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Department of
Mathematics, |
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Mathematics at UVM
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The First Two Centuries |
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Roger Cooke |
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This document sketches the people and
activities that have characterized the Department of Mathematics (now the
Department of Mathematics and Statistics) at the |
Contents
§1. The Early Years, 1801–1825
§ 2. The “Benedictine” Era,
1825–1854
§ 3. A Generation of Struggle, 1854–1885
§ 4. The Beginnings of Growth, 1885–1914
§ 5. Another Generation of Struggle,
1915–1954
§ 6. UVM Moves Toward the Mainstream,
1955–1965
Mathematics at the
By Roger Cooke
Introduction. In 1990, as the bicentennial of the founding
of UVM approached, a committee was formed to organize a proper celebration of
the occasion. This committee
solicited proposals from the Faculty for projects suitable to the celebration. At the suggestion of my colleague Jeff
Dinitz, I proposed to write the history of mathematics at UVM. The following pages are the result of
that proposal. Being a history of mathematics, this history appropriately has
the form of a matrix whose rows are indexed by historical periods and whose
columns are indexed by the areas of relevance to the mathematical history:
personalities, teaching, research, students, and the mathematical environment.
Not every position in this matrix will be occupied, but these are the general
categories that will be discussed.
Sources. The following material was
gleaned from a wide variety of sources. The most important source of documentary
material was of course the University Archives. I am particularly indebted to
David Blow, the University Archivist, for his constant and efficient
help and suggestions in locating the documents I needed. These documents
include the personnel files on the characters involved, the minutes of
the meetings of the Board of Trustees (known originally as the Corporation),
and the University Catalogs. The Wilbur Collection at the Bailey–Howe
Library has also been a valuable source of material. Finally I am indebted to several retired
colleagues (all unfortunately now deceased) for their personal
reminiscences: Heath Riggs, Ivan
Hershner, N. James Schoonmaker, Joseph Izzo, and most especially George
Nicholson,i whose mathematical career spanned more than one-third of
the history of mathematics at UVM.
Overview of the Subject. For the general history of UVM there are
various secondary sources, for example, that of Lindsay (1954). For that reason the general history of
UVM will be touched on only where it is necessary to the narrative, and no
attempt will be made to render judgment on the many issues that arise. Also, before concentrating on the details
of our own mathematical development, it may be well to expend a few words on
the perspective within which this mathematics is to be judged. We might consider UVM in the context of
its educational mission in
Taking the local point of view,
one is pleasantly surprised to find some rather well-educated
mathematicians at UVM, even in the early nineteenth century, teaching
mathematics in some depth to ordinary citizens aspiring to careers as lawyers,
physicians, clergy, or farmers. As
we shall see, some of these mathematicians even conducted a modest level of
independent research. The national
perspective reminds us that, although UVM is the twentieth oldest university in
America, there was already in existence at the time of its founding in 1791 a
sizable number of New England institutions. Indeed, these institutions provided the
early UVM faculty. Harvard, which
was founded before
UVM, in contrast, was significantly smaller
than these European universities and dedicated primarily to teaching. Although UVM was founded in close cooperation
with the government of the State of
§1. The Early Years, 1801–1825
The Curriculum. The
In keeping with the admission requirements, the
curriculum also placed a heavy emphasis on a literary and classical education.
Still, mathematics was not neglected.
In fact, one may well ask who today has an intimate knowledge of all these
particular mathematics courses, some of which have passed out of the standard
curriculum or been absorbed into more comprehensive courses:
Second
year. 1st. Vulgar Arithmetickiii... 2d... Logarithms
and Algebra... 3d... Geometry and
Elements. Plane Trigonometry. Mensuration of superfices
and solids. Gauging. Mensuration of heights and distances...
Third year. 1st.
Surveying and Levelling. Navigation. Conick Sections. Dialling. Spherical
Geometry.
Projections
of the Sphere. Spherical Trigonometry and Spherical Astronomy...
The emphasis on spherical
geometry, trigonometry, and astronomy is explained by the needs of navigation.
These subjects, along with geography (another important obligatory part of the curriculum)
were considered essential parts of one’s knowledge of the world. Without them, one’s appreciation
of even English literature is impoverished.iv Lindsay
(1954, p. 69) expresses some puzzlement that the calculation of eclipses was
universal in all colleges of the time and considers its presence in the
curriculum “probably a hangover from the medieval curriculum.” In fact, the calculation of eclipses has
a very important function in the mundane affairs of
commerce, specifically in navigation: it can be used to determine terrestrial
longitude. Latitude is easy to determine.
One has only to go outdoors at night and observe the elevation of the
pole star. That elevation is the
latitude of the point of observation. This last statement is a slight oversimplification,
of course, since the “pole star” isn’t exactly true north,
but the point is that latitude is easily determined. Any known star can be observed at its
culmination (transit of the local meridian), and the local latitude can then be
worked out from spherical trigonometry. Even if one lives at the bottom of a
mine shaft one can work out latitude; for example, it is the arcsine of the
number of revolutions a Foucault pendulum precesses in one sidereal day, and
the direction of precession will distinguish northern latitude from southern.
Longitude, on the other hand, is more difficult to
calculate. Any two points at the
same latitude will observe the same elevation for each fixed star and planet.
The problem, assuming you know that the origin of the longitude coordinate is
Greenwich, is to determine how many degrees one’s own location is west or
east of the intersection of its circle of latitude with the Greenwich
meridian. Direct measurement is
precluded by the topography of the earth. If one knew what time it was in
Greenwich when it is noon locally, of course, the longitude would be determined
exactly. Each hour of time difference
between local standard time and Greenwich mean time represents a difference
of 15º of
longitude. For that reason the
British government had offered a prize for a clock
that would keep accurate time on board a ship. Setting such a clock to Greenwich time,
one would then always know longitude by comparing local (sun) time with the
clock time. Such clocks were not easy
to develop, and were far from cheap and reliable. The cheap way of calculating
longitude, once astronomy was sufficiently
sophisticated to predict planetary motion with accuracy, was to use a big clock
in the sky. For instance, the moon
undergoes changes of phases. If one
could chart these phases accurately enough, it would only be necessary to look
at the moon to know what time it is in Greenwich. Unfortunately the
moon’s phases change too slowly to permit measuring any changes over the
period of a few hours. When Galileo
discovered the moons of Jupiter, he realized that their configurations,
once worked out, could be a much better “universal clock” than the
moon-related phenomena such as phases or tides. They were actually used for this purpose
in some surveying work. The most
easily observable clock of this type, however, is a lunar eclipse (a solar
eclipses is not visible simultaneously at widely separated places). The use of
eclipses in calculating longitude was known from very early times. A lunar eclipse on September 20, 330 B.
C. E. was observed at both Arbela (47º E, in
the territory of modern Iraq) and Carthage (10ºE, on the northern coast of
Africa), and these observations were used by Ptolemy to calculate the difference
in longitude between the two places (Neugebauer 1975, p. 668). The accurate
mapping of the world was still a matter of pressing practical importance in the
early nineteenth century, and scientific journals were eager to have
accurate observations of eclipses to compare with the predicted times. For that
reason one must disagree with Lindsay’s assessment of this part of the curriculum.
As for the rest of the curriculum, it was, to be sure, not the latest in
research mathematics. The calculus, already 150 years old, was not part of the
curriculum, to say nothing of the researches in mathematical physics due to
Euler, Laplace, Lagrange, and others. On the other hand, Laplace and Lagrange
were still alive at this time; it is not surprising that their work had not
reached Vermont. At least
Laplace’s Mécanique
céleste had reached Boston and inspired a translation by Nathaniel
Bowditch that far excelled the original in clarity of expression.
Instruction
and Equipment. Teaching was by lecture and
demonstrations given by the faculty, since there was very little opportunity
for hands-on laboratory work by the students. What we now call laboratory equipment
used for instruction was known in those days as “philosophical
apparatus,” and described by ex-President John Wheeler in an address at UVM’s
semicentennial celebration in 1852:
Of astronomical and
philosophical apparatus, there was a telescope, planetarium, quadrants, two
sets of 24 inch Globes, and other necessary articles of value, besides seven
hundred dollars worth of instruments purchased of the Rev. Dr. Prince of Salem,
Mass., by individuals, [Wheeler says in a footnote that the latter were Dr.
John Pomeroy, David Russel Esq., and Col. W. C. Harrington—RLC] and
deposited for the use of the University, in the Philosophical Chamber. The
apparatus was more complete, than in any of the Colleges in New England, except
Harvard and Yale. (Wheeler, 1854, p. 2)
Lindsay (1954, p. 106) mentions
a catalog from the Rev. Dr. Prince listing objects for sale, including glass
plates ground so as to make an airtight joint, a flask beam for weighing
air, a pipe of mephitic air (carbon dioxide), a long glass tube with plate and
collar of leather for the Torricellian experiment (showing the decrease of
atmospheric pressure with altitude), a model water pump of brass and glass to
show the action of valves, an electric generator turned by winch, an electrical
cannon for firing hydrogen gas, a battery consisting of nine jars, a
microscope, and many other objects. Unfortunately it is not known which of
these were purchased. Lindsay
reports that only one of these objects remained in the early 1950’s, a
compound magnet encased in brass.
There does not seem to be any record of textbooks,
if any, used for the instruction at this period. Certainly the library was not a rich
source of reading material. The
minutes of the Board of Trustees’ meeting in January 1811 (Vol. 1, p.
154), lists the entire University Library of the time. It consisted of thirty volumes, with a
heavy emphasis on literature and divinity studies. The few science books available were all
devoted to the applied parts of science, such as (Erastus) Darwin’s Zoonomia, Priestley’s Corruption, Priestley’s On Air (5 volumes), and Paine’s Geography.
