Historia Mathematics 10 (1983) 448-457
NOTE
A FORGOTTENCHAPTER IN THE HISTORY OF FELIX KLEIN’S
ERLAh’GERPROGRAMV
BY DAVID E. ROWE
PACE UNIVERSITY,
PLEASANTVILLE, NY 10570
Felix Klein's single most important mathematical accomplishment
was unquestionably his "Erlanger Programm" of 1872 [l].
This work remained a leitmotiv not only for Klein but for much
of the mathematical world throughout his lifetime [2]. The purpose
of this note is to correct a widespread misconception concerning
the original presentation of the "Erlanger Programm"
ad, at the same time, to point out how this error has obscured
a significant event in Klein's early development as an educator
and scientific organizer. Hopefully this will serve not only to
reduce the possibility of future confusion, but also to give a
clearer sense of the manner in which Klein's later pedagogical
activities were related to the interests and ideas of his youth.
By now the error in question has been firmly entrenched in
the secondary literature; in fact, it has become part of the
standard lore in the history of mathematics. Within the last
fifteen years three eminent historians of mathematics have written
Kle
In 1872 Klein became professor at Erlangen. In his
inaugural address he explained the importance of the
group conception for the classification of the different
fields of mathematics. The address, which became known
as the "Erlangen Program," declared.every geometry to be
the theory of invariants of a particular transformation
group. [Struik 1967, 1751
Klein, in a celebrated inaugural address in 1872,
when he became professor at Erlangen, showed how it
[the group concept] could be applied as a convenient
means of characterizing the various geometries that
had appeared during the century. [Bayer 1968, 5923
He gave this characterization in a speech of 1872,
"Vergleichende Betrachtungen ueber neuere geometrische
Forschungen" (A Comparative Review of Recent Researches
in Geometry), on the occasion of his admission to the
University of Erlangen, and the views expressed in it
are known as the Erlanger Programm. [Kline 1972, 9171
Each of these accounts incorrectly identifies the content of
in's speech, his Erlangen Antrittsrede of December 7, 1872,
0315-0860/83 $3.00
Copyright 0 1983by Academic Press,Inc.
All rights of reproduction in anyform reserved.
448
HM 10 Notes 449
with the content of his much more famous Eintrittsprogramm,
"Vergleichende Betrachtungen ueber neuere geometrische Forschungen."
In fact, the two were devoted to wholly different subjects. Klein
himself was perfectly clear on this point in his introductory remarks
to the chapter “Zum Erlanger Programm" in [Klein 1921, 411-
4121, written some fifty years after the event.
According to his own account, the source of the "Erlanger
Programm" can be traced back to ideas presented in his "Ueber
die sogennante Nicht-Euklidische Geometrie, Zweiter Aufsatz'
[Klein 1921, 311-3431, which was completed in early June of 1872
(it appeared the following year in Band 6 of Mathematische dnnalen).
As for the "Erlanger Programm" itself, Klein remarked that it was
written under the influence of two very particular circumstances.
The first of these was the stimulation he received through
numerous conversations with Sophus Lie, who had paid him a visit
during the months of September and October, 1872. Lie had just
developed the fundamental notions underlying his theory of firstorder
partial differential equations and, in particular, his
theory of contact transformations. During this period they met
daily, sharing the excitement of their respective discoveries.
Klein was especially impressed by Lie's thoughts on the grouptheoretic
classification of geometries, a subject they both
investigated with the greatest enthusiasm.
The second and more immediate factor leading to the publication
of the "Erlanger Programm" was actually a matter of obligation
connected with Klein's appointment at the University. As
he himself noted:
"in Erlangen the newly appointed professor, in addition
to a speech through which he introduces himself to
the circle of his colleagues, traditionally submitted
a published program" [Klein 1921, 4111.
