![xii Introduction
with affine connections. There is a wealth of geometries which can be described by
transformation groups in the spirit of the Erlangen program. Several of these ge-
ometries were studied by Klein and Lie; among them we can mention Minkowski
geometry, complex geometry, contact geometry and symplectic geometry. In modern
geometry, besides the transformations of classical geometry which take the form of
motions, isometries, etc., new notions of transformations and maps between spaces
arose. Today, there is a wealth of new geometries that can be described by trans-
formation groups in the spirit of the Erlangen program, including modern algebraic
geometry where, according to Grothendieck’s approach, the notion of morphism is
more important than the notion of space.1
As a concrete example of this fact, one can
compare the Grothendieck–Riemann–Roch theorem with the Hirzebruch–Riemann–
Roch. The former, which concerns morphisms, is much stronger than the latter, which
concerns spaces.
Besides Lie and Klein, several other mathematicians must be mentioned in this
venture. Lie created Lie theory, but others’ contributions are also immense. About
two decades before Klein wrote his Erlangen program, Riemann had introduced new
geometries, namely, in his inaugural lecture, Über die Hypothesen, welche der Geo-
metrie zu Grunde liegen (On the hypotheses which lie at the bases of geometry)
(1854). These geometries, in which groups intervene at the level of infinitesimal
transformations, are encompassed by the program. Poincaré, all across his work,
highlighted the importance of groups. In his article on the Future of mathematics2
, he
wrote: “Among the words that exerted the most beneficial influence, I will point out
the words group and invariant. They made us foresee the very essence of mathemat-
ical reasoning. They showed us that in numerous cases the ancient mathematicians
considered groups without knowing it, and how, after thinking that they were far away
from each other, they suddenly ended up close together without understanding why.”
Poincaré stressed several times the importance of the ideas of Lie in the theory of
group transformations. In his analysis of his own works,3
Poincaré declares: “Like
Lie, I believe that the notion, more or less unconscious, of a continuous group is the
unique logical basis of our geometry.” Killing, É. Cartan, Weyl, Chevalley and many
others refined the structures of Lie theory and they developed its global aspects and
applications to homogeneous spaces. The generalization of the Erlangen program
to these new spaces uses the notions of connections and gauge groups, which were
1See A. Grothendieck, Proceedings of the International Congress of Mathematicians, 14–21 August 1958,
Edinburgh, ed. J.A. Todd, Cambridge University Press, p. 103–118. In that talk, Grothendieck sketched his
theory of cohomology of schemes.
2H. Poincaré, L’Avenir des mathématiques, Revue générale des sciences pures et appliquées 19 (1908)
p. 930–939. [Parmi les mots qui ont exercé la plus heureuse influence, je signalerai ceux de groupe et d’invariant.
Ils nous ont fait apercevoir l’essence de bien des raisonnements mathématiques ; ils nous ont montré dans com-
bien de cas les anciens mathématiciens considéraient des groupes sans le savoir, et comment, se croyant bien
éloignés les uns des autres, ils se trouvaient tout à coup rapprochés sans comprendre pourquoi.]
3Analyse de ses travaux scientifiques, par Henri Poincaré. Acta Mathematica, 38 (1921), p. 3–135. [Comme
Lie, je crois que la notion plus ou moins inconsciente de groupe continu est la seule base logique de notre
géométrie]; p. 127. There are many similar quotes in Poincaré’s works.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-16-320.jpg)
![Chapter 1
Sophus Lie, a giant in mathematics
Lizhen Ji
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Some general comments on Lie and his impact . . . . . . . . . . . . . . . . . . . . . . 2
3 A glimpse of Lie’s early academic life . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 A mature Lie and his collaboration with Engel . . . . . . . . . . . . . . . . . . . . . . 6
5 Lie’s breakdown and a final major result . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 An overview of Lie’s major works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7 Three fundamental theorems of Lie in the Lie theory . . . . . . . . . . . . . . . . . . 13
8 Relation with Klein I: the fruitful cooperation . . . . . . . . . . . . . . . . . . . . . . 15
9 Relation with Klein II: conflicts and the famous preface . . . . . . . . . . . . . . . . . 16
10 Relations with others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11 Collected works of Lie: editing, commentaries and publication . . . . . . . . . . . . . 23
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1 Introduction
There are very few mathematicians and physicists who have not heard of Lie groups
or Lie algebras and made use of them in some way or another. If we treat discrete
or finite groups as special (or degenerate, zero-dimensional) Lie groups, then almost
every subject in mathematics uses Lie groups. As H. Poincaré told Lie [25] in October
1882, “all of mathematics is a matter of groups.” It is clear that the importance of
groups comes from their actions. For a list of topics of group actions, see [17].
Lie theory was the creation of Sophus Lie, and Lie is most famous for it. But Lie’s
work is broader than this. What else did Lie achieve besides his work in Lie theory?
This might not be so well known. The differential geometer S. S. Chern wrote in 1992
that “Lie was a great mathematician even without Lie groups” [7]. What did and can
Chern mean? We will attempt to give a summary of some major contributions of Lie
in 6.
One purpose of this chapter is to give a glimpse of Lie’s mathematical life by
recording several things which I have read about Lie and his work. Therefore, it is
short and emphasizes only a few things about his mathematics and life. For a fairly de-
tailed account of his life (but not his mathematics), see the full length biography [27].
We also provide some details about the unfortunate conflict between Lie and Klein
and the famous quote from Lie’s preface to the third volume of his books on trans-](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-23-320.jpg)
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formation groups, which is usually only quoted without explaining the context. The
fruitful collaboration between Engel and Lie and the publication of Lie’s collected
works are also mentioned.
We hope that this chapter will be interesting and instructive to the reader of this
book and might serve as a brief introduction to the work and life of Lie discussed in
this book.
2 Some general comments on Lie and his impact
It is known that Lie’s main work is concerned with understanding how continuous
transformation groups provide an organizing principle for different areas of math-
ematics, including geometry, mechanics, and partial differential equations. But it
might not be well known that Lie’s collected works consist of 7 large volumes of the
total number of pages about 5600. (We should keep in mind that a substantial por-
tion of these pages are commentaries on his papers written by the editors. In spite
of this, Lie’s output was still enormous.) Probably it is also helpful to keep in mind
that Lie started to do mathematics at the age of 26 and passed away at 57. Besides
many papers, he wrote multiple books, which total over several thousands of pages.
According to Lie, only a part of his ideas had been put down into written form. In an
autobiographic note [9, p. 1], Lie wrote:
My life is actually quite incomprehensible to me. As a young man, I had no idea
that I was blessed with originality, Then, as a 26-year-old, I suddenly realized that
I could create. I read a little and began to produce. In the years 1869–1874, I had
a lot of ideas which, in the course of time, I have developed only very imperfectly.
In particular, it was group theory and its great importance for the differential
equations which interested me. But publication in this area went woefully slow. I
could not structure it properly, and I was always afraid of making mistakes. Not the
small inessential mistakes . . . No, it was the deep-rooted errors I feared. I am glad
that my group theory in its present state does not contain any fundamental errors.
Lie was a highly original and technically powerful mathematician. The recog-
nition of the idea of Lie groups (or transformation groups) took time. In 1870s, he
wrote in a letter [26, p. XVIII]:
If I only knew how to get the mathematicians interested in transformation groups
and their applications to differential equations. I am certain, absolutely certain in
my case, that these theories in the future will be recognized as fundamental. I want
to form such an impression now, since for one thing, I could then achieve ten times
as much.
In 1890, Lie was confident and wrote that he strongly believed that his work would
stand through all times, and in the years to come, it would be more and more appre-
ciated by the mathematical world.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-24-320.jpg)
![1 Sophus Lie, a giant in mathematics 3
Eduard Study was a privatdozent (lecturer) in Leipzig when Lie held the chair in
geometry there. In 1924, the mature Eduard Study summarized Lie as follows [26,
p. 24]:
Sophus Lie had the shortcomings of an autodidact, but he was also one of the most
brilliant mathematicians who ever lived. He possessed something which is not found
very often and which is now becoming even rarer, and he possessed it in abundance:
creative imagination. Coming generations will learn to appreciate this visionary’s
mind better than the present generation, who can only appreciate the mathemati-
cians’ sharp intellect. The all-encompassing scope of this man’s vision, which,
above all, demands recognition, is nearly completely lost. But, the coming gen-
eration [. . . ] will understand the importance of the theory of transformation groups
and ensure the scientific status that this magnificent work deserves.
What Lie studied are infinitesimal Lie groups, or essentially Lie algebras. Given
what H. Weyl and É. Cartan contributed to the global theory of Lie groups starting
around the middle of 1920s and hence made Lie groups one of the most basic and
essential objects in modern (or contemporary) mathematics, one must marvel at the
above visionary evaluation of Lie’s work by Study. For a fairly detailed overview of
the historical development of Lie groups with particular emphasis on the works of
Lie, Killing, É. Cartan and Weyl, see the book [14].
Two months after Lie died, a biography of him appeared in the American Mathe-
matical Monthly [12]. It was written by George Bruce Halsted, an active mathematics
educator and a mathematician at the University of Texas at Austin, who taught famous
mathematicians like R. L. Moore and L. E. Dickson. Reading it more than one hun-
dred years later, his strong statement might sound a bit surprising but is more justified
than before, “[. . . ] the greatest mathematician in the world, Sophus Lie, died [. . . ]
His work is cut short; his influence, his fame, will broaden, will tower from day to
day.”
Probably a more accurate evaluation of Lie was given by Engel in a memorial
speech on Lie [9, p. 24] in 1899:
If the capacity for discovery is the true measure of a mathematician’s greatness, then
Sophus Lie must be ranked among the foremost mathematicians of all time. Only
extremely few have opened up so many vast areas for mathematical research and
created such rich and wide-ranging methods as he [. . . ] In addition to a capacity
for discovery, we expect a mathematician to posses a penetrating mind, and Lie was
really an exceptionally gifted mathematician [. . . ] His efforts were based on tackling
problems which are important, but solvable, and it often happened that he was able
to solve problems which had withstood the efforts of other eminent mathematicians.
In this sense, Lie was a giant for his deep and original contribution to mathematics,
and is famous not for other reasons. (One can easily think of several mathematicians,
without naming them, who are famous for various things besides mathematics). Inci-
dentally, he was also a giant in the physical sense. There are some vivid descriptions
of Lie by people such as É. Cartan [1, p. 7], Engel [27, p. 312], and his physics](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-25-320.jpg)
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colleague Ostwald at Leipzig [27, p. 396]. See also [27, p. 3]. For some interest-
ing discussions on the relations between giants and scientists, see [11, pp. 163–164,
p. 184] and [22, pp. 9–13].
3 A glimpse of Lie’s early academic life
Lie was born on December 17, 1842. His father, Johann Herman Lie, was a Lutheran
minister. He was the youngest of the six children of the family. Lie first attended
school in the town of Moss in South Eastern Norway and on the eastern side of the
Oslo Fjord. In 1857 he entered Nissen’s Private Latin School in Christiania, which
became Oslo in 1925. At that time, he decided to pursue a military career, but his poor
eyesight made this impossible, and he entered University of Christiania to pursue
a more academic life.
During his university time, Lie studied science in a broad sense. He took math-
ematics courses and attended lectures by teachers of high quality. For example, he
attended lectures by Sylow in 1862.1
Though Lie studied with some good mathematicians and did well in most courses,
on his graduation in 1865, he did not show any special ability for mathematics or any
particular liking for it. Lie could not decide what subject to pursue and he gave
some private lessons and also volunteered some lectures for a student union while
trying to make his decision. He knew he wanted an academic career and thought for
a while that astronomy might be the right topic. He also learnt some mechanics, and
wondered about botany, zoology or physics. Lie reached the not-so-tender age of 26
in 1868 and was still not sure what he should pursue as a career. But this year was
a big turning point for him.
In June 1868, the Tenth Meeting of Scandinavian Natural Sciences was held in
Christiania. It attracted 368 participants. Lie attended many lectures and was par-
ticularly influenced by the lecture of a former student of the great French geometer
Michel Chasles, which referred to works of Chasles, Möbius, and Plücker.
It seems that the approaching season, the autumn of 1868, became one long con-
tinuing period of work for Sophus Lie, with his frequent borrowing of books from the
library. In addition to Chasles, Möbius and Plücker, Lie discovered the Frenchman
Poncelet, the Englishman Hamilton, and the Italian Cremona, as well as others who
had made important contributions to algebraic and analytic geometry.
Lie plowed through many volumes of the leading mathematical journals from
Paris and Berlin, and in the Science Students Association he gave several lectures
during the spring of 1869 on what he called his “Theory of the imaginaries”, and
on how information on real geometric objects could be transferred to his “imaginary
objects.”
1Ludwig Sylow (1832–1918) was Norwegian, like Lie. He is now famous and remembered for the Sylow
subgroups. At that time, he was not on the permanent staff of the university of Christiania, but he was substitut-
ing for a regular faculty member and taught a course. In this course, he explained Abel’s and Galois’ work on
algebraic equations. But it seems that Lie did not understand or remember the content of this course, and it was
Klein who re-explained these theories to him and made a huge impact on Lie’s mathematical life.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-26-320.jpg)
![1 Sophus Lie, a giant in mathematics 5
Sophus Lie wrote a paper on his discovery. The paper was four pages long and it
was published at his own expenses.2
After this paper was translated into German, it
was published in the leading mathematics journal of the time, Crelle’s Journal.3
With
this paper, he applied to the Collegium for a travel grant and received it. Then he left
for Berlin in September 1869 and begun his glorious and productive mathematical
career.
There were several significant events in Berlin for Lie on this trip. He met Felix
Klein and they immediately became good friends. They shared common interests
and common geometric approaches, and their influence on each other was immense.
Without this destined (or chance) encounter, Lie and Klein might not have been the
people we know.
Lie also impressed Kummer by solving problems which Kummer was working
on. This gave him confidence in his own power and originality. According to Lie’s
letter to his boyfriend [26, p. XII]:
Today I had a triumph which I am sure you will be interested to hear about. Professor
Kummer suggested that we test our powers on a discussion of all line congruences of
the 3rd degree. Fortunately, a couple of months ago, I had already solved a problem
which was in a way special of the above, but was nevertheless much more general.
. . . I regard this as a confirmation of my good scientific insight that I, from the very
first, understood the value of findings. That I have shown both energy and capability
in connection with my findings; that I know.
In the summer of 1970, Lie and Klein visited Paris and met several important
people such as Jordan and Darboux. The interaction with Jordan and Jordan’s new
book on groups had a huge influence on both of them. This book by Jordan contained
more than an exposition of Galois theory and can be considered as a comprehensive
discussion of how groups were used in all subjects up to that point. For Klein and Lie,
it was an eye opener. Besides learning Galois theory, they started to realize the basic
and unifying role groups would play in geometry and other parts of mathematics. In
some sense, the trips with Klein sealed the future research direction of Lie. Klein
played a crucial role in the formative years of Lie, and the converse is also true. We
will discuss their interaction in more detail in 8 and 9.
At the outbreak of the Franco–Prussian war in July, Klein left, and Lie stayed
for one more month and then decided to hike to Italy. But he was arrested near
Fontainebleau as he was suspected of being a spy and spent one month in jail. Dar-
boux came and freed him. In [8], Darboux wrote:
True, in 1870 a misadventure befell him, whose consequences I was instrumen-
tal in averting. Surprised at Paris by the declaration of war, he took refuge at
2The publication of this paper is unusual also by today’s standards. According to [2], “His first published
paper appeared in 1869. It gives a new representation of the complex plane and uses ideas of Plücker. But Lie
had difficulties in getting these ideas published by the Academy in Christiania. He was impatient. Professor
Bjerknes asked for more time to look at the paper, but Professor Broch returned it after two days – saying he
had understood nothing! However, three other professors – who probably understood the material even less –
supported publication. This happened as a result of influence by friends of Lie.”
3The German version of this paper is still only 8 pages long, but in his collected works edited by Engel and
Heegaard, there are over 100 pages of commentaries devoted to it.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-27-320.jpg)
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Fontainebleau. Occupied incessantly by the ideas fermenting in his brain, he would
go every day into the forest, loitering in places most remote from the beaten path,
taking notes and drawing figures. It took little at this time to awaken suspicion.
Arrested and imprisoned at Fontainebleau, under conditions otherwise very com-
fortable, he called for the aid of Chasles, Bertrand, and others; I made the trip to
Fontainebleau and had no trouble in convincing the procureur impérial; all the notes
which had been seized and in which figured complexes, orthogonal systems, and
names of geometers, bore in no way upon the national defenses.
Afterwards, Lie wrote to his close friend [26, p. XV], “except at the very first,
when I thought it was a matter of a couple of days, I have taken things truly philo-
sophically. I think that a mathematician is well suited to be in prison.”
In fact, while he was in prison, he worked on his thesis and a few months later,
he submitted his thesis, on March 1871. He received his doctorate degree in July
1872, and accepted a new chair at the university of Christiania set up for him by the
Norwegian National Assembly. It was a good thesis, which dealt with the integration
theory of partial differential equations. After his thesis, Lie’s mathematics talent was
widely recognized and his mathematical career was secured.
When Lie worked on his thesis with a scholarship from the University of Christia-
nia, he needed to teach at his former grammar school to supplement his income. With
this new chair, he could devote himself entirely to mathematics. Besides developing
his work on transformation groups and working with Klein towards the formulation
of the Erlangen program, Lie was also involved in editing with his former teacher
Ludwig Sylow the collected works of Abel. Since Lie was not familiar with alge-
bras, especially with Abel’s works, this project was mainly carried out by Sylow. But
locating and gathering manuscripts of Abel took a lot of effort to both men, and the
project took multiple years.
In his personal life, Lie married Anna Birch in 1874, and they had two sons and
a daughter.
Lie published several papers on transformation groups and on the applications to
integration of differential equations and he established a new journal in Christiania
to publish his papers, but these papers did not receive much attention. Because of
this, Lie started to work on more geometric problem such as minimal surfaces and
surfaces of constant curvature.
Later in 1882 some work by French mathematicians on integration of differential
equations via transformation groups motivated Lie to go back to his work on integra-
tion of differential equations and theories of differential invariants of groups.
4 A mature Lie and his collaboration with Engel
There were two people who made, at least contributed substantially to make, Lie the
mathematician we know today. They were Klein and Engel. Of course, his story with
Klein is much better known and dramatic and talked about, but his interaction with
Engel is not less important or ordinary.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-28-320.jpg)
![1 Sophus Lie, a giant in mathematics 7
In the period from 1868 to 1884, Lie worked constantly and lonely to develop
his theory of transformation groups, integration problems, and theories of differential
invariants of finite and infinite groups. But he could not describe his new ideas in an
understandable and convincing way, and his work was not valued by the mathematics
community. Further, he was alone in Norway and no one could discuss with him or
understand his work.
In a letter to Klein in September 1883 [9, p. 9], Lie wrote that “It is lonely, fright-
fully lonely, here in Christiania where nobody understands my work and interests.”
Realizing the seriousness of the situation of Lie and the importance of summa-
rizing in a coherent way results of Lie and keeping him productive, Klein and his
colleague Mayer at Leipzig decided to send their student Friedrich Engel to assist
Lie. Klein and Mayer knew that without help from someone like Engel, Lie could not
produce a coherent presentation of his new novel theories.
Like Lie, Engel was also a son of a Lutheran minister. He was born in 1861, nine-
teen years younger than Lie. He started his university studies in 1879 and attended
both the University of Leipzig and the University of Berlin. In 1883 he obtained his
doctorate degree from Leipzig under Mayer with a thesis on contact transformations.
After a year of military service in Dresden, Engel returned to Leipzig in the spring
of 1884 to attend Klein’s seminar in order to write a Habilitation. At that time, be-
sides Klein, Mayer was probably the only person who understood Lie’s work and
his talent. Since contact transformations form one important class in Lie’s theories
of transformation groups, Engel was a natural candidate for this mission. Klein and
Mayer worked together to obtain a stipend from the University of Leipzig and the
Royal Society of Sciences of Saxony for Engel so that he could travel to work with
Lie in Christiania.
In June 1884, Lie wrote a letter to Engel [9, p. 10],
From 1871–1876, I lived and breathed only transformation groups and integration
problems. But when nobody took any interest in these things, I grew a bit weary
and turned to geometry for a time. Now just in the last few years, I have again taken
up these old pursuits of mine. If you will support me with the further development
and editing of these things, you will be doing me a great service, especially in that,
for once, a mathematician finally has a serious interest in these theories. Here in
Christiania, a specialist like myself is terribly lonely. No interest, no understanding.
According to a letter of Engel in the autumn of 1884 after meeting Lie [27, p. 312]:
The goal of my journey was twofold: on one hand, under Lie’s own guidance, I
should become immersed in his theories, and on the other, I should exercise a sort of
pressure on him, to get him to carry on his work for a coherent presentation of one
of his greater theories, with which I should help him apply his hand.
Lie wanted to write a major comprehensive monograph on transformation groups,
not merely a simple introduction to his new theory. It “should be a systematic and
strict-as-possible account that would retain its worth for a long time” [9, p. 11].
Lie and Engel met twice every day, in the morning at the apartment of Engel and
in the afternoon at Lie’s apartment. They started with a list of chapters. Then Lie](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-29-320.jpg)
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dictated an outline of each chapter and Engel would supply the detail. According to
Engel [9, p. 11],
Every day, I was newly astonished by the magnificence of the structure which Lie
had built entirely on his own, and about which his publications, up to then, gave only
a vague idea. The preliminary editorial work was completed by Christmas, after
which Lie devoted some weeks to working through all of the material in order to lay
down the final draft. Starting at the end of January 1885, the editorial work began
anew; the finished chapters were reworked and new ones were added. When I left
Christiania in June of 1885, there was a pile of manuscripts which Lie figured would
eventually fill approximately thirty printer’s sheets. That it would be eight years
before the work was completed and the thirty sheets would become one hundred and
twenty-five was something neither of us could have imagined at that time.
Lie and Engel worked intensively over the nine month period when Engel was
there. This collaboration was beneficial to both parties. To Engel, it was probably the
best introduction to Lie theories and it served his later mathematical research well.
According to Kowalewski, a student of Lie and Engel, [9, p. 10], “Lie would never
have been able to produce such an account by himself. He would have drowned
in the sea of ideas which filled his mind at that time. Engel succeeded in bringing
a systematic order to this chaotic mass of thought.”
After returning to Leipzig, Engel finished his Habilitation titled “On the defining
equations of the continuous transformation groups” and became a Privatdozent.
In 1886, Klein moved to Göttingen for various reasons. (See [18] for a brief
description of Klein’s career). Thanks to the efforts of Klein, Lie moved to Leipzig
in 1886 to take up a chair in geometry. More description of this will be given in 8
below.
When Lie visited Leipzig in February 1886 to prepare for his move, he wrote to
his wife excitedly [27, p. 320], “to the best of my knowledge, there have been no
other foreigners, other than Abel and I, appointed professor at a German University.
(The Swiss are out of the calculation here.) It is rather amazing. In Christiania I have
often felt myself to be treated unfairly, so I have truly achieved an unmerited honor.”
Leipzig was the hometown of the famous Leibniz and a major culture and aca-
demic center. In comparison to his native country, it was an academic heaven for
Lie.
In April 1886, Lie became the Professor of Geometry and Director of the Math-
ematical Seminar and Institute at the University of Leipzig. Lie and Engel resumed
to work intensively on their joint book again. In 1888, the leading German scientific
publisher Teubner, based in Leipzig, published the first volume of Theory of transfor-
mation groups, which was 632 pages long.
In that year, Engel also became the assistant to Lie after Friedrich Schur left.
When Lie went to a nerve clinic near the end of 1889, Engel gave Lie’s lectures for
him.
The second volume of their joint book was published in 1890 and was 555 pages
long, and the third volume contained 831 pages and was published in 1893.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-30-320.jpg)
![1 Sophus Lie, a giant in mathematics 9
The three big volumes of joint books with Engel would not see the light of day or
even start without the substantial contribution of Engel.
It was a major piece of work. In a 21-page review of the first volume [9, p. 16],
Eduard Study wrote,
The work in question gives a comprehensive description of an extensive theory
which Mr. Lie has developed over a number of years in a large number of indi-
vidual articles in journals . . . Because most of these articles are not well known,
and because of their concise format, the content, in spite of its enormous value, has
remained virtually unknown to the scientific community. But by the same token we
can also be thankful that the author has had the rare opportunity of being able to
let his thoughts mature in peace, to form them in harmony and think them through
independently, away from the breathless competitiveness of our time. We do not
have a textbook written by a host of authors who have worked together to introduce
their theories to a wider audience, but rather the creation of one man, an original
work which, from beginning to end, deals with completely new things [. . . ] We do
not believe that we are saying too much when we claim that there are few areas of
mathematical science which will not be enriched by the fundamental ideas of this
new discipline.
It is probably interesting to note that Engel’s role and effort in this massive work
were not mentioned here. Maybe the help of a junior author or assistant was taken
for granted in the German culture at that time.
In the preface to the third volume, Lie wrote [9, p. 15]:
For me, Professor Engel occupies a special position. On the initiative of F. Klein
and A. Mayer, he traveled to Christiania in 1884 to assist me in the preparation of
a coherent description of my theories. He tackled this assignment, the size of which
was not known at that time, with the perseverance and skill which typifies a man of
his caliber. He has also, during this time, developed a series of important ideas of
his own, but has in a most unselfish manner declined to describe them here in any
great detail or continuity, satisfying himself with submitting short pieces to Mathe-
matische Annalen and, particularly, Leipziger Berichte. He has, instead, unceasingly
dedicated his talent and free time which his teaching allowed him to spend, to work
on the presentation of my theories as fully, as completely and systematically, and
above all, as precisely as is in any manner possible. For this selfless work which
has stretched over a period of nine years, I, and, in my opinion, the entire scientific
world owe him the highest gratitude.
Lie and Engel formed a team both in terms of writing and teaching. Some students
came to study with both Lie and Engel. Engel also contributed to the success of Lie’s
teaching. For example, a major portion of students who received the doctoral degree
at Leipzig was supervised under Lie. Lie also thanked Engel for this in the preface of
the third volume.
But this preface also contained a description of some conflict with Klein, and
hence Engel’s academic future suffered due to this. See 9 for more detail.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-31-320.jpg)
![10 Lizhen Ji
5 Lie’s breakdown and a final major result
After his move to Leipzig, Lie worked hard and was very productive. While Leipzig
was academically stimulating to Lie, it was not stress-free for him, and relations with
others were complicated too. “The pressure of work, problems of collaboration, and
domestic anxieties made him sleepless and depressed, and in 1889 he had a complete
breakdown” [27, p. 328]. Lie had to go to a nerve clinic and stayed there for seven
months. He was given opium, but the treatment was not effective, and he decided to
cure the problem himself.4
He wrote to his friend [26, XXIII]:
In the end I began to sleep badly and finally did not sleep at all. I had to give up my
lecturing and enter a nerve clinic. Unfortunately I have been an impossible patient.