Personnel. In 1807 Mr. James Dean was
deemed suitably qualified and respectable to be a tutor of mathematics
and astronomy, and in 1809 he became the first professor of mathematics
and natural philosophy at UVM. Dean
became a scholar of some note, and his biography appears in Appleton’s Cyclopedia of American Biography. According to information contained in
the University archives in a folder bearing his name, he was born at Windsor,
Vermont on November 26, 1776 and received the A. B. degree from Dartmouth in
1800 and the A. M. degree, also from Dartmouth, in 1806. He left UVM in 1814, when the University
closed and rented its buildings to the American Government for use in the war
against the British. In his years at UVM up to that point his salary, nominally
$400 per year when he was appointed in 1809, had not been regularly paid. The
University accounts in January 1811 show that he was owed $847.15. This was surely not a sum one could
easily afford to forego at the time. The University, however, was not going
to settle easily. In the discussion
leading up to the closing of the University (minutes of the meetings of the
Board of Trustees, Vol. II, p. 55, meeting of March 24, 1814) we read:
Resolved, That the Treasurer in
paying the debts due from the Corporation to the late President and Professors,
pay to them in proportion to their existing debts, provided Professor Dean
withdraws his suit without Cost, at the same time having respect to what
greater proportion anyone has already received. And if said Dean does not so
withdraw his suit, pay said President and Professor Chamberlain first...
He went to Dartmouth as tutor,
but because of the legal dispute that tore Dartmouth apart at this time (in
which he backed the losing side) he came back to UVM as a professor from 1822 until
1824. He was President pro tempore of the University briefly in 1824,
just before leaving to take up a position at Union College. From Union College he received an
honorary Doctor of Divinity degree in 1847. He died in January 1849 and is buried in
Elmwood Cemetery in Burlington.
No portraits of James Dean exist, but his physical
appearance and character, as they appeared to President Wheeler, were described
in considerable detail in President Wheeler’s semicentennial address:
He possessed a mathematical
mind, distinguished for its clearness and accuracy, rather than its depth and
scientific insight. He devoted himself to the life of a student, and
acquired much, and various knowledge, rather than comprehension and profound
principles. He was rigid in his discipline, the sharp lines of which were
perhaps increased, by an occasional irritability of temper, which seemed to
spring from his very peculiar physical constitution. He was inordinately fleshy,
and in such way as to give the appearance rather of disease, than of health.
His influence in the University was marked by adherence to law and order,
in the simple and earnest pursuit of its objects. (Wheeler, 1854, p. 24)
In other words, Dean was rather
a dabbler, one who preferred breadth to depth. These are precisely the qualities needed
in an institution devoted more to teaching than to research, even though
Wheeler’s tone suggests that he thought otherwise. One suspects that Wheeler felt some
antipathy toward Dean.
The modest scholarly reputation
that Dean attained was based on six articles by him listed in the Royal Society Catalogue of Scientific
Papers (Vol. 2, p. 185). Four
of these were published in 1815 in the Memoirs
of the American Academy of Arts and Letters, Vol. 3, pt. 2. One of the
latter is a detailed report of an observation of the solar eclipse of September
17, 1811. Dean gives precise statements of the local solar time for the
beginning and end of the eclipse as observed by himself and his companion (see
the remarks above on the significance of eclipses for geography). The
other three articles published in 1815 are connected with geometric
astronomy. One is a description of
a “cometarium.” Another is “A method of displaying at one
view all the annual cycles of the equation of time in a complete revolution of
the Sun’s apogee.” (The
misnamed “equation of time” is the amount by which mean solar
time—on which clocks are based—differs
from true solar time. It is a
periodic function of time, and Dean’s article provided a small elliptical
piece of paper riveted to a chart.
By turning the ellipse one could determine the amount by which solar
time is ahead of or behind clock time at any given moment.)
The most important of the four articles,
however, was a tour de force of spherical trigonometry, “An investigation
of the apparent motion of the earth viewed from the Moon arising from the
Moon’s librations.” The
last word here is the key to this paper.
It is a commonplace that the moon always turns the same face to the
earth, but this commonplace is not strictly true. The moon rotates on its axis at a
uniform rate, but its orbit about the earth is slightly elliptical. As a result, terrestrial astronomers get
an occasional peek around the edge of the moon into its hidden side. This is one kind of libration. The other kind occurs because the
moon’s axis of rotation is not perpendicular to the plane of its orbit
around the earth, so that its northern and southern poles come alternately into
view to the earth. Looking at these
two kinds of libration from the perspective of the moon, one finds that
the earth does not remain in a fixed location in the sky, as would be the
case if the commonly held view were accurate. Instead the earth describes a small but
complicated closed curve in the sky over a long period of time. Dean gave a careful analysis of this
curve and showed that it is the curve described by a pendulum bob at the bottom
of a Y-shaped string, in which the vertical stroke of the Y is 40 times the distance
from the fork in the Y to the line through its two tips. This article so intrigued the astronomer
Bowditch that he was inspired to perform a detailed mathematical analysis of
such a pendulum. Bowditch’s
article was published in the same issue of the Memoirs of the American Academy in which Dean’s four articles
appeared.v
Besides the papers just
discussed, Dean also published an observation of several meteors in Silliman’s Journalvi in 1823, and an article
“On the diameter of screws,” in the Boston Journal of Philosophy in 1826, which he rewrote and expanded
in the Journal of the Franklin Institute
in 1845, just four years before his death.
His analysis is really an analysis of the properties of a helix,
combined with some frictional considerations, and therefore applies more
properly to bolts than to screws. The problem is to find the diameter
that enables the ratio of power (he seems to mean torque) to weight to be
minimized. Besides these research
papers, he was the author of a gazetteer of Vermont published in Montpelier in
1808, a copy of which can be found in the Wilbur Collection of the University
of Vermont. Among the papers in his
file in the University archives is a letter of November 27, 1833 to the
physicist Joseph Henry, reporting some meteor observations, and speculating on
ways by which it could be proved that the aurora borealis is electrical in
nature.
UVM’s first
professor of mathematics was therefore, at least by American standards of the
times, intellectually respectable.
The remaining question is, how much of his knowledge did he impart to
his students? There is some
evidence that he was an interesting and inspiring lecturer. When he became professor in 1809, he
delivered an inaugural speech (actually on April 24, 1810), in the manner of
the German Antrittsrede entitled
“An Oration on Curiosity.”
In my view, this 19-page summary history of natural philosophy is
exceedingly inaccurate, even for its own time, and contains no memorable
thoughts or sentences. It was,
however, printed and published by the Samuel Mills Press in Burlington in May
1810 at the request of the students!vii His most enduring contribution
to American civilization, however, has been immortalized as “Dean’s
Method” of apportioning the House of Representatives. Those familiar with the paradoxes that
can result from any system of proportional representation will appreciate the
difficulty of framing general principles of representation
that will be fair under all circumstances.
The most notorious such paradox is the so-called Alabama paradox, which
arose after the 1880 census. It was
discovered that if the total number of representatives in Washington was
increased, Alabama would actually be entitled to fewer representatives than it
would get if the number was left unchanged. Dean’s Method is explained in
detail in the book Fair Representation
by M. L. Balinski and Y. Peyton Young (Yale University Press, 1982).
As mentioned above, James Dean left UVM when the
University closed down in 1814 and did not return until 1821. His place as professor of mathematics
during this time was taken by the Rev. Ebenezer Burgess (1815–1817) and
the Rev. Gamaliel Smith Olds (1819–1821). Neither of these men was particularly a
mathematician or scientist. The Historical Catalogue of Brown University,
published in 1914, lists Burgess as a tutor in the period 1811–1813. Born in Wareham, Massachusetts on April
1, 1790, he was Preceptor at the University Grammar School in Providence
1809–1811. He graduated from
Andover Theological Seminary in 1815 and received a D. D. from Middlebury
College in 1835. After leaving UVM
he was an agent of the American Colonization Society for Africa in
1817–18. In 1821 he was
ordained a Congregationalist minister and was pastor of the First Church of
Dedham, Massachusetts for fifty years, from 1820 until his death in 1870.
Somewhat less is known about G. S. Olds
(1777–1848). He wrote a pamphlet now in the Wilbur Collection of UVM
bearing the title, “Statement of Facts Relative to the Appointment of the
Author to the Office of Professor of
Chemistry in Middlebury College and the Termination of his Connexion with that
College,” published by Denio and Phelps, Greenfield, Massachusetts,
1818. As Rev. Olds tells the
story, he was approached by Middlebury College in 1816 and asked to take the
position of professor of chemistry.
Not feeling quite qualified, he asked for a delay of one year in
assuming the position, to which the President of Middlebury assented
orally. Olds was to be paid while
preparing himself by listening to Benjamin Silliman’s course of lectures
in New Haven, Connecticut. When he
wrote requesting his salary, however, the President responded, “that our
treasury is at present entirely empty, and is likely to remain so, I apprehend,
for some time.” Before his
year of preparatory study was expired, Middlebury changed its mind and annulled
the appointment, apparently claiming some impropriety on the part of Rev. Olds. It was the charges against him that
provoked the pamphlet. The minutes
of the UVM Board of Trustees indicate that he was the fourth person to whom the
position of professor of natural philosophy was offered,
the first three chosen having declined the honor.
At the time of James Dean’s
return to the University we find the first University Catalog, for
the year 1822, from which we can form a picture of the course of study followed
by the students and taught by the faculty.
There were only two professors in the Classical College and five
in the Medical School. Prof.
Dean’s companion in the Classical College was Lucas Hubbell, A. M.,
Professor of the Learned Languages.
These two, and perhaps the President (Rev. Daniel Haskel) taught all 40
of the classical students. At the
time, UVM adhered to a three-term school year, the “autumnal term”
extending from September until December, the spring term from March through
mid-May, and the Summer term from mid-May through mid-August.
The
mathematical portion of the curriculum was based entirely on a famous series of
textbooks by the British mathematician Charles Hutton (1737–1823), a
professor at the Royal Military Academy at Woolwich. Hutton’s Arithmetick occupied the second semester
of the Freshman year, Hutton’s Algebra
the first term of the sophomore year, and Hutton’s Geometry, Hutton’s Trigonometry, and Hutton’s Conick Sections the third term of the
sophomore year. That, except for a
little astronomy in the last term of the junior year, was the extent of
mathematics in the curriculum. Pycior
(1988) points out that the American editions of these European works were often
modified by their American editors.
This is the case with Hutton’s work, which was edited by Robert
Adrain (1775–1843), an Irish immigrant who taught at Queen’s
College (Rutgers) and Columbia College.
