Proving the Binomial Theorem in Britain, 1750–1830

This article examines the contested status and evolving proofs of the Binomial Theorem in Britain during the period 1750–1830. Although universally acknowledged as true and widely used in calculus, algebra, and the theory of infinite series, the theorem's general proof remained a source of prolonged mathematical and philosophical debate. The authors investigate why over forty British publications from this era sought to re-prove or reinterpret the theorem, linking this phenomenon to broader shifts in mathematical rigour and the eventual decline of the Newtonian fluxional calculus. The paper analyzes challenges surrounding the multiplicity of binomial forms and exponents, the lack of accepted general principles governing infinite series, and deep unease over Newton's own inductive, non-proof-based approach. Despite its central role in British mathematical education and its celebrated association with Newton, the Binomial Theorem's exact scope and justification remained elusive for decades. The authors argue that the persistence of divergent proofs and unresolved doubts reflects a transitional era in British mathematics—one marked by growing awareness of foundational uncertainty and the influence of more rigorous continental methods. This study thus offers insight into how mathematical authority, legacy, and proof were contested concepts in Enlightenment and post-Enlightenment Britain.