Review of **Joseph Liouville 1809-1822: Master of Pure and Applied
Mathematics.**

By Jesper Lützen.

*Springer-Verlag. New York 1990, 884+xix pages, $98.00*

There is a phenomenon that is noticed particularly as one's hair first turns gray. Called metamophosis or transmigration, it works this way: A very happy few of us get popped up a metalevel. From creatures of flesh and blood, these few are transformed into other forms of existence. Those who are wealthy enough may order a transformation into a Foundation, a Fund, a Chair, or a Building. Those who are of modest means might metamophose into a Carrel, a Lecture Series, or a Prize for Sophomores. Several years ago, when visiting Atlanta, Georgia, I was shocked to find that a man I had known as an undergraduate, a man who was neither a soldier nor a politician, had been transformed into a Superhighway.

Certainly one of the highest accolades that our profession can bestow on its notables is to transform them into adjectives or possessive nouns. Abelian groups, Hilbert spaces, Pascal's triangle, Gaussian distribution, Tschebyscheff polynomials are a few instances of what I have in mind. A complete list of this sort would be finite but uncountable. Higher honors are indicated perhaps, if one is put into lower case (e.g. a sequence is cauchy) or if one achieve entrée into the complimentary Boolean mode (e.g. non-euclidean or non-cantorian).

Every professional mathematician worth his salt can possibly mention at least six theorems or areas that go by the name of Liouville; for example, Liouville transcendentals or phase flow preserves volume. Yet details of the man's private life have remained obscure, and he is underrated in the Pantheon of mathematicians. The work under review, written by Jesper Lützen of the Department of Mathematics at the University of Copenhagen, changes this completely. Lützen has made a significant step toward the rehabilitation from mere adjectival status of one of the finest of 19th-century mathematicians.

Joseph Liouville (1809-1882), member of the Institute and commander of the
Legion of Honor, was a teacher, a prolific researcher (he authored more than
400 publications) a founder and editor of a mathematical journal that still
exists (*Journal de Mathématiques Pures et Appliquées,*
a.k.a. Liouville's Journal), an academician, a scientific and institutional
politician, a Second Republic politician, and a considerable scrapper --- by
which I mean that he got into dog and cat fights with, among others,
Serret, Libri-Carucci, Pontécoulant, and Le Verrier of Neptune
fame. What did he argue about? Everything. Aside from a large core that
was personal or political, the arguments were about such issues as mathematical
validity, methodology, importance, rigot, clarity, aesthetics, plagiarism,
priority. He even got into a fight with one man who asserted that the
value of a definite integral depended on the (dummy) variable of
integration.

Moving from applied interests to pure theory, but never completely abandoning applications, often using applications as sources of inspiration, Liouville made profound contributions to the theories of differentiation of fractional order, integration in finite elementary terms, spectral theory of second-order ordinary differential equations (Sturm-Liouville theory) transcendental numbers, doubly periodic functions, and mechanics. And this is only a partial list.

The energy and productivity of 19th-century figures are often mind-boggling.
Think of all those novels written by Dickens and Trollope. Think of how
Leslie Stephen almost single-handedly produced the *Dictionary of
National Biography*. As a young man, Liouville taught 34 hours a week; his
schedule often included classes on Sunday morning at 8:00 am. (Are professors
spoiled today?) In addition to the 400 or so papers that he wrote and
the journal that he founded, edited, and raised to great prominence,
Liouville left behind almost 50,000 pages of notes. To review and understand
the man's published work is accomplishment enough. To sift through
the *Nachlass*, as it is called, in search of pearls among the dross
would require a lifetime of application. But many pearls have turned up;
Weierstrass appreciate the value of Liouville's unpublished work, especially
his development of elliptic functions from double periodicity.

Lützen has produced a monumental study of the professional man and of his work. His book is divided into two parts; the first, and shorter, part details the career of this mathematician from his student days to the time he spent at the Academy, and does it in such a way that the reader gets an intimate picture of French academic life in the mid-19th century. The competition was fierce, the greasy poles of success were few and high, egos were long and tempers were short, and the institutional politics make our own sound like the gentle lapping of waves on Lake Placid.