Adrain’s Hutton is bound in two volumes of about six hundred pages
each. The material listed above is all in the first of these two
volumes. The second volume covers
spherical trigonometry and the “Doctrine of fluxions,” that
is, the calculus. Apparently the
second volume was not used in instruction at UVM at this time. The volumes now in the stacks of the
Bailey–Howe library belonged to Professor G. W. Benedict, who will be
discussed in the next section.
Although a thorough discussion of the contents of Hutton’s
textbooks would require far more time and space than the current project
allows, certain points are of sufficient
interest to be mentioned.
Hutton’s arithmetic goes far beyond mere computation and considers
many topics that we now regard as algebra, such as compound interest problems,
permutations and combinations, raising to power (called involution), extraction
of roots (called evolution), logarithms, and mixture problems (called
alligation problems and classified as alligation medial and alligation
alternate). Hutton’s algebra, written at a time when elementary algebra
was not yet thoroughly elucidated through the use of complex numbers, betrays a
lack of insight on the part of its author and a consequent lack of motivation
for most of its methods. Algebra is defined as “the science of
computing by symbols.” Some of the purported applications of algebra are
curious, as they are bound to be, since algebra by itself (without its use in
calculus and differential equations) really
has no application to ordinary human life. Under the heading APPLICATION
OF ARITHMETICAL PROGRESSION TO MILITARY AFFAIRS we find
problems of the following sort:
A
detachment having 12 successive days to march, with orders to advance the first
day only two leagues, the second 3
, and so on increasing 1 league each day’s march:
What is the length of the whole march, and what is the last day’s march?
One wonders if students were any better fooled then
by such fraudulent “applications” than they are today. Even the
simplest topic of algebra, the solution of quadratic equations, gives the
impression of rules laboriously and mindlessly memorized, to be passed on to
students without any reflection whatever. Nowhere is the student told that solving
a quadratic equation amounts to finding two numbers when one knows their
sum and product. The rules for
solving cubic equations are mostly approximate. As Hutton puts it,
There are many particular and
prolix rules usually given for the solution of some of the above-mentioned
powers or equations. But they may be all readily solved by the following easy
rule of Double Position, sometimes called Trial-and-error.
A formula for solving cubic
equations is given, called “Carden’s formula.” Girolamo Cardano (1501–1576) was
an Italian mathematician who worked out or plagiarized this rule. Again it is
manifest that the author himself does not understand what he is doing. He gives
the rule, but no proof of it, and never tells the reader that the irreducible
case, in which the formula leads to imaginary numbers, occurs precisely when
the equation has three real roots, even though this fact had been known since
not long after Cardano’s original work.
Hutton’s geometry escapes
all the difficulty of the theory of parallels by simply defining
parallel lines to be lines that lie at the same perpendicular distance from
each other at all their points. By cutting the Gordian knot of the
parallel postulate thus efficiently, he is able to
prove a great deal in very little space, but of course the reader is deprived
of any knowledge of one of the fundamental problems in geometry at the time,
the role of Euclid’s fifth postulate, which is not even stated.viii
Such was the mathematical education available in Vermont
in 1823 for the price of $20.00 per year tuition, plus board at $1.25 per
week. (Rooms in the University
commons were free.) Even when an
allowance is made for inflation, the cost per theorem has probably never
been so low at any time since.
Students.
According to the General Catalogue
of the University of Vermont, 1791–1890, UVM had about 100 living
alumni by the year 1820, of whom 51 were lawyers, 20 were (Protestant)
clergymen, 5 were physicians, 5 were farmers, 3 were in the armed forces, 3 were
merchants, 5 were teachers, 2 were professors, and 1 was a sea captain. No occupations were given for the
others. The most noteworthy alumnus
of the early period from the point of view of mathematics was George Palmer
Williams of the class of 1825, who obtained an LL. D. from Kentucky College in
1849, became an Episcopal priest, and was professor of mathematics and physics
at the University of Michigan from 1841 to 1863. He was president of the faculty at Michigan
in 1845–46 and again in 1848–49. He died in 1881.
On May 27, 1824 UVM suffered
one of the greatest calamities in its frequently precarious existence. The only university building, the
“college edifice” (in its rebuilt form now known as the Old
Mill), was destroyed by fire.
President Haskel suffered a nervous breakdown from this blow and never
fully recovered from it. James Dean
was appointed President pro tempore to
officiate at commencement in August of that year.
However, Dean had decided that three of his students, due to insufficient
scholarship, were not to be promoted with the rest of their class. The three students promised reform and
appealed to the trustees not to be held back. The trustees agreed and thereby
precipitated the resignation of Professor Dean. On this rather sour note, the first
section of our narrative ends.
§ 2. The “Benedictine” Era, 1825–1854
Personnel. The University was fortunate in its
recovery from the loss of its building and its senior professor. A successor to Prof. Dean was found
almost immediately in the person of George Wyllys Benedict, whose biography can
be found in the Vermont Alumni Weekly
of February 17, 1926. G. W.
Benedict was a vigorous and resourceful man, just the person to put things back
together again. It is his copy of
Hutton’s course of mathematics that is now in the stacks of the
Bailey–Howe library. Born in
When I came here, I was an entire
stranger to the Institution and to every person connected with it, to the
region round about, and to all its inhabitants. Why I came, I can hardly tell.
Certainly the inducements held out to me were slight enough. The member of the
Corporation [Board of Trustees—RLC], Hon. Titus Hutchinson, who called
upon me, then a resident in another state, to ask if I would consent to be a
candidate for the professorship of mathematics and natural philosophy, told me
that the college building was burned down, that the Institution had met with
many difficulties, and had poverty to contend with. Through the
generosity of individuals, chiefly the inhabitants of Burlington, a
partial rebuilding was to be commenced as soon as mild weather would justify
such operations, but a slow growth was to be looked for, in his judgement. For
salary, he could promise but $600 per annum, and that not very regularly paid.
There was however a freedom from discouragement in all that he said, and a confidence
in the continued life and ultimate strength of the Institution, which won my
sympathy, and gained my assent to his proposal.
G. W. Benedict’s energy was sorely needed by
the University. It is very largely thanks to his efforts
that a great deal of money was raised to fill the shelves of a very
respectable library. While teaching
mathematics at UVM, he sent off for
publication the one scientific paper listed under his name in the Royal Society Catalogue (Vol. 1, p.
270), an observation of a meteor seen in Burlington on April 14, 1826,
published in Silliman’s Journal,
Vol. XI, (1826), p. 120. His first
innovation was to add calculus to the curriculum, based on a textbook by
Étienne Bézout (1730–1783). Lindsay (1954, pp. 195–200)
recounts Benedict’s construction of the first chemistry laboratory
in any American university (contradicting claims by the University of Michigan,
where a chemistry laboratory was built in 1856). Benedict not only designed and financed
this laboratory, he gave comprehensive lectures on electricity to the
community. Although he taught
mathematics for only four years (he continued to teach chemistry until 1847),
he became treasurer of the University and one of its most articulate advocates
in the world of learning. UVM was
in need of such an advocate, since its curriculum was looked upon unfavorably
by more traditional places. The
classical language requirement for admission was applied only to those who
planned to study the classics.
Those who wished to study science or modern literature were exempt from
it. G. W. Benedict wrote a pamphlet
in defense of UVM’s policy, which was distributed in the name of the
entire faculty. Although UVM
revived rapidly largely because of his labors, those labors took their toll on
him. He resigned in 1847 for health
reasons and went to work for a company bringing a telegraph line from Troy, New
York to Burlington. He soon formed
his own company to bring a telegraph from Boston to Burlington. In 1853, he and his son bought the Burlington Free Press. Both were outspoken abolitionists
and defenders of the civil rights of freed slaves. Appointed historian of the University,
he wrote a comprehensive history of its early days for Vol. XIII of the American Quarterly Register, which
unfortunately is not in the University Library. Prof. Benedict died in 1871,
and is buried in the Benedict family plot in Green Mount Cemetery.
G. W. Benedict taught
mathematics at UVM only until 1829.
In that year George Russell Huntington, a graduate of the class of 1826,
was appointed professor of mathematics and civil engineering, the first
mention of engineering at UVM as a separate discipline. (Rensselaer Polytechnic Institute was
only five years old at the time.)
I have been unable to discover any more information about Prof.
Huntington, except that he died in 1872.
He taught only until 1832, when the chair of mathematics and civil
engineering was taken over by Farrand Northrup Benedict, an 1823 graduate of
Hamilton College and a cousin of G. W. Benedict.
The biography of F. N. Benedict
is given in more detail in an article by Prof. Evan Thomas in the Vermont Alumni Quarterly, but will be
summarized here. He was born in Parsippany, New Jersey in 1803. Upon his
graduation from Hamilton he entered the law, but abandoned it after two years
and began to practice as a civil engineer.
The Royal Society Catalogue
lists one paper under his name, “A method of determining the temperature
of the mercury in a siphon barometer, from the observed upper and lower
readings; and of testing the accuracy of the instrument,” published in Silliman’s Journal, Vol. XL
(1841), pp. 250–263. An earlier
publication, overlooked by the Royal Society, was titled, “On the
sections of a plane, with the solids formed by the revolution of the conic
sections, about axes situated in their planes,” also in Silliman’s Journal, Vol. XXXI
(1837), pp. 258–266. F. N.
Benedict retired rather early from UVM (in 1855) and returned to Parsippany,
New Jersey, apparently for the sake of his wife’s health. He was an ardent conservationist with a
keen interest in the Adirondacks, where he purchased large tracts of land to
ensure their preservation. In 1874
he undertook an arduous survey of the Hudson and Raquette Rivers (albeit for
purposes of which modern environmentalists would not approve—he wanted to
see what potential they had for damming.)
His environmental interests are discussed in an article by Warder H.
