, which touched on fractionaldimensional 
measure, I decided to see what was known about it. To my 
horror, I found everything I knew compressed into two pages, in which a 
large fraction of the space was devoted to references! Although profoundly 
impressed, I decided not to allow myself to be overawed. 

Among nearly contemporary Harvard undergraduates, I suspect that 
Joseph Doob, Arthur Sard, Joel Brenner, Angus Taylor, and Herbert Robbins 
were profiting similarly from their Harvard undergraduate education; 
human minds are at their most receptive during the years from 17 to 22. Although 
expert professorial guidance is doubtless most beneficial when given 
to students planning an academic career, and conditions today are very different 
from those of the 1930s, I think it would be hard to improve on my 
mathematical education!47 It prepared me well for a year as a research student 
at Cambridge University (see §16), after which I was ready to carry on 
three years of free research in Harvard's Society of Fellows (see §17). 

As a junior fellow, I ate regularly in Lowell House for three more years 
with undergraduates, a handful of resident Law School students, and tutors. 
I then participated actively for two more years in dormitory life, as faculty 
instructor and senior tutor of Lowell House, trying to live up to the ideals of 
intellectual communication from which I had myself benefited so much. 


GARRETT BIRKHOFF 

In retrospect, although I had pleasant human relations with my prewar 
undergraduate tutees, I fear I gave some of them an overdose of mathematical 
ideology. They decided (no doubt rightly) that mathematics as I presented 
it was simply not their 'dish of tea'! As senior tutor, I was more popular for 
being otherwise a normal and gregarious human being, and top man on the 
Lowell House squash team (# 1), than for being inspiring mathematically. 

Thesis topics. To be a good mathematics thesis adviser at any level, one 
should be acquainted with a variety of interesting possible thesis topics, and 
the mathematical thinking processes of a variety of students. At Harvard, 
a substantial fraction of theses in the 1930s dealt with such simple topics 
as relating the vibrating string and Fourier series to musical scales and harmony; 
there was (and is) a Wister Prize for excellence in "mathematics and 
music". My tutee Russell ("Rusty") Greenhood, later a financial officer at 
the Massachusetts General Hospital, got a prize for his thesis on "The X2 test 
and goodness of fit", a statistical topic about which I knew nothing. He may 
have discussed his thesis with Huntington, but all students were encouraged 
to work independently. Generally speaking, prospective research mathematicians 
chose advanced thesis topics in very pure mathematics. Thus Harry 
Pollard (A.B. '40, Ph.D. '42) wrote an impressive undergraduate thesis on 
the Riemann zeta function, which may be found in the Harvard archives. 

16. HARVARD, YALE, AND OXBRIDGE 
Harvard and Yale have often been considered as American (New England?) 
counterparts of Cambridge and Oxford universities in "old" England. 
Actually, it was John Harvard of Emmanuel College, Cambridge, who gave to 
Harvard its first endowment. More relevant to this account, the House Plan 
at Harvard and the College Plan at Yale (both endowed by Yale's Edward 
Harkness) were modelled on the educational traditions that had (in 1930) 
been evolving at "Oxbridge" for centuries. Moreover, since the time of Newton, 
Cambridge had been one of the world's greatest centers of mathematics 
and physics, and I formed as a senior the ambition of becoming a graduate 
student there. 

Fortunately for me, Lady Julia Henry endowed in 1932 four choice fellowships, 
one to be awarded by each of these four universities, to support a year's 
study across the ocean. President Lowell in person interviewed the candidates 
applying at Harvard, of whom I was one. He asked me two questions: (i) 
was I more interested in theoretical or applied mathematics? (ii) since most 
candidates seemed to want to go to Cambridge, would I accept a fellowship 
at Oxford? Thinking that being a theorist sounded more distinguished than 
being a problem-solver, I replied that my interests were theoretical. Moreover 
I knew of no famous mathematicians or physicists at Oxford, and stated that 
I would try to find another means of getting to Cambridge. 


MATHEMATICS AT HARVARD, 1836-1944 

As a research student interested in quantum mechanics, I attended Dirac's 
lectures and was given R. H. Fowler as adviser during my first term. Like 
Widder two years later [CMA, p. 82], but far less mature, I also attended the 
brilliant lectures given by Hardy in each of three terms, and sampled several 
other lecture courses. When I first met Hardy, he asked me how my father 
was progressing with his theory of esthetics. I told him with pride that my 
father's book Aesthetic Measure had just appeared. His only comment was: 
"Good! Now he can get back to real mathematics". I was shocked by his lack 
of appreciation! 

The Julia Henry Fellow from Yale was the mathematician Marshall Hall, 
who has since done outstanding work in combinatorial theory. We compared 
impressions concerning the system of Tripos Examinations used at 
Cambridge to rank students, for which Cambridge students were prepared 
by their tutors. We agreed that Cambridge students were better trained than 
we, but thought that the paces they were put through took much of the bloom 
off their originality! 

My course with E. C. Kemble at Harvard had left me with the mistaken impression 
that quantum mechanics was concerned with solving the Schrodinger 
equation in a physical universe containing only atomic nuclei and electrons. 
Dirac's lectures were much more speculative, and it was not until I heard Carl 
Anderson lecture on the newly discovered positron in the spring of 1933 that 
I realized that Dirac's lectures were concerned with a much broader concept 
of quantum mechanics than that postulated by Schrodinger's equations. 

In the meantime, I had decided to concentrate on finite group theory, and 
was transferred to Philip Hall as adviser. By that spring, I had rediscovered 
lattices (the "Dualgruppen" of Dedekind; see §8), which had also been independently 
rediscovered a few years earlier by Fritz Klein who called them 
"Verbiinde". Recognizing their widespread occurrence in "modern algebra" 
and point-set topology, I wrote a paper giving "a number of interesting applications" 
of what I called "lattice theory", and wrote my father about them. 
He mentioned my results to Oystein Ore at Yale, who had taught algebra 
to both Marshall Hall and Saunders Mac Lane. Ore immediately recalled 
Dedekind's prior work, and soon a major renaissance of the subject was under 
way. This has been ably described by H. Mehrtens in his book, Die 
Entstehung der Verbiinde, cf. also [GB, Part I]. 

In retrospect, I think that I was very lucky that Emmy Noether, Artin, 
and other leading German algebraists had not taken up Dedekind's "Dualgruppe" 
concept before 1932. As it was, by 1934 Ore had rediscovered the 
idea of C. S. Peirce (see §8), of defining lattices as partially ordered sets, and 
by 1935 he had done a far more professional job than I in applying them to 
determine the structure ofalgebras -and especially that of "groups with operators" 
(e.g., vector spaces, rings, and modules). However, by that time (in 


GARRETT BIRKHOFF 

continuing correspondence with Philip Hall) I had applied lattices to projective 
geometry, Whitney's "matroids", the logic of quantum mechanics (with 
von Neumann), and set-theoretic topology, as well as to what is now called 
universal algebra, so that my self-confidence was never shattered! 

17. THE SOCIETY OF FELLOWS 
Our modern Ph.D. degree requirements were originally designed in Germany 
to train young scholars in the art of advancing knowledge. The German 
emphasis was on discipline, and Ph.D. advisers might well use candidates as 
assistants to further their own research. Having never "earned" a Ph.D. by 
serving as a research apprentice himself, Lowell was always skeptical of its 
value for the very best minds, somewhat as William James once decried "The 
Ph.D. Octopus". Throughout his academic career, Lowell kept trying to imagine 
the most stimulating and congenial environment in which a select group 
of the most able and original recent college graduates could be free to develop 
their own ideas [Yeo, Ch. XXXII]. In his last decade as Harvard's president, 
he discussed what this environment should be with the physiologist L. J. Henderson 
and the mathematician-turn ed-philosopher A. N. Whitehead, among 
others. 

As successful models for such a select group, these very innovative men 
analyzed the traditions of the prize fellows of Trinity and Kings Colleges at 
Cambridge University, of All Souls College at Oxford, and of the Fondation 
Thiers in Paris. They decided that a group of about 24 young men (a natural 
social unit), appointed for a three year term (with possible reappointment 
for a second term), dining once a week with mature creative scholars called 
senior fellows, and lunching together as a group twice a week, would provide 
a good environment. The only other stated requirement was negative: "not 
to be a candidate for a degree" while a junior fellow. 

The proper attitude of such a junior fellow was defined in the following 
noble "Hippocratic Oath of the Scholar" [SoF, p. 31], read each year before 
the first dinner: 

'You have been selected as a member of this Society for your 
personal prospect of serious achievement in your chosen field, and 
your promise of notable contributions to knowledge and thought. 
That promise you must redeem with your whole intellectual and 
moral force. 

You will practice the virtues, and avoid the snares, ofthe scholar. 
You will be courteous to your elders who have explored to the point 
from which you may advance; and helpful to your juniors who will 
progress farther by reason of your labors. Your aim will be knowledge 
and wisdom, not the reflected glamour of fame. You will 


MATHEMATICS AT HARVARD, 1836-1944 

not accept credit that is due to another, or harbor jealousy of an 

explorer who is more fortunate. 

You will seek not a near, but a distant, objective, and you will 
not be satisfied with what you may have done. All that you may 
achieve or discover you will regard as a fragment of a larger pattern, 
which from his separate approach every true scholar is striving 
to descry. 

To these things, in joining the Society of Fellows, you dedicate 
yourself.' 

Some months later, we were informed frankly that if one out of every four 
of us had an outstanding career, the senior fellows would feel that their 
enterprise had been very successful. 

Like all institutions, Harvard's Society of Fellows has changed with the 
times. Thus junior fellows may now be women, and may use their work to 
fulfill departmental Ph.D. requirements. But the ceremony of reading the 
preceding statement to new junior fellows at their first dinner in the Society's 
rooms has not changed. 

As a junior fellow, I was so absorbed in developing my own ideas and 
in exploring the literature relating to them (especially abstract algebra, settheoretic 
topology, and Banach spaces), that I attended only two Harvard 
courses or seminars. I had studied in 1932-33 Stone's famous Linear Transformations 
in Hilbert Space, one of the three books that established functional 
analysis (the study of operators on "function spaces") as a major area 
of mathematics.48 Moreover, Whitney was rapidly becoming famous as a 
topologist with highly original ideas. Therefore, I audited Stone's course 
(Mathematics 12) on the theory of real functions, which he ran as a seminar, 
in 1933-34, and I participated actively in Whitney's seminar on topological 
groups in 1935-36. 

I also attended the weekly colloquia. At an early one of these, Stone 
announced his theorem that every Boolean algebra is isomorphic to a field 
of sets. Having proved the previous spring that every distributive lattice was 
isomorphic to a ring of sets, I became quite excited. He went on to prove 
much deeper results in the next few years, while I kept on exploring the 
mathematical literature for other examples of lattices. 

There were five mathematical junior fellows during the years 1933-44: 
John Oxtoby, Stan Ulam, Lynn Loomis, Creighton Buck, and myself. In 
addition, the mathematical logician W. V. Quine was (like me) among the 
first six selected, as was the noted psychologist B. F. Skinner. Like many 
other junior fellows, the last four of the six just named joined the Harvard 
faculty, where their influence would be felt for decades. But that is another 
story! 


GARRETT BIRKHOFF 

Ulam and Oxtoby. Instead, I will take up here the accomplishments of 
Ulam and Oxtoby through 1944. Most important was their proof that, in 
the sense of (Baire) category theory, almost every measure-preserving homeomorphism 
of any "regularly connected" polyhedron of dimension r ~ 2 is 
metrically transitive. As they observed in their paper,49 "the effect of the 
ergodic theorem was to replace the ergodic hypothesis (of Ehrenfest) by the 
hypothesis of metric transitivity (of Birkhoff)". Philosophically, therefore, 
they in effect showed that Hamiltonian systems should almost surely satisfy 
the ergodic theorem. This constituted a notable modern extension of the 
tradition of Lagrange, Laplace, Poincare, and G. D. Birkhoff. 

During World War II, like von Neumann (but full-time), Ulam worked 
at Los Alamos. There he is credited as having conceived, independently of 
Edward Teller, the basic idea underlying the H-bomb developed some years 
later. 

Two other junior fellows of the same vintage who applied mathematics to 
important physical problems after leaving Harvard were John Bardeen and 
James Fisk. After joining the Bell Telephone Laboratories in 1938, Bardeen 
went on to win two Nobel prizes. Fisk became briefly director of research of 
the Atomic Energy Commission after the war, and finally vice president in 
charge of research at the Bell Telephone Labs. I hope that these few examples 
will suggest the wisdom and timeliness of the plan worked out by Lowell, 
Whitehead, Henderson, and others, and endowed by Lowell's own fortune. 
Of the first fifty junior fellows, no less than six (Bardeen, Fisk, W. V. Quine, 
Paul Samuelson, B. F. Skinner, and E. Bright Wilson) have received honorary 
degrees from Harvardl 

The Putnam Competition. The aim of Lowell and his brother-in-law 
William Lowell Putnam, to restore undergraduate admiration for intellectual 
excellence (see §7), was given a permanent national impetus in 1938 with the 
administration of the first Putnam Competition by the Mathematical Association 
of America. For a description of the establishment of this competition, 
in which George D. Birkhoff played a major role, and its subsequent history 
to 1965, I refer you to the Amer. Math. Monthly 72 (1965), 469-83. Ofthe 
five prewar Putnam Fellowship winners, Irving Kaplansky is current director 
of the NSF funded Mathematical Sciences Research Institute in Berkeley, 
after a long career as a leading American algebraist; he and Andrew Gleason 
have recently been presidents of the AMS; while Richard Arens and Harvey 
Cohn have also had distinguished and productive research careers. All of 
them except Gleason (who joined the U.S. Navy as a code breaker in 1942) 


MATHEMATICS AT HARVARD, 1836-1944 

contributed through their teaching to the mathematical vitality of Harvard 
in the years 1938-44! 

