Leonhard Euler (1707-1783) was not just a mathematician – he was one of the greatest scholars of the 18th century, period. A physicist, astronomer, logician, engineer, and even a philosopher and theologian, Euler‘s contributions to human knowledge are staggering in both their breadth and depth. Yet he remains best known for his pioneering work in mathematics, a field he revolutionized and propelled into the modern era almost singlehandedly.
Euler‘s collected works run to more than 80 volumes, a testament to his unrivaled productivity. Remarkably, this represents only about three-quarters of his actual total output, as many of his manuscripts were lost in the turmoil of the Napoleonic Wars. Even in his own lifetime, Euler‘s contemporaries struggled to keep pace with his relentless stream of new ideas and theorems. The mathematician François Arago dubbed him "the Mozart of mathematics," capable of composing entire treatises with the same ease that Mozart tossed off sonatas.
A Precocious Prodigy
Leonhard Euler was born on April 15, 1707 in Basel, Switzerland to Paul Euler, a pastor, and Marguerite Brucker, a pastor‘s daughter. From a young age, Leonhard showed a prodigious talent for mathematics, devouring his father‘s math texts and solving problems with astonishing facility. At 13, he enrolled at the University of Basel, receiving his Master‘s degree at 16 with a comparative study of Newton‘s and Descartes‘ philosophies.
But Euler‘s most important education came from his private lessons with the brilliant Johann Bernoulli, then regarded as Europe‘s finest mathematician. Bernoulli had left his professorship in Groningen to return to his native Basel in 1705. Recognizing Euler‘s raw potential, he took the youth under his wing, proclaiming "I have not yet encountered anyone who learned mathematics as swiftly as you." Under Bernoulli‘s guidance, Euler mastered the works of Newton, Leibniz, L‘Hôpital and the Bernoullis and quickly began making original discoveries of his own.
Conquering St. Petersburg
In 1727, when Euler was just 20, Bernoulli‘s sons Daniel and Nicolas helped secure him a position at the prestigious St. Petersburg Academy in Russia. Founded by Peter the Great in 1724, the Academy was then at the forefront of scientific research, drawing top scholars from across Europe. Euler quickly climbed the ranks, becoming a professor of physics in 1730 and the chief mathematician in 1733 when Daniel Bernoulli returned to Basel.
Over his 14 years in St. Petersburg, Euler poured forth a staggering number of seminal works, making major advances in fields like number theory, calculus, geometry, trigonometry, astronomy and mechanics. Some of his notable breakthroughs from this period include:
- Developing the modern notation for mathematical functions, like f(x)
- Discovering the Euler-Mascheroni constant (0.57721…)
- Laying the foundations for the study of topology with his paper on the Seven Bridges of Königsberg
- Virtually inventing the field of analytical mechanics with his two-volume "Mechanica"
Enlightening the Enlightenment
In 1741, at the invitation of Frederick the Great, Euler moved to the Berlin Academy where he would enjoy his most productive period over the next 25 years. Though Euler had already lost sight in his right eye in 1735, it scarcely slowed his phenomenal output. He published over 375 works in Berlin, nearly half his career total, including landmark treatises like:
- "Introductio in analysin infinitorum" (1748), establishing the modern foundations of mathematical analysis
- "Institutiones calculi differentialis" (1755) on differential calculus
- "Vollständige Anleitung zur Algebra" (1770), a seminal algebra textbook
During this time, Euler also made fundamental contributions to number theory, graph theory, logic, and continued his groundbreaking work in calculus, mechanics, fluid dynamics, optics and astronomy. His correspondence with the leading scientists and philosophers of the day helped spread and shape the ideas of the Enlightenment.
However, Euler‘s devout religious convictions and modest personality clashed with the elite, often agnostic intellectual culture of Frederick‘s court. He famously butted heads with the French philosopher Voltaire, one of Frederick‘s closest confidants. According to one popular story, Euler once bested the sharp-tongued Voltaire in a debate by proclaiming: "Sir, (a+b)^n / n = x, hence God exists—reply!" leaving Voltaire dumbfounded by the mathematical non-sequitur.
Mathematical Monk
In 1766, Euler returned to St. Petersburg and became almost completely blind after a cataract operation. Remarkably, his productivity only seemed to increase. With the aid of his scribes and his photographic memory, Euler produced over 400 works in his later years, nearly half his already prodigious lifetime output.
Much like a mathematical monk, Euler‘s life revolved around his research, his faith, and his family. Of his 13 children, only five survived infancy. Euler doted on them, claiming some of his best ideas came to him while bouncing a baby on his knee. He entertained the court of Catherine the Great with amusing mathematical puzzles and demonstrations. Euler also penned philosophical works arguing for the existence of God and rebutting the period‘s fashionable French atheists.
