Arithmetic of Infinities

work by Wallis

Also known as: “Arithmetica Infinitorum”

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June 18, 2026

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John Wallis's Arithmetica Infinitorum (1655), or The Arithmetic of Infinitesimals, was a significant work that contributed to the development of calculus. Wallis extended existing methods for calculating areas and volumes by using numerical values for spatial indivisibles and incorporating negative and fractional exponents. This approach, which treated areas under curves as infinite series, allowed him to achieve impressive results for complex curves. Wallis also introduced the symbol \u221e for infinity in 1655. His work, while sometimes lacking rigorous proof, greatly influenced later mathematicians like Isaac Newton.123

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John Wallis
In his Arithmetica Infinitorum (“The Arithmetic of Infinitesimals”) of 1655, the result of his interest in Torricelli’s work, Wallis extended Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents; thus he did not follow Cavalieri’s geometric approach and instead assigned numerical values…
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In his Arithmetica Infinitorum (“The Arithmetic of Infinitesimals”) of 1655, the result of his interest in Torricelli’s work, Wallis extended Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents; thus he did not follow Cavalieri’s geometric approach and instead assigned numerical values…
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mathematics
The precalculus period
…in his Arithmetica Infinitorum (1655; The Arithmetic of Infinitesimals). Wallis, a successor to Henry Briggs as the Savilian Professor of Geometry at Oxford, was a champion of the new methods of arithmetic algebra that he had learned from his teacher William Oughtred. Wallis expressed the area under a curve as…
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…in his Arithmetica Infinitorum (1655; The Arithmetic of Infinitesimals). Wallis, a successor to Henry Briggs as the Savilian Professor of Geometry at Oxford, was a champion of the new methods of arithmetic algebra that he had learned from his teacher William Oughtred. Wallis expressed the area under a curve as…
Read More