18. FOUR NOTABLE MEETINGS 
I shall now turn to some impressions of the moods of, and Harvard's 
participation in, four notable meetings that took place in the late 1930s: 
the International Topological Congress in Moscow in 1935; the International 
Mathematical Congress in Oslo and Harvard's Tercentenary in 1936; and the 
Semicentennial meeting of the AMS in 1938. 

Lefschetz was a major organizer of the 1935 Congress in Moscow. He, 
von Neumann, Alexander and Tucker went to it from Princeton; Hassler 
Whitney, Marshall Stone, David Widder (informally) and I from Harvard. 
Whitney'S paper [CAM, pp. 97-118] describes the fruitfulness for topology of 
this Congress, an event which Widder also mentions [CAM, p. 82]. For me, it 
provided a marvellous opportunity to get first-hand impressions of the thinking 
of many mathematicians whose work I admired, above all Kolmogoroff, 
but also Alexandroff and Pontrjagin. 

Widder, Stone, and I met in Helsinki, just before the Congress, whence we 
took a wood-fired train to Leningrad. There we were greeted at the station 
by L. Kantorovich and an official Cadillac. By protocol, he took a street-car 
to his home, where he had invited us for tea, while we were driven there in 
the Cadillac. I was astonished! I would have been even more astonished had 
I realized that within two years I would be studying the work of Kantorovich 
on vector lattices (and that of Freudenthal, also at the Congress); that 20 
years later I would be admiring his book with V.I. Krylov on Approximation 
Methods ofHigher Analysis; or that in about 30 years he would get a Nobel 
prize for inventing the simplex method of linear programming, discovered 
independently by George Dantzig in our country somewhat later. 50 

Marshall Stone, infinitely more worldly wise than I, reported privately 
that evening Kantorovich's disaffection with the Stalin regime. I was astonished 
for the third time, having assumed that all well-placed Soviet citizens 
supported their government. Many of my other naive suppositions were corrected 
in Moscow. 

For example, when I expressed to Kolmogoroff my admiration for his 
GrundbegrifJe der Wahrscheinlichkeitsrechnung, he remarked that he considered 
it only an introduction to Khinchine's deeper Asymptotische Gesetze der 
Wahrscheinlichkeitsrechnung. The algebraist Kurosh and I made a limited 
exchange of opinions in German, and I also met at the Congress I. Gelfand, 
who would get an honorary degree at Harvard 50 years later! 


GARRETT BIRKHOFF 

Above all, I was impressed by the crowding and poverty I saw in Moscow 
(the famine had just ended a year earlier), and the inaccessibility of government 
officials behind the Kremlin walls. 

At the International Mathematical Congress in Oslo a year later, I was 
dazzled by the depth and erudition of the invited speakers, and the panorama 
of fascinating areas of research that their talks opened up. I was permitted 
to present three short talks (Marcel Riesz gave four!), and there seemed to be 
an adequate supply of listeners for all the talks presented. Paul Erdos gave 
one talk, and he must have been the only speaker who did not wear a necktie! 

Naturally, I was pleased that the two Fields medallists (Lars Ahlfors and 
Jesse Douglas) were both from Cambridge, Massachusetts, and delighted that 
the 1940 International Congress was scheduled to be held at Harvard, with 
my father as Honorary President! I was also impressed by the efficient organization 
for the Zentralblatt of reviews of mathematical papers displayed by 
Otto Neugebauer (cf. fLAM, §21]). This convinced me of the desirability of 
transplanting his reviewing system to AMS auspices, if funds could be found 
to cover the initial cost. Of course, this was accomplished three years later. 

On both my 1935 and 1936 trips to Europe, I stopped off in Hamburg to 
see Artin in Hamburg. In 1935, I also stopped off in Berlin to meet Erhard 
Schmidt and my future colleague Richard Brauer and his brother Alfred. 
Near Hamburg in 1936, the constant drone of military airplanes made me 
suddenly very conscious of the menace of Hitler's campaign of rearmament! 

The serene atmosphere of Harvard's Tercentenary celebration that September 
was a welcome contrast, and I naturally went to the invited mathematical 
lectures. Among them, Hardy's famous lecture on Ramanujan was most 
popular.51 It did not bother me that the technical content of the others was 
over my head, and I dare say over the heads of the vast majority of the large 
audiences present! 

The summer meeting of the AMS was held at Harvard in conjunction 
with this Tercentenary; its description in the Bull. Amer. Math. Soc. (42, 
761-76) states that: "Among the more than one thousand persons attending 
the meetings ... , approximately eight hundred registered, of whom 443 are 
members of the Society". What a contrast with the Harvard of John Farrar 
and Nathaniel Bowditch, a hundred years earlier! 

A fourth notable mathematical meeting celebrated the Golden Jubilee of 
the AMS at Columbia University in September, 1938. It was to celebrate this 
anniversary that R. C. Archibald wrote the historical review [Arc] on which 
I have drawn so heavily, here and in [LAM], and that my father surveyed 
"Fifty years of American mathematics" from his contemporary standpoint. 

The meeting honored Thomas Scott Fiske of Columbia, who had by then 
attended 164 of the 352 AMS meetings that had taken place. (Of these 352 
meetings, 221 had been held at Columbia.) A review of the occasion was 


MATHEMATICS AT HARVARD, 1836-1944 

published in the Bull. A mer. Math. Soc. 45 (1939), 1-51, including Fiske's 
reminiscence that, in the early days of the AMS, C. S. Peirce was "equally 
brilliant, whether under the influence of liquor or otherwise, and his company 
was prized . .. so he was never dropped . .. even though he was unable to pay 
his dues." 

19. ANOTHER DECADE OF TRANSITION 
In §12 and §13, I recalled the mathematical activity in physics and philosophy 
at Harvard through 1940. I shall now give some impressions of the main 
themes of research and teaching of the Harvard mathematics department 
from 1930 through 1943. 

During these years, it was above all G. D. Birkhoffwho acted as a magnet 
attracting graduate students to Harvard. After getting an honorary degree 
from Harvard in 1933, he served as dean of the faculty under President 
Conant from 1934 to 1938, meanwhile being showered with honorary degrees 
and elected a member of the newly founded Pontifical Academy. He directed 
the theses of C. B. Morrey, D. C. Lewis, G. Baley Price, Hassler Whitney, 
and 12 other Harvard Ph.D.'s after 1930. In 1935, he wrote with Magnus 
Hestenes an important series of papers on natural isoperimetric conditions in 
the calculus of variations, and throughout the 1930s he wrote highly original 
sequels to his earlier papers on dynamical systems, the four color theorem, 
etc., while continuing to lecture to varied audiences also on relativity, his 
ideas about quantum mechanics, and his philosophy of science. 

Meanwhile, Walsh and Widder pursued their special areas of research in 
classical analysis, Walsh publishing many papers as well as a monograph on 
"Approximation by polynomials in the complex domain" in the tradition of 
Montel, Widder his well-known Laplace Transform. Variety within classical 
analysis and its applications was provided at Harvard by Walsh and Widder. 
For example, Joseph Doob and Lynn Loomis wrote theses with Walsh, while 
Ralph Boas and Harry Pollard wrote theses with Widder during these years. 
While Ahlfors was there (from 1935 to 1938), Harvard's national leadership 
in classical analysis was even more pronounced, being further strengthened 
by the presence of Wiener in neighboring MIT. 

Coolidge, Graustein, and Huntington continued to give well-attended 
courses on geometry and axiomatic foundations, keeping these subjects very 
much alive at Harvard. In particular, Coolidge gave a series of Lowell lectures 
on the history of geometry, while Graustein published occasional papers on 
differential geometry, and served as editor of the Transactions Amer. Math. 
Soc. from 1936 until his death in 1941. In his role of associate dean, 

Graustein also worked out a detailed "Graustein plan" which metered skillfully 
the tenure positions available in each department of the Faculty of Arts 
and Sciences, aimed at achieving a roughly uniform age distribution. 


Former Presidents of the Society at Harvard University, September 1936 
Left to right: White, Fiske, Bliss, Osgood, Coble, Dickson, and Birkhoff 



MATHEMATICS AT HARVARD, 1836-1944 

Moreover, every department member performed capably and conscientiously 
his teaching and tutorial duties, undergraduate honors being "based 
on the quality of the student's work in his courses, on his thesis, and on the 
general examination" (the latter a less sophisticated version of the Cambridge 
Tripos). 

New trends. However, this seeming emphasis on classical mathematics 
was deceptive. By 1935, Kellogg had died, Osgood had retired, and Morse 
had gone to the Institute for Advanced Study at Princeton. Their places 
were taken by Marshall Stone, Hassler Whitney, Saunders Mac Lane, and 
myself. (I recall that like Walsh and Widder, Stone and Whitney were Ph.D. 
students ofG. D. Birkhoff.) Stone, already famous as a functional analyst, was 
concentrating on Boolean algebra and its relation to topology. Whitney was 
founding the theories of differentiable manifolds and sphere bundles [CMA, 
pp. 109-117]. Mac Lane was exhibiting great versatility and expository skill 
in papers on algebra and graph theory. 

Before 1936, when I became a faculty instructor after attending all the four 
"notable meetings" described in §17, I had never taught a class. I realized 
that my survival at Harvard depended on my success in interesting freshmen 
in the calculus, and was most grateful for the common sense advice given 
by Ralph Beatley regarding pitfalls to be avoided. "Teach the student, not 
just the subject", and "face the class, not the backboard" were two of his 
aphorisms. All new instructors were "visited" by experienced teachers, who 
reported candidly on what they witnessed at department meetings, usually 
with humor. I was visited by Coolidge, and became so unnerved that I splintered 
a pointer while sliding a blackboard down. I survived the test, and 
became a colleague of Stone, Whitney, and Mac Lane. Thus, after 1938, 
the four youngest members of the Harvard mathematical faculty were primarily 
interested in functional analysis, topology, and abstract algebra. In 
addition, Quine had introduced a new full graduate course in mathematical 
logic (Mathematics 19). This treated general "deductive systems", thus going 
far beyond Huntington's half-course on "fundamental concepts". 

I am happy to say that Stone (Harvard '22), Whitney, and Mac Lane are 
still active, while both Widder and Beatley (Harvard ' 13) are in good health. 
Stone recently managed the AMS conference honoring von Neumann, while 
Whitney, Mac Lane, and Widder are fellow contributors to the series of 
volumes in which this report is being published. 

David Widder and I were put in charge of the Harvard Colloquium in the 
years 1936-40. My father and Norbert Wiener usually sat side by side in the 
front row, and made lively comments on almost every lecture. C. R. Adams 
and Tamarkin often drove up from Brown to attend the colloquium, bringing 
graduate students with them. My role included shopping conscientiously for 
good cookie bargains for these convivial and sociable affairs, where tea was 
served by a faculty wife. Most interesting for me were the talks by Ore, von 


GARRETT BIRKHOFF 

Neumann, and Menger on lattice theory, then my central research interest. In 
1938, these three participated in the first AMS Symposium on lattice theory 
(see Bull. A mer. Math. Soc. 44 (1938), 793-837), with Stone, Stone's thesis 
student Holbrook MacNeille, who would become the first executive director 
of the AMS, and myself. Two years later the AMS published the first edition 
of my book Lattice Theory. 

Beginning in 1937-38, Mac Lane and I taught alternately a new undergraduate 
full course on algebra (Mathematics 6), which immediately became very 
popular. I began the course with sets and ended with groups; in the second 
year, my students included Loomis, Mackey, and Philip Whitman. The next 
year, Mac Lane began with groups and ended with sets; his students included 
Irving Kaplansky. After amicable but sometimes intense discussions, we settled 
on the sequence of topics presented in our Survey of Modern Algebra 
(Macmillan, 1941). In it and in our course, we systematically correlated rigorous 
axiomatic foundations with elementary applications to number theory, 
the theory of equations, geometry, and logic. 

20. END OF AN ERA 
Meanwhile, war clouds were getting more and more threatening! Germany 
and Russia invaded and absorbed Poland in 1939, and the International 
Mathematical Congress scheduled to be held at Harvard was postponed indefinitely. 
After the fall of France in the spring of 1940, Germany's invasion 
of Russia, and Pearl Harbor, it became clear that our country would have to 
devote all its strength to winning a war against totalitarian tyranny. 

