This paper, authored by R. Courant, K. Friedrichs, and H. Lewy, explores the algebraic problems that arise when linear differential equations in mathematical physics are replaced by difference equations in a rectangular grid. The authors focus on the behavior of solutions as the grid spacing approaches zero, particularly in the simplest but typical cases. They treat boundary value and eigenvalue problems for elliptic difference equations and initial value problems for hyperbolic and parabolic equations, proving that solutions of the difference equations converge to those of the corresponding differential equations. The paper also discusses the convergence conditions for elliptic equations, showing that higher-order difference quotients tend to the corresponding differential quotients. For hyperbolic equations, convergence is generally only possible if the ratios of grid spacings in different directions satisfy certain inequalities determined by the characteristics of the grid. The authors highlight the importance of the potential theory's Dirichlet problem as a typical example in the elliptic case and explore its connection to the well-known problem of finding shortest paths in statistics.This paper, authored by R. Courant, K. Friedrichs, and H. Lewy, explores the algebraic problems that arise when linear differential equations in mathematical physics are replaced by difference equations in a rectangular grid. The authors focus on the behavior of solutions as the grid spacing approaches zero, particularly in the simplest but typical cases. They treat boundary value and eigenvalue problems for elliptic difference equations and initial value problems for hyperbolic and parabolic equations, proving that solutions of the difference equations converge to those of the corresponding differential equations. The paper also discusses the convergence conditions for elliptic equations, showing that higher-order difference quotients tend to the corresponding differential quotients. For hyperbolic equations, convergence is generally only possible if the ratios of grid spacings in different directions satisfy certain inequalities determined by the characteristics of the grid. The authors highlight the importance of the potential theory's Dirichlet problem as a typical example in the elliptic case and explore its connection to the well-known problem of finding shortest paths in statistics.