It has always been my belief that the doctors did not understand my illness. I have
been treated with opium, in enormous dose, to calm my nerves, but it did not help.
Also sleeping draughts.
Three to four weeks ago I got tired of staying at the nerve clinic. I decided to try
to see what I could do myself to regain my equilibrium and the ability to sleep. I have
now done what the doctors say no one can endure, that is to say I have completely
stopped taking opium. It has been a great strain. But now, on a couple of days,
against the doctors’ advice, I have taken some exercise.
I hope now that in a week’s time I shall have completely overcome the harmful
effects of the opium cure. I think myself that the doctors have only harmed me with
opium.
My nerves are very strained, but my body has still retained its horsepower. I
shall cure myself on my own. I shall walk from morning to evening (the doctors say
it is madness). In this way I shall drive out all the filth of the opium, and afterwards
my natural ability to sleep will gradually return. That is my hope.
Finally he thought that he had recovered, and was released. Actually he was not
cured at the time of release. Instead, in the reception book of the clinic, his condition
at that time was recorded as “a Melancholy not cured” [27, p. 328]. His friends
and colleagues found changes in Lie’s attitudes towards others and his behaviors:
mistrusting and accusing others for stealing his ideas. Indeed, according to Engel
[27, p. 397], Lie did recover his mathematical ability, but “not as a human being. His
mistrust and irritability did not dissipate, but rather they grew more and more with
the years, such that he made life difficult for himself and all his friends. The most
painful thing was that he never allowed himself to speak openly about the reasons for
his despondency.”
When he was busy teaching and working out his results, he did not have much
time to pick up new topics. While at the nerve clinic, he worked again on the so-
called Helmholtz problem on the axioms of geometry5
and wrote two papers about it.
4According to the now accepted theory, Lie suffered from the so-called pernicious anemia psychosis, an
incurable disease at that time. People also believe that his soured relations with Klein and others were partly
caused by this disease. See the section on the period 1886–1898 in [9] and the reference [29] there.
5Lie’s work on the Helmholtz problem was apparently well known at the beginning of the 20th century.
According to [5, Theorem 16.7], the Lie–Helmholtz Theorem states that spaces of constant curvature, i.e., the
Euclidean space, the hyperbolic space and the sphere, can be characterized by abundance of isometries: for every](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-32-320.jpg)
![1 Sophus Lie, a giant in mathematics 11
Lie had thought of and worked on this problem for a long time and had also
criticized the work of Helmholtz and complained to Klein about it. According to [27,
p. 380–381],
Very early on, Lie was certainly clear that the transformation theory he was devel-
oping was related to non-Euclidean geometry, and in a letter to Mayer as early as
1875, Lie had pointed out that von Helmhotz’s work on the axioms of geometry from
1868, were basically and fundamentally an investigation of a class of transformation
groups: “I have long assumed this, and finally had it verified by reading his work.”
Klein too, in 1883, has asked Lie what he thought of von Helmhotz’s geomet-
ric work. Lie replied immediately that he found the results correct, but that von
Helmhotz operated with a division between the real and imaginary that was hardly
appropriate. And a little later, after having studied the treatise more thoroughly, he
communicated to Klein that von Helmhotz’s work contained “substantial shortcom-
ings”, and he thought it positively impossible to overcome these shortcomings by
means of the elementary methods that von Helmhotz had applied. Lie went on to
complete and simplify von Helmhotz’s spatial theory [. . . ]
In 1884, Lie wrote to Klein [26, XXVI]:
If I ever get as far as to definitely complete my old calculations of all groups and
point transformations of a three-dimensional space I shall discuss in more detail
Helmholtz’s hypothesis concerning metric geometry from a purely analytical aspect.
According to [27, p. 381],
Lie did further work with von Helmhotz’s space problem, and confided to Klein
in April 1887, that the earlier works on the problem had now come to a satisfying
conclusion – at least when one was addressing finite dimensional transformation
groups, and therein, a limited number of parameters. What remained was to deduce
some that extended across the board such that infinite-dimensional groups could be
included.
Lie’s work on the Helmholtz problem led him to being awarded the first Lo-
batschevsky prize in 1897. Klein wrote a very strong report on his work, and this
report was the determining factor for this award.
6 An overview of Lie’s major works
As mentioned before, Lie was very productive and he wrote many thousands of pages
of papers and multiple books. His name will be forever associated with Lie groups
and Lie algebras and several other dozen concepts and definitions in mathematics
two congruently ordered triples of points, there is an isometry of the space that moves one triple to the other,
where two ordered triples of points .v1; v2; v3/, .v0
1; v0
2; v0
3/ are congruent if the corresponding distances are
equal, d.vi ; vj / D d.v0
i ; v0
j / for all pairs i; j. References to papers of Weyl and Enriques on this theorem
were given in [5].](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-33-320.jpg)
![12 Lizhen Ji
(almost all of them involve Lie groups or Lie algebras in various ways). One natural
question is what exactly Lie had achieved in Lie theory. The second natural question
is: besides Lie groups and Lie algebras, what else Lie had done.
It is not easy to read and understand Lie’s work due to his writing style. In a pref-
ace to a book of translations of some papers of Lie [21] in a book series Lie Groups:
History, Frontiers and Applications, which contain also some classical books and
papers by É. Cartan, Ricci, Levi-Civita and also other more modern ones, Robert
Hermann wrote:
In reading Lie’s work in preparation for my commentary on these translations, I was
overwhelmed by the richness and beauty of the geometric ideas flowing from Lie’s
work. Only a small part of this has been absorbed into mainstream mathematics. He
thought and wrote in grandiose terms, in a style that has now gone out of fashion,
and that would be censored by our scientific journals! The papers translated here and
in the succeeding volumes of our translations present Lie in his wildest and greatest
form.
We nevertheless try to provide some short summaries. Though articles in Encyclope-
dia Britannica are targeting the educated public, articles about mathematicians often
give fairly good summaries. It might be informative and interesting to take a look at
such an article about Lie before the global theory of Lie groups were developed by
Weyl and Cartan. An article in Encyclopedia Britannica in 1911 summarized Lie’s
work on Lie theory up to that time:
Lie’s work exercised a great influence on the progress of mathematical science dur-
ing the later decades of the 19th century. His primary aim has been declared to be
the advancement and elaboration of the theory of differential equations, and it was
with this end in view that he developed his theory of transformation groups, set forth
in his Theorie der Transformationsgruppen (3 vols., Leipzig, 1888–1893), a work of
wide range and great originality, by which probably his name is best known. A spe-
cial application of his theory of continuous groups was to the general problem of
non-Euclidean geometry. The latter part of the book above mentioned was devoted
to a study of the foundations of geometry, considered from the standpoint of B. Rie-
mann and H. von Helmholtz; and he intended to publish a systematic exposition of
his geometrical investigations, in conjunction with Dr. G. Scheffers, but only one
volume made its appearance (Geometrie der Berührungstransformationen, Leipzig,
1896).
The writer of this article in 1911 might not have imagined the wide scope and multi-
faceted applications of Lie theory. From what I have read and heard, a list of topics
of major work of Lie is as follows:
1. Line complexes. This work of Lie was the foundation of Lie’s future work on
differential equations and transformation groups, and hence of Lie theory [13].
It also contains the origin of toric varieties.
2. Lie sphere geometry and Lie contact structures. Contact transformations are
closely related to contact geometry, which is in many ways an odd-dimensional
counterpart of symplectic geometry, and has broad applications in physics.
Relatively recently, it was applied to low-dimensional topology.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-34-320.jpg)
![1 Sophus Lie, a giant in mathematics 15
1. The first theorem should be stated as: a Lie group homomorphism is deter-
mined locally by a Lie algebra homomorphism.
2. The second theorem says that any Lie group homomorphism induces a Lie al-
gebra homomorphism. Conversely, given a Lie algebra homomorphism, there
is a local group homomorphism between corresponding Lie groups. If the do-
main of the locally defined map is a simply connected Lie group, then there is
a global Lie group homomorphism.
3. The third theorem says that given a Lie algebra g, there exists a Lie group G
whose Lie algebra is equal to g. (Note that there is no group action here and
hence this statement is different from the statements above.)
8 Relation with Klein I: the fruitful cooperation
There are many differences and similarities between Lie and Klein. Lie was a good
natured, sincere great mathematician. For example, he gave free lectures in the sum-
mer to USA students to prepare them for his later formal lectures. He went out of his
way to help his Ph.D. students. He was not formal, and his lectures were not polished
and could be messy sometimes.
Klein was a good mathematician with a great vision and he was also a powerful
politician in mathematics. He was a noble, strict gentleman. His lectures were always
well-organized and polished.
Lie and Klein first met in Berlin in the winter semester of 1869–1870 and they be-
came close friends. It is hard to overestimate the importance of their joint work and
discussions on their mathematical works and careers. For example, it was Klein who
helped Lie to see the analogy between his work on differential equations and Abel’s
work on the solvability of algebraic equations, which motivated Lie to develop a gen-
eral theory of differential equations that is similar to the Galois theory for algebraic
equations, which lead to Lie theory. On the other hand, it was Lie who provided
substantial evidence to the general ideas in the Erlangen program of Klein that were
influential on the development of that program.
Klein also helped to promote Lie’s work and career in many ways. For example,
when Klein left Leipzig, he secured the vacant chair for Lie in spite of many objec-
tions. Klein drafted the recommendation of the Royal Saxon Ministry for Cultural
Affairs and Education in Dresden to the Philosophical Faculty of the University of
Leipzig, and the comment on Lie run as follows [9, p. 12]:
Lie is the only one who, by force of personality and in the originality of his thinking,
is capable of establishing an independent school of geometry. We received proof of
this when Kregel von Sternback’s scholarship was to be awarded. We sent a young
mathematician – our present Privatdocent, Dr. Engel – to Lie in Chtristiania, from
where he returned with a plethora of new ideas.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-37-320.jpg)
![16 Lizhen Ji
It is also helpful to quote here a letter written by Weierstrass at that time [9, p. 12]:
I cannot deny that Lie has produced his share of good work. But neither as a scholar
nor as a teacher is he so important that there is a justification in preferring him,
a foreigner, to all of those, our countrymen, who are available. It now seems that he
is being seen as a second Abel who must be secured at any cost.
One particular fellow countryman Weierstrass had in mind was his former student
Hermann Schwarz, who was also a great mathematician.
Another crucial contribution of Klein to Lie’s career was to send Engel to help
Lie to write up his deep work on transformation groups. Without Engel, Lie’s contri-
butions might not have been so well known and hence might not have had the huge
impacts on mathematics and physics that they have now. It is perhaps sad to note
that Engel was punished by Klein in some way for being a co-author of Lie after the
breakup between Lie and Klein. One further twist was that Klein made Engel edit
Lie’s collected works carefully after Lie passed away.
9 Relation with Klein II: conflicts and the famous preface
The breakup between Lie and Klein is famous for one sentence Lie put down in the
preface of the third volume of his joint book with Engel on Lie transformation groups
published in 1893: “I am not a student of Klein, nor is the opposite the case, even if
it perhaps comes closer to the truth.”
This is usually the only sentence that people quote and say. It sounds quite strong
and surprising, but there are some reasons behind it. The issue is about the formula-
tion and credit of ideas in the Erlangen program, which was already famous at that
time. It might be helpful to quote more from the foreword of Lie [9, p. 19]:
F. Klein, whom I kept abreast of all my ideas during these years, was occasioned
to develop similar viewpoints for discontinuous groups. In his Erlangen Program,
where he reports on his and on my ideas, he, in addition, talks about groups which,
according to my terminology, are neither continuous or discontinuous. For example,
he speaks of the group of all Cremona transformations and of the group of distor-
tions. The fact that there is an essential difference between these types of groups
and the groups which I have called continuous (given the fact that my continuous
groups can be defined with the help of differential equations) is something that has
apparently escaped him. Also, there is almost no mention of the important concept
of a differential invariant in Klein’s program. Klein shares no credit for this con-
cept, upon which a general invariant theory can be built, and it was from me that
he learned that each and every group defined by differential equations determines
differential invariants which can be found through integration of complete systems.
I feel these remarks are called for since Klein’s students and friends have repeat-
edly represented the relationship between his work and my work wrongly. Moreover,
some remarks which have accompanied the new editions of Klein’s interesting pro-
gram (so far, in four different journals) could be taken the wrong way. I am no](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-38-320.jpg)
![1 Sophus Lie, a giant in mathematics 17
student of Klein and neither is the opposite the case, though the latter might be
closer to the truth.
By saying all this, of course, I do not mean to criticize Klein’s original work in
the theory of algebraic equations and function theory. I regard Klein’s talent highly
and will never forget the sympathetic interest he has taken in my research endeav-
ors. Nonetheless, I don’t believe he distinguishes sufficiently between induction and
proof, between a concept and its use.
According to [27, p. 317], in the same preface,
Lie’s assertion was that Klein did not clearly distinguish between the type of groups
which were presented in the Erlangen Programme – for example, Cremonian trans-
formations and the group of rotations, which in Lie’s terminology were neither con-
tinuous nor discontinuous – and the groups Lie had later defined with the help of
differential equations:
“One finds almost no sign of the important concept of differential invariants in
Klein’s programme. This concept, which first of all a common invariant theory could
be build upon, is something Klein has no part of, and he has learned from me that
every group that it defined by means of differential equations, determines differential
invariants, which can be found through the integration of integrable systems.”
. . . Lie continued, in their investigations of geometry’s foundation, Klein, as
well as von Helmhotz, de Tilly, Lindemann, and Killing, all committed gross errors,
and this could largely be put down to their lack of knowledge of group theory.
Maybe some explanations are in order to shed more light on these strong words
of Lie. According to [26, pp. XXIII–XXIV],
Sophus Lie gradually discovered that Felix Klein’s support for his mathematical
work no longer conformed with his own interests, and the relationship between the
two friends became more reserved. When, in 1892, Felix Klein wanted to republish
the Erlangen program and explain its history, he sent the manuscript to Sophus
Lie for a comment. Sophus Lie was dismayed when he saw what Felix Klein has
written, and got the impression that his friend now wanted to have his share of what
Sophus Lie regarded as his life’s work. To make things quite clear, he asked Felix
Klein to let him borrow the letters he had sent him before the Erlangen program was
written. When he learned that these letters no longer existed, Sophus Lie wrote to
Felix Klein, November 1892.
The letter from Lie to Klein in November 1892 goes as follows [9, p. XXIV]:
I am reading through your manuscript very thoroughly. In the first place, I am afraid
that you, on your part, will not succeed in producing a presentation that I can accept
as correct. Even several points which I have already criticized sharply are incorrect,
or at least misleading, in your current presentation. I shall try as far as possible
to concentrate my criticism on specific points. If we do not succeed in reaching
agreement, I think that it is only right and reasonable that we each present our views
independently, and the mathematical public can then form their own opinion.
For the time being I can only say how sorry I am that you were capable of
burning my so significant letters. In my eyes this was vandalism; I had received
your specific promise that you would take care of them.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-39-320.jpg)
![18 Lizhen Ji
I have already told you that my period of naiveness is now over. Even if I still
firmly retain good memories from the years 1869–1872, I shall nevertheless try to
keep myself that which I regard as my own. It seems that you sometimes believe
that you have shared my ideas by having made use of them.
The comprehensive biography of Lie [27, p. 371] gives other details on the origin of
this conflict:
The relationship between them [Lie and Klein] had certainly cooled over recent
years, although they continued to exchange letters the same way, although not as
frequently as earlier. But it was above all professional divergencies that were cen-
tral to the fact that Lie now broke off relationship. Following Lie’s publication of
the first volume of his Theorie der Transformationsgruppen, Klein judged that there
was sufficient interest to have his Erlangen Programme been republished. But before
Klein’s text from 1872 was printed anew, Klein had contacted Lie to find out how
the working relationship and exchange of ideas between them twenty years earlier
should be presented. Lie had made violent objections to the way in which Klein
had planned to portray the ideas and the work. But Klein’s Erlangen Programme
was printed, and it came out in four different journals, in German, Italian, English
and French – without taking into account Lie’ commentary on his assistance in for-
mulating this twenty-year-old programme. More and more in mathematical circles,
Klein’s Erlangen Programme was spoken of as central to the paradigm shift in ge-
ometry that occurred in the previous generation. A large part of the third volume of
Lie’s great work on transformation groups was devoted to a deepening discussion
on the hypotheses or axioms that ought to be set down as fundamental to a geom-
etry, that – whether or not it accepted Euclid’s postulates – satisfactorily clarified
classical geometry as well as the non-Euclidean geometry of Gauss, Lobachevsky,
Bolyai, and Riemann.
The information that spread regarding the relationship between Klein’s and
Lie’s respective work, was, according to Lie, both wrong and misleading. Lie con-
sidered he had been side-lined but was eager to “set things right”, and grasped the
first and best opportunity. In front of the professional substance of his work he
placed his twenty-page foreword. The power-charge that liquidated their friendship
and sent shock-waves through the mathematical milieu was short, if not sweet.
Klein was the king of German mathematics and probably also of the European
mathematics at that time. What was people’s reaction to the strong preface of Lie?
Maybe a letter from Hilbert to Klein in 1893 will explain this [26, p. XXV]: “In his
third volume, his megalomania spouts like flames.”
Lie probably did not suffer too much professionally from this conflict with Klein
since he had the chair at Leipzig. But this was not the case with Engel. Since Engel’s
name also appeared on the book, he had to pay for this. Engel was looking for a job,
and a position of professorship was open at that time at the University of Königsberg,
the hometown and home institution of Hilbert where he held a chair in mathematics,
and this open position was a natural and likely choice for Engel. In the same letter to
Klein, Hilbert continued [26, p. XXV], “I have excluded Engel completely. Although
he himself has not made any comment in the preface, I hold him to some extent co-
responsible for the incomprehensible and totally useless personal animosity which
the third volume of Lie’s work on transformation groups is full off.”](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-40-320.jpg)
![1 Sophus Lie, a giant in mathematics 19
Engel could not get an academic job for several years,6
and Klein arranged Engel
to edit the collected works of Grassmann and then later the collected works of Lie;
on the latter he worked for several decades.
Another consequence of this conflict with Klein was that Lie could not finish
another proposed joint book with Engel on applications of transformation groups to
differential equations. According to [27, p. 390–391], after the publication of these
three volumes,
The next task that Lie saw for himself was to make refinements and applications of
what had now been completely formulated. But this foreword [of the third volume]
with its sharp accusations against Klein, caused hindrances to the further work. Be-
cause Lie in the same foreword had praised Engel to the skies for his “exact” and
“unselfish activity”, it now became difficult for Engel to continue to collaborate with
Lie – consequently as well, nothing came of the announced work on, among other
things, differential invariants and infinite-dimensional continuous groups. As for
Engel, his career outlook certainly now lay in other directions than Lie’s. Accord-
ing to Lie’s German student, Gerhard Kowalewski, relations between Lie and Engel
gradually became so cool that they were seldom to be seen in the same place.
It should be pointed out that relations between Lie and Engel had some hard time
before this foreword came out. It was caused by the fight between Lie and Killing due
to some overlap in their work on Lie theories, in particular, Lie algebras. For some
reason, at the initial stage, Killing communicated with Engel and cited some papers of
Engel instead of Lie’s, and Lie felt than Engel betrayed his trust. For a more detailed
discussion on this issue, see [27, pp. 382–385, p. 395].7
After Lie’s death, Engel
continued and carried out his mentor’s work in several ways. See 11 for example.
He was a faithful disciple and was justly awarded with the Norwegian Order of St
Olaf and an honorary doctorate from the University of Oslo.
Maybe there is one contributing factor to these conflicts.8
It is the intrinsic mad-
ness of all people who are devoted to research and are doing original work, in partic-
6On the other hand, all things ended well with Engel. In 1904, he accepted the chair of mathematics at the
University of Greifswald when his friend Eduard Study resigned, and in 1913, took the chair of mathematics
at the University of Giessen. Engel also received a Lobachevsky Gold Medal. The Lobachevsky medal is
different from the Lobatschevsky prize won by his mentor Lie and his fellow countryman Wilhelm Killing. The
medal was given on a few occasions to the referee of a person nominated for the prize. For instance, Klein also
received, in 1897, a the Gold medal, for his report on Lie. See [28].
7Manfred Karbe pointed out that in his autobiography [20, pp. 51–52], Kowalewski speculates about the
mounting alienation of Lie and Engel, and reports about Lie’s dislike of his three-volume Transformation
Groups. When Lie needed some material of his own in his lectures or seminars, he never made use of these
books but only of the papers in Math. Ann. And Kowalewski continues on page 52, line -5:
“Von hier aus kann man es vielleicht verstehen, dass die Abneigung gegen das Buch sich auf den Mitarbeiter
bertrug, dem er doch so sehr zu Dank verpflichtet war.” (From this one may perhaps understand that the aversion
to the book is transferred to the collaborator to whom he was so much indebted.)
8There has been an explanation of Lie’s behavior in this conflict with Klein by establishing a relation between
genius and madness. According to [27, p. 394], after Lie’s death, “In Göttingen, Klein made a speech that gave
rise to much rumor, not least because here, in addition to all his praise for his old friend, Klein suggested that
the close relationship between genius and madness, and that Lie had certainly been struck by a mental condition
that was tinged with a persecution complex – at least, by assessing the point from notes that Klein made for his
speech, it seems that this was the expression he used.”](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-41-320.jpg)
![20 Lizhen Ji
ular mathematicians and scientists. According to a comment of Lie’s nephew, Johan
Vogt, a professor at the University of Oslo in economics, and also a translator, writer
and editor, made in 1930 on his uncle [27, p. 397],
We shall avail ourselves of a popular picture. Every person has within himself some
normality and some of what may be called madness. I believe that most of my col-
leagues possess ninety-eight percent normality and two percent madness. But So-
phus Lie certainly had appreciably more of the latter. The merging of a pronounced
scientific gift and an impulsiveness that verges on the uncontrollable, would certainly
describe many of the greatest mathematicians. In Sophus Lie this combination was
starkly evident.
It might be helpful to point out that later at the request of the committee of Lo-
batschevsky prize, Klein wrote a very strong report about the important work of Lie
contained in this third volume on transformation groups, and this report was instru-
mental in securing the inaugural Lobatschevsky prize for Lie.
It might also be helpful to quote from Klein on Lie’s work related to this conflict.
The following quote of Klein [19, pp. 352–353], its translation and information about
it were kindly provided by Hubert Goenner:
I will now add a personal remark. The already mentioned Erlangen Program is about
an outlook which – as already stated in the program itself – I developed in personal
communication with Lie (now professor in Leipzig, before in Christiana). Lie who
has been engaged particularly with transformation groups, created a whole theory of
them, which finds its account in a larger œuvre “Theory of Transformation Groups”,
edited by Lie and Engel, Vol. I 1888, Vol. II 1890. In addition, a third volume
will appear, supposedly in not all too distant a time. Obviously, we cannot think
about responding now to the contents of Lie’s theories [. . . ]. My remark is limited
to having called attention to Lie’s theories.
The above comment was made by Klein in the winter of 1889 or at the beginning
of 1890, but Klein backed its publication until 1893, the year of the ill-famed preface
to the third volume by Lie and Engel.
Further details about this unfortunate conflict and the final reconciliation between
these two old friends are also given in [27, pp. 384–394]. See also the article [18]
for more information about Klein and on some related discussion on the relationship
between Lie and Klein.
The above discussion showed that the success of the Erlangen program was one
cause for the conflict between Lie and Klein. A natural question is how historians
of mathematics have viewed this issue. Given the fame and impacts of the Erlangen
program, it is not surprising that there have been many historical papers about it.
Two papers [15] [3] present very different views on the contributions of Klein and
Lie to the success and impacts of this program. The paper [15] argues convincingly
that Lie’s work in the period 1872–1892 made the Erlangen program a solid program
with substantial results, while the paper [3] was written to dispute this point of view.
It seems that the authors are talking about slightly different things. For example, [3]
explains the influence of Klein and the later contribution of Study, Cartan and Weyl,](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-42-320.jpg)
![1 Sophus Lie, a giant in mathematics 21
but most of their contributions were made after 1890. The analysis of the situation in
[10, p. 550] seems to be fair and reasonable:
It seems that the Erlangen Program met with a slow reception until the 1890s, by
which time Klein’s status as a major mathematician at the University of Göttingen
had a great deal to do with its successful re-launch. By that time too a number of
mathematicians had done considerable work broadly in the spirit of the programme,
although the extent to which they were influenced by the programme, or were even
aware of it, is not at all clear [. . . ] Since 1872 Lie had gone on to build up a vast
theory of groups of continuous transformations of various kinds; but however much
it owned to the early experiences with Klein, and however much Klein may have
assisted Lie in achieving a major professorship at Leipzig University in 1886, it
is doubtful if the Erlangen Program had guided Lie’s thoughts. Lie was far too
powerful and original a mathematician for that.
10 Relations with others
As mentioned, both Klein and Engel played crucial roles in the academic life of
Lie. Another important person to Lie is Georg Scheffers, who obtained his Ph.D.
in 1890 under Lie. Lie thought highly of Scheffers. In a letter to Mittag–Leffler [27,
p. 369], Lie wrote “One of my best pupils (Scheffers) is sending you a work, which
he has prepared before my eyes, and who has taken his doctorate here in Leipzig with
a dissertation that got the best mark . . . Scheffers possesses an unusually evident
talent and his calculations are worked through with great precision, and bring new
results.”
After the collaboration between Lie and Engel unfortunately broke off, Scheffers
substituted for Engel and edited two of Lie’s lecture notes in the early 1890s. They
are Lectures on differential equations with known infinitesimal transformations of
568 pages, and Lectures on Continuous groups of 810 pages. Later in 1896, they
also wrote a book together, Geometry of contact transformations of 694 pages. All
these book projects of Lie with others indicate that Lie might not have been able to
efficiently write up books by himself. For example, he only wrote by himself a book
of 146 pages and a program for a course in Christiania in 1878.
In 1896 Scheffers became docent at the Technical University of Darmstadt, where
he was promoted to professor in 1900. The collaboration with Lie stopped after this
move. From 1907 to 1935, when he retired, Scheffers was a professor at the Technical
University of Berlin.
According to a prominent American mathematician, G. A. Miller, from the end of
the nineteenth century, “The trait of Lie’s character which impressed me most forcibly
when I first met him in the summer of 1895 was his extreme openness and lack of
effort to hide ignorance on any subject.”
Though he was motivated by discontinuous groups (or rather finite groups) taught
by Sylow and kept on studying a classical book on finite substitution groups by Jor-](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-43-320.jpg)
![22 Lizhen Ji
dan, he could never command the theory of finite groups. Miller continued, “In fact
he frequently remarked during his lectures that he always got stuck when he entered
upon the subject of discontinuous groups.”