It was clear to me that our war effort was unlikely to be helped by any of 
the beautiful ideas about "modern" algebra, topology, and functional analysis 
that had fascinated me since 1932, and so from 1942 until the war ended, 
I concentrated my research efforts on more relevant topics. Most interesting 
of these scientifically was trying to predict the underwater trajectories of airlaunched 
torpedoes, a problem on which I worked with Norman Levinson 
and Lynn Loomis, a study in which my father also took an interest. I believe 
that our work freed naval research workers in the Bureau of Ordnance to 
concentrate on more urgent and immediate tasks. 

George D. Birkhoff. During these years, my father continued to think 
about natural philosophy, much as Simon Newcomb and C. S. Peirce had. 
He lectured on a broad range of topics at the Rice Institute, and also in South 
America and Mexico, where he and my mother were good will ambassadors 
cooperating in Nelson Rockefeller's effort to promote hemispheric solidarity 
against Hitler. 

My father finally succeeded in constructing a relativistic model of gravitation 
which was invariant under the Lorentz group, yet predicted the "three 
crucial effects" whose explanation had previously required Einstein's general 


MATHEMATICS AT HARVARD, 1836-1944 

theory of relativity. Because it assumed Minkowski's four-dimensional flat 
space-time, the model also accommodated electromagnetic phenomena such 
as the relativistic motion of particles in electron and proton accelerators. 52 

The exploration of this theory and other ideas he had talked about provided 
an important stimulus to the development of the National University 
of Mexico into a significant research center. The honorary degree that I received 
there in 1955, as well as my honorary membership in the Academy 
of Sciences in Lima, were in large part tributes to his influence on the two 
oldest universities in the Western Hemisphere. 

The department pamphlet of 1942-43. In spite of the war, the pamphlet 
of the Harvard mathematics department for 1942-43 gives the illusion of a 
balance of mathematical activities that had been fairly constant for nearly 
a decade. Although President Conant had gone to Washington to run the 
National Defense Research Council with Vannevar Bush, he had left intact 
the plan of undergraduate education worked out by Lowell. 

Perhaps suggestive of future trends, Beatley was in charge of three sections 
of freshman calculus, Chuck (c. E.) Rickart (then a B.P.) of two; only 
Whitney's and Mac Lane's sections were taught by tenured research faculty 
members. Stone, Kaplansky, and I taught second-year calculus; of the three 
of us, Kaplansky was the most popular teacher. Advanced calculus was taught 
by Whitney and my father, geometry by Coolidge, and undergraduate algebra 
(Mathematics 6) by Ed Hewitt. Real and complex analysis (our main introductory 
graduate courses) were taught by Loomis and Widder, respectively; 
ordinary differential equations (a full course) by my father; and mechanics by 
van Vleck. Graustein had died, but differential geometry was taught by Kaplansky; 
topology was taught by Mac Lane. Widder's student Harry Pollard 
and I taught Mathematics lOb and lOa, respectively. 

Applied mathematics. The only "applied" touch visible in this 194243 
pamphlet was my changed wording for the description of Mathematics 
lOa: I announced that it would treat "the computation of [potential] fields 
in special cases of importance in physics and airfoil theory", and that "In 
1942-43, analogous problems for compressible non-viscous flow will also be 
treated, and emphasis ... put on airfoil theory and air resistance to bullets". 
Also, two courses in "mechanics" were listed: Mathematics 4 to be taught by 
van Vleck, and Mathematics 8 by Kemble. Actually, van Vleck and Dean 
Westergard of the Engineering School had agreed with me that we should 
teach Mathematics 4 (= Engineering Science 6) in rotation. When my turn 
came, John Tate (in naval uniform) was in the class. 

Moreover, appreciation for "applied" mathematics as such was reviving in 
the Harvard Engineering School, with whose faculty I was getting acquainted 
as part of my "continuing education". Though they did not worry about 
Weierstrassian rigor, let alone Cantorian set theory or symbolic logic, Richard 


GARRETT BIRKHOFF 

von Misess3 and my friend Howard Emmons knew infinitely more about real 
flows around airfoils than I. Associated with von Mises were his coauthor 
Philipp Frank, by then primarily interested in the philosophy of science, and 
Stefan Bergman of "kernel function" fame, as well as Hilda Geiringer von 
Mises at Wheaton and Will Prager at Brown. After emigrating together from 
Berlin to Istanbul to escape Hitler, all of these distinguished mathematicians 
had come to New England,S4 greatly enhancing its role in Continuum 
Mechanics, including especially the mathematical analysis of fluid motions, 
elastic vibrations, and plastic deformations. 

But most important for the post-war era, the Gordon McKay bequest of 
1903, which Nathaniel Shaler had labored so hard to secure for Harvard, 
was about to become available. In addition, the 1940 bequest of $125,000, 
given by Professor A. E. Kennelly because "the great subject of mathematics 
applied to electric engineering, together with its study and teaching, have 
throughout my life been an inspiration in my work", was being used to pay 
the salary of Howard Aiken, while he worked at IBM on the development of 
a programmable computer. Harvard was getting ready for the dawn of the 
computer age! 

NOTES 

Further Supplementary Notes and references for this essay, identified by letters, 
will be deposited in the Harvard Archives. 

IFor Peirce's career and influence, see [Pei] and [DAB 14,393-7]. 

2See John Pickering's Eulogy of Nathaniel Bowditch, Little Brown, 1868; 
[DAB 2, 496-8]; [EB 4, p. 31], and [Bow, vol. 1, pp. 1-165]. 

3See p. 69 of Pickering's Eulogy. The accepted value today is (a -b)/a = 
1/297. 

4Benjamin Peirce senior also wrote a notable history of Harvard, recording 
the many benefactions made to it before the American Revolution. 

SSee [Cat, 1835]. 

6See [TCH, p. 220] and [Qui]. Kirkland was succeeded by Josiah Quincy, 
who would be followed in 1846 by Edward Everett. 

7For Lovering's scientific biography, by B. O. Peirce, see [NAS 2: 327-44]. 
He was president of the American Academy from 1880 to 1892. 

8For William Bond's biography, see [DAB 2, 434-5]. His son George succeeded 
him as director of the Harvard Observatory. For more information, 


MATHEMATICS AT HARVARD, 1836-1944 

see The Harvard College Observatory: the first four directorships, 1839-1919, 
by Bessie Z. Jones and Lyle G. Boyd, Harvard University Press, 1971. 

9See Simon Newcomb's autobiography, Reminiscences ofan Astronomer for 
colorful details about his life, and [DAB 13, 452-5] for a biographical survey. 
Hill's first substantial paper was published in Runkle's Mathematical 
Monthly. For his later work, see [NAS 8: 275-309], by E.W. Brown, and 
[DAB 9, 32-3]. 

!OSee [DAB 7, 447-9] for biographies of Gould (Harvard '44), who founded 
the Astronomical Journal, and his father of the same name. 

II[DAB 14, 393-7]. As superintendent, he received $4000/yr, which must 
have doubled his salary. 

12Runkle was MIT President from 1870 to 1878. 

13For the model used, see Newcomb's Popular Astronomy, 5th ed., Part IV, 
Ch. III. Until nuclear energy was discovered, the source of the sun's energy 
was a mystery. W. E. Story was Byerly's classmate. 

14These are associated with systems of linear DE's of the form dxddt 
Laij(t)xj. Hamilton had discovered quaternions in 1843, while Cayley's famous 
paper on matrices was published in 1853. 

ISU.S. government employees helped to prepare Peirce's manuscript for lithographing. 


16Crelle's J. fur Math. 84 (1878), 1-68. 

17See [HH, p. 42], Eliot's article on "The New Education" in the Atlantic 
Monthly 23 (1869), expresses Eliot's opinions before he became president; 
his inaugural address is reprinted in [Mor, pp. lix-Ixxviii]. 

18See When MIT was Boston Tech., by Samuel C. Prescott, MIT Press, 1954. 

20 Byerly was also active in promoting Radcliffe (Harvard's "Female Annex"), 
where Byerly Hall is named for him; see [DAB, Suppl., pp. 145-6]. Elizabeth 
Cary Agassiz was its president. For the Radcliffe story, see [HH, pp. 193-7]. 

21 Cf. [S-G, p. 69]. Oliver Wendell Holmes Sr. wittily observed that professorial 
chairs in "astronomy and mathematics" and "geology and zoology", like 
those of Louis Agassiz and his classmate Benjamin Peirce, should be called 
"settees, not chairs". 


GARRETT BIRKHOFF 

22See §7 (pp. 32-4) of my article in [Tar, pp. 25-78], and pp. 293-5 of my 
father's article in [AMS, pp. 270-315], reprinted in [GDB, vol. iii, pp. 60552]. 
A biography of Osgood by J. L. Walsh will be included in this volume. 
For "The Scientific Work of Maxime Bacher", see my father's article in the 
Bull. AMS 25 ( 1919), 197-215, reprinted in [GDB, vol. iii, pp. 227-45]. 

23For Klein's great influence on American mathematics, see the Index of 
[Arc]; also [Tar, pp. 30-32], and §10 of my article with M. K. Bennett in 
Wm. Aspray and Philip Kitcher (eds.), History and Philosophy of Modern 
Mathematics, University of Minnesota Press, 1988. 

24Bull. Amer. Math. Soc. 5 (1898), 59-87, and vol. 7 of the AMS Colloquium 
Publications (1914). 

25See Bouton's Obituary in Bull. Amer. Math. Soc. 28 (1922), 123-4. 

26For an appreciative account of Coolidge's career, see the Obituary by D. J. 
Struik in the Amer. Math. Monthly 62 (1955), 669-82. Ref. 60 there to a 
biography of Graustein by Coolidge seems not to exist. 

27Ann. ofMath. 10 (1909),181-92. 

28Senate Document # 304 (41 pp.), U.S. Printing Office, 1940. See also EVH 
in Quart. A mer. Statist. Assn. (1921), 859-70, and Trans. Amer. Math. 
Soc. 30 (1928), 85-110. 

29[Yeo, p. 67]. Owen Wister's book Philosophy Four gives an amusing description 
of the "Zeitgeist" at Harvard in those years. 

30 Lowell's The Government ofEngland and (his friend) James Bryce's American 
Commonwealth were the leading books on these two important subjects. 
See [Yeo, p. Ill]. Lord Bryce, when British ambassador to the United States, 
gave Lowell's manuscript a helpful critical reading. 

31In the two volumes [Low] and [Yeo]. 

32Cf. [LAM, §12]. For many years, the Putnams graciously hosted dinner 
meetings of the visiting committee, to which all the members of the mathematics 
department were invited. 

33See [GDB, pp. xv-xxi] for Veblen's recollections and appraisal of my father's 
work. The grandson of a Norwegian immigrant, Veblen had graduated 
at 18 from the University of Iowa before going to Harvard. See [Arc, pp. 
206-18], for biographies of Veblen and my father. 


MATHEMATICS AT HARVARD, 1836-1944 

34See G. G. Lorentz, K. Jetter, and S. D. Riemenschneider, Birkhoff Interpolation, 
Addison-Wesley, 1983. 

35This classic is currently being republished by the American Physical Society 
in translated form, prefaced by an excellent historical introduction by Daniel 
Goroff. 

36The outline of these (Bull. A mer. Math. Soc. 27, 67-69) includes many 
topics of general interest that were not included in the printed volume. These 
include from the first lecture: (7) methods of computation and their validity, 

(8) relativistic dynamics, and (9) dissipative systems. The last lecture was 
entitled "The significance of dynamical systems for general scientific theory", 
and dealt with (1) the dynamical model in physics, (2) modern cosmogony 
and dynamics, (3) dynamics and biological thought, and (4) dynamics and 
philosophical speculation. My father's interest in relativity presumably dates 
from a course he took with A. A. Michelson at Chicago around 1900; see his 
review "Books on relativity", Bull. Amer. Math. Soc. 28 (1922), 213-21. 
37Cf. [GDB, III, pp. 365-81], reprinted from the Ann. of Math. 33 (1932), 
329-45, and the Fifth Yearbook (1930) of the NCTM. 

38Harvey Davis, after teaching mathematics (as a graduate student), physics, 
and engineering [Mor, p. 430] at Harvard, became president of the Stevens 
Institute of Technology. Conant, of course, was President Lowell's successor 
at Harvard. 

39Mrs. William Lowell Putnam lent her summer home to the Birkhoffs during 
the summer of 1927; see also §17. 

40See [Whi], in which pp. 125-65 contain an essay by Quine on "Whitehead 
and the rise of modern logic". 

41 Dunham Jackson had been Secretary of the Division since 1913. Other 
losses were: the differential geometer Gabriel Marcus Green (cf. Bull. A mer. 
Math. Soc. 26, pp. 1-13), and Leonard Bouton (in 1921). 

42 Actually, Walsh had asked Osgood to supervise his thesis, but Osgood declined. 
Like Coolidge and Huntington ('95), Graustein (' 1 0) and Walsh (' 16) 
had both been Harvard undergraduates. 

43Closely related to Haar functions, these would prove very useful for signal 
processings in the 1970s. 

44For a charming description of Morse and his contributions, see Raoul Bott, 
Bull. Amer. Math. Soc. (N.s.) 3 (1980), 907-50. 


