## Lost Geometry It is a great mistake to suppose geometry any substitute for logic. - Coleridge When I was small, growing up in Wisconsin, I loved to walk
along the railroad tracks. As I walked, I would watch the steel rails grow from
a point in the distance ahead of me, sweep around me, and then disappear again
in the distance over my shoulder, converging slowly back to a point. The pure
geometry of it was breathtaking. What impressed me the most, however, was the
powerful metaphor that it suggested: How wide the present seemed, simply
because of my presence there; how small the future and the past. And yet, I
could move along the tracks, imagining myself expanding and contracting the
infinite timeline of history. I could move ahead until any previous place along
that continuum had shrunk to insignificance, and I could, despite the
relentless directionality that I imagined moving along the tracks like so many
schedule-bound trains, drift backwards as easily as I could let myself be
carried forward. Sometimes I would just stop. I would crouch down in some
nondescript other time and widen the place out for a closer look. It seemed
marvelous how uniquely full of life I went on to spend my undergraduate days at MIT, whose
long corridors reminded me more than a little of the railroad tracks that I had
left behind in Wisconsin. At night, the corridors were usually empty, and I
felt at home as I walked along them, watching the walls expand and contract.
There were places where laboratories were filled with incomprehensible
collections of equipment, where bulletin boards were posted with pages of
beautiful scribbles, and where doors stood ajar to dimly lit offices in which
old men pored over stacks of scattered paper. Walking those corridors, I was
filled again with the idea that I could stop One night, my wanderings took me to the cases of
mathematical models that lined the corridors of the Mathematics Department. I
had no particular plans to study mathematics at the time — I was going to be an
astronomer. And yet, I paused there much longer than usual, drawn in by the
stark dignity that the models maintained beneath their old brown bulb; taken,
finally, by the unspeakable complexities that their forms suggested. The models
cast a spell on me. I came back to that same place, night after night. I stood
there and stared. The models, it said, were I became a mathematician, of course. Not a geometer, but a
logician — at least that is the pedigree I inherited during my graduate
education. I’ve wondered about that choice many times. Not about being a
mathematician, for I can’t imagine being anything else, but about choosing
logic over geometry. What made me leave behind the soft curves of the geometric
models, nestled in their velvet display cloth and incandescent half-light, for
the razor sharp lines and fluorescent lighting of the weekly logic seminar? I
thought in those kinds of images, and I felt that I made the choice
consciously: I wanted to live in the daylight; I wanted order. There is something
very personal in trying to understand that decision, but it is a personal
version of something more universal: trying to understand the mathematical
structures that we all impose on a non-mathematical life. Suffice it to say
that I went on to find geometry in other places. I came to the University of Arizona in the fall of 1994,
and one of the fist things I noticed was the Math Department’s display of
mathematical models. It looked very neglected. There were some things in it
that I had never seen before, some fantastic things, and yet students and
faculty seemed to skirt around the display without ever giving it a glance. It
was The models at the University of Arizona as they appeared
ca. 1995. There seemed to be something a little sad about the U of
A’s collection, back then. The models were not, in general, of the finest
quality. Many of them, you could tell, had been constructed by students whose
mathematical enthusiasm did not quite translate to their hands. Several of
those models were in the process of self-destructing, or had long since done
so. There were also many commercially produced models, slick and smooth, but
they were mostly of very simple surfaces and polyhedra, and the comical
pretense of applying such high production values to such elementary ideas made
them seem quaint. Other models were of indeterminate origin. Some of them
seemed to represent the most arcane geometric propositions in the display, but
their bases were covered with clumsy layers of thick paint, and their curves
strove for an unfulfilled elegance. It was as if their deepest desires could
not quite be verbalized. I began to imagine that the spell the case was casting
on me was only the meek collective voice of the models, asking to be told their
story. I had to admit: They I started asking around the department, and it did not
take long to find out that there was very little knowledge of the models’
origins. I began with David Gay, who had some of the more interesting models in
his office, and was still using them to teach math education courses. He told
me that “Dan Madden and I last worked on the case in the early 80s, and even
then there was no one in the department who was here when the models were
bought.” John Leonard, who told me that he had used “quite a few in various classes,” even repairing some of the ones
that were falling apart, pushed the model’s origins further into the past: They
had been a “given” when he arrived in 1966. Finally, Dan Madden told me that
“Most of the stuff in the case dates back to before 1950. The history of most
of it is long gone. Too bad.” I tracked down Harvey Cohn, who had been chairman during
the 1960s, and who was, as far as I could determine, one of the oldest living
persons with any ties to the department[1].