When Lie first arrived in Leipzig, teaching was a challenge for him due to both
lack of students and the amount of time needed for preparation. In a letter to a friend
from the youth, Lie wrote [26, p. XXI], “While, in Norway, I hardly spent five
minutes a day on preparing the lectures, in Germany I had to spend an average of
about 3 hours. The language is always a problem, and above all, the competition
implies that I had to deliver 8–10 lectures a week.”
When Lie and his assistant Engel decided to present their own research on trans-
formation groups, students from all over the world poured in, and the Ecole Normale
sent its best students to study with Lie. It was a big success. According to the recol-
lection of a student of Lie [26, p. XXVII]:
Lie liked to teach, especially when the subject was his own ideas. He had vivid
contact with his students, who included many Americans, but also Frenchmen, Rus-
sians, Serbs and Greeks. It was his custom to ask us questions during the lectures
and he usually addressed each of us by name.
Lie never wore a tie. His full beard covered the place where the tie would have
been, so even the most splendid tie would not have shown to advantage. At the
start of a lecture he would take off his collar with a deft movement, saying: “I love
to be free”, and he would then begin his lecture with the words “Gentlemen, be
kind enough to show me your notes, to help me remember what I did last time.”
Someone or other on the front bench would immediately stand up and hold out the
open notebook, whereupon Lie, with a satisfied nod of the bead, would say “Yes,
now I remember.” In the case of difficult problems, especially those referring to Lie’s
complex integration theories, it could happen that the great master, who naturally
spoke without any kind of preparation whatsoever, got into difficulties and, as the
saying goes, became stuck. He would then ask one of his elite students for help.
Lie had many students. Probably one of the most famous was Felix Hausdorff.
Lie tried to convince Hausdorff to work with him on differential equations of the first
order without success. Of course, Hausdorff became most famous for his work on
topology. See [27, p. 392–393] for a description of Hausdorff and his interaction
with Lie.
Throughout his life, Lie often felt that he was under-recognized and under-appre-
ciated. This might be explained by his late start in mathematics and his early isolation
in Norway. He paid careful attention to other people’s reaction to his work. For
example, Lie wrote about Darboux in a letter to Klein [25] in October 1882:
Darboux has studied my work with remarkable thoroughness. This is good insofar
as he has given gradually more lectures on my theories at the Sorbonne, for example
on line and sphere geometry, contact transformation, and first-order partial differ-
ential equations. The trouble is that he continually plunders my work. He makes
inessential changes and then publishes these without mentioning my name.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-44-320.jpg)
![1 Sophus Lie, a giant in mathematics 23
11 Collected works of Lie:
editing, commentaries and publication
Since Lie theories are so well known and there are many books on different aspects of
Lie groups and Lie algebras, Lie’s collected works are not so well known to general
mathematicians and students. The editing and publication of Lie’s collected works
are both valuable and interesting to some people. In view of this, we include some
relevant comments.
Due to his death at a relatively young age, the task of editing Lie’s work com-
pletely fell on others. It turned out that editing and printing the collected work of
Sophus Lie was highly nontrivial and a huge financial burden on the publisher. The
situation is well explained in [6]:
Twenty-three years after the death of Sophus Lie appears the first volume to be
printed of his collected memoirs. It is not that nothing has been done in the mean-
time towards making his work more readily available. A consideration of the matter
was taken up soon after his death but dropped owing to the difficulties in the way
of printing so large a collection as his memoirs will make. An early and unsuc-
cessful effort to launch the enterprise was made by the officers of Videnskapssel-
skapet i Kristiania; but plans did not take a definite form till 1912; then through the
Mathematisch-physische Klasse der Leipziger Akademie and the publishing firm of
B. G. Teubner steps were taken to launch the project. Teubner presented a plan for
raising money by subscription to cover a part of the cost of the work and a little
later invitations to subscribe were sent out. The responses were at first not encour-
aging; from Norway, the homeland of Lie, only three subscriptions were obtained in
response to the first invitations.
In these circumstances, Engel, who was pressing the undertaking, resorted to
an unusual means. He asked the help of the daily press of Norway. On March
9, 1913, the newspaper TIDENS TEGN of Christiania carried a short article by
Engel with the title Sophus Lies samlede Afhandlinger in which was emphasized
the failure of Lie’s homeland to respond with assistance in the work of printing
his collected memoirs. This attracted the attention of the editor and he took up the
campaign: two important results came from this, namely, a list of subscriptions from
Norway to support the undertaking and an appropriation by the Storthing to assist
in the work. By June the amount of support received and promised was sufficient to
cause Teubner to announce that the work could be undertaken; and in November the
memoirs for the first volume were sent to the printer, the notes and supplementary
matter to be supplied later.
The Great War so interfered with the undertaking that it could not be contin-
ued, and by the close of the war circumstances were so altered that the work could
not proceed on the basis of the original subscriptions and understandings and new
means for continuing the work had to be sought. Up to this time the work had been
under the charge of Engel as editor. But it now became apparent that the publication
of the memoirs would have to become a Norwegian undertaking. Accordingly, Poul
Heegaard became associated with Engel as an editor. The printing of the work be-
came an enterprise not of the publishers but of the societies which support them in
this undertaking. Under such circumstances the third volume of the series, but the
first one to be printed, has now been put into our hands. “The printing of further vol-](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-45-320.jpg)
![24 Lizhen Ji
umes will be carried through gradually as the necessary means are procured; more
I cannot say about it,” says Engel, “because the cost of printing continues to mount
incessantly.”
The first volume was published in 1922, and the sixth volume was published in 1937.
The seventh volume consisted of some unpublished papers of Lie and was published
only in 1960 due to the World War II and other issues. This was certainly a major
collected work in the last 100 years.
The collected works of Lie are very well done with the utmost dedication and
respect thanks to the efforts of Engel and Heegaard. This can be seen in the editor’s
introduction to volume VI of Lie’s Collected Works,
If one should go through the whole history of mathematics, I believe that he will
not find a second case where, from a few general thoughts, which at first sight do
not appear promising, has been developed so extensive and wide-reaching a theory.
Considered as an edifice of thought Lie’s theory is a work of art which must stir
up admiration and astonishment in every mathematician who penetrates it deeply.
This work of art appears to me to be a production in every way comparable with
that [. . . ] of a Beethoven [. . . ] It is therefore entirely comprehensible if Lie [. . . ]
was embittered that ‘deren Wesen, ja Existenz, den Mathematikern fort-während
unbekannt zu sein scheint’ (p. 680). This deplorable situation, which Lie himself
felt so keenly, exists no longer, at least in Germany. In order to do whatever lies in
my power to improve the situation still further, [. . . ] I have sought to clarify all the
individual matters (Einzelheiten) and all the brief suggestions in these memoirs.
Each volume contains a substantial amount of notes, commentaries and supplemen-
tary material such as letters of Lie, and “This additional material has been prepared
with great care and with the convenience of the reader always in mind.” For example,
as mentioned before, Lie’s first paper was only 8 pages long, but the commentary
consisted of over 100 pages. According to [4],
Although Engel was himself an important and productive mathematician he has
found his place in the history of mathematics mainly because he was the closest stu-
dent and the indispensable assistant of a greater figure: Sophus Lie, after N. H. Abel
the greatest Norwegian mathematician. Lie was not capable of giving to the ideas
that flowed inexhaustibly from his geometrical intuition the overall coherence and
precise analytical form they needed in order to become accessible to the mathemat-
ical world [. . . ] Lie’s peculiar nature made it necessary for his works to be eluci-
dated by one who knew them intimately and thus Engel’s “Annotations” completed
in scope with the text itself.
Acknowledgments. I would like to thank Athanase Papadopoulos for carefully reading
preliminary versions of this article and his help with references on the Lie–Helmholtz
Theorem, and Hubert Goenner for several constructive and critical suggestions and
references. I would also like to thank Manfred Karbe for the reference [20] and
a summary of a possible explanation in [20] about the alienation between Lie and
Engel.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-46-320.jpg)
![Bibliography 25
Bibliography
[1] M. A. Akivis and B. A. Rosenfeld, Elie Cartan (1869–1951). Translated from the Russian
manuscript by V. V. Goldberg. Translations of Mathematical Monographs, 123. American
Mathematical Society, Providence, RI, 1993.
[2] N. Baas, Sophus Lie. Math. Intelligencer 16 (1994), no. 1, 16–19.
[3] G. Birkhoff and M. K. Bennett, Felix Klein and his ”Erlanger Programm”. History and phi-
losophy of modern mathematics (Minneapolis, MN, 1985), 145–176, Minnesota Stud. Philos.
Sci., XI, Univ. Minnesota Press, Minneapolis, MN, 1988.
[4] H. Boerner, Friedrich Engel, Complete Dictionary of Scientific Biography. Vol. 4. Detroit:
Charles Scribner’s Sons, 2008. p. 370–371.
[5] H. Busemann, Local metric geometry. Trans. Amer. Math. Soc. 56 (1944). 200–274.
[6] R. Carmichael, Book Review: Sophus Lie’s Gesammelte Abhandlungen. Bull. Amer. Math.
Soc. 29 (1923), no. 8, 367–369; 31 (1925), no. 9–10, 559–560; 34 (1928), no. 3, 369–370; 36
(1930), no. 5, 337.
[7] S. S. Chern, Sophus Lie and differential geometry. The Sophus Lie Memorial Conference
(Oslo, 1992), 129–137, Scand. Univ. Press, Oslo, 1994.
[8] G. Darboux, Sophus Lie. Bull. Amer. Math. Soc. 5 (1899), 367–370.
[9] B. Fritzsche, Sophus Lie: a sketch of his life and work. J. Lie Theory 9 (1999), no. 1, 1–38.
[10] J. Gray, Felix Klein’s Erlangen Program, ‘Comparative considerations of recent geometrical
researches’. In Landmark Writings in Western Mathematics 1640–1940, Elsevier, 2005, 544–
552.
[11] J. Gribbin, The Scientists: A History of Science Told Through the Lives of Its Greatest Inven-
tors. Random House Trade Paperbacks, 2004.
[12] G. Halsted, Sophus Lie. Amer. Math. Monthly 6 (1899), no. 4, 97–101.
[13] T. Hawkins, The birth of Lie’s theory of groups. The Sophus Lie Memorial Conference (Oslo,
1992), 23–50, Scand. Univ. Press, Oslo, 1994.
[14] T. Hawkins, Emergence of the theory of Lie groups. An essay in the history of mathemat-
ics 1869–1926. Sources and Studies in the History of Mathematics and Physical Sciences.
Springer-Verlag, New York, 2000.
[15] T. Hawkins, The Erlanger Programm of Felix Klein: reflections on its place in the history of
mathematics. Historia Math. 11 (1984), no. 4, 442–470.
[16] S. Helgason, Sophus Lie, the mathematician. The Sophus Lie Memorial Conference (Oslo,
1992), 3–21, Scand. Univ. Press, Oslo, 1994.
[17] L. Ji, A summary of topics related to group actions. Handbook of group actions, ed. L. Ji,
A. Papadopoulos and S.-T. Yau, Vol.1, Higher Education Press, Beijing and International
Press, Boston, 2015, 33–187.
[18] L. Ji, Felix Klein: his life and mathematics. In Sophus Lie and Felix Klein: The Erlangen
program and its impact in mathematics and physics, ed. L. Ji and A. Papadopoulos, European
Mathematical Society Publishing House, Zürich, 2015, 27–58.
[19] F. Klein, Nicht-Euklidische Geometrie, I. Vorlesung, Wintersemester 1889/90. Ausgearbeitet
von F. Schilling, Göttingen, 1893.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-47-320.jpg)
![26 Lizhen Ji
[20] G. Kowalewski, Bestand und Wandel. Meine Lebenserinnerungen zugleich ein Beitrag zur
neueren Geschichte der Mathematik. Verlag Oldenbourg, München 1950.
[21] S. Lie, Sophus Lie’s 1880 transformation group paper. In part a translation of “Theorie der
Transformations-gruppen”’by S. Lie [Math. Ann. 16 (1880), 441–528]. Translated by Michael
Ackerman. Comments by Robert Hermann. Lie Groups: History, Frontiers and Applications,
Vol. I. Math. Sci. Press, Brookline, Mass., 1975.
[22] R. Merton, On the Shoulders of Giants: A Shandean Postscript. University of Chicago Press,
1993.
[23] G. Miller, Some reminiscences in regard to Sophus Lie. Amer. Math. Monthly 6 (1899), no.
8–9, 191–193.
[24] C. Reid, Hilbert. Reprint of the 1970 original. Copernicus, New York, 1996.
[25] D. Rowe, Three letters from Sophus Lie to Felix Klein on Parisian mathematics during the
early 1880s. Translated from the German by the author. Math. Intelligencer 7 (1985), no. 3,
74–77.
[26] E. Strom, Sophus Lie. The Sophus Lie Memorial Conference (Oslo, 1992), Scand. Univ. Press,
Oslo, 1994.
[27] A. Stubhaug, The mathematician Sophus Lie. It was the audacity of my thinking. Translated
from the 2000 Norwegian original by Richard H. Daly. Springer-Verlag, Berlin, 2002.
[28] A. Vassilief, Prox Lobachevsky (premier concours), Nouv. Ann. Math., 3e sér. 17 (1898),
137–139.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-48-320.jpg)
![Chapter 2
Felix Klein: his life and mathematics
Lizhen Ji
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 A nontrivial birth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Three people who had most influence on Klein . . . . . . . . . . . . . . . . . . . . . . 32
5 Academic career . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 As a teacher and educator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7 Main contributions to mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8 The Evanston Colloquium Lectures and the resulting book . . . . . . . . . . . . . . . 42
9 A summary of the book “Lectures on mathematics” and Klein’s conflicts with Lie . . . 45
10 The ambitious encyclopedia in mathematics . . . . . . . . . . . . . . . . . . . . . . . 51
11 Klein’s death and his tomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
12 Major mathematicians and mathematics results in 1943–1993 from Klein’s perspective 53
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1 Introduction
Felix Klein was not only one of the great mathematicians but also one of the great ed-
ucators and scientific writers of the nineteenth and the early twentieth century. He was
a natural born leader and had a global vision for mathematics, mathematics education
and mathematical development. He had the required ambition, drive and ability to
remove obstacles on his way and to carry out his plans. He was a benign and no-
ble dictator, and was the most kingly mathematician in the history of mathematics.
He had a great deal of influence on German mathematics and the world mathematics
community. Indeed, it was Klein who turned Göttingen into the leading center of
mathematics in the world.
I have heard of Felix Klein for a long time but have not really tried to look up
information about him. On the other hand, when I became interested in him and
wanted to learn more about him, for example, the exact mathematical content of his
competition with Poincaré, his relation with Lie and their conflicts, and his contri-
bution towards the Erlangen program, there was no biography of Klein in English
which was easily available. Later we found some extensive writings about him in
obituary notices [18], and in books [13] [12]. After reading various sources about
Klein which were available to me in English, I found him an even more interesting](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-49-320.jpg)
![28 Lizhen Ji
person than I thought. One purpose in writing this chapter was to share with the
reader a summary of what I read about this incredible mathematician with some em-
phasis on his lectures in the USA around 1893 and their influence on the development
of the Americian mathematics community, and some thoughts which occurred to me
during this reading process. Another purpose is to give a brief outline of the rich life
of Klein and to supplement other more scientific papers in this book.1
Probably, Klein is famous to different groups of people for different reasons. For
example, for people working in Lie theories and geometry, the Erlangen program he
proposed had far-reaching consequences; for people working on discrete subgroups of
Lie groups and automorphic forms, Kleinian groups and his famous books with Fricke
on discrete groups of linear fractional transformations (or Möbius transformations)
and automorphic forms have had a huge impact; many other people enjoy his books
on elementary mathematics and history of mathematics, and of course high school and
college teachers also appreciate his books and points of view on education. A special
feature of Klein is his broad command of and vision towards mathematics. We will
discuss more on this last point from different aspects.
In terms of mathematical contribution, Klein was probably not of the same rank
as his predecessors Gauss or Riemann and his younger colleagues Hilbert and Weyl
in Göttingen, but he also made real contributions and was a kingly mathematician,
more kingly than any of the others mentioned above. He could command respect
from others like a king or even like a god. In some sense, Klein earned this. On the
influence of Klein on the mathematics in Göttingen, Weyl [7, p. 228] said that “Klein
ruled mathematics there like a god, but his godlike power came from the force of
his personality, his dedication, and willingness to work, and his ability to get things
done.”
But kings are kings and can be tough and remote from the ordinary people. In
1922, the eminent analyst Kurt O. Friedrichs was young and visited Klein [7, p. 229]:
“I was amazed, simply swept off my feet by Klein’s grace and charm. ... He could be
very charming and gentlemanly when all went his way but with anyone who crossed
him he was a tyrant.”
Klein made important original contributions to mathematics early in his career.
But his research was cut short due to exhaustion and the nervous breakdown caused
by his competition with Poincaré on Fuchsian groups and the uniformization of Rie-
mann surfaces.2
He fell down but picked it up. Very few people can do this. Since
he could not do mathematical research anymore, he devoted his time and energy to
1
Most mathematicians have heard of Klein and have been influenced by his mathematics. On the other hand,
various writings and stories about him are scattered in the literature, and we feel that putting several snapshots
from various angles could convey a vague global picture of the man and his mathematics. It might be helpful to
quote [18, p. i]: “After his death there appeared, one after another, a number of sketches of the man and his work
from the pens of many of his pupils. But, just as a photograph of a man of unusual personality, or of a place of
striking beauty, conveys little to one not personally acquainted with the original, so it is, and so it must be, with
these sketches of Klein.” For the reader now, these many sketches are probably not easily accessible.
2
It is interesting to see a different explanation of the breakdown of Klein in [18, p. vii]: “His breakdown
was probably accelerated by the antagonism he experienced at Leipzig. He was much younger than his col-
leagues, and they resented his innovating tendencies. In particular, there was opposition to his determination to
avail himself of the vaunted German “Lehrfreiheit”, and to interpret the word “Geometry” in its widest sense,
beginning his lectures with a course on the Geometric Theory of Functions.”](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-50-320.jpg)
![2 Felix Klein: his life and mathematics 29
education and mathematical writing, and more importantly to building and providing
a stimulating environment for others. For example, he brought Hilbert to Göttingen
and turned this small college city into the leading center of mathematics, which at-
tracted people from all over the world. His lecture tours in the USA near the end of
the nineteenth century also played a pivotal role in the emergence of the US mathe-
matics community as one of the leading ones in the world. In this sense, Klein was
also a noble mathematician and had a far-reaching and long-lasting impact.
His approach to mathematics emphasizes the big picture and connections between
different subjects without paying too much attention to details or work to substantiate
his vision. He valued results and methods which can be applied to a broad range of
topics and problems. For example, he proposed the famous Erlangen program but
did not really work on it to substantiate it. Instead, his friend Lie worked to make it
an important concrete program. As Courant once commented, Klein tended to soar
above the terrain that occupied ordinary mathematicians, taking in and enjoying the
vast view of mathematics, but it was often difficult for him to land and to do the hard
boring work. He had no patience for thorny problems that require difficult technical
arguments. What counted for him was the big pictures and the general pattern behind
seemingly unrelated results. According to Courant [13, p. 179], Klein had certainly
understood “that his most splendid scientific creations were fundamentally gigantic
sketches, the completion of which he had to leave to other hands.”
Klein was a master writer and speaker. In some sense, he was a very good sales-
man of mathematics, but he drew a lot of criticism for this. Because of this, he had
a lot of influence on mathematics and the mathematics community. He is a unique
combination of an excellent mathematician, master teacher and efficient organizer.
People’s responses to these features of Klein were not all positive. According to
a letter from Mittag-Leffler to Hermite in 1881 [7, p. 224]:
You asked me what are the relations between Klein and the great Berliners . . .
Weierstrass finds that Klein is not lacking in talent but is very superficial and even
sometimes rather a charlatan. Kronecker finds that he is quite simply a charlatan
without real merit. I believe that is also the opinion of Kummer.
In 1892, when the faculty of Berlin University discussed the successor to Weier-
strass, they rejected Klein as a “dazzling charlatan” and a “complier.” His longtime
friend Lie’s opinion was also harsh but more concrete [7, p. 224]:
I rank Klein’s talents highly and shall never forget the ready sympathy with which
he always accompanied me in my scientific attempts; but I opine that he does not,
for instance, sufficiently distinguish induction from proof, and the introduction of
a concept from its exploitation.
In spite of all these criticisms and reservations, Klein was the man who was largely
responsible for restoring Göttingen’s former luster and hence for initiating a process
that transformed the whole structure of mathematics at German universities and also
in some other parts of the world. Among people from Europe, he had the most im-
portant influence on the emergence of mathematics in the USA. There is no question](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-51-320.jpg)
![30 Lizhen Ji
of the fact as to he was the most dynamic mathematical figure in the world in the last
quarter of the nineteenth century.
Klein represented an ideal German scholar in the nineteenth century. He strove
for and attained an extraordinary breadth of knowledge, much of which he acquired
in his active interaction with other mathematicians. He also freely shared his insights
and knowledge with his students and younger colleagues, attracting people towards
him from around the world.
Unlike most mathematicians who only affected certain parts of mathematics and
the mathematics community, the influence of Klein is also global and foundational.
The legacy of Klein lives on as it is witnessed in the popularity of his books, the con-
tinuing and far-reaching influence of the general philosophy of the Erlangen program,
interaction between mathematics and physics, and the theory of Kleinian groups. In
some sense, Klein did not belong to his generation but was ahead of time.
2 A nontrivial birth
It is customary that kings are born at special times and places. Maybe their destiny
gives them something extra to start with. The mostly kingly mathematician, Felix
Klein, was born in the night of April 25, 1849, in Düsseldorf in the Rhineland when
[18, p. i]
there was anxiety in the house of the secretary to the Regierungspräsident. Without,
the canon thundered on the barricades raised by the insurgent Rhinelanders against
their hated Prussian rulers. Within, although all had been prepared for flight, there
was no thought of departure; [. . . ] His birth was marked by the final crushing of
the revolution of 1848; his life measured the domination of Prussia over Germany,
typifies all that was best and nobest in that domination.
Shortly after his birth, his hometown and the nearby region became the battle-
ground of the last war of the 1848 Revolution in the German states.
In the twenty years that followed Klein’s birth, Prussia became a major power in
Europe, and there were almost constantly conflicts and turmoils, culminating in the
Franco–Prussian war, with a crushing victory over France.
Later Klein served in the voluntary corps of emergency workers and he witnessed
firsthand the battle sites of Metz and Sedan, where the Empire of Louis-Napoléon
Bonaparte (Napoléon III) finally collapsed and was replaced by the Third Republic.
In Germany the Second German Empire began, with Otto von Bismarck as a powerful
first chancellor.3
3Otto von Bismarck was a conservative German statesman who dominated European affairs from the 1860s
to his dismissal in 1890. After a series of short victorious wars he unified numerous German states into a power-
ful German Empire under Prussian leadership, then created a “balance of power” that preserved peace in Europe
from 1871 until 1914.](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-52-320.jpg)
![32 Lizhen Ji
4 Three people who had most influence on Klein
There are three people who played a crucial role in the informative years of Klein.
The first person was Plücker, his teacher during his college years. Physics and the
interaction between mathematics and physics had always played an important role in
the mathematical life of Klein. It is reasonable to guess that this might have something
to do with the influence of Plücker. For most mathematicians, Plücker is well known
for Plücker coordinates in projective geometry. But he started as a physicist. In fact,
In 1836 at the age of 35, he was appointed professor of physics at the University of
Bonn and he started investigations of cathode rays that led eventually to the discovery
of the electron. Almost 30 years later, he switched to and concentrated on geometry.
Klein had written many books, some of which are still popular. Probably the most
original book by him is “Über Riemann’s Theorie der Algebraischen Functionen und
ihre Integrale”, published in 1882, where he tried to explain and justify Riemann’s
work on functions on Riemann surfaces, in particular, the Dirichlet principle, using
ideas from physics. Klein wrote [12, p. 178],
in modern mathematical literature, it is altogether unusual to present, as occurs in my
booklet, general physical and geometrical deliberations in naive anschaulicher form
which later find their firm support in exact mathematical proofs. [. . . ] I consider
it unjustifiable that most mathematicians suppress their intuitive thoughts and only
publish the necessary, strict (and mostly arithmetical) proofs [. . . ] I wrote my work
on Riemann precisely as a physicist, unconcerned with all the careful considerations
that are usual in a detailed mathematical treatment, and, precisely because of this, I
have also received the approval of various physicists.
In a biography of Klein [6], Halsted wrote: “The death of Plücker on May 22nd
1868 closed this formative period, of which the influence on Klein cannot be overes-
timated. So mighty is the power of contact with the living spirit of research, of taking
part in original work with a master, of sharing in creative authorship, that anyone who
has once come intricately in contact with a producer of first rank must have had his
whole mentality altered for the rest of his life. The gradual development, high attain-
ment, and then continuous achievement of Felix Klein are more due to Plücker than to
all other influences combined. His very mental attitude in the world of mathematics
constantly recalls his great maker.”
The second person was Alfred Clebsch, another important teacher of Klein. Cleb-
sch could be considered as a postdoctor mentor of Klein. After obtaining his Ph.D. in
1868, Klein went to Göttingen to work under Clebsch for eight months. When Klein
first met him, Clebsch was only 35 years old and was already a famous teacher and
leader of a new school in algebraic geometry.
Clebsch made important contributions to algebraic geometry and invariant theory.