56 GARRETT BIRKHOFF 

45The majority of students, not being interested in a mathematical career, 
presumably had very different impressions. 

46Though original, my ideas were not new. Tamarkin kindly softened the 
blow by writing that my paper "showed promise". Six months later, I published 
a revised and very condensed paper containing my sharpest results in 
Bull. Amer. Math. Soc. 39 (1933),601-7. 

471n Stone's words [Tex, p. 15], "the Harvard of my student days could not 
have offered more opportunity or encouragement to a student eager for study 
and learning." 

48The others were von Neumann's Mathematische Grundlagen der Quantenmechanik 
and Banach's Theorie des Operations Lineaires; cf. Historia Math. 
11 (1984), 258-321. 

49Ann. ofMath. 42 (1941), 874-920. See also Ulam's charming Adventures 
ofa Mathematician (Scribners, 1976) for other aspects of his life. 

50For the story of the independent discoveries of linear programming by 
Kantorovich (1939), Frank Hitchcock (1941), T. C. Koopmans ('"'-' 1944), 
and G. B. Dantzig (1946), see Robert Dorfman, Ann. Hist. Comput. 6 
(1984), 283-95. 

51 Published in the Amer. Math. Monthly 44 (1937), 137-55, and as Ch. lof 
Hardy's book Ramanujan (Cambridge University Press, 1940). 

52See [GDB, pp. 920-83], and the article by Carlos Graef Fernandez in pp. 
167-89 of the AMS Symposium Orbit Theory (G. Birkhoffand R. E. Langer, 
eds.), Amer. Math. Soc., 1959. 

531n a very different way, von Mises' book Probability, Statistics and Truth 
was a famous contribution to the foundations of probability theory, which 
are shaky because sequential frequencies are not countably additive. 

54Minkowski's son-in-law Reinhold Rudenberg had also come from the University 
of Berlin to Harvard, while Hans Reissner had come to MIT. 

REFERENCES 

[Ahl] Lars V. Ahlfors, Collected Papers, vol. 1. Birkhiiuser, Boston, 1982. 

[AMM] American Mathematical Monthly, published by the Math. Assn. of America. 


[AMS] Semicentennial Addresses, American Mathematical Society Centennial 
Publications, vol. II, Amer. Math. Soc., 1938; Arno Press reprint, 1980. 


MATHEMATICS AT HARVARD, 1836-1944 

[Arc] Raymond Clare Archibald, A Semicentennial History ofthe American Mathematical 
Society, Amer. Math. Soc., 1940. 

[Bow] Nathaniel Bowditch, translator and annotator, "The Mecanique Celeste of 
the Marquis de La Place", 4 vols., Boston, 1829-39. References in this paper will be 
to the Chelsea reprint (1963). 

[Bre] Joel L. Brenner, "Student days -1930", A mer. Math. Monthly 86 (1979), 
350-6. 
[Cat] Harvard College Catalogues, published annually since 1835. Copies on display 
in Harvard Archives. 
[CMA] A Century of Mathematics in America, Peter Duren et aI., (eds.), vol. 1, 
Amer. Math. Soc., 1988. 
[DAB] Dictionary of American Biography, Allen Johnson and Dumas Malone 
(eds.), Scribners, 1928-1936. 
[DMP] Harvard Division of Mathematics Pamphlets from 1891 through 1941. 
Copies in Harvard Archives. 
[DSB] Dictionary of Scientific Biography, Charles C. Gillispie (ed.). Scribner's, 
1970-. 
[Dup] A. Hunter Dupree, Science in the Federal Government, Harvard University 
Press, 1957. (Reprinted by Johns Hopkins University Press, 1986.) 
[EB] Encyclopaedia Britannica, 1971 edition. 
[GB] Garrett Birkhoff, Selected Papers in Algebra and Topology, J. Oliveira and 
G.-c. Rota, eds., Birkhauser, 1987. 
[GDB] George David Birkhoff, Collected Papers, 3 vols. Amer. Math. Soc., 1950. 
[HH] Hugh Hawkins, Between Harvard and America. The Educational Leadership 
of Charles W. Eliot, Oxford University Press, 1972. 
[JLC] Julian Lowell Coolidge, "Three Hundred Years of Mathematics at Harvard", 
A mer. Math. Monthly 50 (1943), 347-56. 
[LAM] "Some leaders in American mathematics: 1891-1941", by Garrett Birkhoff. 
This is [Tar, pp. 29-78]. 
[LIn] Edward Weeks, The Lowells and their Institute, Little Brown, 1963. 
[Love] James Lee Love, "The Lawrence Scientific School. .. 1847-1906", Burlington, 
N.C. 1944. Copy available in Harvard Archives. 
[Low] Abbott Lawrence Lowell, At War with Academic Traditions in America, 
Harvard University Press, 1934. 
[May] Kenneth O. May, The Mathematical Association ofAmerica: the First Fifty 
Years, Math. Assn. of America, 1972. 
[Mor] Samuel E. Morison (ed.), The Development of Harvard University, 18691929, 
Harvard University Press, 1930. 
[NAS] Biographical Memoirs of the National Academy of Sciences, vols. 1-. 

U.S. National Academy of Sciences, 
[Pei] "Benjamin Peirce", four "Reminiscences", followed by a Biographical Sketch 
by R.C. Archibald. A mer. Math. Monthly 32 (1925), 1-30. 
[Pit] A.E. Pitcher, A History ofthe Second Fifty Years ofthe American Mathematical 
Society, 1939-1988, Amer. Math. Soc., 1988. 


GARRETT BIRKHOFF 

[Put] "The William Lowell Putnam Competition", by G. Birkhoff and L. E. Bush, 
Amer. Math. Monthly 72 (1965), 469-83. 
[Qui] Josiah Quincy, History ofHarvard University, 2 vols., Cambridge, 1840. 
[S-G] David Eugene Smith and Jekuthiel Ginsburg, A History ofMathematics in 
America before 1900, Math. Assn. America, 1934. Arno Press reprint, 1980. 
[SoF] The Society ofFellows, by George C. Homans and Orville T. Bailey, Harvard 
University Press, 1948. Revised (Crane Brinton, ed.), 1959. 
[Tar] Dalton Tarwater (ed.), The Bicentennial Tribute to American Mathematics: 
1776-1976, Math. Assn. America, 1977. 
[TCH] Samuel E. Morison, Three Centuries of Harvard: 1636-1936, Harvard 
University Press, 1936. 
[Tex] Men and Institutions in America, Dalton Tarwater (ed.), Graduate Studies, 
Texas Tech. University, vol. 13 (1976). 
[Van] H. S. Vandiver, "Some of my recollections of George D. Birkhoff', J. Math. 
Anal. Appl. 7 (1963), 272-83. 
[Whi] Paul A. Schilpp (ed.), The Philosophy of Alfred North Whitehead, Northwestern 
University Press, 1941. 
[Yeo] Henry A. Yeomans, Abbott Lawrence Lowell, Harvard University Press, 
1948. 


THE SCIENTIFIC WORK OF MAXIME BOCHER. 


BY PROFESSOR GEORGE D. BIRKHOFF. 

WITH the recent death of Professor Maxime Bocher at only 
fifty-one years of age American mathematics has suffered a 
heavy loss. Our task in the following pages is to review and 
appreciate his notable mathematical work.* 

His researches cluster about Laplace's equation Au = 0, 
which is the very heart of modern analysis. Here one stands 
in natural contact with mathematical physics, the theory of 
linear differential equations both total and partial, the theory 
of functions of a complex variable, and thus directly or indirectly 
with a great part of mathematics. 

His interest in the field of potential theory began in undergraduate 
days at Harvard University through courses given 
by Professors Byerly and B. O. Peirce. There is still on file 
at the Harvard library an undergraduate honor thesis entitled 
" A thesis on three systems of parabolic coordinates," written 
by him in 1888. Under the circumstances it was inevitable 
that he should use formal methods in dealing with his topic, 
but a purpose to penetrate further is found in the concluding 
sentences. No better opportunity for fulfilling such a purpose 
could have been granted than was given by his graduate work. 
under Felix Klein at Gottingen (1888-1891). 

In the lectures on Lame's functions which Klein delivered 
in the winter of 1889-1890 his point of departure was the 
cyclidic coordinate system of Darboux. This sytem of coordinates 
was known to be so general as to include nearly all 
of the many types of coordinates useful in potential theory, 
and Wangerin had shown (1875-1876) how solutions of Laplace's 
equation existed in the form of triple products, each 
factor being a function of one of the three cyclidic coordinates. 
After presenting this earlier work Klein extended his "oscillation 
theorem" for the case of elliptic coordinates (1881) to 
the more general cyclidic coordinates. By this means he was 
able to attack the problem of setting up a potential function 
taking on given values over the surface of a solid bounded by 

*An account of his life and service by Professor Osgood will appearin a later number of the BULLETIN. 
Reprinted with permission from the Bulletin ofthe American Mathematical Society, Volume 
25, pp. 197-215. 

59 


Maxime Bacher 


THE SCIENTIFIC WORK OF MAXINE BOCHER 

six or fewer confocal cyclides. This function was given by a 
series of the triple" Lame's" products discovered by Wangerin. 

Klein also aimed to get at the various forms of series and 
integrals previously employed in potential theory as actual 
limiting cases, and thus to bring out the underlying unity in 
an extensive field of mathematics. 

The task which Bocher undertook was to carry through the 
program sketched by Klein. He did this admirably in his 
first mathematical paper "Ueber die Reihenentwickelungen 
der Potentialtheorie," which appeared in 1891 and which 
served both as a prize essay and as his doctor's dissertation 
at Gottingen.* But the space available was so brief that 
he was only able to outline results without giving their proofs. 

One must look to his book with the same title, t published 
three years later, for an adequate treatment of the subject. 
Here is also to be found original work not outlined in his dissertation. 
It was characteristic that he did not call attention 
explicitly to the new advances although these formed 
his most important scientific work in the years 1891-1894. 
We turn now to a consideration of this book, which thus contains 
nearly all that he did before 1895. 

Besides giving the classification of all types of confocal 
cyclides in the real domain and of the corresponding Lame's 
products, as sketched by Klein, Bocher determined to what 
extent the theorem of oscillation holds in the degenerate cases 
and found an interesting variety of possibilities. 

The difficulties presented by these degenerate cases are decidedly 
greater than those of the general case when the singular 
points e. (i = 1, 2, 3, 4, 5) of the Lame's linear differential 
equation are regular with exponents 0, 1/2. A very simple degenerate 
case is that arising when two such points coincide in 
a single point and one of the two intervals (ml, m2), (n!, n2) 
under consideration ends at this point. By an extension of 
Klein's geometric method, he proved that the theorem of 
oscillation fails to hold even here. 

More specifically, the facts are as follows. In the general 
case the oscillation theorem states that for any choice of integers 
m, n (m, n ~ 0) there is a unique choice of the two ac


* This paper appears as (2) in the chronological Hst of papers given at 
the end of the present article. Hereafter footnote references to papers 
will be made by number. 
t (15). 


GEORGE D. BIRKHOFF 

cessory parameters in the differential equation, yielding solutions 
Ul, U2 such that Ul vanishes at ml and m2, and m times 
for ml < x < m2, while U2 vanishes at nl and n2, and n times 
for nl < x < n2. If now, for instance, ml lies at the double 
singular point el = e2, while ml < m2 < e3 < e4 < nl < n2 
< es, there exist such solutions Ul, U2 only if n> Tm where Tm 
is an integer increasing indefinitely with m. But, to compensate 
for this deficiency of solutions of the boundary value 
problem, Bacher found it necessary to introduce solutions 
Ulk, U2k dependent on n and a continuous real parameter k 
such that Ul vanishes at m2 and infinitely often for ml < x 
< m2 although remaining finite, while U2 vanishes at nl and 
n2, and n times for nl < x < n2. 

The corresponding expansion in Lame's products presents 
a remarkable form under these circumstances, for it is made 
up of a series and an integral component. In another 
case this type of expansion takes the form of an integral 
augmented by a finite number of complementary terms, as 
he had pointed out in an important paper "On some applications 
of Bessel's functions with pure imaginary index,"* 
published in 1892 in the Annals of Mathematics. 

Although dealing satisfactorily with the oscillation theorem 
in the case specified above and other similar cases, Bocher did 
not discuss adequately the case in which three or more singular 
points unite to form an irregular singular point. t Indeed 
it appears that he fell into an error of reasoning as follows. 
If the irregular point be taken at t = + 00 the Lame's equation 
has the form 


where in the case under consideration <p has a limit <Po 9= 0 
as t becomes infinite. The lemma which Bacher then sought 
to provet was that there always exists a solution y finite for 
t 2 T and not identically zero. His proof for the case <Po > 0 
is essentially correct. Here he interpreted the equation above 
as the equation of motion of a particle distant y from a point o of its line of motion and repelled from it with a force <py. 

* (7). In passing, attention may also be called to a slightly earlier article 
(3) 
on Bessel's functions. 
t See (15), p. 179. 
t See (15), p. 177. 