His comments echoed in the same large, empty space. He wrote to me that: ...these models were here before I came (1958) so I can not give you much information on them. I recall that when I quit as head of the department (1967) there was a general feeling of euphoria on the demise of the ancient regime, and the past history was mostly obliterated. I am therefore pleasantly surprised that when the natives became revolting they did not also destroy the artifacts. Harvey Cohn was the first to tell me the story, which I
would hear repeated in various versions, with fingers pointing in various
directions, that a secretary had thrown away all of the documentation for the models
in a fit of “spring cleaning.” Others in the department assured me that the
story was apocryphal. The documentation, nevertheless, had disappeared. Whatever threads still connected
the models to their past had obviously grown very tenuous. Like the
deteriorating threads that wove some of the older string models into promises
of higher geometry, many of the threads that I attempted to follow backwards
into the past evaporated when they were scrutinized too closely, leaving only
an enigmatic puff of dust. Harvey Cohn had suggested one such tenuous thread,
writing, “incidentally,” that “some of the models were collected by Charles
Merchant, my administrative assistant.” When asked, a number of people in the
department remembered Charles Merchant. He had, during Harvey Cohn’s tenure in
the 1960s, been a lecturer in the department as well as an administrative
assistant. There was agreement among those who remembered him that he While discussing Charles Merchant, another name — Joe
Foster — came up many times[2].
Joe Foster had been an instructor in the department from 1942 until 1969. He
had degrees in both mathematics and astronomy, and, apparently, quite an
interest in both the models and the calculating machines. During World War II
he had taught spherical trigonometry and the principles of navigation to the
university’s ROTC students. The models, by several accounts, were a regular
presence in his classes, and it seems likely that during his time with the
department he was the primary caretaker of the collection. In fact, he seems to
have enlarged it. His legacy, however, is as enigmatic as that of Charles
Merchant. Richard Thompson remembers that Joe Foster would have his students
build models to be displayed in the case, but Except for the models that were produced by his students, no one remembers Joe Foster ever talking about obtaining new models for the collection — only of “using” the ones that were already there. And so the origins of the department’s collection, at least the department’s collection of commercially produced models, slipped further into the past: Before World War II, and beyond the reach of even the most tenuous threads of the department’s oral history. Alaina Levine, the undergraduate mathematics major who is responsible for putting this catalogue together, checked through old university bulletins to see if the models might have been mentioned in earlier course descriptions (in mathematics, engineering, or art), signaling the date of their arrival on campus. Alas, there was nothing there, and the light looking backwards through the tunnel seemed to blink out. This is when John Brillhart cheerfully started sharing his
books with me. John seemed to know quite a lot about the early history of
research universities in the United States, and his suggestions hinted gently
at a small piece of wisdom: When trying to solve for X, Y, and Z, it is helpful
to know your ABCs. Those readings, among many others to be discovered as the
inter-woven threads of history are followed backwards through library books,
dusty journal articles, microfilms, and inter-library loans, are included in
the bibliography at the end of this catalogue. In the stories that are told
there, the particular threads leading up to the U of A’s collection of
mathematical models unwind backwards into an historical past; and the past
moves forward, inevitably, to describe the weave of the present. If one is
comfortable moving back and forth along the timeline, it is possible to tie
together some of the threads, and something of a greater story begins to emerge
from the metaphors. Nineteenth century America, I learned, had been swept up
in one educational reform movement after another. All of those movements,
however, seemed to share a common prescription for what was ailing the public
schools: Buy mathematical models! Early in the century, sets of mathematical
models, sold by catalogue, provided a way for the newly established common
schools to feel as if they were participating in a common curriculum.
