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SOPHUS LIE. 



THE 

AMERICAN 
MATHEMATICAL MONTHLY. 

Entered at the Post-office at Springfield, Missouri, as Second-class Mail Matter. 

Vol. VI. APRIL, 1899. No. 4. 

BIOGRAPHY. 



SOPHUS LIE. 



BY GEORGE BRUCE HALSTED. 

J the eighteenth of February, 1899, the greatest mathematician in the 
world, Sophus Lie, died at Christiania in Norway. 

He was essentially a geometer, though applying his splendid powers of 
space creation to questions of analysis. From Lie comes the idea that 
every system of geometry is characterized by its group. 

In ordinary geometry a surface is a locus of points ; in Lie's Kugel-geome- 
trie it is the aggregate of spheres touching this surface. By a simple correlation 
of this sphere-geometry with Pluecker's line-geometry, Lie reached results as 
unexpected as elegant. The transition from this line-geometry to this sphere- 
geometry was an example of contact-transformations. 

Now contact-transformations find application in the theory of partial dif- 
ferential equations, whereby this theory is vastly clarified. Old problems were 
settled as sweepingly as new problems were created and solved. Again, with 
his Theorie der Transformationsgruppen, Lie changed the very face and fashion of 
modern mathematics. 

A magnificent application of his theory of continuous groups is to the gen- 
eral problem of nun-Euclidean geometry as formulated by Helmholtz. To this 
was awarded the great Lobachevski Prize. Not even this award could sufficient- 



98 

ly emphasize the epoch-making importance of Lie's work in the evolution 
of geometry. 

Moreover, the foundations of all philosophy are involved. To know the 
non-Euclidean geometry involves abandonment of the position that axioms as to 
their concrete content are necessities of the inner intuition ; likewise abandon- 
ment of the position that axioms are derivable from experience alone. 

Lie said that in the whole of modern mathematics the weightiest part is 
the theory of differential equations, and, true to this conviction, it has always 
been his- aim to deepen and advance this theory. 

Now it may justly be maintained that in his theory of transformation 
groups Lie has himself created the most important of the newer departments of 
mathematics. 

By the introduction of his concept of continuous groups of transformations 
he put the isolated integration theories of former mathematicians upon a com- 
mon basis. 

The masterly reach of Lie's genius is illustrated by his encompassment of 
the fundamentally important theory of differential invariants associated with the 
English names Cayley, Cockle, Sylvester, Forsyth. 

Thirteen years ago Sylvester announced his conception of 'Reciprocants,' 
a body of differential invariants not for a group, but for a mere interchange of 
variables. A number of Englishmen thereupon took up investigations about or- 
thogonal, linear and projective groups, groups in whose transformations inter- 
changes of variables occur as particular cases, and whose differential invariants 
are consequently classes of reciprocants, and of the analogues of reciprocants, 
when more variables than two are considered. 

Now all these investigations were long subsequent to Lie's consideration 
of the groups in question as leading cases of a general conception. Thus they 
were merely secondary investigations ! 

Again the theory of complex numbers appears as a part of the great 
'Theorie der Transformationsgruppen.' Indeed, this continent of 'transforma- 
tions' opened up and penetrated with such giant steps by Lie represents 
the most remarkable advance which mathematics in all its entirety has made in 
this latter part of the century. 

Sophus Lie it was who made prominent the importance of the notion of 
group, and gave the present form to the theory of continuous groups. This idea, 
like a brilliant dye, has now so permeated the whole fabric of mathematics that 
Poincare actually finds that in Euclid 'the idea of the group was potentially pre- 
< existent,-' and that he had 'some obscure instinct for it, without reaching a dis- 
tinct notion of it.' Thus the last shall be first and the first last. 

In personal character Lie was our ideal of a genius, approachable, out- 
spoken, unconventional, yet at times fierce, intractable. 

His work is cut short ; his influence, his fame, will broaden, will tower 
from day to day. 



99 
SOPHUS LIE.* 



BY PROFESSOR GASTON DARBOUX. 

Sophus Lie was born on the 17th of December, 1842, at Nordfjordeid (near 
Floro) where his father, John Herman Lie, was pastor. The studies of his 
childhood and youth did not reveal in him that exceptional aptitude for mathe- 
matics which is signalized so early in the lives of the great geometers : Gauss, 
Abel, and many others. Even on leaving the University of Christiania in 1865, 
he still hesitated between philology and mathematics. It was the works 
of Pliicker on modern geometry which first made him fully conscious of 
his mathematical abilities and awakened within him an ardent desire to conse- 
crate himself to mathematical research. Surmounting all difficulties and work- 
ing with indomitable energy he published his first work in 1869, and we can say 
that from 1870 on he was in possession of the ideas which were to direct 
his whole career. 

At this time I frequently had the pleasure of meeting and conversing with 
him in Paris where he had come with his friend F. Klein. A course of lectures 
by Sylow revealed to Lie all the importance of the theory of substitution groups; 
the two friends studied this theory in the great treatise of our colleague Jordan ; 
they saw fully the essential r6ie which it would be called upon to play in all the 
branches of mathematics to which it had then not been applied. They have both 
had the good fortune to contribute by their works to impressing upon mathemat- 
ical studies the direction which appeared to them to be the best. 