THE SCIENTIFIC WORK OF MAXINE BOCHER 

The gist of the argument employed is that one can find an 
initial velocity of projection toward 0 just sufficient to carry 
it into that point as a limiting position. This part of the 
lemma constitutes a very simple and interesting theorem 
concerning a special type of irregular point. In the case 
CPo < 0, however, using a similar dynamical interpretation, he 
argued* "we have infinitely many oscillations as we approach 
t = + 00, and since the attractive force is not infinitely weak, 
the amplitudes of the oscillations remain finite." This argument 
appears insufficient although the lemma as stated for 
Lame's equation is probably correct.t To satisfactorily complete 
the discussion it would seem to be necessary to call in 
the explicit analytic theory of the irregular singular point, since 
the corresponding theory of the regular singular point is required 
in the simpler cases.t 

In his book Bacher considered the boundary problem under 
periodic conditions, when the interval between two adjacent 
singular points is taken an even number of times and is regarded 
as closed; this case arises, for example, when the solid 
in the potential problem is a complete ellipsoid. Here the 
function cP in the linear differential equation above written is 
an even doubly periodic functioIi with real period. By the 
aid of these properties of cp he reduced the new boundary problem 
to one of the ordinary type. 

Likewise in treating the roots of Lame's polynomials he 
made a distinct advance by extending the dynamical method 
of Stieltjes from the real axis to the complex plane. Thus he 
was able to prove that the roots of these polynomials lie within 
the triangle whose vertices are the three finite singular points 
of the corresponding Lame's equation. 

Finally we may note that at the end of his book he obtained 
all Lame's products satisfying the equation <lu + Pu = 0. 

The determinative effect of the dissertation and book upon 
the direction of Bacher's later researches was very great. In 
the first place he had used sphere geometry and the algebra of 
elementary divisors as essential tools in analysis; his resulting 
interest in the fundamental parts of geometry and algebra 
never subsided, and some of his research lies in these fields. 

. (15),,P. 178. The translation is not literal. 
t In this connection see (7), p. 150, footnotes. 
t Since the above was written Professor Os~ood has disposed of the 
question at issue by elementary means. See his note in this number of 
the BULLETIN. 


GEORGE D. BIRKHOFF 

But, more important still, he was brought into contact with 
open mathematical questions. The most vital of these questions 
from the purely mathematical point of view was doubtless 
the very difficult analytical question of convergence and 
representation presented by the series of Lame's products. 
This was the outstanding problem which Klein emphasized, * 
but to which Bacher seems never to have given particular attention. 
Another more practical direction of effort was afforded 
by the task of giving rigorous and accessible form to 
the work of Sturm and Klein on the real solutions of ordinary 
linear differential equations and then going on further in this 
overlooked but attractive field of research. It was primarily 
to this task that he now turned. 

In 1897 he published an article in the BULLETINt showing the 
immediate usefulness of Sturm's theorems for fixing the distribution 
of the roots of Bessel's functions with real index. A 
year later in the same place he presented the fundamentals of 
Sturm's work in simplified rigorous form, and gave the first 
analytic proof of Klein's theorem of oscillation.t 

Reference should also be made to his article on the boundary 
problems of ordinary differential equations which appeared in 
the German mathematical encyclopedia in 1900. This article 
together with his address on "Boundary problems in one dimension" 
before the Fifth International Congress of Ma thematicians 
in 1912 give an excellent account of this field to the 
latter date. 

Bacher wrote a considerable number of other papeIS in this 
same field.§ Perhaps the most important of these are the 
three to which we will refer first and which appeared in the 
beginning volumes of the Transactions. 

His paper" Application of a method of d'Alembert to the 
proof of Sturm's theorems of comparison" (1900) contained 
an elegant proof of what Bacher had called the theorems of 
comparison. His method was entirely different from Sturm's, 
being based on the Riccati's resolvent equation, and was very 
simple. 

In the second of these papers "On certain pairs of trans


* See the concluding pages of his 1889-1890 lectures on Lame's function" 
and his preface to BOcher's book (15). 

t (26). See (38) also. 

t (30), (31), (35). 

§ (32), (38), (42), (46), (48), (49). (50), (55), (65), (81), (85), (92), (93), 

(100). 


THE SCIENTIFIC WORK OF MAXINE BOCHER 

cendental functions whose roots separate each other"~ (1901) 
his starting point was the linear differential equation 

y" + py' + qy = 0, 

and a pair of linear forms in y, y', 

<P = 'P2Y' -'PlY, 'lr = 1/t2y' -V'lY· 

These latter satisfy a "homogeneous Riccati's equation" 

('Pl1/t2 -'P21/t1) (<p''lr -<P'lr') + A<p2 + B<p'lr + C'lr2 = 0, 

and Bacher considered the relation of the roots of <P, 'lr. 

He notes first that <P, 'lr cannot vanish together unless 
'P11/t2 -'P21/t1 = 0, for otherwise y = y' = 0. In order thut 
<P, 'lr cannot vanish together it is thus sufficient to assume 
'P11/t2 -'P21/t1 =f 0. Also if <P = 0, then <p' =f °if C =f 0, by the 
above equation. A like remark holds for 'lr. Hence the roots 
of <P, 'lr are simple if A =f 0, C =f 0. 

Under these hypotheses between any pair of adjacent roots 
of <P there must be a root of 'lr. For if 'lr has no such root the 
homogeneous Riccati's equation at these roots shows that <p' 
has one and the same sign at both roots, which is impossible. 
Likewise between any pair of adjacent roots of 'lr there must 
be a root of <P. 

Hence the roots of <P, 'lr separate each other if 

'P11/t2 -'P21/t1 =f 0, A =f 0, C =f 0. 

This is the third theorem of the paper. The sixth theorem 
gives similar conditions sufficient to ensure cyclical separation 
of the roots of three linear forms. 

Here Bacher not only achieved greater generality and simplicity 
than Sturm but, as I wish to point out, he has reached 
a maximum of generality. 

For, let Y1> Y2 be any pair of linearly independent solutions 
yielding the values <P1, <P2 and 'lr1> 'lr2 of <P and 'lr. Then 

<P = CI<PI + C2<P2, 'lr = Cl'lrl + C2'lr2 

are the general values of <P, 'lr. If <P1> <P2 are regarded as the 
homogeneous coordinates of a point P in the projective line, 
<P vanishes if P coincides with E == (-C2, CI); similarly 'lr 
vanishes if Q = ('lr1> 'lr2) coincides with the same point E. 

*See also (100). 

GEORGE D. BIRKHOFF 

Clearly the roots of cJ>, 'I' will only be distinct for all values 
of CI> C2 if CPlt/;2 -CP2t/;1 9= o. Moreover, if these roots are to 
separate each other for all values of C1. C2, the points P, Q must 
pass any point E in alternation. This is only possible if P, Q 
never reverse their direction of motion; in other words the 
Wronskians of cJ>b cJ>2 and of '1'1. '1'2 must be of invariant signs. 
Taking into account the fact that YIY2' -Yl'Y2 is not zero, 
this gives precisely the conditions A 9= 0, C 9= o. 

This same geometric interpretation shows a similar generality 
in the other theorems. 

Of like completeness is the third paper" On the real solutions 
of systems of two homogeneous linear differential equations 
of the first order" (1902), where he treated analogous 
questions and also derived comparison theorems. 

It was a matter of primary interest with him to vary proofs 
of known theorems as well as to discover new theorems. An 
illustration in point is afforded by his treatment of the elementary 
separation theorem for the roots of linearly independent 
solutions Y1. Y2 of an ordinary linear differential 
equation of the second order. 

Here he first gave a very brief proof* based on the function 
Yl/Y2: if Yl vanishes at a and b but not for a < x < b, 
while Y2 is not zero for a ~ x ~ b, then the derivative of Yl/Y2 
is of one sign for a < x < b since YIY2' -Yl'Y2 9= o. This is 
impossible. By this argument and a like argument based on 
Y2/Yl it follows that the roots of Y1. Y2 separate each other. In 
the same placet he isolates a geometric proof implicitly given 
by Klein depending on the fact that if Yb Y2 be taken as homogeneous 
coordinates of a point in the projective line then 
YIY2' -Yl'Y2 9= 0 is the condition that this point moves continually 
in one sense. Later he gave a second analytic proof 
based on the function 

Yl' _ Y2' ,t 
Yl Y2 

and also a second geometric proof* based on the vector 

Yl + ~-1Y2 in the complex plane which will rotate continually 

in one sense if YIY2' -Y2Yl' 9= o. 

. (26), p. 210. 
t Footnote, p. 210. 
l (48). 

§ (99), pp. 46-47. 


THE SCIENTIFIC WORK OF MAXINE BOCHER 

It was not easy for him to believe that the methods of 
Sturm were inadequate to deal with any particular boundary 
problem in one dimension. The problem for periodic conditions, 
which had been formulated by him in his encyclopedia 
article, was first successfully attacked by Mason in 
1903-1904 by means of the calculus of variations. In a 
very interesting note published in 1905,* Bacher showed that 
the principal result fell out immediately by the methods of 
Sturm, and that these methods were applicable under much 
more general conditions. Likewise in his address before the 
Fifth International Congress of Mathematicians alluded to 
above he noted that the equation 

d( du)

dX k dx + (Xg -l)u = 0, l < 0, 

(X a parameter) comes directly under the case treated by 
Sturm after division by IXIeven if g changes sign. This simple 
remark disposed of the necessity of treating this case separately, 
as had been done earlier. 

Bacher was interested in all phases of the theory of ordinary 
linear differential equations with real independent variable. 
Having seen the gap in the theory of the regular singular 
point for real independent variable when the coefficients are 
not analytic, he proved that theorems analogous to those 
given by Fuchs in the complex domain are true.t It was 
necessary here to replace the power series treatment by a variation 
of the method of successive approximation which has 
been seen later to afford a new approach to the theory of the 
regular singular point in the complex domain. 

He also did some work in the field of fundamental existence 
theorems for linear differential equations.! He showed 
that it is sufficient to impose the condition of integrability 
(joined with other conditions) upon the coefficients in place 
of Peano's condition of continuity,§ and thus advanced beyond 
Peano. Bacher seems also to have been the first to prove 
that the solutions of a linear differential system are continuous 
functionals of the coefficients.II 

,. (65). 

t (37), (40), (41). 

t (32), (37), (56). 

§ (56), p. 311.

I! (56), p. 315; (55), p. 208. 


GEORGE D. BIRKHOFF 

In 1901 he published a paper on "Green's functions in space 
of one dimension," in which he pointed out that the Green's 
function for the equation of Laplace in one dimension y" = 0, 
exhibited by Burkhardt in 1894, might be extended to the 
general nth order ordinary linear differential equation with 
fairly general boundary conditions. These extended Green's 
functions have turned out to be of great importance. Later 
he returned to the subject of Green's functions with the most 
general linear boundary conditions and set up these functions 
for linear difference equations.* Also he extended the notion 
of adjoint boundary conditions to very general cases. t 

We have now referred briefly to the most important of 
his researches on ordinary linear differential equations with 
real independent variable. In this domain his best work 
is perhaps to be found. Directly springing from this field 
were his researches on linear dependence of functions of a 
single real variablet-an important topic which he was the 
first to isolate sufficiently from the field of linear differential 
equations. 

His paper on "The roots of polynomials which satisfy certain 
linear differential equations of the second order" § lies in 
the field of ordinary linear differential equations with a complex 
variable. Here he generalizes further the extension of 
the method of Stieltjes which he had employed in dealing 
with Lame's polynomials. 

The series arising in mathematical physics had been Bocher's 
point of departure. Indeed it is the existence of these series 
which constitutes the main importance of the boundary value 
problems of linear differential equations. Nevertheless he 
gave special attention only to Fourier's series which he took 
up in an expository article in the Annals of Mathematics for 
1906.11 Here he called attention to the remarkable phenomenon 
exhibited by a Fourier's series near a point of discontinuity, 
previously noted by Gibbs and called" Gibbs's phenomenon" 
by Bocher who gave the first adequate treatment of it., 

His contributions to the theory of the harmonic function 
in two dimensions are elegant and distinctly important. 

. (81).
t (85). 
t (43), (45), (47), (51), (97). 
§ (29). 
II (67). See also (89). 
(j Reference may also be made here to the short note on infinite series 
(60). 


THE SCIENTIFIC WORK OF MAXINE BOCHER 

The first of these occurs incidentally in his paper" Gauss's 
third proof of the fundamental theorem of algebra."* It consists 
in a proof of the average value theorem by means of 
Gauss's theorem for the circle, which in polar coordinates r, cp 
IS 

r27< ou dcp = 0.

Jo or 

Integrating with respect to r from °to a and reversing the 
order of integration, we get 

127< (u(a, cp) -u(O, cp))dcp = 0, 

whence the average value theorem follows at once. This 
very neat proof was probably suggested by the artifice used 
by Gauss in his third proof of the fundamental theorem of 
algebra. 

The" Note on Poisson's integral" (1898) gives a more natural 
interpretation of Poisson's integral than had been stated 
before. By the average value theorem a harmonic function 
is the average of its values on any circle with its center at the 
given point. He generalized this theorem in the spirit of 
the geometry of inversion and thus reached a visual interpretation 
of Poisson's integral which may be formulated as follows: 
The value of a harmonic function at any point within 
a circle is the average of its values as read by an observer at 
the point who turns with uniform angular velocity, if the rays 
of light to his eye take the form of circular arcs orthogonal to 
the given circle. 