Before Göttingen, he taught in Berlin and Karlsruhe. His collaboration with Paul Gor-
dan in Giessen led to the introduction of the Clebsch–Gordan coefficients for spher-
ical harmonics, which are now widely used in representation theory of compact Lie
groups and in quantum mechanics, and to find the explicit direct sum decomposition
of the tensor product of two irreducible representations into irreducible representa-](https://image.slidesharecdn.com/2613171-250605135730-4dca65eb/85/Sophus-Lie-And-Felix-Klein-The-Erlangen-Program-And-Its-Impact-In-Mathematics-And-Physics-1st-Edition-Lizhen-Ji-54-320.jpg)
Sophus Lie And Felix Klein The Erlangen Program And Its Impact In Mathematics And Physics 1st Edition Lizhen Ji
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- 5. IRMA Lectures in Mathematics and Theoretical Physics 23 Edited by Christian Kassel and Vladimir G. Turaev Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France
- 6. IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature 7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) 8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) Volumes 1–5 are available from De Gruyter (www.degruyter.de)
- 7. Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics Lizhen Ji Athanase Papadopoulos Editors
- 8. Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France Editors: Lizhen Ji Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109-1043 USA 2010 Mathematics Subject Classification: 01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40 Key words: Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group, rigidity, Galois theory, symmetries of partial differential equations, mathematical physics ISBN 978-3-03719-148-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2015 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1
- 9. Preface The Erlangen program provides a fundamental point of view on the place of trans- formation groups in mathematics and physics. Felix Klein wrote the program, but Sophus Lie also contributed to its formulation, and his writings are probably the best example of how this program is used in mathematics. The present book gives the first modern historical and comprehensive treatment of the scope, applications and impact of the Erlangen program in geometry and physics and the roles played by Lie and Klein in its formulation and development. The book is also intended as an introduc- tion to the works and visions of these two mathematicians. It addresses the question of what is geometry, how are its various facets connected with each other, and how are geometry and group theory involved in physics. Besides Lie and Klein, the names of Bernhard Riemann, Henri Poincaré, Hermann Weyl, Élie Cartan, Emmy Noether and other major mathematicians appear at several places in this volume. A conference was held at the University of Strasbourg in September 2012, as the 90th meeting of the periodic Encounter between Mathematicians and Theoretical Physicists, whose subject was the same as the title of this book. The book does not faithfully reflect the talks given at the conference, which were generally more specialized. Indeed, our plan was to have a book interesting for a wide audience and we asked the potential authors to provide surveys and not technical reports. We would like to thank Manfred Karbe for his encouragement and advice, and Hubert Goenner and Catherine Meusburger for valuable comments. We also thank Goenner, Meusburger and Arnfinn Laudal for sending photographs that we use in this book. This work was supported in part by the French program ANR Finsler, by the GEAR network of the National Science Foundation (GEometric structures And Rep- resentation varieties) and by a stay of the two editors at the Erwin Schrödinger Insti- tute for Mathematical Physics (Vienna). Lizhen Ji and Athanase Papadopoulos Ann Arbor and Strasbourg, March 2015
- 11. Contents Preface v Introduction xi 1 Sophus Lie, a giant in mathematics . . . . . . . . . . . . . . . . . . . . . 1 Lizhen Ji 2 Felix Klein: his life and mathematics . . . . . . . . . . . . . . . . . . . . 27 Lizhen Ji 3 Klein and the Erlangen Programme . . . . . . . . . . . . . . . . . . . . 59 Jeremy J. Gray 4 Klein’s “Erlanger Programm”: do traces of it exist in physical theories? 77 Hubert Goenner 5 On Klein’s So-called Non-Euclidean geometry . . . . . . . . . . . . . . . 91 Norbert A’Campo, Athanase Papadopoulos 6 What are symmetries of PDEs and what are PDEs themselves? . . . . . 137 Alexandre Vinogradov 7 Transformation groups in non-Riemannian geometry . . . . . . . . . . 191 Charles Frances 8 Transitional geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Norbert A’Campo, Athanase Papadopoulos 9 On the projective geometry of constant curvature spaces . . . . . . . . 237 Athanase Papadopoulos, Sumio Yamada 10 The Erlangen program and discrete differential geometry . . . . . . . . 247 Yuri B. Suris 11 Three-dimensional gravity – an application of Felix Klein’s ideas in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Catherine Meusburger 12 Invariances in physics and group theory . . . . . . . . . . . . . . . . . . 307 Jean-Bernard Zuber
- 12. viii Contents List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
- 13. Sophus Lie.
- 14. Felix Klein.
- 15. Introduction The Erlangen program is a perspective on geometry through invariants of the auto- morphism group of a space. The original reference to this program is a paper by Felix Klein which is usually presented as the exclusive historical document in this matter. Even though Klein’s viewpoint was generally accepted by the mathematical commu- nity, its re-interpretation in the light of modern geometries, and especially of modern theories of physics, is central today. There are no books on the modern developments of this program. Our book is one modest step towards this goal. The history of the Erlangen program is intricate. Klein wrote this program, but Sophus Lie made a very substantial contribution, in promoting and popularizing the ideas it contains. The work of Lie on group actions and his emphasis on their impor- tance were certainly more decisive than Klein’s contribution. This is why Lie’s name comes first in the title of the present volume. Another major figure in this story is Poincaré, and his role in highlighting the importance of group actions is also critical. Thus, groups and group actions are at the center of our discussion. But their importance in mathematics had already been crucial before the Erlangen program was formulated. From its early beginning in questions related to solutions of algebraic equations, group theory is merged with geometry and topology. In fact, group actions existed and were important before mathematicians gave them a name, even though the for- malization of the notion of a group and its systematic use in the language of geometry took place in the 19th century. If we consider group theory and transformation groups as an abstraction of the notion of symmetry, then we can say that the presence and importance of this notion in the sciences and in the arts was realized in ancient times. Today, the notion of group is omnipresent in mathematics and, in fact, if we want to name one single concept which runs through the broad field of mathematics, it is the notion of group. Among groups, Lie groups play a central role. Besides their mathematical beauty, Lie groups have many applications both inside and out- side mathematics. They are a combination of algebra, geometry and topology. Besides groups, our subject includes geometry. Unlike the word “group” which, in mathematics has a definite significance, the word “geometry” is not frozen. It has several meanings, and all of them (even the most recent ones) can be encompassed by the modern interpretation of Klein’s idea. In the first version of Klein’s Erlangen program, the main geometries that are em- phasized are projective geometry and the three constant curvature geometries (Eu- clidean, hyperbolic and spherical), which are considered there, like affine geometry, as part of projective geometry. This is due to the fact that the transformation groups of all these geometries can be viewed as restrictions to subgroups of the transfor- mation group of projective geometry. After these first examples of group actions in geometry, the stress shifted to Lie transformation groups, and it gradually included many new notions, like Riemannian manifolds, and more generally spaces equipped
- 16. xii Introduction with affine connections. There is a wealth of geometries which can be described by transformation groups in the spirit of the Erlangen program. Several of these ge- ometries were studied by Klein and Lie; among them we can mention Minkowski geometry, complex geometry, contact geometry and symplectic geometry. In modern geometry, besides the transformations of classical geometry which take the form of motions, isometries, etc., new notions of transformations and maps between spaces arose. Today, there is a wealth of new geometries that can be described by trans- formation groups in the spirit of the Erlangen program, including modern algebraic geometry where, according to Grothendieck’s approach, the notion of morphism is more important than the notion of space.1 As a concrete example of this fact, one can compare the Grothendieck–Riemann–Roch theorem with the Hirzebruch–Riemann– Roch. The former, which concerns morphisms, is much stronger than the latter, which concerns spaces. Besides Lie and Klein, several other mathematicians must be mentioned in this venture. Lie created Lie theory, but others’ contributions are also immense. About two decades before Klein wrote his Erlangen program, Riemann had introduced new geometries, namely, in his inaugural lecture, Über die Hypothesen, welche der Geo- metrie zu Grunde liegen (On the hypotheses which lie at the bases of geometry) (1854). These geometries, in which groups intervene at the level of infinitesimal transformations, are encompassed by the program. Poincaré, all across his work, highlighted the importance of groups. In his article on the Future of mathematics2 , he wrote: “Among the words that exerted the most beneficial influence, I will point out the words group and invariant. They made us foresee the very essence of mathemat- ical reasoning. They showed us that in numerous cases the ancient mathematicians considered groups without knowing it, and how, after thinking that they were far away from each other, they suddenly ended up close together without understanding why.” Poincaré stressed several times the importance of the ideas of Lie in the theory of group transformations. In his analysis of his own works,3 Poincaré declares: “Like Lie, I believe that the notion, more or less unconscious, of a continuous group is the unique logical basis of our geometry.” Killing, É. Cartan, Weyl, Chevalley and many others refined the structures of Lie theory and they developed its global aspects and applications to homogeneous spaces. The generalization of the Erlangen program to these new spaces uses the notions of connections and gauge groups, which were 1See A. Grothendieck, Proceedings of the International Congress of Mathematicians, 14–21 August 1958, Edinburgh, ed. J.A. Todd, Cambridge University Press, p. 103–118. In that talk, Grothendieck sketched his theory of cohomology of schemes. 2H. Poincaré, L’Avenir des mathématiques, Revue générale des sciences pures et appliquées 19 (1908) p. 930–939. [Parmi les mots qui ont exercé la plus heureuse influence, je signalerai ceux de groupe et d’invariant. Ils nous ont fait apercevoir l’essence de bien des raisonnements mathématiques ; ils nous ont montré dans com- bien de cas les anciens mathématiciens considéraient des groupes sans le savoir, et comment, se croyant bien éloignés les uns des autres, ils se trouvaient tout à coup rapprochés sans comprendre pourquoi.] 3Analyse de ses travaux scientifiques, par Henri Poincaré. Acta Mathematica, 38 (1921), p. 3–135. [Comme Lie, je crois que la notion plus ou moins inconsciente de groupe continu est la seule base logique de notre géométrie]; p. 127. There are many similar quotes in Poincaré’s works.
- 17. Introduction xiii closely linked to new developments in physics, in particular, in electromagnetism, phenomena related to light, and Einstein’s theory of general relativity. Today, instead of the word “geometry” we often use the expression “geometric structure”, and there is a wealth of geometric structures which can be described by transformation groups in the spirit of the Erlangen program. We mention in particular the notion of .G; X/ structure introduced by Charles Ehresmann in the 1930s, which is of paramount importance. Here X is a homogeneous space and G a Lie group acting transitively on G. A .G; X/ structure on a manifold M is then an atlas whose charts are in X and whose coordinate changes are restrictions of elements of G acting on X. Ehresmann formulated the notions of developing map and of holonomy trans- formations, which are basic objects in the study of these structures and their moduli spaces. .G; X/ structures have several variants and they have been developed and adapted to various settings by Haefliger, Kuiper, Benzécri, Thurston, Goldman and others to cover new structures, including foliations and singular spaces. The most spectacular advancement in this domain is certainly Thurston’s vision of the eight ge- ometries in dimension three, his formulation of the geometrization conjecture and the work around it, which culminated in the proof of the Poincaré conjecture by Perel- man. We talked about mathematics, but the Erlangen program also encompasses physics. In fact, geometry is closely related to physics, and symmetry is essential in modern physics. Klein himself investigated the role of groups in physics, when he stressed the concept of geometric invariants in his description of Einstein’s theories of special and general relativity, in particular by showing the importance of the Lorentz group, and also in his work on the conservation laws of energy and momentum in general relativity. Another milestone that led to conceptual clarifications and made it possible to systematically exploit the notion of symmetry in physics was E. Noether’s work that related symmetries of physical systems to conserved quantities. In conclusion, the central questions that are behind the present volume are: What is geometry? What is the relation between geometry and physics? How are groups used in physics, especially in contemporary physics? Let us now describe briefly the content of this volume. Chapters 1 and 2, written by Lizhen Ji, are introductions to the lives and works on Lie and Klein. Even though Klein was a major mathematician, surprisingly enough, there is no systematic English biography of him. The author’s aim is to fill this gap to a certain extent. Besides providing convenient short biographies of Lie and Klein, the author wishes to convince the reader of the importance of their works, especially those which are in close relation with the Erlangen program, and also to show how close the two men were in their ideas and characters. They both learned from each other and they had a profound influence on each other. This closeness, their ambitiousness, the competition among them and their disputes for priority of some discoveries were altogether the reasons that made them split after years of collaboration and friend- ship. The conflict between them is interesting and not so well known. The author describes this conflict, also mentions the difficulties that these two men encountered
- 18. xiv Introduction in their professional lives and in their relations with other mathematicians. Both of them experienced nervous breakdowns.4 The chapter on Lie also contains an out- line of his important theories as well as statements of some of his most significant theorems. In particular, the author puts forward in modern language and comments on three fundamental theorems of Lie. Concerning Klein, it is more difficult to pick out individual theorems, because Klein is known for having transmitted ideas rather than specific results. The author explains how Klein greatly influenced people and the world around him through his lectures and conversations, his books, the journals he edited, and he also recalls his crucial influence in shaping up the university of Göttingen to be the world’s most important mathematics center. In these surveys, the author also mentions several mathematicians who were closely related in some way or another to Lie and Klein, among them Hilbert, Hausdorff, Engel, Plücker, Sylow, Schwarz and Poincaré. The chapter on Lie also reviews other aspects of Lie’s work besides Lie groups. Chapter 3, by Jeremy Gray, is a historical commentary on the Erlangen program. The author starts by a short summary of the program manifesto and on the circum- stances of its writing, mentioning the influence of several mathematicians, and the importance of the ideas that originate from projective geometry (specially those of von Staudt). He then brings up the question of the impact of this program on the views of several mathematicians, comparing the opinions of Birkhoff and Bennet and of Hawkins. In Chapter 4, Hubert Goenner presents a critical discussion of the general impact and of the limitations of the Erlangen program in physics. He starts by recalling that the influence of the Erlangen program in physics was greatly motivated by the geometrization of special relativity by H. Minkowski, in which the Lorentz group appears as one of the main objects of interest, but he stresses the fact that the no- tion of field defined on a geometry – and not the notion of geometry itself – is then the central element. He comments on the relation of Lie transformations with theo- ries of conservation laws and the relations of the Erlangen program with symplectic geometry, analytical mechanics, statistical physics, quantum field theories, general relativity, Yang–Mills theory and supergravity. The paper has a special section where the author discusses supersymmetry. In a final section, the author mentions several generalizations of the notion of Lie algebra. In Chapter 5, Norbert A’Campo and Athanase Papadopoulos comment on the two famous papers of Klein, Über die sogenannte Nicht-Euklidische Geometrie (On the so-called non-Euclidean geometries), I and II. The two papers were written respec- tively one year and a few months before the Erlangen program, and they contain in essence the main ideas of this program. We recall that the 19th century saw the birth of non-Euclidean geometry by Lobachevsky, Bolyai and Gauss, and at the same time, the development of projective geometry by Poncelet, Plücker, von Staudt and others, and also of conformal geometry by Liouville and others. Groups made the first link between all these geometries, and also between geometry and algebra. Klein, in the 4Klein’s nervous breakdown was probably due to overwork and exhaustion, caused in part by his rude compe- tition with Poincaré on Fuchsian functions, whereas Lie’s nervous breakdown was the consequence of a chronic illness, pernicious anemia, related to a lack in vitamin B12, which at that time was incurable.
- 19. Introduction xv papers cited above, gives models of the three constant-curvature geometries (hyper- bolic, Euclidean and spherical) in the setting of projective geometry. He defines the distance functions in each of these geometries by fixing a conic (the “conic at infin- ity”) and taking a constant multiple of the logarithm of the cross ratio of four points: the given two points and the two intersection points of the line joining them with the conic at infinity. The hyperbolic and spherical geometries are obtained by using real and complex conics respectively, and Euclidean geometry by using a degenerate one. The authors in Chapter 5 comment on these two important papers of Klein and they display relations with works of other mathematicians, including Cayley, Beltrami, Poincaré and the founders of projective geometry. Klein’s interaction with Lie in their formative years partly motivated Lie to de- velop Lie’s version of Galois theory of differential equations and hence of Lie trans- formation theory.5 In fact, a major motivation for Lie for the introduction of Lie groups was to understand differential equations. This subject is treated in Chapter 6 of this volume. The author, Alexandre Vinogradov, starts by observing that Lie initi- ated his work by transporting the Galois theory of the solvability of algebraic equa- tions to the setting of differential equations. He explains that the major contribution of Lie in this setting is the idea that symmetries of differential equations are the basic elements in the search for their solutions. One may recall here that Galois approached the problem of solvability of polynomial equations through a study of the symmetries of their roots. This is based on the simple observation that the coefficients of a poly- nomial may be expressed in terms of the symmetric functions of their roots, and that a permutation of the roots does not change the coefficients of the polynomial. In the case of differential equations, one can naively define the symmetry group to be the group of diffeomorphisms which preserve the space of solutions, but it is not clear how such a notion can be used. There is a differential Galois theory which is parallel to the Galois theory of polynomial equations. In the differential theory, the question “what are the symmetries of a (linear or nonlinear, partial or ordinary) differential equation?” is considered as the central question. Chapter 6 also contains reviews of the notions of jets and jet spaces and other constructions to explain the right setup for formulating the question of symmetry, with the goal of providing a uniform frame- work for the study of nonlinear partial differential equations. The author is critical of the widely held view that each nonlinear partial differential equation arising from geometry or physics is special and often requires its own development. He believes that the general approach based on symmetry is the right one. The author mentions developments of these ideas that were originally formulated by Lie and Klein in works of E. Noether, Bäcklund, É. Cartan, Ehresmann and others. A lot of questions in this domain remain open, and this chapter will certainly give the reader a new perspective on the geometric theory of nonlinear partial differential equations. In Chapter 7, Charles Frances surveys the modern developments of geometric structures on manifolds in the lineage of Klein and Lie. The guiding idea in this 5 Lie has had a course at Oslo by Sylow on Galois and Abel theory before he meets Klein, but it is clear that Klein also brought some of his knowledge to Lie.
- 20. xvi Introduction chapter is the following question: When is the automorphism group of a geometric structure a Lie group, and what can we say about the structure of such a Lie group? The author considers the concept of Klein geometry, that is, a homogeneous space acted upon by a Lie group, and a generalization of this notion, leading to the con- cept of a Cartan geometry. (Cartan used the expression espace généralisé.) Besides the classical geometries, like constant curvature spaces (Euclidean, Lobachevsky and spherical) as well as projective geometry which unifies them, the notion of Cartan ge- ometry includes several differential-geometric structures. These notions are defined using fiber bundles and connections. They describe spaces of variable curvature and they also lead to pseudo-Riemannian manifolds, conformal structures of type .p; q/, affine connections, CR structures, and the so-called parabolic geometries. The author presents a series of important results on this subject, starting with the theorem of My- ers and Steenrod (1939) saying that the isometry group of any Riemannian manifold is a Lie group, giving a bound on its dimension, and furthermore, it says that this group is compact if the manifold is compact. This result gave rise to an abundance of developments and generalizations. The author also explains in what sense pseudo- Riemannian manifolds, affine connections and conformal structures in dimensions 3 are rigid, symplectic manifolds are not rigid, and complex manifolds are of an intermediate type. Thus, two general important questions are addressed in this survey: What are the possible continuous groups that are the automorphism groups of a geometry on a compact manifold? What is the influence of the automorphism group of a structure on the topology or the diffeomorphism type of the underlying manifold? Several examples and recent results are given concerning Cartan geometries and in particular pseudo-Riemannian conformal structures. Chapter 8, by Norbert A’Campo and Athanase Papadopoulos, concern transitional geometry. This is a family of geometries which makes a continuous transition be- tween hyperbolic and spherical geometry, passing through Euclidean geometry. The space of transitional geometry is a fiber space over the interval Œ1; 1 where the fiber above each point t is a space of constant curvature t2 if t 0 and of constant curva- ture t2 if t 0. The fibers are examples of Klein geometries in the sense defined in Chapter 7. The elements of each geometry are defined group-theoretically, in the spirit of Klein’s Erlangen program. Points, lines, triangles, trigonometric formulae and other geometric properties transit continuously between the various geometries. In Chapter 9, by Athanase Papadopoulos and Sumio Yamada, the authors intro- duce a notion of cross ratio which is proper to each of the three geometries: Euclidean, spherical and hyperbolic. This highlights the relation between projective geometry and these geometries. This is in the spirit of Klein’s view of the three constant cur- vature geometries as part of projective geometry, which is the subject of Chapter 5 of the present volume. Chapter 10, by Yuri Suris, concerns the Erlangen program in the setting of dis- crete differential geometry. This is a subject which recently emerged, whose aim is to develop a theory which is the discrete analogue of classical differential geometry.
- 21. Introduction xvii It includes discrete versions of the differential geometry of curves and surfaces but also higher-dimensional analogues. There are discrete notions of line, curve, plane, volume, curvature, contact elements, etc. There is a unifying transformation group approach in discrete differential geometry, where the discrete analogues of the clas- sical objects of geometry become invariants of the respective transformation groups. Several classical geometries survive in the discrete setting, and the author shows that there is a discrete analogue of the fact shown by Klein that the transformation groups of several geometries are subgroups of the projective transformation group, namely, the subgroup preserving a quadric. Examples of discrete differential geometric geometries reviewed in this chapter include discrete line geometry and discrete line congruence, quadrics, Plücker line geometry, Lie sphere geometry, Laguerre geometry and Möbius geometry. Important notions such as curvature line parametrized surfaces, principal contact element nets, discrete Ribeaucour transformations, circular nets and conical nets are discussed. The general underlying idea is that the notion of transformation group survives in the dis- cretization process. Like in the continuous case, the transformation group approach is at the same time a unifying approach, and it is also related to the question of “multi- dimensional consistency” of the geometry, which says roughly that a 4D consistency implies consistency in all higher dimensions. The two principles – the transforma- tion group principle and consistency principle – are the two guiding principles in this chapter. Chapter 11 by Catherine Meusburger is an illustration of the application of Klein’s ideas in physics, and the main example studied is that of three-dimensional gravity, that is, Einstein’s general relativity theory6 with one time and two space variables. In three-dimensions, Einstein’s general relativity can be described in terms of cer- tain domains of dependence in thee-dimensional Minkowski, de Sitter and anti de Sitter space, which are homogeneous spaces. After a summary of the geometry of spacetimes and a description of the gauge invariant phase spaces of these theories, the author discusses the question of quantization of gravity and its relation to Klein’s ideas of characterizing geometry by groups. Besides presenting the geometrical and group-theoretical aspects of three-dimen- sional gravity, the author mentions other facets of symmetry in physics, some of them related to moduli spaces of flat connections and to quantum groups. Chapter 12, by Jean-Bernard Zuber, is also on groups that appear in physics, as group invariants associated to a geometry. Several physical fields are mentioned, including crystallography, piezzoelectricity, general relativity, Yang–Mills theory, quantum field theories, particle physics, the physics of strong interactions, electro- magnetism, sigma-models, integrable systems, superalgebras and infinite-dimensional algebras. We see again the work of Emmy Noether on group invariance principles in variational problems. Representation theory entered into physics through quan- tum mechanics, and the modern theory of quantum group is a by-product. The au- thor comments on Noether’s celebrated paper which she presented at the occasion of Klein’s academic Jubilee. It contains two of her theorems on conservation laws. 6 We recall by the way that Galileo’s relativity theory is at the origin of many of the twentieth century theories.
- 22. xviii Introduction Today, groups are omnipresent in physics, and as Zuber puts it: “To look for a group invariance whenever a new pattern is observed has become a second nature for particle physicists”. We hope that the various chapters of this volume will give to the reader a clear idea of how group theory, geometry and physics are related to each other, the Erlangen program being a major unifying element in this relation. Lizhen Ji and Athanase Papadopoulos
- 23. Chapter 1 Sophus Lie, a giant in mathematics Lizhen Ji Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Some general comments on Lie and his impact . . . . . . . . . . . . . . . . . . . . . . 2 3 A glimpse of Lie’s early academic life . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 A mature Lie and his collaboration with Engel . . . . . . . . . . . . . . . . . . . . . . 6 5 Lie’s breakdown and a final major result . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 An overview of Lie’s major works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 Three fundamental theorems of Lie in the Lie theory . . . . . . . . . . . . . . . . . . 13 8 Relation with Klein I: the fruitful cooperation . . . . . . . . . . . . . . . . . . . . . . 15 9 Relation with Klein II: conflicts and the famous preface . . . . . . . . . . . . . . . . . 16 10 Relations with others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 11 Collected works of Lie: editing, commentaries and publication . . . . . . . . . . . . . 23 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1 Introduction There are very few mathematicians and physicists who have not heard of Lie groups or Lie algebras and made use of them in some way or another. If we treat discrete or finite groups as special (or degenerate, zero-dimensional) Lie groups, then almost every subject in mathematics uses Lie groups. As H. Poincaré told Lie [25] in October 1882, “all of mathematics is a matter of groups.” It is clear that the importance of groups comes from their actions. For a list of topics of group actions, see [17]. Lie theory was the creation of Sophus Lie, and Lie is most famous for it. But Lie’s work is broader than this. What else did Lie achieve besides his work in Lie theory? This might not be so well known. The differential geometer S. S. Chern wrote in 1992 that “Lie was a great mathematician even without Lie groups” [7]. What did and can Chern mean? We will attempt to give a summary of some major contributions of Lie in 6. One purpose of this chapter is to give a glimpse of Lie’s mathematical life by recording several things which I have read about Lie and his work. Therefore, it is short and emphasizes only a few things about his mathematics and life. For a fairly de- tailed account of his life (but not his mathematics), see the full length biography [27]. We also provide some details about the unfortunate conflict between Lie and Klein and the famous quote from Lie’s preface to the third volume of his books on trans-
- 24. 2 Lizhen Ji formation groups, which is usually only quoted without explaining the context. The fruitful collaboration between Engel and Lie and the publication of Lie’s collected works are also mentioned. We hope that this chapter will be interesting and instructive to the reader of this book and might serve as a brief introduction to the work and life of Lie discussed in this book. 2 Some general comments on Lie and his impact It is known that Lie’s main work is concerned with understanding how continuous transformation groups provide an organizing principle for different areas of math- ematics, including geometry, mechanics, and partial differential equations. But it might not be well known that Lie’s collected works consist of 7 large volumes of the total number of pages about 5600. (We should keep in mind that a substantial por- tion of these pages are commentaries on his papers written by the editors. In spite of this, Lie’s output was still enormous.) Probably it is also helpful to keep in mind that Lie started to do mathematics at the age of 26 and passed away at 57. Besides many papers, he wrote multiple books, which total over several thousands of pages. According to Lie, only a part of his ideas had been put down into written form. In an autobiographic note [9, p. 1], Lie wrote: My life is actually quite incomprehensible to me. As a young man, I had no idea that I was blessed with originality, Then, as a 26-year-old, I suddenly realized that I could create. I read a little and began to produce. In the years 1869–1874, I had a lot of ideas which, in the course of time, I have developed only very imperfectly. In particular, it was group theory and its great importance for the differential equations which interested me. But publication in this area went woefully slow. I could not structure it properly, and I was always afraid of making mistakes. Not the small inessential mistakes . . . No, it was the deep-rooted errors I feared. I am glad that my group theory in its present state does not contain any fundamental errors. Lie was a highly original and technically powerful mathematician. The recog- nition of the idea of Lie groups (or transformation groups) took time. In 1870s, he wrote in a letter [26, p. XVIII]: If I only knew how to get the mathematicians interested in transformation groups and their applications to differential equations. I am certain, absolutely certain in my case, that these theories in the future will be recognized as fundamental. I want to form such an impression now, since for one thing, I could then achieve ten times as much. In 1890, Lie was confident and wrote that he strongly believed that his work would stand through all times, and in the years to come, it would be more and more appre- ciated by the mathematical world.
