Definición y uso

Anwendungen,

Leipzig, Teubner, 1893, and

Geometrie der Beriihrungstrans­

formationen,

Leipzig, Teubner, 1896. The striking similarities oflanguage, and even of style, of all six books and of Lie's papers suggest that in all cases he was the chief writer, or that his influence was so great that it even determined the style of the writing.

It can be said that Lie was one of the last great mathematicians of the nineteenth century. There was something of Gauss or Riemann in his scientific profile (although in human terms Gauss, Riemann, and Lie were quite different personalities). Like his great precursors, Lie hardly needed a milieu: of course he valued pupils, but he took nothing from them and gave his ideas generously to the young mathematicians he met on his way.

The nineteenth century gave birth to the legend of the lonely genius-com­

poser, philosopher, mathematician, or writer-creating values far away from people, by the force of spirit alone.

285

Of course there were really no such great hermits even in the nineteenth century and even the likes of Gauss and Balzac, say, were greatly influenced by their times. On the other hand, it was no accident that the image of the ivory-tower philosopher was particularly dear to the people of the nineteenth century.

As for Klein, he had nothing in common with this nineteenth-century image.

We have already mentioned that immediately after the Franco-Prussian War Klein went to live in Gottingen, to which he was attracted, above all, by his friendship with A. Clebsch and W. Weber; but he did not long remain there. In 1872 there was an opening for a professor at the newly organized mathematics department at Erlangen University, and the influential Clebsch, who held Klein in high esteem, recommended him for the post.

Felix Klein

In Germany at that time a prospective professor was required to deliver a public lecture to the Academic Board of the university on a subject chosen by the candidate himself. The decision whether to offer the post to the candidate was made after the lecture was discussed. The twenty-three-year-old Klein chose as his subject a

Comparative review of recent research in geometry,286

(just as, in a similar situation, eighteen years before, Riemann had spoken

On the hypotheses that lie at the foundations of geometry).

287 The principal ideas of Klein's lecture were described in Chapter 7 above. The lecture soon became known as The Erlangen Program, a title which underscores both the broad vistas opened by Klein for further progress in geometry and his clear stand­

point. It greatly enhanced the author's prestige.

The starting point for the Erlangen program and, at the same time, the application of its ideas, were provided by Klein's and Lie's previous concrete geometric works, beginning with the paper on W-curves and ending with Klein's broad vision of non-Euclidean geometries (these could be spoken of in the plural after the paper "Dber die sogennante nicht-Euklidische Geome­

trie"; see Chapter 4). At the present time all works in this area are considered from the viewpoint of the Erlangen Program. At one time, geometric research in this area was very popular and was dealt with in great detail in university geometry textbooks, particularly German ones.288

Klein's Erlangen years (1872-1875) were remarkably productive in the

130 Felix Klein and Sophus Lie

(a) (b)

FIGURE 34

scientific sense. As a result he received a very flattering invitation to the Technische Hochschule in Munich, which enjoyed a high reputation in Ger­

many and where he worked for five more years. In 1880 Klein joined the geometry department of Leipzig University; in 1886 he yielded his post there to Lie and moved to Gottingen, where he would remain till the end of his life.

The period of Klein's greatest scientific productivity was his time in Munich and his first years in Leipzig. His works dealt with geometry, mechanics, and the theory of functions of a complex variable (theory of automorphic func­

tions). He worked with particular intensity in 1 880-1882, when he developed the (geometric) theory of automorphic functions, following ideas "combining Galois and Riemann," as he explained later, i.e., attempting to imbue Rie­

mann's geometric approaches with group-theoretic ideas derived from Galois.

A significant role in Klein's research was played by pictures of the type shown in Figs. 34(a) or (b) and by (discrete) transformation groups (linear fractional transformations of a complex variable) related to such pictures (transforming Figs. 34(a) and (b) into themselves). These groups proved to be closely linked to certain rectilinear polyhedra and the solution of algebraic equations in radicals (one of Klein's first books was devoted to that range of questions).

289

However, Klein failed to notice that the groups he considered could be interpreted as (discrete) subgroups of the groups of isometries of the Loba­

chevskian plane .P modeled by the interior .Jf' of a circle (or as the half-plane

£). In this model the role of isometries is played by Mobius's circle trans­

formations sending .Jf' (or £) into itself; the "straight lines" of .P are parts of circles and straight lines in ^.Jf' perpendicular to its boundary (or perpendicular to the boundary of the half-plane £); "angles" are ordinary angles, etc. (see Fig. 35, which shows "straight lines" of the Lobachevskian plane passing through the point

A

and not intersecting line

a

)

.290

At the most intense period of his scientific work Klein came across a cycle of articles published in French journals and dealing with much the same subjects. They were written by the young French mathematician Henri

