1698 – 1746 · Defender of the Calculus
The Scottish prodigy who became a professor at 19, provided the most rigorous defence of Newton's calculus, and extended Taylor's series into a powerful general tool.
Born in February 1698 in Kilmodan, Argyllshire, Scotland. His father, a minister, died when Colin was six weeks old; his mother died when he was nine. He was raised by his uncle, also a minister.
Despite this difficult start, Maclaurin entered the University of Glasgow at the extraordinary age of 11. He earned his M.A. at 14 with a thesis on the power of gravity.
At 19, he was appointed Professor of Mathematics at Marischal College, Aberdeen — making him one of the youngest professors in history. This appointment was won by competitive examination.
At Glasgow, Maclaurin discovered Euclid's Elements on his own and mastered the first six books within days, without instruction.
Born into the Gaelic-speaking Scottish Highlands, Maclaurin's rise to the pinnacle of British mathematics was a remarkable journey from remote beginnings.
His M.A. thesis already showed the geometric instincts that would characterize his entire mathematical career — always grounding algebra in geometric intuition.
After Aberdeen, Maclaurin traveled to London where he met Isaac Newton, beginning a relationship that would shape his career. Newton was so impressed that he personally recommended (and offered to pay part of the salary for) Maclaurin's appointment as deputy to James Gregory at Edinburgh in 1725.
At Edinburgh, Maclaurin became the dominant figure in Scottish mathematics. He won prizes from the Académie des Sciences in Paris (1724, 1740), was elected FRS, and produced a stream of important papers.
His masterwork, A Treatise of Fluxions (1742), was the most rigorous defence of Newtonian calculus ever written — directly prompted by Bishop Berkeley's devastating critique.
During the Jacobite rising of 1745, Maclaurin organized the defence of Edinburgh. The physical toll of this effort, combined with a difficult flight to York after the city fell, broke his health. He died on 14 June 1746, aged 48.
Geometria Organica (1720), Treatise of Fluxions (1742), Treatise of Algebra (posth. 1748)
Newton's personal endorsement and financial support cemented Maclaurin's position as the leading British mathematician of his generation.
Maclaurin's defense of Edinburgh during the '45 rebellion was a physical effort that ultimately cost him his life — a mathematician who died for his city.
Maclaurin worked in the shadow of Berkeley's challenge, the aftermath of the Newton-Leibniz dispute, and the Scottish Enlightenment.
In 1734, Bishop George Berkeley published The Analyst, attacking the logical foundations of calculus. He called infinitesimals "ghosts of departed quantities." This provoked Maclaurin's great defence.
Edinburgh in Maclaurin's era was becoming a center of intellectual life. He was a contemporary of David Hume and contributed to the flourishing of Scottish science and philosophy.
After Newton's death (1727), British mathematics needed a champion. Maclaurin stepped into this role, becoming the foremost Newtonian of his generation.
While Maclaurin defended Newton's geometric methods, Euler on the Continent was developing a more algebraic, analytic approach that would eventually dominate.
The 1745 Jacobite rebellion disrupted Scottish intellectual life. Maclaurin's active role in Edinburgh's defence shows how political turmoil touched even mathematicians.
The Hanoverian succession and relative political stability enabled the growth of scientific institutions, but the calculus controversy remained a persistent intellectual crisis.
The Maclaurin series is the special case of the Taylor series expanded around a = 0:
f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …
While Taylor stated the general theorem, Maclaurin made extensive and systematic use of the a=0 case, applying it to compute specific function expansions with great skill.
Key examples he explored:
Partial sums S1, S3, S5 converging to ex. Each additional term extends the region of accurate approximation.
Modern students sometimes view the Maclaurin series as merely "Taylor at zero" — a trivial specialization. But Maclaurin's actual contribution was far deeper.
In the Treatise of Fluxions, he used the series expansion at zero as part of a systematic approach to computing limits, extrema, and curvatures of curves. He developed tests for convergence and applied series methods to problems in geometry and physics.
His Maclaurin-Cauchy integral test for series convergence anticipated work usually attributed to Cauchy by nearly a century. He also established the first rigorous treatment of maxima and minima using higher-order derivatives.
Unlike many contemporaries, Maclaurin was careful about convergence. He understood that series could diverge and took pains to identify the range of valid approximation.
Maclaurin's systematic treatment of extrema (using f''(x) for the second derivative test) was the first rigorous account, predating standard textbook treatments by decades.
Every algebraic result in the Treatise is accompanied by geometric justification — Maclaurin refused to let algebra float free from geometric meaning.
In 1734, Bishop George Berkeley published The Analyst, a devastating critique of the logical foundations of calculus. He argued that if mathematicians ridiculed religious faith, their own infinitesimals were no less mysterious.
Maclaurin's response, the Treatise of Fluxions (1742), was a masterpiece of mathematical rigor — over 750 pages long. It attempted to place Newton's calculus on a foundation as rigorous as Greek geometry.
Key innovations in the Treatise:
Independently discovered by both Maclaurin (1742) and Euler (1732–1738), this remarkable formula connects discrete sums to continuous integrals:
Σ f(k) ≈ ∫ f(x)dx + [f(a)+f(b)]/2 + Σ B2k/(2k)! · [f(2k−1)(b) − f(2k−1)(a)]
where B2k are Bernoulli numbers. The formula expresses the difference between a sum and an integral as a series of correction terms involving derivatives at the endpoints.
This provides a systematic method for evaluating sums approximately, computing numerical integrals with correction terms, and analyzing the asymptotic behavior of series.