According to Riemann's program, the theory of harmonic 
functions requires a development independent of the theory 
of functions of a complex variable. In 1905 Bocher demonstratedt 
that a harmonic function could not become infinite 
at a point unless it was of the form Clog r + v, where C is a 
constant, r is the distance from a variable point to the given 
point and v is harmonic at that point. This theorem corresponds 
to the fundamental theorem in functions of a complex 
variable which states that if fez) becomes infinite at the isolated 
singular point z = a, thenf(z) is of the form (z -at-rg(z) 
where r is a positive integer and g(z) is analytic and not zero 

. (17), p. 206. 
t (59). 

GEORGE D. BIRKHOFF 

at z = a. He demonstrated further that a similar theorem 
holds for large classes of linear partial differential equations. 

Another extremely interesting paper "On harmonic functions 
in two dimensions" appeared in 1906. Here he defines 
u to be harmonic if it is single valued and continuous with 
continuous first partial derivatives and satisfies Gauss's theorem 
for every circle. If u possessed continuous second partial 
derivatives also it would then follow at once by Green's 
theorem that u is harmonic in the customary sense. But it 
is the merit of Bacher's paper to have proved that u is harmonic 
in the ordinary sense without further assumptions. On 
the basis of the definition made, the average value theorem is 
first deduced as outlined above. Also if s', n' are the new 
variables s, n after an inversion (taking circles into circles) 
we have 

o= Jau ds = j' au ds' 
an . an' 

along corresponding circles, since ds'ldn' = dsldn (the inversion 
being conformal). Thus u is "harmonic" in the transformed 
plane also, so that the definition is invariant under 
inversion. Hence Poisson's integral formula, which comes 
from the average value theorem by inversion, also holds, and 
u is harmonic in the ordinary sense. 

He also determined the precise region of convergence of 
the real power series in x, y for any harmonic function u(x, y).* 

In connection with his papers on harmonic functions in two 
dimensions it is natural to call to mind his early paper" On 
the differential equation Au +k2u = 0" (1893), which is taken 
in two dimensions. The" u-functions" so defined give a generalization 
of harmonic functions which he treated by means 
of the fact that u(x, y)ekz satisfies Laplace's equation in three 
dimensions. A similar method had been employed earlier by 
Klein. 

Practically , none of Bacher's work lies directly in the field 
of functions of a complex variable. t 

We have still to consider his contributions in the fields of 
algebra and geometry. In the early paper on the fundamental 
theorem of algebra cited above he made clear how, by taking 
for granted a few theorems in functions of a complex variable, 

* (74). 
t See (78), however. 

THE SCIENTIFIC WORK OF MAXINE BOCHER 

an immediate proof could be given; and then he went on to 
show that by elimination of these theorems, the proof could 
be given a second more fundamental form and finally a third 
form due to Gauss and involving only distinctly elementary 
theorems. In a second paper* he simplified Gauss's proof 
very considerably by replacing Gauss's auxiliary function 
'4'If by 1 If. Here f = 0 is the given equation. 

Here and elsewhere he succeeded in simplifying an apparently 
definitive proof. This kind of work was congenial to 
Bacher, who believed that mathematics was capable of almost 
indefinite simplification, and that such simplification was of 
the highest consequence. 

In the paper with the title "A problem in statics and its 
relation to certain algebraic invariants" (1904) he employed a 
dynamical method similar to his extension of the method of 
Stieltjes in order to develop an interpretation of the roots of 
covariants as the positions of equilibrium of particles in the 
complex plane. Thus if fl,f2 are polynomials of the same degree 
in the homogeneous variables Xl, X2, the vanishing of 
their Jacobian determines the points of equilibrium in the 
field of force under the inverse first power law due to particles 
of "mass" 1 at the roots of fl and of "mass" -1 at the roots 
of f2 in the XdX2 plane. 

We shall not refer to his geometrical paperst save to mention 
the one entitled "Einige Siitze fiber projective Spiegelung" 
(1893) in which he proves that conics in different planes 
may be projectively reflected into each other through a pair of 
lines in four ways, and also that the general ('ollineation of 
space may be represented as the product of a rigid motion 
and a projective reflection through a pair of lines. 

Besides this original research he undertook various more or 
less didactic articles with characteristic unselfishness.t However, 
just as in the article on Fourier's series, matter of an 
original cast is nearly always present. 

The same may be said of his books,§ even of the most elementary. 
We have already considered his book on the series 
of potential theory. Of the others, the most significant are 
his Algebra, where a satisfactory exposition of the elementary 

. (18).
t (6), (8), (12), (13), (53). 
t (14), (20), (24), (39), (66), (67). (70), (73), (83), (92). 
§ (15), (71), (77), (94), (95), (99). 



GEORGE D. BIRKHOFF 

divisor theory is given, his Cambridge tract on integral equations,*
and his Paris 1913-14 lectures "Le~onssur les Methodes 
de Sturm." In the last is given the first complete discussion of 
the convergence of the series used in the method of successive 
approximations. This furnishes another good instance of 
Bocher's power to seize on important theorems which have 
been missed although near at hand. In concluding this brief 
survey it is worth while noting that a few of his papers are 
fairly popular in character. t 

In a recent one of these, "Mathematiques et mathematiciens 
Fran~ais" (1914), while speaking of the characteristics of 
American creative work in all fields (page 9), Bocher says" Ce 
qu'il y a de plus caracteristique dans la meilleure production 
intellectuelle americaine, c'est la finesse et Ie controle voulu des 
moyens et des effets. La faute la plus commune dans ce que 
nous avons fait de mieux, ce n'est pas l'exces de force, ma-is 
plutot son defaut" and later (page 10) "Ce que je viens de 
dire serapporte aussi bien aux mathematiques qu's. toute 
autre branche de la production intellectuelle en Amerique." 
There can be no doubt that this characterization is applicable 
to his own mathematical production. His papers excel 
in simplicity and elegance, and nearly all of them treat subjects 
of great importance to marked advantage. The U8efulness 
of his papers is exceptional,t 

In amount and quality his production exceeds that of any 
American mathematician of earlier date in the field of pure 
mathematics. 

Because of this fact and the weight he has added to our mathematical 
traditions in other ways, Maxime Bocher will ever 
remain a memorable personality in American mathematics. 

LIST OF BOCHER'S WRITINGS.I! 

1888. 

(1) 
The meteorological labors of Dove, Redfield and Espy. American 
Meteorological Journal, vol. 5, No.1, pp. 1-13, May. 
* In connection with this, attention should be called to a short note on 
integral equations listed as (84) below. 
t (1), (9), (11), (82), (90), (91). His first paper "On the meteorological 
labors of Dove, Redfield and Espy" was a youthful essay written about 
the time of his graduation from Harvard UnIversity. 

: This is brought out clearly in Professor Osgood's Lehrbuch der 
Funktionentheorie, vol. 1.
II Substanti ally as compiled by him. 


THE SCIENTIFIC WORK OF MAXINE BOCHER 

1891. 

(2) 
tiber die Reihenentwickelungen der Potentialtheorie. Gekronte 
Preisschrift und Dissertation. Gottingen, Kastner. 4 + 66 pp. 
1892. 

(3) On Bessel's functions of 
the second kind. Annals of Mathematics, 
vol. 6, No.4, RP. 85-90, Jan. 
(4) 
Pockels on the differential equation t.u + k2u = 0 [Review]. Annals 
of Mathematics, vol. 6, No.4, pp. 90-92, Jan. 
(5) Geometry 
not mathematics [Letter to editor]. Nation, vol. 54, No. 
1390, p. 131, Feb. 
(6) On a nine-point conic. 
Annals of Mathematics, vol. 6, No.5, p. 132, 
March. 
(7) On some applications of Bessel's functions with pure imaginary index. 
Annals of Mathematics, vol. 6, No.6, pp. 137-160, May. 
(8) Note on the nine-point conic. Annals of Mathematics, vol. 6, No.7, 
p. 178, June. 
(9) Collineation 
ail a mode of motion. Bulletin of the New York M athematical 
Society, vol. 1, No. 10, pp. 225-231, July. 
1893. 

(10) 
On the differential equation t.u + k2u = O. A merican Journal of 
Mathematics, vol. 15, No.1, pp. 78-83, Jan. 
(11) 
A bit of mathematical history. Bulletin of the New York Mathematical 
Society, vol. 2, No.5, pp. 107-109, Feb. 
(12) 
Some propositions concerning the geometric representation of imaginaries. 
Annals of Mathematics, vol. 7, No.3, pp. 70-72, March. 
(13) 
Einige Siitze tiber projective Spiegelung. Mathematische Annalen, 
vol. 43, No.4, pD. 598-600. 
(14) 
Chapter IX, Historical Summary, pp. 267-275. An Elementary Treatise 
on Fourier's Series and Spherical, Cylindrical and Ellipsoidal 
Harmonics. By W. E. Byerly. Boston, Ginn. 
1894. 

(15) 
tiber die Reihenentwickelungen der Potentialtheorie. Mit einem 
Vorwort von Felix Klein. Leipzig, Teubner, 8 +258 pp. 
1895. 

(16) 
Hayward's Vector Algebra [Review]. Bulletin of the American Mathematical 
Society, ser. 2, vol. 1, No.5, pp. 111-115, Feb. 
(17) 
Gauss's third proof of the fundamental theorem of algebra. Bulletin 
of the American Mathematical Society, ser. 2, vol. 1, No.8, pp. 205209, 
May. 
(18) 
Simplification of Gauss's third proof that every algebraic equation 
has a root. American Journal of Mathema;ics, vol. 17, No.3, pp. 
266-268, July. 
(19) 
General equation of the second degree [Set of formulas on a card]. 
Harvard University Press. 
189(>' 

(20) 
On Cauchy's theorem concerning complex integrals. Bulletin of the 
American Mathematical Society, ser. 2, vol. 2, No.5, pp. 146-149, 
Feb. 
(21) Bessel's functions [Review]. 
Bulletin of the American Mathematical 
Society, ser. 2, vol. 2, No.8, pp. 255-265, May. 

GEORGE D. BIRKHOFF 

(22) Linear.differential equations and their applicat~ons. [Report by T. 
S. FIske of a lecture at the Buffalo ColloquIUm). Bulletin of the 
American Mathematical Society, ser. 2, vol. 3, No.2, pp. 52-55, Nov. 
(23) 
Heffter's Linear Differential Equations [Review). Bulletin of the 
American Mathematical Society, ser. 2, vol. 3, No.2, pp. 86-92, Nov. 
(24) 
Regular points of linear differential equations of the second order. 
Cambridge, Harvard University Press, 23 pp. 
1897. 

(25) 
Schlesinger's Linear Differential Equations [Review). Bulletin of the 
American Mathematical Society, ser. 2, vol. 3, No.4, pp.146-153, Jan. 
(26) On certain methods of Sturm and their application to the roots of 
. Bessel's functions. 
Bulletin of the American Mathematical Society, 
ser. 2, vol. 3, No.6, pp. 205-213, March. 

(27) 
Review of Bailey and Woods: Plane and Solid Analytic Geometry. 
Bulletin of the American Mathematical Society, ser. 2, vol. 3, No. 
9, pp. 351-352, June. 
1898. 

(28) 
Examples of the construction of Riemann's surfaces for the inverse 
of rational functions by the method of conformal representation. 
By C. L. Bouton with an introduction by Maxime B6cher. Annals 
of Mathematics, vo!. 12, No.1, pp. 1-26, Feb. 
(29) 
The roots of polynomials which satisfy certain linear differential 
equations of the second order. Bulletin of the American Mathematical 
Society, ser. 2, vol. 4, No.6, pp. 256-258, March. 
{30) 
The theorems of oscillation of Sturm and Klein (first paper). Bulletin 
of the American Mathematical Society, ser. 2, vol. 4, No.7, pp. 295313, 
April. 

(31) 
The theorems of oscillation of Sturm and Klein (second paper). Bulletin 
of the American Mathematical Society, ser. 2, vol. 4, No.8, 
pp. 365-376, May. 
(32) 
Note on some points in the theory of linear differential equations. 
Annals of Mathematics, vol. 12, No.2, pp. 45-53, May. 
(33) 
Note on Poisson's integral, Bulletin of the American Mathematical 
Society, ser. 2, vol. 4, No.9, pp. 424-426, June. 
(34) 
Niewenglowski's Geometry [Review). Bulletin of the American Mathematical 
Society, ser. 2, vol. 4, No.9, pp. 448-452, June. 
(35) 
The theorems of oscillation of Sturm and Klein (third paper). Bulletin 
of the American Mathematical Society, ser. 2, vol. 5, No.1, 
pp. 22-43, Oct. 
1899. 