A short note of Lie "Sur une transformation geometrique," presented to 
our Academy in October, 1870, contains an extremely original discovery. Noth- 
ing resembles a sphere less than a straight line and yet, by using the ideas of 
Pliicker, Lie found a singular transformation which makes a sphere correspond 
to a straight line, and which consequently makes possible the derivation of a 
theorem relative to an ensemble of spheres from every theorem relative to an ag- 
gregate of straight lines, and vice versa. It is true that if the lines are real, the 
corresponding spheres are imaginary. But such difficulties are not sufficient to 
deter geometers. In this curious method of transformation, each property rela- 
tive to asymptotic lines of a surface is transformed into a property relative to 
lines of curvature. The name of Lie will remain attached to these concealed re- 
lations which connect the two essential and fundamental elements of geometric 
investigation, the straight line and sphere. He has developed them in detail in 
a memoir full of new ideas which appeared in 1872 in the Mathematische Annalen. 

The works following this brilliant beginning fully confirmed all the hopes 
to which it gave birth. Since the year 1872 Lie has put forth a series of memoirs 
upon the most difficult and most advanced parts of the integral calculus. He 

*From the Bulletin of the American Mathematical Society. Translated by Edgar Odell Lovett from 
Compte8 rendu8 . 



100 

commences by a profound study of the works of Jacobi on the partial differential 
equations of the first order and at first cooperates with Mayer in perfecting this 
theory in an essential point. Then, by continuing the study of this beautiful 
subject, he is led to construct progressively that masterful theory of continuous 
transformation groups which constitutes his most important work and in which, 
at least at the start, he was aided by no one. The detailed analysis of this vast 
theory would require too much space here. It is proper, however, to point out 
particularly two elements wholly essential to these researches : first, the use of 
contact transformations which throws such a vivid and unexpected light upon 
the most difficult and obsure parts of the theories relative to the integration of 
partial differential equations ; second, the use of infinitesimal transformations. 
The introduction of these transformations is due entirely to Lie ; their use, like 
that of Lagrange's variation, naturally greatly extends both the notion of differ- 
ential and the applications of the infinitesimal calculus. 

The construction of so extended a theory did not satisfy Lie's activity. In 
order to show its importance he has applied it to a great number of particular 
subjects, and each time he has had the good fortune of meeting with new and el- 
egant properties. I find my preference in the researches which he has published 
since 1876 on minimal surfaces. The theory of these surfaces, the most attrac- 
tive perhaps that presents itself in geometry, still awaits, and may await a long 
time, the complete solution of the first problem to be proposed in it, namely, the 
determination of a minimal surface passing through a given contour. But, in re- 
turn, it has been enriched by a great number of interesting propositions due to a 
multitude of geometers. In 1866 Weierstrass made known a very precise and 
simple system of formulae which has called forth a whole series of new studies on 
these surfaces. In his works Lie returns simply to the formulae of Monge ; he 
gives their geometric interpretation and shows how their use can lead to the 
most satisfactory theory of minimal surfaces. He makes known methods which 
permit of determining all algebraic minimal surfaces of given class and order. 
Finally, he studies the following problem : to determine all algebraic minimal 
surfaces inscribed in a given algebraical developable surface. He gives the com- 
plete solution for the case where only one of these surfaces inscribed in the de- 
velopable is known. 

Of great interest also are the researches which we owe to him on the sur- 
faces of constant curvature, in the study of which he makes use of a theorem of 
Bianchi on geodesic lines and circles, likewise those on surfaces of translation, on 
the surfaces of Weingarten, on the equations of the second order having two in- 
dependent variables, et cetera. I should reproach myself for forgetting, even in 
so rapid a resume, the applications which Lie has made of his theory of groups 
to the non-Euclidean geometry and to the profound study of the axioms which 
lie at the basis of our geometric knowledge. 

These extensive works quickly attracted to the great geometer the atten- 
tion of all those who cultivate science or are interested in its progress. In 1877 
a new chair of mathematics was created for him at the University of Christiania, 



101 

and the foundation of a Norwegian review enabled him to pursue his work and 
publish it in full. In 1886, he accepted the honor of a call to the University of 
Leipzig; he taught in this university with the rank of ordinary professor from 
1886 to 1898. To this period of his life is to be referred the publication of his 
didactic works, in which he has coordinated all his researches. Six months ago 
he returned to his native land to assume at Christiania the chair which had been 
especially reserved for him by the Norwegian parliament, with the exceptional 
salary of ten thousand crowns. Unfortunately, excess of work had exhausted 
his strength and he died of cerebal ansemia at the age of fifty-six years. 

Nowhere is his loss felt more keenly than in our country, where he had 
so many friends. True, in 1870 a misadventure befell him, whose consequences 
I was instrumental in averting. Surprised at Paris by the declaration of war, he 
took refuge at 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 imprisioned at Fontainebleau, under 
conditions otherwise very comfortable, he called for the aid of Chasles, Bertrand, 
and others ; I made the trip to Fontainebleau and had no trouble in convincing 
the procureur imperial ; 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. Lie was released ; his high and generous spirit bore no 
grudge against our country. Not only did he return voluntarily to visit it but 
he received with great kindness French students, scholars of our Nicole Normale 
who would go to Leipzig to follow his lectures. It is to the ficole Normale that 
he dedicated his great work on the theory of transformation groups, A number 
of our thesis at the Sorbonne have been inspired by his teaching and dedicated 
to him. 

The admirable works of Sophus Lie enjoy t(?e distinction, to-day quite 
rare, of commanding the common admiration of geometers as well as analysts. 
He has discovered fundamental propositions which will preserve his name from 
oblivion, he has created methods and theories which, for a long time to come, 
will exercise their fruitful influence on the development of mathematics. The 
land where he was born and which has known how to honor him can place with 
pride the name of Lie beside that of Abel, of whom he was a worthy rival and 
whose approaching centenary he would have been so happy in celebrating.