- 25. 1 Sophus Lie, a giant in mathematics 3 Eduard Study was a privatdozent (lecturer) in Leipzig when Lie held the chair in geometry there. In 1924, the mature Eduard Study summarized Lie as follows [26, p. 24]: Sophus Lie had the shortcomings of an autodidact, but he was also one of the most brilliant mathematicians who ever lived. He possessed something which is not found very often and which is now becoming even rarer, and he possessed it in abundance: creative imagination. Coming generations will learn to appreciate this visionary’s mind better than the present generation, who can only appreciate the mathemati- cians’ sharp intellect. The all-encompassing scope of this man’s vision, which, above all, demands recognition, is nearly completely lost. But, the coming gen- eration [. . . ] will understand the importance of the theory of transformation groups and ensure the scientific status that this magnificent work deserves. What Lie studied are infinitesimal Lie groups, or essentially Lie algebras. Given what H. Weyl and É. Cartan contributed to the global theory of Lie groups starting around the middle of 1920s and hence made Lie groups one of the most basic and essential objects in modern (or contemporary) mathematics, one must marvel at the above visionary evaluation of Lie’s work by Study. For a fairly detailed overview of the historical development of Lie groups with particular emphasis on the works of Lie, Killing, É. Cartan and Weyl, see the book [14]. Two months after Lie died, a biography of him appeared in the American Mathe- matical Monthly [12]. It was written by George Bruce Halsted, an active mathematics educator and a mathematician at the University of Texas at Austin, who taught famous mathematicians like R. L. Moore and L. E. Dickson. Reading it more than one hun- dred years later, his strong statement might sound a bit surprising but is more justified than before, “[. . . ] the greatest mathematician in the world, Sophus Lie, died [. . . ] His work is cut short; his influence, his fame, will broaden, will tower from day to day.” Probably a more accurate evaluation of Lie was given by Engel in a memorial speech on Lie [9, p. 24] in 1899: If the capacity for discovery is the true measure of a mathematician’s greatness, then Sophus Lie must be ranked among the foremost mathematicians of all time. Only extremely few have opened up so many vast areas for mathematical research and created such rich and wide-ranging methods as he [. . . ] In addition to a capacity for discovery, we expect a mathematician to posses a penetrating mind, and Lie was really an exceptionally gifted mathematician [. . . ] His efforts were based on tackling problems which are important, but solvable, and it often happened that he was able to solve problems which had withstood the efforts of other eminent mathematicians. In this sense, Lie was a giant for his deep and original contribution to mathematics, and is famous not for other reasons. (One can easily think of several mathematicians, without naming them, who are famous for various things besides mathematics). Inci- dentally, he was also a giant in the physical sense. There are some vivid descriptions of Lie by people such as É. Cartan [1, p. 7], Engel [27, p. 312], and his physics
- 26. 4 Lizhen Ji colleague Ostwald at Leipzig [27, p. 396]. See also [27, p. 3]. For some interest- ing discussions on the relations between giants and scientists, see [11, pp. 163–164, p. 184] and [22, pp. 9–13]. 3 A glimpse of Lie’s early academic life Lie was born on December 17, 1842. His father, Johann Herman Lie, was a Lutheran minister. He was the youngest of the six children of the family. Lie first attended school in the town of Moss in South Eastern Norway and on the eastern side of the Oslo Fjord. In 1857 he entered Nissen’s Private Latin School in Christiania, which became Oslo in 1925. At that time, he decided to pursue a military career, but his poor eyesight made this impossible, and he entered University of Christiania to pursue a more academic life. During his university time, Lie studied science in a broad sense. He took math- ematics courses and attended lectures by teachers of high quality. For example, he attended lectures by Sylow in 1862.1 Though Lie studied with some good mathematicians and did well in most courses, on his graduation in 1865, he did not show any special ability for mathematics or any particular liking for it. Lie could not decide what subject to pursue and he gave some private lessons and also volunteered some lectures for a student union while trying to make his decision. He knew he wanted an academic career and thought for a while that astronomy might be the right topic. He also learnt some mechanics, and wondered about botany, zoology or physics. Lie reached the not-so-tender age of 26 in 1868 and was still not sure what he should pursue as a career. But this year was a big turning point for him. In June 1868, the Tenth Meeting of Scandinavian Natural Sciences was held in Christiania. It attracted 368 participants. Lie attended many lectures and was par- ticularly influenced by the lecture of a former student of the great French geometer Michel Chasles, which referred to works of Chasles, Möbius, and Plücker. It seems that the approaching season, the autumn of 1868, became one long con- tinuing period of work for Sophus Lie, with his frequent borrowing of books from the library. In addition to Chasles, Möbius and Plücker, Lie discovered the Frenchman Poncelet, the Englishman Hamilton, and the Italian Cremona, as well as others who had made important contributions to algebraic and analytic geometry. Lie plowed through many volumes of the leading mathematical journals from Paris and Berlin, and in the Science Students Association he gave several lectures during the spring of 1869 on what he called his “Theory of the imaginaries”, and on how information on real geometric objects could be transferred to his “imaginary objects.” 1Ludwig Sylow (1832–1918) was Norwegian, like Lie. He is now famous and remembered for the Sylow subgroups. At that time, he was not on the permanent staff of the university of Christiania, but he was substitut- ing for a regular faculty member and taught a course. In this course, he explained Abel’s and Galois’ work on algebraic equations. But it seems that Lie did not understand or remember the content of this course, and it was Klein who re-explained these theories to him and made a huge impact on Lie’s mathematical life.
- 27. 1 Sophus Lie, a giant in mathematics 5 Sophus Lie wrote a paper on his discovery. The paper was four pages long and it was published at his own expenses.2 After this paper was translated into German, it was published in the leading mathematics journal of the time, Crelle’s Journal.3 With this paper, he applied to the Collegium for a travel grant and received it. Then he left for Berlin in September 1869 and begun his glorious and productive mathematical career. There were several significant events in Berlin for Lie on this trip. He met Felix Klein and they immediately became good friends. They shared common interests and common geometric approaches, and their influence on each other was immense. Without this destined (or chance) encounter, Lie and Klein might not have been the people we know. Lie also impressed Kummer by solving problems which Kummer was working on. This gave him confidence in his own power and originality. According to Lie’s letter to his boyfriend [26, p. XII]: Today I had a triumph which I am sure you will be interested to hear about. Professor Kummer suggested that we test our powers on a discussion of all line congruences of the 3rd degree. Fortunately, a couple of months ago, I had already solved a problem which was in a way special of the above, but was nevertheless much more general. . . . I regard this as a confirmation of my good scientific insight that I, from the very first, understood the value of findings. That I have shown both energy and capability in connection with my findings; that I know. In the summer of 1970, Lie and Klein visited Paris and met several important people such as Jordan and Darboux. The interaction with Jordan and Jordan’s new book on groups had a huge influence on both of them. This book by Jordan contained more than an exposition of Galois theory and can be considered as a comprehensive discussion of how groups were used in all subjects up to that point. For Klein and Lie, it was an eye opener. Besides learning Galois theory, they started to realize the basic and unifying role groups would play in geometry and other parts of mathematics. In some sense, the trips with Klein sealed the future research direction of Lie. Klein played a crucial role in the formative years of Lie, and the converse is also true. We will discuss their interaction in more detail in 8 and 9. At the outbreak of the Franco–Prussian war in July, Klein left, and Lie stayed for one more month and then decided to hike to Italy. But he was arrested near Fontainebleau as he was suspected of being a spy and spent one month in jail. Dar- boux came and freed him. In [8], Darboux wrote: True, in 1870 a misadventure befell him, whose consequences I was instrumen- tal in averting. Surprised at Paris by the declaration of war, he took refuge at 2The publication of this paper is unusual also by today’s standards. According to [2], “His first published paper appeared in 1869. It gives a new representation of the complex plane and uses ideas of Plücker. But Lie had difficulties in getting these ideas published by the Academy in Christiania. He was impatient. Professor Bjerknes asked for more time to look at the paper, but Professor Broch returned it after two days – saying he had understood nothing! However, three other professors – who probably understood the material even less – supported publication. This happened as a result of influence by friends of Lie.” 3The German version of this paper is still only 8 pages long, but in his collected works edited by Engel and Heegaard, there are over 100 pages of commentaries devoted to it.
- 28. 6 Lizhen Ji Fontainebleau. Occupied incessantly by the ideas fermenting in his brain, he would go every day into the forest, loitering in places most remote from the beaten path, taking notes and drawing figures. It took little at this time to awaken suspicion. Arrested and imprisoned at Fontainebleau, under conditions otherwise very com- fortable, he called for the aid of Chasles, Bertrand, and others; I made the trip to Fontainebleau and had no trouble in convincing the procureur impérial; all the notes which had been seized and in which figured complexes, orthogonal systems, and names of geometers, bore in no way upon the national defenses. Afterwards, Lie wrote to his close friend [26, p. XV], “except at the very first, when I thought it was a matter of a couple of days, I have taken things truly philo- sophically. I think that a mathematician is well suited to be in prison.” In fact, while he was in prison, he worked on his thesis and a few months later, he submitted his thesis, on March 1871. He received his doctorate degree in July 1872, and accepted a new chair at the university of Christiania set up for him by the Norwegian National Assembly. It was a good thesis, which dealt with the integration theory of partial differential equations. After his thesis, Lie’s mathematics talent was widely recognized and his mathematical career was secured. When Lie worked on his thesis with a scholarship from the University of Christia- nia, he needed to teach at his former grammar school to supplement his income. With this new chair, he could devote himself entirely to mathematics. Besides developing his work on transformation groups and working with Klein towards the formulation of the Erlangen program, Lie was also involved in editing with his former teacher Ludwig Sylow the collected works of Abel. Since Lie was not familiar with alge- bras, especially with Abel’s works, this project was mainly carried out by Sylow. But locating and gathering manuscripts of Abel took a lot of effort to both men, and the project took multiple years. In his personal life, Lie married Anna Birch in 1874, and they had two sons and a daughter. Lie published several papers on transformation groups and on the applications to integration of differential equations and he established a new journal in Christiania to publish his papers, but these papers did not receive much attention. Because of this, Lie started to work on more geometric problem such as minimal surfaces and surfaces of constant curvature. Later in 1882 some work by French mathematicians on integration of differential equations via transformation groups motivated Lie to go back to his work on integra- tion of differential equations and theories of differential invariants of groups. 4 A mature Lie and his collaboration with Engel There were two people who made, at least contributed substantially to make, Lie the mathematician we know today. They were Klein and Engel. Of course, his story with Klein is much better known and dramatic and talked about, but his interaction with Engel is not less important or ordinary.
- 29. 1 Sophus Lie, a giant in mathematics 7 In the period from 1868 to 1884, Lie worked constantly and lonely to develop his theory of transformation groups, integration problems, and theories of differential invariants of finite and infinite groups. But he could not describe his new ideas in an understandable and convincing way, and his work was not valued by the mathematics community. Further, he was alone in Norway and no one could discuss with him or understand his work. In a letter to Klein in September 1883 [9, p. 9], Lie wrote that “It is lonely, fright- fully lonely, here in Christiania where nobody understands my work and interests.” Realizing the seriousness of the situation of Lie and the importance of summa- rizing in a coherent way results of Lie and keeping him productive, Klein and his colleague Mayer at Leipzig decided to send their student Friedrich Engel to assist Lie. Klein and Mayer knew that without help from someone like Engel, Lie could not produce a coherent presentation of his new novel theories. Like Lie, Engel was also a son of a Lutheran minister. He was born in 1861, nine- teen years younger than Lie. He started his university studies in 1879 and attended both the University of Leipzig and the University of Berlin. In 1883 he obtained his doctorate degree from Leipzig under Mayer with a thesis on contact transformations. After a year of military service in Dresden, Engel returned to Leipzig in the spring of 1884 to attend Klein’s seminar in order to write a Habilitation. At that time, be- sides Klein, Mayer was probably the only person who understood Lie’s work and his talent. Since contact transformations form one important class in Lie’s theories of transformation groups, Engel was a natural candidate for this mission. Klein and Mayer worked together to obtain a stipend from the University of Leipzig and the Royal Society of Sciences of Saxony for Engel so that he could travel to work with Lie in Christiania. In June 1884, Lie wrote a letter to Engel [9, p. 10], From 1871–1876, I lived and breathed only transformation groups and integration problems. But when nobody took any interest in these things, I grew a bit weary and turned to geometry for a time. Now just in the last few years, I have again taken up these old pursuits of mine. If you will support me with the further development and editing of these things, you will be doing me a great service, especially in that, for once, a mathematician finally has a serious interest in these theories. Here in Christiania, a specialist like myself is terribly lonely. No interest, no understanding. According to a letter of Engel in the autumn of 1884 after meeting Lie [27, p. 312]: The goal of my journey was twofold: on one hand, under Lie’s own guidance, I should become immersed in his theories, and on the other, I should exercise a sort of pressure on him, to get him to carry on his work for a coherent presentation of one of his greater theories, with which I should help him apply his hand. Lie wanted to write a major comprehensive monograph on transformation groups, not merely a simple introduction to his new theory. It “should be a systematic and strict-as-possible account that would retain its worth for a long time” [9, p. 11]. Lie and Engel met twice every day, in the morning at the apartment of Engel and in the afternoon at Lie’s apartment. They started with a list of chapters. Then Lie
- 30. 8 Lizhen Ji dictated an outline of each chapter and Engel would supply the detail. According to Engel [9, p. 11], Every day, I was newly astonished by the magnificence of the structure which Lie had built entirely on his own, and about which his publications, up to then, gave only a vague idea. The preliminary editorial work was completed by Christmas, after which Lie devoted some weeks to working through all of the material in order to lay down the final draft. Starting at the end of January 1885, the editorial work began anew; the finished chapters were reworked and new ones were added. When I left Christiania in June of 1885, there was a pile of manuscripts which Lie figured would eventually fill approximately thirty printer’s sheets. That it would be eight years before the work was completed and the thirty sheets would become one hundred and twenty-five was something neither of us could have imagined at that time. Lie and Engel worked intensively over the nine month period when Engel was there. This collaboration was beneficial to both parties. To Engel, it was probably the best introduction to Lie theories and it served his later mathematical research well. According to Kowalewski, a student of Lie and Engel, [9, p. 10], “Lie would never have been able to produce such an account by himself. He would have drowned in the sea of ideas which filled his mind at that time. Engel succeeded in bringing a systematic order to this chaotic mass of thought.” After returning to Leipzig, Engel finished his Habilitation titled “On the defining equations of the continuous transformation groups” and became a Privatdozent. In 1886, Klein moved to Göttingen for various reasons. (See [18] for a brief description of Klein’s career). Thanks to the efforts of Klein, Lie moved to Leipzig in 1886 to take up a chair in geometry. More description of this will be given in 8 below. When Lie visited Leipzig in February 1886 to prepare for his move, he wrote to his wife excitedly [27, p. 320], “to the best of my knowledge, there have been no other foreigners, other than Abel and I, appointed professor at a German University. (The Swiss are out of the calculation here.) It is rather amazing. In Christiania I have often felt myself to be treated unfairly, so I have truly achieved an unmerited honor.” Leipzig was the hometown of the famous Leibniz and a major culture and aca- demic center. In comparison to his native country, it was an academic heaven for Lie. In April 1886, Lie became the Professor of Geometry and Director of the Math- ematical Seminar and Institute at the University of Leipzig. Lie and Engel resumed to work intensively on their joint book again. In 1888, the leading German scientific publisher Teubner, based in Leipzig, published the first volume of Theory of transfor- mation groups, which was 632 pages long. In that year, Engel also became the assistant to Lie after Friedrich Schur left. When Lie went to a nerve clinic near the end of 1889, Engel gave Lie’s lectures for him. The second volume of their joint book was published in 1890 and was 555 pages long, and the third volume contained 831 pages and was published in 1893.
- 31. 1 Sophus Lie, a giant in mathematics 9 The three big volumes of joint books with Engel would not see the light of day or even start without the substantial contribution of Engel. It was a major piece of work. In a 21-page review of the first volume [9, p. 16], Eduard Study wrote, The work in question gives a comprehensive description of an extensive theory which Mr. Lie has developed over a number of years in a large number of indi- vidual articles in journals . . . Because most of these articles are not well known, and because of their concise format, the content, in spite of its enormous value, has remained virtually unknown to the scientific community. But by the same token we can also be thankful that the author has had the rare opportunity of being able to let his thoughts mature in peace, to form them in harmony and think them through independently, away from the breathless competitiveness of our time. We do not have a textbook written by a host of authors who have worked together to introduce their theories to a wider audience, but rather the creation of one man, an original work which, from beginning to end, deals with completely new things [. . . ] We do not believe that we are saying too much when we claim that there are few areas of mathematical science which will not be enriched by the fundamental ideas of this new discipline. It is probably interesting to note that Engel’s role and effort in this massive work were not mentioned here. Maybe the help of a junior author or assistant was taken for granted in the German culture at that time. In the preface to the third volume, Lie wrote [9, p. 15]: For me, Professor Engel occupies a special position. On the initiative of F. Klein and A. Mayer, he traveled to Christiania in 1884 to assist me in the preparation of a coherent description of my theories. He tackled this assignment, the size of which was not known at that time, with the perseverance and skill which typifies a man of his caliber. He has also, during this time, developed a series of important ideas of his own, but has in a most unselfish manner declined to describe them here in any great detail or continuity, satisfying himself with submitting short pieces to Mathe- matische Annalen and, particularly, Leipziger Berichte. He has, instead, unceasingly dedicated his talent and free time which his teaching allowed him to spend, to work on the presentation of my theories as fully, as completely and systematically, and above all, as precisely as is in any manner possible. For this selfless work which has stretched over a period of nine years, I, and, in my opinion, the entire scientific world owe him the highest gratitude. Lie and Engel formed a team both in terms of writing and teaching. Some students came to study with both Lie and Engel. Engel also contributed to the success of Lie’s teaching. For example, a major portion of students who received the doctoral degree at Leipzig was supervised under Lie. Lie also thanked Engel for this in the preface of the third volume. But this preface also contained a description of some conflict with Klein, and hence Engel’s academic future suffered due to this. See 9 for more detail.
- 32. 10 Lizhen Ji 5 Lie’s breakdown and a final major result After his move to Leipzig, Lie worked hard and was very productive. While Leipzig was academically stimulating to Lie, it was not stress-free for him, and relations with others were complicated too. “The pressure of work, problems of collaboration, and domestic anxieties made him sleepless and depressed, and in 1889 he had a complete breakdown” [27, p. 328]. Lie had to go to a nerve clinic and stayed there for seven months. He was given opium, but the treatment was not effective, and he decided to cure the problem himself.4 He wrote to his friend [26, XXIII]: In the end I began to sleep badly and finally did not sleep at all. I had to give up my lecturing and enter a nerve clinic. Unfortunately I have been an impossible patient. It has always been my belief that the doctors did not understand my illness. I have been treated with opium, in enormous dose, to calm my nerves, but it did not help. Also sleeping draughts. Three to four weeks ago I got tired of staying at the nerve clinic. I decided to try to see what I could do myself to regain my equilibrium and the ability to sleep. I have now done what the doctors say no one can endure, that is to say I have completely stopped taking opium. It has been a great strain. But now, on a couple of days, against the doctors’ advice, I have taken some exercise. I hope now that in a week’s time I shall have completely overcome the harmful effects of the opium cure. I think myself that the doctors have only harmed me with opium. My nerves are very strained, but my body has still retained its horsepower. I shall cure myself on my own. I shall walk from morning to evening (the doctors say it is madness). In this way I shall drive out all the filth of the opium, and afterwards my natural ability to sleep will gradually return. That is my hope. Finally he thought that he had recovered, and was released. Actually he was not cured at the time of release. Instead, in the reception book of the clinic, his condition at that time was recorded as “a Melancholy not cured” [27, p. 328]. His friends and colleagues found changes in Lie’s attitudes towards others and his behaviors: mistrusting and accusing others for stealing his ideas. Indeed, according to Engel [27, p. 397], Lie did recover his mathematical ability, but “not as a human being. His mistrust and irritability did not dissipate, but rather they grew more and more with the years, such that he made life difficult for himself and all his friends. The most painful thing was that he never allowed himself to speak openly about the reasons for his despondency.” When he was busy teaching and working out his results, he did not have much time to pick up new topics. While at the nerve clinic, he worked again on the so- called Helmholtz problem on the axioms of geometry5 and wrote two papers about it. 4According to the now accepted theory, Lie suffered from the so-called pernicious anemia psychosis, an incurable disease at that time. People also believe that his soured relations with Klein and others were partly caused by this disease. See the section on the period 1886–1898 in [9] and the reference [29] there. 5Lie’s work on the Helmholtz problem was apparently well known at the beginning of the 20th century. According to [5, Theorem 16.7], the Lie–Helmholtz Theorem states that spaces of constant curvature, i.e., the Euclidean space, the hyperbolic space and the sphere, can be characterized by abundance of isometries: for every
- 33. 1 Sophus Lie, a giant in mathematics 11 Lie had thought of and worked on this problem for a long time and had also criticized the work of Helmholtz and complained to Klein about it. According to [27, p. 380–381], Very early on, Lie was certainly clear that the transformation theory he was devel- oping was related to non-Euclidean geometry, and in a letter to Mayer as early as 1875, Lie had pointed out that von Helmhotz’s work on the axioms of geometry from 1868, were basically and fundamentally an investigation of a class of transformation groups: “I have long assumed this, and finally had it verified by reading his work.” Klein too, in 1883, has asked Lie what he thought of von Helmhotz’s geomet- ric work. Lie replied immediately that he found the results correct, but that von Helmhotz operated with a division between the real and imaginary that was hardly appropriate. And a little later, after having studied the treatise more thoroughly, he communicated to Klein that von Helmhotz’s work contained “substantial shortcom- ings”, and he thought it positively impossible to overcome these shortcomings by means of the elementary methods that von Helmhotz had applied. Lie went on to complete and simplify von Helmhotz’s spatial theory [. . . ] In 1884, Lie wrote to Klein [26, XXVI]: If I ever get as far as to definitely complete my old calculations of all groups and point transformations of a three-dimensional space I shall discuss in more detail Helmholtz’s hypothesis concerning metric geometry from a purely analytical aspect. According to [27, p. 381], Lie did further work with von Helmhotz’s space problem, and confided to Klein in April 1887, that the earlier works on the problem had now come to a satisfying conclusion – at least when one was addressing finite dimensional transformation groups, and therein, a limited number of parameters. What remained was to deduce some that extended across the board such that infinite-dimensional groups could be included. Lie’s work on the Helmholtz problem led him to being awarded the first Lo- batschevsky prize in 1897. Klein wrote a very strong report on his work, and this report was the determining factor for this award. 6 An overview of Lie’s major works As mentioned before, Lie was very productive and he wrote many thousands of pages of papers and multiple books. His name will be forever associated with Lie groups and Lie algebras and several other dozen concepts and definitions in mathematics two congruently ordered triples of points, there is an isometry of the space that moves one triple to the other, where two ordered triples of points .v1; v2; v3/, .v0 1; v0 2; v0 3/ are congruent if the corresponding distances are equal, d.vi ; vj / D d.v0 i ; v0 j / for all pairs i; j. References to papers of Weyl and Enriques on this theorem were given in [5].
- 34. 12 Lizhen Ji (almost all of them involve Lie groups or Lie algebras in various ways). One natural question is what exactly Lie had achieved in Lie theory. The second natural question is: besides Lie groups and Lie algebras, what else Lie had done. It is not easy to read and understand Lie’s work due to his writing style. In a pref- ace to a book of translations of some papers of Lie [21] in a book series Lie Groups: History, Frontiers and Applications, which contain also some classical books and papers by É. Cartan, Ricci, Levi-Civita and also other more modern ones, Robert Hermann wrote: In reading Lie’s work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie’s work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form. We nevertheless try to provide some short summaries. Though articles in Encyclope- dia Britannica are targeting the educated public, articles about mathematicians often give fairly good summaries. It might be informative and interesting to take a look at such an article about Lie before the global theory of Lie groups were developed by Weyl and Cartan. An article in Encyclopedia Britannica in 1911 summarized Lie’s work on Lie theory up to that time: Lie’s work exercised a great influence on the progress of mathematical science dur- ing the later decades of the 19th century. His primary aim has been declared to be the advancement and elaboration of the theory of differential equations, and it was with this end in view that he developed his theory of transformation groups, set forth in his Theorie der Transformationsgruppen (3 vols., Leipzig, 1888–1893), a work of wide range and great originality, by which probably his name is best known. A spe- cial application of his theory of continuous groups was to the general problem of non-Euclidean geometry. The latter part of the book above mentioned was devoted to a study of the foundations of geometry, considered from the standpoint of B. Rie- mann and H. von Helmholtz; and he intended to publish a systematic exposition of his geometrical investigations, in conjunction with Dr. G. Scheffers, but only one volume made its appearance (Geometrie der Berührungstransformationen, Leipzig, 1896). The writer of this article in 1911 might not have imagined the wide scope and multi- faceted applications of Lie theory. From what I have read and heard, a list of topics of major work of Lie is as follows: 1. Line complexes. This work of Lie was the foundation of Lie’s future work on differential equations and transformation groups, and hence of Lie theory [13]. It also contains the origin of toric varieties. 2. Lie sphere geometry and Lie contact structures. Contact transformations are closely related to contact geometry, which is in many ways an odd-dimensional counterpart of symplectic geometry, and has broad applications in physics. Relatively recently, it was applied to low-dimensional topology.