Throughout the rest of the century, the market for these models was maintained
with fervent calls for, successively, “Object-Oriented Instruction,” “Technical
Training,” “Art Education,” and “Exact Thinking.” Many of the advocates of
these “object-oriented” pedagogies, and the “ocular demonstrations” on which
they relied, were the very same people who were producing the models. They sat
on local school boards and they supervised instruction for entire regions of
the country. Business was so good that it diversified into much more lucrative
catalogues of “Mathematical Apparatus,” which included everything from
finely-crafted orreries and tellurians to the latest in elegant “Pointing
Rods.” There was definitely money to be made in the boom market for models, and
before the century was over the Gilded Age had left its mark on schools and
universities across the country. In America, it has always been difficult to just buy some A case in point was W. W. Ross, who served as a
superintendent of public schools in northwestern Ohio for forty-two years in
the late 1800s. He was, at various times, president of the Ohio State Teacher’s
Association, president of the Tri-State Teacher’s Association for Ohio,
Michigan, and Indiana, and president of the Ohio State Board of Examiners. He
also made and sold mathematical models. Ross’ specialty was wooden models that
could be dissected to show how the familiar formulas for areas and volumes in
plane and solid geometry could be constructed from simpler pieces. His model
designs developed into elaborately hinged production numbers that were sure
crowd-pleasers[5]. Yet Ross
was undoubtedly more than just a shrewd manipulator of the educational market.
He earnestly covered his otherwise beautifully crafted models with detailed
explanations, in tiny print, and then gave them a heavy shellacking for the
ages. He spoke with conviction of his belief in “object-oriented” instruction,
arguing that the use of models was the only way to make the formulas of
geometry “the permanent property of the reason rather than the uncertain
possession of the memory.” In a tack that was sure to generate opponents and
hurt his sales, Ross claimed that his “ocular demonstrations” were more valuable
to students than formal proofs. Nevertheless, business was doing quite well.
The annual report of the Ohio state commissioner for the common schools showed
regular, significant expenditures on apparatus during the period that Ross was
promoting his ideas. His models eventually began to show up in more prestigious
colleges and universities outside of Ohio: Hood College in Maryland, Wesleyan
University in Connecticut, and, sure enough, the University of Arizona. How the
models came to be at the U of A[6],
who might have purchased them, and what educational reforms members of the
department may have believed in, passionately or dispassionately, is unknown.
But there they are. I stare at them and can almost hear the opposing faculty
members not talking to each other. By the end of the 1800s, the
model business had shifted its attention to the country’s emerging colleges and
universities. University mathematics departments in the United States were
still very focused on teaching, and the American research community hadn’t
really developed yet[7].
As in the common schools, a fine set of models, proudly displayed, could
provide instant educational cachet for a department: They said that the faculty
were up-to-date with the latest mathematical discoveries Monge, of course, is known as the father of differential
geometry, and his efforts in the early 1800s to classify surfaces by the
motions of lines, along with his “descriptive geometry” for representing
three-dimensional surfaces in two-dimensions, led naturally to the construction
of elaborate models made of tightly stretched strings. One of his students, Théodore
Olivier, built some of the most beautiful mathematical models ever made — and
made some money in the process. The models were expensive. He sold his models
to the emerging technical schools in the United States, which were doing their
best to emulate the example of Monge and the École Polytechnic. Klein came
along later in the century, making it his special mission to promote the use
and construction of mathematical models in graduate education, and the first
research universities in the United States once again did their best to follow
the European lead. Klein and his colleague Ludwig Brill established a
Laboratory for the Construction of Mathematical Models in Munich, and the
labors of their graduate students were sold world-wide. The prices, once again,
reflected America’s awe of European craftsmanship[8]. Eventually, the spell of the mathematical models lost its
hold on the mathematical community. Economic realities in the early part of the
20th century made their acquisition an increasingly difficult proposition, and
the market for finely-crafted models, as well as the finely-crafted theories of
education that went with them, fell off. Educators returned to an emphasis on
“fundamental skills,” and the misguided “over-emphasis” on mathematical models
and visualization was triumphantly discredited by those who had always been
disciples of a more Cartesian discipline in mathematics education. Model
makers, at least what was left of them, sensed the shift in the market, and the
more enterprising ones jumped in with cheap knock-offs of the European models.
