Review of Leonhard Euler: Mathematical Genius in the Enlightenment by Ronald S. Callinger. Princeton U. Press, 2016. xvii+669 pps.

In the pantheon of great mathematicians, Leonhard Euler (1707-1783) is one of the supreme deities. It is not possible even to outline his accomplishments in the word limits of this review. Euler was the founding father of the calculus of variations and of graph theory. He did pioneering work in calculus, differential equations, complex number theory, number theory, and differential geometry; and also in celestial mechanics, continuum mechanics, and optics. He invented the constant e. There is Euler's formula e^{i θ} = cos(θ) + i sin(θ), and the other Euler's formula V+F-E = 2. There are Euler's totient function φ(n) , Euler's constant γ, Eulerian angles, Eulerian paths, and the Eulerian formulation of continuum mechanics. There are the Euler numbers, not to be confused with the Eulerian numbers. Euler was one of the leading figures in the victory of Newton's physics over Descartes' physics, and in establishing that all of the behaviors of the solar system then known were a consequence of Newton's law of gravity. And on and on. His collected works fill eighty volumes; their translation into English is an ongoing project.

Euler was also central in establishing the scientific institutions of his time. In particular, he was one of the leading figures in the creation of the Berlin (Royal Prussian) Academy of Sciences under Frederick II.

However, in the popular mythology of mathematics that celebrates the pantheon, Euler cuts a rather gray figure. Few interesting stories are told about Euler, and few interesting sayings are quoted. There are no anecdotes that show how, though an amazing mathematician, he was also a regular guy, and very few that show how he was a unusual guy. One collection of mathematical quotations [1] contains one apocryphal story of how Euler flummoxed Diderot with a bogus algebraic proof of the existence of God, one rather pedestrian quote about maxima and minima, and another about the distribution of primes. The single striking personal anecdote in Calinger's 699 page book is about how boring he could be:

Her [the queen mother of Prussia] efforts to draw Euler into the sprited conversation failed. He responded only to queries in monosyllables. The exasperated queen chided him asking, "Why do you not wish to speak to me?" Euler who remembered the brutality of the Bironovschina period in Russia responded, "Madame, it is because I have just come from a country where a man's words can get him hanged." p. 187

As the subtitle indicate, Ronald Calinger's new biography of Euler places him in the context of the Enlightenment. He recounts in detail Euler's close interactions with fellow mathemacians and scientists, such as the Bernoullis, Maupertuis, and d'Alembert, and his more superficial interactions with Enlightenment figures such as Diderot and Voltaire. The biography is very thorough and deeply researched; it includes a 50 page bibliography, and a glossary/index of about 500 names, practically a who's-who of the eighteeenth century Enlightenment, particularly its scientific side.

1. Mathematically Speaking: A Dictionary of Quotations, Carl Gaither and Alma E. Cavazos-Gaither, Insitute of Physics Pubs.