Thus, what today is referred to as the "Erlanger Programm"
was, in fact, only the written portion of Klein's original inaugural
presentation; his public lecture, naturally enough, was
reserved for matters more accessible to a general university
audience. Indeed, Klein made explicit mention of the lecture
topic he had chosen on this occasion in the passage quoted above
which continues:
I regard this custom, despite all the discomfort for
the participants, as very valuable, since it allows the
new professor the unusual opportunity to address his
colleagues on matters he feels are significant, while
also compelling him to clarify his own thoughts regarding
these concerns. In my case, the public lecture
took place on December 7, at which time the published
work was distributed to the listeners. 1 chose, it might
450 Notes HM 10
be noted in passing, as the theme for my lecture, the
pedagogical principles and goals for my future academic
activity.... [Klein 1921, 411-4121
The most recent conflation of Klein's "Erlanger Programm"
with his public lecture occurs in [Portnoy 19821, where the author
makes two erroneous inferences, both due to this very misconception,
She writes that the "Erlanger Programm" was first published in an
Italian translation of 1890 (Fano's translation in d,nnali di Matematica,
2(17)) and cites the 1893 German version that appeared in
Mathematische Annalen [Klein 18931. But the latter was only a
republication of the original [Klein 18721, to which Klein added
a series of remarks. Obviously this mistake would have been
avoided had the author known that the "Vergleichende Betrachtungen
ueber neuere geometrische Forschungen" was a "published work...
distributed to the listeners" at the time of Klein's inaugural
lecture.
Portnoy's second erroneous inference occurs when she invites
comparison between Klein's "Erlanger Programm" and Biemann's
equally famous Probevorlesung of 1854, "Ueber die Hypothesen
welche der Geometrie zu Grunde liegen." The latter was, of course,
the written version of Riemann's oral presentation delivered before
the faculty in G&tingen. Portnoy suggests that, by comparison
with Klein's "Erlanger Programm," Riemann's classic was a
"poor candidate for publication, especially while its author was
relatively unknown" [Portnoy 1982, 131. Her conclusion here may
be perfectly correct, but since Klein's paper was neither delivered
orally nor written as a publishable version of a public presentation,
it is hard to see how comparing the two can lend any support
to her statement. The circumstances surrounding both works were
in fact so dissimilar that it is difficult to imagine any meaningful
basis for comparison between them.
AS for Klein's dntrittsrede, its theme, as already noted, was
mathematics education. Pedagogy I it might be recalled, was one
of Klein's central interests during his later career at GGttingen
(1886-19131, especially after 1892, when H. A. Schwarz was called
from GGttingen to Berlin [3]. Klein's lectures for Lehramtskandidaten,
published under the title Elementarmathematik vom hzheren
Standpunkte aus [Klein 1924-19281, became exceptionally popular,
were translated into several foreign languages, and were repeatedly
issued in numerous editions. Moreover, in 1908 Klein was
elected president of the Internationale Mathematische UnterrichtsKommission,
and during his tenure he oversaw publication of several
large tomes devoted to the history of mathematics education
prior to World War I. All told, he published over thirty books
and articles dealing with educational matters.
Considering Klein's intense interest and active participation
in pedagogical problems later in his career, as well as his many
accomplishments within this sphere, it certainly is surprising
HM 10 Notes 451
that historians of mathematics have generally overlooked the fact
that the "Erlanger Programm" was accompanied by an educational
program as well. This omission is especially curious considering
Klein's own, very extensive discussion of his Erlangen Antrittsrede
in his autobiographical sketch:
In my inaugural speech in December, I set forth a
detailed program for my teaching plans while asserting
that the unity of all knowledge and the ideal of a complete
education should not be neglected because of
specialized studies. Accordingly, humanistic and mathematical-scientific
education belong together, and
should not be placed in opposition to one another. On
the other hand, it is necessary to cultivate applied
mathematics as well as pure, in order to preserve the
connection between neighboring disciplines such as
physics and technology. Furthermore, along with the
development of logical capability, equal account must
be taken of the need to develop intuition and, more
generally, mathematical imagination, out of which creative
originality can arise. Finally the universities
should pay heed to the preparatory teaching in the
schools, and thus place particular emphasis on the education
of school teachers. Here the organizations in
the technical high schools could be examined, and in
many respects taken as a model.
On the basis of these considerations, the following
practical steps were suggested. Regularly repeated elementary
lectures should be held along with the special
lectures intended for a small number of serious students
of science, both backed up by means of exercises
and seminar activity. Moreover, courses in descriptive
geometry should be offered that stress drawing capability.
Furthermore a reading room and library with
open-stacks should be established, which would make it
possible for students to study the published literature,
while extensive model collections would aid them
in developing their mathematical intuition.
