Book: A Treatise on the Calculus of Finite Differences

Overview
George Boole's A Treatise on the Calculus of Finite Differences presents a systematic and algebraic study of the discrete analogue of differential calculus. It frames finite differences as operators acting on sequences and functions defined on integers, develops techniques for manipulating those operators, and forges a conceptual bridge between discrete and continuous analysis. The exposition emphasizes symbolic methods and general principles rather than isolated computational tricks.

Foundations and Operators
The text introduces basic notions such as forward and backward differences, notation for successive differences, and the shift operator. Boole treats the difference operator Δ and the shift operator E algebraically, exploring their commutation relations and the consequences of treating them as formal entities. This operator viewpoint enables compact derivations of identities and provides a toolkit for forming and solving discrete equations.

Analogies with Classical Calculus
A central theme is the parallel between differentiation/integration and differencing/summation. Boole develops the idea of an "indefinite sum" as an inverse to the difference operator, mirroring antiderivatives, and formulates discrete analogues of series expansions and remainder terms. By comparing finite difference formulas with Taylor series and differential operators, the treatise elucidates how discrete methods reproduce and diverge from continuous intuition.

Techniques and Results
Boole presents systematic methods for interpolation, notably Newton's forward and divided difference techniques, and derives general expansions for functions defined on lattices. He investigates factorial polynomials and their role as natural bases for discrete calculus, and connects these to combinatorial coefficients. The work includes procedures for summation of series, transformation of sequences, and reduction of higher-order difference equations by operator factorization, yielding explicit solution formulas in many cases.

Applications
Practical applications receive attention through problems in interpolation, numerical summation, and the evaluation of series that arise in analysis and applied mathematics of the period. The operator methods facilitate the summation of complicated expressions and the derivation of closed forms for sequence values, which proved useful in numerical computation, finite table-making, and the emerging needs of applied science. Boole also hints at probabilistic and combinatorial interpretations where discrete operators naturally act on counting functions.

Legacy and Influence
The treatise consolidates finite difference theory into an algebraic, operator-based framework that influenced later developments in operator calculus, umbral methods, and discrete analysis. Its emphasis on symbolic manipulation presaged techniques used by later mathematicians in difference equations, numerical analysis, and combinatorics. The clarity and generality of Boole's approach helped legitimize the calculus of finite differences as a coherent branch of mathematical analysis and provided a foundation for both theoretical exploration and practical computation.