- 35. 1 Sophus Lie, a giant in mathematics 13 3. The integration theory of differential equations. This subject has died and re- covered in a strong way in connection with integrable systems and hidden sym- metries. 4. The theory of transformation groups (or Lie groups). This has had a huge impact through the development, maturing and applications of Lie groups. The theory of transformation groups reached its height in the 1960–1970s. But the theory of Lie groups is becoming more important with the passage of time and will probably stay as long as mathematics is practiced. 5. Infinitesimal transformation groups (or Lie algebras). Lie algebras are simpler than Lie groups and were at first used as tools to understand Lie groups, but they are important in their own right. For example, the infinite-dimensional Kac–Moody Lie algebras are natural generalizations of the usual finite di- mensional Lie groups, and their importance and applications are now well- established. Though they also have the corresponding Kac–Moody Lie groups, it is not clear how useful they can be. 6. Substantial contribution to the Erlangen program, which was written and for- mally proposed by Felix Klein and whose success and influence was partially responsible for the breakup of the friendship between Lie and Klein. Lie con- tributed to the formulation and also the development of this program, and his role has been recognized more and more by both historians of mathematics and practicing mathematicians. 7. The Helmholtz space problem: determine geometries whose geometric proper- ties are determined by the motion of rigid bodies. See footnote 4. The solution of this problem led Lie to be awarded the Lobachevsky prize. Lie’s work on this problem also had a big impact on Poincaré’s work on geometry. 8. Minimal surfaces. In 1878, building on the work of Monge on integration of the Euler–Lagrange equations for minimal surfaces, Lie assigned each mini- mal surface a complex-analytic curve. This was the starting point of a fruitful connection between minimal surfaces and analytic curves. Together with the work of Weierstrass, Riemann, Schwarz, and others, this introduced the wide use of methods and results of complex function theory in the theory of minimal surfaces at the end of the 19th century. 7 Three fundamental theorems of Lie in the Lie theory When people talk about Lie’s work, they often mention three fundamental theorems of Lie. His second and third fundamental theorems are well known and stated in many textbooks on Lie theories. On the other hand, the first fundamental theorem is not mentioned in most books on Lie groups and Lie algebras. The discussion below will explain the reasons:
- 36. 14 Lizhen Ji 1. It addresses a basic problem in transformation group theory rather than a prob- lem in abstract Lie theory. 2. It is such a basic result that people often take it for granted. We will first discuss these theorems in the original setup of transformation groups and later summarize all three theorems in the modern language. The first theorem says that a local group action on a manifold is determined by the induced vector fields on the manifold. Now the space of vector fields of the manifold forms a Lie algebra. So the study of Lie group actions is reduced to the study of Lie algebras. This is a deep insight of Lie and is one of the reasons for people to say that Lie reduced the study of Lie groups to Lie algebras, and hence reduced a nonlinear object to a linear one. In the case of a one-parameter group of local diffeomorphisms of a manifold, the action is determined by one vector field on the manifold. Conversely, given a vector field, the existence of the corresponding local solution should have been well known in Lie’s time. The proper definition of manifold was not known then, but no notion of manifolds was needed since the action of a Lie group in Lie’s work is local and hence can be considered on Rn . In Lie’s statement, the key point is to show how the vector fields on a manifold M associated with a Lie group G is determined by a homomorphism from g D TeG to the space .M/ of vector fields on M. (One part of the theorem is that g D TeG.) Lie’s second theorem says that given a Lie algebra homomorphism g D TeG ! .M/, then there is a local action. One important point is that there is already a Lie group G whose Lie algebra is g. One special case of Lie’s Fundamental Theorems 1 and 2 is that a one-parameter group of diffeomorphisms 't of a manifold M amounts to a vector field X on the manifold. This has two components: 1. The family 't induces a vector field X by taking the derivative, and 't is uniquely determined by X. The uniqueness follows from the fact that 't satis- fies an ODE. 2. Given a vector field X, there is an one-parameter family of local diffeomor- phisms 't which induces X. If M is compact, then the diffeomorphisms 't are global. This amounts to integrating a vector field on a manifold into a flow. The third theorem says that given any abstract Lie algebra g, and a Lie algebra homomorphism g ! .M/, then there is a local Lie group (or the germ of a Lie group) G and an action of G on M which induces the homomorphism g ! .M/. Lie was interested in Lie group actions. Now people are more interested in the theory of abstract Lie groups and usually reformulate these results in terms of ab- stract Lie groups and Lie algebras. If we generalize and put these three fundamental theorems in the modern language of Lie theory, then they can be stated as follows and can be found in most books on Lie groups and algebras:
- 37. 1 Sophus Lie, a giant in mathematics 15 1. The first theorem should be stated as: a Lie group homomorphism is deter- mined locally by a Lie algebra homomorphism. 2. The second theorem says that any Lie group homomorphism induces a Lie al- gebra homomorphism. Conversely, given a Lie algebra homomorphism, there is a local group homomorphism between corresponding Lie groups. If the do- main of the locally defined map is a simply connected Lie group, then there is a global Lie group homomorphism. 3. The third theorem says that given a Lie algebra g, there exists a Lie group G whose Lie algebra is equal to g. (Note that there is no group action here and hence this statement is different from the statements above.) 8 Relation with Klein I: the fruitful cooperation There are many differences and similarities between Lie and Klein. Lie was a good natured, sincere great mathematician. For example, he gave free lectures in the sum- mer to USA students to prepare them for his later formal lectures. He went out of his way to help his Ph.D. students. He was not formal, and his lectures were not polished and could be messy sometimes. Klein was a good mathematician with a great vision and he was also a powerful politician in mathematics. He was a noble, strict gentleman. His lectures were always well-organized and polished. Lie and Klein first met in Berlin in the winter semester of 1869–1870 and they be- came close friends. It is hard to overestimate the importance of their joint work and discussions on their mathematical works and careers. For example, it was Klein who helped Lie to see the analogy between his work on differential equations and Abel’s work on the solvability of algebraic equations, which motivated Lie to develop a gen- eral theory of differential equations that is similar to the Galois theory for algebraic equations, which lead to Lie theory. On the other hand, it was Lie who provided substantial evidence to the general ideas in the Erlangen program of Klein that were influential on the development of that program. Klein also helped to promote Lie’s work and career in many ways. For example, when Klein left Leipzig, he secured the vacant chair for Lie in spite of many objec- tions. Klein drafted the recommendation of the Royal Saxon Ministry for Cultural Affairs and Education in Dresden to the Philosophical Faculty of the University of Leipzig, and the comment on Lie run as follows [9, p. 12]: Lie is the only one who, by force of personality and in the originality of his thinking, is capable of establishing an independent school of geometry. We received proof of this when Kregel von Sternback’s scholarship was to be awarded. We sent a young mathematician – our present Privatdocent, Dr. Engel – to Lie in Chtristiania, from where he returned with a plethora of new ideas.
- 38. 16 Lizhen Ji It is also helpful to quote here a letter written by Weierstrass at that time [9, p. 12]: I cannot deny that Lie has produced his share of good work. But neither as a scholar nor as a teacher is he so important that there is a justification in preferring him, a foreigner, to all of those, our countrymen, who are available. It now seems that he is being seen as a second Abel who must be secured at any cost. One particular fellow countryman Weierstrass had in mind was his former student Hermann Schwarz, who was also a great mathematician. Another crucial contribution of Klein to Lie’s career was to send Engel to help Lie to write up his deep work on transformation groups. Without Engel, Lie’s contri- butions might not have been so well known and hence might not have had the huge impacts on mathematics and physics that they have now. It is perhaps sad to note that Engel was punished by Klein in some way for being a co-author of Lie after the breakup between Lie and Klein. One further twist was that Klein made Engel edit Lie’s collected works carefully after Lie passed away. 9 Relation with Klein II: conflicts and the famous preface The breakup between Lie and Klein is famous for one sentence Lie put down in the preface of the third volume of his joint book with Engel on Lie transformation groups published in 1893: “I am not a student of Klein, nor is the opposite the case, even if it perhaps comes closer to the truth.” This is usually the only sentence that people quote and say. It sounds quite strong and surprising, but there are some reasons behind it. The issue is about the formula- tion and credit of ideas in the Erlangen program, which was already famous at that time. It might be helpful to quote more from the foreword of Lie [9, p. 19]: F. Klein, whom I kept abreast of all my ideas during these years, was occasioned to develop similar viewpoints for discontinuous groups. In his Erlangen Program, where he reports on his and on my ideas, he, in addition, talks about groups which, according to my terminology, are neither continuous or discontinuous. For example, he speaks of the group of all Cremona transformations and of the group of distor- tions. The fact that there is an essential difference between these types of groups and the groups which I have called continuous (given the fact that my continuous groups can be defined with the help of differential equations) is something that has apparently escaped him. Also, there is almost no mention of the important concept of a differential invariant in Klein’s program. Klein shares no credit for this con- cept, upon which a general invariant theory can be built, and it was from me that he learned that each and every group defined by differential equations determines differential invariants which can be found through integration of complete systems. I feel these remarks are called for since Klein’s students and friends have repeat- edly represented the relationship between his work and my work wrongly. Moreover, some remarks which have accompanied the new editions of Klein’s interesting pro- gram (so far, in four different journals) could be taken the wrong way. I am no
- 39. 1 Sophus Lie, a giant in mathematics 17 student of Klein and neither is the opposite the case, though the latter might be closer to the truth. By saying all this, of course, I do not mean to criticize Klein’s original work in the theory of algebraic equations and function theory. I regard Klein’s talent highly and will never forget the sympathetic interest he has taken in my research endeav- ors. Nonetheless, I don’t believe he distinguishes sufficiently between induction and proof, between a concept and its use. According to [27, p. 317], in the same preface, Lie’s assertion was that Klein did not clearly distinguish between the type of groups which were presented in the Erlangen Programme – for example, Cremonian trans- formations and the group of rotations, which in Lie’s terminology were neither con- tinuous nor discontinuous – and the groups Lie had later defined with the help of differential equations: “One finds almost no sign of the important concept of differential invariants in Klein’s programme. This concept, which first of all a common invariant theory could be build upon, is something Klein has no part of, and he has learned from me that every group that it defined by means of differential equations, determines differential invariants, which can be found through the integration of integrable systems.” . . . Lie continued, in their investigations of geometry’s foundation, Klein, as well as von Helmhotz, de Tilly, Lindemann, and Killing, all committed gross errors, and this could largely be put down to their lack of knowledge of group theory. Maybe some explanations are in order to shed more light on these strong words of Lie. According to [26, pp. XXIII–XXIV], Sophus Lie gradually discovered that Felix Klein’s support for his mathematical work no longer conformed with his own interests, and the relationship between the two friends became more reserved. When, in 1892, Felix Klein wanted to republish the Erlangen program and explain its history, he sent the manuscript to Sophus Lie for a comment. Sophus Lie was dismayed when he saw what Felix Klein has written, and got the impression that his friend now wanted to have his share of what Sophus Lie regarded as his life’s work. To make things quite clear, he asked Felix Klein to let him borrow the letters he had sent him before the Erlangen program was written. When he learned that these letters no longer existed, Sophus Lie wrote to Felix Klein, November 1892. The letter from Lie to Klein in November 1892 goes as follows [9, p. XXIV]: I am reading through your manuscript very thoroughly. In the first place, I am afraid that you, on your part, will not succeed in producing a presentation that I can accept as correct. Even several points which I have already criticized sharply are incorrect, or at least misleading, in your current presentation. I shall try as far as possible to concentrate my criticism on specific points. If we do not succeed in reaching agreement, I think that it is only right and reasonable that we each present our views independently, and the mathematical public can then form their own opinion. For the time being I can only say how sorry I am that you were capable of burning my so significant letters. In my eyes this was vandalism; I had received your specific promise that you would take care of them.
- 40. 18 Lizhen Ji I have already told you that my period of naiveness is now over. Even if I still firmly retain good memories from the years 1869–1872, I shall nevertheless try to keep myself that which I regard as my own. It seems that you sometimes believe that you have shared my ideas by having made use of them. The comprehensive biography of Lie [27, p. 371] gives other details on the origin of this conflict: The relationship between them [Lie and Klein] had certainly cooled over recent years, although they continued to exchange letters the same way, although not as frequently as earlier. But it was above all professional divergencies that were cen- tral to the fact that Lie now broke off relationship. Following Lie’s publication of the first volume of his Theorie der Transformationsgruppen, Klein judged that there was sufficient interest to have his Erlangen Programme been republished. But before Klein’s text from 1872 was printed anew, Klein had contacted Lie to find out how the working relationship and exchange of ideas between them twenty years earlier should be presented. Lie had made violent objections to the way in which Klein had planned to portray the ideas and the work. But Klein’s Erlangen Programme was printed, and it came out in four different journals, in German, Italian, English and French – without taking into account Lie’ commentary on his assistance in for- mulating this twenty-year-old programme. More and more in mathematical circles, Klein’s Erlangen Programme was spoken of as central to the paradigm shift in ge- ometry that occurred in the previous generation. A large part of the third volume of Lie’s great work on transformation groups was devoted to a deepening discussion on the hypotheses or axioms that ought to be set down as fundamental to a geom- etry, that – whether or not it accepted Euclid’s postulates – satisfactorily clarified classical geometry as well as the non-Euclidean geometry of Gauss, Lobachevsky, Bolyai, and Riemann. The information that spread regarding the relationship between Klein’s and Lie’s respective work, was, according to Lie, both wrong and misleading. Lie con- sidered he had been side-lined but was eager to “set things right”, and grasped the first and best opportunity. In front of the professional substance of his work he placed his twenty-page foreword. The power-charge that liquidated their friendship and sent shock-waves through the mathematical milieu was short, if not sweet. Klein was the king of German mathematics and probably also of the European mathematics at that time. What was people’s reaction to the strong preface of Lie? Maybe a letter from Hilbert to Klein in 1893 will explain this [26, p. XXV]: “In his third volume, his megalomania spouts like flames.” Lie probably did not suffer too much professionally from this conflict with Klein since he had the chair at Leipzig. But this was not the case with Engel. Since Engel’s name also appeared on the book, he had to pay for this. Engel was looking for a job, and a position of professorship was open at that time at the University of Königsberg, the hometown and home institution of Hilbert where he held a chair in mathematics, and this open position was a natural and likely choice for Engel. In the same letter to Klein, Hilbert continued [26, p. XXV], “I have excluded Engel completely. Although he himself has not made any comment in the preface, I hold him to some extent co- responsible for the incomprehensible and totally useless personal animosity which the third volume of Lie’s work on transformation groups is full off.”
- 41. 1 Sophus Lie, a giant in mathematics 19 Engel could not get an academic job for several years,6 and Klein arranged Engel to edit the collected works of Grassmann and then later the collected works of Lie; on the latter he worked for several decades. Another consequence of this conflict with Klein was that Lie could not finish another proposed joint book with Engel on applications of transformation groups to differential equations. According to [27, p. 390–391], after the publication of these three volumes, The next task that Lie saw for himself was to make refinements and applications of what had now been completely formulated. But this foreword [of the third volume] with its sharp accusations against Klein, caused hindrances to the further work. Be- cause Lie in the same foreword had praised Engel to the skies for his “exact” and “unselfish activity”, it now became difficult for Engel to continue to collaborate with Lie – consequently as well, nothing came of the announced work on, among other things, differential invariants and infinite-dimensional continuous groups. As for Engel, his career outlook certainly now lay in other directions than Lie’s. Accord- ing to Lie’s German student, Gerhard Kowalewski, relations between Lie and Engel gradually became so cool that they were seldom to be seen in the same place. It should be pointed out that relations between Lie and Engel had some hard time before this foreword came out. It was caused by the fight between Lie and Killing due to some overlap in their work on Lie theories, in particular, Lie algebras. For some reason, at the initial stage, Killing communicated with Engel and cited some papers of Engel instead of Lie’s, and Lie felt than Engel betrayed his trust. For a more detailed discussion on this issue, see [27, pp. 382–385, p. 395].7 After Lie’s death, Engel continued and carried out his mentor’s work in several ways. See 11 for example. He was a faithful disciple and was justly awarded with the Norwegian Order of St Olaf and an honorary doctorate from the University of Oslo. Maybe there is one contributing factor to these conflicts.8 It is the intrinsic mad- ness of all people who are devoted to research and are doing original work, in partic- 6On the other hand, all things ended well with Engel. In 1904, he accepted the chair of mathematics at the University of Greifswald when his friend Eduard Study resigned, and in 1913, took the chair of mathematics at the University of Giessen. Engel also received a Lobachevsky Gold Medal. The Lobachevsky medal is different from the Lobatschevsky prize won by his mentor Lie and his fellow countryman Wilhelm Killing. The medal was given on a few occasions to the referee of a person nominated for the prize. For instance, Klein also received, in 1897, a the Gold medal, for his report on Lie. See [28]. 7Manfred Karbe pointed out that in his autobiography [20, pp. 51–52], Kowalewski speculates about the mounting alienation of Lie and Engel, and reports about Lie’s dislike of his three-volume Transformation Groups. When Lie needed some material of his own in his lectures or seminars, he never made use of these books but only of the papers in Math. Ann. And Kowalewski continues on page 52, line -5: “Von hier aus kann man es vielleicht verstehen, dass die Abneigung gegen das Buch sich auf den Mitarbeiter bertrug, dem er doch so sehr zu Dank verpflichtet war.” (From this one may perhaps understand that the aversion to the book is transferred to the collaborator to whom he was so much indebted.) 8There has been an explanation of Lie’s behavior in this conflict with Klein by establishing a relation between genius and madness. According to [27, p. 394], after Lie’s death, “In Göttingen, Klein made a speech that gave rise to much rumor, not least because here, in addition to all his praise for his old friend, Klein suggested that the close relationship between genius and madness, and that Lie had certainly been struck by a mental condition that was tinged with a persecution complex – at least, by assessing the point from notes that Klein made for his speech, it seems that this was the expression he used.”
- 42. 20 Lizhen Ji ular mathematicians and scientists. According to a comment of Lie’s nephew, Johan Vogt, a professor at the University of Oslo in economics, and also a translator, writer and editor, made in 1930 on his uncle [27, p. 397], We shall avail ourselves of a popular picture. Every person has within himself some normality and some of what may be called madness. I believe that most of my col- leagues possess ninety-eight percent normality and two percent madness. But So- phus Lie certainly had appreciably more of the latter. The merging of a pronounced scientific gift and an impulsiveness that verges on the uncontrollable, would certainly describe many of the greatest mathematicians. In Sophus Lie this combination was starkly evident. It might be helpful to point out that later at the request of the committee of Lo- batschevsky prize, Klein wrote a very strong report about the important work of Lie contained in this third volume on transformation groups, and this report was instru- mental in securing the inaugural Lobatschevsky prize for Lie. It might also be helpful to quote from Klein on Lie’s work related to this conflict. The following quote of Klein [19, pp. 352–353], its translation and information about it were kindly provided by Hubert Goenner: I will now add a personal remark. The already mentioned Erlangen Program is about an outlook which – as already stated in the program itself – I developed in personal communication with Lie (now professor in Leipzig, before in Christiana). Lie who has been engaged particularly with transformation groups, created a whole theory of them, which finds its account in a larger œuvre “Theory of Transformation Groups”, edited by Lie and Engel, Vol. I 1888, Vol. II 1890. In addition, a third volume will appear, supposedly in not all too distant a time. Obviously, we cannot think about responding now to the contents of Lie’s theories [. . . ]. My remark is limited to having called attention to Lie’s theories. The above comment was made by Klein in the winter of 1889 or at the beginning of 1890, but Klein backed its publication until 1893, the year of the ill-famed preface to the third volume by Lie and Engel. Further details about this unfortunate conflict and the final reconciliation between these two old friends are also given in [27, pp. 384–394]. See also the article [18] for more information about Klein and on some related discussion on the relationship between Lie and Klein. The above discussion showed that the success of the Erlangen program was one cause for the conflict between Lie and Klein. A natural question is how historians of mathematics have viewed this issue. Given the fame and impacts of the Erlangen program, it is not surprising that there have been many historical papers about it. Two papers [15] [3] present very different views on the contributions of Klein and Lie to the success and impacts of this program. The paper [15] argues convincingly that Lie’s work in the period 1872–1892 made the Erlangen program a solid program with substantial results, while the paper [3] was written to dispute this point of view. It seems that the authors are talking about slightly different things. For example, [3] explains the influence of Klein and the later contribution of Study, Cartan and Weyl,
- 43. 1 Sophus Lie, a giant in mathematics 21 but most of their contributions were made after 1890. The analysis of the situation in [10, p. 550] seems to be fair and reasonable: It seems that the Erlangen Program met with a slow reception until the 1890s, by which time Klein’s status as a major mathematician at the University of Göttingen had a great deal to do with its successful re-launch. By that time too a number of mathematicians had done considerable work broadly in the spirit of the programme, although the extent to which they were influenced by the programme, or were even aware of it, is not at all clear [. . . ] Since 1872 Lie had gone on to build up a vast theory of groups of continuous transformations of various kinds; but however much it owned to the early experiences with Klein, and however much Klein may have assisted Lie in achieving a major professorship at Leipzig University in 1886, it is doubtful if the Erlangen Program had guided Lie’s thoughts. Lie was far too powerful and original a mathematician for that. 10 Relations with others As mentioned, both Klein and Engel played crucial roles in the academic life of Lie. Another important person to Lie is Georg Scheffers, who obtained his Ph.D. in 1890 under Lie. Lie thought highly of Scheffers. In a letter to Mittag–Leffler [27, p. 369], Lie wrote “One of my best pupils (Scheffers) is sending you a work, which he has prepared before my eyes, and who has taken his doctorate here in Leipzig with a dissertation that got the best mark . . . Scheffers possesses an unusually evident talent and his calculations are worked through with great precision, and bring new results.” After the collaboration between Lie and Engel unfortunately broke off, Scheffers substituted for Engel and edited two of Lie’s lecture notes in the early 1890s. They are Lectures on differential equations with known infinitesimal transformations of 568 pages, and Lectures on Continuous groups of 810 pages. Later in 1896, they also wrote a book together, Geometry of contact transformations of 694 pages. All these book projects of Lie with others indicate that Lie might not have been able to efficiently write up books by himself. For example, he only wrote by himself a book of 146 pages and a program for a course in Christiania in 1878. In 1896 Scheffers became docent at the Technical University of Darmstadt, where he was promoted to professor in 1900. The collaboration with Lie stopped after this move. From 1907 to 1935, when he retired, Scheffers was a professor at the Technical University of Berlin. According to a prominent American mathematician, G. A. Miller, from the end of the nineteenth century, “The trait of Lie’s character which impressed me most forcibly when I first met him in the summer of 1895 was his extreme openness and lack of effort to hide ignorance on any subject.” Though he was motivated by discontinuous groups (or rather finite groups) taught by Sylow and kept on studying a classical book on finite substitution groups by Jor-
- 44. 22 Lizhen Ji dan, he could never command the theory of finite groups. Miller continued, “In fact he frequently remarked during his lectures that he always got stuck when he entered upon the subject of discontinuous groups.” When Lie first arrived in Leipzig, teaching was a challenge for him due to both lack of students and the amount of time needed for preparation. In a letter to a friend from the youth, Lie wrote [26, p. XXI], “While, in Norway, I hardly spent five minutes a day on preparing the lectures, in Germany I had to spend an average of about 3 hours. The language is always a problem, and above all, the competition implies that I had to deliver 8–10 lectures a week.” When Lie and his assistant Engel decided to present their own research on trans- formation groups, students from all over the world poured in, and the Ecole Normale sent its best students to study with Lie. It was a big success. According to the recol- lection of a student of Lie [26, p. XXVII]: Lie liked to teach, especially when the subject was his own ideas. He had vivid contact with his students, who included many Americans, but also Frenchmen, Rus- sians, Serbs and Greeks. It was his custom to ask us questions during the lectures and he usually addressed each of us by name. Lie never wore a tie. His full beard covered the place where the tie would have been, so even the most splendid tie would not have shown to advantage. At the start of a lecture he would take off his collar with a deft movement, saying: “I love to be free”, and he would then begin his lecture with the words “Gentlemen, be kind enough to show me your notes, to help me remember what I did last time.” Someone or other on the front bench would immediately stand up and hold out the open notebook, whereupon Lie, with a satisfied nod of the bead, would say “Yes, now I remember.” In the case of difficult problems, especially those referring to Lie’s complex integration theories, it could happen that the great master, who naturally spoke without any kind of preparation whatsoever, got into difficulties and, as the saying goes, became stuck. He would then ask one of his elite students for help. Lie had many students. Probably one of the most famous was Felix Hausdorff. Lie tried to convince Hausdorff to work with him on differential equations of the first order without success. Of course, Hausdorff became most famous for his work on topology. See [27, p. 392–393] for a description of Hausdorff and his interaction with Lie. Throughout his life, Lie often felt that he was under-recognized and under-appre- ciated. This might be explained by his late start in mathematics and his early isolation in Norway. He paid careful attention to other people’s reaction to his work. For example, Lie wrote about Darboux in a letter to Klein [25] in October 1882: Darboux has studied my work with remarkable thoroughness. This is good insofar as he has given gradually more lectures on my theories at the Sorbonne, for example on line and sphere geometry, contact transformation, and first-order partial differ- ential equations. The trouble is that he continually plunders my work. He makes inessential changes and then publishes these without mentioning my name.
- 45. 1 Sophus Lie, a giant in mathematics 23 11 Collected works of Lie: editing, commentaries and publication Since Lie theories are so well known and there are many books on different aspects of Lie groups and Lie algebras, Lie’s collected works are not so well known to general mathematicians and students. The editing and publication of Lie’s collected works are both valuable and interesting to some people. In view of this, we include some relevant comments. Due to his death at a relatively young age, the task of editing Lie’s work com- pletely fell on others. It turned out that editing and printing the collected work of Sophus Lie was highly nontrivial and a huge financial burden on the publisher. The situation is well explained in [6]: Twenty-three years after the death of Sophus Lie appears the first volume to be printed of his collected memoirs. It is not that nothing has been done in the mean- time towards making his work more readily available. A consideration of the matter was taken up soon after his death but dropped owing to the difficulties in the way of printing so large a collection as his memoirs will make. An early and unsuc- cessful effort to launch the enterprise was made by the officers of Videnskapssel- skapet i Kristiania; but plans did not take a definite form till 1912; then through the Mathematisch-physische Klasse der Leipziger Akademie and the publishing firm of B. G. Teubner steps were taken to launch the project. Teubner presented a plan for raising money by subscription to cover a part of the cost of the work and a little later invitations to subscribe were sent out. The responses were at first not encour- aging; from Norway, the homeland of Lie, only three subscriptions were obtained in response to the first invitations. In these circumstances, Engel, who was pressing the undertaking, resorted to an unusual means. He asked the help of the daily press of Norway. On March 9, 1913, the newspaper TIDENS TEGN of Christiania carried a short article by Engel with the title Sophus Lies samlede Afhandlinger in which was emphasized the failure of Lie’s homeland to respond with assistance in the work of printing his collected memoirs. This attracted the attention of the editor and he took up the campaign: two important results came from this, namely, a list of subscriptions from Norway to support the undertaking and an appropriation by the Storthing to assist in the work. By June the amount of support received and promised was sufficient to cause Teubner to announce that the work could be undertaken; and in November the memoirs for the first volume were sent to the printer, the notes and supplementary matter to be supplied later. The Great War so interfered with the undertaking that it could not be contin- ued, and by the close of the war circumstances were so altered that the work could not proceed on the basis of the original subscriptions and understandings and new means for continuing the work had to be sought. Up to this time the work had been under the charge of Engel as editor. But it now became apparent that the publication of the memoirs would have to become a Norwegian undertaking. Accordingly, Poul Heegaard became associated with Engel as an editor. The printing of the work be- came an enterprise not of the publishers but of the societies which support them in this undertaking. Under such circumstances the third volume of the series, but the first one to be printed, has now been put into our hands. “The printing of further vol-
- 46. 24 Lizhen Ji umes will be carried through gradually as the necessary means are procured; more I cannot say about it,” says Engel, “because the cost of printing continues to mount incessantly.” The first volume was published in 1922, and the sixth volume was published in 1937. The seventh volume consisted of some unpublished papers of Lie and was published only in 1960 due to the World War II and other issues. This was certainly a major collected work in the last 100 years. The collected works of Lie are very well done with the utmost dedication and respect thanks to the efforts of Engel and Heegaard. This can be seen in the editor’s introduction to volume VI of Lie’s Collected Works, If one should go through the whole history of mathematics, I believe that he will not find a second case where, from a few general thoughts, which at first sight do not appear promising, has been developed so extensive and wide-reaching a theory. Considered as an edifice of thought Lie’s theory is a work of art which must stir up admiration and astonishment in every mathematician who penetrates it deeply. This work of art appears to me to be a production in every way comparable with that [. . . ] of a Beethoven [. . . ] It is therefore entirely comprehensible if Lie [. . . ] was embittered that ‘deren Wesen, ja Existenz, den Mathematikern fort-während unbekannt zu sein scheint’ (p. 680). This deplorable situation, which Lie himself felt so keenly, exists no longer, at least in Germany. In order to do whatever lies in my power to improve the situation still further, [. . . ] I have sought to clarify all the individual matters (Einzelheiten) and all the brief suggestions in these memoirs. Each volume contains a substantial amount of notes, commentaries and supplemen- tary material such as letters of Lie, and “This additional material has been prepared with great care and with the convenience of the reader always in mind.” For example, as mentioned before, Lie’s first paper was only 8 pages long, but the commentary consisted of over 100 pages. According to [4], Although Engel was himself an important and productive mathematician he has found his place in the history of mathematics mainly because he was the closest stu- dent and the indispensable assistant of a greater figure: Sophus Lie, after N. H. Abel the greatest Norwegian mathematician. Lie was not capable of giving to the ideas that flowed inexhaustibly from his geometrical intuition the overall coherence and precise analytical form they needed in order to become accessible to the mathemat- ical world [. . . ] Lie’s peculiar nature made it necessary for his works to be eluci- dated by one who knew them intimately and thus Engel’s “Annotations” completed in scope with the text itself. Acknowledgments. I would like to thank Athanase Papadopoulos for carefully reading preliminary versions of this article and his help with references on the Lie–Helmholtz Theorem, and Hubert Goenner for several constructive and critical suggestions and references. I would also like to thank Manfred Karbe for the reference [20] and a summary of a possible explanation in [20] about the alienation between Lie and Engel.
