With Euler’s tricentennial still being celebrated — and properly so — the book under review appears as a tribute to the titan about whom Laplace, not usually given to modesty, famously said, “He is the master of us all.” Debnath quotes Laplace in this connection on the first page of his preface and then goes on to cite Gauss and Weil; respectively: “... the study of Euler’s work will remain the best school for different fields of mathematics and nothing else can replace it...” and “No mathematician ever attained such a position of undisputed leadership in all branches of mathematics, pure and applied, as Euler did for the best part of the eighteenth century.” So Debnath’s contribution is not only timely but apposite.
The book comes equipped with a rather detailed and certainly very useful chronology of Euler’s life (1707–1783), properly citing his two golden periods in Russia (both in St. Petersburg, 1727–1741 and 1776–1783). During the latter interval Euler completed almost half of his life’s work, a truly remarkable fact on three counts (at least): he was nearly blind after unsuccessful cataract surgery in 1771, he covered an incredible spectrum of subjects in not only mathematics but science (e.g. chemistry, naval science, the motion of planets), and he was in his late 60s at the start of these miraculous final years. It’s a marvel to witness such a counterexample to the conceit that mathematics is a young man’s game, as G.H. Hardy would have it (in his ultimately ever so tragic A Mathematician’s Apology).
We also come across, in crisp and clear form, citations of many of the well-known contributions by the master, now all but household items in each mathematician’s kitchen/workshop/atelier/laboratory: big gamma in 1729, zeta in 1730, Euler-MacLaurin summation in 1732, infinitely many primes bis in the 1730s, the Adventure of the Königsberg Bridges in 1736, little gamma in 1740, the identity exp(ix) = cos(x) + isin(x) in 1743, zeta at the positive even integers in 1749, V - E + F = 2 in 1758, &c. All terrific stuff, of course. The great and good man died on 18 September, 1783, of a stroke; he was playing with his grandson at the time.
The book proper is divided into fourteen chapters, the first concerning the state of the mathematical world before Euler and the second a biographical sketch of his life. The other twelve chapters focus on Euler’s work in distinct (but naturally often overlapping) areas of mathematics and science: number theory and algebra, geometry and spherical trigonometry, topology (polyhedra) and graph theory, calculus and analysis, infinite series, infinite products, differential equations, fluid mechanics, solid mechanics and elasticity, probability, ballistics, astronomy and physics. Manifestly Debnath is working on a big canvas. But this is not to say that he does not do justice to both Euler’s contributions in these disparate areas and to their connections with other (and subsequent) mathematical themes. Quite the contrary.
For example, Debnath’s coverage of the Euler-Lagrange equation, in the relatively early biographical section of the book, beautifully fits the discussion in a historical framework, focused on the calculus of variations as such. It all started with Euler’s move to Berlin in 1741 (at age 34), his publication in 1744 of “the solution of the isoperimetric problem in its broadest sense,” thereby siring the calculus of variations (with the variational principle at center stage), and Euler’s natural philosophy centered on his “theological convictions.” Says Debnath: “The following 1744 statement of Euler is characteristic of the philosophical origin of what is known as the principle of least action as a guiding principle in nature: ‘As the construction of the universe is the most perfect possible, being the handiwork of [the] all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed. There is, consequently, no doubt but all the effects of the world can be derived by the method of maxima and minima from their final courses as well as from their efficient ones.’ ” Debnath goes on to describe the prehistory of this principle (Maupertuis) and its subsequent effects (Lagrange’s Analytical Mechanics) leading into William Rowan Hamilton’s 19th century work in analytical dynamics “which gave a new and very appealing form to the Lagrange equations” and in connection with which we find ourselves face to face with nothing less than the Hamiltonian operator, the life’s blood of both classical and quantum mechanics, and then quantum electrodynamics, and then quantum field theory, and lo, we have a trajectory from Euler to the hypermodern intersection of mathematics and theoretical physics. Can it be any more appropriate, given Euler’s own ecumenism in this regard?
Debnath isn’t done yet with the calculus of variations (it isn’t even chapter 9 yet): “The first necessary condition for the existence of an extremum of a functional in a domain leads to the celebrated Euler-Lagrange equation... [T]he fundamental concepts of the calculus of variations were developed in the eighteenth century in order to obtain the differential equations of applied mathematics and mathematical physics. During its early major developments, the problems of the calculus of variations were reduced to questions of... existence [of solutions of] differential equations until David Hilbert developed a method in which the existence of a minimizing function was established directly as the limit of a sequence of approximations....” He then goes on to discuss the indicated work of Weierstrass, Jacobi, Hilbert once more, and George David Birkhoff.
The point is, of course, that Debnath has a good bird’s eye view of the flow of the mathematics he is dealing with. He does a good job in explicating everything from both a historical and technical perspective. This is indeed a remarkable achievement, not least because we’re dealing with the huge scope of Euler’s creativity and discoveries: only a huge canvas will do.
Thus, the foregoing sample from the calculus of variations is just one of any number of examples of this sort (in this case I posit that my students are right: an example does make a proof). It is certainly the case that Debnath’s Tricentennial Tribute conveys serious and high-level scholarship. It’s also well-written and flows well. It’s a major achievement.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.