This paper by Courant, Friedrichs, and Lewy investigates algebraic problems arising from replacing differential quotients with difference quotients in a grid, leading to problems with clear structure. The main focus is on how solutions behave as the grid spacing approaches zero. The authors consider simple, typical cases to demonstrate the applicability of their methods to more general difference equations and those with multiple variables. They treat boundary value and eigenvalue problems for elliptic difference equations and initial value problems for hyperbolic and parabolic difference equations. They prove that the limit process is always possible, showing that solutions of difference equations converge to solutions of the corresponding differential equations. For elliptic equations, they show that higher-order difference quotients approach the corresponding differential quotients. While for elliptic equations convergence is independent of the grid choice, for hyperbolic equations, convergence is only guaranteed if the grid spacing ratios satisfy certain conditions determined by the characteristics of the equation. The typical example for elliptic equations is the boundary value problem in potential theory, which has been studied extensively. However, previous studies often rely on specific properties of the potential equation, limiting the method's applicability to other problems. In addition to the main goal, the authors discuss the connection between elliptic boundary value problems and the well-known problem of random walks in statistics. The paper focuses on the elliptic case, discussing the grid structure and the convergence properties of solutions.This paper by Courant, Friedrichs, and Lewy investigates algebraic problems arising from replacing differential quotients with difference quotients in a grid, leading to problems with clear structure. The main focus is on how solutions behave as the grid spacing approaches zero. The authors consider simple, typical cases to demonstrate the applicability of their methods to more general difference equations and those with multiple variables. They treat boundary value and eigenvalue problems for elliptic difference equations and initial value problems for hyperbolic and parabolic difference equations. They prove that the limit process is always possible, showing that solutions of difference equations converge to solutions of the corresponding differential equations. For elliptic equations, they show that higher-order difference quotients approach the corresponding differential quotients. While for elliptic equations convergence is independent of the grid choice, for hyperbolic equations, convergence is only guaranteed if the grid spacing ratios satisfy certain conditions determined by the characteristics of the equation. The typical example for elliptic equations is the boundary value problem in potential theory, which has been studied extensively. However, previous studies often rely on specific properties of the potential equation, limiting the method's applicability to other problems. In addition to the main goal, the authors discuss the connection between elliptic boundary value problems and the well-known problem of random walks in statistics. The paper focuses on the elliptic case, discussing the grid structure and the convergence properties of solutions.