- 47. Bibliography 25 Bibliography [1] M. A. Akivis and B. A. Rosenfeld, Elie Cartan (1869–1951). Translated from the Russian manuscript by V. V. Goldberg. Translations of Mathematical Monographs, 123. American Mathematical Society, Providence, RI, 1993. [2] N. Baas, Sophus Lie. Math. Intelligencer 16 (1994), no. 1, 16–19. [3] G. Birkhoff and M. K. Bennett, Felix Klein and his ”Erlanger Programm”. History and phi- losophy of modern mathematics (Minneapolis, MN, 1985), 145–176, Minnesota Stud. Philos. Sci., XI, Univ. Minnesota Press, Minneapolis, MN, 1988. [4] H. Boerner, Friedrich Engel, Complete Dictionary of Scientific Biography. Vol. 4. Detroit: Charles Scribner’s Sons, 2008. p. 370–371. [5] H. Busemann, Local metric geometry. Trans. Amer. Math. Soc. 56 (1944). 200–274. [6] R. Carmichael, Book Review: Sophus Lie’s Gesammelte Abhandlungen. Bull. Amer. Math. Soc. 29 (1923), no. 8, 367–369; 31 (1925), no. 9–10, 559–560; 34 (1928), no. 3, 369–370; 36 (1930), no. 5, 337. [7] S. S. Chern, Sophus Lie and differential geometry. The Sophus Lie Memorial Conference (Oslo, 1992), 129–137, Scand. Univ. Press, Oslo, 1994. [8] G. Darboux, Sophus Lie. Bull. Amer. Math. Soc. 5 (1899), 367–370. [9] B. Fritzsche, Sophus Lie: a sketch of his life and work. J. Lie Theory 9 (1999), no. 1, 1–38. [10] J. Gray, Felix Klein’s Erlangen Program, ‘Comparative considerations of recent geometrical researches’. In Landmark Writings in Western Mathematics 1640–1940, Elsevier, 2005, 544– 552. [11] J. Gribbin, The Scientists: A History of Science Told Through the Lives of Its Greatest Inven- tors. Random House Trade Paperbacks, 2004. [12] G. Halsted, Sophus Lie. Amer. Math. Monthly 6 (1899), no. 4, 97–101. [13] T. Hawkins, The birth of Lie’s theory of groups. The Sophus Lie Memorial Conference (Oslo, 1992), 23–50, Scand. Univ. Press, Oslo, 1994. [14] T. Hawkins, Emergence of the theory of Lie groups. An essay in the history of mathemat- ics 1869–1926. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2000. [15] T. Hawkins, The Erlanger Programm of Felix Klein: reflections on its place in the history of mathematics. Historia Math. 11 (1984), no. 4, 442–470. [16] S. Helgason, Sophus Lie, the mathematician. The Sophus Lie Memorial Conference (Oslo, 1992), 3–21, Scand. Univ. Press, Oslo, 1994. [17] L. Ji, A summary of topics related to group actions. Handbook of group actions, ed. L. Ji, A. Papadopoulos and S.-T. Yau, Vol.1, Higher Education Press, Beijing and International Press, Boston, 2015, 33–187. [18] L. Ji, Felix Klein: his life and mathematics. In Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics, ed. L. Ji and A. Papadopoulos, European Mathematical Society Publishing House, Zürich, 2015, 27–58. [19] F. Klein, Nicht-Euklidische Geometrie, I. Vorlesung, Wintersemester 1889/90. Ausgearbeitet von F. Schilling, Göttingen, 1893.
- 48. 26 Lizhen Ji [20] G. Kowalewski, Bestand und Wandel. Meine Lebenserinnerungen zugleich ein Beitrag zur neueren Geschichte der Mathematik. Verlag Oldenbourg, München 1950. [21] S. Lie, Sophus Lie’s 1880 transformation group paper. In part a translation of “Theorie der Transformations-gruppen”’by S. Lie [Math. Ann. 16 (1880), 441–528]. Translated by Michael Ackerman. Comments by Robert Hermann. Lie Groups: History, Frontiers and Applications, Vol. I. Math. Sci. Press, Brookline, Mass., 1975. [22] R. Merton, On the Shoulders of Giants: A Shandean Postscript. University of Chicago Press, 1993. [23] G. Miller, Some reminiscences in regard to Sophus Lie. Amer. Math. Monthly 6 (1899), no. 8–9, 191–193. [24] C. Reid, Hilbert. Reprint of the 1970 original. Copernicus, New York, 1996. [25] D. Rowe, Three letters from Sophus Lie to Felix Klein on Parisian mathematics during the early 1880s. Translated from the German by the author. Math. Intelligencer 7 (1985), no. 3, 74–77. [26] E. Strom, Sophus Lie. The Sophus Lie Memorial Conference (Oslo, 1992), Scand. Univ. Press, Oslo, 1994. [27] A. Stubhaug, The mathematician Sophus Lie. It was the audacity of my thinking. Translated from the 2000 Norwegian original by Richard H. Daly. Springer-Verlag, Berlin, 2002. [28] A. Vassilief, Prox Lobachevsky (premier concours), Nouv. Ann. Math., 3e sér. 17 (1898), 137–139.
- 49. Chapter 2 Felix Klein: his life and mathematics Lizhen Ji Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 A nontrivial birth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Three people who had most influence on Klein . . . . . . . . . . . . . . . . . . . . . . 32 5 Academic career . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 As a teacher and educator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7 Main contributions to mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8 The Evanston Colloquium Lectures and the resulting book . . . . . . . . . . . . . . . 42 9 A summary of the book “Lectures on mathematics” and Klein’s conflicts with Lie . . . 45 10 The ambitious encyclopedia in mathematics . . . . . . . . . . . . . . . . . . . . . . . 51 11 Klein’s death and his tomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 12 Major mathematicians and mathematics results in 1943–1993 from Klein’s perspective 53 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1 Introduction Felix Klein was not only one of the great mathematicians but also one of the great ed- ucators and scientific writers of the nineteenth and the early twentieth century. He was a natural born leader and had a global vision for mathematics, mathematics education and mathematical development. He had the required ambition, drive and ability to remove obstacles on his way and to carry out his plans. He was a benign and no- ble dictator, and was the most kingly mathematician in the history of mathematics. He had a great deal of influence on German mathematics and the world mathematics community. Indeed, it was Klein who turned Göttingen into the leading center of mathematics in the world. I have heard of Felix Klein for a long time but have not really tried to look up information about him. On the other hand, when I became interested in him and wanted to learn more about him, for example, the exact mathematical content of his competition with Poincaré, his relation with Lie and their conflicts, and his contri- bution towards the Erlangen program, there was no biography of Klein in English which was easily available. Later we found some extensive writings about him in obituary notices [18], and in books [13] [12]. After reading various sources about Klein which were available to me in English, I found him an even more interesting
- 50. 28 Lizhen Ji person than I thought. One purpose in writing this chapter was to share with the reader a summary of what I read about this incredible mathematician with some em- phasis on his lectures in the USA around 1893 and their influence on the development of the Americian mathematics community, and some thoughts which occurred to me during this reading process. Another purpose is to give a brief outline of the rich life of Klein and to supplement other more scientific papers in this book.1 Probably, Klein is famous to different groups of people for different reasons. For example, for people working in Lie theories and geometry, the Erlangen program he proposed had far-reaching consequences; for people working on discrete subgroups of Lie groups and automorphic forms, Kleinian groups and his famous books with Fricke on discrete groups of linear fractional transformations (or Möbius transformations) and automorphic forms have had a huge impact; many other people enjoy his books on elementary mathematics and history of mathematics, and of course high school and college teachers also appreciate his books and points of view on education. A special feature of Klein is his broad command of and vision towards mathematics. We will discuss more on this last point from different aspects. In terms of mathematical contribution, Klein was probably not of the same rank as his predecessors Gauss or Riemann and his younger colleagues Hilbert and Weyl in Göttingen, but he also made real contributions and was a kingly mathematician, more kingly than any of the others mentioned above. He could command respect from others like a king or even like a god. In some sense, Klein earned this. On the influence of Klein on the mathematics in Göttingen, Weyl [7, p. 228] said that “Klein ruled mathematics there like a god, but his godlike power came from the force of his personality, his dedication, and willingness to work, and his ability to get things done.” But kings are kings and can be tough and remote from the ordinary people. In 1922, the eminent analyst Kurt O. Friedrichs was young and visited Klein [7, p. 229]: “I was amazed, simply swept off my feet by Klein’s grace and charm. ... He could be very charming and gentlemanly when all went his way but with anyone who crossed him he was a tyrant.” Klein made important original contributions to mathematics early in his career. But his research was cut short due to exhaustion and the nervous breakdown caused by his competition with Poincaré on Fuchsian groups and the uniformization of Rie- mann surfaces.2 He fell down but picked it up. Very few people can do this. Since he could not do mathematical research anymore, he devoted his time and energy to 1 Most mathematicians have heard of Klein and have been influenced by his mathematics. On the other hand, various writings and stories about him are scattered in the literature, and we feel that putting several snapshots from various angles could convey a vague global picture of the man and his mathematics. It might be helpful to quote [18, p. i]: “After his death there appeared, one after another, a number of sketches of the man and his work from the pens of many of his pupils. But, just as a photograph of a man of unusual personality, or of a place of striking beauty, conveys little to one not personally acquainted with the original, so it is, and so it must be, with these sketches of Klein.” For the reader now, these many sketches are probably not easily accessible. 2 It is interesting to see a different explanation of the breakdown of Klein in [18, p. vii]: “His breakdown was probably accelerated by the antagonism he experienced at Leipzig. He was much younger than his col- leagues, and they resented his innovating tendencies. In particular, there was opposition to his determination to avail himself of the vaunted German “Lehrfreiheit”, and to interpret the word “Geometry” in its widest sense, beginning his lectures with a course on the Geometric Theory of Functions.”
- 51. 2 Felix Klein: his life and mathematics 29 education and mathematical writing, and more importantly to building and providing a stimulating environment for others. For example, he brought Hilbert to Göttingen and turned this small college city into the leading center of mathematics, which at- tracted people from all over the world. His lecture tours in the USA near the end of the nineteenth century also played a pivotal role in the emergence of the US mathe- matics community as one of the leading ones in the world. In this sense, Klein was also a noble mathematician and had a far-reaching and long-lasting impact. His approach to mathematics emphasizes the big picture and connections between different subjects without paying too much attention to details or work to substantiate his vision. He valued results and methods which can be applied to a broad range of topics and problems. For example, he proposed the famous Erlangen program but did not really work on it to substantiate it. Instead, his friend Lie worked to make it an important concrete program. As Courant once commented, Klein tended to soar above the terrain that occupied ordinary mathematicians, taking in and enjoying the vast view of mathematics, but it was often difficult for him to land and to do the hard boring work. He had no patience for thorny problems that require difficult technical arguments. What counted for him was the big pictures and the general pattern behind seemingly unrelated results. According to Courant [13, p. 179], Klein had certainly understood “that his most splendid scientific creations were fundamentally gigantic sketches, the completion of which he had to leave to other hands.” Klein was a master writer and speaker. In some sense, he was a very good sales- man of mathematics, but he drew a lot of criticism for this. Because of this, he had a lot of influence on mathematics and the mathematics community. He is a unique combination of an excellent mathematician, master teacher and efficient organizer. People’s responses to these features of Klein were not all positive. According to a letter from Mittag-Leffler to Hermite in 1881 [7, p. 224]: You asked me what are the relations between Klein and the great Berliners . . . Weierstrass finds that Klein is not lacking in talent but is very superficial and even sometimes rather a charlatan. Kronecker finds that he is quite simply a charlatan without real merit. I believe that is also the opinion of Kummer. In 1892, when the faculty of Berlin University discussed the successor to Weier- strass, they rejected Klein as a “dazzling charlatan” and a “complier.” His longtime friend Lie’s opinion was also harsh but more concrete [7, p. 224]: I rank Klein’s talents highly and shall never forget the ready sympathy with which he always accompanied me in my scientific attempts; but I opine that he does not, for instance, sufficiently distinguish induction from proof, and the introduction of a concept from its exploitation. In spite of all these criticisms and reservations, Klein was the man who was largely responsible for restoring Göttingen’s former luster and hence for initiating a process that transformed the whole structure of mathematics at German universities and also in some other parts of the world. Among people from Europe, he had the most im- portant influence on the emergence of mathematics in the USA. There is no question
- 52. 30 Lizhen Ji of the fact as to he was the most dynamic mathematical figure in the world in the last quarter of the nineteenth century. Klein represented an ideal German scholar in the nineteenth century. He strove for and attained an extraordinary breadth of knowledge, much of which he acquired in his active interaction with other mathematicians. He also freely shared his insights and knowledge with his students and younger colleagues, attracting people towards him from around the world. Unlike most mathematicians who only affected certain parts of mathematics and the mathematics community, the influence of Klein is also global and foundational. The legacy of Klein lives on as it is witnessed in the popularity of his books, the con- tinuing and far-reaching influence of the general philosophy of the Erlangen program, interaction between mathematics and physics, and the theory of Kleinian groups. In some sense, Klein did not belong to his generation but was ahead of time. 2 A nontrivial birth It is customary that kings are born at special times and places. Maybe their destiny gives them something extra to start with. The mostly kingly mathematician, Felix Klein, was born in the night of April 25, 1849, in Düsseldorf in the Rhineland when [18, p. i] there was anxiety in the house of the secretary to the Regierungspräsident. Without, the canon thundered on the barricades raised by the insurgent Rhinelanders against their hated Prussian rulers. Within, although all had been prepared for flight, there was no thought of departure; [. . . ] His birth was marked by the final crushing of the revolution of 1848; his life measured the domination of Prussia over Germany, typifies all that was best and nobest in that domination. Shortly after his birth, his hometown and the nearby region became the battle- ground of the last war of the 1848 Revolution in the German states. In the twenty years that followed Klein’s birth, Prussia became a major power in Europe, and there were almost constantly conflicts and turmoils, culminating in the Franco–Prussian war, with a crushing victory over France. Later Klein served in the voluntary corps of emergency workers and he witnessed firsthand the battle sites of Metz and Sedan, where the Empire of Louis-Napoléon Bonaparte (Napoléon III) finally collapsed and was replaced by the Third Republic. In Germany the Second German Empire began, with Otto von Bismarck as a powerful first chancellor.3 3Otto von Bismarck was a conservative German statesman who dominated European affairs from the 1860s to his dismissal in 1890. After a series of short victorious wars he unified numerous German states into a power- ful German Empire under Prussian leadership, then created a “balance of power” that preserved peace in Europe from 1871 until 1914.
- 53. 2 Felix Klein: his life and mathematics 31 Klein’s academic life practically coincided with the rise and fall of the second Reich. All these historical events influenced Klein’s character and his perspective on mathematics and the mathematics community. 3 Education Overall, Klein had a fairly normal life and uninterrupted education. He attended the gymnasium in Düsseldorf, and did not find the Latin and Greek classics exciting. Klein entered the University of Bonn in 1865 at the age of 16 and found the courses there, with emphasis on natural sciences, ideally suited to him. His university education at Bonn contributed significantly to his universalist outlook with a wide variety of subjects including mathematics, physics, botany, chemistry, zoology, and mineralogy, and he participated in all five sections of the Bonn Natural Sciences Seminar. In mathematics, Klein took some courses with the distinguished analyst Rudolf Lipschitz, including analytic geometry, number theory, differential equations, me- chanics and potential theory. But Lipschitz was just an ordinary teacher to Klein. When he entered the University of Bonn, Klein aspired to become a physicist and studied with Plücker, a gifted experimental physicist and geometer. Plücker picked Klein to be an assistant for the laboratory courses in physics when Klein was only in his second semester. The interaction with Plücker probably had the most important influence on Klein in his formative years. By the time Klein met Plücker in 1866, Plücker’s interests had returned to geom- etry after having worked exclusively on physics for nearly twenty years, and he was writing a two-volume book on line geometry titled “Neue Geometrie des Raumes.” When he died unexpectedly in May 1868, Plücker had only finished the first vol- ume. As a student of Plücker, the death of Plücker provided a uniquely challenging opportunity for Klein: to finish the second volume and edit the work of his teacher. Originally, the rising and inspiring geometer Clebsch at Göttingen was respon- sible for completing the book of Plücker. But he delegated this task to Klein. This seemingly impossible task changed the life of Klein in many ways. First, it gave Klein a good chance to learn line geometry solidly, which played an important role in his future work with Lie and eventually in the Erlangen program. Second, it also brought him into close contact with Clebsch and his school which in- cluded many distinguished mathematicians such as Gordan, Max Noether, Alexander von Brill, etc. Through them, Klein learned and worked on Riemann’s theory of func- tions, which eventually became Klein’s favorite subject. He also became Clebsch’s natural successor in many other ways. For example, he took over many students of Clebsch and the journal Mathematische Annalen started by Clebsch. Klein obtained his Ph.D. degree in December 1868 with Rudolf Lipschitz as a joint (or nominal) advisor.
- 54. 32 Lizhen Ji 4 Three people who had most influence on Klein There are three people who played a crucial role in the informative years of Klein. The first person was Plücker, his teacher during his college years. Physics and the interaction between mathematics and physics had always played an important role in the mathematical life of Klein. It is reasonable to guess that this might have something to do with the influence of Plücker. For most mathematicians, Plücker is well known for Plücker coordinates in projective geometry. But he started as a physicist. In fact, In 1836 at the age of 35, he was appointed professor of physics at the University of Bonn and he started investigations of cathode rays that led eventually to the discovery of the electron. Almost 30 years later, he switched to and concentrated on geometry. Klein had written many books, some of which are still popular. Probably the most original book by him is “Über Riemann’s Theorie der Algebraischen Functionen und ihre Integrale”, published in 1882, where he tried to explain and justify Riemann’s work on functions on Riemann surfaces, in particular, the Dirichlet principle, using ideas from physics. Klein wrote [12, p. 178], in modern mathematical literature, it is altogether unusual to present, as occurs in my booklet, general physical and geometrical deliberations in naive anschaulicher form which later find their firm support in exact mathematical proofs. [. . . ] I consider it unjustifiable that most mathematicians suppress their intuitive thoughts and only publish the necessary, strict (and mostly arithmetical) proofs [. . . ] I wrote my work on Riemann precisely as a physicist, unconcerned with all the careful considerations that are usual in a detailed mathematical treatment, and, precisely because of this, I have also received the approval of various physicists. In a biography of Klein [6], Halsted wrote: “The death of Plücker on May 22nd 1868 closed this formative period, of which the influence on Klein cannot be overes- timated. So mighty is the power of contact with the living spirit of research, of taking part in original work with a master, of sharing in creative authorship, that anyone who has once come intricately in contact with a producer of first rank must have had his whole mentality altered for the rest of his life. The gradual development, high attain- ment, and then continuous achievement of Felix Klein are more due to Plücker than to all other influences combined. His very mental attitude in the world of mathematics constantly recalls his great maker.” The second person was Alfred Clebsch, another important teacher of Klein. Cleb- sch could be considered as a postdoctor mentor of Klein. After obtaining his Ph.D. in 1868, Klein went to Göttingen to work under Clebsch for eight months. When Klein first met him, Clebsch was only 35 years old and was already a famous teacher and leader of a new school in algebraic geometry. Clebsch made important contributions to algebraic geometry and invariant theory. Before Göttingen, he taught in Berlin and Karlsruhe. His collaboration with Paul Gor- dan in Giessen led to the introduction of the Clebsch–Gordan coefficients for spher- ical harmonics, which are now widely used in representation theory of compact Lie groups and in quantum mechanics, and to find the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representa-
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- 56. MacNALLY, LEONARD (1752-1820), Irish informer, was born in Dublin, the son of a merchant. In 1776 he was called to the Irish, and in 1783 to the English bar. He supported himself for some time in London by writing plays and editing the Public Ledger. Returning to Dublin, he entered upon a systematic course of informing against the members of the revolutionary party, for whom his house had become the resort. He also betrayed to the government prosecutors political clients whom he defended eloquently in the courts. He made a fine defence for Robert Emmet and cheered him in his last hours, although before appearing in court he had sold, for £200, the contents of his brief to the lawyers for the Crown. After living a professed Protestant all his life, he received absolution on his deathbed from a Roman Catholic priest. He died on the 13th of February 1820. MACNEE, SIR DANIEL (1806-1882), Scottish portrait painter, was born at Fintry in Stirlingshire. At the age of thirteen he
- 57. was apprenticed, along with Horatio Macculloch and Leitch the water-colour painter, to John Knox, a landscapist of some repute. He afterwards worked for a year as a lithographer, was employed by the Smiths of Cumnock to paint the ornamental lids of their planewood snuff-boxes, and, having studied in Edinburgh at the “Trustees’ Academy,” supporting himself meanwhile by designing and colouring book illustrations for Lizars the engraver, he established himself as an artist in Glasgow, where he became a fashionable portrait painter. He was in 1829 admitted a member of the Royal Scottish Academy; and on the death of Sir George Harvey in 1876 he was elected president, and received the honour of knighthood. From this period till his death, on the 18th of January 1882, he resided in Edinburgh, where his genial social qualities and his inimitable powers as a teller of humorous Scottish anecdote rendered him popular. MACNEIL, HERMON ATKINS (1866- ), American sculptor, was born at Chelsea, Massachusetts. He was an instructor in industrial art at Cornell University in 1886-1889, and was then a pupil of Henri M. Chapu and Falguière in Paris. Returning to America, he aided Philip Martiny in the preparation of sketch models for the Columbian exposition, and in 1896 he won the Rinehart scholarship, passing four years (1896-1900) in Rome. In 1906 he became a National Academician. His first important work was “The Moqui Runner,” which was followed by “A Primitive Chant,” and “The Sun
- 58. Vow,” all figures of the North-American Indian. A “Fountain of Liberty,” for the St Louis exposition, and other Indian themes came later; his “Agnese” and his “Beatrice,” two fine busts of women, also deserve mention. His principal work is the sculpture for a large memorial arch, at Columbus, Ohio, in honour of President McKinley. In 1909 he won in competition a commission for a large soldiers’ and sailors’ monument in Albany, New York. His wife, Carol Brooks MacNeil, also a sculptor of distinction, was a pupil of F. W. MacMonnies. McNEILE, HUGH (1795-1879), Anglican divine, younger son of Alexander McNeile (or McNeill), was born at Ballycastle, Co. Antrim, on the 15th of July 1795. He graduated at Trinity College, Dublin, in 1810. His handsome presence, and his promise of exceptional gifts of oratory, led a wealthy uncle, Major-General Daniel McNeill, to adopt him as his heir; and he was destined for a parliamentary career. During a stay at Florence, Hugh McNeile became temporarily intimate with Lord Byron and Madame de Staël. On returning home, he determined to abandon the prospect of political distinction for the clerical profession, and was disinherited. In 1820 he was ordained, and after holding the curacy of Stranorlar, Co. Donegal, for two years, was appointed to the living of Albury, Surrey, by Henry Drummond.
- 59. Edward Irving endeavoured, not without success at first, to draw McNeile into agreement with his doctrine and aims. Irving’s increasing extravagance, however, soon alienated McNeile. His preaching now attracted much attention; in London he frequently was heard by large congregations. In 1834 he accepted the incumbency of St Jude’s, Liverpool, where for the next thirty years he wielded great political as well as ecclesiastical influence. He repudiated the notion that a clergyman should be debarred from politics, maintaining at a public meeting that “God when He made the minister did not unmake the citizen.” In 1835 McNeile entered upon a long contest, in which he was eventually successful, with the Liverpool corporation, which had been captured by the Whigs, after the passing of the Municipal Reform Act. A proposal was carried that the elementary schools under the control of the corporation should be secularized by the introduction of what was known as the Irish National System. The threatened withdrawal of the Bible as the basis of denominational religious teaching was met by a fierce agitation led by McNeile, who so successfully enlisted public support that before the new system could be introduced every child was provided for in new Church of England schools established by public subscriptions. At the same time he conducted a campaign which gradually reduced the Whig element in the council, till in 1841 it almost entirely disappeared. To his influence was also attributed the defeat of the Liberal parliamentary candidates in the general election of 1837, followed by a long period of Conservative predominance in Liverpool politics. McNeile had the Irish Protestant’s horror of Romanism, which he constantly denounced in the pulpit and on the platform; and Macaulay, speaking in the House of Commons on the Maynooth endowment in April 1845, singled him out for attack as the most powerful representative of uncompromising Protestant
- 60. opinion in the country. As the Tractarian movement in the Church of England developed, he became one of its most zealous opponents and the most conspicuous leader of the evangelical party. In 1840 he published a volume of Lectures on the Church of England, and in 1846 (the year after Newman’s secession to Rome) The Church and the Churches, in which he maintained with much dialectical skill the evangelical doctrine of the “invisible Church” in opposition to the teaching of Newman and Pusey. Hugh McNeile was in close sympathy with the philanthropic work as well as the religious views of the 7th earl of Shaftesbury, who more than once tried to persuade Lord Palmerston to raise him to the episcopal bench. But although Palmerston usually followed the advice of Shaftesbury in the appointment of bishops, he would not consent to the elevation to the House of Lords of so powerful a political opponent as McNeile, whom Lord John Russell had accused of frustrating for thirty years the education policy of the Liberal party. In 1860 he was appointed a canon of Chester; and in 1868 Disraeli appointed him dean of Ripon. This preferment he resigned in 1875, and he lived in retirement at Bournemouth till his death on the 28th of January 1879. McNeile married, in 1822, Anne, daughter of William Magee, archbishop of Dublin, and aunt of William Connor Magee, archbishop of York, by whom he had a large family. Although a vehement controversialist, Hugh McNeile was a man of simple and sincere piety of character. Sir Edward Russell, an opponent alike of his religious and his political opinions, bears witness to the deep spirituality of his teaching, and describes him as an absolutely unique personality. “He made himself leader of the Liverpool people, and always led with calm and majesty in the most excited times. His eloquence was grave, flowing, emphatic—had a dignity in delivery, a perfection of elocution, that only John Bright
- 61. equalled in the latter half of the 19th century. Its fire was solemn force. McNeile’s voice was probably the finest organ ever heard in public oratory. His action was as graceful as it was expressive. He ruled an audience.” See J. A. Picton, Memorials of Liverpool, vol. i. (1873); Sir Edward Russell, “The Religious Life of Liverpool,” in the Sunday Magazine (June 1905); Charles Bullock, Hugh McNeile and Reformation Truth. (R. J. M.) MACNEILL, HECTOR (1746-1818), Scottish poet, was born near Roslin, Midlothian, on the 22nd of October 1746, the son of an impoverished army captain. He went to Bristol as a clerk at the age of fourteen, and soon afterwards was despatched to the West Indies. From 1780 to 1786 he acted as assistant secretary on board the flagships of Admiral Geary and Sir Richard Bickerton (1727- 1792). Most of his later life was spent in Scotland, and it was in the house of a friend at Stirling that he wrote most of his songs and his Scotland’s Skaith, or the History of Will and Jean (1795), a narrative poem intended to show the deteriorating influences of whisky and pothouse politics. A sequel, The Waes of War, appeared next year. In 1800 he published The Memoirs of Charles Macpherson, Esq., a novel understood to be a narrative of his own hardships and adventures. A complete edition of the poems he wished to own
- 62. appeared in 1812. His songs “Mary of Castlecary,” “Come under my plaidy,” “My boy Tammy,” “O tell me how for to woo,” “I lo’ed ne’er a lassie but ane,” “The plaid amang the hether,” and “Jeanie’s black e’e,” are notable for their sweetness and simplicity. He died at Edinburgh on the 15th of March 1818. MACOMB, a city and the county-seat of McDonough county, Illinois, U.S.A., in the W. part of the state, about 60 m. S.W. of Peoria. Pop. (1890), 4052; (1900), 5375 (232 foreign-born); (1910), 5774. Macomb is served by the Chicago, Burlington Quincy, and the Macomb Western Illinois railways. The city is the seat of the Western Illinois state normal school (opened in 1902), and has a Carnegie library and a city park. Clay is found in the vicinity, and there are manufactures of pottery, bricks, c. The city was founded in 1830 as the county-seat of McDonough county, and was called Washington by the settlers, but the charter of incorporation, also granted in 1830, gave it the present name in honour of General Alexander Macomb. Macomb was first chartered as a city in 1856.