For a while, they were able to supply the diminishing demand. The increasingly
clumsy constructions could no longer capture the mathematical community’s
collective imagination, however, and dust began to settle on the country’s
collections of mathematical models. It is from this twilight era of the golden age of
mathematical models that we are able to pick up a final thread that leads
forward again to the models in the display case at the U of A; and leads me,
backwards, to a new appreciation of my boyhood fascinations at MIT. This is the
thread: “R. P. Baker,” in an inelegant script, like something from a scrap of
paper along the railroad tracks. It is painted on a large number of models in
the U of A’s collection. Who was s/he? A student? A faculty member? Both seemed
like good guesses, for the models, though they were of more ambitious design
and generally higher production value than the majority of the models in the
case, still seemed too clumsy to have been produced commercially. They were
odd. The models that showed the maker’s mark most clearly were an obsessive
sequence of plaster surfaces, each one only slightly different from the next,
as if the maker had been struggling with an ideal form that had taken shape in
their head, but could not quite realize it. Whatever mathematical form may or
may not have been captured, the It was in the course of reading an article by Peggy
Kidwell, the curator of mathematical models at the National Museum of American
History at the Smithsonian Institution in Washington, that I found out R. P.
Baker’s true identity — at least, the relevant names and dates that often pass
for true identity in historical accounts. Richard P. Baker (1866-1937) was an
Englishman, educated at Oxford and the University of London. Around the turn of
the century he moved to Chicago, apparently with the idea of becoming a player
in the burgeoning university model-making business. His first catalogue, in
1905, contained descriptions of 100 models, which he hoped to craft in a
made-to-order fashion that would set him apart from the other model-makers and
their standardized collections. Some of his more exotic offerings included
Riemann surfaces, surfaces from the theories of optics and thermodynamics, and
representations of statistical distributions. Something changed his plans,
however — either a lack of business or an unsatisfied desire to realize the
forms in his catalogue descriptions — for he soon accepted a teaching position
at Iowa State University and took up graduate studies in mathematics, at a
distance, at the University of Chicago. He completed his Ph.D. in 1910, but
remained devoted to model-making for the rest of his life. Ambitiously, he
tried to compete with the Germans. The outbreak of WWI must have been good for
business: because of the disruption in shipping, but also because of America’s
souring attitude toward Germany. By 1931 his catalogue had mushroomed to over
500 offerings. Unfortunately, his ambitions were rising just as the stock
market was crashing. Baker was nevertheless able to realize almost 300 of his
conceptions before he died. I learned all of this from Peggy Kidwell at the
Smithsonian. She was very interested to learn that the U of A had a collection
of Baker models. She knew of only a few collections in the East, at Brown
University and at the Smithsonian itself, and was unaware of any such
collections in the West. When Alaina Levine put pictures of the U of A’s entire
collection of models up on the Web, Peggy Kidwell was able to confirm that many
of the other models — particularly, many of the string models — were also
Baker’s creations. It seems that the U of A has a fairly large and uncommon
collection of One of the most satisfying outcomes of finding, and telling,
the story of the Baker models was having the Smithsonian provide the department
with a replacement copy of its “lost” catalogue: The 1931 edition gives Baker’s
description of every one of models in the U of A’s collection. It made me happy, to tie together a thread so nicely[9],
but it also, ultimately, made me feel as if I had lost something. I had stared
with such wonder at those surfaces, when I finally noticed them. Now their
subtle variations were “explained” by a changing parameter in the Fresnel
equation. I knew I had to recognize the necessity of that: That’s math, order,
what we call history. It is also what catalogues are all about It is time for me to move on. The U of A’s display case is
going through the process of a complete re-furbishing, and the models will be
displayed with a renewed dignity for some time to come. I hope, though, that
students will continue to press their noses against the glass. I hope that they
will read the new information about the models’ origins and realize that there
are still a myriad stories to tell. I hope that the models will speak to them,
in the remarkable language of mathematics, which has cast its spell up and down
the timeline of history. I hope they will wonder. I know it has taken me a long
way.
REFERENCES 1.
