1692 – 1770 · The Venetian Mathematician
The Jacobite exile who mastered the art of approximation, gave us tools to tame the factorial, and built bridges between discrete and continuous mathematics.
Born in 1692 in Garden, Stirlingshire, Scotland, into a family with deep Jacobite (Stuart loyalist) sympathies. His father, Archibald Stirling, was a supporter of the exiled Stuart dynasty.
Stirling entered Balliol College, Oxford around 1710, likely on a scholarship arranged for promising Jacobite Scots. He showed exceptional mathematical talent, coming to the attention of Newton's circle.
However, his Jacobite connections proved dangerous. After the failed 1715 rising, Stirling's political associations made his position at Oxford untenable. By 1717, he had left England for Venice, where he would spend nearly a decade in exile.
In Venice, Stirling studied at the University of Padua and continued his mathematical research, publishing his first major work while abroad.
The Stirlings of Garden were committed Jacobites. James's political identity shaped his entire early career, forcing him into exile and away from the British mathematical establishment.
At Oxford, Stirling absorbed the Newtonian mathematical tradition. His talent was recognized by John Keill and other Newtonians, who supported his early work.
Venice was a haven for Jacobite exiles. Stirling's years there gave him access to Italian mathematical traditions and a quiet environment for deep research.
Stirling's career took an unusual arc. While in Venice (c.1717–1725), he published Lineae Tertii Ordinis Neutonianae (1717), extending Newton's classification of cubic curves. This work earned him Newton's personal admiration.
Returning to London around 1725, Stirling entered the mathematical circle around Newton, de Moivre, and Maclaurin. He was elected FRS in 1726 and corresponded extensively with the leading mathematicians of the day.
His masterwork, Methodus Differentialis (1730), was a landmark treatise on interpolation, series summation, and numerical methods. It contained the famous factorial approximation and the Stirling numbers.
In 1735, Stirling made an unexpected career change, becoming manager of the Leadhills silver mine in Scotland. He spent the remaining 35 years of his life as an industrial manager, largely abandoning mathematics. He died on 5 December 1770, aged 78.
Lineae Tertii Ordinis (1717), Methodus Differentialis (1730)
Newton was impressed enough by Stirling's cubic curves work to recommend him for the Royal Society. Their correspondence reveals mutual respect.
At Leadhills, Stirling applied his mathematical skills to surveying, drainage, and industrial optimization — an early example of applied mathematics in industry.
Stirling's letters with Euler, Maclaurin, and de Moivre reveal a mathematician of the first rank who was deeply engaged with the major problems of his era.
Stirling's life was shaped by Jacobite politics, the Newton-Leibniz divide, and the early industrial revolution.
The Jacobite movement sought to restore the Catholic Stuart dynasty to the British throne. Stirling's family allegiance forced him into exile and ultimately led him away from academic mathematics entirely.
Stirling arrived in London's mathematical scene just as Newton was in his final years (d. 1727). He was among the last generation to interact personally with Newton.
Stirling corresponded with the young Euler, who was transforming Continental mathematics. Their exchange of letters on series and interpolation was mutually productive.
Stirling's move to the Leadhills mine in 1735 placed him at the intersection of mathematics and the early industrial revolution. Scotland's mineral wealth demanded technical expertise.
The 18th century demanded methods for computing values of functions from tables. Navigation, astronomy, and commerce all required interpolation — Stirling's specialty.
Stirling, along with Maclaurin, represented the remarkable flowering of Scottish mathematics in the early 18th century, despite political upheaval and limited institutional support.
The factorial function n! = 1 × 2 × 3 × … × n grows extraordinarily fast. Stirling's approximation provides a remarkably accurate continuous approximation:
n! ≈ √(2πn) · (n/e)n
The history of this formula involves both de Moivre and Stirling. De Moivre first derived the form n! ≈ C · nn+1/2 · e−n in 1733 but could not determine the constant C. Stirling then showed that C = √(2π), completing the formula.
The relative error is remarkably small: only about 0.8% for n=10 and decreasing as 1/(12n) for large n. A refined version includes correction terms:
n! ≈ √(2πn)(n/e)n(1 + 1/12n + 1/288n² + …)
The factorial function grows so fast that exact computation quickly becomes impractical. 100! has 158 digits. 1000! has 2568 digits. Stirling's formula makes these computable.
Taking logarithms gives an even more useful form:
ln(n!) ≈ n ln(n) − n + ½ ln(2πn)
This log-Stirling formula is ubiquitous in statistical mechanics, information theory, and combinatorics. It transforms multiplicative problems into additive ones.
The approximation connects to the gamma function Γ(n+1) = n!, extending the factorial to non-integer values and enabling continuous analysis of discrete quantities.
Boltzmann's entropy formula S = k ln(W) requires computing logarithms of combinatorial quantities. Stirling's approximation makes the entire field of statistical mechanics tractable.
Shannon's entropy, channel capacity, and source coding theorems all use Stirling's approximation to evaluate binomial coefficients in the limit of large block lengths.
Stirling's formula is the prototype of asymptotic approximation — the art of finding simpler expressions that become arbitrarily accurate in a limiting regime.
Stirling's determination that C = √(2π) was a tour de force of 18th-century analysis, connecting the factorial to the circle through the Wallis product.
Stirling numbers of the first kind s(n,k) count the number of permutations of n elements with exactly k cycles. They appear when expanding the falling factorial:
x(x−1)(x−2)…(x−n+1) = Σ s(n,k) xk
Stirling numbers of the second kind S(n,k) count the number of ways to partition n elements into exactly k non-empty subsets:
xn = Σ S(n,k) x(x−1)…(x−k+1)
Together, they form a pair of inverse triangular arrays — the matrices [s(n,k)] and [S(n,k)] are inverses of each other. They translate between ordinary powers and falling factorials.