(36) 
Burkhardt's Theory of Functions [Review). Bulletin of the American 
Mathematical Society, ser. 2, vol. 5, No.4, pp. 181-185, Jan. 
(37) 
On singular points of linear differential equations with real coefficients. 
Bulletin of the American Mathematical Society, ser. 2, vol. 5, No.6, 
pp. 275-281, March. 
(38) 
An elementary proof that Bessel's functions of the zeroth order have 
an infinite number of real roots. Bulletin of the American Mathematical 
Society, ser. 2, vol. 5, No.8, pp. 385-388, May. 
(39) 
Examples in the theory of functions. Annals of Mathematics, ser. 2, 
vol. 1, No.1, pp. 37-40, Oct. 
1900. 

(40) 
On regular singular points of linear differential equations of the second 
order whose coefficients are not necessarily analytic. Transactions 
of the American Mathematical Society, vol. 1, No.1, pp. 40-52, Jan. j 
also No.4, p. 507, Oct. 

THE SCIENTIFIC WORK OF MAXINE BOCHER 

(41) 
Some theorems concerning linear differential equations of the second 
order. Bulletin of the American Mathematical Society, ser. 2, vol. 
6, No.7, pp. 279-280, April. 
(42) 
Application of a method of d'Alembert to the proof of Sturm's theorems 
of comparison. Transactions of the American Mathematical 
Society, vol. 1, No.4, pp. 414-420, Oct. 
(43) 
On linear dependence of functions of one variable. Bulletin of the 
American Mathematical Society, ser. 2, vol. 7, No.3, pp. 120--121, 
Dec. 
(44) 
Randwertaufgaben bei gewohnlichen Differentialgleichungen. Encyklopiidie 
der mathematischen Wissenschaften, II A 7a, pp. 437463, 
Leipzig, Teubner. 
1901. 

(45) The theory of linear dependence. 
Annals of Mathematics, ser. 2, vol. 
2, No.2, pp. 81-96, Jan. 
(46) Green's functions in space of one dimension. 
Bulletin of the American 
MatherT'atical Society, ser. 2, vol. 7, No.7, pp. 297-299, April. 
(47) 
Certain cases in which the vanishing of the Wronskian is a sufficient 
condition for linear dependence. Transactions of the American 
Mathematical Society, vol. 2, No.2, pp. 139-149, April. 
(48) 
An elementary proof of a theorem of Sturm. Transactions of the 
American Mathemafical Society, vol. 2, No.2, pp. 150--151, April. 
(49) N on-oscillatory linear differential equations of the second order. Bulletin 
of the American Mathematical Society, ser. 2, vol. 7, No.8, 
pp. 333-340, May. 
(50) 
On certain pairs of transcendental functions whose roots separate 
each other. Transactions of the American Mathematical Society, 
vol. 2, No.4, pp. 428-436, Oct. 
(51) 
On Wronskians of functions of a real variable. Bulletin of the American 
Mathe~/latical Society, ser. 2, vol. 8, No.2, pp. 53-63, Nov. 
(52) 
Picard's Traite d'Analyse [Review]. Bulletin of the American Mathematical 
Society, ser. 2, vol. 8, No.3, pp. 124-128, Dec. 
1902. 

(53) 
Some applications of the method of abridged notation. Annals of 
Mathematics, ser. 2, vol. 3, No.2, pp. 45-54, Jan. 
(54) Review of Schlesinger: Einfiihrung in die Theorie der Differentialgleichungen 
mit einer unabhii.ngigen Variabeln. Bulletin of the 
AmericanMathematicalSociety,ser. 2, vol. 8, No.4, pp.168-169,Jan. 
(55) On the real solutions of two homogeneous linear differential equations 
of the first order. Transactions of the American Mathematical Society, 
vol. 3, No.2, pp. 196--215, April. 
(56) On systems of linear differential equations of the first order, .4 merican 
Journal of Mathematics, vol. 24, No.4, pp. 311-318, Oct. 
(57) Review of Gauss' Wissenschaftliches Tagebuch. Bulletin of the American 
Mathematical Society, ser. 2, vol. 9, No.2, pp. 125-126, Nov. 
1903. 

(58) 
The Elements of Plane Analytic Geometry. By George R. Briggs. 
Revised and enlarged by Maxime B6cher. New York, Wiley, 4 + 
191 p. 
(59) 
Singular points of functions which satisfy partial differential e,:/uations 
of the elliptic type. Bulletin of the American Mathe11latical 
Society, ser. 2, vol. 9, No.9, pp. 455-465, June. 

GEORGE D. BIRKHOFF 

(60) On 
the uniformity of the convergence of certain absolutely convergent 
series. Annals of Mathematics, ser. 2, vol. 4, No.4, pp. 159160, 
July. 
1904. 

(61) 
Contribution to Sprechsaal fUr die Encyklopiidie der Mathematischen 
Wissenschaften. Archiv der Mathematik und Physik, vol. 7, No. 
3. p. 181, Feb. 
(62) 
The'fundamental conceptions and methods of mathematics. Address 
delivered before the Department of Mathematics of the International 
Congress of Arts and Science, St. Louis, Sept. 20,1904. Bulletin 
of the American Mathematical Society, ser. 2, vol. 11, No.3, 
pp. 115-135, Dec. Also in Congress of Arts and Science Universal 
Exposition, St. Louis, 1904, vol.l. Boston, Houghton and Mifflin, 
1905, pp. 456-473. 
(63) 
A problem in statics and its relation to certain algebraic invariants. 
Proceedings of the American Academy of .'irts and Sciences, vol. 40, 
No. 11, pp. 469-484, Dec. 
1905. 

(64) 
Linear differential equations with discontinuous coefficients. Annals 
of Mathematics, ser. 2, vol. 6, No.3, pp. 97-111 (49-B3), April. 
(65) 
Sur les equations differentielles lineaires du second ordre a solution, 
periodique. Comptes Rendus de I' Academie des Sciences, vol. 140" 
No. 14, pp. 928-931, April. 
(66) 
A problem in analytic geometry with a moral. Annols of Mathematics, 
ser. 2, vol. 7, No.1, pp. 44-48, Oct. 
1906. 

(67) 
Introduction to the theory of Fourier's series. Annals of Mathematics, 
vel. 7, No.2, and No.3, pp. 81-152, Jan. and April. 
(68) 
On harmonic functions in two dimensions. Proceedings of the Am~ican 
Academy of Arts and Sciences, vol. 41, No. 26, pp. 577-583, 
March. 
(69) Review of Picard: Sur Ie Developpement de I'Analyse, etc. Science, 
n. s., vol. 23, No. 598, p. 912, June. 
(70) 
Another proof of the theorem concerning artificial singularities. --I.nnals 
of Mathematics, ser. 2, vol. 7, No.4, pp. 163-164, July. 
1907. 

(71) Introduction to Higher Algebra. By Maxime Bocher. Prepared for 
publication with the cooperation of E. P. R. Duval. New York, 
Macmillan, 11 + 321 pp.. 
1908. 

(72) Review of Bromwich: Quadratic Forms 
and their Classification by 
Means of Invariant Factors. Bulletin of the American Mathematical 
Society, ser. 2, vol. 14, No.4, pp. 194-195, Jan. 
(73) On the small forced vibrations of systems with one degree of freedom. 
Annals of Mathematics, ser. 2, vo!' 10, No.1, pp. 1-8, Oct. 
1909. 

(74) On the regions of convergence of power-series which represent twodimensional 
harmonic functions. Transactions of the American 
Mathematical Society, vol. 10, No.2, pp. 271-278, April. 
. A German translation appeared in 1909: EinfUhrung in die h6here 
Algebra. Deutsch von Hans Beck. Mit einem Geleitwort von Eduard 
Study. Leipzig, Teubner, 12 + 348 pp. 

THE SCIENTIFIC WORK OF MAXINE BOCHER 

(75) 
Review of Runge: Analytische Geometrie der Ebene. Bulletin of the 
American Mathematical Society, ser. 2, vol. 16, No.1, pp. 30-33, Oct. 
(76) 
Review of d'AdMmar: Exercices et Lec;:ons d'Analyse. Bulletin of the 
American Mathematical Society, ser. 2, vol. 16, No.2, pp. 87-88, Nov. 
(77) 
An introduction to the study of integral equations. Cambridge 
Tracts in Mathematics and Mathematical Physics, No. 10, Cambridge, 
England, University Press, 72 pp.. 
1910. 

{78) On semi-analytic functions of two variables. A nnals of Mathematics, 
ser. 2, vol. 12, 'No.1, pp. 18-26, O~t. 

(79) 
Kowalewski's Determinants [Review]. Bulletin of the American 
Mathematical Society, ser. 2, vol. 18, No.3, pp. 120-140, Dec. 
1911. 

(80) 
The published and unpublished work of Charles Sturm on algebraic 
and differential equations. Presidential address delivered before 
the American Mathematical Society, April 28, 1911. Bulletin of 
the American Mathematical Society, ser. 2, vol. 18, No.1, pp. 1-18, 
Oct. 
(81) 
Boundar)' problems and Green's functions for linear differential and 
difference equations, Annals of Mathematics, ser. 2, vol. 13, No.2, 
pp. 71-88, Dec. 
(82) 
Graduate work in mathematics in universities and in other institutions 
of like grade in the United States. General rpport. United States 
Bureau of Education Bulletin, No.6, pp. 7-20. Also in Bulletin of the 
American Mathematical Society, ser. 2, vol. 18, No.3, pp. 122-137, 
Dec. 
1912. 

(83) 
On linear equations with an infinite number of variables. By Maxime 
Bocher and Louis Brand. Annals of MathematlCS, ser. 2, vol. 13, 
No.4, pp. 167-186, June. 
(84) 
A simple proof of a fundamental theorem in the theory of integral 
equations. Annals of Mathematics, ser. 2, vol. 14, No.2, pp. 8485, 
Dec. 
1913. 

(85) 
Ap;>lications and generalizations of the conception of adjoint systems, 
Transactions of the American Mathe1Mtical Society, vol. 14, No.4, 
pp. 403-420, Oct. 
(86) 
Doctorates conferred by American universities [Letter to the editor], 
Science, n. s., vol. 38, No. 981, p. 546, Oct. 
(87) 
Boundary problems in one dimension [A lecture delivered Aug. 27, 
1912). Proceedings of the Fifth International Congress of Mathematicians, 
Cambridge, England, University Press, vol. 1, pp. 
163-195. 
1914. 

(88) 
The infinite regions of various geometries. Bulletin of the American 
Mathematical Society, ser. 2, vol. 20, No.4, pp. 185-200, Jan.t 
(89) 
On Gibbs's phenomenon. Journal fur die reine und angewandte Mathew.
atik, vol. 144, No.1, pp. 41-47, JaIl . 
. A second edition appeared in 1914. 
t See vol. 22, (1915) No.1, p. 40, Oct. 

GEORGE D. BIRKHOFF 

(90) 
MatMmatiques et matMmaticiens frant;ais. Rerue Internationale de 
I'Enseignement, vol. 67, No.1, pp. 20-31, Jan. 
(91) 
Charles Sturm et les matMmatiques modernes. Revue du Mois, 
vol. 17, No. 97, pp. 88-104, Jan. 
(92) On a small variation which renders a linear differential system incompatible. 
Bulletin of the American Mathematical Sllciety, ser. 2, vol. 
21, No.1, pp. 1-6, Oct. 
(93) The smallest characteristic number in 
a certain exceptional case. 
Bulletin of the American Mathematical Society, ser. 2, vol. 21, No.1, 
pp. 66-99, Oct. 
1915. 

(94) 
Trigonometry with the theory and use of logarithms. By Maxime 
Bacher and H. D. Gaylord, New York, Holt, 9 + 142 pp. 
(95) 
Plane analytic geometry with introductory chapters on the differential 
calculus. New York, Holt, 13 + 235 pp. 
1916. 

(96) 
Review of Gibb: A Course in Interpolation etc. and Carse and Shearer: 
A Course in Fourier's Analysis etc. Bulletin of the American M athematical 
Society, ser. 2, vol. 22, No.7, pp. 359-361, April. 
(97) 
On the Wronskian test for linear dependence, Annals of Mathematics, 
ser. 2, vol. 17, No.4, pp. 167-168, June. 
(98) 
Syllabus of a Brief Course in Solid Analytic Geometry. Lancaster, 
New Era Press, 10 p. 
(99) 
Let;ons sur les methodes de Sturm dans la theorie des equations differentielles 
lineaires et leurs developpements modernes. Professees a la Sorbonne en 1913-14. Recueillies et redigees par G. Julia. 
Paris, Gauthier-Villars, 6 + 118 pp. 
1917. 

(100) Note supplementary to the paper "On certain pairs of transcendental 
functions whose roots separate each other." Transactions of the 
A merican Mathematical Society, vol. 18, No.4, pp. 519-521, Oct. 
1.918 
(101) Concerning direction cosines 
and Hesse's normal form. American 
Mathematical Monthly, vol. 25, No.7, pp. 308-310, Sept. 

Joseph L. Walsh (1895-1973) was educated at Harvard, receiving a bachelor's 
degree in 1916 and a Ph.D. in 1920. His thesis adviser was G. D. Birkhoff. He 
was on the Harvard Jaculty Jrom 1921 until his retirement in 1966, when he 
moved to a special chair at the University ojMaryland. He did basic research 
in complex approximation theory, conJormal mapping, harmonic Junctions, 
and orthogonal expansions. The Walsh Junctions, a complete orthonormal 
extension ojthe Rademacher Junctions, became important in digital communication. 
Walsh was President oj the AMS and a member oj the National 
Academy ojSciences. His biographical sketch ojOsgood is published here Jor 
the first time by permission ojthe Harvard University Archives. 