Cadbury titled, “A Foot-Note to John Todd’s Long Lake,” Back
Log Camp Press, Sobail, New York, 1957, part of a collection about Long Lake
published by Howard I. Becker.ix
F. N. Benedict died in 1880. Historians are supposed to maintain a
nonjudgmental aloofness from the characters they write about, but I cannot help
revealing that of all the former professors I have encountered in doing the research
for this paper, he is my favorite, the one I would most like to have met. As Prof. Thomas points out, he was not
only a competent, if slightly eccentric, mathematician, he was also a man of
warm human sympathies. In his file in the UVM archives there is a letter
written a few months before his death, to the daughters of one of his former
pupils, a Rev. Parker. It begins as follows:
Although suffering from great
weakness and loss of sight, I cannot refrain, without violence to my feelings,
from assuring you of my deep sympathy in the loss of your dear father...
Curriculum. The curriculum and entrance requirements
at UVM changed slightly during the 1830’s, as already mentioned in
connection with G. W. Benedict. The
1837 catalog reveals that UVM had gone from a three-term academic year to a
two-term system, with semesters referred to as the “Autumnal Term”
and the “Summer Term.”
Commencement was on the first Wednesday in August, and the
“summer vacation” was the months of January and February, for obvious
climatic reasons. As the catalog
shows, the students studied algebra during the Autumnal Term of the first
year and geometry, including solid and spherical geometry, during the Summer
Term of that year. In the second
year they took up conic sections, plane and spherical trigonometry (Autumnal
Term) and surveying, navigation, projections, differential
and integral calculus, and civil engineering (Summer Term). Science in the third year was devoted to
physics and chemistry, especially mechanics, optics, and crystallography. Finally, in the fourth year the study of
algebra and calculus was resumed, and astronomy was added. This was quite a respectable curriculum. The textbooks used were
Bonnycastle’s Algebra,
Legendre’s Geometry in a
translation by Charles Davies (1798–1876), Herschel’s Outline of Astronomy, and the
“Cambridge Course,” (a series of textbooks used at Harvard), all of
which were a vast improvement on the earlier ones. John Bonnycastle (ca. 1760–1821),
like Hutton, a professor at the Royal Military Academy at Woolwich, was a much
clearer writer who seemed to have real insight into his subject. As for Adrien-Marie Legendre
(1752–1833), he is one of the greatest mathematicians of all time, and
his textbook of geometry is pellucid.
At the very least, he discussed the problem of parallels and stated the
parallel postulate, which he made a futile effort to prove. John Herschel (1792–1871) was the
son of the famous astronomer William Herschel.
Further indication of the content of these courses
is fortunately available from two sets of notes that have been preserved. One
is a set of notes on the course in analytic geometry from the Autumnal Term of
1848 kept by Matthew Buckham, later president of UVM. The bulk of the course is devoted to
applications of coordinates to the study of conic sections, whose equations are
given in rectangular, oblique, and polar coordinates. The second notebook is from the 1849
course in central forces. These
notes, which were kept by McKendree Petty, the successor of F. N. Benedict as
professor of mathematics, show clearly that the course was an elaboration of
Newton’s Principia, Book I,
Sections II–III and XII (on the orbits of particles and the attraction of
spherical bodies). No attempt is made to find simpler or clearer proofs
of the propositions than Newton himself gave.
Students. Among the mathematical students of this
period one should mention Joel Tyler Benedict (apparently a nephew of F. N.
Benedict) of the class of 1843, who later became a professor of civil
engineering and mathematics at the New York Free Academy. He wrote an algebra text in 1857 that
conceals the true nature of algebra every bit as effectively
as the books used earlier at UVM. The
most distinguished was undoubtedly Selim Hobart Peabody, who graduated in 1852
and became a professor of mathematics and civil engineering at the Agricultural
College of Pennsylvania in 1854 and obtained a Ph. D. in 1877 and an LL. D. in
1881 from the University of Iowa.
He became professor of mechanical engineering at Illinois Industrial
University in 1878 and president of the University of Illinois in 1880. Less well-known was Otis David Smith of
the class of 1853, who became professor of English literature and mathematics
(sic!) at the Agricultural and Mechanical College of Alabama in Auburn.
The Mathematical Environment. During this period the University
matured into an institution with an established and well-deserved reputation of
competence, and even excellence in some respects. The 1843 catalog boasts that the library
“contains upwards of seven thousand volumes, chiefly selected. It is open to the Senior and Junior
classes every Saturday—to the Sophomore and Freshman classes every
Wednesday—at noon, during term time... ” The catalog of the library holdings from
1843 (University Press, Burlington, printed by Chauncey Goodrich) bears out
this claim. Of the 7,000 books some
250 are on mathematics and physics, and these contain some impressive works,
such as Colson’s translation of Donna Maria Agnesi’s Analytical Institutions (London 1801),
George Airy’s Mathematical Tracts
on Physical Astronomy (Cambridge 1826), Arbogast’s Du calcul des dérivations
(Strasbourg 1800), as well as works by Biot, the Bernoulli brothers, Lazare
Carnot, Roger Cotes, Delambre, Euler, Gauss, La Croix, MacLaurin, Newton,
Poisson, and Young. The chief deficiency
that a modern researcher would notice is an absence of mathematical
journals: no Comptes rendus, no Liouville’s
Journal, no Crelle’s Journal.
These were the lifeblood of researchers in out-of-the-way parts of Europe, entirely
missing at UVM. To be sure, the
library did have many back numbers of the Transactions
of the Royal Society in an abridgement due to Charles Hutton, but these
dried specimens of mathematics were no substitute for the living plant.
Instruction was undoubtedly by lecture and textbook
reading. There were apparently no quizzes or midterm examinations. According to
the 1844 Catalog,
the students are examined at
the close of each study by the Faculty, and also annually by the Faculty and a
committee, during the three weeks immediately preceding Commencement, in all
the studies pursued under the direction of the Faculty. The examinations are
intended to be exact and thorough, and the results in the case of each student
are noted and recorded.
The University was apparently
forced to retreat on some of its academic innovations, and the requirement of
Greek for entrance was restored in 1839. The University also returned to a
three-term academic year, with the eight-week vacation falling in December and
January.
Part of the University’s prosperity at this
time was due to the generosity of one Azarias (or Azariah) Williams, a native
of Sheffield, England, who in 1839 gave the University land
estimated in value at $25,000 in return for an annuity during the remainder of
his life. Upon his death in 1849
the University became full possessor of this land. At its meeting of May 10,
1849 the Trustees adopted the following resolutions:
Resolved, by the
Corporation of the University of Vermont, that a suitable monument of durable
material and imposing structure with an inscription commemorating the great
liberality of the deceased to this Institution be erected on some conspicuous
site at the expense of this Corporation.
Resolved, that if the consent of all persons authorized to grant it can
be obtained, the mortal remains of the late Azarias Williams be removed from
their present resting place and deposited under the proposed monument.
Like so many good resolutions, these seem not to
have been carried out. Instead, in 1853 the Trustees substituted a more modest
effort:
Resolved, that out of respect
to the memory of Azariah Williams, the largest donor as yet to this University,
the Professorship of Mathematics be styled the Williams Professorship of
Mathematics.
And it was done: F. N. Benedict, in his last year
of service, became the first Williams Professor of Mathematics. It was a
chair having all the appearance of an endowed chair, but no funds were ever set
aside to endow it. Mr. Williams’ name continues in the catalog to this
day, even though the title of Williams Professor of Mathematics was vacant from
the 1950s until the 1990s.x
§ 3. A Generation of Struggle, 1854–1885
Personnel. The
second Williams Professor of Mathematics, the Rev. McKendree Petty, was
appointed at a time of gathering danger to America, as the Civil War drew near.
The University Archives contain Rev. Petty’s diary, begun in 1844, when
he was just 17. There are not many entries, however, until 1855. (The diaries
are arranged as the days of a single year, but were used over many years, with
the year being noted at the beginning of each entry.) McKendree Petty was born in North Dorset
on July 4, 1827, graduated from UVM in 1849, then taught in an academy in
Castleton for one year, while preparing to study for the law. He journeyed to
Louisiana and obtained his law degree from LSU in 1852. He became Williams
Professor at UVM in 1854, and was ordained a Methodist minister in 1859.
Reading the Petty diaries one gets the impression
of a rather melancholy man with an overly developed conscience. Many of the
entries are full of self-reproach for his lack of achievement, for “evil
thoughts,” and other faults. It may be, however, that the diaries were
the safety valve for his gloomier side. Certainly he was much loved by his
students, who rained tributes on him after his death in 1887 and for years
afterward. The external conditions of the times, of course, were conducive to
depression. The diary entries from May 29, 1856 contain powerful emotional
outbursts evoked by the burning of Lawrence, Kansas, and the vicious attack on
Senator Sumner by Preston Brooks.
Petty, who had lived in the South, describes the horror of a slave
auction. He was strongly abolitionist, as was most of Vermont. Petty’s reflections on the
state of the world mix with his own personal problems in an entry of December
30, 1862 (a very bleak time for the Union):
The appointment received of the
corporation of the University of Vermont in 1854 is still held. The
anticipations then enjoyed have not been realized, and today the office
is one of beggary rather than of ease and competence. Thus the War which desolates
the South by the ravages of mighty armies disturbs the various relations of business
at the North and changes posts of honorable independence into undesirable
places of meager sustenance...
It
should be noted that Petty had six children at the time.
With Petty’s religious vocation, it is not to
be expected that he would make original contributions to mathematics. Only a few diary entries relate to
mathematics, and these are rather elementary parts of planned lectures. Unlike his three predecessors, he did
not teach civil engineering, which seems to have passed out of the curriculum
at this time. Despite his having
been a much-loved teacher, he sounds rather burned out in this diary entry from
June 4 (no year given):
Engaged all morning in
Differential Calculus. Find it very difficult to bring students to any
desirable interest in the study. Some succeed, many are willing to fail and do
so without any proper notion of the nature of their neglect. We are hampered by
a class of students that are here they know not for what—idle, restless,
mischief-making bodies—that bring more or less [illegible] into all
College exercises. These dead limbs, in my opinion, as well for their own good
as for our prosperity, should, after due trial, be “lopped off” and
allowed to fall to their more appropriate spheres.
McKendree Petty was forced to retire in 1885 by a
degenerative neural disease. He
lingered on for another 18 months after retirement and died in Burlington in
September 1887. Tributes to him were many. The first volume (1888) of
UVM’s yearbook, The Ariel, was
dedicated to him. In 1910, his
portrait was presented by grateful former students to Lambda Iota fraternity.