- 63. MACOMER, a village of Sardinia in the province of Cagliari, from which it is 95 m. N.N.W. by rail, and the same distance S.W. of Golfo degli Aranci. Pop. (1901), 3488. It is situated 1890 ft. above sea-level on the southern ascent to the central plateau (the Campeda) of this part of Sardinia; and it is the junction of narrow- gauge lines branching from the main line eastwards to Nuoro and westwards to Bosa. The old parish church of S. Pantaleone has three Roman mile-stones in front of it, belonging to the Roman high-road from Carales to Turris Libisonis. The modern high-road follows the ancient. The district, especially the Campeda, is well fitted for grazing and horse and cattle breeding, which is carried on to a considerable extent. It is perhaps richer in nuraghi than any other part of Sardinia. MACON, NATHANIEL (1758-1837), American political leader, was born at Macon Manor, Warren county, North Carolina, on the 17th of December 1758. He studied at the college of New Jersey
- 64. (now Princeton University) from 1774 to 1776, when the institution was closed on account of the outbreak of the War of Independence; served for a short time in a New Jersey militia company; studied law at Bute Court-house, North Carolina, in 1777-1780, at the same time managing his tobacco plantation; was a member of a Warren county militia company in 1780-1782, and served in the North Carolina Senate in 1781-1785. In 1786 he was elected to the Continental Congress, but declined to serve. In 1791-1815 he was a member of the national House of Representatives, and in 1815-1828 of the United States Senate. Macon’s point of view was always local rather than national. He was essentially a North Carolinian first, and an American afterwards; and throughout his career he was an aggressive advocate of state sovereignty and an adherent of the doctrines of the “Old Republicans.” He at first opposed the adoption of the Federal constitution of 1787, as a member of the faction led by Willie Jones (1731-1801) of Halifax, North Carolina, but later withdrew his opposition. In Congress he denounced Hamilton’s financial policy, opposed the Jay Treaty (1795) and the Alien and Sedition Acts, and advocated a continuance of the French alliance of 1778. His party came into power in 1801, and he was Speaker of the house from December 1801 to October 1807. At first he was in accord with Jefferson’s administration; he approved the Louisiana Purchase, and as early as 1803 advocated the purchase of Florida. For a number of years, however, he was politically allied with John Randolph.1 As speaker, in spite of strong opposition, he kept Randolph at the head of the important committee on Ways and Means from 1801 to 1806; and in 1805-1808, with Randolph and Joseph H. Nicholson (1770-1817) of Maryland, he was a leader of the group of about ten independents, called the “Quids,” who strongly criticized Jefferson and opposed the presidential candidature
- 65. of Madison. By 1809, however, Macon was again in accord with his party, and during the next two years he was one of the most influential of its leaders. In December 1809 he introduced resolutions which combined the ideas of Peter Early (1773-1817) of Georgia, David R. Williams (1776-1830) of South Carolina, and Samuel W. Dana (1757-1830) of Connecticut with his own. The resolutions recommended the complete exclusion of foreign war vessels from United States ports and the suppression of illegal trade carried on by foreign merchants under the American flag. The substance of these resolutions was embodied in the “Macon Bill, No. 1,” which passed the House but was defeated in the Senate. On the 7th of April 1810 Macon reported from committee the “Macon Bill, No. 2,” which had been drawn by John Taylor (1770-1832) of South Carolina, and was not actively supported by him. This measure (amended) became law on the 1st of May, and provided for the repeal of the Non-Intercourse Act of 1809, authorized the president, “in case either Great Britain or France shall before the 3rd day of March next so revoke or modify her edicts as that they shall cease to violate the neutral commerce of the United States,” to revive non- intercourse against the other, and prohibited British and French vessels of war from entering American waters. In 1812 Macon voted for the declaration of war against Great Britain, and later was chairman of the Congressional committee which made a report (July 1813) condemning Great Britain’s conduct of the war. He opposed the Bank Act of 1816, the “internal improvements” policy of Calhoun (in the early part of his career) and Clay, and the Missouri Compromise, his speech against the last being especially able. In 1824 Macon received the electoral vote of Virginia for the vice- presidency, and in 1826-1828 was president pro tempore of the Senate. He was president of the North Carolina constitutional
- 66. convention in 1835, and was an elector on the Van Buren ticket in 1836. He died at his home, Buck Springs, Warren county, North Carolina, on the 29th of June 1837. See William E. Dodd, The Life of Nathaniel Macon (Raleigh, N.C., 1903); E. M. Wilson, The Congressional Career of Nathaniel Macon (Chapel Hill, N.C., 1900). 1 Their names are associated in Randolph-Macon College, named in their honour in 1830. MÂCON, a town of east-central France, capital of the department of Saône-et-Loire, 45 m. N. of Lyons on the Paris-Lyon railway. Pop. (1906), 16,151. Mâcon is situated on the right bank of the Saône facing the plain of the Bresse; a bridge of twelve arches connects it with the suburb of St Laurent on the opposite bank. The most prominent building is the modern Romanesque church of St Pierre, a large three-naved basilica, with two fine spires. Of the old cathedral of St Vincent (12th and 13th centuries), destroyed at the Revolution, nothing remains but the Romanesque narthex, now used as a chapel, the façade and its two flanking towers. The hôtel de ville contains a library, a theatre and picture-gallery. Opposite to it stands a statue of the poet Alphonse Lamartine, a native of the town. Mâcon is the seat of a prefecture, and has tribunals of first
- 67. instance and of commerce, and a chamber of commerce. There are lycées and training colleges. Copper-founding is an important industry; manufactures include casks, mats, rope and utensils for the wine-trade. The town has a large trade in wine of the district, known as Mâcon. It is a railway centre of considerable importance, being the point at which the line from Paris to Marseilles is joined by that from Mont Cenis and Geneva, as well as by a branch from Moulins. Mâcon (Matisco) was an important town of the Aedui, but under the Romans it was supplanted by Autun and Lyons. It suffered a succession of disasters at the hands of the Germans, Burgundians, Vandals, Huns, Hungarians and even of the Carolingian kings. In the feudal period it was an important countship which in 1228 was sold to the king of France, but more than once afterwards passed into the possession of the dukes of Burgundy, until the ownership of the French crown was established in the time of Louis XI. In the 16th century Mâcon became a stronghold of the Huguenots, but afterwards fell into the hands of the League, and did not yield to Henry IV. until 1594. The bishopric, created by King Childebert, was suppressed in 1790. MACON, a city and the county-seat of Bibb county, Georgia, U.S.A., in the central part of the state, on both sides of the Ocmulgee river (at the head of navigation), about 90 m. S.S.E. of
- 68. Atlanta. Pop. (1900), 23,272, of whom 11,550 were negroes; (1910 census) 40,665. Macon is, next to Atlanta, the most important railway centre in the state, being served by the Southern, the Central of Georgia, the Georgia, the Georgia Southern Florida, the Macon Dublin Savannah, and the Macon Birmingham railways. It was formerly an important river port, especially for the shipment of cotton, but lost this commercial advantage when railway bridges made the river impassable. It is, however, partially regaining the river trade in consequence of the compulsory substitution of drawbridges for the stationary railway bridges. The city is the seat of the Wesleyan female college (1836), which claims to be the first college in the world chartered to grant academic degrees to women; Mercer university (Baptist), which was established in 1833 as Mercer Institute at Penfield, became a university in 1837, was removed to Macon in 1871, and controls Hearn Academy (1839) at Cave Spring and Gibson Mercer Academy (1903) at Bowman; the state academy for the blind (1852), St Stanislaus’ College (Jesuit), and Mt de Sales Academy (Roman Catholic) for women. There are four orphan asylums for whites and two for negroes, supported chiefly by the Protestant Episcopal and Methodist Churches, and a public hospital. Immediately east of Macon are two large Indian mounds, and there is a third mound 9 m. south of the city. Situated in the heart of the “Cotton Belt,” Macon has a large and lucrative trade; it is one of the most important inland cotton markets of the United States, its annual receipts averaging about 250,000 bales. The city’s factory products in 1905 were valued at $7,297,347 (33.8% more than in 1900). In the vicinity are large beds of kaolin, 30 m. wide, reaching nearly across the state, and frequently 35 to 70 ft. in depth. Macon is near the fruit-growing region of Georgia, and large quantities of peaches and of garden products are annually shipped from the city.
- 69. Macon (named in honour of Nathaniel Macon) was surveyed in 1823 by order of the Georgia legislature for the county-seat of Bibb county, and received its first charter in 1824. It soon became the centre of trade for Middle Georgia; in 1833 a steamboat line to Darien was opened, and in the following year 69,000 bales of cotton were shipped by this route. During the Civil War the city was a centre for Confederate commissary supplies and the seat of a Treasury depository. In July 1864 General George Stoneman (1822- 1894) with 500 men was captured near the city by the Confederate general, Howell Cobb. Macon was finally occupied by Federal troops under General James H. Wilson (b. 1837) on the 20th of April 1865. In 1900-1910 the area of the city was increased by the annexation of several suburbs. MACPHERSON, SIR DAVID LEWIS (1818-1896), Canadian financier and politician, was born at Castle Leathers, near Inverness, Scotland, on the 12th of September 1818. In 1835 he emigrated to Canada, settling in Montreal, where he built up a large fortune by “forwarding” merchandise. In 1853 he removed to Toronto, and in the same year obtained the contract for building a line of railway from Toronto to Sarnia, a project from which sprang the Grand Trunk railway, in the construction of which line he greatly increased his wealth. In 1864 he was elected to the Canadian parliament as member of the Legislative Council for Saugeen, and on
- 70. the formation of the Dominion, in 1867, was nominated to the Senate. In the following years he published a number of pamphlets on economic subjects, of which the best-known is Banking and Currency (1869). In 1880 he was appointed Speaker of the Senate, and from October 1883 till 1885 was minister of the interior in the Conservative cabinet. In 1884 he was knighted by Queen Victoria. He died on the 16th of August 1896. MACPHERSON, JAMES (1736-1796), Scottish “translator” of the Ossianic poems, was born at Ruthven in the parish of Kingussie, Inverness, on the 27th of October 1736. He was sent in 1753 to King’s College, Aberdeen, removing two years later to Marischal College. He also studied at Edinburgh, but took no degree. He is said to have written over 4000 lines of verse while a student, but though some of this was published, notably The Highlander (1758), he afterwards tried to suppress it. On leaving college he taught in the school of his native place. At Moffat he met John Home, the author of Douglas, for whom he recited some Gaelic verses from memory. He also showed him MSS. of Gaelic poetry, supposed to have been picked up in the Highlands, and, encouraged by Home and others, he produced a number of pieces translated from the Gaelic, which he was induced to publish at Edinburgh in 1760 as Fragments of Ancient Poetry collected in the Highlands of Scotland. Dr Hugh Blair, who was a firm believer in the authenticity
- 71. of the poems, got up a subscription to allow Macpherson to pursue his Gaelic researches. In the autumn he set out to visit western Inverness, the islands of Skye, North and South Uist and Benbecula. He obtained MSS. which he translated with the assistance of Captain Morrison and the Rev. A. Gallie. Later in the year he made an expedition to Mull, when he obtained other MSS. In 1761 he announced the discovery of an epic on the subject of Fingal, and in December he published Fingal, an Ancient Epic Poem in Six Books, together with Several Other Poems composed by Ossian, the Son of Fingal, translated from the Gaelic Language, written in the musical measured prose of which he had made use in his earlier volume. Temora followed in 1763, and a collected edition, The Works of Ossian, in 1765. The genuineness of these so-called translations from the works of a 3rd-century bard was immediately challenged in England, and Dr Johnson, after some local investigation, asserted (Journey to the Western Islands of Scotland, 1775) that Macpherson had only found fragments of ancient poems and stories, which he had woven into a romance of his own composition. Macpherson is said to have sent Johnson a challenge, to which Johnson replied that he was not to be deterred from detecting what he thought a cheat by the menaces of a ruffian. Macpherson never produced his originals, which he refused to publish on the ground of the expense. In 1764 he was made secretary to General Johnstone at Pensacola, West Florida, and when he returned, two years later, to England, after a quarrel with Johnstone, he was allowed to retain his salary as a pension. He occupied himself with writing several historical works, the most important of which was Original Papers, containing the Secret History of Great Britain from the Restoration to the Accession of the House of Hanover; to which are prefixed Extracts from the Life of
- 72. James II., as written by himself (1775). He enjoyed a salary for defending the policy of Lord North’s government, and held the lucrative post of London agent to Mahommed Ali, nabob of Arcot. He entered parliament in 1780, and continued to sit until his death. In his later years he bought an estate, to which he gave the name of Belville, in his native county of Inverness, where he died on the 17th of February 1796. After Macpherson’s death, Malcolm Laing, in an appendix to his History of Scotland (1800), propounded the extreme view that the so-called Ossianic poems were altogether modern in origin, and that Macpherson’s authorities were practically non-existent. For a discussion of this question see Celt: Scottish Gaelic Literature. Much of Macpherson’s matter is clearly his own, and he confounds the stories belonging to different cycles. But apart from the doubtful morality of his transactions he must still be regarded as one of the great Scottish writers. The varied sources of his work and its worthlessness as a transcript of actual Celtic poems do not alter the fact that he produced a work of art which by its deep appreciation of natural beauty and the melancholy tenderness of its treatment of the ancient legend did more than any single work to bring about the romantic movement in European, and especially in German, literature. It was speedily translated into many European languages, and Herder and Goethe (in his earlier period) were among its profound admirers. Cesarotti’s Italian translation was one of Napoleon’s favourite books. Authorities.—For Macpherson’s life, see The Life and Letters of James Macpherson ... (1894, new ed., 1906), by T. Bailey Saunders, who has laboured to redeem his character from the suspicions generally current with English readers. The antiquity
- 73. of the Ossianic poems was defended in the introduction by Archibald Clerk to his edition of the Poems of Ossian (1870). Materials for arriving at a decision by comparison with undoubtedly genuine fragments of the Ossianic legend are available in The Book of the Dean of Lismore, Gaelic verses, collected by J. McGregor, dean of Lismore, in the early 16th century (ed. T. McLauchlan, 1862); the Leabhar na Feinne (1871) of F. J. Campbell, who also discusses the subject in Popular Tales of the Western Highlands, iv. (1893). See also L. C. Stern, “Die ossianische Heldenlieder” in Zeitschrift für vergleichende Litteratur-geschichte (1895; Eng. trans. by J. L. Robertson in Trans. Gael. Soc. of Inverness, xxii., 1897-1898); Sir J. Sinclair, A Dissertation on the Authenticity of the Poems of Ossian (1806); Transactions of the Ossianic Society (Dublin, 1854-1861); Cours de littérature celtique, by Arbois de Jubainville, editor of the Revue celtique (1883, c.); A. Nutt, Ossian and the Ossianic Literature (1899), with a valuable bibliographical appendix; J. S. Smart, James Macpherson: an Episode in Literature (1905). McPHERSON, JAMES BIRDSEYE (1828-1864), American soldier, was born at Sandusky, Ohio, on the 14th of November 1828. He entered West Point at the age of twenty-one, and graduated (1853) at the head of his class, which included Sheridan, Schofield
- 74. and Hood. He was employed at the military academy as instructor of practical military engineering (1853). A year later he was sent to engineer duty at New York, and in 1857, after constructing Fort Delaware, he was sent as superintending engineer to San Francisco, becoming 1st lieutenant in 1858. He was promoted captain during the first year of the Civil War, and towards the close of 1861 became lieutenant-colonel and aide-de-camp to General Halleck, who in the spring of 1862 sent him to General Grant as chief engineer. He remained with Grant during the Shiloh campaign, and acted as engineer adviser to Halleck during the siege operations against Corinth in the summer of 1862. In October he distinguished himself in command of an infantry brigade at the battle of Corinth, and on the 8th of this month was made major-general of volunteers and commander of a division. In the second advance on Vicksburg (1863) McPherson commanded the XVII. corps, fought at Port Gibson, Raymond and Jackson, and after the fall of Vicksburg was strongly recommended by Grant for the rank of brigadier-general in the regular army, to which he was promoted on the 1st of August 1863. He commanded at Vicksburg until the following spring. He was about to go on leave of absence in order to be married in Baltimore when he received his nomination to the command of the Army of the Tennessee, Grant’s and Sherman’s old army, which was to take part under Sherman’s supreme command in the campaign against Atlanta (1864). This nomination was made by Sherman and entirely approved by Grant, who had the highest opinion of McPherson’s military and personal qualities. He was in command of his army at the actions of Resaca, Dallas, Kenesaw Mountain and the battles about Atlanta. On the 22nd of July, when the Confederates under his old classmate Hood made a sudden and violent attack on the lines held by the Army of the Tennessee, McPherson rode up, in the
- 75. woods, to the enemy’s firing line and was killed. He was one of the most heroic figures of the American Civil War, and Grant is reported to have said when he heard of McPherson’s death, “The country has lost one of its best soldiers, and I have lost my best friend.” MACQUARIE, a British island in the South Pacific Ocean, in 54° 49′ S. and 159° 49′ E. It is about 20 m. long, and covered with a grassy vegetation, with some trees or shrubs in the sheltered places which afford food to a parrot of the genus Cyanorhamphus, allied to those of the Auckland Islands. Although it has no settled population, Macquarie is constantly visited by sailors in quest of the seals which abound in its waters. MACRAUCHENIA, a long-necked and long-limbed, three- toed South American ungulate mammal, typifying the suborder Litopterna (q.v.).
- 76. MACREADY, WILLIAM CHARLES (1793-1873), English actor, was born in London on the 3rd of March 1793, and educated at Rugby. It was his intention to go up to Oxford, but in 1809 the embarrassed affairs of his father, the lessee of several provincial theatres, called him to share the responsibilities of theatrical management. On the 7th of June 1810 he made a successful first appearance as Romeo at Birmingham. Other Shakespearian parts followed, but a serious rupture between father and son resulted in the young man’s departure for Bath in 1814. Here he remained for two years, with occasional professional visits to other provincial towns. On the 16th of September 1816, Macready made his first London appearance at Covent Garden as Orestes in The Distressed Mother, a translation of Racine’s Andromaque by Ambrose Philips. Macready’s choice of characters was at first confined chiefly to the romantic drama. In 1818 he won a permanent success in Isaac Pocock’s (1782-1835) adaptation of Scott’s Rob Roy. He showed his capacity for the highest tragedy when he played Richard III. at Covent Garden on the 25th of October 1819. Transferring his services to Drury Lane, he gradually rose in public favour, his most conspicuous success being in the title-rôle of Sheridan Knowles’s William Tell (May 11, 1825). In 1826 he completed a successful engagement in America, and in 1828 his performances met with a very flattering reception in Paris. On the 15th of December 1830 he
- 77. appeared at Drury Lane as Werner, one of his most powerful impersonations. In 1833 he played in Antony and Cleopatra, in Byron’s Sardanapalus, and in King Lear. Already Macready had done something to encourage the creation of a modern English drama, and after entering on the management of Covent Garden in 1837 he introduced Robert Browning’s Strafford, and in the following year Bulwer’s Lady of Lyons and Richelieu, the principal characters in which were among his most effective parts. On the 10th of June 1838 he gave a memorable performance of Henry V., for which Stanfield prepared sketches, and the mounting was superintended by Bulwer, Dickens, Forster, Maclise, W. J. Fox and other friends. The first production of Bulwer’s Money took place under the artistic direction of Count d’Orsay on the 8th of December 1840, Macready winning unmistakable success in the character of Alfred Evelyn. Both in his management of Covent Garden, which he resigned in 1839, and of Drury Lane, which he held from 1841 to 1843, he found his designs for the elevation of the stage frustrated by the absence of adequate public support. In 1843-1844 he made a prosperous tour in the United States, but his last visit to that country, in 1849, was marred by a riot at the Astor Opera House, New York, arising from the jealousy of the actor Edwin Forrest, and resulting in the death of seventeen persons, who were shot by the military called out to quell the disturbance. Macready took leave of the stage in a farewell performance of Macbeth at Drury Lane on the 26th of February 1851. The remainder of his life was spent in happy retirement, and he died at Cheltenham on the 27th of April 1873. He had married, in 1823, Catherine Frances Atkins (d. 1852). Of a numerous family of children only one son and one daughter survived. In 1860 he married Cecile Louise Frederica Spencer (1827-1908), by whom he had a son.
- 78. Macready’s performances always displayed fine artistic perceptions developed to a high degree of perfection by very comprehensive culture, and even his least successful personations had the interest resulting from thorough intellectual study. He belonged to the school of Kean rather than of Kemble; but, if his tastes were better disciplined and in some respects more refined than those of Kean, his natural temperament did not permit him to give proper effect to the great tragic parts of Shakespeare, King Lear perhaps excepted, which afforded scope for his pathos and tenderness, the qualities in which he specially excelled. With the exception of a voice of good compass and capable of very varied expression, Macready had no especial physical gifts for acting, but the defects of his face and figure cannot be said to have materially affected his success. See Macready’s Reminiscences, edited by Sir Frederick Pollock, 2 vols. (1875); William Charles Macready, by William Archer (1890). MACROBIUS, AMBROSIUS THEODOSIUS, Roman grammarian and philosopher, flourished during the reigns of Honorius and Arcadius (395-423). He himself states that he was not a Roman, but there is no certain evidence whether he was of Greek or perhaps African descent. He is generally supposed to have been praetorian praefect in Spain (399), proconsul of Africa (410), and
- 79. lord chamberlain (422). But the tenure of high office at that date was limited to Christians, and there is no evidence in the writings of Macrobius that he was a Christian. Hence the identification is more than doubtful, unless it be assumed that his conversion to Christianity was subsequent to the composition of his books. It is possible, but by no means certain, that he was the Theodosius to whom Avianus dedicates his fables. The most important of his works is the Saturnalia, containing an account of the discussions held at the house of Vettius Praetextatus (c. 325-385) during the holiday of the Saturnalia. It was written by the author for the benefit of his son Eustathius (or Eustachius), and contains a great variety of curious historical, mythological, critical and grammatical disquisitions. There is but little attempt to give any dramatic character to the dialogue; in each book some one of the personages takes the leading part, and the remarks of the others serve only as occasions for calling forth fresh displays of erudition. The first book is devoted to an inquiry as to the origin of the Saturnalia and the festivals of Janus, which leads to a history and discussion of the Roman calendar, and to an attempt to derive all forms of worship from that of the sun. The second book begins with a collection of bons mots, to which all present make their contributions, many of them being ascribed to Cicero and Augustus; a discussion of various pleasures, especially of the senses, then seems to have taken place, but almost the whole of this is lost. The third, fourth, fifth and sixth books are devoted to Virgil, dwelling respectively on his learning in religious matters, his rhetorical skill, his debt to Homer (with a comparison of the art of the two) and to other Greek writers, and the nature and extent of his borrowings from the earlier Latin poets. The latter part of the third book is taken up with a dissertation upon luxury and the sumptuary laws intended
- 80. to check it, which is probably a dislocated portion of the second book. The seventh book consists largely of the discussion of various physiological questions. The value of the work consists solely in the facts and opinions quoted from earlier writers, for it is purely a compilation, and has little in its literary form to recommend it. The form of the Saturnalia is copied from Plato’s Symposium and Gellius’s Noctes atticae; the chief authorities (whose names, however, are not quoted) are Gellius, Seneca the philosopher, Plutarch (Quaestiones conviviales), Athenaeus and the commentaries of Servius (excluded by some) and others on Virgil. We have also two books of a commentary on the Somnium Scipionis narrated by Cicero in his De republica. The nature of the dream, in which the elder Scipio appears to his (adopted) grandson, and describes the life of the good after death and the constitution of the universe from the Stoic point of view, gives occasion for Macrobius to discourse upon many points of physics in a series of essays interesting as showing the astronomical notions then current. The moral elevation of the fragment of Cicero thus preserved to us gave the work a popularity in the middle ages to which its own merits have little claim. Of a third work, De differentiis et societatibus graeci latinique verbi, we only possess an abstract by a certain Johannes, identified with Johannes Scotus Erigena (9th century). See editions by L. von Jan (1848-1852, with bibliog. of previous editions, and commentary) and F. Eyssenhardt (1893, Teubner text); on the sources of the Saturnalia see H. Linke (1880) and G. Wissowa (1880). The grammatical treatise will be found in Jan’s edition and H. Keil’s Grammatici latini, v.; see also G. F. Schömann, Commentatio macrobiana (1871).
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