1616 – 1703 • Arithmetica Infinitorum
Savilian Professor of Geometry at Oxford for over 50 years, Wallis bridged the gap between classical geometry and the calculus, pioneering infinite products, interpolation, and the symbol for infinity.
John Wallis was born November 23, 1616, in Ashford, Kent, England. His father, a minister, died when Wallis was six. He was educated at local schools and then at Martin Holbeach's school in Felsted, Essex.
Wallis entered Emmanuel College, Cambridge, in 1632, studying theology, medicine, and natural philosophy. Mathematics was not formally taught at Cambridge at the time; Wallis later claimed he learned arithmetic from his brother during a Christmas holiday.
He was ordained in 1640 and served as chaplain to several Parliamentarian figures during the English Civil War. His remarkable ability to decipher Royalist coded messages brought him to prominence.
In 1649, Wallis was appointed Savilian Professor of Geometry at Oxford — a position he held for 54 years until his death. Despite being essentially self-taught in mathematics, he became one of England's leading mathematicians.
His Arithmetica Infinitorum (1656) was a landmark work that extended Cavalieri's indivisibles into a systematic arithmetic of infinite processes. Newton read it as a student and credited it as a primary inspiration for his development of calculus.
Wallis was a founding member of the Royal Society and served as its secretary. He was known for his phenomenal mental calculation abilities and his irascible temper, particularly in disputes with Hobbes.
Wallis sided with Parliament in the Civil War (1642–1651), using his mathematical talents to break Royalist ciphers. His reward was the prestigious Oxford professorship, awarded by the Parliamentary authorities.
The founding of the Royal Society in 1660 formalized England's scientific community. Wallis was a charter member and key figure, helping establish the institutional framework for British science.
Wallis worked in the generation just before Newton and Leibniz. His Arithmetica Infinitorum systematized the techniques of Cavalieri, Fermat, and others, setting the stage for the full development of calculus.
"I did endeavour to give a good account of those infinite processes; and to show that they were not merely conjectural, but demonstrative."
— John Wallis, Treatise on Algebra (1685)Wallis discovered this remarkable infinite product in 1655, published in Arithmetica Infinitorum (1656).
Wallis derived his product by a brilliant use of interpolation. He computed the integrals:
∫_0^1 (1−x^(1/p))^q dx
for integer values of p and q, obtaining ratios involving factorials. He then interpolated to find the value at q = 1/2, which gives the area of a quarter-circle.
The key insight was that the ratio of successive integral values followed a pattern that could be extended to half-integer arguments. The Wallis product emerges from this interpolation applied to what we now call the Beta function.
Wallis's approach was daring: he assumed that patterns observed for integers would continue to hold for fractions — a leap of faith that was not rigorously justified but produced correct results.
Modern derivations use:
Newton's reading of Wallis's interpolation technique directly inspired his discovery of the generalized binomial theorem.
Wallis introduced the lemniscate symbol ∞ for infinity in his De Sectionibus Conicis (1655).
Before Wallis, exponents were limited to positive integers: x², x³, x⁴. Wallis systematically extended this notation:
x^(1/2) = √x (square root as exponent 1/2)x^(1/3) = ³√x (cube root as exponent 1/3)x^(−1) = 1/x (negative exponent = reciprocal)x^(−n) = 1/x^n (general negative exponents)x^0 = 1 (zeroth power)This unification was crucial: it meant that the laws of exponents x^a · x^b = x^(a+b) worked for all rational exponents, not just positive integers.
Newton seized on Wallis's generalization. If exponents could be fractions, why not extend the binomial theorem to fractional powers?
(1+x)^(1/2) = 1 + (1/2)x − (1/8)x² + ...
This was Newton's generalized binomial theorem, which he discovered by reading Wallis's Arithmetica Infinitorum in 1664–65. It became the foundation of Newton's calculus.
Wallis also introduced the concept of interpolation between known values — finding the value of a function at non-integer arguments by pattern recognition. This technique, central to numerical analysis, was essentially invented by Wallis.
Wallis's greatest conceptual contribution was replacing Cavalieri's geometric "indivisibles" with arithmetic processes. Where Cavalieri compared areas by comparing their slices geometrically, Wallis computed sums of infinite series.
He showed that for integers n:
∫_0^1 x^n dx = 1/(n+1)
by computing the limit of the sum (0^n + 1^n + 2^n + ... + N^n) / (N^n · N) as N approaches infinity. This was essentially Riemann integration avant la lettre.
