What contributions did Leonhard Euler, Joseph-Louis Lagrange, and Carl Friedrich Gauss make to numerical analysis?

Leonhard Euler, Joseph-Louis Lagrange, and Carl Friedrich Gauss were among the most influential mathematicians of the 18th and 19th centuries, each making foundational contributions to the development of numerical analysis.

Leonhard Euler

Euler was renowned for his skillful and imaginative use of algorithmic devices—computational procedures—to solve complex problems. His work in calculus and differential equations provided essential techniques for solving problems in physics and engineering. Notably, he developed a partial solution to the three-body problem, which assisted in calculating lunar tables for determining longitude at sea.1

Joseph-Louis Lagrange

Lagrange sought abstract generality and aimed to establish infinite processes on a sound basis. His work in mechanics, particularly his Mécanique analytique, introduced generalized coordinates and the Lagrangian equations, which are fundamental to modeling classical mechanical systems.12

Carl Friedrich Gauss

Gauss made significant advancements in the numerical side of science, most notably through his invention of the method of least squares. This method allowed him to accurately calculate the orbit of the asteroid Ceres after it had vanished from view. Additionally, his work in geodesy, cartography, and potential theory involved extensive numerical computations and rigorous analysis.34