Leonhard Euler: Life and Thought

Ronald S. Calinger    Department of History, Catholic University of America, Washington, DC, 20064, USA

As the European Enlightenment began in the 1720s, few new accomplishments in mathematics were expected. Although mathematics had not yet become a profession in the previous century, when most of its practitioners came from the aristocracy or positions in medicine or law, that period culminating in the inventions of differential calculus by Isaac Newton and Gottfried Leibniz was considered a great age in mathematics, leaving little to be developed. But some scholars anticipated a fecund era for the field.1 Above all, the research of Leonhard Euler would prove them right. The Swiss-born Euler was to be one of the four preeminent mathematical scientists in history, the other three being Archimedes, Newton, and Carl Friedrich Gauss. Only for Newton and Euler did Gauss reserve the term summus.2

Driven by a passion for mathematics and natural science, a commitment to build a strong institutional base for them, and an insistent defense of reform Christianity, Euler made seminal contributions across the mathematical sciences and was arguably the most prolific mathematician in history. At the core of his research were infinitary analysis, or differential calculus, and rational mechanics. Along with celestial mechanics, he made them the sciences par excellence of the eighteenth century. He was the principal creator of the calculus of variations and differential equations, and he pioneered the differential geometry of surfaces. In mechanics Euler, not Newton, formulated most of the fundamental differential equations before William Rowan Hamilton. Operating within Enlightenment rivalries, in his case with Jean d’Alembert, Alexis Clairaut, Daniel Bernoulli, and Colin Maclaurin, he led in transforming mechanics and astronomy into modern exact sciences based on calculus. Euler founded continuum mechanics and advanced the study of ballistics, cartography, dioptrics, the theory of elasticity, hydraulics, hydrodynamics, music theory, number theory, optics, and ship theory. Massive and fearless computations, an extraordinary application of analysis and analogies, an appeal to his near unerring instinct, and clarity in writing characterize his work. Not since Claudius Ptolemy had a single geometer so dominated all branches of the mathematical sciences. During the eighteenth century four royal science institutions, in Paris, London, St. Petersburg, and Berlin, eclipsed universities in scientific research. It was largely Euler’s efforts that made the St. Petersburg and Berlin Academies of Science prominent European centers. The more than 810 of his articles and books, which fill seventy-four large volumes in the first three series of his Opera omnia, include approximately one-third of the entire corpus of research in mathematics, theoretical physics, and engineering mechanics published from 1726 to 1800, while the equivalent of research articles fill his extensive correspondence.3

1 Lineage, Youth, and Formal Education

Leonhard Euler was born on Friday April 15, 1707 (n. s.), in Basel, Switzerland. While most of Protestant and Orthodox Europe followed the Julian calendar or old style, the city had adopted the current Gregorian style in 1701. Euler’s birth house was probably located in the neighborhood around St. Martin’s Church near the center of the city close to the market quarter and ship landing on the Rhine River. He was the first child of Paul Euler, an Evangelical-Reformed minister, and Margaretha née Brucker. While “reformed” generally refers to the Protestantism of the Calvinists and Lutherans, Basel’s variety was of a pietism stressing love and the inner life. Leonhard’s mother Margaretha, the daughter of a hospital minister, was from a distinguished line of artists and humanistic scholars. Their son was baptized two days after his birth in the same St. Martin’s church as his father had been.