The second part of the book (more than 500 pages) is devoted to 10 very substantial essays describing areas in which Liouville worked, providing a close analysis of what he did in the context of the times and carrying the mathematical material forward a bit in time so the reader cann learn how it all "turned out". Each of the essays is almost a small book in its own right.

My own interest in Liouville, and my reason for wanting to review this book,
stem from my undergraduate thesis. As a student I got interested in the
question of integration in finite terms. For example, can the indefinite
integral of exp(x^{2}) be expressed in terms of a finite number
of combinations of elementary functions or their combinations? (The
answer is no.) Professor David V. Widder, to whom I expressed this interest,
put me onto the fundamental work of Liouville on this question, and I wrote
my thesis based on Liouville's original papers.

The French was easy enough, but old material always presents difficulties.
The background, the notations, the assumptions, the unwritten portions
are not what you think they are, based on your knowledge of what the subject
subsequently became. Then there are the expository merits or deficiencies
of an individual author. Joseph Gergonne, editor of
the *Annales de Mathé matiques Pures et Appliquées,*
to whom Liouville had submitted one of his first papers, came up with this
assessment:

"I do not want to dispute the mathematical capacity of Mr. Liouville. But what purpose does such a capacity serve if if is not accompanied by the art of arranging the material and the art of writing something that can be read, understood, and enjoyed. Unfortunately there are today too many young persons, otherwise of great merit, who consider as an immaterial accessory this art which I regard as the essential merit, the merit par excellence for the lack of which all the rest is worthless." --- Gergonne, 1830. (One hundred sixty years later, let old and young, theorem-provers and editors alike, profit from this admonition.)

Nonetheless, I was able to work through Liouville's methods to my own satisfaction, and the experience left me with great admiration for the ingenuity of the man. Years later, I am left with great admiration and sympathy for those historians of our subject (such as Lützen) who laboriously try to make sense of archival material. This review, then, lays a palm on the grave of an old thesis advisor, as I have come to think of Liouville.

And how did the chapter of mathematics known as "integration in finite terms" turn out after all this time? In its day, although it had obvious connections to the emerging Galois theory, it seemed an oddball topic. Lützen reveals it all --- from Pappus of Alexandria to the 20th century, in which the contributors include such names as G.H. Hardy, Morduchai-Boltovsky, J.F. Ritt, A. Ostrowski, M. Rosenlicht, and R.H. Risch. To quote Lützen, "The greatest triumph of the modern theory is Risch's complete solution of Liouville's main problem: given an explicit finite function, determine in a finite number of steps if its integral is also a finite function." Many symbolic manipulation packages now have finite integrators built in; the extent to which they are useful in scientific analysis or computation is another matter.

What is clear from the comprehensiveness of this work and from the loving care devoted to it is the author's admiration, if not idolization, of his subject. As part of his appreciative reevaluation, he puts forward claims --- very likely justified --- that Liouville was responsible for or anticipated in a substantial way a number of important concepts usually ascribed to later writers. Examples include certain results on stability, which were rediscovered 40 years later by Poincaré and Liapunoff independently, and the first published existence proof in ordinary differential equations, in which Liouville, scooping Cauchy, used the method of successive approximations generally ascribed to Picard (1890). Claims are made for priorities in the spectral theory of symmetric integral operators as well as for the Dirichlet principle.

In these days of bedroom-bugging biography, all that is missing is an in-depth
treatment of Liouville the private man. Lützen was probably correct
not to vent what he knows on this topic. Through such personal information
(including events, letters, diaries) as has been included here, it is
easy enough to extrapolate to the fuller life. I would assert that, unlike
Voitaire, who became a major intellectual and biographic figure but whose
works (other than *Candide*) are now never read, or unlike the
mathematician Alan Turing, who became the scandal-ridden hero of a West
End and Broadway play, Liouville will remain a mathematician. He appears
neither pure nor simple, nor to have been exclusively devoted to his subject
with obsessive passion, but at least he will not longer be a mere adjective.

*Philip J. Davis is a professor in the Division of Applied Mathematics
at Brown University. *