Leonhard Euler: Mathematical Genius in the Enlightenment

  • Ronald S. Calinger
Princeton University Press: 2016. 9780691119274 | ISBN: 978-0-6911-1927-4

The Swiss mathematician Leonhard Euler (1707–83) was all but blind when he moved to St Petersburg in 1766 for a second stint as the star of the Russian Imperial Academy of Sciences. He had lost vision in his right eye 28 years before; a cataract was claiming his left. Yet Euler, then 59, boasted in one letter that the loss of sight meant “one more distraction removed”. From then on, his productivity increased: more than half of his eventual output of 866 publications was published either over the remaining 17 years of his life or posthumously.

The publication of Euler's collected works, which began in 1911, is still unfinished; it will fill more than 80 large volumes. There are also hundreds of letters — many with as much scientific content as the best of his papers. Leonhard Euler, written by historian of mathematics Ronald Calinger, is perhaps the first biography that attempts to offer a panoramic view of this immense body of work.

Leonhard Euler's analysis gave physics and astronomy their modern mathematical shape. Credit: RIGB/SPL

Euler dominated almost all branches of mathematics, as well as physics, astronomy and engineering, during the Enlightenment era. Euler's mathematics was often ahead of his time: he foreshadowed the use of groups of symmetries, the topology of networks, decision theory and the theory of sets (he was, for instance, the first to draw Venn diagrams). Nearly alone among his contemporaries, he advocated for the beauty and importance of number theory. His work on prime numbers, in particular, set the stage for a golden age of mathematics that would follow decades later.

However, Euler's greatest legacy, in both pure and applied mathematics, was the field of analysis. Seventeenth-century mathematicians, culminating with Isaac Newton and his arch-enemy Gottfried Wilhelm Leibniz, had founded calculus — the study of the rates of change of quantities in time (differentials or derivatives) and the intimately related idea of areas between curves (integrals). Euler's analysis turned calculus into a powerful science and endowed mathematics and physics with their modern language and appearance.

The founders of calculus often grasped at concepts that they could not fully understand. The field relied on infinitesimals, which had a metaphysical aura so controversial that they were in part responsible for getting Galileo Galilei in hot water with the Catholic Church, according to historian of mathematics Amir Alexander (Infinitesimal (Oneworld, 2014); see Nature http://doi.org/9hz; 2014).

In Euler's time, that controversy was still far from resolved. There were no rigorous definitions of limits or of the continuum of the real numbers; neither was put onto solid foundations until the nineteenth century.

Just as he was unfazed by blindness, Euler did not let these troubles hinder his mathematical creativity. In his treatment of infinitesimals — used in differential and integral calculus and in adding up infinite series — he took an approach that Calinger describes as “happy-go-lucky”. Euler's pragmatism is reminiscent of the 'shut up and calculate' attitude of the vast majority of twentieth-century physicists towards quantum mechanics, setting problematic foundations aside to allow enormous progress in applications (D. Kaiser Nature 505, 153–155; 2014). Euler's powers of intuition and his method of testing his hypotheses on special cases using his unparalleled calculation skills meant that his results were almost always right, says Calinger.

At the centre of analysis, Euler placed the concept of differential equations: those that link a function and its derivatives, and in which the solution consists of calculating the function itself. (In celestial mechanics, for example, the functions can represent the trajectories of planets.) He came to be regarded as the “principal inventor” of the field, Calinger writes, and his work on analysis “displaced synthetic Euclidean geometry from its two-millennium primacy”.

Euler demonstrated the power of this innovative science when he applied it to physical problems, such as the laws of the mechanics of solid bodies. In particular, he solved what many in the eighteenth century considered the most important open problem in astronomy: reconciling the complex motions of the Moon with Newton's universal law of gravitation. This 'three-body problem' involves the interactions of the Sun, Moon and Earth, and is much harder than predicting one planet's motion around the Sun. Some, including Euler, had suspected that Newton's inverse-square law would break down in this crucial test, demanding the formulation of another theory. The problem had enormous practical importance: lunar motions could be used to calculate a vessel's longitude at sea, and Euler was in the race to find a reliable method of doing so. (Eventually, precise timekeeping turned out to be a better solution.)

I have one quibble. The book's strict chronological order means that it often reads as a sequence of disconnected summaries of Euler's papers and correspondence, jumping from fundamental problems in algebra to ordering ink for his academy's printing presses, often in the same paragraph. Still, fragmented as the narrative is, we manage to glimpse a personality. He was a man of integrity who — with few exceptions — gave credit where it was due and maintained a belief in “a harmony between written revelation and natural phenomena”. And although Calinger remarks on Euler's perceived lack of “courtly manners”, we infer that this was really just a dearth of interest in flattering the nobility.

As a result, Euler never became head of the academies at which he worked, in Frederick the Great's Berlin or Catherine the Great's St Petersburg. No matter: his importance in the evolution of mathematics is clear. This impressively researched tome will be of great value to anyone with a serious interest in the history of mathematics and the Enlightenment.