These points of view and practical suggestions grew
out of the numerous observations I had made during my
years of travel. They have also remained the essential
guiding principles for my later activity; only in practice,
due to the extent of the demands placed on the
students, have I modified this somewhat by separating
the elementary lectures into beginning and intermediate
courses. [4] [Klein 1923, 181
It is important to see this speech in its proper perspective.
Klein had just been called to Erlangen as an Ordinarius (i.e.,
452 Notes HM 10
full professor) at the unusually young age of twenty-three. His
Antrittsrede was given before the faculty senate, a body to which
only the Ordinarien were allowed membership. It would be interesting
to know how the esteemed senate responded to this address,
but one can well imagine that many probably dismissed it as
nothing more than the wild-eyed idealism of a mere youth.
On the other hand, it must not be overlooked that by 1923
Klein's recollection of the speech he gave over fifty years earlier
was apparently rather vague. This becomes quite obvious
when comparing his remarks with the actual content of the Erlangen
Antrittsrede (a transcription of Klein's Antrittsrede together
with commentary will be presented in a forthcoming issue) [5].
Nevertheless, it cannot be denied that Felix Klein was very much
interested in educational issues from the very beginning of his
career. The fact that he had to overcome several decades of
steadfast resistance to his efforts on behalf of educational reform
may, quite possibly, serve to obscure rather than illuminate
his activities in this arena [6]. In light of this and the obvious
significance Klein himself attached to his Erlangen Antrittsrede,
it is important to establish the actual facts and circumstances
surrounding this event and its relation to the "Erlanger
Programm" by giving Klein's speech the attention it deserves as
an early statement of his pedagogical objectives-ones that
would develop and mature over the remainder of his life.
NOTES
1. According to Richard Courant, the "Erlanger Programm" was
perhaps the best read and most influential mathematical work of
the sixty-year period 1865-1925 [Courant 1925, 7661. One finds
it mentioned even in such unlikely places as Spengler's Decline
of the West (New York: Modern Library, p. 68).
2. Klein returned to it often, but nowhere in more striking
fashion than in his last work on relativity theory, where group
invariants play an important role. See [Klein 1921, 533-6121
and especially [Klein 1967, Vol. II]. For a general assessment
of the significance of the "Erlanger Programm," see [Carathgodory
19191.
3. Schwarz succeeded Weierstrass who retired in 1892. This
gave Klein the freedom to develop his scientific and educational
reforms at G&tingen [Klein 1923, 231. For a discussion of the
behind the scenes rivalry between Klein and the Berlin school
(including Schwarz), see [Biermann 1973, 129-1311.
4. Because this work is fairly inaccessible (the only library
in this country known to the author that holds it being Harvard
University), the original German is made available here:
In meiner Antrittsrede im Dezember legte ich ein
ausfi.ihrliches Programm meiner geplanten Unterrichtstztigkeit
vor. Ueber den Spezialstudien darf die
HM 10 Notes 453
Einheit aller Wissenschaft und das Ideal einer Gesamtbildung
nicht vergessen werden. Daher geh&en such
humanistische und nz~chematisch-naturwissenschaftliche
Bildung zusammen und dirfen nicht in Gegensatz gebracht
werden. Anderseits ist neben der reinen such die an.gewandte
Mathematik zu pflegen, urn den Zusammenhang mit
den angrenzenden Wissenschaftsgebieten wie Physik und
Technik zu wahren. Ferner muss in der Mathematik neben
den logischen Fzhigkeiten die Anschauung als gleichberechtiger
Faktor und Gberhaupt die mathematische
Phantasie und die aus ihr entspringende Selbsttztigkeit
entwickelt werden. Schliesslich hat die Universitzt
auchxden vorbereitenden Unterricht in den Schulen zu
beachten und daher besonderes Gewicht auf die Ausbildung
der Lehramtskandidaten zu legen, wobei die Einrichtungen
der technischen Hochschulen in mancher Beziehung als
vorbildlich betrachtet werden k&nen. Aus diesen Gesichtpunkten
ergaben sich folgende praktische Forderungen:
Es sind regelm%sig wiederholte Elementarvorlesungen
und daneben Spezialvorlesungen f& eine kleinere Zahl
wissenschaftlich Interessierter abzuhalten, die sich
beide auf Uebungen und Seminarbetrieb stitzen. Daneben
m$sen Kurse in darstellender Geometrie mit Betonung des
zeichnerischen 6nnens stattfinden. Ferner ist ein
Lesezimmer mit Pr&enzbibliothek einzurichten, welches
den Studierenden das Studium der einschligigen Litteratur
&glich macht, wghrend ausgedehnte Modellsammlungen fzr
die Ausbildung der mathematischen Anschauung zu sorgen
haben. Diese Gesichtspunkte und praktischen Vorschlzge
hatte ich mir wzhrend der Jahre des Wanderns in mannigfather
Beobachtung gesammelt. Sie sind such die wesentlichen
Richtlinien f:r meine sp&ere Wirksamkeit geblieben
nur bin ich unter dem Druck der Praxis von der Hghe der
an die Zuh&er gestellten Forderungen etwas abgegangen
und habe sp.ater die Elementar-Vorlesungen jn Anfangs- und
Kursusvorlesungen getrennt.