This portrait now hangs in the John Dewey Lounge of the Old Mill.xi
Curriculum. It is
not surprising that, amid the general poverty and disorganization created by
the Civil War, the mathematics curriculum shrank from its previous
dimensions. By 1867, mathematics
was confined to the first two years of study in the classical
course and its highest level was a two-month long study of calculus. The textbooks used during Petty’s
time were a series by Elias Loomis (1811–1889), professor at the City
University of New York. They are
competently written, though it is hard to see in what way they are an
improvement over those used previously.
Davies’ version of Legendre, for example, gives the parallel
postulate in Playfair’s form: Through
a point not on a given line, there can be drawn one and only one line parallel
to the given line. Loomis
adopts a slightly different wording: Two intersecting lines cannot both be
parallel to the same line. This
formulation does not rule out the possibility that parallel lines do not exist,
but that possibility is ruled out by the unspoken assumption that a line is totally
ordered and separates the plane into two disjoint half-planes. A person who wants to know only
one-variable calculus might do just as well to study Loomis’ text as
anything written since. It would
not do as preparation for engineers, however. Many subjects now covered, such as
vector calculus, were not yet invented, and no scientist should nowadays
attempt to get by without multivariable calculus.
The
Mathematical Environment. The change of greatest moment
at UVM during this period was its incorporation as a land-grant institution in
November 1865, when, under the terms of the Morrill Act, it became the
University of Vermont and State Agricultural College, offering
instruction in engineering and agriculture as well as the classical academic
subjects. (As we have seen,
however, engineering had been taught at UVM from the earliest days.) Norwich and Middlebury Colleges were offered
the chance to unite with UVM in one large institution at this time, but
declined the honor. To fulfill
its part of the law the State of Vermont was to contribute $8,000 per year to
the maintenance of the new institution. Vermont’s Senator Morrill, the
author of the act that created the land-grant colleges, protested that this was
a paltry investment, considering that the State received in return half the profits
from various gifts to the University, including $100,000 from John P. Howard,
$20,000 from John N. Pomeroy, and $200,000 from Frederick Billings. However, the State was not inclined to
hear him (Lindsay, 1954, p. 223).
One result of this new structure, shown in the 1866–67 catalog,
was the creation of the College of Agriculture, containing Sections of
Engineering, Mining and Metallurgy, and Analytical and Agricultural
Chemistry. The Agriculture College
listed among its faculty McKendree Petty and Samuel Huntington, Jr., C. E. as
Instructor in Civil Engineering.
Huntington was succeeded in the 1869–70 catalog by Volney Giles
Barbour, who left a considerable mark on engineering at UVM. (His portrait now
hangs in the Votey Engineering Building.) The entrance requirements for the new
unit do not appear particularly onerous:
Applicants for admission to the
Agricultural College must be at least 15 years of age, and must bring
satisfactory testimonials of good character, and be able to sustain an
examination in all the parts of a common school education, and particularly in
English Grammar, Geography, Arithmetic, and Algebra as far as quadratic equations...
The new
sections had some influence on the curriculum, chiefly the addition
of courses in mechanical drawing and projections. Another change, of great significance
was proclaimed in the 1871–72 catalog:
By the recent action of the
Trustees, the Academic and Scientific departments of the University are
open to young women on and after the first day of the Spring Term of the
present Academic year. Young women are admitted to all the courses of the department,
subject to the same conditions as young men. They are required to board and
room in private families approved by the Faculty.
The first women students
turned out to be of very high caliber, and two of them (Lida A. Mason and Ellen
A. Hamilton) were admitted—after some controversy—to ΦBK when
they graduated in 1875. Their
portraits now hang in the Memorial Lounge of Waterman Building, alongside that
of George Washington Henderson of the class of 1877, who was born into slavery
in Virginia, yet graduated at the top of his class. (The UVM chapter of ΦBK was
chartered in 1848.)
Yet another innovation in this period was a sort of
high school contest sponsored by UVM, not only in mathematics, but also in
other areas of science and in the classical languages. According to the
1873–74 catalog (p. 23):
By the liberality of Mrs. M. C.
Wheeler of Burlington $100 was offered last year in prizes to those who should
best sustain the entrance examinations. This was evenly divided between the
Classical and Scientific Departments, two prizes ($30 and $20) being
offered in each. A competitive examination in writing resulted as follows: In
the Classical Department, the first prize was awarded to Miss Ethel
Persis Sherman of Montgomery, a graduate of the New London (N. H.) Institution;
the second to Miss Lettie Estella Durant, of Montpelier, a graduate of the
Montpelier High School. In the
Scientific Department, the first prize was taken by Charles Wayland
Drew, of Burlington, a graduate of Burlington High School; the second by
William Thompson of Greenwich, N. Y., who was fitted for College at the
High School in that place. The questions
are reproduced here, as serving in some sort to indicate the kind and degree of
preparation which is desired in candidates for admission.
The reference to the desired preparation reflects
the fact that students no longer had to be examined by the UVM faculty in order
to be admitted. The University was
willing to accept the certification of a high school. Here are a few
sample mathematical questions. Notice that mathematical preparation was
demanded of students in both classical and scientific divisions.
ALGEBRA:
Classical Division
5. Divide
by 
.
8. Solve the
equation 
.
ARITHMETIC:
Scientific Division
5. Cube root of 
?
ALGEBRA:
Scientific Division
3. Reduce 
.
The goal of the mathematical curriculum was stated
in the 1878–79 catalog:
The high importance of thorough
mathematical training, both as a logical gymnastic and as preparation for
practical life, is constantly kept in view. Instruction is given by means of
Recitations and Lectures in Pure and Mixed [Applied] Mathematics during the first
two years. More advanced practical courses are pursued in the Department of
Science.
A Digression: Religion at UVM. Although mathematics is not directly affected
by the religious character of the institution, it may suffer
indirectly if the professors of mathematics are also clergymen, whose time
outside of their teaching duties is likely to be spent on their religious
duties rather than on the cultivation of mathematical research. Such was certainly the case with the
Rev. Ebenezer Burgess, and with McKendree Petty, who taught an entire
generation of UVM students.
Mathematics also suffers if
part of the population is excluded from the opportunity to study it on
religious grounds, as was certainly the case throughout most of the history of
UVM. The position of religion at
UVM during the nineteenth century is bound to seem anomalous to the twentieth
century. Although founded as a
secular institution and granted public lands for its sustenance, the University
was unabashedly sectarian in its administration. The by-laws of the University from 1885,
Chapter III, Sect. 2, p. 11, state:
There shall be public prayer in
the Chapel every week-day morning at such hour as the Faculty shall direct,
preceded by the reading of a portion of the Scripture, and other devotional
exercises at their discretion, which service it shall be the duty of the
Faculty and students regularly to attend.
Sect. 3. Every student shall
attend public worship on the Lord’s day. During that day nothing shall be
done which would disturb the quiet, or interfere with the religious observance
of the day. All days devoted to religious purposes by the Government of the
land shall be observed in such manner as may be prescribed by the Faculty.
These rules were surely a
deterrent to the enrollment of Catholics, for whom deliberate participation in
Protestant worship was a mortal sin until the late 1960’s. This may account for the paucity of
French surnames among the students and faculty at UVM, despite the fact that
the French formed a significant portion of the population of
Vermont. One cannot help
wondering whether the people who made these rules ever gave any thought to the
proposition that it might be unjust to take taxable land from the entire
population and bestow it for the benefit of one particular religious
group. The most likely explanation
is that they were simply afflicted
with the kind of cultural blindness that never really recognizes that
reasonable people can differ radically from
oneself in such matters. Such
blindness persists even today; one frequently reads statements by public figures
which take it for granted that everyone in America is Protestant, Catholic, or
Jewish. The relaxing of these rigid
rules came about piecemeal. The
Faculty, in its wisdom, exempted itself from compulsory Chapel in 1892. Medical students, apparently being
regarded as incorrigible, were exempted in 1912. Only in 1916 was the existence of
non-Protestants rather grudgingly recognized, with a statement that,
“Students who, because of their religious affiliation,
object to attendance at Chapel, are excused by the President upon request being
made to him.” Compulsory
Chapel was (at last!) abolished in 1920.
§ 4. The Beginnings of Growth, 1885–1914
Personnel. On a
national scale, one of the more interesting phenomena in the period from the
end of the Civil War until World War I was an exodus of well-to-do young Americans
to France and Germany in pursuit of an education. The German universities,
which were considered the finest in the world, were particularly
hospitable to young Americans, and such mathematicians as Felix Klein and Carl
Neumann had many American students. The long-term effect of
this Germanization of American universities was a decreased emphasis on the
British-style education for public service and an increased emphasis on
research as a duty of the professor, an effect
that was reinforced later by the large number of scholars who were refugees
from the Nazis. The full effect was
felt in the last half of the twentieth century. The first generation of
European-educated Americans for the most part did very little research
themselves. Instead they acquainted
their students with the contemporary trends in the most advanced centers of
mathematical research.
Among the young Americans who obtained part of
their education in Europe during this period was McKendree Petty’s
successor as the Williams Professor of Mathematics, Archibald Lamont
Daniels. He was born in Hudson,
Michigan in 1849 and obtained the A. B. degree at the University of Michigan in
1876. He then went to Gőttingen
and Berlin, where he took courses from such great mathematicians as Schwarz and
Weierstrass. He returned to America
in 1881 and worked on a fellowship at Johns Hopkins until 1883. Hopkins at the time was only slightly
inferior to the great European universities. It had the best of the early American
research mathematicians: Simon
Newcomb (actually a Canadian), W. E. Story, Fabian Franklin, and Thomas Craig;
and it had enjoyed the services of the great British mathematicians J. J.
Sylvester and Arthur Cayley. In
1883 Daniels became instructor at Princeton University, taking over the
teaching duties of Henry Burchard Fine, who had gone on leave. He received the D. Sc. degree from
Princeton in 1885. He became
instructor in mathematics at UVM that year, and the following year, upon the
retirement of McKendree Petty, he became the third Williams Professor of
Mathematics.