William Fogg Osgood 

J. L. WALSH 
William Fogg Osgood (March 10, 1864-July 22, 1943) was born in Boston, 
Massachusetts, the son of William and Mary Rogers (Gannett) Osgood. He 
prepared for college at the Boston Latin School, entered Harvard in 1882, 
and was graduated with the A.B. degree in 1886, second in his class of 286 
members. He remained at Harvard for one year of graduate work in mathematics, 
received the degree of A.M. in 1887, and then went to Germany to 
continue his mathematical studies. During Osgood's study at Harvard, the 
great Benjamin Peirce (1809-1880), who had towered like a giant over the 
entire United States, was no longer there. James Mills Peirce (1834-1906), 
son of Benjamin, was in the Mathematics Department, and served also later 
(1890-1895) as Dean of the Graduate School and (1895-1898) as Dean of 
the Faculty of Arts and Sciences. William Elwood Byerly was also a member 
of the Department (1876-1913), and is remembered for his excellent 
teaching and his texts on the Calculus and on Fourier's Series and Spherical 
Harmonics. Benjamin Osgood Peirce (1854-1914) was a mathematical 
physicist, noted for his table of integrals and his book on Newtonian Potential 
Theory. Osgood was influenced by all three of those named -they were 
later his colleagues in the department -and also by Frank Nelson Cole. 

Reproduced with permission of Harvard University Archives, from the papers of Joseph 

L. Walsh. 
79 


William Fogg Osgood 


WILLIAM FOGG OSGOOD 

Cole graduated from Harvard with the Class of 1882, studied in Leipzig from 
1882 to 1885, where he attended lectures on the theory of functions by Felix 
Klein, and then returned to Harvard for two years, where he too lectured on 
the theory of functions, following Klein's exposition. 

Felix Klein left Leipzig for Gottingen in 1886, and Osgood went to Gottingen 
in 1887 to study with him, Klein (Ph.D., Gottingen, 1871) had become 
famous at an early age, especially because of his Erlanger Program, in which 
he proposed to study and classify geometries (Euclidean, hyperbolic, projective, 
descriptive, etc.) according to the groups of transformations under 
which they remain invariant; thus Euclidean geometry is invariant under the 
group of rigid motions. The group idea was a central unifying concept that 
dominated research in geometry for many decades. Klein was also interested 
in the theory of functions, following the great Gottingen tradition, especially 
in automorphic functions. Later he took a leading part in organizing the 
Enzyklopiidie der Mathematischen Wissenchaften, the object of which was to 
summarize in one collection all mathematical research up to 1900. Klein also 
had an abiding interest in elementary mathematics, on the teaching of which 
he exerted great influence both in Germany and elsewhere. 

The mathematical atmosphere in Europe in 1887 was one of great activity. 
It included a clash of ideals, the use of intuition and arguments borrowed 
from physical sciences, as represented by Bernhard Riemann (1826-1865) 
and his school, versus the ideal of strict rigorous proof as represented by 
Karl Weierstrass (1815-1897), then active in Berlin. Osgood throughout his 
mathematical career chose the best from the two schools, using intuition in 
its proper place to suggest results and their proofs, but relying ultimately on 
rigorous logical demonstrations. The influence of Klein on "the arithmetizing 
of mathematics" remained with Osgood during the whole of his later life. 

Osgood did not receive his Ph.D. from Gottingen. He went to Erlangen 
for the year 1889-1890, where he wrote a thesis, "Zur Theorie der zum 
algebraischen Gebilde ym = R(x) gehorigen Ableschen Functionen." He 
received the degree there in 1890 and shortly after married Theresa Ruprecht 
of Gottingen, and then returned to Harvard. 

Osgood's thesis was a study of Abelian integrals of the first, second, and 
third kinds, based on previous work by Klein and Max Noether. He expresses 
in the thesis his gratitude to Max Noether for aid. He seldom mentioned the 
thesis in later life; on the one occasion that he mentioned it to me he tossed 
it off with "Oh, they wrote it for me." Nevertheless, it was part of the theory 
of functions, to which he devoted so much of his later life. 

In 1890 Osgood returned to the Harvard Department of Mathematics, and 
remained for his long period of devotion to the science and to Harvard. At 
about this time a large number of Americans were returning from graduate 


J. L. WALSH 
work in Germany with the ambition to raise the scientific level of mathematics 
in this country. There was no spirit of research at Harvard then, except 
what Osgood himself brought, but a year later Maxime Bacher (A.B., Harvard, 
1888; Ph.D., Gottingen, 1891) joined him there, also a student greatly 
influenced by Felix Klein, and a man of mathematical background and ideals 
similar to those of Osgood. They were very close friends both personally and 
in scientific work until Bacher's death in 1918. 

Osgood's scientific articles are impressive as to their high qUality. In 1897 
he published a deep investigation into the subject of uniform convergence of 
sequences of real continuous functions, a topic then as always of considerable 
importance. He found it necessary to correct some erroneous results on the 
part of du Bois Reymond, and established the important theorem that a 
bounded sequence of continuous functions on a finite interval, convergent 
there to a continuous function, can be integrated term by term. Shortly 
thereafter, A. Schoenflies was commissioned by the Deutsche MathematikerVereinigung 
to write a report on the subject of Point Set Theory. Schoenflies 
wrote to Osgood, a much younger and less illustrious man, that he did not 
consider Osgood's results correct. The letter replied in the spirit that he 
was surprised at Schoenflies' remarkable procedure, to judge a paper without 
reading it. When Schoenflies' report appeared (1900), it devoted a number 
of pages to an exposition of Osgood's paper. Osgood's result, incidentally, as 
extended to non-continuous but measurable functions, became a model for 
Lebesgue in his new theory of integration (1907). 

In 1898 Osgood published an important paper on the solutions of the 
differential equation y' = f(x, y) satisfying the prescribed initial conditions 
y(a) = b. Until then it had been hypothesised that f(x, y) should satisfy a 
Lipschitz condition in y: If(x, yd -f(x, ydl s MIYl -Y21, from which it 
follows that a unique solution exists. Osgood showed that if f(x, y) is merely 
continuous there exists at least one solution, and indeed a maximal solution 
and a minimal solution, which bracket any other solution. He also gave a 
new sufficient condition for uniqueness. 

In 1900 Osgood established, by methods due to H. Poincare, the Riemann 
mapping theorem, namely that an arbitrary simply connected region of the 
plane with at least two boundary points, can be mapped uniformly and conformally 
onto the interior of a circle. This is a theorem of great importance, 
stated by Riemann and long conjectured to be true, but without a satisfactory 
proof. Some of the greatest European mathematicians (e.g., H. Poincare, H. 

A. Schwarz) had previously attempted to find a proof but without success. 
This theorem remains as Osgood's outstanding single result. 
Klein had invited Osgood to collaborate in the writing of the Enzyklopiidie, 
and in 1901 appeared Osgood's article "Allgemeine Theorie der analytischen 
Funktionen a) einer und b) mehrerer komplexen Grossen." This was a deep, 


WILLIAM FOGG OSGOOD 

scholarly, historical report on the fundamental processes and results of mathematical 
analysis, giving not merely the facts but including numerous and 
detailed references to the mathematical literature. The writing of it gave 
Osgood an unparalleled familiarity with the literature of the field. 

In 1901 and 1902 Osgood published on sufficient conditions in the Calculus 
of Variations, conditions which are still important and known by his name. 
He published in 1903 an example of a Jordan curve with positive area, then 
a new phenomenon. In 1913 he published with E. H. Taylor a proof of the 
one-to-oneness and continuity on the boundary of the function mapping a 
Jordan region onto the interior of a circle; this fact had been conjectured from 
physical considerations by Osgood in his Enzyklopiidie article, but without 
demonstration. The proof was by use of potential theory, and a simultaneous 
proof by functional-theoretic methods was given by C. Caratheodory. 

In 1922 Osgood published a paper on the motion ofthe gyroscope, in which 
he showed that intrinsic equations for the motion introduce simplifications 
and make the entire theory more intelligible. 

From time to time Osgood devoted himself to the study of several complex 
variables; this topic is included in his Enzyklopiidie article. He published a 
number of papers, gave a colloquium to the American Mathematical Society 
(1914) on the subject, and presented the first systematic treatment in his 
Funktionentheorie. He handled there such topics as implicit function theorems, 
factorization, singular points of analytic transformations, algebraic 
functions and their integrals, uniformization in the small and in the large. 

It will be noted that Osgood always did his research on problems that 
were both intrinsically important and classical in origin -"problems with a 
pedigree," as he used to say. He once quoted to me with approval a German 
professor's reply to a student who had presented to him an original question 
together with the solution, which was by no means trivial: "Ich bestreite 
Ihnen das Recht, ein beliebiges Problem zu stellen und aufzuI6sen." 

Osgood loved to teach, at all levels. His exposition was not always thoroughly 
transparent, but was accurate, rigorous, and stimulating, invariably 
with emphasis on classical problems and results. This may have been due in 
some measure to his great familiarity with the literature through writing the 
Enzyklopiidie article. He also told me on one occasion that his own preference 
as a field of research was real variables rather than complex, but that 
circumstances had constrained him to deal with the latter; this may also have 
been a reference to the Enzyklopiidie. 

Osgood's great work of exposition and pedagogy was his Funktionentheorie, 
first published in 1907 and of which four later editions were published. 
Its purpose was to present systematically and thoroughly the fundamental 
methods and results of analysis, with applications to the theory of functions 
of a real and of a complex variable. It was more systematic and more rigorous 


J. L. WALSH 
that the French traites d'analyse, also far more rigorous than, say, Forsyth's 
theory of functions. It was a moment to the care, orderliness, rigor, and 
didactic skill of its author. When G. P6lya visited Harvard for the first time, 
I asked him whom he wanted most to meet. He replied "Osgood, the man 
from whom I learned function theory" -even though he knew Osgood only 
from his book. Osgood generously gives Bacher part of the credit for the 
Funktionentheorie, for the two men discussed with each other many of the 
topics contained in it. The book became an absolutely standard work wherever 
higher mathematics was studied. 

Osgood had previously (1897) written a pamphlet on Infinite Series, in 
which he set forth much of the theory of series needed in the Calculus, and 
his text on the Calculus dates from 1907. This too was written in a careful 
exact style, that showed on every page that the author knew profoundly the 
material he was presenting and its background both historically and logically. 
It showed too that Osgood knew the higher developments of mathematics 
and how to prepare the student for them. The depth of Osgood's interest 
in the teaching of the calculus is indicated also by his choice of that topic 
for his address as retiring president of the American Mathematical Society 
in 1907. 

Osgood wrote other texts for undergraduates, in 1921 an Analytic Geometry 
with W. C. Graustein, which again was scholarly and rigorous, and in 
1921 a revision of his Calculus, now called Introduction to the Calculus. In 
1925 he published his Advanced Calculus, a masterly treatment of a subject 
that he had long taught and that had long fascinated him. He published a text 
on Mechanics in 1937, the outgrowth of a course he had frequently given, 
and containing a number of novel problems from his own experience. 

After Osgood's retirement from Harvard in 1933 he spent two years (19341936) 
teaching at the National University of Peking. Two books in English of 
his lectures there were prepared by his students and published there in 1936: 
Functions ofReal Variables and Functions ofa Complex Variable. Both books 
borrowed largely from the Funktionentheorie. 

Osgood did not direct the Ph.D. theses of many students; the theses he 
did direct were those of C. W. Mcg. Blake, L. D. Ames, E. H. Taylor, and 
(with C. L. Bouton) G. R. Clements. I asked him in 1917 to direct my own 
thesis, hopefully on some subject connected with the expansion of analytic 
functions, such as Borel's method of summation. He threw up his hands, "I 
know nothing about it." 

Osgood's influence throughout the world was very great, through the soundness 
and depth of his Funktionentheorie, through the results of his own research, 
and through his stimulating yet painstaking teaching of both undergraduates 
and graduate students. He was intentionally raising the scientific 
level of mathematics in America and elsewhere, and had a great part in this 


WILLIAM FOGG OSGOOD 

process by his productive work, scholarly textbooks, and excellent classroom 
teaching. 

Osgood's favorite recreations were touring in his motor car, and smoking 
cigars. For the latter, he smoked until little of the cigar was left, then inserted 
the small blade of a penknife in the stub so as to have a convenient way to 
continue. 

Osgood was a kindly man, somewhat reserved and formal to outsiders, 
but warm and tender to those who knew him. He had three children by Mrs. 
Teresa Ruprecht Osgood: William Ruprecht, Freida Bertha (Mrs. Walter Sitz, 
now deceased), Rudolph Ruprecht. His years of retirement were happy ones. 
He married Mrs. Celeste Phelpes Morse in 1932, and died in 1943. He was 
buried in Forest Hills Cemetery, Boston.