The Euler (Äuler, Ewler, Öwler) family came from the town of Lindau on Lake Constance (the Bodensee) in the German Swiss Canton. Au is the diminutive of Äule, which refers to a small, wet field or meadow. Au appears in the names of many small German towns, such as Dessau and Nassau. The owner of an Äule was an Äuler (oyler). The Eulers were variously called Euler-Schoelpin, signifying squint-eyed, which suggests that they were susceptible to an eye malady. The first written record of an Euler appeared in 1287, but a documented continuous line did not commence until 1458. Lindau, though on the far side of the Canton from Basel, had many close economic, political, and religious ties with the town. Hans Georg Euler, the great-great grandfather of Leonhard and the grandson of the German-speaking patriarch Hans Euler, moved to Basel in 1594. Hans Georg obtained citizenship in Basel, became a comb- and brush-maker, fathered fifteen children in two marriages, and lived to be ninety.4 Apparently the next three generations were artisans, most of them comb-makers or tradesmen belonging to hospitality guilds. They built the family’s modest financial base. In the fourth generation, four of the fourteen male cousins were able to become Basler Evangelical-Reformed ministers. These included Paul Euler, who matriculated at the University of Basel in 1685 at the age of fifteen. While at the university, Paul resided at the home of Jacob Bernoulli, under whose direction he wrote his senior thesis on ratios and proportions. He shared rooms with the young Johann I Bernoulli. Paul Euler completed his theological studies in 1693.

Leonhard did not spend his early youth in Basel. In June 1708 his father was named pastor-designate to St. Martin’s church in nearby Riehen-Bettingen. In November he was installed and the family moved to Riehen, about five kilometers northeast of Basel. It and Bettingen combined had a population of fourteen hundred. Supported by a sub-Mediterranean climate, these small villages were known for their rich vegetation, especially the white blossoms of the cherry trees in the late spring and the gold and red leaves on the grapes in the vineyards. The Eulers lived in a two-room parsonage until it was enlarged in 1712. One room was a study and the other living quarters. Of Leonhard’s two younger sisters, Anna Marie was born in 1708 and Marie Magdalena in 1711. His paternal grandmother lived with the family to her death in 1712. Johann Heinrich, the fourth child of the Eulers, was not born until 1719, after his brother departed for studies at the Basel Gymnasium. Leonhard was a talented child, apparently cheerful and sociable. The simplicity of rural life together with the model of his parents has its reflection in the forthright nature and even disposition of the adult Euler.

Leonhard’s parents were his first teachers. Familiar with the humanistic tradition, his mother Margaretha introduced him to Greek and Roman classics. The elementary instruction that his father Paul offered included mathematics, conceived as a subject underlying all natural knowledge. Paul began not with a geometry text but with Christoff Rudolff’s Coss, or algebra, the German equivalent of the Italian cosa or unknown, a two-part work that Michael Stifel had expanded from 208 to 484 pages. Paul possibly employed a reprint from 1615 of the first edition published in 1553. After explaining place-value notation and the four basic arithmetical operations, it examines in verbal form first, second, and third degree equations. Euler’s unfinished autobiography notes that he diligently studied the text for several years and made progress in solving its 434 problems, almost all of them first- or second-degree equations.5 He did this before moving to lodge with his maternal grandmother in Basel and enroll in the city’s Gymnasium, probably at age eight. Only an exceptional child of this age could have advanced previously through the difficult Coss.

The Basel Gymnasium, a Latin school, was in a pitiful state. Students were taught the Latin language and selections from ancient classics. Greek was optional. Teachers did not spare the rod, and fistfights broke out in the classroom. Like most parents, Euler’s hired a tutor, in their case a young theologian named Johann Burckhardt, who sided with Johann I Bernoulli in disputes with British geometers and natural philosophers, especially Brook Taylor. Burckhardt taught Euler the humanities and mathematics, a subject earlier struck from the curriculum by a vote of the townspeople.

In 1720 Euler matriculated at the University of Basel into the Philosophical Faculty, essentially the school of arts and sciences. He was thirteen, at the time roughly the normal age for entering a university. The university was in decline. Its enrollment had fallen from over a thousand students a century earlier to just above a hundred. It had only nineteen professors, underpaid and most of them mediocre. The exception was Johann I Bernoulli. The Philosophical Faculty provided the general education preparatory to choosing a specialty for a higher degree. Through industry and a powerful memory, Euler mastered all his subjects. Apparently he skipped the dry introductory mathematics lectures of Bernoulli, as Charles Darwin and Albert Einstein would later avoid sessions of tedious college courses. At fourteen Euler gave a speech titled “Declamatio: De Arithmetica et...