Daniels’ research record
is consistent with that of other “first-generation” American
scholars. It consists of three expository
articles in the American Journal of
Mathematics, Vols. VI and VII (1884–85) explaining Weierstrass’
theory of elliptic functions for the benefit of Americans. Daniels has the distinction of being the
only UVM nineteenth-century faculty member at UVM listed in Poggendorf’s Biographisch-Literarisches Handwőrterbuch
zur Geschichte der exacten Wissenschaften (Bd. 4, 1904, p. 295). Poggendorf credits him with 4 years of
study in Gőttingen, 2 in Berlin, and 1 at the Johns Hopkins University,
“Cambridge, Mass.” A.
L. Daniels preserved the notes from Weierstrass’ course on analytic
function theory, which were not published with Weierstrass’ collected
works and are still of great interest to historians of mathematics. These notes passed to his grandson, R.
V. (Bill) Daniels, who recently retired from UVM after a career of distinction
as a specialist in Soviet history and a period of service as a state Senator
from
Daniels lost no time in
revamping the curriculum to suit his own background, using the textbooks of
Simon Newcomb (1835–1909) as the base. This period being a time of expansion
for the University, Daniels regularly had the assistance of an instructor in
teaching the courses. The most
interesting of these was Frederick Merritt Corse, who received his A. B. at UVM
in 1888, then became Curator of Buildings, Secretary of the Faculty, Registrar,
Instructor in Mathematics, and Instructor in Political Economy over the period
1891–1894 while also obtaining a Master’s degree from
Mr. Corse was followed by two
other instructors, one of whom, Allison Wing Slocum, was actually a professor
of physics, but taught courses in the Mathematics Department as well, leaving
A. L. Daniels free to teach more advanced courses. A. W. Slocum (1866–1933) was born
in Dartmouth, Massachusetts, obtained a bachelor’s and master’s
degree from Haverford, then went on to Harvard to obtain a second
master’s degree. He was Tyndal scholar at Harvard in 1890–1891 and
Morgan scholar at the University of Berlin in 1891–92.
There are two other instructors
in mathematics during the 1890’s, Warren Gardiner Bullard (1896–97)
and Arthur Dexter Butterfield (1897–1900). I have not found any information on W.
G. Bullard. A. D. Butterfield
became an assistant professor of mathematics in the Engineering Department in
1900, the first official indication that UVM
now had two mathematics departments, which were bound to be rivals for
personnel and resources. Those resources had increased during the
1890’s. The minutes of the
Board of Trustees’ meeting of December 1, 1891 (Vol. IV, p. 224) note
The bequest of fifty
thousand dollars to the University to found a Professorship of Mathematical,
Natural, or Technic Science, by the will of the late Hon. Edwin Flint of Mason
City, Iowa, and alumnus of the class of 1836, was announced by the President.
The first Flint Professor
of Mechanics and Bridge Engineering was Volney Giles Barbour. Only once has this professorship been
held by a mathematician (Percy Fraleigh, during the 1950’s).
A. D. Butterfield was a hydraulic engineer,
born at Dunstable, Massachusetts in 1870.
He received a B. S. degree from Worcester Polytechnic Institute in 1893,
and an M. S. (also from WPI) in 1898.
He went on leave from UVM in 1904 to obtain an A. M. from Columbia. He finally obtained a Doctor of
Engineering degree from WPI in 1945, after retiring from UVM in 1942. In 1945 he became director of
Veteran’s Education at UVM, a post he resigned in 1948 because of a
desire to engage in other work. He
was an acquaintance of Atwater Kent, from whom he requested money to build an
observatory at UVM. (Apparently, the money was not forthcoming, since the
observatory was not built.)
Curriculum.
Despite the turnover in personnel, one has a sense of stagnation in the
curriculum during the 1890’s.
As mentioned above, the elementary mathematics courses were taught by
Prof. A. W. Slocum of the physics Department, leaving Prof. Daniels free to offer
advanced courses in geometric function theory with a sketch of the theory of
elliptic functions and integrals.
He also offered a course in
ordinary and partial differential equations and
one in projective geometry, taught from the textbook of Cremona. At this point the UVM curriculum seemed
to be acquiring real sophistication.
Daniels’ course on analytic function theory was supplemented with
readings from Durège’s book on Riemann’s theory of complex
variables. Heinrich Durège
(1821–1893) was a professor at the University of Prague. His book went through at least four
German editions before being translated into English. Although the notation has
changed considerably and is now couched in terms of manifolds, the classical
examples upon which an intuitive understanding of Riemann surfaces must be
based are discussed with great clarity in this book. This course, however, was offered
irregularly. The best mathematics offered
during this period was taught in the Physics Department. Slocum offered a
course in mathematical physics using as texts Riemann’s Partielle Differentialgleichungen—would
we dare to use a textbook in a foreign language nowadays?—and
Fourier’s Analytical Theory of Heat.
He also used Maxwell’s Heat and
Duhem’s Potentiel thermodynamique.
In fairness, however, it must be said that Daniels reciprocated by offering a
course in “Heat, Magnetism, and Electricity.”
For unknown reasons, Prof. Daniels ceased to teach
the more advanced courses after 1904. The academic students (exclusive of
Medical students) numbered about 350 at this time, 164 of whom were first-year
students. These figures
suggest a high dropout rate. Whether mathematics students contributed to this
high failure rate is unclear.
Certainly the admission standard for the University (as distinct from
the Agricultural College) in 1905 was rigorous enough, consisting of
(1) Arithmetic, including the metric
system; (2) Algebra, including the four species, factoring, largest common
divisor and lowest common multiple, fractions, theory of exponents, involution,
elementary forms of binomial theorem, evolution, surds, simple equations with
one, two and three unknown quantities, simple quadratic equations. In the
instruction the aim should be the formation of the habit of clear and concise
expression, and to this end the class room work should be largely oral.
II. Solid and Spherical
Geometry.
The following year these
requirements were expanded to include trigonometry, simple permutations and
combinations, determinants, linear equations, graphical treatment of equations,
Descartes’ rule of signs, and Horner’s method “but not
Sturm’s functions or multiple roots.” (Descartes’ rule of
signs gives an upper bound on the number of positive and negative roots a
polynomial with real coefficients can have. Sturm’s functions, modifications
of the remainders that arise in finding the greatest common divisor of a
polynomial and its derivative, can be used to find the exact number of
real roots in a given interval, provided the polynomial has no repeated
roots.) Admission by high school
certification is now supplemented by the College Board Examinations.
The engineering mathematics
department began to grow in 1905, adding assistant professor George Monroe
Brett to the faculty. (I have not
been able to obtain any information on Brett.) In 1908 the engineering mathematics
faculty gained professor Evan Thomas, whose articles in the Vermont Alumni Magazine are my main
source for the biographies I have sketched here. As one might guess from his names, he
was born in Wales, in the Rhondda Valley made famous in the novel How Green was my Valley. In 1867, at the age of 14, he was
apprenticed to a firm of clothiers, and his parents emigrated to
Ohio. Visiting his parents at the
age of 18, he decided to enroll in Dennison University, where he obtained the
B. S. degree in 1876. He became a
Congregationalist minister at Vershire, Vermont in 1880, then spent 3 years as
a pastor in the Ludlow/Plymouth area, where he also managed the local
newspaper. He came to Essex
Junction as pastor in 1886. In 1892
he taught a semester of mathematics at UVM. Eventually he became head of engineering
mathematics and mechanics at UVM, retiring in 1928. He was the author of several articles on
pedagogy, as well as Chapters XVII and XVIII in the second edition of Walter
Hill Crockett’s History of Lake
Champlain, published by McAuliffe Paper
Company, Burlington, 1936. Prof.
Thomas’ chapters tell of the raising of two of Benedict Arnold’s
ships, which were sunk by the British in October of 1776, and of the
construction of the bridges across Lake Champlain. He died in 1947.
Growth continued in the Engineering Mathematics
Department until 1914. There seems
to have been some general reorganization of UVM around this time. After 1911 the catalogs refer to the
“College of Arts and Sciences” and the “College of
Engineering,” where previously these units had been referred to as the
“Departments of Arts and Sciences” and the “Department of
Engineering.” In 1914 A. L.
Daniels retired from the Academic Mathematics Department, bringing an end to
the first phase of UVM’s hesitant steps toward membership in the
worldwide mathematical research community.
§ 5. Another Generation of Struggle, 1915–1954
Personnel. A. L. Daniels’ successor,
the fourth Williams Professor of Mathematics, was Elijah Swift, like Daniels a part
of the generation of young Americans who obtained their education in Europe.
Being about twenty years later than Daniels, however, his mathematical career
was correspondingly more prominent. He was born October 23, 1882 in Buffalo,
New York, and graduated from Harvard (A. B.) in 1903. In that year he presented
a paper before the American Mathematical Society, “On the condition that
a point transformation of the plane be a projective transformation,” published
in the 1904 Bulletin of the American
Mathematical Society. He then
went to Gőttingen, where he studied under Hilbert. (His notes from Hilbert’s
1905–1906 course in Integral Equations are now kept in the Mathematics
Department.)xii He
received the Ph. D. degree Magna cum Laude from Gőttingen University in 1907 for a dissertation,
Über die Form und Stabilität gewisser
Flűssigkeitstropfen,” (On the form and stability of drops of certain
liquids). Then, like A. L. Daniels,
he took up a position at Princeton, from which he was hired by UVM. Although I
have not searched exhaustively for his publications, it is clear that he has
several, in various areas of mathematics.
In the Bulletin of the American
Mathematical Society, Vol. XIV (1908), he published a “Note on the
second variation in an isoperimetric problem,” and in the American Journal of Mathematics, Vol. L
(1928), he published, “Canonical forms for ordinary homogeneous linear differential
equations of the second order with periodic coefficients.” Clearly, in Elijah Swift UVM had found
one of the best American mathematicians available at the time. In 1931, he became Dean of the College
of Arts and Sciences, retiring from that position in 1948. He died July 21,
1957.
