section_path: Contributions to mathematics and physics/Physics, astronomy, and engineering
section_name: Physics, astronomy, and engineering
paragraph_text: Physics, astronomy, and engineering
  
  Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants e and π, continued fractions, and integrals. He integrated Leibniz 's differential calculus with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler–Maclaurin formula.
  
  Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the Sun. His calculations contributed to the development of accurate longitude tables.
  
  Euler made important contributions in optics. He disagreed with Newton's corpuscular theory of light, which was the prevailing theory of the time. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.
  
  In fluid dynamics, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines. In 1757 he published an important set of equations for inviscid flow in fluid dynamics, that are now known as the Euler equations.
  
  Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.