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Euler in the Age of Enlightenment

By James Case

Leonhard Euler: Mathematical Genius in the Enlightenment. By Ronald S. Calinger. Courtesy of Princeton University Press.
Leonhard Euler: Mathematical Genius in the Enlightenment. By Ronald S. Calinger, Princeton University Press, Princeton, NJ, 2016, 696 pages, $55.00.

As a graduate student in the history of science at the University of Chicago during the 1960s, Ronald Calinger considered writing his doctoral thesis on the life of Leonhard Euler. However, he bowed to advice from Saunders Mac Lane that the time was not yet right for such a project, since a massive effort to catalogue and translate Euler’s complete works was just getting underway. Only as that poject was nearing completion did he resume work on the book he had always intended to write.

Calinger’s new book is neither a mathematical nor scientific biography. As the subtitle suggests, it is largely an attempt to position Euler within the pantheon of Enlightenment figures, including thinkers such as Voltaire, Rousseau, and Montesquieu, as well as rulers like Charles II of England, Louis XIV of France, and Frederick the Great of Prussia, who chartered the royal scientific institutions that eclipsed the research performance of traditional universities for a time. There were about seventy such institutions by 1789.

The Enlightenment seems to have begun as an extension of the Copernican Revolution, during which educated Europeans discarded traditional ideas about the natural world in favor of reason and experience. Shortly thereafter, they began applying similar thinking to social questions, including the rightful purpose and proper form of both government and religion. 

Not very much is known of Leonhard Euler’s early life. His father Paul III served as pastor of Saint Martin’s Church in the village of Riehen-Bettingen, some five kilometers from Basel, Switzerland. Aware of their son’s extraordinary intellectual ability, Euler’s parents soon sent him to live with his maternal grandmother in Basel, where he could attend the city’s gymnasium. To supplement its rather meager offerings, they also arranged for him to be tutored by Johannes Burckhardt, a mathematically-inclined theologian. In 1720, at the age of twelve, Euler enrolled at the University of Basel to prepare for a career in the clergy. There he encountered Johann Bernoulli, who had single-handedly turned Basel into a leading center for mathematics. From the age of sixteen, Euler devoted the bulk of his attention to that discipline. At eighteen, he wrote a short paper on isochronal curves, which appeared a year later in Leipzig’s Acta Eruditorum. That publication, together with the support of Bernoulli, led to an appointment at the recently-founded St. Petersburg Academy of Sciences.

In April 1727, 10 days before his 20th birthday, Euler left Basel aboard a Rhine riverboat, never to return. Arriving in St. Petersburg seven weeks later, he was greeted by a host of German speakers, among them Christian Goldbach, who was to stimulate Euler’s interest in the (then-unfashionable) subject of number theory. The two would correspond for more than thirty years.

Members of the new academy were expected to produce practical as well as scholarly results. While mastering Russian, Euler performed studies of navigation and ship design, flood control, and map making. He  also participated in the operation of the academy’s workshop and sawmill, served as an examiner for the military cadet corps, worked in the office of weights and measures, and helped to modify the St. Petersburg tariff laws. Before long he drew up plans for a rudimentary water turbine—destined to replace overshot waterwheels as the power source of choice in 19th century mills—a single cylinder steam engine, and a propeller with which to drive a steamship.

In 1733, Euler met and married Katharina, daughter of Swiss-born painter Georg Gsell. The marriage was apparently a happy one, despite the fact that only five of thirteen children survived to adulthood. The marriage lasted nearly forty years, with almost half of them spent in St. Petersburg. 

Even before mounting the Prussian throne, Frederick the Great had dreamed of turning Berlin into an “Athens on the [River] Spree.” Voltaire, among others, convinced him of the need for a royal academy of science, comparable to the ones in London, Paris, and St. Petersburg, and assured him that Euler’s appointment to the new institution would bestow much glory on it. Although not very interested initially, Euler became receptive when a movement to purge the Russian court of foreign influences caused him to fear for his personal safety.

In 1741, he moved to Berlin. Berlin academy director Pierre-Louis Maupertuis soon became a fast friend, and helped to make Euler’s early years in Berlin both happy and productive. However, following the conclusion of the Seven Years’ War in 1763, during part of which Berlin was occupied by foreign troops, Frederick made it clear that Euler would not succeed Maupertuis as director. Thus, when Catherine the Great made him a princely offer to return to St. Petersburg in 1766, Euler jumped at the chance.

When Euler arrived in St. Petersburg for the first time, Newtonian theory was still encountering resistance from partisans of the older Cartesian and Leibnizian alternatives; the Paris academy was a stronghold of the former. The age-old practice of mingling science with mysticism did not die easily, in Europe or elsewhere. Newton himself studied alchemy, was never sure that his laws would suffice to explain all celestial motion, and suspected that divine intervention would occasionally be required to avoid potential collisions. Likewise Euler, when asked if a comet could ever strike Earth, replied that it was indeed possible but would never happen because any such catastrophe would violate God’s biblical promise to protect mankind. On the other hand, critics of Newtonian theory objected to the occult nature of the “action at a distance” implicit in his law of gravitation.

In time, the debate surrounding Newtonian theory came to focus on five presumably-verifiable predictions, namely those concerning tidal motion, the shape of the planet Earth, the orbits of comets, planetary and lunar motion, and fluid dynamics. Euler eventually addressed all five issues, most prominently those concerning comets and planetary and lunar motion. He postponed his study of fluid dynamics for almost twenty years, possibly to avoid duplicating the efforts of his close friend Daniel Bernoulli. In time, the Paris Academy became a bastion of Newtonian theory.

Perhaps the most decisive episode in the debate surrounding Newtonian theory concerned irregularities in the orbit of the moon. At one point, mathematician Alexis Clairault suggested that a correction to Newton’s inverse square law of gravitation involving an inverse fourth power was necessary to fully account for the observed anomalies. Several years passed before Euler was able to confirm to the satisfaction of his contemporaries that Newton’s law suffices to explain all that could then be observed. It was by no means the most acrimonious dispute in which Euler became involved.   

In 1751, Maupertuis’ claim of priority for the principle of least action was disputed by mathematician Samuel König, who produced a copy of a 1707 letter from Leibniz to a colleague, in which such a principle was clearly enunciated. Rather than claiming priority for himself (as he might well have done), Euler magnanimously took the side of his friend Maupertuis and personally accused König of forgery before the Berlin Academy. That verdict remained more or less intact until, some 150 years later, additional copies of the 1707 letter surfaced in the Bernoulli archives.

Anyone seeking a concise account of Euler’s mathematics, or particular parts thereof, should look elsewhere. Many excellent expositions are available, including [1, 2, and 3]. Calinger describes the age in which the man lived, along with the intellectual currents and political realities that drew his attention to particular problems. Is that not enough to ask of any book?


[1] Dunam, W. (1999). Euler: The Master of Us All. Washington, D.C.: Mathematical Association of America.

[2] Havil, J. (2009). Gamma: Exploring Euler’s Constant. Princeton, NJ: Princeton University Press.

[3] Richeson, D.S. (2012). Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Princeton, NJ: Princeton University Press.

James Case writes from Baltimore, Maryland. 
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