Along with Prof. Swift, the
Mathematics Department of the College of Arts and Sciences hired James Edward
Donahue. These two were the core of the Mathematics Department for the next 17
years. J. E. Donahue was born in Fairfield, Vermont on April 25,
1880. He graduated from Burlington
High School in 1897 and from UVM in 1902.
He obtained the M. A. degree from Harvard in 1910 and remained there
until 1912, when he became an instructor at Washington University in St.
Louis. After three years at UVM he
joined the navy for the duration of America’s participation in World War
I, after which he returned to the Mathematics Department. In 1930 he went on leave from UVM and
obtained a doctoral degree from Columbia University in 1931 for a dissertation
“Concerning the geometry of the second derivative of a polygenic
function,” written under the direction of Edward Kasner. Tragically, he lived only one year after
obtaining this degree, dying of a cerebral hemorrhage while on vacation in
Maine in August 1932.
Swift and Donahue were joined
by a succession of instructors, and gradually more and more professors were
added. In 1920–21,
Instructors Howard Guy Millington and Fred Walter Householder were added, both
of whom later became assistant professors. Millington also had an appointment
in the Engineering Mathematics Department.
H. G. Millington was born August 28, 1887 and received the B. S. degree
from Rensselaer Polytechnic Institute.
He became Assistant Coordinator of Civilian Pilot Training at UVM in
1942. He retired from UVM in 1954 and died on February 25, 1965. F. W. Householder was born in Jackson,
Tennessee, April 7, 1884. He
received the B. A., M. A., and LL. B. degrees at the University of Texas. He was actually an historian who
happened to know some mathematics.
In those post-War years, as Swift later explained in a letter to President
Millis, mathematicians were hard to find. Householder seems to have worked out
well at first, but suffered
from “burn-out” in the 30’s. (Heath Riggs, who took courses from him,
confirmed that this judgment of Swift’s is accurate.) Householder went to California to work
in the shipbuilding industry at the beginning of World War II, and Swift took
advantage of this situation, together with the University’s precarious financial
position (all salaries had just been reduced by 25%) to urge him to resign. Householder
had been replaced by Douglas T. McClay, a Harvard graduate, about whom I have
not been able to learn anything. I
do not know the date of Householder’s death.
The year 1923 saw the addition
of the longest-serving member of the UVM faculty. George Hubert Nicholson, who was born on
Prince Edward Island, attended Mount Allison University in New Brunswick, where
he received the A. B. in 1922. When
the Canadian scholarship he had been hoping for was preferentially awarded to a
veteran of the war, he went to Harvard to obtain a master’s degree. At UVM he taught a heavy load (12 hours
per week) for fifty years. (Although
he officially retired in 1963, he continued to teach part
time until 1973.) With such a heavy
teaching load, he had no time for research, although in 1940 he did write a
brochure on mathematical instruments, which is still in his file in the
UVM Archives. Shortly after he came to UVM some students asked him to coach a
hockey team. He did so, spending
$300 of his own money to flood a field near the site of the present
library, and thus became the founder of the UVM Hockey Team. When he retired for good in 1973, the
old Lucy Ann Abbott house, then the home of the Mathematics Department, was
renamed the Nicholson Building in his honor.
Another instructor, added in
1924, was Horace Alpheus Giddings, who was born in Farnworth, New Hampshire in
1902. He had obtained the B. S.
degree at the University of New Hampshire in 1923. He remained at UVM until 1930.
In 1928 Percy Austin Fraleigh became
an assistant professor, bringing the total faculty to 7 (four professors and
three instructors). Fraleigh was
born April 12, 1895 at Hyde Park, New York and received the degrees of A. B.
(1917), A. M. (1918), and Ph. D. (1927) at Cornell. He remained at UVM until 1963. During his last thirteen years at UVM he
was the Flint Professor of Mathematics, the only mathematician to occupy this
endowed chair. He died in the early
1980’s (I am uncertain of the exact date). The next new professor, James Atkins
Bullard, like Millington, was appointed in both the Arts and Sciences and
Engineering Mathematics Department, replacing Evan Thomas in the latter. He was born in Parsippany, New Jersey on
February 3, 1887 and received the Ph. D. from Clark University in 1914. He was an instructor at Worcester Polytechnic
Institute until coming to UVM in 1928.
Sometime during the twenties he was co-author (with Arthur Kiernan) of a
trigonometry text. He remained at
UVM until 1953, becoming the fifth and last Williams Professor of
Mathematics in 1944 upon the resignation of Dean Elijah Swift from the
Mathematics Department. He came out
of retirement for two years in the mid-50’s. He died in Parsippany, New Jersey on
April 10, 1959.
The last of the new faculty,
Myron Ellis Witham, a civil engineer, was hired in 1932. He was born October 29, 1880 in
Rockport, Massachusetts. He was a prominent football player at Dartmouth,
indeed the hero of its 1903 team.
In addition to teaching mathematics, he also coached the UVM football
team and taught physical education. He died in Burlington in 1972.
Throughout the economically
lean years of the 30’s and the war years the faculty members just listed
kept mathematics going at the University of Vermont. There was little, if any, innovation in
curriculum during this time and essentially no research. Not until the postwar years did the
faculty begin to increase again. In
the new expansion of the department we find the first woman to
teach mathematics at UVM. She was
Ruth Gertrude Simond, who was born March 7, 1904. She received the B. A. and M. A. degrees
from Boston University and the Ph. D. from the University of Michigan. She then taught at Hampton Institute,
Berea College, Heidelberg College (Tiffin,
Ohio), and Morningside College (Sioux City, Iowa) and served as a cryptoanalyst
for the Navy during the war before coming to UVM as assistant professor of
mathematics in 1948. She died
September 15, 1958, having apparently resigned her position because of ill
health the previous June. She is
buried in Franklin, New Hampshire.
The University hired an assistant professor,
William Thompson Fishback, and an associate professor, William Scribner
Kimball, in 1951. I was able to
contact Professor Fishback through my colleague Professor Dan Archdeacon, who
had been his student at
Curriculum. The curriculum was revised when Swift and Donahue
took over, but still remained quite rudimentary. Calculus was followed by Synthetic Geometry
and Theory of Functions. That was
the entire curriculum. The
Engineering Mathematics Department continued to teach mechanics and differential
equations and duplicated the teaching of analytic geometry and calculus. (There was a great deal of duplication
at this time; the Engineering College even had its own English
Department.) The large growth in
faculty discussed above reflects the growth in the size of the student
body, rather than any growth in curriculum. The courses that were added during this
time, while of practical value to the students, no doubt, were not by any means
advanced mathematics courses. Mathematics of finance was added in 1932,
and a course in the teaching of algebra and geometry in 1934. The admission requirements of 30 years
earlier remain in the catalog unchanged.
The impression of stagnation created by this curriculum is confirmed
by a person who actually took the courses.
Heath Riggs, who graduated from UVM in 1940, told me that he took mostly
physics courses, since there wasn’t much of interest in the Mathematics
Department. He describes Fraleigh
and Nicholson as excellent teachers and competent mathematicians, but confirms
what other sources suggest, that Bullard and Householder were incoherent lecturers.
In summary, this was a period when the University
was overworking the teaching faculty in order to serve a large number of
students, leaving the faculty little time to develop new courses and no time
for research. Not until the end of World
War II did some real updating of the curriculum begin, and then only
slowly. Courses in advanced
calculus, differential equations,
complex variables, and infinite series were added in 1944. A course covering Lebesgue integration
appears for the first time in 1946.
The first step into graduate education appears in 1947, with the
addition of a Master’s thesis course.
The
Mathematical Environment. As already noted, teaching loads were
too heavy to allow mathematical research.
Mathematics was solely a service discipline at this time. In fact, the entire department was
merged with the Engineering Mathematics Department to form a single Department
of Mathematics and Mechanics in the College of Technology in 1946. (An oral tradition that I have not
verified asserts that this move was made partly because of the terms of the
Wilbur Fund. This fund, which
provided money for the College of Arts and Sciences, specified that that
College not grow beyond a fixed number of students. With mathematics majors being counted as
engineers, the College had some room to grow in other areas.) The new department consisted of
professors Bullard and Fraleigh and assistant professors McClay, Millington,
Nicholson, and Larivee. (Jules
Alphonse Larivee is the only professor of apparent French or French-Canadian
descent ever to belong to the Mathematics Faculty. I have not been able to learn much more
about him. Professor Fishback told me that Larivee came from
Students. It becomes increasingly difficult to
trace the present whereabouts of UVM alumni in the twentieth century, partly
because of their large numbers, and partly because the University has not
published a general catalog of them since 1902. One who absolutely must be mentioned,
however, is John Francis Kenney of the class of 1920. He became a professor at Northwestern
and the University of Wisconsin. He
was a mathematical statistician who published many articles in this area and
wrote a definitive two-volume textbook The Mathematics of Statistics, published by Van Nostrand, which
went through at least two editions.
The second edition was published in 1947. Volume two (1951) was written
jointly with E. S. Keeping. After
retiring from the University of Wisconsin, he came back to Brandon to
live. In 1967 he gave the
University $5,000 to establish an annual prize in memory of his parents for the
best graduate work in mathematics.
The Kenney prize has been awarded annually ever since.
§ 6. UVM Moves Toward the Mainstream, 1955–1965
Personnel.
Universities grow like arthropods, taking in more students until the
shell bursts, then shedding the old exoskeleton and growing a new one. The year 1954 saw the retirement of J.
A. Bullard and G. H. Millington. It
was also the last year for Assistant Professor Larivee and Associate Professor
W. S. Kimball. It was time for both
expansion and renewal. The new
faculty was begun with the hiring of Heath Kenyon Riggs (d. 2011) and Ivan
Raymond Hershner, Jr. (d. 2005) in 1953.
Riggs, a 1940 graduate of UVM, had spent a year as Research Assistant in
the Department of Mathematics, then served as Director of Admissions before
departing for graduate study at the University of Chicago. They soon set out renovating the
curriculum. The following year
Julius Solomon Dwork (d. 2002) was hired as associate professor and Roland
Frederick Smith as assistant professor.