The formula is used to compute zeta function values, derive Stirling's approximation, and forms the basis of many numerical quadrature methods.
The appearance of Bernoulli numbers B2k = {1/6, −1/30, 1/42, −1/30, ...} connects the formula to deep number theory and the Riemann zeta function.
Euler and Maclaurin arrived at the formula independently — Euler algebraically, Maclaurin geometrically. This dual discovery highlights the formula's naturalness.
Essential in computational physics, asymptotic analysis, and analytic number theory. It appears in the theory of modular forms and string theory.
From his earliest work, Geometria Organica (1720, written when he was 21), Maclaurin was a master of algebraic curves. He studied their properties, intersections, and constructions with remarkable skill.
The Trisectrix of Maclaurin is an algebraic curve defined in polar coordinates by:
r = a(1 + 2cosθ) or equivalently r = a · sec(θ/3)
This curve can be used to trisect any angle — a problem famously impossible with straightedge and compass alone, but achievable with more general curves.
Maclaurin also proved Maclaurin's theorem on conics: if a triangle is inscribed in a conic and two sides pass through fixed points, the third side envelopes another conic.
This early work on curves generated by mechanical linkages impressed Newton himself and led to Maclaurin's election to the Royal Society at age 21.
Maclaurin independently discovered Cramer's rule for solving systems of linear equations using determinants, publishing it in his posthumous Treatise of Algebra (1748).
A generalization of the AM-GM inequality involving elementary symmetric means: S1 ≥ S21/2 ≥ S31/3 ≥ …
Maclaurin proved that a rotating fluid body in equilibrium takes the form of an oblate ellipsoid — confirming Newton's prediction about the Earth's shape.
Maclaurin's approach was unique: he insisted on geometric rigour while developing powerful algebraic tools.
Ground every concept in geometric intuition
Use Archimedean methods to establish limits
Derive results using Newton's fluxional calculus
Apply to physics, astronomy, or pure geometry
Maclaurin was the last great practitioner of the Newtonian geometric style. Where Euler would prove results algebraically, Maclaurin always sought a diagram, a construction, a visual justification. This made his work harder to extend but gave it an unmatched solidity.
The Treatise of Fluxions was explicitly designed to silence Berkeley's critiques. Maclaurin showed that every result of the calculus could be derived without appealing to infinitely small quantities — using only finite geometric arguments and limits.
The Maclaurin series was known long before Maclaurin. Special cases appear in the work of Madhava (14th century India), Gregory, and Taylor himself. Stigler's law of eponymy applies: the series bears the name of someone who did not first discover it but who popularized and systematized its use.
Some historians argue that Maclaurin's insistence on geometric methods — while admirable for rigor — reinforced British mathematics' resistance to the more powerful Continental analytic methods, contributing to a century of relative stagnation.
While Maclaurin answered Berkeley's challenge more thoroughly than anyone, some argue he didn't fully resolve the foundational issues. True resolution had to wait for Cauchy, Weierstrass, and the epsilon-delta formalism of the 19th century.
Newton's financial involvement in Maclaurin's Edinburgh appointment raised eyebrows. Was it pure merit, or was Newton installing a loyal ally in a key Scottish position during the priority dispute?
Maclaurin's legacy is complex: he is both underrated and overrated, depending on the context.
Maclaurin series, Euler-Maclaurin formula, Trisectrix of Maclaurin, Maclaurin's inequality, Maclaurin spheroid, Maclaurin expansion.
Maclaurin helped establish Edinburgh as a mathematical center. The tradition he built at the university continued through the centuries.
Maclaurin represents the culmination of Newtonian geometric mathematics. After him, even in Britain, the analytic methods of Euler and Lagrange prevailed.
Maclaurin expansions of transfer functions are used to design and analyze filters. The first few terms give the low-frequency behavior of any system.
The Euler-Maclaurin formula is the basis of the trapezoidal rule with endpoint corrections, yielding surprisingly accurate results for smooth functions.
The Euler-Maclaurin formula provides the analytic continuation of the Riemann zeta function, essential in prime number theory and mathematical physics.
Maclaurin spheroids — the equilibrium shapes of rotating fluid bodies — model neutron stars, planetary formation, and galactic dynamics.
Series expansion at zero is the default operation in systems like Mathematica and Maple. Maclaurin's approach is embedded in every symbolic computation engine.
The second-derivative test for extrema that Maclaurin systematized is fundamental to optimization algorithms, from simple calculus to modern machine learning.
A Treatise of Fluxions (1742) — Maclaurin's masterwork; dense but rewarding. Available in digitized form.
Geometria Organica (1720) — his early work on algebraic curves and mechanical constructions.
S. Mills, "The Collected Letters of Colin Maclaurin" (1982) — essential primary source for understanding his intellectual world.
E. Sageng, "Colin Maclaurin and the Foundations of the Method of Fluxions," PhD thesis (1989).
N. Guicciardini, The Development of Newtonian Calculus in Britain (1989) — places Maclaurin in the broader story of British mathematics.
J. Grabiner, "Was Newton's Calculus a Dead End?" American Mathematical Monthly (1997).
D. Jesseph, Berkeley's Philosophy of Mathematics (1993) — the definitive study of the philosophical challenge that provoked Maclaurin's greatest work.
"The supposition of an infinitely little magnitude is too bold a postulate for such a discipline as geometry; and it becomes mathematicians to demonstrate what they advance, not by a mere calculation, but by an investigation."
— Colin Maclaurin, A Treatise of Fluxions, 1742Colin Maclaurin (1698–1746)