R. P. Baker, 2.
D. Bressoud, 3.
F. Cajori, 4.
________, 5.
H. S. M. Coxeter, 6.
________, P. DuVal, H.T. Flather, and J.F. Petrie, 7.
T. Everard, 8.
H. Eves, 9.
J. Ewing, ed., 10.
S. Greenblatt, 11.
H. Hahn, The Crisis in Intuition, in 12.
W. Hawney, 13.
J. Henle, Classical Mathematics, this MONTHLY 14.
D. Hilbert and S. Cohn-Vossen, 15.
J. Holbrook, 16.
O. Impey and A. Macgregor, eds., 17.
A. Jackson, Mathematical Treasures of the Smithsonian
Institution, 18.
P. S. Jones, ed., 19.
M. Joswig, K. Polthier, eds., Electronic Geometry
Models, http://www-sfb288.math.tu-berlin.de/eg-models, 2003. 20.
C. Kapetanya, unpublished, untitled draft of a
dissertation on Geoffrey Thomas Bennett, University of Cambridge, 2003. 21.
J. Kenseth, ed., 22.
P. Kidwell, American Mathematics Viewed Objectively: The
Case of Geometric Models, in 23.
S. G. Kohlstedt, Parlors, Primers and Public Schooling:
Education for Science in Nineteenth Century America, 24.
A. Lugli, Inquiry as Collection: The Athanasius Kircher
Museum in Rome, 25.
A. M. Luyendijk-Elshout, Death Enlightened: A Study
of Frederik Ruysch, 26.
N. Luzin, Function: Part I, this MONTHLY 27.
A. MacGregor, ed., 28.
P. Mauries, 29.
W. McCallum, ed., 30.
R. McCarney, The Evolution of Mathematics at Hood College,
unpublished article, 1993. 31.
W. Mueller, Book Review: Fragments of Infinity: A
Kaleidoscope of Math and Art, by Ivars Peterson, this MONTHLY 32.
________, Mathematical Wunderkammern, this MONTHLY 33. ________,
Reform Now, Before It’s Too Late!, this MONTHLY
34.
R. P. Multhauf, ed., 35.
K. H. Parshall, D. E. Rowe, 36.
_______, Embedded in the Culture: Mathematics at the
World's Columbian Exposition of 1893, 37.
B. Pascal, 38.
W. D. Reeve, ed., 39.
D. L. Roberts, Albert Harry Wheeler (1873-1950): A
Case Study in the Stratification of American Mathematical Activity, 40.
_______, Selected Bibliography on Geometric Models, Visualization,
and the Teaching of Mathematics, unpublished bibliography, 1992. 41.
W. W. Ross, 42.
A. Seba, 43.
B.M. Stafford and F. Terpak, 44.
J. J. Sylvester, James Joseph, Presidential Address to the
mathematical and physical section of the British Association for the
Advancement of Science, Exeter, 1869. 45.
A. Vierling, 46.
D. J. Warner, Commodities for the
Classroom: Apparatus for Science and Education in the Ante-bellum America,
47.
L. Weschler,
[1] I was told that a department head from the 1940s —
Greyson — might still be living in Tucson, but I was unable to track him down. [2] A third name — Roy Whitman — was mentioned a few
times. He may have constructed some models for the case in the 1970s, but I was
unable to find much information about him. [3] He said that he learned this “trick” from an older
faculty member, but the name of that more distant predecessor seems to have
been forgotten. [4] Most of these have now been moved to the math
education classroom for future teachers to contemplate. [5] Not unlike the “transformer” toys so popular with
children in recent years. [6] They probably arrived in the late 1890s, when Ross
was going national with his business. [7] Most of the “dissertations” of this era were solutions to calculus problems. [8] I wonder if Joe Foster’s students felt as if they
were weaving the thread of this once powerful tradition into the present or if
they were being waved from its frayed ends. [9] In fact, I did this quite literally when I
re-strung one of Baker’s models. Many of the string models in the collection
have deteriorated badly over the years, and could use this kind of attention. I
recommend it to anyone with the patience to appreciate, on the scale of a
needlepoint, the inspirations and miscalculations of Baker’s designs. |