In 1955 Harry Lighthall, Jr. (d. 1975) was hired as an instructor, and
in 1956, when Prof. Smith left, Joseph Anthony Izzo, Jr. (d. 1999) was hired as
assistant professor. At that point
Professor Hershner left to work for the Pentagon. (He retired in 1980 at the
rank of Colonel and taught at George Mason University until 1985, when he
retired irrevocably. From speaking with him on the telephone in August of 1990
I had the impression of a very vigorous man; he was about to leave for a
vacation in Moscow.) Professor
Hershner’s replacement as head of the Mathematics Department was N. James
Schoonmaker (d. 2009). In 1959
Lighthall became assistant professor, and Donald E. Moser (d. 2003) joined the
Department as associate professor.
When Professor Fraleigh retired in 1963, Erling William Chamberlain was
hired as an assistant professor.xiii The following year Bruce Meserve (d.
2008), a specialist in mathematical education, was hired as full
professor. All these professors had
been educated at excellent graduate schools. They began the rather formidable
task of moving UVM into the mainstream of contemporary mathematics, a process
that continues down to the present day.
At this point we leave the story of the UVM
personnel. The first step
toward the mainstream, as it turned out, was an enormous expansion in the
faculty. In the year 1966 no fewer
than seven new assistant professors were hired, including the first
specialists in statistics and computer science. Since that story is still being written
by people at work at UVM today, it must wait for a future historian. In very brief summary, when the
University was reorganized in 1973, the faculty of the Department resisted a
proposal that Mathematics be moved into the College of Arts and Sciences,
preferring to remain part of the College of Technology, which was renamed the
College of Engineering, Mathematics and Business Administration. Since various departments all over
campus had found need of statistics and were hiring their own statisticians to
teach this subject, a Program in Statistics was organized, to run in parallel
with a Program in Computer Science.
Five years later, in an administrative reorganization of the College,
Computer Science was merged with Electrical Engineering and Statistics with
Mathematics, forming the present Department of Mathematics and Statistics. As of the present writing it seems
possible that the next reorganization will once again divide the two areas and
create a Department of Statistics, though this change may not come for several
years.xiv
Curriculum. Professor Hershner recalled that the
major task facing the Department was to rebuild the curriculum. Riggs, who obtained a Ph. D. degree from
the University of Chicago, where he studied under Marshall Stone, Irving Kaplansky,
and Antoni Zygmund, was particularly interested in expanding the algebra offerings. A course in groups, rings, and fields
was taught for the first time in 1955. The following year a senior problems
course was added, along with courses in computers and numerical analysis. Computers and numerical analysis, of course,
go hand in hand. Professor Riggs
spent a leave of absence studying computational mathematics at the University
of California at Berkeley in 1965–66, afterwards introducing the course
in numerical analysis that is one of the core elements of the computational
part of the curriculum.xv
The curriculum expanded enormously during the early 1960’s, with
courses being added on group theory, Galois theory, probability, topology, differential
geometry, number theory, foundations of geometry, computers, and numerical
analysis.
Research. While the curriculum was being
modernized it was logical to bring UVM into the mainstream of mathematical
research. This movement was slower,
since research had never been an important component of UVM’s educational
mission in general. The Graduate
College at UVM was formed only in 1953; at that time only the Medical College offered a
doctoral degree. Prof. Hershner
told me that there was only one master’s degree student in mathematics
when he came in 1953. When he left
in 1956 there were several. The master’s
programs continued to expand throughout the 50’s and 60’s, and
planning was begun for a Ph. D. in applied mathematics and statistics.xvi At any
rate, research, which had not previously been an issue in tenure and promotion
cases, was strongly weighted by Clint Cook (d. 1969), the Vice-President for
Academic Affairsxvii during the 1960’s. It has assumed increasing importance
with every change of administration since.
Conclusion
Strictly
speaking, one cannot come to the conclusion of a story that is not yet
over. The heading of this section
indicates only that the narration has reached a conclusion. To anyone who is disappointed that
I have not carried the story down to the present day, I offer the
triple excuse that (1) the story is still being written by people who would
probably disagree about its meaning if asked, (2) the story is enormously more
complicated from 1965 on because of the increase in personnel and activity, and
(3) I am myself a participant in that activity, hence unqualified to judge
it. The most I will say is that UVM
has followed the vast majority of American universities in moving away from an
academic model of organization and in the direction of a corporate model. Along with that change has come an increased
emphasis on obtaining funding from external sources for faculty research as an
obligation of tenure-track faculty.
Indeed, many faculty feel that this obligation trumps everything else in
tenure, promotion, and salary decisions.
Along with it comes a movement away from the traditional academic ideal
of a community of knowledge, to be shared with all, and toward the concept of
knowledge as a commodity in which the University has a proprietary interest.
References
LINDSAY,
JULIAN IRA
1954 Tradition Looks Forward. The University of
Vermont: A History 1791–1904, published by the University of Vermont.
NEUGEBAUER, OTTO
1975 History of Ancient Mathematical Astronomy,
3 Vols., Springer, New York.
PYCIOR, HELENA
1988
“British Synthetic vs. French Analytic Styles of Algebra in the Early American
Republic,” in: The History of Modern
Mathematics. Vol. I: Ideas and their
Reception, Edited by David E. Rowe and John McCleary, Academic Press,
Boston, pp. 125–156.
WHEELER,
JOHN
1854 Historical Discourse, Free Press Print,
Burlington.
i (Note added in October 2007)
Prof. Nicholson died at the age of 97 in 1995.
ii A student who could meet these
requirements nowadays would be considered as already having completed a good
portion of a classics major!
iii Vulgar arithmetic means the arithmetic of common fractions, as
opposed to decimal fractions.
iv Speaking
only for myself, I have found that the chief difficulty in coming to an
appreciation of Milton’s Paradise Lost was occasioned by the
numerous allusions to rivers and mountains from ancient mythology, geography
that was familiar to educated people in Milton’s day but meant nothing to
me.
v My late
colleague from the Physics Department, Professor Al Crowell, gave a detailed
discussion of this paper of Dean’s in 1971 (see the Burlington Free Press, March 11, 1971).
vi Benjamin
Silliman (1779–1864) founded this journal, officially known as the American
Journal of Science, in 1818.
Silliman’s son assisted him in the editing after 1838.
vii Perhaps
the explanation is the paucity of reading material available to students in
those days.
viii Another
mathematician of even more naive views, J. J. Callahan, President of Duquesne
University in the 1930’s, announced that he had proved the parallel
postulate. His “proof,” published in a book titled Euclid or Einstein, cuts through the difficulty
in exactly the same way. Apparently
Callahan never realized that Hutton had anticipated him. He of course failed to
show that his definition of parallel lines was equivalent to
Euclid’s. His book, by the
way, is mostly a scurrilous personal attack on Einstein.
ix (Note added in October 2007) F. N. Benedict is believed by most
authorities to be the author of an anonymous article “The Wilds of
Northern New York,” published in Putnam’s
Monthly: A Magazine of Literature, Science, and Art, Vol. IV, No. XXI,
September 1854. That article was
reprinted by Purple Mountain Press, Fleischmanns, New York in 2001, with a
preface by Sandra Weber.
x (Note added in October 2007) In the year 2000, through
an amusing error, the present author was honored with this august title. The error was only recently discovered
by my friend Jeff Dinitz. The
position of Williams Professor was not
vacant at the time. It had been
bestowed some years earlier on my close friend and colleague Ken Gross, who
richly deserved it. The final irony
is that the illegal transfer of that honor to me was championed by none other
than Ken Gross himself, who, like the rest of the Department, had forgotten
that he held the title. The
Williams Chair has now been properly and legally and deservedly presented to
our colleague Ken Golden.
xi (Note added in October 2007) In 2004, Rev.
Petty’s great-grandson Edward H. Worthen wrote and published a beautiful
and detailed biography of McKendree Petty, including a very complete genealogy
of the Petty family, under the title McKendree,
a copy of which he kindly presented to me.
xii (Note added in October 2007) I presented those notes to the Rare
Books Room at the library in 2003.
At the moment, they do not seem to be in the library’s catalog,
however.
xiii (Note added in October 2007) Professor Chamberlain
retired in 1996.
xiv (Note added in October 2007) The Department remains
united, as the Department of Mathematics and Statistics. The College has reorganized since the
departure of Business Administration to form its own school. It is now known as the College of
Engineering and Mathematical Sciences and includes engineering, mathematics and
statistics, and computer science.
xv The
story of computing at UVM deserves more space and more prominent attention than
it can get in a general history of mathematics. In brief, here is what happened: In 1955 IBM announced plans to build its
704 computer at MIT and share the time with other New England Colleges. One-third of the time was reserved for
IBM, one-third for MIT, and the remaining third was apportioned among the other
New England Colleges. Prof. Riggs
accordingly spent some time in Cambridge learning about computing in general
and the 704 in particular. This was
in the days before FORTRAN, and programming had to be done in symbolic
language, very close to machine language.
Later Prof. Dwork wrote a
discretized version of the differential equations that describe the movement of
weather patterns, and Prof. Riggs and the head of the Weather Service at
Burlington took these programs to MIT to run, thus carrying out some of the first
attempts at computerized weather forecasting.
In 1960 President Fey of UVM
offered to buy an IBM 1620 for the University, provided he could be assured the
faculty would use it. Prof. Riggs
obtained affidavits from 35 faculty members asserting their intention to
use the proposed machine, which was accordingly bought. Prof. Riggs then gave
an Evening Division course in computing to a broad group of engineers,
teachers, and government workers from all over Vermont and thereby launched
Vermont into the computer era. From
this small beginning, UVM (like every office in the world) has been completely
permeated by computers. There are
now thousands of powerful computers on the campus, counting PC’s, work
stations, and mainframes.
xvi Such a
program was begun in 1970, and about six doctoral degrees were granted before
the program was closed in 1976. A carefully
prepared plan for a Ph. D. program was approved at all levels except the Board
of Trustees in 1990, but then-President Davis, engaged in planning for fiscal
austerity, refused to present it to the Board.
(Note
added in October 2007). The program
was approved in the early 1990s, and UVM has since produced a number of very
respectable Ph.D.s, whose dissertations have been published in reputable, even
prestigious, journals.
xvii (Note added in October 2007) This office is now
called Provost.