John Wallis: The cryptographer who gave us the symbol for infinity
In short: Before he became a foundational figure in calculus, John Wallis served as a chief codebreaker for the English Parliament. His mathematical genius emerged from a desire to formalize the infinite, leading him to introduce the symbol ∞ and develop techniques that bridged the gap between geometry and modern analysis.
Born in 1616, John Wallis was a polymath whose career spanned theology, linguistics, and mathematics. Though initially trained for medicine, his true passion lay in the abstract. His appointment in 1649 as the Savilian Professor of Geometry at Oxford was initially driven by political connections rather than a pre-existing mathematical reputation. However, he quickly proved his worth, spending 54 years in the role and producing work that significantly advanced the field of infinitesimal calculus.
Wallis is perhaps best remembered for his 1656 masterpiece, Arithmetica Infinitorum. In this work, he systematized the methods of René Descartes and Bonaventura Cavalieri, extending their reach by introducing standard notation for powers, including rational and negative exponents. He famously defined conic sections analytically and, in his quest to calculate the area under curves, introduced the symbol ∞ to represent infinity, alongside 1/∞ for an infinitesimal. His innovative use of interpolation allowed him to derive the famous Wallis product, an infinite sequence that provides a value for π.
Beyond pure mathematics, Wallis was a formidable cryptographer. From 1643 to 1689, he deciphered Royalist dispatches, recognizing early on that ciphers based on variable keys were far more secure than those relying on secret algorithms. His intellectual reach was vast, though not without controversy; he was accused of plagiarism by William Holder regarding techniques for teaching deaf individuals to speak. Despite such disputes, his contributions to the development of calculus and his early analytical treatment of curves cemented his legacy as one of the greatest intellectuals of the early modern period.
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