The unsigned Stirling numbers of the first kind |s(n,k)| count permutations of {1,...,n} with exactly k cycles. For example, |s(4,2)| = 11 because there are 11 permutations of 4 elements with exactly 2 cycles. They connect to the harmonic numbers: |s(n,1)| = (n−1)!.
S(n,k) counts set partitions. S(4,2) = 7 because {1,2,3,4} can be split into 2 non-empty subsets in 7 ways. The Bell numbers B(n) = Σ S(n,k) count all partitions. These appear throughout combinatorics and algebra.
Stirling numbers convert between the two natural bases for polynomials: ordinary powers {1, x, x², ...} and falling factorials {1, x, x(x−1), x(x−1)(x−2), ...}. This is fundamental to the calculus of finite differences.
Stirling numbers appear in the analysis of algorithms (hashing, sorting), in algebraic topology (simplicial complexes), in quantum field theory (Feynman diagrams), and in the study of random permutations in probability.
Stirling's Methodus Differentialis, sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (1730) — "The Differential Method, or a Treatise on the Summation and Interpolation of Infinite Series" — was his masterwork.
The book systematically developed:
While Newton used forward differences, Stirling showed that central differences δf = f(x+h/2) − f(x−h/2) give more symmetric and often more accurate interpolation formulas.
Stirling developed methods to speed up the convergence of series like Σ1/n², enabling computation of constants like π²/6 to high precision.
Stirling's correspondence with Euler (1730s) was mutually enriching. Stirling communicated results on series summation; Euler responded with the Euler-Maclaurin formula.
The interpolation methods in Methodus Differentialis were used by astronomers and navigators for over a century to compute intermediate values from tables.
Stirling combined algebraic ingenuity with computational pragmatism, always seeking formulas that could be applied to real calculations.
Compute function values at equally spaced points
Form the table of central differences
Apply Stirling's formula for intermediate values
Use summation methods for series and integrals
Stirling was a computational mathematician avant la lettre. His methods were designed for practical calculation, optimizing accuracy while minimizing the number of operations. His interpolation formula uses fewer table entries than Newton's for comparable accuracy.
Stirling was one of the first to think systematically about asymptotic behavior — what happens as n becomes large. His factorial approximation is the paradigmatic example of an asymptotic expansion, a concept that would not be formalized until Poincaré in the 1880s.
The factorial approximation is called "Stirling's approximation" despite de Moivre doing most of the work. De Moivre derived the entire formula except for the constant √(2π). Stirling determined the constant — important, but arguably the smaller contribution. Yet history gave Stirling the credit.
Stirling's years in Venice (c.1717–1725) remain poorly documented. What was he doing there? Some sources suggest he had access to trade secrets about glass-making or navigation that he was not meant to learn, and that he was asked to leave.
Stirling's decision to become a mine manager in 1735 — at the height of his mathematical powers — puzzles historians. Was it financial necessity? Jacobite political constraints? Disillusionment? He produced almost no mathematics after this career change.
Stirling's Jacobite connections may have prevented him from obtaining the academic positions his talent warranted. Like de Moivre, he was shut out of the university system — in Stirling's case by politics rather than nationality.
Despite abandoning mathematics at 43, Stirling left a legacy that pervades modern science and computation.
Stirling's approximation, Stirling numbers (1st and 2nd kind), Stirling's interpolation formula, Stirling's series, Stirling cycle numbers, Stirling permutation numbers.
Had Stirling continued in mathematics after 1735, he might have rivaled Euler in contributions. His early work shows a mind of the highest order, but the mine consumed his remaining decades.
Ian Tweddle's 2003 translation and commentary finally made this masterwork accessible to modern readers, revealing the full depth of Stirling's achievements.
Computing entropy S = k·ln(W) requires ln(N!) for astronomical N. Stirling's formula makes this tractable, enabling the derivation of Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions.
Stirling numbers count permutations by cycle type, essential for analyzing randomized algorithms, hash tables, and sorting networks. The expected number of cycles in a random permutation uses Stirling numbers of the first kind.
Stirling's approximation appears in computing log-likelihoods of multinomial and Poisson models, in variational inference, and in the analysis of large language models' tokenization statistics.
Stirling numbers of the second kind count ways to partition molecular binding sites into groups, essential for modeling cooperative binding and drug design.
Channel coding theorems and rate-distortion theory require evaluating binomial coefficients for large n. Stirling's formula converts these into tractable expressions involving entropy functions.
Stirling numbers appear in the representation theory of the symmetric group, which governs the behavior of identical quantum particles and underlies quantum error correction codes.
I. Tweddle, James Stirling's Methodus Differentialis: An Annotated Translation of Stirling's Text (Springer, 2003) — the definitive modern edition with extensive commentary. Essential for understanding Stirling's actual mathematics.
I. Tweddle, James Stirling: This About Series and Such Things (Scottish Academic Press, 1988) — the only full-length biography, reconstructing Stirling's life from fragmentary sources.
R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics (2nd ed., 1994) — Chapter 6 provides an excellent modern treatment of Stirling numbers and their applications.
J. Riordan, An Introduction to Combinatorial Analysis (1958) — classic treatment of Stirling numbers in combinatorics.
N. Guicciardini, The Development of Newtonian Calculus in Britain (1989) — places Stirling in the broader context of British mathematics.
V. Katz, A History of Mathematics (3rd ed., 2009) — good coverage of Stirling's interpolation work.
"The summation and interpolation of series is of the utmost importance in all parts of mathematics and natural philosophy; for by its means we are enabled to express general laws in a finite form, and to compute the values of quantities which would otherwise be beyond our reach."
— James Stirling, preface to Methodus Differentialis, 1730James Stirling (1692–1770)