George Boole's "A Treatise on the Calculus of Finite Differences" is a seminal work in mathematics, published in 1860 as a companion to his "Treatise on Differential Equations." This book systematically explores the calculus of finite differences, which deals with the differences between consecutive values of a sequence or function, contrasting it with the infinitesimal calculus.

The treatise begins by laying out the fundamental nature and direct theorems of finite differences, including the analysis of elementary functions and generating functions. It then delves into practical applications such as interpolation, mechanical quadrature, and the finite integration and summation of series. A significant portion of the book is dedicated to the study of series, examining their convergence and divergence, and methods for approximate summation, notably involving Bernoulli's numbers.

The latter half of the book focuses on various types of difference equations: those of the first order (both first and higher degree), linear equations with constant and variable coefficients, and mixed and partial difference equations. Boole's work also includes a discussion on the calculus of functions and provides geometrical applications, illustrating the practical utility of these mathematical concepts. Throughout the book, Boole emphasizes the connection and analogy between the calculus of finite differences and differential calculus. It became a classic textbook, influencing the development of discrete mathematics and serving as a foundational text for fields like computer science and digital logic. The book includes numerous exercises and their solutions, making it a comprehensive resource for students.