5. The unpublished manuscript of the Erlangen Antrittsrede
is part of the Klein Nachlass housed at the Nieders%hsische
Staats- und Universit%sbibliothek Ggttingen.
6. IManegold 19701 has a good deal on the opposition to
Klein's educational ideas (especially with regard to applied
mathematics) and how it was eventually overcome. He also discusses
Klein's Erlangen Antrittsrede (pp. 92-93).
REFERENCES
Biermann, K. R. 1973. Die Mathematik und ihre Dozenten an der
Berliner Universitst. 1810-1920. Berlin: Akademie-Verlag.
Boyer, C. 1968. A history of mathematics. New York: Wiley.
Carathgodory, C. 1919. Die Bedeutung des Erlanger Programms.
Naturwissenschaften .7, Sonderheft Klein, 297-300.
454 Notes HM 10
Courant, R. 1925. Felix Klein. Naturwissenschaften 13, 765-
772. (Gedzchtnisrede of July 31, 1925.)
Klein, F. 1872. "Vergleichende Betrachtungen ueber neuere
geometrische Forschungen," Programm zum Eintritt in die
philosophische Fakultgt und den Senat der K. FriedrichAlexanders-Universit&
zu Erlangen. Erlangen: Deichert.
[In Klein 1921, 46-497.1 See also [Klein 18933.
1873. Ueber die sogennante Nicht-Euklidische Geometrie.
Zweiter Aufsatz. Mathematische Annalen 6, 112-145. [In
Klein 1921, 311-343.1
1893. Vergleichende Betrachtungen ueber neuer geometrische
Forschungen. Mathematische Annalen 43, 63-100.
1921. Gesammelte Mathematische Abhandlungen, Bd. I,
R. Fricke and A. Ostrowski, Eds. Berlin: Julius Springer.
1923 Gijttinger Professoren. Lebensbilder von eigener
Hand. Mitteilungen Universititsbund G&tingen 5, 11-36.
1924-1928. Elementarmathematik vom h:heren Standpunkte
aus, 3 vols. Berlin: Julius Springer.
1967. Vorlesungen ueber die Entwicklung der Mathematik
im 19. Jahrhundert, 2 ~01s. in one. New York: Chelsea,
Reprint of the Berlin edition of 1926-1927.
Kline, M. 1972. Mathematical thought from ancient to modern
times. New York: Oxford University Press.
Manegold, K. 1970. Universit2, Technische Hochschule, und
Industrie. Ein Beitrag zur Emanzipation der Tech&k im 19.
Jahrhundert unter besonderer Ber&ksichtigung der Bestrebungen
Felix Kleins. Berlin: Dunker 6i Humbolt.
Portnoy, E. 1982. Riemann's contribution to differential geometry.
Historia Mathematics 9, 1-18.
Reid, C. 1970. Hilbert. New York: Springer.
1976. Courant in GEttingen and New York. New York:
Springer.
Struik, D. 1967. A concise history of mathematics, 3rd ed.
New York: Dover.
Tobies, R. 1981. Felix Klein. Leipzig: Teubner. (In Biographen
hervorragender Naturwissenschaftler, Techniker und Mediziner,
Bd. 50.)
Wussing, H. 1968. Zur Fntstehungsgeschichte des Erlanger Programmes.
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DDR 1, 23-40.