1) Childhood and Youth

Leonhard Euler was born on 15 April 1707 in Basel, Switzerland. He began his education at an early age and showed interest in fields such as mathematics and philosophy from a young age. His father initially planned for him to become a clergyman; however, Euler’s passion for mathematics soon became dominant. At the age of thirteen, he was admitted to the University of Basel, where he met mathematician Johann Bernoulli and began his studies under his guidance. This period played a decisive role in shaping Euler’s scientific orientation.

2) Career Path

In 1727, Euler moved to St. Petersburg to take up a position at the Russian Academy of Sciences. In 1741, he joined the Berlin Academy of Sciences. He returned to St. Petersburg in 1766 and continued his work there. During these periods, he focused on various topics in mathematics, physics, and engineering. He also played a key role in popularizing several notations used in scientific writing.

3) Major Achievements

One of Euler’s most notable contributions is his work in complex analysis. Equations such as Euler’s formula and Euler’s identity have become foundational pillars of mathematics. Moreover, Euler laid the groundwork for a new mathematical field, graph theory, thereby contributing to the development of modern computer science. His work has had a broad impact across a wide range of disciplines, from mathematical analysis to topology, and from physical theories to engineering, expanding the boundaries of mathematical thought.

4) Contributions

Euler made numerous contributions to mathematics, physics, engineering, astronomy, and other scientific fields. His work in mathematical analysis and function theory, particularly on differential equations, found extensive applications in engineering and physics. He also pioneered the development of modern mathematical fields such as topology and graph theory. Euler formulated important theories in number theory and fluid mechanics, and developed countless fundamental principles across various branches of the natural sciences.

Mathematical Analysis

Euler worked on functions, series, and integral calculus.

• Euler’s formula (Complex analysis): Euler’s formula is a mathematical expression in complex analysis that establishes the relationship between complex exponential functions and trigonometric functions. It is written as:

• Euler’s identity: This formula symbolizes the beauty and depth of mathematics by bringing together fundamental mathematical constants (e, π, i, 1, and 0).

• Euler’s Integral Calculations: In the 18th century, Euler developed methods for solving integrals of various types of functions. He focused particularly on techniques for integrating rational, trigonometric, and exponential functions, defining special functions and generalizing certain integral expressions. His work in this area made it possible to analyze functions analytically and laid the foundation for methods that later became standard in mathematical analysis.

Number Theory

Euler conducted research in number theory, particularly focusing on prime numbers and modular arithmetic.

• Euler’s Totient Function: Euler studied the totient function (φ(n)), which calculates the number of positive integers less than n that are coprime to n. The function is expressed in terms of n’s prime factors (P₁, P₂, …) as follows:

• Euler’s Prime Theorem: Euler developed theories on the distribution of prime numbers. Euler’s Prime Theorem provides a mathematical result used in the analysis of prime numbers.

Graph Theory

Euler conducted foundational work that established the key concepts of graph theory. The Seven Bridges of Königsberg problem is one of the most famous examples in this field.

• Seven Bridges of Königsberg Problem: Euler examined whether it was possible for a person to cross each of the seven bridges in Königsberg exactly once in a single route. This work established one of the fundamental principles of graph theory: for such a path to exist, every vertex in the graph must have an even degree.

Music and Acoustics

Euler performed mathematical analyses in the fields of music theory and acoustics. He developed mathematical models for the frequencies of sound waves.

• Acoustics: Euler studied the propagation of sound waves and developed mathematical models in acoustics. These models include equations that describe the physical properties of sound. The equations he developed continue to be used in engineering and physics today.

Physics

Euler contributed to physical theories such as mechanics and fluid dynamics. The equations he developed formed the fundamental mathematical models of these fields. Euler also made significant contributions to physical theories, particularly through the equations he formulated for mechanics and fluid dynamics, which became the cornerstones of these disciplines.

• Euler’s Equations of Motion (Mechanics): Euler formulated equations describing the motion of rigid bodies. These equations are used to analyze rotational motion and equilibrium states. The following equation expresses the relationship between moment of inertia, torque, and angular velocity:

Euler Equations (Fluid Dynamics): Euler developed fundamental equations to model the motion of fluids. These equations describe the dynamic relationship between pressure, velocity, density, and external forces:

Astronomy

Euler worked on the mathematical modeling of planetary motion. Using differential equations, he provided more precise descriptions of the movements of celestial bodies.

• Motion of Planets: Euler used differential equations to analyze planetary orbits. The following equation models the motion of a planet under the gravitational attraction of a central body (such as the Sun):

Engineering and Structural Dynamics

Euler conducted theoretical work in structural engineering. The models he developed were used in the analysis of beam deformation and strength.

• Euler’s Beam Theory: Euler developed a theory examining the mechanical behavior of beams. This theory explains the relationship between force, elastic modulus, and bending moment. The following equation forms the basis of this theory:

Logic

Euler developed visual analytical methods in mathematical logic. Euler diagrams are tools used to represent relationships and intersections between sets. These diagrams were developed to visualize logical relationships and illustrate how sets interact and overlap.

5) Works and Projects

Euler published numerous works on mathematics, mechanics, and astronomy. His major publications include:

• "Introductio in analysin infinitorum" (Introduction to the Analysis of the Infinite) – A study on functions and mathematical analysis.
• "Institutiones calculi differentialis" (Foundations of Differential Calculus) – A treatise on the theory of differential equations.
• "Mechanica" – Mathematical models of mechanical systems.
• "Theoria motus" – Mathematical analysis of planetary motion.

These theoretical works later found applications in engineering and physics.

6) Personal Life

In 1734, Euler married Katharina Gsell and had five children. He lost sight in his right eye in the 1740s and became completely blind in 1776. Despite this, he continued his work, dictating his mathematical calculations to assistants.

7) Daily Life

Euler’s daily life was largely centered on his academic work. He was actively involved in scientific circles in St. Petersburg and Berlin, spending long hours solving mathematical problems. Even after losing his sight, he maintained his productivity through mental calculations and verbal dictation of his work.

8) Final Years

Euler suffered from serious health problems beginning in 1771 and died in St. Petersburg in 1779. Even in his final years, he continued developing mathematical theories. His works remain foundational reference sources in physics, engineering, and mathematics.