- Leonhard Euler: Mathematical Genius in the Enlightenment by Ronald Calinger
This book represents the culmination of a lifetime of research by an eminent historian of mathematics. The result is a very rich biography that goes far beyond that field. Euler is among the most prolific mathematicians of all time, which means that despite Calinger's work being almost 700 pages long, he can only touch briefly on a large number of topics. The result is a very interesting book that may serve as a jumping-off place for a large number of deeper studies.
Calinger's book may best be approached in terms of being like fruitcake, a rich mosaic of fascinating topics bound together within the dough of Euler's personal story. The most plentiful tidbits are the mathematical vignettes. This makes sense: Euler's life was defined by his mathematical interests, and Calinger is very well equipped to understand them. He sees the eighteenth-century discussion of mathematics as a dynamic between an older geometrical, conceptual approach and a newer analytic, abstract one. Newton and Leibniz may be seen as the (rival) founders of the latter, which for Euler was transmitted more directly through his early teacher, Johann Bernoulli. Throughout his life, Euler maintained that mathematics epitomized reason's clear and accurate understandings even as he moved beyond the geometrical models on which that characterization was founded and into analysis. In his mathematical researches, Euler was constantly seeking "to perfect thought," which was not the same as insisting on rigor (129). His interests ranged from astronomy to shipbuilding to differential equations to number theory. He approached these interests through reason and calculation but was also open to other approaches, including "empirical mathematics, in which results of careful experiments guide the formulation of method" (161). Calinger does a very good job of presenting the issues that concerned Euler and masterfully follows the twists and turns of his work. Euler's output was so large, however, that his biographer cannot go into any particular topic in great depth. This is a book that will engage [End Page 351] mathematicians and historians of mathematics for decades, as they seek to build upon Calinger's foundations.
The book's importance extends beyond the mathematical, however. Readers who push the specifics of Euler's mathematics aside will find a satisfying variety of other nuggets. Euler was born and educated in Basel; he pursued his professional life first at the St. Petersburg Academy of Sciences and then as a member of Frederick II's Academy of Science in Berlin. Throughout his life he was engaged in defining what it meant to be a mathematical courtier. In his subtitle, Calinger accurately describes Euler as a man of the Enlightenment, but the world that he opens up goes beyond the traditional French-focused one. Philosophes such as Voltaire and mathematicians such as d'Alembert do appear in Euler's world—the first as an agent provocateur in Friedrich's court, the second as a brilliant upstart—but both are essentially occasional visitors from afar. The result is a refreshingly different story of Enlightenment, which stands to open our views of the period considerably.
There is yet more within this book. Euler's interests may have been primarily mathematical, but a mathematics pursued as a way to "perfect thought" is wonderfully expansive. Calinger's efforts to follow his subject through all of these twists and turns have resulted in multiple little essays on subjects that from a twenty-first-century point of view are at best peripherally related to mathematics. So, for example, presenting Euler's arithmetic textbook leads to a quick recognition of the ways in which the German language and orthography were being regularized in the middle of the eighteenth century; his study of harmony, to a discussion of the relations between acoustics, aesthetics, optics, and harmonic theory in the mid-eighteenth century; and his study of artillery, to considerations of eighteenth-century military strategy. These little essays may be seen as enticing treats for those who wish to avoid the...