Wallis then boldly extended this to fractional exponents, claiming:
∫_0^1 x^(p/q) dx = 1/(p/q + 1) = q/(p+q)
This gave him the tools to compute areas under curves like y = x^(1/2) (the parabola) and y = (1−x^2)^(1/2) (the circle).
The ratio of the area under the circle to the area of the enclosing square involved π, which led to the Wallis product. Thus his interpolation technique connected infinite products to the quadrature of the circle — the oldest problem in mathematics.
Evaluate for integer cases
Arrange values in tables
Find patterns for half-integers
State the infinite product/series
Wallis reasoned by bold induction from patterns. He freely assumed that formulas valid for integers would extend to fractions. While not rigorous by modern standards, his intuitions were almost always correct and enormously productive.
Wallis deliberately chose arithmetic (sums, products, ratios) over geometric methods (proportions, similar figures). This arithmetization of the infinite was a crucial step toward the algebraic calculus that Newton and Leibniz would develop.
Thomas Hobbes, the famous philosopher, claimed to have squared the circle — a classical problem already suspected to be impossible. Wallis publicly demolished Hobbes's proofs, beginning a feud that lasted 25 years (1655–1679).
Wallis published devastating refutations of Hobbes's mathematical errors in increasingly acerbic pamphlets. Hobbes, unable to accept his mistakes, kept producing new (equally flawed) proofs. The exchange became one of the most famous intellectual feuds in history.
While Wallis was mathematically correct, modern historians note his attacks were often personally cruel and politically motivated — Hobbes was a Royalist, Wallis a Parliamentarian.
Hobbes made elementary errors in his geometric proofs, confusing ratios and misapplying proportions. Wallis gleefully catalogued each mistake, using Hobbes as an example of what happens when philosophers attempt mathematics without proper training.
Wallis also engaged in priority disputes regarding Newton's calculus, publishing Newton's early letters in his Treatise on Algebra (1685) without full permission — contributing to the Newton-Leibniz priority controversy.
Wallis's codebreaking service to the government was controversial. He broke codes for both Parliamentarians and Royalists at different times, earning accusations of political opportunism alongside admiration for his skill.
The Wallis product was the first in a long line of infinite product representations for constants: Euler's product for sine, Weierstrass products, Hadamard factorization. Infinite products are now central to complex analysis and number theory.
Wallis's interpolation technique evolved into the modern theory of interpolation: Newton's divided differences, Lagrange interpolation, and spline methods used throughout numerical analysis and data science.
Wallis's interpolation of factorials to half-integer arguments directly inspired Euler's gamma function Γ(n), which extends the factorial to all complex numbers and appears throughout mathematics and physics.
Newton explicitly credited Wallis's Arithmetica Infinitorum as the work that inspired his binomial series and fluxions. Without Wallis, the development of calculus might have been significantly delayed.
Wallis's extension of exponents to rational and negative values is used universally in mathematics, science, and engineering. Every expression like x^(-2) or x^(3/2) uses Wallis's notation.
Wallis's infinity symbol has become one of the most recognizable symbols in all of mathematics, appearing in calculus, set theory, topology, and popular culture. It transcends mathematics as a cultural icon.
Wallis's interpolation methods are ancestors of modern numerical techniques: polynomial interpolation, splines, and approximation theory used in computer graphics, finite element analysis, and scientific computing.
The Wallis product and related infinite products appear in quantum field theory (regularization of path integrals), statistical mechanics (partition functions), and the computation of quantum amplitudes.
Infinite products related to Wallis's work appear in filter design and the theory of orthogonal polynomials used in signal processing, spectral analysis, and communications engineering.
Wallis was one of history's great codebreakers. His methods — systematic pattern analysis, frequency counting, and mathematical structure exploitation — are ancestors of modern cryptanalytic techniques.
John Wallis (tr. Jacqueline Stedall, 2004). First English translation of Arithmetica Infinitorum with extensive commentary. Essential for understanding Wallis's methods.
Jacqueline Stedall (2002). Excellent scholarly account of English algebra in the 17th century, with detailed treatment of Wallis's contributions and their context.
Douglas Jesseph (1999). Detailed account of the Hobbes-Wallis controversy, illuminating both the mathematical and political dimensions of their famous feud.
William Dunham (2005). Places Wallis's work in the broader story of calculus development, showing how his ideas influenced Newton, Leibniz, and subsequent generations.
"I find that most of the general theorems of this kind may be obtained from a table of simple cases by induction, without any previous knowledge of the truth of those theorems."
— John Wallis, Arithmetica Infinitorum (1656)John Wallis (1616–1703)
Savilian Professor • Codebreaker • Pioneer of the Infinite