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 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 Y2 be any pair of linearly independent solutions
yielding the values 'lr2 of

'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.