Poin-FIGURE 35

care (1854-1912), who was hardly known at the time.29 1 These articles made a profound impression on Klein. In his very first letter to Poincare Klein communicated to his younger and less famous colleague everything he knew in the field both were interested in-in particular the results of papers he had not yet published. This behavior was in sharp contrast to that of many leading scientists confronted by rivals working in the same field (no need to list them here). But Klein's later activities were marked by acute rivalry with the remarkable French mathematician who was developing the same range of questions at the same time. In particular it was Poincare who first noticed the relationship between circular transformations of the plane and Lobachevsky's non-Euclidean geometry. It is after him that the "models" of hyperbolic geometry292 in the circle and the half-plane Jt', whose "isometries" are circular transformations sending Jt' into itself, are called

Poincare models of Loba­

chevskian geometry.293

The discovery of a connection between the theory of automorphic functions and non-Euclidean geometry impressed Poincare greatly and provided him with a "geometric key" to the entire theory, 294 which he of course immediately put to good use. This circumstance gave Poincare a certain advantage over Klein. Besides, the difference in age was to have a telling effect. Pioncare was five years younger than Klein, and mathematics is the domain of the young. In any case, as a result of the stress of this acute scientific rivalry-which was certainly not conducive to a calm creative effort -Klein suffered a serious nervous breakdown, brought on by exhaustion.

This illness enabled Poincare to celebrate a victory that is reflected in his memoirs of that period, and, in particular, in his famous report on mathe­

matical discovery delivered at the Paris psychological society in 1908.295 Especially popular is Poincare's story of how the idea of the connection between non-Euclidean isometries and linear fractional transformations z' =

(az + b)f(cz + d)

of the complex variable z (i.e., Mobius's circle transforma­

tions296 sending a fixed circle into itself) came to him at the moment he put his foot on the footboard of an omnibus, seemingly without any link to his previous thoughts. On the other hand, Poincare's victory echoes in the

dis-132 Felix Klein and Sophus Lie

appointment exuding from Klein's recollections about this rivalry in

Vorle­

sungen iiber die Entwicklung der Mathematik.

Today, however, conditioned by a long period when the attitude towards joint research is quite different from the attitude typical for the period between the seventeenth and nineteenth centuries, we are perfectly aware of the fact that there are no winners or losers in scientific contests. (Was there a winner in the competition between Newton and Leibniz for priority in the founding of the calculus? It is incredible how much ardor was expended on a rivalry which to us seems senseless.) The view, advanced some twenty years ago by the Moscow mathematician Israel Mois­

seievich Gelfand (b. 1913) in his seminar, to the effect that Klein achieved greater success than Poincare since the two mathematicians' most important ideas and theorems were derived from Klein, seemed paradoxical at the time;

but that viewpoint (as debatable from our vantage point as the opposite view) now finds support in the ubiquity of the term "Klein groups" in modern mathematics and in the frequent references in the latest research297 to Klein's old works on the theory of automorphic functions.

Klein's illness, the result of his exhaustion, unfortunately had a marked effect on his scientific activity, which never again reached the level of the early 1880s. Klein became a mature scientist at an early age-not a rare case in our discipline. At seventeen he was Plucker's assistant at the university; at eighteen he was faced with the formidable task of publishing his teacher's unfinished book, a task assigned to him by the also quite young but already very influential Clebsch. Klein also published his own first independent research at that time. In this respect Klein was different from Lie, who developed relatively late as a professional mathematician and hesitated for a long time before choosing his future occupation. But Klein's creative period was much shorter than Lie's (compare their collected works). In this sense Klein is probably closer to Poncelet (of whom he wrote with such regret), and differs sharply from, say, Mobius, whose productivity as a mathematician hardly seemed to decrease over the years.

But being very active by nature, Klein immediately made up for his reduced creative potential by truly extensive teaching, and by literary, organizational, and administrative activities. In 1872 Clebsch, then thirty-nine years old, suddenly died of diphtheria, and Klein immediately took over

M athematische Annalen,

the journal founded and headed by Clebsch. Klein became the journal's

de facto

editor, and in 1876 its formal editor. Under Klein's leader­

ship M

athematische Annalen

soon gained the reputation of being the world's leading mathematics journal. Klein's books began to appear, beginning in 1882:

Vorlesungen iiber Riemann's Theorie der algebraischen Funktionen und

ihrer lntegrale

(1882),

Vorlesungen iiber das Jkosaeder und die Atiflosung der

Gleichungen vom fonften Grade

(Leipzig, Teubner, 1884; see Note 289), four volumes, written jointly with Karl Immanuel Robert Fricke (1861-1930), Klein's pupil and collaborator, of

Vorlesungen iiber die Theorie der elliptischen

Modulfunktionen

(Bd. 1, 2, Leipzig, Teubner, 1 890 and 1892) and

Vorlesungen

iiber die Theorie der automorphen Funktionen

(Bd. 1, 2, Teubner, 1897 and

1901, 191 1, 1912, the second volume of the book was published in three separate issues); four volumes, written jointly with Arnold Sommerfeld, of

Uber die Theorie des Kreisels

(Leipzig, Teubner, 1897-1910), and others. And throughout his time in Gottingen, mimeographed versions of his lecture courses were regularly published, and many of these were subsequently­

often posthumously-printed as books and translated into many languages.

Thus, for example, the courses mentioned many times in the present book,

In document Guía de referencia básica Ada, 2005 (página 53-57)