Leonhard Euler

Lore & Background

Leonhard Euler was born in Basel on 15 April 1707 to Paul III Euler, a pastor of the Reformed Church, and Marguerite (née Brucker). He was the oldest of four children, with two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. Soon after his birth, the Eulers moved from Basel to Riehen, Switzerland, where his father became pastor. From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli. Around age eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel, also receiving private tutoring from Johannes Burckhardt. In 1720, at age 13, Euler enrolled at the University of Basel. There he studied under Johann Bernoulli, who advised him to work through difficult mathematical books and offered free access every Saturday afternoon to comment on difficulties. With Bernoulli's backing, Euler obtained his father's consent to become a mathematician. He received a Master of Philosophy in 1723 with a dissertation comparing the philosophies of René Descartes and Isaac Newton. In 1726, Euler completed a dissertation on the propagation of sound titled De Sono.

Reader's Guide

Euler's significance lies in his foundational contributions across mathematics and applied science. He founded graph theory and topology, partly through solving the problem of the Seven Bridges of Königsberg, which is also considered the first practical application of topology. He discovered that for a polyhedron with no holes, the number of vertices plus faces minus edges equals 2, now known as the Euler characteristic. He introduced key mathematical notations: π for the ratio of a circle's circumference to its diameter, f(x) for the value of a function, i for the imaginary unit √-1, Σ for summations, Δ for finite differences, and lowercase letters for triangle sides with capitals for angles. He gave the current definition of the constant e, the base of the natural logarithm, now known as Euler's number. In physics, he reformulated Isaac Newton's laws of motion into new laws in his two-volume work Mechanica to better explain the motion of rigid bodies. He formulated the partial differential equations for the motion of inviscid fluid and laid the mathematical foundations of potential theory. His three volumes on optics contributed to the design of microscopes and telescopes, and his study of ships helped navigation. With 866 publications collected in the Opera Omnia Leonhard Euler, he is regarded as arguably the most prolific contributor in the history of mathematics and science. Pierre-Simon Laplace said, "Read Euler, read Euler, he is the master of us all"; Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."

Did You Know?

• Euler entered the Paris Academy prize competition 15 times, winning 12 of them.
• He mastered Russian and took on an additional job as a medic in the Russian Navy after arriving in Saint Petersburg in 1727.
• Euler's father had taken courses from Jacob Bernoulli at the University of Basel some years before Euler's birth.
• In 1731, Euler left the Russian Navy, refusing a promotion to lieutenant.
• Euler's correspondence with academy secretary Johann Daniel Schumacher comprised more than 300 letters, continuing from 1730 until 1757.

A Universal Mind Across Disciplines

Euler's intellectual reach defied the boundaries of any single field. A Swiss-born polymath, he worked as a mathematician, physicist, astronomer, logician, geographer, music theorist, and engineer over a career that spanned Saint Petersburg and Berlin. His output was staggering: 866 publications, later gathered in the Opera Omnia Leonhard Euler, cementing his reputation as arguably the most prolific contributor in the history of mathematics and science. Beyond pure abstraction, he tackled applied problems with equal vigor. His two-volume Mechanica recast Newton's laws of motion into a framework better suited to rigid bodies, while his studies of elastic deformations of solid objects, beam bending, and column critical loads gave engineers practical tools. He formulated the partial differential equations governing inviscid fluid motion and laid the groundwork for potential theory. Three volumes on optics informed the design of microscopes and telescopes, and his analysis of ships aided navigation. Contemporaries and successors alike called him a "universal genius" possessed of "almost unlimited powers of imagination, intellectual gifts and extraordinary memory."

The Language of Modern Mathematics

Euler did not merely solve problems; he built the symbolic vocabulary through which later generations would think. He popularized the lowercase Greek letter π to represent the ratio of a circle's circumference to its diameter, and he was the first to write f(x) to denote the value of a function at a given input. He assigned the letter i to the imaginary unit, the square root of negative one, introduced capital Σ to compactly express summations, and used capital Δ for finite differences. In geometry, he established the convention of labeling triangle sides with lowercase letters and angles with capitals. He also supplied the modern definition of the constant e, the base of the natural logarithm, now universally called Euler's number. These choices transformed how mathematical ideas could be communicated, compressed, and extended.

Breakthroughs That Reshaped Pure Mathematics

Among Euler's most celebrated contributions is the founding of graph theory, sparked by the famous Seven Bridges of Königsberg puzzle. That solution is also regarded as the first practical application of topology, a field Euler co-founded. In number theory and analysis, he cracked several long-standing unsolved problems, most notably the Basel problem. He also discovered that for any polyhedron without holes, the number of vertices plus the number of faces minus the number of edges always equals two; this invariant is now called the Euler characteristic. His work spanned analytic number theory, complex analysis, and infinitesimal calculus, and he is widely considered the greatest mathematician of the eighteenth century. The depth of his influence is captured in the words of those who came after him. Pierre-Simon Laplace urged students to "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss went further, declaring that "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."

From a Pastor's Household to the Bernoulli Circle

Leonhard Euler entered the world on 15 April 1707 in Basel, the oldest of four children born to Paul III Euler, a Reformed Church pastor, and Marguerite (née Brucker). The family soon relocated to Riehen, where the young Leonhard spent most of his childhood. His earliest mathematical instruction came from his father, who had himself studied under Jacob Bernoulli at the University of Basel. Around age eight, Euler moved in with his maternal grandmother and enrolled in a Latin school, while also receiving private lessons from a young theologian named Johannes Burckhardt, who shared a passion for mathematics. In 1720, at just thirteen, he entered the University of Basel, where Johann Bernoulli, Jacob's younger brother, taught elementary mathematics. Bernoulli, too busy for regular private lessons, instead encouraged Euler to wrestle with difficult texts independently and reserved Saturday afternoons for resolving the student's accumulated difficulties. Euler later recalled that resolving one objection would make ten others vanish. With Bernoulli's backing, Euler persuaded his father to let him pursue mathematics rather than the ministry. In 1723, Euler received a Master of Philosophy with a dissertation comparing the philosophies of René Descartes and Isaac Newton. In 1726, he completed a dissertation on the propagation of sound titled De Sono.