When he wasn‘t doing math, Euler could be found studying the Bible, reading the latest scientific journals, or playing his beloved clavier. He was said to be able to recite the entire Aeneid by heart. Yet this Renaissance man remained above all a mathematician‘s mathematician, once declaring: "I do not think any pleasure can surpass that of a new discovery in this science."
Enduring Contributions
Euler made groundbreaking discoveries in so many areas it‘s impossible to summarize them all. In pure mathematics, some of his most celebrated results include:
- Euler‘s identity e^(i*pi) + 1 = 0, considered the most beautiful equation in math
- Euler‘s formula V-E+F=2 relating vertices, edges and faces of a polyhedron
- The Euler totient function phi(n) counting relatively prime numbers
- Euler‘s constant e, the base of the natural logarithm
- Establishing the use of i to represent the imaginary unit
- Standardizing many modern math notations like Sigma for summations, f(x) for functions, etc.
Beyond math, Euler also advanced the state of the art in:
- Astronomy: Calculating the orbits of comets and planets, explaining the tides, observing the aberration of starlight
- Optics: Improving the design of telescopes and microscopes, studying the wave theory of light
- Physics: Developing the kinetic theory of gases, explaining capillary action, researching elasticity
- Cartography: Improving map projections, measurement techniques and longitude calculations
- Naval science: Designing ships and propellors, studying floating bodies and fluid dynamics
- Music theory: Applying mathematical techniques to composition and acoustics
- Philosophy: Arguing against the dominant Leibnizian monadism in favor of Newtonian mechanics and a Christian worldview
By the Numbers
The sheer volume and scope of Euler‘s works is unprecedented and may never be equaled. Based on the current publication of his collected "Opera Omnia" by the Swiss Euler Committee:
- Euler wrote 866 books and papers totalling over 25,000 pages
- These are spread over 14 disciplinary fields including math, mechanics, astronomy, physics and philosophy
- He authored 55 book-length treatises and over 800 shorter works
- About a third of his work was written in Latin, another third in French, and the remainder in German
- It‘s estimated Euler was responsible for a third of all physics and mathematics publications in Europe from 1726 to 1800
- So far 76 volumes of his Opera Omnia have been published, with several more still to come
- His combined published output fills an estimated 60 to 80 quarto volumes
An Eternal Legacy
Euler died of a brain hemorrhage on September 18, 1783 while playing with his grandson. He was 76 years old. His passing was widely mourned in scientific circles across Europe. The French mathematician and philosopher Marquis de Condorcet eulogized him:
"He ceased to calculate and to breathe."
Yet in a sense, Euler has never stopped inspiring and enlightening mathematicians. Entire branches of mathematics still rest on foundations he built. Each year, new references and applications of his work come to light, from computer algorithms and 3D modelling software to physics experiments and engineering marvels.
Numerous awards, equations, constants and concepts bear his name in tribute to his genius:
- Euler‘s Number e, the base of the natural logarithm
- The Euler Line connecting orthocenter, centroid and circumcenter of a triangle
- Euler‘s Constant γ ≈ 0.577
- The Euler Characteristic for polyhedra
- Euler diagrams for illustrating set relationships
- Euler angles for describing 3D rotations
- Euler‘s Formula e^(ix) = cos x + i sin x
- Euler-Lagrange Equations in calculus of variations
- Euler Polynomials and Euler Numbers in combinatorics
- And many more…
Euler‘s singular impact on mathematics was perhaps best captured by Pierre-Simon Laplace, one of his greatest 18th century successors, who declared:
"Read Euler, read Euler, he is the master of us all."
From the most abstract reaches of number theory to the everyday applications of algebra, Euler‘s vast legacy continues to permeate almost every branch of modern mathematics and undergird much of our 21st century technology. He was not only a brilliant practitioner of math and science, but a profoundly decent and humble human being — a true exemplar of the Enlightenment ideal of a life dedicated to the pursuit of truth, beauty and progress. In the pantheon of history‘s greatest minds, Euler undoubtedly belongs in the first rank.
Sources
- Calinger, Ronald. "Leonhard Euler: The First St. Petersburg Years (1727–1741)." Historia Mathematica 23.2 (1996): 121-166.
- Dunham, William. Euler: The Master of Us All. Vol. 22. Mathematical Association of America, 1999.
- Euler, Leonhard. Opera Omnia. The Euler Committee, 1911-present.
- Gautschi, Walter. "Leonhard Euler: His Life, the Man, and His Works." SIAM Review 50.1 (2008): 3-33.
- Knobloch, Eberhard, ed. A Tribute to Euler 1707-1783: Papers from the Euler Tercentenary Celebration. Halle/Saale, July 20, 1983. No. 421. Birkhäuser, 1984.
- Varadarajan, V. S. "Euler and His Work on Infinite Series." Bulletin of the American Mathematical Society 44.4 (2007): 515-539.
